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nowadays the family of iron pnictides is a well - established and important prototype system for unconventional high - temperature superconductivity . starting with the first famous compound @xcite in 2008 , today several different sub - families with a wide structural variety are known . all different groups of iron pnictides share some common physical properties , such as their interesting and sometimes puzzling magnetic behavior . most compounds show a phase transition at low temperatures from a tetragonal to an orthorhombic crystal symmetry which is typically accompanied by the formation of long - range antiferromagnetic order.@xcite it is common believe that the suppression of these phase transitions for example by chemical substitution is crucial for the emergence of unconventional superconductivity.@xcite although it is obvious that an understanding of the magnetic fluctuations in the iron pnictides is mandatory to unveil the physics underlying the superconductivity , this task has proven to be more complex than anticipated.@xcite for example , there was discussion in the literature whether the magnetic moments are better described by an itinerant@xcite or a localized@xcite model and there is up to now no consensus concerning the role of correlation effects@xcite . furthermore , the magnitude of the magnetic moments is difficult to reproduce within density functional theory ( dft ) and it is known to be quite sensitive to computational parameters.@xcite one of the most important experimental tools to get insight into the electronic structure of the iron pnictides is angle - resolved photoemission spectroscopy ( arpes ) . there are numerous publications on this topic , although it was shown that dft calculations have typically problems to reproduce all features of the arpes spectra correctly.@xcite this is often ascribed to strong correlation effects , although this question is still under discussion.@xcite another important difficulty which so far is often ignored is the connection between the magnetic phase of the iron pnictides and the resulting consequences for arpes . this is due to the formation of twinned crystals during the phase transition from tetragonal to orthorhombic and it results in mixed magnetic domains which are orthogonal to each other . macroscopic tools like arpes or transport measurements can so only see the averaged information , while information on the anisotropy is lost.@xcite this is a huge drawback considering a comprehensive study of the electronic structure in the iron pnictides , as it is known that the in - plane anisotropy plays a significant role.@xcite in experiment it is possible to effectively detwin the crystals by applying uniaxial stress during the measurement . this was already done successfully for the 122-prototype in the undoped and in the co - doped case . however , such measurements are connected with several technical difficulties and consequently they are rarely done.@xcite yet , to fully understand the electronic properties of the iron pnictide superconductors in a comprehensive way and to get a deeper insight concerning the influence of the in - plane anisotropy in the magnetic phase such studies are absolutely mandatory . although there is nowadays experimental data on detwinned crystals showing clearly the anisotropy in the fermi surface there is hardly any theoretical work focusing on this problem of magnetic anisotropy in arpes data . in this work this issue is addressed by a comprehensive dft study on the magnetic phase of and on the corresponding arpes spectra . the computational results can be directly compared to the available experimental arpes data on detwinned crystals.@xcite in order to deal with this complex situation the korringa - kohn - rostoker - green function ( kkr - gf ) approach is used , which was already shown to be indeed a very useful and accurate tool to deal with the iron pnictides.@xcite the impact of disorder due to substitution is dealt with by means of the coherent potential approximation ( cpa ) , giving results fully compatible to supercell calculations and more reliable than those based on the virtual crystal approximation ( vca).@xcite all calculations have been performed self - consistently and fully relativistically within the four component dirac formalism , using the munich spr - kkr program package.@xcite the orthorhombic , antiferromagnetic phase of is investigated in its experimentally observed stripe spin state using a full 4-fe unit cell . this implies antiferromagnetic chains along the @xmath1- and @xmath2-axes and ferromagnetic chains along the @xmath3-axis . the lattice parameters where chosen according to experimental x - ray data and the experimental as position @xmath4.@xcite to account for the influence of substitution in a linear interpolation for the lattice parameters with respect to the concentration @xmath0 is used based on available experimental data@xcite and vegard s law@xcite . more details on the procedure can be found in a previous publication.@xcite the treatment of disorder introduced by substitution is dealt with by means of the cpa . the basis set considered for a @xmath5 including @xmath6 , @xmath7 , @xmath8 , @xmath9 and @xmath10 orbitals . for the electronic structure calculations the local density approximation ( lda ) exchange - correlation potential with the parameterization given by vosko , wilk and nusair was applied.@xcite the spectroscopical analysis is based on the fully relativistic one - step model of photoemission in its spin density matrix formulation . for more technical details on these calculations see ref.@xcite . the geometry of the spectroscopy setup was taken from experiment including a tilt of the sample around either the @xmath1 or @xmath3 axis . the incident light hit the sample under a constant polar angle @xmath11 and an azimuthal angle @xmath12 of either @xmath13 or @xmath14 . these geometries are referred to as @xmath15 and @xmath16 , meaning the direction of the incident light is either parallel to the antiferromagnetic or the ferromagnetic in - plane directions . the corresponding electrons were collected with an angle @xmath17 of @xmath18 or @xmath19 and a varying angle @xmath20 between @xmath21 and @xmath22 . this geometry is in line to the experimental setup . if not indicated otherwise , an as - terminated surface was chosen . however , the question of surface termination will be discussed in more detail in the following . to describe the anisotropy of the iron pnictides in arpes calculations reasonably well one needs first to ensure that the spin - dependent potentials from the self - consistent field ( scf ) calculations are accurate enough . obviously , the magnetic ordering plays a significant role concerning the anisotropy of the electronic structure and hence the quality of the theoretical description of the arpes spectra is determined by the quality of the spin - dependent potentials . the most meaningful indication for a proper description of the magnetic state is good agreement with experimental data on the magnetic order . for the iron pnictides this is known to be a non - trivial task as the magnetic moments are often overestimated by dft.@xcite for the undoped mother compound a total magnetic moment of @xmath23 was obtained . experiment reports a total magnetic moment of approximately @xmath24 from neutron diffraction@xcite while mssbauer spectroscopy@xcite and @xmath25sr spectroscopy@xcite coherently give a value of around @xmath26 . hence , the calculated total magnetic moment is found in good agreement with experiment and captures the proper order of magnitude accurately.@xcite more importantly , the cpa allows to evaluate the substitution dependent self - consistent evolution of the magnetic moments with increasing co concentration in . the corresponding results are shown in fig . [ fig_magnmom ] , where the results for spin and orbital magnetic moments are given in an atom - resolved way . the total magnetic moment is calculated as substitutionally averaged sum over all contributions . in agreement with experiment the total magnetic moments shows a nearly linear decay until the long - range magnetic order disappears.@xcite in the calculations the critical co substitution for the disappearance of antiferromagnetic order occurs for @xmath27 , which is in reasonably good agreement with the experimental value @xmath28.@xcite it should be mentioned that the results in fig . [ fig_magnmom ] are slightly improved with respect to experiment in comparison with our previous work@xcite due to the higher @xmath29 expansion used here . however , the trends in the magnetic moments and the resulting conclusions are the same . as the one - step model of photoemission fully accounts for matrix - elements as well as for surface effects the resulting spectra can be directly compared to experimental arpes data . as stressed before , it is extremely difficult to see the magnetic anisotropy correctly in experimental spectra because of the twinning of crystals . here reference is made especially to the work of yi _ et al._@xcite , who did remarkable measurements on detwinned single crystals of and by applying uniaxial stress to the crystals . similar results were obtained for example by kim _ et al._@xcite . in this context it is important to note that the brillouin zone ( bz ) of the magnetic 4-fe spin - density - wave ( sdw ) state is only half the size compared to the bz in the nonmagnetic 2-fe state . for that reason it is most appropriate to use in the following the notation for the 4-fe sdw bz where the information of @xmath30 and @xmath31 from the nonmagnetic bz is down - folded to one @xmath32 point.@xcite in fig . [ fig_cal_afmain ] the fermi surface around the @xmath32 point is shown in the sdw bz as calculated from the spin - dependent potentials for a photon energy of @xmath33ev . the overlay of black points corresponds to the experimentally measured bz , reproduced from the work of yi _ et al._@xcite . as can be seen , the agreement of the calculated fermi surface and the experimental data is remarkably good . characteristic are the bright intensity spots along the @xmath34-direction ( i.e. along @xmath1 ) , corresponding directly to the antiferromagnetic order along the @xmath1-axis and the bigger pedal - like structures along the @xmath35-direction ( i.e. along @xmath3 ) which corresponds to the ferromagnetic order along the @xmath3-axis . ( 12,10.0 ) ( 2.20,1.2 ) in the 4-fe sdw bz for a photon energy of @xmath33ev . the overlay of black points is a reconstruction of the sdw bz from experimental arpes data , reproduced from the work of yi _ et al._@xcite.,title="fig : " ] ( 1.2,1.35)(0,1.95)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 1.85,0.9)(1.95,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 0.2,4.0 ) ( 5.2,-0.1)@xmath36 $ ] ( 10.7,1.2 ) in the 4-fe sdw bz for a photon energy of @xmath33ev . the overlay of black points is a reconstruction of the sdw bz from experimental arpes data , reproduced from the work of yi _ et al._@xcite.,title="fig : " ] ( 11.0,9.5)(0,0)max ( 11.0,1.0)(0,0)min it should be noted that in fig . [ fig_cal_afmain ] the intensity over two different light polarizations was averaged , namely for the direction of the incident light either parallel to the antiferromagnetic @xmath1-axis @xmath15 or parallel to the ferromagnetic @xmath3-axis @xmath16 . all features of the electronic structure are visible for both polarizations of light . however , the intensity patterns vary notably with the polarization due to matrix element effects , indicating strong multiorbital character , just as seen in experiment.@xcite if not indicated otherwise this averaging will be applied in the following . for comparison the two contributions to the total fermi surface for @xmath33ev are shown polarization - resolved in fig . [ fig_cal_afpol ] , for the incident light direction being either parallel to the @xmath3-axis @xmath16(fig . [ fig_cal_afpol ] ( a ) ) or parallel to the @xmath1-axis @xmath15(fig . [ fig_cal_afpol ] ( b ) ) . ( 5.9,8 ) ( 1.10,1.2)-axis ( a ) or parallel to the antiferromagnetic @xmath1-axis ( b ) . , title="fig : " ] ( 0.2,1.35)(0,1.3)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 0.75,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.6,3.0 ) ( 2.7,-0.1)@xmath36 $ ] ( 0.65,7.0)(a ) ( 2.75,7.0)@xmath16 ( 7.3,8 ) ( 1.20,1.2)-axis ( a ) or parallel to the antiferromagnetic @xmath1-axis ( b ) . , title="fig : " ] ( 0.85,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.8,-0.1)@xmath36 $ ] ( 0.65,6.95)(b ) ( 2.75,6.95)@xmath15 ( 7.0,1.2)-axis ( a ) or parallel to the antiferromagnetic @xmath1-axis ( b ) . , title="fig : " ] ( 7.3,6.95)(0,0)max ( 7.3,1.0)(0,0)min it can be seen that for @xmath16 the intensity of the bright spots along the @xmath1-axis is significantly enhanced while for @xmath15 the intensity around the inner circle of @xmath32 is enhanced . this polarization dependence is again in full agreement with the experimental findings.@xcite at this point it was shown that the detwinned , antiferromagnetic fermi surface obtained by the calculations agrees very well with experiment . one may also ask how the fermi surface of a twinned , antiferromagnetic crystal should look like and how does it differ from the nonmagnetic case . therefore , the calculated fermi surfaces for both cases are shown in fig . [ fig_cal_twin ] and compared with experimental arpes data@xcite of twinned crystals at @xmath37k ( a ) and @xmath38k ( b ) . please note that the transition from a paramagnetic to an antiferromagnetic state occurs at around @xmath39k , accordingly the experimental data shown as overlay of black points corresponds to the nonmagnetic and the twinned antiferromagnetic state , respectively . the representation of the twinned fermi surface is based on a superposition of spectra obtained independently for antiferromagnetic states rotated by @xmath19 against each other which is supposed to be a good approximation for twinned crystals , see for example the work of tanatar _ et al._@xcite . ( 5.9,8 ) ( 1.10,1.2 ) against each other . the overlay of black points is reproduced from the experimental arpes data of yi _ et al._@xcite , title="fig : " ] ( 0.2,1.35)(0,1.3)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 0.75,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.6,3.0 ) ( 2.7,-0.1)@xmath36 $ ] ( 0.65,7.0)(a ) ( 2.1,7.0)nonmagnetic ( 7.3,8 ) ( 1.20,1.2 ) against each other . the overlay of black points is reproduced from the experimental arpes data of yi _ et al._@xcite , title="fig : " ] ( 0.85,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.8,-0.1)@xmath36 $ ] ( 0.65,6.95)(b ) ( 2.75,6.95)twinned ( 7.0,1.2 ) against each other . the overlay of black points is reproduced from the experimental arpes data of yi _ et al._@xcite , title="fig : " ] ( 7.3,6.95)(0,0)max ( 7.3,1.0)(0,0)min comparing the nonmagnetic and the twinned antiferromagnetic state with each other it is obvious that there is a significant difference in the shape of the fermi surface which is due to the underlying change of the electronic structure during the magnetic phase transition . however , the twinned fermi surface is in principle isotropic along @xmath34 and @xmath35 due to the fact that the in - plane anisotropy cancels almost completely for two magnetic domains that are rotated by @xmath19 against each other . this means that although some influence of the magnetic ordering can be seen for twinned crystals , it is not possible to deduce information about the important in - plane anisotropy from the corresponding spectra . this stresses again the importance of arpes measurements and calculations on detwinned crystals to investigate the magnetic structure correctly . to summarize , the agreement of the calculations with the experimental data is altogether quite well for the nonmagnetic as well as for the twinned magnetic state . ( 5.9,8.8 ) ( 1.1,1.2 ) dispersion as seen by arpes along the both in - plane real space axes @xmath1 ( a ) and @xmath3 ( b ) . the black lines mark the photon energies where the alternation of @xmath40 and @xmath41 can be seen along @xmath42 . notably , the vertical intensity stripes at @xmath43 in ( a ) seem almost independent on @xmath42 , indicating some connection to a surface related phenomenon . , title="fig : " ] ( 0.4,1.95)(0,1.55)[l]2.5,3.0,3.5,4.0 ( 0.88,0.9)(1.28,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.5,3.7 ) ( 2.8,-0.1)@xmath44 $ ] ( 0.65,8.3)(a ) ( 2.85,8.2)@xmath1-axis ( 7.0,8.8 ) ( 1.1,1.2 ) dispersion as seen by arpes along the both in - plane real space axes @xmath1 ( a ) and @xmath3 ( b ) . the black lines mark the photon energies where the alternation of @xmath40 and @xmath41 can be seen along @xmath42 . notably , the vertical intensity stripes at @xmath43 in ( a ) seem almost independent on @xmath42 , indicating some connection to a surface related phenomenon . , title="fig : " ] ( 0.88,0.9)(1.28,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.8,-0.1)@xmath45 $ ] ( 0.65,8.3)(b ) ( 2.85,8.2)@xmath3-axis ( 6.9,1.2 ) dispersion as seen by arpes along the both in - plane real space axes @xmath1 ( a ) and @xmath3 ( b ) . the black lines mark the photon energies where the alternation of @xmath40 and @xmath41 can be seen along @xmath42 . notably , the vertical intensity stripes at @xmath43 in ( a ) seem almost independent on @xmath42 , indicating some connection to a surface related phenomenon . , title="fig : " ] ( 7.25,8.2)(0,0)max ( 7.25,1.0)(0,0)min going back to the original study of detwinned antiferromagnetic crystals the @xmath42 dispersion is shown along the @xmath1- and @xmath3-axes in fig . [ fig_cal_kz ] . the difference between @xmath40 and @xmath41 manifests itself mainly by the alternating intensity distributions . we find @xmath40 for photon energies of @xmath4622ev and @xmath47ev respectively , while @xmath41 can be found at @xmath48 . this is in good agreement with literature which reports @xmath41 at @xmath49ev and @xmath40 at @xmath50ev.@xcite the anisotropic features , namely the bright spots along @xmath34 and the pedals along @xmath35 seem quite independent on @xmath42 , which agrees with the experimental reports on the detwinned crystals.@xcite for further discussion the fermi surfaces for @xmath51ev and @xmath47ev respectively are shown in fig . [ fig_cal_3448 ] . the important aspect to note is that the anisotropic features are preserved independent on @xmath42 , meaning they are preserved for @xmath40 as well as for @xmath41 . however , the most striking anisotropy between the @xmath1- and @xmath3-directions seen in the @xmath42 dispersion are the almost vertical intensity lines along the @xmath1-axis for @xmath43 . they are surprisingly robust concerning the @xmath42 dispersion , already indicating possible surface related phenomena , as will be discussed later in more detail . ( 5.9,8 ) ( 1.10,1.2)ev and @xmath47ev , corresponding to either @xmath41 ( a ) or to @xmath40 ( b ) . it can be seen that the topology of interest , namely the anisotropic features are principally independent on @xmath42.,title="fig : " ] ( 0.2,1.35)(0,1.3)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 0.75,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.6,3.0 ) ( 2.7,-0.1)@xmath36 $ ] ( 0.65,7.0)(a ) ( 2.45,7.0)@xmath51ev ( 7.3,8 ) ( 1.20,1.2)ev and @xmath47ev , corresponding to either @xmath41 ( a ) or to @xmath40 ( b ) . it can be seen that the topology of interest , namely the anisotropic features are principally independent on @xmath42.,title="fig : " ] ( 0.85,0.9)(1.3,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.8,-0.1)@xmath36 $ ] ( 0.65,6.95)(b ) ( 2.45,6.95)@xmath47ev ( 7.0,1.2)ev and @xmath47ev , corresponding to either @xmath41 ( a ) or to @xmath40 ( b ) . it can be seen that the topology of interest , namely the anisotropic features are principally independent on @xmath42.,title="fig : " ] ( 7.3,6.95)(0,0)max ( 7.3,1.0)(0,0)min ( 6.8,9.5 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.4,3.5 ) ( -0.50,8.9)@xmath52 ( 1.55,8.3)(a ) ( 3.25,8.3)@xmath1-axis ( 6.8,8.8 ) ( 1.1,1.2 ) ( 0.12,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 1.55,8.3)(b ) ( 2.7,8.3)nonmagnetic ( 7.8,8.8 ) ( 1.1,1.2 ) ( 0.12,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.55,8.3)(c ) ( 3.25,8.3)@xmath3-axis ( 7.3,1.2 ) ( 7.6,8.2)(0,0)max ( 7.6,1.0)(0,0)min + ( 6.8,9.1 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.4,3.5 ) ( 3.2,-0.1)@xmath44 $ ] ( -0.50,8.9)@xmath53 ( 1.55,8.3)(d ) ( 3.25,8.3)@xmath1-axis ( 6.8,8.8 ) ( 1.1,1.2 ) ( 0.12,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath45 $ ] ( 1.55,8.3)(e ) ( 2.7,8.3)nonmagnetic ( 7.8,8.8 ) ( 1.1,1.2 ) ( 0.12,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath45 $ ] ( 1.55,8.3)(f ) ( 3.25,8.3)@xmath3-axis ( 7.3,1.2 ) ( 7.6,8.2)(0,0)max ( 7.6,1.0)(0,0)min to complete the study of the in - plane anisotropy in the undoped compound the spin - dependent bands are investigated polarization - dependent along the two in - plane directions @xmath1 and @xmath3 for @xmath33ev and for comparison the isotropic bands of the nonmagnetic case . anisotropies due to the orthorhombic lattice distortion for the nonmagnetic case are very small and have no significant influence , as shown also in earlier work.@xcite the nonmagnetic bands for the polarizations @xmath16 and @xmath15 are shown in fig . [ fig_cal_bs00 ] ( b ) and ( e ) , respectively . already for the nonmagnetic case it becomes obvious , that more information can be deduced for light with a polarization parallel to the ferromagnetic chains . for a perpendicular light polarization the intensity for some bands decrease so strongly that they practically seem to vanish . this is however not due to a vanishing of the bands but only due to the strong intensity variation , i.e. matrix element effects , as already mentioned before and as seen in experiment . for the spin - polarized band structure with antiferromagnetic order along the @xmath1-axis the corresponding cases for @xmath16 and @xmath15 are shown in fig . [ fig_cal_bs00 ] ( a ) and ( d ) , respectively . the green solid lines are guides to the eye which emphasize the two important anisotropic bands in these spectra . first of all , there is some significant reorientation of the bands compared to the nonmagnetic case . most striking is the appearance of a steep double - u shaped band which was not visible in the nonmagnetic case . this new appearance is most likely due to down - folding of the brillouin zone when going from the 2-fe cell to a magnetic 4-fe cell . the second important band is a pure hole pocket which is compared to the nonmagnetic case shifted to higher binding energies . it should be noted that the intensity of this band is extremely polarization dependent . it is the dominating band for @xmath16 while it is barely visible for @xmath16 . comparing with the polarization dependent fermi surface in fig . [ fig_cal_afpol ] it is obvious that this band is also part of the bright intensity spots in the fermi surface along the @xmath1-direction and very characteristic for the anisotropy . it is also noteworthy that these two significant bands cross each other exactly at the fermi level . this crossing is also reported in experiment as can be seen from the black points in fig . [ fig_cal_bs00 ] ( a ) , which are reproduced from the experimental arpes data of yi _ et al._@xcite . thus , the experimental arpes data could be again well reproduced by the calculations . the situation for the magnetic bands with ferromagnetic order along the @xmath3-axis as shown in fig . [ fig_cal_bs00 ] ( c ) and ( f ) is in many aspects similar to that for the bands along the @xmath1-direction . one can identify two prominent anisotropic bands , one with a double - u like shape which has a higher intensity for a light polarization of @xmath15 , while the other band marked with the solid green line is the dominating one for light @xmath16 . the important difference is that these bands do not touch each other as they are significantly shifted away in binding energy . note that also no crossing is reported in experiment.@xcite why these steep bands with the double - u shape can not be seen in experiment for the @xmath3-direction gets also clear : the responsible band is simply completely shifted above the fermi level . this observation can in principle be compared to the band splitting in ferromagnets . note that along the @xmath3-axis there is ferromagnetic coupling while along the @xmath1-axis the magnetic order is antiferromagnetic . thus , for one sees along the ferromagnetic chains a band splitting of approximately @xmath54ev for a magnetic moment around @xmath55 . this is comparable for example to ni which shows a band splitting of approximately @xmath56ev for a moment of approximately @xmath57.@xcite consequently , for decreasing magnetic moments upon alloying one expects a reduced band splitting together with a continuous matching of the anisotropic bands . to investigate this issue in further detail one has to look at the evolution of the arpes band structure for increasing co substitution on the fe position which goes in hand with the reduction of the magnetic moments . ( 6.6,9.3 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.4,3.5 ) ( 3.2,-0.1)@xmath44 $ ] ( -0.50,8.9)*_a_-axis * ( 1.55,8.3)(a ) ( 3.0,8.3)@xmath58 ( 6.6,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath36 $ ] ( 1.55,8.3)(b ) ( 3.0,8.3)@xmath59 ( 6.6,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath36 $ ] ( 1.55,8.3)(c ) ( 3.0,8.3)@xmath60 ( 7.0,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath36 $ ] ( 1.55,8.3)(d ) ( 3.0,8.3)@xmath61 ( 7.0,1.2 ) ( 7.3,8.2)(0,0)max ( 7.3,1.0)(0,0)min + ( 6.6,9.3 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.4,3.5 ) ( 3.2,-0.1)@xmath45 $ ] ( -0.50,8.9)*_b_-axis * ( 1.55,8.3)(e ) ( 3.0,8.3)@xmath58 ( 6.6,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath45 $ ] ( 1.55,8.3)(f ) ( 3.0,8.3)@xmath59 ( 6.6,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath45 $ ] ( 1.55,8.3)(g ) ( 3.0,8.3)@xmath60 ( 7.0,8.8 ) ( 1.1,1.2 ) ( 0.25,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 1.05,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.2,-0.1)@xmath45 $ ] ( 1.55,8.3)(h ) ( 3.0,8.3)@xmath61 ( 7.0,1.2 ) ( 7.3,8.2)(0,0)max ( 7.3,1.0)(0,0)min substitution of fe with co in is one of the common ways to induce superconductivity in by electron doping . the substitution does consequently diminish the strength of the antiferromagnetic coupling within this compound until the long - range magnetic order collapses and superconductivity emerges.@xcite as the strength of the magnetic order decreases with co doping , in experiment as well as in the calculations , it can be assumed that the strong in - plane anisotropy does also decrease . the breakdown of the long - range antiferromagnetic order in fig . [ fig_magnmom ] appears for a somewhat higher co concentrations than in experiment . thus , the breakdown of the anisotropy is expected at higher doping levels . this is true for a concentration of @xmath61 in which is the first substitution level where the magnetic order did completely vanish in the self - consistent calculation . considering the evolution of the magnetic moments in fig . [ fig_magnmom ] one can see that the initial magnetic moment decreased to approximately @xmath62 , @xmath63 and @xmath64 of the original value for co substitutions of @xmath58 , @xmath65 and @xmath66 respectively . to investigate the impact of alloying in detail , the arpes band structure for these concentrations including the already nonmagnetic @xmath61 for both directions @xmath1 and @xmath3 is presented in fig . [ fig_cal_bsco ] . the calculations are performed for @xmath33ev comparable to fig . [ fig_cal_bs00 ] but they are only shown for a light polarization of @xmath16 because is was already clarified that the anisotropic bands can be best seen with this specific polarization . the black dashed lines in fig . [ fig_cal_bsco ] are shown for comparison and they correspond to the band position of the anisotropic bands in the undoped compound with the highest anisotropy , seen in fig . [ fig_cal_bs00 ] ( a ) and ( c ) . the green solid lines are guides to the eye to identify more easily the corresponding anisotropic bands for the specific co concentrations . the difference between black dashed lines and green solid lines is thus the change of the original anisotropy with increasing co substitution . for the case of the nonmagnetic with @xmath61 shown in fig . [ fig_cal_bsco ] ( d ) and ( h ) the anisotropy has completely vanished and the band structures coincide with each other . this could be expected from experiment and it is reproduced in the calculations . it should be noted again at this point , that the crystal lattice is still orthorhombic , however , the lattice anisotropy is indeed too weak to be visible in the band structure.@xcite some other interesting findings can be deduced from the evolution of the band structure upon co substitution . first of all the intensity of the double - u shaped band decreases continuously , however , it only completely disappears after the collapse of the long - range antiferromagnetic order . the change in anisotropy for the antiferromagnetic order along the @xmath1-axis is mostly characterized with the consequent shift of the hole - pocket to lower binding energies . this is also experimentally reported for a decrease in the magnetic coupling strength , either induced through co doping or increasing temperature.@xcite concerning this situation for the ferromagnetic order along the @xmath3-axis the most prominent feature is the shift of the double - u shaped band to lower binding energies . what can be seen in fig . [ fig_cal_bs00 ] ( e ) to ( h ) is that the energy difference of these two main anisotropic bands does strongly and continuously decrease . this is in agreement with the assumption of a smaller band splitting for decreasing ferromagnetic coupling strength . in summary one can say that for the antiferromagnetic order along the @xmath1-axis mostly the hole - pocket changes while the double - u shaped band stays more or less constant . for the ferromagnetic order along the @xmath3-axis it is the other way round . the double - u shaped band undergoes the strongest change while the other band stays more or less unchanged in energy and shape . the final result is the same in both cases , a matching of the bands and a consequent isotropic in - plane band structure . this detailed analysis allows to follow the change from the strong in - plane anisotropy of the undoped compound to the isotropic behavior in the co substituted system in a continuous way with direct correspondence to arpes . thus , this approach based on kkr - cpa proves its advantages for investigating the iron pnictide superconductors at regions of interest which are difficult to evaluate by means of other band structure methods . c|c ( 6.2,9.3 ) ( 1.1,1.2 ) ( 0.2,1.15)(0,1.48)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.10,0.9)(1.45,0)[l]-0.2 , 0.0 , 0.2 ( -0.5,2.8 ) ( 3.0,-0.0)@xmath44 $ ] ( 4.50,8.6)*as - terminated * ( 2.5,7.5)(a ) ( 5.5,7.5)fermi surface ( 7.0,1.2 ) ( 7.1,7.2)(0,0)max ( 7.1,1.0)(0,0)min ( 7.5,8.8 ) ( 1.1,1.2 ) ( 2.10,0.9)(1.45,0)[l]-0.2 , 0.0 , 0.2 ( 3.0,-0.0)@xmath36 $ ] ( 7.0,1.2 ) ( 7.1,7.2)(0,0)max ( 7.1,1.0)(0,0)min & ( 6.2,9.3 ) ( 1.1,1.2 ) ( 0.2,1.15)(0,1.48)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 2.10,0.9)(1.45,0)[l]-0.2 , 0.0 , 0.2 ( 3.0,-0.0)@xmath44 $ ] ( 4.50,8.6)*ba - terminated * ( 2.5,7.5)(c ) ( 5.5,7.5)fermi surface ( 7.0,1.2 ) ( 7.1,7.2)(0,0)max ( 7.1,1.0)(0,0)min ( 7.5,8.8 ) ( 1.1,1.2 ) ( 2.10,0.9)(1.45,0)[l]-0.2 , 0.0 , 0.2 ( 3.0,-0.0)@xmath36 $ ] ( 7.0,1.2 ) ( 7.1,7.2)(0,0)max ( 7.1,1.0)(0,0)min + ( 6.55,9.3 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 0.9,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( -0.4,3.5 ) ( 3.0,-0.0)@xmath44 $ ] ( 2.0,8.5)(b ) ( 3.0,8.5)band structure along @xmath1-axis ( 7.05,1.2 ) ( 7.25,8.2)(0,0)max ( 7.25,1.1)(0,0)min ( 7.4,8.8 ) ( 1.1,1.2 ) ( 0.9,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.0,-0.0)@xmath36 $ ] ( 7.0,1.2 ) ( 7.25,8.2)(0,0)max ( 7.22,1.0)(0,0)min & ( 6.55,9.3 ) ( 1.1,1.2 ) ( 0.2,1.25)(0,0.96)[l]-0.4,-0.3,-0.2,-0.1 , 0.0 , 0.1 , 0.2 , 0.3 ( 0.9,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.0,-0.0)@xmath44 $ ] ( 2.0,8.5)(d ) ( 3.0,8.5)band structure along @xmath1-axis ( 7.05,1.2 ) ( 7.25,8.2)(0,0)max ( 7.25,1.1)(0,0)min ( 7.4,8.8 ) ( 1.1,1.2 ) ( 0.9,0.9)(1.25,0)[l]-0.4,-0.2 , 0.0 , 0.2 , 0.4 ( 3.0,-0.0)@xmath36 $ ] ( 7.0,1.2 ) ( 7.25,8.2)(0,0)max ( 7.22,1.0)(0,0)min using the one - step model of photoemission one can identify different surface states and can thus clarify the origin of surface bands . the reason for the occurrence of surface - states has long been developed in multiple scattering theory , which is the underlying basis of the spr - kkr method.@xcite the so - called determinant condition uses the reflection matrices of the bulk crystal @xmath67 and of the surface barrier potential @xmath68 , which connects the inner potential of the bulk crystal with the vacuum level . the appearance of a surface state is given by the following condition : @xmath69 for better visualization we plot @xmath70 in the following . if this expression is bigger than approximately @xmath71 we speak about a surface state . for values between @xmath72 and @xmath71 the state is defined as a so - called surface resonance . for values below one has bulk states . more details can be found in the overview by braun and donath.@xcite the application of this determinant approach is demonstrated in fig . [ fig_cal_det ] . here one can see the fermi surfaces and the band structures along the @xmath1-axis for an as - terminated and a ba - terminated surface , respectively , together with the corresponding plot of @xmath70 on the right hand side of each picture . the bands are shown for @xmath33ev and they are averaged over the two light polarizations @xmath16 and @xmath15 in order to make all relevant contributions equally visible . it should be noted , that the determinant condition itself and without a high intensity in the corresponding band structure plot is only an indication for a surface state or a surface resonance . only if a high intensity in the @xmath70 plot coincides with a band in the band structure one can associate this band with a clear surface character . for example has the bright octagon shape of the determinant plot of the fermi surface in fig . [ fig_cal_det ] ( a ) not a corresponding counterpart in the band structure plot . the two high intensity spots along the @xmath1-axis which are equally visible in the fermi surface as well as in the determinant condition are in clear contrast to this behavior . thus , this feature has a surface related origin , more specifically a surface state as the intensity of @xmath70 is in the order of @xmath73 . this surface state can be also identified in the band structure along the @xmath1-axis as shown in fig . [ fig_cal_det ] ( b ) where a strong intensity in the determinant plot coincides with the steep bands that cut the fermi level and which are part of the already discussed double - u shape . consequently , these bands can be identified as surface states . this is in accordance with the earlier findings for the @xmath42-dispersion in fig . [ fig_cal_kz ] ( a ) where the vertical intensities at @xmath43 were independent on @xmath42 , indicating a connection to a surface related phenomenon . another verification for the surface related origin of these bands is shown in fig . [ fig_cal_det ] ( c ) and ( d ) , where the corresponding fermi surface and band structure are shown for a ba - terminated surface . as already indicated before all other calculations presented in this paper are under the assumption of an as - terminated surface . obviously , the assumed surface termination has also an influence on surface related phenomena and surface states might be shifted significantly in energy . indeed , the surface states discussed for the as - terminated surface have completely vanished in the ba - terminated case . the corresponding high intensities are missing in the @xmath70 plots and in the band structure of fig . [ fig_cal_det ] ( d ) the characteristic steep bands from the as - terminated surface have also vanished . no intensities in the determinant plot coincide with features in the band structure and thus surface effects have been removed by the ba termination . overall , the fermi surface and the band structure have undergone significant changes for the altered surface termination . the characteristic anisotropic features of the fermi surface in fig . [ fig_cal_det ] ( a ) are hardly visible in the ba - terminated case in fig . [ fig_cal_det ] ( c ) . it seems like the ba layer on top acts as some kind of damping layer which reduces the intensity and blurs the electronic states which are clearly visible in an as - terminated surface . in particular one has to note that the agreement with experimental arpes data is significantly better for an as - terminated surface compared to the ba - terminated one . especially the steep bands along the @xmath1-axis are seen in experiment@xcite and they could be successfully identified as surface states which are only visible for an as - terminated surface . this result can be used for conclusions on the most likely surface termination of . interestingly , the surface termination in this material is still not clear and under debate , although several experimental measurements and theoretical calculations exist.@xcite according to first principle calculations only three possibilities for the surface termination exist , namely a fully as - terminated or a fully ba - terminated surface as well as an as surface covered with half of the stoichiometric ba atoms.@xcite there are experimental scanning tunneling microscopy ( stm ) and low - energy electron - diffraction ( leed ) measurements which indicate a ba - terminated surface@xcite . however , there are also experimental stm + leed data which clearly favor an as - terminated surface@xcite . the arpes calculations clearly favor an as - terminated surface as it was shown that agreement with experiment is considerably better compared to the ba - terminated one . this can not rule out the possibility of some partial covering with few ba adatoms but one can state that every ba atoms on top muffles the electronic structure seen in experiment and that this structure is due to an as - terminated compound . so one would expect a more or less clean as - terminated surface as most probable surface termination for . additional covering with some ba atoms might be possible but also might have some degrading influence on the quality of the arpes measurement . the munich spr - kkr package was used for self - consistent and fully relativistic calculations of orthorhombic in its experimentally observed stripe antiferromagnetic ground - state for @xmath74 up to @xmath61 . the substitutional disorder induced by co on fe positions was dealt with on a cpa level which was earlier shown to be fully equal to a comprehensive supercell calculation.@xcite magnetic moments of @xmath23 for undoped were reproduced and additionally a reasonable magnetic behavior for increasing co substitution with a continuous decrease of the magnetic moments until a collapse of the antiferromagnetic order at 15% co concentration was reached . this is in good agreement with experimental behavior.@xcite concerning arpes most experimental data available is actually insufficient to talk about in - plane anisotropy due to twinning effects during the phase transition from the nonmagnetic tetragonal to the antiferromagnetic orthorhombic phase . a complicated detwinning process , typically with uniaxial stress on the single crystal , is necessary to gain anisotropic data of the electronic structure.@xcite referring to the available experimental data it was possible to reproduce the electronic structure of in very good agreement with experiment . the fermi surface shows all important anisotropic features , namely some bright spots of intensity along the antiferromagnetic order along the @xmath1-axis and more blurred pedals along the ferromagnetic order along the @xmath3-axis . also in agreement with experiment a strong dependence on the polarization of light was found , been either parallel to the ferromagnetic or to the antiferromagnetic order , indicating the strong multiorbital character . for comparison the fermi surface of the nonmagnetic phase as well as a hypothetical fermi surface for a twinned arpes measurement as a superposition of two antiferromagnetic cells rotated by @xmath19 to each other was shown . both were again in agreement with experiment . in addition to the anisotropic @xmath42 dispersion some focus was put on the anisotropic band structure along the @xmath1- and @xmath3-axes and it was compared to the nonmagnetic case . one could identify the important anisotropic bands and these could be interpreted in terms of band splitting for the ferromagnetic chains along the @xmath3-axis , principally comparable to typical ferromagnetic band splitting as observed for example in ni . in addition the evolution of these anisotropic bands for small steps of @xmath0 in was presented until the breakdown of long - range antiferromagnetic order . the decreasing band splitting and a continuous matching of the anisotropic bands could be reproduced in great detail and consistent with experimental findings . finally the so - called determinant condition @xmath70 was used to evaluate possible surface states of the band structure . it was possible to identify steep bands along the @xmath1-axis as surface states . these are at least partially responsible for the characteristic bright intensity spots in the electronic structure along the @xmath1-axis and can also be seen in experiment . interestingly , these surface states are only visible near the fermi level for an as - terminated surface . it was shown that a ba - terminated top - layer acts as some kind of damping which moves the surface states far away and blurs the electronic structure . significantly better agreement with experimental data is found for an as - terminated surface . this leads to the conclusion that an as - terminated surface would be most likely , an issue that is in fact experimentally not convincingly clarified.@xcite some ba adatoms might be still possible but one would expect a negative influence on the quality of the measurements . to conclude , this publication was successful in reproducing the strong in - plane anisotropy of and its behavior under substitution in very good agreement with experiment using arpes calculations . these calculations allow even predictions on possible surface terminations . we acknowledge the financial support from the deutsche forschungsgemeinschaft dfg ( projects for 1346 ) and from the bundesministerium fr bildung und forschung bmbf ( project 05k13wma ) . we further thank for the support from centem plus ( l01402 ) . j. fink , s. thirupathaiah , r. ovsyannikov , h. a. drr , r. follath , y. huang , s. de jong , m. s. golden , y .- z . zhang , h. o. jeschke , r. valent , c. felser , s. dastjani farahani , m. rotter , and d. johrendt , phys . b * 79 * , 155118 ( 2009 ) . l. x. yang , y. zhang , h. w. ou , j. f. zhao , d. w. shen , b. zhou , j. wei , f. chen , m. xu , c. he , y. chen , z. d. wang , x. f. wang , t. wu , g. wu , x. h. chen , m. arita , k. shimada , m. taniguchi , z. y. lu , t. xiang , and d. l. feng , phys . lett . * 102 * , 107002 ( 2009 ) . h. gretarsson , a. lupascu , j. kim , d. casa , t. gog , w. wu , s. r. julian , z. j. xu , j. s. wen , g. d. gu , r. h. yuan , z. g. chen , n .- l . wang , s. khim , k. h. kim , m. ishikado , i. jarrige , s. shamoto , j .- h . chu , i. r. fisher , and y .- j . kim , phys . b * 84 * , 100509 ( 2011 ) . p. vilmercati , a. fedorov , f. bondino , f. offi , g. panaccione , p. lacovig , l. simonelli , m. a. mcguire , a. s. m. sefat , d. mandrus , b. c. sales , t. egami , w. ku , and n. mannella , phys . b * 85 * , 220503 ( 2012 ) . v. b. zabolotnyy , d. s. inosov , d. v. evtushinsky , a. koitzsch , a. a. kordyuk , g. l. sun , j. t. park , d. haug , v. hinkov , a. v. boris , c. t. lin , m. knupfer , a. n. yaresko , b. buchner , a. varykhalov , r. follath , and s. v. borisenko , nature * 457 * , 569 ( 2009 ) . d. v. evtushinsky , d. s. inosov , v. b. zabolotnyy , a. koitzsch , m. knupfer , b. bchner , m. s. viazovska , g. l. sun , v. hinkov , a. v. boris , c. t. lin , b. keimer , a. varykhalov , a. a. kordyuk , and s. v. borisenko , phys . b * 79 * , 054517 ( 2009 ) . d. v. evtushinsky , d. s. inosov , v. b. zabolotnyy , m. s. viazovska , r. khasanov , a. amato , h .- h . klauss , h. luetkens , c. niedermayer , g. l. sun , v. hinkov , c. t. lin , a. varykhalov , a. koitzsch , m. knupfer , b. bchner , a. a. kordyuk , and s. v. borisenko , new journal of physics * 11 * , 055069 ( 2009 ) . d. v. evtushinsky , v. b. zabolotnyy , t. k. kim , a. a. kordyuk , a. n. yaresko , j. maletz , s. aswartham , s. wurmehl , a. v. boris , d. l. sun , c. t. lin , b. shen , h. h. wen , a. varykhalov , r. follath , b. bchner , and s. v. borisenko , phys . b * 89 * , 064514 ( 2014 ) . m. yi , d. lu , j .- h . chu , j. g. analytis , a. p. sorini , a. f. kemper , b. moritz , s .- k . mo , r. g. moore , m. hashimoto , w .- s . lee , z. hussain , t. p. devereaux , i. r. fisher , and z .- x . shen , proc . ac . sci . us . * 108 * , 6878 ( 2011 ) . a. a. aczel , e. baggio - 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by means of one - step model calculations the strong in - plane anisotropy seen in angle - resolved photoemission of the well - known iron pnictide prototype compounds and in their low - temperature antiferromagnetic phases is investigated . the fully - relativistic calculations are based on the korringa - kohn - rostoker - green function approach combined with the coherent potential approximation alloy theory to account for the disorder induced by co substitution on fe sites in a reliable way . the results of the calculations can be compared directly to experimental spectra of detwinned single crystals . one finds very good agreement with experiment and can reveal all features of the electronic structure contributing to the in - plane anisotropy . in particular the local density approximation can capture most of the correlation effects for the investigated system without the need for more advanced techniques . in addition , the evolution of the anisotropy for increasing co concentration @xmath0 in can be tracked almost continuously . the results are also used to discuss surface effects and it is possible to identify clear signatures to conclude about different types of surface termination .

arxiv

the m31 globular cluster ( gc ) b158 ( a.k.a . bo 158 ) , named following the revised bologna catalogue v3.4 @xcite , contains a bright x - ray source that was discovered by the einstein observatory @xcite , and has shown up in every x - ray observation of the region since ; we call this x - ray source xb158 . xb158 exhibited strong intensity modulation on a 10017@xmath050 s ( @xmath12.78 hr ) period during the 2002 january xmm - newton observation @xcite . @xcite found similar variation in the folded lightcurves from a 1991 june rosat observation and a 2000 june xmm - newton observation ; they found that the amplitude of modulation decreased with increasing source intensity . they found no such modulation in the 2001 june xmm - newton observation , setting a 2@xmath3 upper limit of 10% modulation . assuming that this represents the orbital period , @xcite found that this is probably a neutron star binary with a low mass donor with a separation @xmath510@xmath6 cm ( i.e. a low mass xb - ray binary , lmxb ) . however , analysis of the unfolded 2000 june xmm - newton lightcurve revealed a single deep dip at the end of the observation , with no evidence for dips in the two previous orbital cycles @xcite . furthermore , @xcite analyzed 3 proprietary xmm - newton observations over 2004 july 1719 , finding @xmath1100% dipping for one orbital cycle , and zero evidence for dips in other cycles ; we concluded that the disk was precessing , with dips only visible for some part of the super - orbital cycle . such behavior is associated with the `` superhump '' phenomenon that is observed in accreting binaries where the mass ratio is smaller than @xmath10.3 @xcite . superhumps were first identified in the superoutbursts of the su uma subclass of cataclysmic variables ( accreting white dwarf binaries ) , manifesting as a periodic increase in the optical brightness on a period that is slightly longer than the orbital period @xcite . su umas are a subclass of dwarf novae with orbital periods @xmath72 hr , that exhibit occasional superoutbursts that last @xmath85 times as long as the normal outbursts @xcite . @xcite proposed that these superoutbursts are enhanced by a tidal instability that occurs when the outer disk crosses the 3:1 resonance with the secondary ; the additional tidal torque causes the disk to elongate and precess , and also greatly enhances the loss of angular momentum ( and therefore the accretion rate ) . the disk precession is prograde in the rest frame , and the secondary repeats its motion with respect to the disk on the beat period between the orbital period and the precession period , a few percent longer than the orbital period . the secondary modulates the disk s viscous dissipation on this period , giving rise to the maxima in the optical lightcurve known as superhumps . some short period , persistently bright cvs exhibit permanent superhumps @xcite . cccccccccccc obs & @xmath9 & ra & dec & xrt exp & @xmath10/@xmath11 & counts + + 00032702001 & 0.0 & 00 43 00.81 & + 41 17 00.2 & 2234 & 10.0 & 57 + 00032702002 & 0.9 & 00 43 06.53 & + 41 16 54.6 & 2510 & 9.7 & 41 + 00032702003 & 2.0 & 00 43 03.73 & + 41 16 40.8 & 2299 & 9.5 & 13 + 00032702004 & 3.0 & 00 43 06.35 & + 41 16 51.3 & 2299 & 9.6 & 40 + 00032702005 & 3.1 & 00 43 13.10 & + 41 16 30.4 & 2489 & 9.2 & 43 + 00032702006 & 4.4 & 00 43 03.22 & + 41 13 58.5 & 2572 & 7.0 & 65 + 00032702007 & 5.4 & 00 43 10.44 & + 41 15 10.9 & 2511 & 7.9 & 51 + 00032702008 & 6.4 & 00 43 06.44 & + 41 16 06.4 & 2635 & 8.9 & 57 + 00032702009 & 7.3 & 00 43 11.97 & + 41 16 27.7 & 2476 & 9.1 & 29 + 00032702010 & 8.8 & 00 43 03.76 & + 41 16 15.5 & 2644 & 9.1 & 39 + 00032702011 & 9.5 & 00 43 08.40 & + 41 17 27.6 & 2405 & 10.2 & 53 + 00032702012 & 10.1 & 00 43 07.20 & + 41 16 29.8 & 2502 & 9.2 & 43 + 00032702013 & 11.2 & 00 43 03.63 & + 41 17 36.2 & 2514 & 10.5 & 48 + 00032702014 & 12.4 & 00 43 04.82 & + 41 17 00.3 & 2411 & 9.8 & 38 + 00032702015 & 13.2 & 00 43 07.68 & + 41 15 24.0 & 2490 & 8.2 & 11 + 00032702016 & 14.2 & 00 43 01.12 & + 41 15 11.3 & 2511 & 8.2 & 27 + 00032702017 & 15.1 & 00 43 06.35 & + 41 14 46.6 & 2273 & 7.6 & 54 + 00032702018 & 16.8 & 00 43 02.79 & + 41 17 00.9 & 2314 & 9.9 & 55 + 00032702019 & 17.1 & 00 43 08.59 & + 41 15 26.4 & 1992 & 8.2 & 45 + 00032702020 & 18.1 & 00 43 04.88 & + 41 17 31.5 & 1389 & 10.3 & 13 + 00032702021 & 19.4 & 00 43 03.24 & + 41 17 28.0 & 2690 & 10.3 & 18 + 00032702022 & 20.2 & 00 43 02.68 & + 41 17 26.6 & 2686 & 10.3 & 42 + 00032702023 & 21.1 & 00 43 03.07 & + 41 17 07.0 & 2821 & 10.0 & 65 + 00032702024 & 22.3 & 00 43 04.56 & + 41 16 19.5 & 2659 & 9.2 & 50 + 00032702025 & 23.2 & 00 43 10.49 & + 41 16 54.1 & 2711 & 9.6 & 46 + 00032702026 & 24.3 & 00 43 07.86 & + 41 15 05.1 & 2740 & 7.8 & 35 + 00032702027 & 25.1 & 00 43 06.42 & + 41 16 41.0 & 2424 & 9.5 & 18 + 00032702028 & 26.1 & 00 43 10.04 & + 41 16 54.2 & 2550 & 9.6 & 49 + 00032702029 & 27.2 & 00 43 10.52 & + 41 17 33.9 & 2524 & 10.2 & 59 + 00032702030 & 28.4 & 00 43 06.50 & + 41 17 25.9 & 2331 & 10.2 & 41 + 4u 1916@xmath12053 is a high inclination neutron star lmxb with an x - ray period of [email protected] min and an optical period [email protected] min @xcite . it exhibits periodic x - ray intensity dips due to absorption by material in the outer disk ; the amplitude of these dips varies over @xmath10 to @xmath1100% on a @xmath14 day period , the precession period of the disk @xcite . @xcite showed that ns lmxbs with orbital periods shorter than @xmath14.2 hr are likely to exhibit superhumps , and identified 4u 1916 - 053 as a persistent superhumping source . xb158 appears to be somewhat analogous to 4u 1916@xmath12053 @xcite . in @xcite we modeled the 2004 july 17 xmm - newton pn spectrum of xb158 with a blackbody and a power law , finding @xmath13 = [email protected] kev , the photon index @xmath14 = [email protected] , @xmath4/dof = 19/19 and the 0.310 kev luminosity was [email protected]@xmath15 erg s@xmath16 ; fitting a single power law emission model yielded a photon index of [email protected] , which is harder than any black hole spectrum @xcite , meaning that the accretor is a neutron star . we conducted three dimensional smoothed particle hydrodynamical ( sph ) modeling , assuming a 1.4 @xmath17 ns , a 2.78 hr period , a total system mass of 1.8 @xmath17 , and a luminosity @xmath180% of the eddington limit . as a result , we estimated the disk precession period to be 29@xmath01 times the orbital period , i.e. 81@xmath03 hr . in @xcite we examined chandra observations of @xmath130 m31 globular cluster x - ray sources spanning @xmath112 years ( 89 acis and 45 hrc ) , including xb158 . we observed 0.310 kev luminosities @xmath1420@xmath18 erg s@xmath16 , and proposed that this is due to varying accretion rates over the disk precession cycle . since we expected a super - orbital period for xb158 over time - scales of a few days , we obtained a series of 30 swift observations , each with 2.5 ks exposure and spaced @xmath11 day apart . in this work we present our analysis of these swift results , and find evidence for a super - orbital period of @xmath16 days we made 30 swift observations over 2013 february 8 march 9 ( obsids 00327020010032702030 , pi r. barnard ) . xb158 was one of two main targets for this survey , and the pointing was chosen to optimize the results from both targets . however , the actual pointings varied significantly over the 30 observations . a journal of swift observations is provided in table [ journ ] ; for each observation we give the time relative to the first observation , pointing , xrt exposure , and off - axis angle , along with the net source counts . for each observation , we placed circular regions around xb158 and a suitable background region . we obtained the net source counts using the `` counts in regions '' tool in the ds9 image viewer , and estimated the intensity by dividing the net counts by the exposure time . these data were obtained in order to determine the extent to which the varying off - axis angles affected our results . we also created spectra from the same extraction regions , created appropriate ancillary response files using xrtmkarf , and found the appropriate response file using the quzcif tool . none of the spectra were suitable for free spectral fitting , hence we obtained luminosity estimates by assuming a model obtained from previous observations . for chandra acis observations with @xmath2200 net source photons , the mean line - of - sight absorption ( @xmath19 ) was 9@xmath04@xmath20 atom @xmath21 ( @xmath4/dof = 7/11 ) , and the mean power law index ( @xmath14 ) was [email protected] ( @xmath4/dof = 5/11 ) . this is consistent with the best absorbed power law fit to the 2004 july 17 xmm - newton observation ( @xmath19 = 0.1 , @xmath14 = [email protected] , @xmath4/dof = 30/24 * ? ? ? as we noted earlier , that xmm - newton spectrum was best described by a blackbody + power law model , but neither the chandra nor swift spectra were sufficient to constrain the two - component emission model . however , we were able to estimate the luminosity for each observation by assuming a fixed emission model , allowing only the normalization to vary . we fitted each spectrum using xspec 12.8.2b , fixing @xmath19 = 9@xmath20 atom @xmath21 and @xmath14 = 0.52 , to find the normalization required to make absorbed model intensity 1.00 count s@xmath16 . we then calculated the unabsorbed flux for this model , allowing us to convert from intensity to flux . multiplying the conversion by the background - subtracted intensity provided by xspec yielded the instrument - corrected , background - subtracted source flux . the luminosity was calculated from the flux assuming a distance of 780 kpc @xcite . we fitted the lightcurve with constant and sinusoidal components using the qdp program provided in heatools , performing a simple search for periodicity . @xcite created a periodogram suitable for unbinned data with a mean of zero that produces exactly equivalent results to such least - squares fitting of sinewaves , but also allows comparison of the best period with other periods . the likelihood that any peak in the periodogram is real is given by the false alarm probability ( @xmath22 ) , where a low value of @xmath22 indicates that the peak is likely to be significant . if the highest frequency to be probed is @xmath23 times higher than the lowest frequency , then the power , @xmath24 , required for a false alarm probability @xmath22 is given by @xmath24 = @xmath25 $ ] ; for small @xmath22 , @xmath26 @xcite . we present our 30 day , 0.310 kev swift xrt lightcurve of xb158 in figure [ swiftlc ] . we see that the 0.310 kev luminosity varied by a factor @xmath15 ; the luminosity dropped from @xmath12.3@xmath15 erg s@xmath16 to @xmath15@xmath18 erg s@xmath16 in @xmath12 days , assuming the mean chandra absorbed power law model . we note that these swift observations are non - contiguous , spacing the 2.5 ks observing time over several hours ; the low intensities ( @xmath10.0050.025 count s@xmath16 ) meant that there was no appreciable variability within each observation . we find no evidence for a dependence of luminosity on off - axis angle ; the luminosities for the observations when xb158 have the largest and smallest off - axis angles have consistent values , while observations at an off - axis angle of 8.2@xmath11 resulted in a factor @xmath15 range in luminosity . the conversion factor for translating 1 count s@xmath16 into 0.310 kev flux ranged over 1.091.29@xmath27 erg @xmath21 count@xmath16 for all observations except the second one , where it was 1.79@xmath27 erg @xmath21 count@xmath16 . this @xmath110% variation about the mean in instrumental correction is clearly not sufficient to account for the factor @xmath15 variation in luminosity . fitting our lightcurve with a constant intensity yielded a best fit luminosity of @xmath11.3@xmath15 erg s@xmath16 , with @xmath4 = 181 for 29 degrees of freedom ( dof ) . however , adding a sinusoidal variation component yielded a much improved fit , with @xmath4/dof = 43/26 ; for this model , the period is [email protected] days , with an amplitude of [email protected]@xmath18 erg s@xmath16 around a mean luminosity of [email protected]@xmath15 erg s@xmath16 , and a phase of [email protected] degrees . all uncertainties in this work are quoted at the 1@xmath3 level . this sinusoidal variability yielded @xmath28 = 138 for 30 bins , with 3 extra free parameters ; f - testing showed that the probability for this improvement being due to chance was 3@xmath29 , equivalent to a @xmath25@xmath3 detection . in figure [ swift_per ] we show our swift lightcurve folded on a 5.65 day period and fitted with the best fit sinusoid . we present our lomb - scargle periodogram for the 0.310 kev swift xrt luminosity lightcurve of xb158 in figure [ ls ] , and also indicate the power required for false alarm probabilities @xmath22 = 0.5 , 0.05 , and 0.005 for reference . we tested 30 frequencies , and oversampled each frequency by a factor of 50 ; this resulted in a periodogram covering a wider range of periods than is interesting ( up to @xmath11500 days ) , so we show only part of the periodogram here . the periodogram shows a single strong peak , with the maximum power of 10.4 at a period of 5.60 days , corresponding to @xmath22 = 0.0009 ; the range of periods which are detected at a @xmath23@xmath3 level is 5.45.8 days . while there is a small peak at the 3 day period , @xmath22 = 1 for this peak xb158 is a high inclination x - ray binary associated with the m31 globular cluster b158 . it exhibits deep intensity dips on a 2.78 hr period in some observations but not others , prompting @xcite to suggest that the disk is precessing , caused by the `` superhumping '' phenomenon observed in low mass ratio systems where the disk crosses the 3:1 resonance with the donor star . @xcite predicted a disk precession period of 29@xmath01 times the orbital period , i.e. 81@xmath03 hr , inspiring a month of daily monitoring of the m31 central region by swift . fitting a sinusoid to the lightcurve revealed a [email protected] day super - orbital period ( 1@xmath3 uncertainties ) ; the 0.310 kev luminosity varied by a factor @xmath15 , which is consistent with the range of luminosities observed in the acis observations of our 13 + year chandra monitoring campaign @xcite . the peak of the lomb - scargle periodogram is at 5.6 days , consistent with that obtained from least squares fitting of the lightcurve with a sinusoid . the suggested super - orbital period is @xmath170% longer than predicted by our 3d sph simulations @xcite . none of the authors of the current paper are experts in sph ; however , j. r. murray stated in a private communication that the longer than expected super - orbital period is likely due to the mass ratio of the donor to the accretor being lower than assumed . the mass of a roche lobe filling main sequence star , @xmath30 , may be approximated as @xmath30 @xmath31 0.11 @xmath32 , where @xmath32 is the orbital period in hours @xcite , and we originally assumed a donor mass of @xmath10.30 @xmath17 . for the accretor , we assumed a 1.4 @xmath33 neutron star . a power law emission model fit to the 2004 july 16 xmm - newton spectrum of b158 yielded a photon index of [email protected] @xcite , which is considerably harder than any spectrum emitted by a black hole binary . however , some neutron stars have masses @xmath22@xmath17 ( see e.g. * ? ? ? * ; * ? ? ? * ) , and it is possible that xb158 contains a particularly massive neutron star . we noted in @xcite that other xbs such as xb146 exhibited strong luminosity fluctuations between fairly consistent maxima and minima during our chandra monitoring observations , and suggested that this long - term behavior could be indicative of a short period / low mass ratio system . our new findings support this hypothesis . we thank the anonymous referees for suggesting improvements to this work , in particular prompting more rigorous estimation of the super - orbital period . we thank the swift team for making this work possible . this work was funded by swift grant nnx13aj76 g .

the m31 globular cluster x - ray binary xb158 ( a.k.a . bo 158 ) exhibits intensity dips on a 2.78 hr period in some observations , but not others . the short period suggests a low mass ratio , and an asymmetric , precessing disk due to additional tidal torques from the donor star since the disk crosses the 3:1 resonance . previous theoretical 3d smoothed particle hydrodynamical modeling suggested a super - orbital disk precession period 29@xmath01 times the orbital period , i.e. @xmath181@xmath03 hr . we conducted a swift monitoring campaign of 30 observations over @xmath11 month in order to search for evidence of such a super - orbital period . fitting the 0.310 kev swift xrt luminosity lightcurve with a sinusoid yielded a period of [email protected] days , and a @xmath25@xmath3 improvement in @xmath4 over the best fit constant intensity model . a lomb - scargle periodogram revealed that periods 5.45.8 days were detected at a @xmath23@xmath3 level , with a peak at 5.6 days . we consider this strong evidence for a 5.65 day super - orbital period , @xmath170% longer than the predicted period . the 0.310 kev luminosity varied by a factor @xmath15 , consistent with variations seen in long - term monitoring from chandra . we conclude that other x - ray binaries exhibiting similar long - term behaviour are likely to also be x - ray binaries with low mass ratios and super - orbital periods .

arxiv

the methods used to calculate the energy and the lifetime of a resonance state are numerous @xcite and , in some cases , has been put forward over strong foundations @xcite . however , the analysis of the numerical results of a particular method when applied to a given problem is far from direct . the complex scaling ( complex dilatation ) method @xcite , when the hamiltonian @xmath5 allows its use , reveals a resonance state by the appearance of an isolated complex eigenvalue on the spectrum of the non - hermitian complex scaled hamiltonian , @xmath6 @xcite . of course in an actual implementation the rotation angle @xmath7 must be large enough to rotate the continuum part of the spectrum beyond the resonance s complex eigenvalue . moreover , since most calculations are performed using finite variational expansions it is necessary to study the numerical data to decide which result is the most accurate . to worsen things the variational basis sets usually depend on one ( or more ) non - linear parameter . for bound states the non - linear parameter is chosen in order to obtain the lowest variational eigenvalue . for resonance states things are not so simple since they are embedded in the continuum . the complex virial theorem together with some graphical methods @xcite allows to pick the best numerical solution of a given problem , which corresponds to the stabilized points in the @xmath7 trajectories @xcite . other methods to calculate the energy and lifetime of the resonance , based on the numerical solution of complex hamiltonians , also have to deal with the problem of which solutions ( complex eigenvalues ) are physically acceptable . for example , the popular complex absorbing potential method , which in many cases is easier to implement than the complex scaling method , produces the appearance of nonphysical complex energy stabilized points that must be removed in order to obtain only the physical resonances @xcite . the aforementioned issues explain , at some extent , why the methods based only in the use of real @xmath2 variational functions are often preferred to analyze resonance states . these techniques reduce the problem to the calculation of eigenvalues of real symmetric matrices @xcite . of course , these methods also have its own drawbacks . one of the main problems was recognized very early on ( see , for example , the work by hol@xmath8ien and midtdal @xcite ) : if the energy of an autoionizing state is obtained as an eigenvalue of a finite hamiltonian matrix , which are the convergence properties of these eigenvalues that lie in the continuum when the size of the hamiltonian matrix changes ? but in order to obtain resonance - state energies it is possible to focus the analysis in a global property of the variational spectrum : the density of states ( dos)@xcite , being unnecessary to answer this question . the availability of the dos allows to obtain the energy and lifetime of the resonance in a simple way , both quantities are obtained as least square fitting parameters , see for example @xcite . despite its simplicity , the determination of the resonance s energy and width based in the dos is far from complete . there is no a single procedure to asses both , the accuracy of the numerical findings and its convergence properties , or which values to pick between the several `` candidates '' that the method offers @xcite . recently , pont _ et al _ @xcite have used _ finite size scaling _ arguments @xcite to analyze the properties of the dos when the size of the hamiltonian changes . they presented numerical evidence about the critical behavior of the density of states in the region where a given hamiltonian has resonances . the critical behavior was signaled by a strong dependence of some features of the density of states with the basis - set size used to calculate it . the resonance energy and lifetime were obtained using the scaling properties of the density of states . however , the feasibility of the method to calculate the resonance lifetime laid on the availability of a known value of the lifetime , making the whole method dependent on results not provided by itself . the dos method relies on the possibility to calculate the ritz - variational eigenfunctions and eigenvalues for many different values of the non - linear parameter @xmath9 ( see kar and ho @xcite ) . for each basis - set size , @xmath3 , used , there are @xmath3 variational eigenvalues @xmath10 . each one of these eigenvalues can be used , at least in principle , to compute a dos , @xmath11 , resulting , each one of these dos in an approximate value for the energy , @xmath12 , and width , @xmath13 , of the resonance state of the problem . if the variational problem is solved for many different basis - set sizes , there is not a clear cut criterion to pick the `` better '' result from the plethora of possible values obtained . this issue will be addressed in section [ model ] . in this work , in order to obtain resonance energies and lifetimes , we calculate all the eigenvalues for different basis - set sizes , and present a recipe to select adequately certain values of @xmath3 , and one eigenvalue for each @xmath3 elected , that is , we get a series of variational eigenvalues @xmath14 . the recipe is based on some properties of the variational spectrum which are discussed in section [ some - properties ] . the properties seem to be fairly general , making the implementation of the recipe feasible for problems with several particles . actually , because we use scaling properties for large values of @xmath3 , the applicability of the method for systems with more than three particles could be restricted because the difficulties to handle large basis sets . the set of approximate resonance energies , obtained from the density of states of a series of eigenvalues selected following the recipe , shows a very regular behaviour with the basis set size . this regular behaviour facilitates the use of finite size scaling arguments to analyze the results obtained , in particular the extrapolation of the data when @xmath15 . the extrapolated values are the most accurate approximation for the parameters of the resonance state that we obtain with our method . this is the subject of section [ recipe ] , where we present results for models of one and two particles . following the same prescription to choose particular solutions of the variational problem we obtain a set of approximate widths in section [ golden - rule ] . using the scaling function that characterizes the behaviour of the approximate energies as a guide , it is possible to find a very good approximation to the resonance width since , again , the data generated using our prescription seems to converge when @xmath4 . finally , in section [ discusion ] we summarize and discuss our results . when one is dealing with the variational spectrum in the continuum region , some of its properties are not exploited to obtain more information about the presence of resonances , usually the focus of interest is the stabilization of the individual eigenvalues . the stabilization is achieved varying some non - linear variational parameter . if @xmath9 is the inverse characteristic decaying length of the variational basis functions , then the spectrum of the kinetic energy scales as @xmath16 , moreover , for potentials that decay fast enough , the spectrum of the whole hamiltonian _ also _ scales as @xmath16 for large ( or small ) enough values of @xmath9 ( see appendix ) . this is so , since the variational eigenfunctions are @xmath2 approximations to plane waves _ except _ when @xmath9 belongs to the stabilization region . when @xmath9 belong to the stabilization region of a given variational eigenvalue , say @xmath17 , then @xmath18 ( where @xmath0 is the resonance energy ) and the variational eigenfunction @xmath19 has the localization length of the potential well . we intend to take advantage of the changes of the spectrum when @xmath9 goes from small to large enough values . the variational spectrum satisfies the hylleras - undheim theorem or variational theorem : if @xmath3 is the basis set size , and @xmath20 is the @xmath21 eigenvalue obtained with a variational basis set of size @xmath3 , then @xmath22 actually , since the threshold of the continuum is an accumulation point , then for small enough values of @xmath9 and a given @xmath23 there is always a @xmath24 such that @xmath25 for the kinetic energy variational eigenvalues , and for fixed @xmath26 , and @xmath27 , if the ordering given by equation ( [ ordering ] ) holds for some @xmath9 then it is true for all @xmath9 . of course this is not true for a hamiltonian with a non zero potential that support resonance states . so , we will take advantage of the variational eigenvalues such that for @xmath9 small enough satisfies equation ( [ ordering ] ) but , for @xmath9 large enough @xmath28 despite its simplicity , the arguments above give a complete prescription to pick a set of eigenstates that are particularly affected by the presence of a resonance . choose @xmath3 and @xmath29 arbitrary , and then look for the smaller values of @xmath30 and @xmath27 such that the two inequalities , equations . ( [ ordering ] ) and ( [ reverse ] ) are fulfilled . so far , all the examples analyzed by us show that if the inequalities are satisfied for some @xmath30 and @xmath27 then they are satisfied too by the eigenvalues @xmath31 , for @xmath32 . to illustrate how our prescription works we used two different model hamiltonians . the first model , due to hellmann @xcite , is a one particle hamiltonian that models a @xmath33-electron atom . the second one is a two particle model that has been used to study the low energy and resonance states of two electrons confined in a semiconductor quantum dot @xcite . the details of the variational treatment of both models will be kept as concise as possible . the one particle model has been used before for the determination of critical nuclear charges for @xmath33-electron atoms @xcite , it also gives reasonable results for resonance states in atomic anions @xcite and continuum states @xcite . the interaction of a valence electron with the atomic core is modeled by a one - particle potential with two asymptotic behaviours . the potential behaves correctly in the regions where electron is far from the atomic core ( @xmath34 electrons and the nucleus of charge @xmath35 ) and when it is near the nucleus . the hamiltonian , in scaled coordinates @xmath36 , is @xmath37 where @xmath38 and @xmath39 is a range parameter that determines the transition between the asymptotic regimes , for distances near the nucleus @xmath40 and in the case @xmath41 the nucleus charge is screened by the @xmath34 localized electrons and @xmath42 . another advantage of the potential comes from its analytical properties . in particular this potential is well behaved and the energy of the resonance states can be calculated using complex scaling methods . so , besides its simplicity , the model potential allows us to obtain the energy of the resonance by two independent methods and check our results . the two particle model that we considered describes two electrons interacting via the coulomb repulsion and confined by an external potential with spherical symmetry . we use a short - range potential suitable to apply the complex scaling method . the hamiltonian @xmath5 for the system is given by @xmath43 where @xmath44 , @xmath45 the position operator of electron @xmath46 ; @xmath47 and @xmath48 determine the range and depth of the dot potential . after re - scaling with @xmath47 , in atomic units , the hamiltonian of equation ( [ hamiltoniano ] ) can be written as @xmath49 where @xmath50 . the variational spectrum of the two particle model , equation ( [ hamil2part ] ) , and all the necessary algebraic details to obtain it , has been studied with great detail in reference @xcite so , until the end of this section , we discuss the variational solution of the one particle model given by equation ( [ hamil ] ) . the discrete spectrum and the resonance states of the model given by equation ( [ hamil ] ) can be obtained approximately using a real @xmath51 truncated basis set @xmath52 to construct a @xmath53 hamiltonian matrix @xmath54 . we use the rayleigh - ritz variational method to obtain the approximations @xmath55 @xmath56 for bound states this functions are variationally optimal . the functions @xmath57 are @xmath58 and @xmath59 are the associated laguerre polynomials of @xmath60 order and degree @xmath61 . the non - linear parameter @xmath9 is used for eigenvalue stabilization in resonance analysis @xcite . note that @xmath9 plays a similar role that the finite size of the box in spherical box stabilization procedures @xcite , as stated by kar _ et . @xcite . resonance states are characterized by isolated complex eigenvalues , @xmath62 , whose eigenfunctions are not square - integrable . these states are considered as quasi - bound states of energy @xmath0 and inverse lifetime @xmath1 . for the hamiltonian equation ( [ hamil ] ) , the resonance energies belong to the positive energy range @xcite . using the approximate solutions of hamiltonian ( [ hamil ] ) we analyze the dos method @xcite that has been used extensively to calculate the energy and lifetime of resonance states , in particular we intend to show that 1 ) the dos method provides a host of approximate values whose accuracy is hard to assess , and 2 ) if the dos method is supplemented by a new optimization rule , it results in a convergent series of approximate values for the energy and lifetime of resonance states . the dos method relies on the possibility to calculate the ritz - variational eigenfunctions and eigenvalues for many different values of the non - linear parameter @xmath9 ( see kar and ho @xcite ) . the localized dos @xmath63 can be expressed as @xcite @xmath64 since we are dealing with a numerical approximation , we calculate the energies in a discretization @xmath65 of the continuous parameter @xmath9 . in this approximation , equation ( [ densidad_sin_suma ] ) can be written as @xmath66 where @xmath67 is the @xmath30-th eigenvalue of the @xmath53 matrix hamiltonian with @xmath68 and @xmath39 fixed . in complex scaling methods the hamiltonian is dilated by a complex factor @xmath69 . as was pointed out long time ago by moiseyev and coworkers @xcite , the role played by @xmath9 and @xmath70 are equivalent , in fact , our parameter @xmath9 corresponds to @xmath71 . besides , the dos attains its maximum at optimal values of @xmath9 and @xmath0 that could be obtained with a self - adjoint hamiltonian without using complex scaling methods @xcite . so , locating the position of the resonance using the maximum of the dos is equivalent to the stabilization criterion used in complex dilation methods that requires the approximate fulfillment of the complex virial theorem @xcite . the values of @xmath72 and @xmath73 are obtained performing a nonlinear fitting of @xmath63 , with a lorentzian function , @xmath74}.\ ] ] one of the drawbacks of this method results evident : for each pair @xmath75 there are several @xmath11 , and since each @xmath11 provides a value for @xmath76 and @xmath77 one has to choose which one is the best . kar and ho @xcite solve this problem fitting all the @xmath11 and keeping as the best values for @xmath0 and @xmath1 the fitting parameters with the smaller @xmath78 value . at least for their data the best fitting ( the smaller @xmath78 ) usually corresponds to the larger @xmath30 . this fact has a clear interpretation , if the numerical method approximates @xmath0 with some @xmath79 , where @xmath3 is the basis set size of the variational method , a large @xmath30 means that the numerical method is able to provide a large number of approximate levels , and so the continuum of positive - energy states is `` better '' approximated . in a previous work @xcite we have shown that a very good approximation to the energy of the resonance state is obtained considering just the energy value where @xmath11 attains its maximum . we denote this value as @xmath76 . figure [ prefig1 ] shows the approximate resonance energy @xmath76 for different basis set size @xmath3 , where @xmath29 is the index of the variational eigenvalue used to calculate the dos . we used the values @xmath80 and @xmath81 corresponding to the ones used before @xcite in the analysis of @xmath82 resonances . the figure [ prefig1 ] also shows the value calculated using complex scaling . it is clear that the accuracy of all the values shown is rather good ( all the values shown differ in less than 6@xmath83 ) , and that larger values on @xmath84 provide better values for the resonance energy . these facts are well known from previous works , _ i.e. _ almost all methods to calculate the energy of the resonance give rather stable and accurate results for @xmath0 . however , the practical importance of this fact is reduced : these are uncontrolled methods , so the accuracy of the values obtained from the dos can not be assessed ( without a value independently obtained ) and these values do not seem to converge to the value obtained using complex scaling when @xmath3 is increased and @xmath84 is kept fixed . there is another fact that potentially could render the whole method useless : for small or even moderate @xmath29 , the values @xmath76 become _ unstable _ ( see figure [ prefig1 ] ) when @xmath3 is large enough . this last point has been pointed previously @xcite . in the problem that we are considering is rather easy to obtain a large number of variational eigenvalues in the interval where the resonances are located , allowing us to calculate @xmath76 up to @xmath85 , but this situation is far from common see , for example , references @xcite . so far we have presented only results about the behaviour of the one particle hamiltonian , from now on we will discuss both models , equations ( [ hamil ] ) and [ hamil2part ] . it is known that the variational eigenvalues @xmath17 do not present crossings when they are calculated for some fixed values of @xmath3 , _ i.e _ the variational spectrum is non - degenerate for any finite hamiltonian matrix . as a matter of fact the avoided crossings between successive eigenvalues in the variational spectrum are the watermark of a resonance . an interesting feature emerges when the variational spectrum for many different basis set sizes @xmath3 are plotted together versus the parameter @xmath9 . besides the places where @xmath86 attains its minimum value , which correspond to the stabilization points , there are some gaps which correspond to crossings between eigenvalues obtained with different basis set sizes , see figure [ prefig2 ] . moreover , the crossings corresponds to eigenvalues with different index @xmath29 , and are the states that satisfy the inequalities equations ( [ ordering]),and ( [ reverse ] ) . it is worth to remark that the main features shown by figure [ prefig2 ] are independent of the number of particles of the hamiltonian and the particular values of the threshold of the continuum . figure [ bundle2p ] shows the behaviour of the variational eigenvalues obtained for the two particle hamiltonian equation ( [ hamil2part ] ) . in this case the ionization threshold is not the asymptotic value of the potential , but it is given by the energy of the one particle ground state . the resonance state came from the two - particle ground state that becomes unstable and enters into the continuum of states when the quantum dots becomes `` too small '' to accommodate two electrons . for more details about the model , see reference @xcite . the left panel of figure [ prefig3 ] shows the behaviour of the maximum value of the dos , @xmath87 , for the one particle hamiltonian , obtained for different basis - set sizes and fixed @xmath29 ( in this case @xmath85 ) , and the @xmath88 obtained choosing a `` bundle '' of states that are linked by a crossing , these states have @xmath89 and @xmath90 respectively . from our numerical data , the maximum value of the dos scales with the basis - set size following two different prescriptions . for @xmath29 fixed , @xmath91 , with @xmath92 , while when the pair @xmath93 is chosen from the set of pairs that label a bundle of states @xmath94 , with @xmath95 . in particular , for @xmath85 we get that @xmath96 , and @xmath97 when @xmath98 . of course we can pick sets of states that are not related by a crossing . for instance , we also picked sets with a simple prescription as follows : choose a given initial pair @xmath99 and form a set of states with the states labeled by @xmath100 and so on . figure [ prefig3](a ) shows two examples obtained choosing @xmath101 , @xmath102 and @xmath103 and @xmath104 both with @xmath105 . quite interestingly , the data in figure [ prefig3 ] show that the scaled maxima of the dos for a bundle and two different sets seem to converge to the _ same _ value when @xmath106 , but only for the bundle the scaling function is @xmath107 . the advantage obtained from picking those eigenvalues @xmath17 that belong to a given bundle is still more evident when the corresponding dos and @xmath0 are calculated . the right panel of figure [ prefig3 ] shows the @xmath0 obtained from the dos whose maxima are shown in the left panel . it is rather evident that these values now seem to converge , besides , the extrapolation to @xmath4 results in a more accurate approximate value for @xmath0 . in contradistinction , the values for @xmath0 corresponding to a fixed index @xmath29 ( the values shown in the figure [ prefig3 ] correspond to @xmath85 ) do not seem to converge anywhere close to the value obtained using complex rotation . figure [ er2par ] shows the resonance energies obtained from the bundles of states shown in figure [ bundle2p ] for the two - particle model . since the numerical solution of this model is more complicated than the solution of the one - particle model the number of approximate values is rather reduced . however , it seems that the data also supports a linear scaling of @xmath76 with @xmath108 . many real algebra methods to calculate resonance energies use a golden rule - like formula to calculate the resonance width . in this section we will use the formula and stabilization procedure proposed by tucker and truhlar @xcite that we will describe briefly for completeness . this projection formula seems to work better for one - particle models . for two - particle models its utility has been questioned @xcite , so to analyze the width of the resonance states of the quantum dot model we fitted the corresponding dos using equation ( [ lorentz ] ) . the method of tucker and truhlar @xcite is implemented by the following steps . choose a basis @xmath109 where @xmath9 is a non - linear parameter . diagonalize the hamiltonian using up to @xmath3 functions of the basis . look for the stabilization value @xmath110 and its corresponding eigenfunction @xmath111 which are founded for some value @xmath112 . define the projector @xmath113 where @xmath114 is the normalized projection of @xmath111 onto the basis @xmath115 for any other @xmath9 . diagonalize the hamiltonian @xmath116 in the basis @xmath115 , again as a function of @xmath9 , and find a value @xmath117 of @xmath9 such that @xmath118 where @xmath119 denotes eigenvalue @xmath84 of the projected hamiltonian for the scale factor @xmath117 , and @xmath120 is the corresponding eigenfunction . with the previous definitions and quantities , the resonance width @xmath1 is given by @xmath121 where @xmath122.\ ] ] despite some useful insights , the procedure sketched above does not determine all the intervening quantities , for instance there are many solutions to equation ( [ seg - estabili ] ) and , of course , the stabilization method provides several good candidates for @xmath111 and @xmath112 . we are able to avoid some of the indeterminacies associated to the tucker and truhlar procedure using a bundle of states associated to a crossing , so @xmath111 and @xmath112 are given by any of the eigenfunctions associated to a bundle and @xmath123 comes from the stabilization procedure . then we construct projectors @xmath124 where @xmath19 is one of the variational eigenstates that belong to a bundle of states . with the projectors @xmath125 we construct hamiltonians @xmath126 , and find the solutions to the problem @xmath127 since there is not an a priori criteria to choose one particular solution of equation ( [ tercera - estabili ] ) we show our numerical findings for several values of @xmath84 . figure [ prefig4 ] shows the behaviour of the resonance width calculated with equation ( [ golden2 ] ) , where we have used @xmath128 as @xmath111 and @xmath129 , where @xmath130 and @xmath131 . despite that the different sets corresponding to different values of @xmath84 do not converge to any definite value , for @xmath3 large enough all the sets scale as @xmath132 , with @xmath133 . since the resonance energy scales as @xmath107 , at least when a bundle of states with a crossing is chosen to calculate approximations ( see figure [ prefig3 ] ) , we suggest that the right scaling for @xmath1 is given by @xmath134 . of course for a given basis size , particular variational functions , stabilization procedures and so on , we can hardly expect to find a proper set of @xmath1 whose scaling law would be @xmath107 . instead of this we propose that the data in the right panel can be fitted by @xmath135 then the best approximation for the resonance width is obtained fitting the curve and selecting the @xmath136 as @xmath137 for @xmath138 the closest value to one . as pointed in reference @xcite , the projection technique to calculate the width of a resonance can be implemented if a suitable form of the projection operator can be found . as this procedure is marred by several issues we used the dos method to obtain the approximate widths of a resonance state of the two particle model . figure [ gama2par ] shows the widths calculated associated to the energies shown in figure [ er2par ] , the parameters of the hamiltonian are exactly the same . there is no obvious scaling function that allows the extrapolation of the data but , even for moderate values of @xmath3 , it seems as the data converge to the value obtained using complex scaling . in this work we analyzed the convergence properties of real @xmath51 basis - set methods to obtain resonance energies and lifetimes . the convergence of the energy with the basis - set size for bound states is well understood , the larger the basis set the better the results and these methods converge to the exact values for the basis - set size going to infinite ( complete basis set ) . this idea is frequently applied to resonance states . the increase of the basis - set size in some commonly used methods does not improve the accuracy of the value obtained for the resonance energy @xmath0 , as showed in figure [ prefig1 ] . this undesirable behavior comes from the fact that the procedure is not variational as in the case of bound states . moreover , the exact resonance eigenfunction does not belong to the hilbert space expanded by the complete basis set . in this work we presented a prescription to pick a set or bundle of states that has linear convergence properties for small width resonances . this procedure is robust because the choice of different bundles results in very similar convergence curves and energy values . in fact , in the method described here , the pairs @xmath93 of the bundles play the role of a second stabilization parameter together with the variational parameter @xmath9 . of a second we tested the method in others one and two particle systems and the general behavior of them is the same . the results are very good in all cases leading to an improvement in the calculation of the resonance energies . nevertheless we have to note that the method could no be applied in cases where two or more resonance energies lie very close because the overlapping bundles . the lifetime calculation is more subtle . the use of golden - rule - like formulas , as we applied here , always give several possible outcomes for the width @xmath139 , corresponding to different pseudo - continuum states @xmath140 . the projection technique , equation ( [ golden2 ] ) , is not the exception and it is not possible to select _ a priori _ which value of @xmath141 is the most accurate . the linear convergence of the dos with basis - set size suggests that the scaling in the lifetime value , in accordance with the energy scaling , should be linear . regrettably , the projection method gives discrete sets of values which can not be tuned to obtain an exact linear convergence . our recipe is to choose the set @xmath139 whose scaling is closest to the linear one , then the best estimation for the resonance width is obtained from extrapolation . many open questions remain on the analysis of the different convergence properties of resonance energy and lifetime . the method presented here to obtain the resonance energy from convergence properties works very well , but the appearance of bundles in the spectrum is not completely understood . even there is not a rigorous proof , the numerical evidence supports the idea that the behaviour of the systems studied here is quite general . in this appendix we give arguments that support our assumptions on the scaling of the eigenenergies with the basis - set parameter @xmath9 . we present our argument for one body hamiltonians , but it is straightforward to generalize to more particles with pair interactions decaying fast enough at large distances . 1 . let @xmath142 be an @xmath143 matrix with all its matrix elements having the form @xmath144 , where @xmath145 , then if @xmath146 the eigenvalues of @xmath142 scales with @xmath147 : @xmath148=0\;\rightarrow det[a(1)-\frac{\lambda(\eta)}{f(\eta)}\,i]=0 \rightarrow \lambda(1)=\frac{\lambda(\eta)}{f(\eta ) } \,.\ ] ] 2 . let @xmath149 @xmath143 be symmetric matrices with @xmath150 , and @xmath151 the eigenvalues of @xmath149 and @xmath152 respectively , in nondecreasing order , then , by the minimax principle @xcite consider a spherical one - particle potential with compact support , @xmath154 if @xmath155 , and finite , @xmath156 ( both conditions could be relaxed , but we adopt them for simplicity ) . let the basis - set functions be of the form @xmath157 with @xmath158 and @xmath159 , then the coefficients take the form @xmath160 . and then , by equation ( [ asa ] ) , all the eigenvalues of the kinetic energy have the same scaling with @xmath16 . we have to show that , in both limits , @xmath162 and @xmath163 , for all the potential matrix elements hold @xmath164 , and then , by the wielandt - hoffman theorem @xcite , the eigenenergies are a perturbation of the eigenvalues of the kinetic energy . a u hazi and h s taylor , phys . rev . a * 14 * , 2071 ( 1976 ) . a ferrn , o osenda and p serra , phys . a * 79 * , 032509 ( 2009 ) . h feschbach , ann . ny * 19 * , 627 ( 1962 ) n moiseyev , c corcoran , phys . rev . a * 20 * , 814 ( 1979 ) . m e fisher , in _ critical phenomena _ , proceedings of the 51st enrico fermi summer school , varenna , italy , m. s. green , ed . ( academic press , new york 1971 ) . s e tucker and d. g. truhlar , j.chem . * 86 * , 6251 ( 1987 ) . f r manby and g doggetty , j. phys . b : at . mol . . phys . * 30 * , 3343 ( 1997 ) j. h. wilkinson , _ the algebraic eigenvalue problem _ , oxford university press , london , ( 1965 ) .

the resonance states of one- and two - particle hamiltonians are studied using variational expansions with real basis - set functions . the resonance energies , @xmath0 , and widths , @xmath1 , are calculated using the density of states and an @xmath2 golden rule - like formula . we present a recipe to select adequately some solutions of the variational problem . the set of approximate energies obtained shows a very regular behaviour with the basis - set size , @xmath3 . indeed , these particular variational eigenvalues show a quite simple scaling behaviour and convergence when @xmath4 . following the same prescription to choose particular solutions of the variational problem we obtain a set of approximate widths . using the scaling function that characterizes the behaviour of the approximate energies as a guide , it is possible to find a very good approximation to the actual value of the resonance width .

arxiv

the theme of @xmath0 ( parity time ) symmetric systems was initiated in the works of bender and collaborators @xcite as an alternative to the standard quantum theory , where the hamiltonian is postulated to be hermitian . the principal conclusion of these works was that @xmath0-invariant hamiltonians , which are not necessarily hermitian , may still give rise to completely real spectra , thus being appropriate for the description of physical settings . in terms of the schrdinger - type hamiltonians , which include the usual kinetic - energy operator and the potential term , @xmath3 , the @xmath0-invariance admits complex potentials , subject to constraint that @xmath4 . recent developments in optics have resulted in an experimental realization of the originally theoretical concept of the @xmath0-symmetric hamiltonians , chiefly due to the work by christodoulides and co - workers christo1 ( see also @xcite ) . it has been demonstrated that the controllable imposition of symmetrically set and globally balanced gain and loss may render optical waveguiding arrays a fertile territory for the construction of @xmath0-symmetric complex potentials . the first two such realizations made use of couplers composed of two waveguides with and without loss @xcite ( so - called passive @xmath5couplers ) , or , in more standard " form , a pair of coupled waveguides , one carrying gain and the other one loss @xcite . in fact , more general models of linearly coupled active ( gain - carrying ) and passive ( lossy ) intrinsically nonlinear waveguides , without imposing the condition of the gain - loss balance , were considered earlier , and stable solitons were found in them @xcite , including exact solutions @xcite ( see also a brief review in ref . recently , an electronic analog of such settings has also been implemented @xcite . configurations with a hidden @xmath6 symmetry have been identified also in fine - tuned parameter regions of microwave billiards @xcite . effects of the nonlinearity in a gross - pitaevski equation on the @xmath7 properties of a bose - einstein condensate have been analyzed in @xcite . the possibility to engineer @xmath0-symmetric _ oligomers _ ( coupled complexes of a few loss - and gain - carrying elements ) @xcite , which may include nonlinearity , was an incentive to a broad array of additional studies on both the few - site systems and entire @xmath0-symmetric lattices @xcite . more recently , nonlinear @xmath0-symmetric systems , incorporating @xmath0-balanced nonlinear terms , have drawn considerable interest too @xcite-@xcite . most of the @xmath0-invariant systems considered thus far have been one - dimensional ( 1d ) in their nature , although the stability of solitons in 2d periodic @xmath0-symmetric potentials has also been recently investigated @xcite . actually , 2d arrays of optical waveguides can be readily built @xcite ( the same is true about other quasi - discrete systems , including electrical ones ) , hence , a natural question is whether @xmath0-symmetric oligomers ( and ultimately lattices built of such building blocks ) can be created in a 2d form . this work aims to make a basic step in this direction , by introducing fundamental 2d plaquettes consisting , typically , of four sites ( in one case , it will be a five - site cross ) . these configurations , illustrated by fig . [ modes ] , are inspired by earlier works on 2d hamiltonian lattices described by discrete nonlinear schrdinger equations @xcite , where diverse classes of modes , including discrete solitary vortices malomed , pelin , have been predicted and experimentally observed moti , yuri . the plaquettes proposed herein should be straightforwardly accessible with current experimental techniques in nonlinear optics , as a straightforward generalization of the coupler - based setting reported in ref . we start from the well - established hamiltonian form of such plaquettes in the conservative form , gradually turning on the gain - loss parameter ( @xmath2 ) , as the strength of the @xmath0-invariant terms , to examine stationary states supported by the plaquettes , studying their stability against small perturbations and verifying the results through direct simulations . actually , in this work we focus on those ( quite diverse , although , obviously , not most generic ) modes that can be found in an analytical form , while their stability is studied by means of numerical methods . the analytical calculations and the manifestations of interesting features , such as a potential persistence past the critical point of the linear @xmath0 symmetry , are enabled by the enhanced symmetry of the modes that we consider below . it is conceivable that additional asymmetric modes may exist too within these 2d configurations . our principal motivation for studying the above systems stems from the fact that realizations of @xmath0-symmetry e.g. within the realm of nonlinear optics will be inherently endowed with nonlinearity . hence , it is only natural to inquire about the interplay of the above type of linear systems with the presence of nonlinear effects . in addition to this physical argument , there exists an intriguing mathematical one which concerns the existence , stability and dynamical fate of the nonlinear states in the presence of @xmath0-symmetric perturbations . in particular , previous works @xcite point to the direction that neither the existence , nor the stability of @xmath0-symmetric nonlinear states mirrors that of their linear counterparts ( or respects the phase transition of the latter generically ) . the presentation of our results is structured as follows . section [ symmetry ] contains a part of the analytical results , including a detailed analysis of the @xmath8symmetry properties of the nonlinear schrdinger type model , as well as the spectral properties of the linear hamiltonian subsystems . section [ numerics ] is devoted to the existence , stability and dynamics of stationary modes in the nonlinear systems . beside analytical results , it contains a detailed presentation of the numerical findings . in section [ conclu ] we summarize conclusions and discuss directions for future studies . the dynamics of the 2d plaquettes that we are going to consider is described by a multicomponent nonlinear schrdinger equation ( nlse ) @xmath9built over a transposition - symmetric linear @xmath10 hermitian matrix hamiltonian @xmath11 and an additional nonlinear @xmath10 matrix operator , @xmath12 . to understand the symmetry properties of this nlse , we first analyze the associated linear problem @xmath13and check then whether the symmetry is preserved by the nonlinear term , @xmath14 . the analysis can be built , in a part , on techniques developed for other nonlinear dynamical systems with symmetry preservation nonlin-1,nonlin-2,nonlin-3,nonlin-4,nonlin-5,nonlin-6,nonlin-7,nonlin-8 . for the present setups , the time reversal operation @xmath15 can be defined as the combined action of a scalar - type complex conjugation @xmath16 , @xmath17 , and the sign change of time , @xmath18 , in full accordance with wigner s original work which introduced these concepts @xcite . for the linear schrdinger equation ( [ a2 ] ) and its solutions @xmath19this implies @xmath20where the overbar denotes complex conjugation . from the actual form of the gain - loss arrangements in the 2d plaquettes we can conjecture the existence of certain plaquette - dependent parity operators @xmath21 , with @xmath22 , which will render the hamiltonians @xmath8symmetric , @xmath23=0 $ ] . to find an explicit representation of these parity operators @xmath21 , we use the following ansatz , @xmath24=0 \label{a2 - 3}\]]together with the pseudo - hermiticity condition @xmath25the latter follows trivially from eq . ( [ a2 - 3 ] ) , @xmath23=0 $ ] , and the transposition symmetry , @xmath11 . these parity operators @xmath21 will be used to check whether the corresponding nonlinear terms @xmath26 satisfy the same @xmath27symmetry . in contrast to linear setups with the @xmath28symmetry being either exact ( @xmath29=0 $ ] , @xmath30 ) or spontaneously broken ( @xmath29=0 $ ] , @xmath31 ) , the nonlinear setups considered in the present paper allow for sectors of exact @xmath28symmetry ( @xmath32=0 $ ] , @xmath30 ) and of broken @xmath28symmetry ( @xmath31@xmath33@xmath34\neq 0 $ ] ) , as it is common for nonlinear @xmath6-symmetric systems @xcite-@xcite . we start from the 2d plaquette of 0 + 0- type depicted as configuration ( a ) in fig . [ modes ] . this plaquette has only two ( diagonally opposite ) nodes carrying the gain and loss , while the other two nodes bear no such effects . the corresponding dynamical equations for the amplitudes at the four sites of this oligomer are @xmath35where @xmath36 is the above - mentioned gain - loss coefficient , and @xmath37 is a real coupling constant . the nonlinearity coefficients are scaled to be @xmath38 ( we use time @xmath39 as the evolution variable , although in the mathematically equivalent propagation equations for optical waveguides @xmath39 has to be identified with the propagation distance , @xmath40 ) . denoting @xmath41 , the matrices @xmath42 and @xmath26 in ( [ a1 ] ) take the form of @xmath43to find the parity matrix @xmath21 which renders the linear hamiltonian @xmath8symmetric , @xmath23=0 $ ] , we use the pseudo - hermiticity condition ( [ a2 - 4 ] ) and notice that @xmath44obviously , the following relations should hold : @xmath45=0 \nonumber \\ \mathcal{p}h_{l,1}\mathcal{p}=-h_{l,1 } & \qquad \longrightarrow \qquad & \left\ { \mathcal{p},h_{l,1}\right\ } = 0 . \label{a6}\end{aligned}\]]the first of these conditions together with @xmath22 , @xmath46 reduces the possible form of the parity transformation to one of the three types , @xmath47where @xmath48 are the usual pauli matrices . taking into account that @xmath49 , the anti - commutativity condition in eqs . ( [ a6 ] ) singles out the only possible parity matrix : @xmath50for the linear transformation , i.e. , the matrix interchanging @xmath51 and @xmath52 , as well as @xmath53 and @xmath54 . one then immediately checks that @xmath55hence , in contrast to the linear component @xmath42 , the nonlinear terms @xmath26 corresponding to eqs . ( [ zpzm1 ] ) are not @xmath8symmetric , in the usual matrix sense . rather , the symmetry properties of the nonlinear terms have to be considered in the context of the nonlinear schrdinger equation itself . acting with @xmath56 on eq . ( [ a1 ] ) we observe that @xmath57 , \notag \\ i\partial _ { t}(\mathcal{p}\mathbf{t}\mathbf{u } ) & = & h_{l}\mathcal{p}\mathbf{t}\mathbf{u}+h_{nl}(\mathcal{p}\mathbf{t}\mathbf{u})(\mathcal{p}\mathbf{t}\mathbf{u})\label{a13}.\end{aligned}\ ] ] hence , the full nlse system ( [ zpzm1 ] ) remains invariant if we define the @xmath58 transformation of the vectorial wave function obeying this system as follows : @xmath59 this is in full analogy to the condition of exact @xmath27symmetry for the corresponding linear schrdinger equation , nonlinearity matrices @xmath60 of more general type than that in ( [ nl-1 ] ) and ( [ nl-2 ] ) may be envisioned which may produce @xmath61symmetric solutions @xmath62 with less simple time dependence . a detailed analysis of such systems will be presented elsewhere . ] . but the condition of _ spontaneously broken _ @xmath28symmetry ( @xmath29=0 $ ] , @xmath31 ) is replaced by the condition of _ completely broken _ @xmath28symmetry ( @xmath31@xmath63@xmath64\neq 0 $ ] ) . in contrast to the present 2d plaquettes , which are mainly motivated by feasible experimental realizations , one can envision more sophisticated setups with @xmath65 . this will lead to a new type of partial ( or intermediate ) @xmath28symmetry ( to be considered elsewhere ) , which for solutions @xmath66 with broken @xmath28symmetry will keep the nonlinear term @xmath26 explicitly @xmath8symmetric ( @xmath67pseudo - hermitian ) in the matrix sense , but not @xmath28symmetric ( under inclusion of the explicit time reversal @xmath18 ) in the sense of the nlse system . for configurations ( b ) and ( c ) in fig . [ modes ] , @xmath68 and @xmath26 are still given by eqs . ( [ a3 ] ) and ( [ a5 ] ) , but with @xmath69respectively . hence , relations ( [ a6 ] ) are valid for both configurations ( b ) and ( c ) as well . using ( [ a7 ] ) in @xmath70 , we find a richer variety of parity operators @xmath21 than for configuration ( a ) . configuration ( b ) allows for @xmath71whereas configuration ( c ) may be associated with @xmath72for configuration ( d ) , we have @xmath73and simple computer algebra gives again two possible parity operators : @xmath74 in strong structural analogy to configuration ( b ) . one verifies that @xmath75 holds also for configurations ( b ) , ( c ) and ( d ) , hence all 2d plaquettes considered in the present paper are not @xmath28symmetric in the usual matrix sense . next , we turn to the eigenvalue problems of the linear setups associated with plaquettes ( a ) - ( d ) , i.e. , to solutions of the equation @xmath76from the corresponding characteristic polynomials , @xmath77 , @xmath78 = 0 , \notag \\ ( \mathrm{b}):\qquad \qquad & ( e^{2}+\gamma ^{2})\left [ e^{2}-(4k^{2}-\gamma ^{2})\right ] = 0 , \notag \\ ( \mathrm{c}):\qquad \qquad & e^{4}-2(2k^{2}-\gamma ^{2})e^{2}+\gamma ^{4}=0 , \notag \\ ( \mathrm{d}):\qquad \qquad & e(e^{2}+\gamma ^{2})\left [ e^{2}-(4k^{2}-\gamma ^{2})\right ] = 0,\end{aligned}\]]we find @xmath79obviously , the matrix hamiltonians @xmath42 for plaquettes ( a ) and ( d ) are not of full rank . for plaquette ( a ) we find @xmath80 , and @xmath42 has a two - dimensional kernel space , @xmath81 . for plaquette ( d ) we find @xmath82 and @xmath83 , where @xmath84 . moreover , we see that the spectrum for plaquette ( d ) , up to the additional eigenvalue @xmath85 , coincides with that for ( b ) . the different eigenvalues of the 4-node plaquettes displayed in eqs . ( [ a15 - 2 ] ) show that these plaquettes are also physically not equivalent . equivalence classes of nonlinear 4-node plaquettes with isospectral linear hamiltonians @xmath86 but different pairwise couplings have been considered , e.g. , in @xcite . for plaquettes ( a ) , ( b ) and ( d ) an exceptional point ( ep ) occurs at @xmath87 , being associated with a branching of the eigenvalue pair @xmath88 @xmath89 in the case of plaquette ( a ) , all four eigenvalues are involved in the branching at @xmath90 , where @xmath91 . via jordan decomposition ( e.g. , with the help of the corresponding linear algebra tool of mathematica ) we find that @xmath92i.e . , a spectral degeneration of the type @xmath93 in arnold s notation @xcite , or , in other words , a third - order ep with a single decoupled fourth mode . hence , plaquettes of type ( a ) may serve as an easily implementable testground for the experimental investigation of third - order eps ( see e.g. @xcite ) . for plaquettes ( b ) and ( d ) we have second - order eps at @xmath87 , similar as for plaquette ( c ) where a pair of second - order eps occurs at @xmath94 with @xmath95 , @xmath96 . from the eigenvalues in ( [ a15 - 2 ] ) we read off the @xmath27symmetry content of the four types of plaquettes . the sector of exact @xmath28symmetry ( i.e. , the sector with all eigenvalues purely real , @xmath97 ) corresponds to @xmath98i.e . , for plaquettes ( b ) and ( d ) the @xmath28symmetry is spontaneously broken as soon as the gain - loss coupling is switched on , namely for @xmath99 . in this section , we seek stationary solutions of the type @xmath100constructed over constant vectors @xmath101 . according to ( [ a13 ] ) and ( [ a13 - 2 ] ) such solutions will be @xmath61symmetric provided it holds @xmath102 for some @xmath103 . we will test these symmetry properties for the solutions to be obtained . we note that restricting the explicit analysis to stationary solutions of the type ( [ e ] ) we by construction exclude from this analysis @xmath61violating solutions with @xmath104 which are necessarily non - stationary . a useful technical tool to facilitate the explicit derivation of stationary solutions @xmath105 are conservation equations of the type @xmath106constructed from eq . ( [ a1 ] ) and its adjoint , where @xmath107 denotes an arbitrary constant matrix . the most simplest of them can be found via eqs . ( [ a2 - 4 ] ) , ( [ a3 ] ) and eq . ( [ a5 ] ) to be @xmath108 \mathbf{u}. \label{a18 - 2}\end{aligned}\]]for stationary equations @xmath109 the time - dependent phase factors @xmath110 cancel so that the left - hand - sides of these relations vanish , yielding simple algebraic constraints on the right - hand - sides . from eq . ( [ a18 - 2 ] ) we see that for stationary solutions the @xmath6 inner product quantum mechanics ( ptqm ) the @xmath7 inner product was introduced first by znojil in @xcite in 2001 . immediately afterwards , it was interpreted by japaridze as indefinite inner product @xcite in a krein space and generalized by mostafazadeh to the @xmath111metric in the context of pseudo - hermitian hamiltonians @xcite . finally , it was used by bender , brody and jones in 2002 to construct the positive definite @xmath112 inner product @xcite . for oligomer settings ( of plaquettes or other few site configurations ) , it can be employed , e.g. , to derive a simple algebraic constraint or as a criterion of the numerical accuracy of the evolutionary dynamics ( especially since the solutions rapidly acquire very large amplitudes when unstable , as will be seen below ) . it also turned out useful in @xcite . ] will remain conserved ( @xmath113const ) regardless of the violated @xmath114pseudo - hermiticity , @xmath115 , characteristic for of our specific nonlinear plaquette couplings ( see eq . ( [ nl-2 ] ) ) . subsequently , we first derive classes of stationary solutions @xmath105 explicitly . then , we analyze the stability of small perturbations over these stationary solutions by the linearization , via ansatz @xmath116 + o(\delta ^{2}),\qquad |\delta |\ll 1,\]]where @xmath117 is the small amplitude of the perturbation . exponents @xmath118 can be defined as wick - rotated eigenvalues from the corresponding @xmath119 perturbation matrix @xmath120 ( see , e.g. , @xcite for more details ) : @xmath121 where @xmath122\mathbf{u}\]]characterizes the stationary problem , and the elements of the matrix @xmath120 are evaluated at @xmath123 . linear stability is ensured for @xmath124 , whereas @xmath125 corresponds to growing and decaying modes , i.e. , exponential instabilities . substituting ansatz ( [ e ] ) for the stationary solutions in eqs . ( [ a1 ] ) , ( [ a18 - 1 ] ) and ( [ a18 - 2 ] ) we obtain the following algebraic equations : @xmath126@xmath127and @xmath128 \mathbf{u , } \notag \label{a31 } \\ 0 & = & \left ( |a|^{2}-|c|^{2}\right ) ( \bar{a}c-\bar{c}a)+\left ( equations can be analyzed via the madelung substitution ( i.e. , via amplitude - phase decomposition ) , @xmath129without loss of generality , we may fix @xmath130 . for arbitrary phase factors in ( [ aa ] ) , eqs . ( [ a30 ] ) and ( [ a31 ] ) are satisfied by @xmath131 and @xmath132 . using this condition in eq . ( [ zpzm2 ] ) and dividing each equation ( [ zpzm2 ] ) by the phase factor on its left - hand side , one obtains the imaginary parts of the resulting equations : @xmath133 = 2kb\sin \left ( \frac{\phi _ { b}+\phi _ { d}}{2}-\phi _ { a}\right ) \cos \left ( \frac{\phi _ { b}-\phi _ { d}}{2}\right ) , \notag \\ \gamma b & = & ka\left [ \sin ( \phi _ { a}-\phi _ { b})+\sin ( \phi _ { c}-\phi _ { b})\right ] = 2ka\sin \left ( \frac{\phi _ { a}+\phi _ { c}}{2}-\phi _ { b}\right ) \cos \left ( \frac{\phi _ { a}-\phi _ { c}}{2}\right ) , \notag \\ 0 & = & kb\left [ \sin ( \phi _ { b}-\phi _ { c})+\sin ( \phi _ { d}-\phi _ { c})\right ] = 2kb\sin \left ( \frac{\phi _ { b}+\phi _ { d}}{2}-\phi _ { c}\right ) \cos \left ( \frac{\phi _ { b}-\phi _ { d}}{2}\right ) , \notag \\ -\gamma b & = & ka\left [ \sin ( \phi _ { a}-\phi _ { d})+\sin ( \phi _ { c}-\phi _ { d})\right ] = 2ka\sin \left ( \frac{\phi _ { a}+\phi _ { c}}{2}-\phi _ { d}\right ) \cos \left ( \frac{\phi _ { a}-\phi _ { c}}{2}\right ) . \notag \\ & & \label{zpzm_add}\end{aligned}\]]for @xmath130 the first of these equations implies @xmath134 , hence either @xmath135 ( case 1 ) or @xmath136 ( case 2 ) . in case 1 , we conclude from the third equation that either @xmath137 and @xmath138 ( case 1a ) , or @xmath139 and @xmath140 is arbitrary ( case 1b ) . in case 2 the third equation is satisfied automatically . in all the three cases , the second and the fourth equation are compatible . they give @xmath141returning to the phase - factor divided equations ( [ zpzm2 ] ) and considering their real parts , we find @xmath142 + a^{3 } , \notag \label{a33 } \\ eb & = & ka\left [ \cos ( \phi _ { a}-\phi _ { b})+\cos ( \phi _ { c}-\phi _ { b})\right ] + b^{3 } , \notag \\ ea & = & kb\left [ \cos ( \phi _ { b}-\phi _ { c})+\cos ( \phi _ { d}-\phi _ { c})\right ] + a^{3 } , \notag \\ eb & = & ka\left [ \cos ( \phi _ { a}-\phi _ { d})+\cos ( \phi _ { c}-\phi _ { d})\right ] + b^{3}.\end{aligned}\]]the pairwise compatibility of the first and third , as well as of the second and fourth equations requires @xmath143for case 1a , these conditions are trivially satisfied , whereas for the remaining cases they lead to further restrictions : @xmath144 in this way the phase angles are fixed for all the three cases and we can turn to the amplitudes . the corresponding equation sets reduce to @xmath145 in the latter two cases ( 1b and 2 ) the amplitudes and phases completely decouple and we have @xmath146case 1a allows for a richer behavior . equating the terms @xmath147 in the upper two equations ( [ a36 ] ) leads to the constraint @xmath148which can be resolved by @xmath149 ( case 1aa ) as well as by @xmath150 ( case 1ab ) . the analysis of these two cases can be completed with the help of the relation @xmath151 from eq . ( [ a32 ] ) . as result we obtain the following set of stationary solutions : @xmath152 from eq . ( [ a31 ] ) , it can also be seen that either @xmath131 or if @xmath153 , then @xmath154 must be true . here , we use the information available so far to check the @xmath28symmetry content of the solutions ( [ a39 - 1aa ] ) - ( [ a39 - 2 ] ) explicitly . for stationary solutions @xmath155 , @xmath156 the @xmath28symmetry condition ( [ a13 - 2 ] ) implies @xmath157taking into account that @xmath16 acts as complex conjugation , we see from the explicit structure of @xmath158 in eq . ( a8 ) that a stationary solution is @xmath28symmetric if , with @xmath130 , it has @xmath138 and @xmath159 ( up to a common phase shift ) . additionally , the amplitudes have to coincide pairwise : @xmath131 , @xmath132 . for eqs . ( [ a39 - 1aa ] ) - ( [ a39 - 2 ] ) this means that all case-1 stationary solutions with @xmath138 are @xmath28symmetric in their present form . the case-2 mode becomes explicitly @xmath8symmetric after a global @xmath160 multiplication by a phase factor : @xmath161 ^{t } , \notag \\ \mathcal{p}\mathcal{t}\mathbf{v}_{0 } & = & \mathbf{v}_{0},\end{aligned}\]]where @xmath162 has to be chosen in eq . ( [ a39 - 2 ] ) . we note that this procedure is effectively equivalent to a redefinition of the original phase constraint : @xmath163 at the very beginning of the calculations in eq . ( [ aa ] ) . the linear stability analysis was performed numerically . subsequently we present corresponding graphical results . the plaquettes ( b ) - ( d ) can be analyzed in a similar way . for brevity s sake , in fig . [ figzpzm1 ] we present only the basic numerical results , by means of the following symbols : [ figzpzm1 ] * case 1aa with @xmath164 blue circles ; * case 1aa with @xmath165 red crosses ; * case 1ab green stars ; * case 2 black squares ; * case 1b is not depicted explicitly because it corresponds to point configurations without gain - loss ( @xmath166 ) and to exceptional point configurations @xmath90 . figure [ figzpzm1 ] presents the mode branches ( their amplitudes , phases , and also their stability ) over the gain - loss parameter @xmath2 , starting from the conservative system at @xmath166 . the same symbols are used in fig . [ figzpzm2 ] , which displays typical examples of the spectral plane @xmath167 for stability eigenvalues @xmath168 of the linearization ; recall that the modes are unstable if they give rise to @xmath169 . explicitly we observe the following behavior . * case 1aa with @xmath164 blue circlesaccording to fig . [ figzpzm2 ] , the present solution is stable . notice that , although featuring a phase profile , it can not be characterized as a vortex state ( the same is true for some other configurations carrying phase structure ) . interestingly , the relevant configuration is generically stable bearing two imaginary pairs of eigenvalues . * case 1aa with @xmath165 red crosses.obviously , this kind of solutions as well as the previous one exist up to the exceptional point @xmath90 of the @xmath0-symmetry breaking in the linear system , where the two branches collide and disappear ( leave the stationary regime and become nonstationary ) . as seen in fig . figzpzm2 , the present branch has two eigenvalue pairs which are purely imaginary for small @xmath2 , but become real ( rendering the configuration unstable ) at @xmath170 and then @xmath171 , respectively . ultimately , these pairs of unstable eigenvalues collide at the origin of the spectral plane with those of the previous branch ( blue circles ) . * case 1ab green stars.this stationary solution has a number of interesting features . firstly , it is the only one among the considered branches which has two unequal amplitudes . secondly , it exists past the critical point @xmath90 of the linear system , due to the effect of the nonlinearity ( the extension of the existence region for nonlinear modes was earlier found in 1d couplers @xcite and oligomers @xcite ) . furthermore , this branch has three non - zero pairs of stability eigenvalues , two of which form a quartet for small values of the gain - loss parameter , while the third is imaginary ( i.e. , the configuration is unstable due to the real parts of the eigenvalues within the quartet ) . at @xmath172 , the eigenvalues of the complex quartet collapse into two imaginary pairs , rendering the configuration stable , in a narrow parametric interval . at @xmath173 , the former imaginary pair becomes real , destabilizing the state again , while subsequent bifurcations of imaginary pairs into real ones occur at @xmath174 and @xmath175 ( at the latter point , all three non - zero pairs are real ) . shortly thereafter , two of these pairs collide at @xmath176 and rearrange into a complex quartet , which exists along with the real pair past that point . * case 2 black squares . in contrast to all other branches , this one is _ always _ unstable . one of the two nonzero eigenvalue pairs is always real ( while the other is always imaginary ) , as seen in fig . this branch also terminates at the exceptional point @xmath90 , as relation @xmath177 can not hold at @xmath178 . this branch collides with the two previous ones via a very degenerate bifurcation ( that could be dubbed a double saddle - center " bifurcation ) , which involves 3 branches instead of two as in the case of the generic saddle - center bifurcation , and two distinct eigenvalue pairs colliding at the origin of the spectral plane . by means of direct simulations , we have also examined the dynamics of the modes belonging to different branches in fig . [ stabzpzm ] . the stable blue - circle branch demonstrates only oscillations under perturbations . this implies that , despite the presence of the gain - loss profile , none of the perturbation eigenmodes grows in this case . nevertheless , the three other branches ultimately manifest their dynamical instability , which is observed through the growth of the amplitude at the gain - carrying site [ b , in fig . [ modes](a ) ] at the expense of the lossy site ( d ) . that is , the amplitude of the solution at the site with the gain grows , while the amplitude of the solution at the dissipation site loses all of its initial power . depending on the particular solution , passive sites ( the ones without gain or loss , such as a and c ) may be effectively driven by the gain ( as in the case of the black - square - branch , where the site a is eventually amplified due to the growth of the amplitude at site b ) or by the loss ( red - cross and green - star branches , where , eventually , the amplitudes at both a and c sites lose all of their optical power ) . we now turn to the generalized ( not exactly @xmath56-symmetric ) configurationsymmetry , spontaneously broken @xmath61symmetry and completely broken @xmath61symmetry see the discussion of eqs . ( [ a13 ] ) and ( [ a13 - 2 ] ) . ] featuring the alternation of the gain and loss along the plaquette in panel ( b ) of fig . [ modes ] . indeed , the absence of @xmath0-symmetry in this case is mirrored in the existence of imaginary eigenvalues in the linear problem of eqs . ( [ a15 - 2 ] ) , as soon as @xmath99 . the corresponding nonlinear solutions ( with @xmath104 ) are not covered by the stationary solution ansatz ( [ e ] ) . stationary solutions ( with @xmath179 ) solely belong to dynamical regimes below the concrete @xmath61thresholds . apart from the two @xmath61symmetry violating solutions , there should exist at least two stationary solutions which we construct in analogy to [ cf . ( [ zpzm2 ] ) ] from @xmath180 substituting the madelung representation ( [ aa ] ) and setting @xmath181 ( for illustration purposes , we focus here only on this simplest case ) , we obtain @xmath182further , fixing @xmath183 , eqs . ( [ phi1 ] ) and ( [ phi2 ] ) yield @xmath184obviously , the solution terminates at point @xmath90 . similar to what was done above , the continuation of this branch and typical examples of its linear stability are shown in figs . [ figpmpm2 ] and [ figpmpm3 ] , respectively . from here it is seen that the blue - circle branch , which has a complex quartet of eigenvalues , is always unstable . in fact , the gain - loss alternating configuration is generally found to be more prone to the instability . the red - cross branch is also unstable via a similar complex quartet of eigenvalues . this quartet , however , breaks into two real pairs for @xmath185 , and , eventually , the additional imaginary eigenvalue pair becomes real too at @xmath186 , making the solution highly unstable with three real eigenvalue pairs . the manifestation of the instability is shown in fig . [ stabpmpm ] , typically amounting to the growth of the amplitudes at one or more gain - carrying sites . [ figpmpm3 ] we now turn to the plaquette in fig . [ modes](c ) , which involves parallel rows of gain and loss . in this case , the stationary equations are @xmath187 in this case too , we focus on symmetric states of the form of @xmath181 [ see eq . ( [ aa ] ) ] , which gives rise to two solutions displayed in fig . figppmm , represented by the following analytical solutions : @xmath188 @xmath189the analysis demonstrates that the branch with the upper sign in eq . ( + -1 ) is always unstable ( through two real pairs of eigenvalues ) , as shown by blue circles in fig . [ figppmm ] . on the other hand , the branch denoted by the red crosses , which corresponds to the lower sign in eq . ( [ + -1 ] ) is stable up to @xmath190 , and then it gets unstable through a real eigenvalue pair . the black - squares branch with the upper sign in eq . ( + -3 ) is always stable , while the green - star branch with the lower sign in eq . ( [ + -3 ] ) is always unstable . at the linear-@xmath0-symmetry breaking point @xmath191 , we observe a strong degeneracy , since all the three pairs of eigenvalues for two of the branches ( in the case of the blue circles , two real and one imaginary , and in the case of red crosses one real and two imaginary ) collapse at the origin of the spectral plane . on the other hand , the black - squares branch is always stable with three imaginary eigenvalue pairs , while the green - star branch has two imaginary and one real pair of eigenvalues . between the latter two , there is again a collision of a pair at the origin at the critical condition , @xmath191 . direct simulations , presented for @xmath192 in fig . [ stabppmm ] , demonstrate the stability of the lower - sign black - squares branch , while the instability of the waveform associated with the blue circles and the green stars leads to the growth and decay of the amplitudes at the sites carrying , respectively , the gain and loss . notice that at the parameter values considered here , the red - cross branch is also dynamically stable as shown in the top right panel of fig . [ stabppmm ] . [ figppmm ] lastly , motivated by the existence of known cross "- shaped discrete - vortex modes in 2d conservative lattices , in addition to the fundamental discrete solitons @xcite , we have also examined the five - site configuration , in which the central site does not carry any gain or loss , while the other four feature a @xmath0-balanced distribution of the gain and loss , as shown in panel ( d ) of fig . [ modes ] . seeking for stationary states with propagation constant , @xmath193 [ instead of @xmath194 in eq . ( [ e ] ) , as in this case we reserve label @xmath194 for one of the sites of the 5-site plaquette in fig . [ modes](d ) ] , we get : @xmath195similarly as before , we use the madelung decomposition @xmath196 , cf . ( [ aa ] ) , and focus on the simplest symmetric solutions with @xmath197 . without the loss of generality , we set @xmath138 , reducing the equations to @xmath198 we report here numerical results for parameters @xmath199 ( smaller @xmath193 yields similar results but with fewer solution branches ) . we have identified five different solutions in this case , see figs . [ figpmzpm1 ] and figpmzpm2 for the representation of the continuation of the different branches , and for typical examples of their stability ( the latter is shown for @xmath200 , @xmath201 and @xmath202 ) . there are two branches ( green stars and black squares ) that only exist at @xmath203 colliding and terminating at that point . one of them has three real eigenvalue pairs and one imaginary pair , while the other branch has two real and two imaginary pairs . two real pairs and one imaginary pair of green stars collide with two real pairs and one imaginary pair of black squares , respectively , while the final pairs of the two branches ( one imaginary for the green stars and one real for the black squares ) collide at the origin of the spectral plane . these collisions take place at @xmath204 , accounting for the saddle - center bifurcation at the point where those two branches terminate . on the other hand , there exist two more branches ( red crosses and magenta diamonds in fig . [ figpmzpm1 ] ) , which collide at @xmath205 . one of these branches ( the less unstable one , represented by magenta diamonds ) bears only an instability induced by an eigenvalue quartet , while the highly unstable branch depicted by the red crosses has four real pairs ( two of which collide on the real axis and become complex at @xmath206 ) . last but not least , the blue circles branch does not terminate at @xmath207 , but continues to larger values of the gain - loss parameter , @xmath208 . it is also unstable ( as the one represented by the magenta diamonds ) due to a complex quartet of eigenvalues . the dynamics of the solutions belonging to these branches is shown in fig . [ stabpmzpm ] . for the branches depicted by black squares and green stars ( recall that they disappear through the collision and the first saddle - center bifurcation at @xmath204 ) , the perturbed evolution is fairly simple : the amplitudes grow at the gain - carrying sites and decay at the lossy ones , while the central passive site ( c ) stays almost at zero amplitude . for the other branches , the amplitudes also grow at the two gain - carrying sites and decay at the lossy elements , while the passive site may be drawn to either the growth or decay . [ figpmzpm1 ] [ figpmzpm2 ] in the present work , we have proposed generalizations of the one - dimensional @xmath0-symmetric nonlinear oligomers into two - dimensional plaquettes , which may be subsequently used as fundamental building blocks for the construction of @xmath0-symmetric two - dimensional lattices . in this context , we have introduced four basic types of plaquettes , three of which in the form of four - site squares . the final one was in the form of the five - site cross , motivated by earlier works on cross - shaped ( alias rhombic or site - centered ) vortex solitons in the discrete nonlinear schrdinger equation . our analysis was restricted to modes which could be found in the analytical form , while their stability against small perturbations was analyzed by means of numerical methods . even within the framework of this restriction , many effects have been found , starting from the existence of solution branches that terminate at the critical points of the respective linear @xmath0-symmetric systems e.g. , in the settings corresponding to plaquettes ( a ) and ( c ) in fig . [ modes ] . the bifurcation responsible for the termination of the pair of branches may take a complex degenerate form [ such as the one in the case of setting ( a ) ] . other branches were found too , that continue to exist , due to the nonlinearity , past the critical points of the underlying linear systems . in addition , we have identified cases [ like the gain - loss alternating pattern ( b ) or the cross plaquette of type ( d ) ] when the @xmath0 symmetry is broken immediately after the introduction of the gain - loss pattern . the spectral stability of the different configurations was examined . most frequently , the stationary modes are unstable , although stable branches were found too [ e.g. , in settings ( a ) and ( c ) ] . we have also studied the perturbed dynamics of the modes . the evolution of unstable ones typically leads to the growth of the amplitudes at the gain - carrying sites and decay at the lossy ones . it was interesting to observe that the passive sites , without gain or loss , might be tipped towards growth or decay , depending on the particular solution ( and possibly on specific initial conditions ) . the next relevant step of the analysis may be to search for more sophisticated stationary modes ( that plausibly can not be found in an analytical form ) , produced by the _ symmetry breaking _ of the simplest modes considered in this work , cf . ref . the difference of such modes from the @xmath0-symmetric ones considered in the present work is the fact that modes with the unbroken symmetry form a continuous family of solutions , with energy @xmath194 depending on the solution s amplitude , see eq . ( [ e ] ) . this feature , which is generic to conservative nonlinear systems , is shared by @xmath0-symmetric ones , due to the automatic " balance between the separated gain and loss . on the other hand , the breaking of the symmetry gives rise to the typical behavior of systems with competing , but not explicitly balanced , gain and loss , which generate a single or several _ attractors _ , i.e. , _ isolated _ solutions with a single or several values of the energy , rather than a continuous family . a paradigmatic example of the difference between continuous families of solutions in conservative models and isolated attractors in their ( weakly ) dissipative counterparts is the transition from the continuous family of solitons in the usual nlse to a pair of isolated soliton solutions , one of which is an attractor ( and the other is an unstable solution playing the role of the separatrix between attraction basins , the stable soliton and the stable zero solution ) in the complex ginzburg - landau equation , produced by the addition of the cubic - quintic combination of small dissipation and gain terms to the nlse @xcite . as concerns the systems considered in the present work , in the context of the breaking of the @xmath0 symmetry it may also be relevant to introduce a more general nonlinearity , which includes @xmath0-balanced cubic gain and loss terms , in addition to their linear counterparts ( cf . @xcite and @xcite ) . nevertheless , it should also be noted that the issue of potential existence of isolated solutions versus branches of solutions in @xmath0-symmetric systems is already starting to be addressed in the relevant literature ( including in plaquette - type configurations ) , as in the very recent work of @xcite . moreover , the present work may pave the way to further considerations of two - dimensional @xmath0-symmetric lattice systems , and even three - dimensional ones . in this context , the natural generalization is to construct periodic two - dimensional lattices of the building blocks presented here , and to identify counterparts of the modes reported here in the infinite lattices , along with new modes which may exist in that case . on the other hand , in the three - dimensional realm , the first step that needs to be completed would consist of the examination of a @xmath0-symmetric cube composed of eight sites , and the nonlinear modes that it can support . this , in turn , may be a preamble towards constructing full three - dimensional @xmath0-symmetric lattices . these topics are under present consideration and will be reported elsewhere . ug thanks holger cartarius and eva - maria graefe for useful discussions . pgk gratefully acknowledges support from the national science foundation under grant dms-0806762 and cmmi-1000337 , as well as from the alexander von humboldt foundation and the alexander s. onassis public benefit foundation . pgk and bam also acknowledge support from the binational science foundation under grant 2010239 . 99 c. m. bender and s. boettcher , phys . rev . lett . * 80 * , 5243 ( 1998 ) ; c. m. bender , s. boettcher and p. n. meisinger , j. math . phys . * 40 * , 2201 ( 1999 ) . z. h. musslimani , k. g. makris , r. el - ganainy and d. n. christodoulides , phys . lett . * 100 * , 030402 ( 2008 ) ; k. g. makris , r. el - ganainy , d. n. christodoulides and z. h. musslimani , phys . a * 81 * , 063807 ( 2010 ) . b. a. malomed and h. g. winful , phys . rev . e * 53 * , 5365 ( 1996 ) ; h. sakaguchi and b. a. malomed , physica d * 147 * , 273 ( 2000 ) ; w. j. firth and p. v. paulau , eur . j. d * 59 * , 13 ( 2010 ) ; p. v. paulau , d. gomila , p. colet , n. a. loiko , n. n. rosanov , t. ackemann , and w. j. firth , opt . express * 18 * , 8859 ( 2010 ) ; a. marini , d. v. skryabin , and b. a. malomed , _ ibid_. * 19 * , 6616 ( 2011 ) ; p. v. paulau , d. gomila , p. colet , b. a. malomed , and w. j. firth , phys . rev . e * 84 * , 036213 ( 2011 ) . m. znojil , rendic . circ . mat . palermo , ser . ii , suppl . * 72 * ( 2004 ) , 211 - 218 , math - ph/0104012 . g. s. japaridze , j. phys . a : math . theor . * 35 * , 1709 - 1718 ( 2002 ) , quant - ph/0104077 . a. mostafazadeh , j. math . phys . * 43 * , 205 - 214 ( 2002 ) , math - ph/0107001 . c. m. bender , d. c. brody , and h. f. jones , phys . lett . * 89 * , 270401 ( 2002 ) , quant - ph/0208076 .

we introduce four basic two - dimensional ( 2d ) plaquette configurations with onsite cubic nonlinearities , which may be used as building blocks for 2d @xmath0-symmetric lattices . for each configuration , we develop a dynamical model and examine its @xmath1symmetry . the corresponding nonlinear modes are analyzed starting from the hamiltonian limit , with zero value of the gain - loss coefficient , @xmath2 . once the relevant waveforms have been identified ( chiefly , in an analytical form ) , their stability is examined by means of linearization in the vicinity of stationary points . this reveals diverse and , occasionally , fairly complex bifurcations . the evolution of unstable modes is explored by means of direct simulations . in particular , stable localized modes are found in these systems , although the majority of identified solutions is unstable .

arxiv

the cataclysmic variable hu aqr currently consists of a 0.80 white dwarf that accretes from a 0.18 main - sequence companion star . the transfer of mass in the tight @xmath12 orbit is mediate by the emission of gravitational waves and the strong magnetic field of the accreting star . since its discovery , irregularities of the observed - calculated variations have led to a range of explanations , including the presence of circum - binary planets . detailed timing analysis has eventually led to the conclusion that the cv is orbited by two planets @xcite , a 5.7 planet in a @xmath13 orbit with an eccentricity of @xmath14 and a somewhat more massive ( 7.6 ) planet in a wider @xmath15 and eccentric @xmath16 orbit @xcite . although , the two - planet configuration turned out to be dynamically unstable on a 100010,000 year time scale ( * ? ? ? * see also [ sect : stability ] ) , a small fraction of the numerical simulations exhibit long term dynamical stability ( for model b2 in * ? ? ? * see tab.[tab : huaqr ] for the parameters ) . it is peculiar to find a planet orbiting a binary , in particular around a cv . while planets may be a natural consequence of the formation of binaries @xcite , planetary systems orbiting cvs could also be quite common . in particular because of recently timing residual in nn serpentis , dp leonis and qs virgo @xcite were also interpreted is being caused by circum - cv planets . although the verdict on the planets around hu aqr ( and the other cvs ) remains debated ( tom marsh private communication , and * ? ? ? * ) , we here demonstrate how a planet in orbit around a cv , and in particular two planets , can constrain the cv evolution and be used to reconstruct the history of the inner binary . we will use the planets to perform a precision reconstruction of the binary history , and for the remaining paper we assume the planets to be real . because of their catastrophic evolutionary history , cvs seem to be the last place to find planets . the original binary lost probably more than half its mass in the common - envelope phase , which causes the reduction of the binary separation by more than an order of magnitude . it is hard to imagine how a planet ( let alone two ) can survive such turbulent past , but it could be a rather natural consequence of the evolution of cvs , and its survival offers unique diagnostics to constrain the origin and the evolution of the system . after the birth of the binary , the primary star evolved until it overflowed it roche lobe , which initiated a common - envelope phase . the hydrogen envelope of the primary was ejected quite suddenly in this episode @xcite , and the white dwarf still bears the imprint of its progenitor : the mass and composition of the white dwarf limits the mass and evolutionary phase of its progenitor star at the moment of roche - lobe overflow ( rlof ) . for an isolated binary the degeneracy between the donor mass at the moment of rlof ( @xmath17 ) , its radius @xmath18 and the mass of its core @xmath19 can not be broken . the presence of the inner planet in orbit around hu aqr @xcite allows us to break this degeneracy and derive the rate of mass loss in the common - envelope phase . the outer planet allows us to validate this calculation and in addition to determine the conditions under which the cv was born . the requirement that the initial binary must have been dynamically stable further constrains the masses of the two stars and their orbital separation . during the cv phase little mass is lost from the binary system @xmath20constant ( but see * ? ? ? * ) , and the current total binary mass ( @xmath21 ) was not affected by the past ( and current ) cv evolution @xcite . the observed white dwarf mass then provides an upper limit to the mass of the core of the primary star at the moment of roche - lobe contact , and therefore also provides a minimum to the companion mass via @xmath22 . with the mass of the companion not being affected by the common envelope phase , we constrain the orbital parameters at the moment of rlof by calculating stellar evolution tracks to measure the core mass @xmath19 and the corresponding radius @xmath23 for stars with zero - age main - sequence mass @xmath24 . in fig.[fig : amcoreformzams3msun ] we present the evolution of the radius of a 3 star as a function of @xmath19 , which is a measure of time we adopted the henyey stellar evolution code mesa @xcite to calculate evolutionary track of stars from @xmath25 to 8 using amuse @xcite to run mesa and determine the mass of the stellar core . the latter is measured by searching for the mass - shell in the stellar evolution code for which the relative hydrogen fraction @xmath26 . at the moment of rlof the core mass is @xmath19 and the stellar radius @xmath27 . via the relation for the roche radius @xcite , we can now calculate the orbital separation at the moment of rlof @xmath28 as a function of @xmath17 . this separation is slightly larger than the initial ( zero - age ) binary separation @xmath29 due to the mass lost by the primary star since its birth @xmath30 . the long ( main - sequence ) time scale in which this mass is lost guarantees an adiabatic response to the orbital separation , i.e. @xmath31 constant . for each @xmath24 we now have a range of possible solutions for @xmath28 as a function of @xmath19 and @xmath32 . this reflects the assumption that the total mass ( @xmath33 ) in the observed binary with mass @xmath34 is conserved throughout the evolution of the cv . in fig.[fig : amcoreformzams3msun ] we present the corresponding stellar radius @xmath18 and @xmath35 as a function of @xmath19 for @xmath36 . this curve for @xmath28 is interrupted when rlof would already have been initiated earlier for that particular orbital separation . we calculate this curve by first measuring the size of the donor for core mass @xmath19 , and assuming that the primary fills its roche - lobe we calculate the orbital separation at which this happens . during the common envelope phase the primary s mantle is blown away beyond the orbit of the planets . the latter responds to this by migrating from the orbits in which they were born ( semi - major axis @xmath37 and eccentricity @xmath38 , the subscript @xmath39 indicates the inner planet , we adopt a @xmath40 to indicate the outer planet ) to the currently observed orbits . using first order analysis we recognize two regimes of mass loss : fast and slow . in the latter case the orbit expands adiabatically without affecting the eccentricity : the minimum possible expansion of the planet s orbit is achieved when the common envelope is lost adiabatically . fast mass loss leads to an increase in the eccentricity as well and may even cause the planet to escape @xcite . a planet born at the shortest possible orbital separation to be dynamically stable will have @xmath41 @xcite , which is slightly smaller than the distance at which circum binary planets tend to form @xcite . in fig.[fig : amcoreformzams3msun ] we present a minimum to the semi - major axis for a planet that was born at @xmath42 and migrated by the adiabatic loss of the hydrogen envelope from the primary star in the common - envelope phase . the planet can have migrated to a wider orbit , but not to an orbit smaller than the solid black curve ( indicated with @xmath43 ) in fig.[fig : amcoreformzams3msun ] . for the 3 star , presented in fig.[fig : amcoreformzams3msun ] , rlof can successfully result in the migration of the planet to the observed separation in hu aqr for @xmath44 , which occurs for @xmath45 . a core mass @xmath46 would , for a 3 primary star , result in an orbital separation that exceeds that of the inner planet in hu aqr ; in this case the core mass of the primary star must have been smaller than 0.521 . another constraint on the initial binary orbit is provided by the requirement that the mass transfer in the post common - envelope binary should be stable when the companion starts to overfill its roche lobe . to guarantee stable mass transfer we require that @xmath47 . the thick part of the red curve in fig.[fig : amcoreformzams3msun ] indicates the valid range for the initial orbital separation and core - mass for which the observed planet can be explained ; the thin parts indicate where these criteria fail . we repeat the calculation presented in fig.[fig : amcoreformzams3msun ] for a range of masses from @xmath48 to 8 with steps of 0.02 , the results are presented as the shaded region in fig.[fig : am0_distribution_hu ] . the response of the orbit of the planet to the mass loss depends on the total amount of mass lost in the common envelope and the rate at which it is lost . numerical common - envelope studies indicate that for an in - spiraling binary @xmath49 @xcite . at this rate the entire envelope @xmath505.8 is expelled well within one orbital period of the inner planet , which leads to an impulsive response and the possible loss ( for @xmath51 ) of the planet . the fact that the hu aqr is orbited by a planet indicates that at the distance of the planet @xmath52 . the eccentricity of the inner planet in hu aqr ( see tab.[tab : huaqr ] ) can be used to further constrain the rate at which the common - envelope was lost from the planetary orbit . the higher eccentricity of the outer planet indicates a more impulsive response , which is a natural consequence of its wider orbits with the same @xmath53 . this regime between adiabatic and impulsive mass loss is hard to study analytically @xcite . [ cols="<,<,<,<,<,<,<,<",options="header " , ] we calculate the effect of the mass loss on the orbital parameters by numerically integrating the planet orbit . the calculations are started by selecting initial conditions for the zero - age binary hu aqr @xmath24 , @xmath29 and consequently @xmath19 from the available parameter space ( shaded area ) in fig.[fig : am0_distribution_hu ] , and integrate the equations of motion of the inner planet with time . planets ware assumed to be born in a circular orbit ( @xmath54 ) in the binary plane with semi - major axis @xmath37 . the equations of motions are integrated using the high - order symplectic integrator huayno @xcite via the amuse framework . during the integration we adopt a constant mass - loss rate @xmath53 applied at every 1/100th of an orbit , and we continued the calculation until the entire envelope is lost ( see [ sect : ce ] and fig.[fig : am0_distribution_hu ] ) , at which time we measure the final semi - major axis and eccentricity of the planetary orbit . during the integration we allow the energy error to increase up to at most @xmath55 . by repeating this calculation while varying @xmath37 and @xmath56 we iterate ( by bisection ) until the result is within 1% of the observed @xmath57 and @xmath58 of the inner planet observed in hu aqr . the converged results of these simulations are presented in fig.[fig : am0_distribution_hu ] ( circles ) , and these represent the range of consistent values for the inner planet s orbital separation @xmath59752 as a function of @xmath608 and consistently reproduce the observed inner planet when adopting @xmath610.267/yr . the highest value for @xmath53 is reached for @xmath62 at an initial orbital separation of @xmath63 . the orbital solution for the inner planet is insensitive to the semi - major axis of the zero - age binary @xmath29 ( for a fixed @xmath24 ) , and each of these solutions were tested for dynamical stability , which turned out to be the case irrespective of the initial binary semi - major axis ( as discussed in [ sect : stability ] ) . we now adopt the in [ sect : innerplanet ] measured value of @xmath53 to integrate the orbit of the outer planet . the effect of the mass outflow on the planet is proportional to the square of the density in the wind at the location of the planet . we correct for this effect by reducing the mass loss rate in the common envelope that affects the outer planet by a factor @xmath64 . we use the same integrator and assumptions about the initial orbits as in [ sect : innerplanet ] , but we adopt the value of @xmath53 from our reconstruction of the inner planet ( see [ sect : innerplanet ] ) . to reconstruct the initial orbital separation of the outer planet @xmath65 , we vary this value ( by bisection ) until the final semi - major axis is within 1% of the observed orbit ( see tab.[tab : huaqr ] ) . the results are presented in fig.[fig : am0_distribution_hu ] ( triangles ) . the post common - envelope eccentricity of the outer planet then turn out to be @xmath66 . after having reconstructed the initial conditions of the binary system with its two planets we test its dynamical stability by integrating the entire system numerically for 1myr using the huayno integrator @xcite . to test the stability we check the semi - major axis and eccentricity of both planets every 100years . if any of these parameters change by a factor of two compared to the initial values or if the orbits cross we declare the system unstable , otherwise they are considered stable . the calculations are repeated with the 4th order hermite predictor - corrector integrator ph4 @xcite within amuse to verify that the results are robust , which turned out to be the case . we then repeated this calculation ten times with random inital orbital phases and again with a 1% gaussian variation in the initial planetary semi - major axes . in fig.[fig : am0_distribution_hu ] we present the resulting stable systems by coloring them red ( circled ) and blue ( triangles ) , the unstable systems are represented by open symbols . from the wide range of possible systems that can produce hu aqr only a small range around @xmath67 turns out to be dynamically stable . the eccentricity of the outer orbit of the stable systems ( which ware stable for initial conditions within 1% ) @xmath68 , which is somewhat smaller than the observed value for hu aqr ( * ? ? ? * @xmath69 ) . these values are obtained with @xmath70 . the small uncertainty in the derived value of @xmath53 is a direct consequence of its sensitivity to @xmath38 and the small error on @xmath24 from the requirement that the initial system is dynamically stable . we have adopted the suggestive results from the timing analysis of hu aqr , that the cv is orbited by two planets , to reconstruct the evolution of this complex system . a word of caution is well placed in that these observations are not confirmed , and currently under debate ( tom marsh private communication , and comments by the referee ) . however , the predictive power that such an observation would entail is interesting . the possibility to reconstruct the initial conditions of a cv by measuring the orbital parameters of two circum binary planets is a general result that can be applied to other binaries . for cvs in particular it enables us to constrain the value of fundamental parameters in the common - envelope evolution . this in itself makes it interesting to perform this theoretical exercise , irrespective of the uncertainty in the observations . on the other hand , the consistency between the observations and the theoretical analysis give some trust to the correctness of these observations . the presence of one planet in an eccentric orbit around a cv allow us to calculate the rate at which the common - envelope was lost from the inner binary . a single planet provides insufficient information to derive the initial mass of the primary star , but allows us to derive the initial binary separation and planetary orbital separation to within about factor of 5 , and the initial rate of mass loss from the common envelope to about a factor 2 . a second planet can be used to further constrain these parameters to a few per cent accuracy and allows us to make a precision reconstruction of the evolution of the cv . we have used the observed two planets in orbit around the cv hu aqr to reconstruct its evolution , to derived its initial conditions ( primary mass , secondary mass , orbital separation , and the orbital separations of both planets ) and to measure the rate of mass lost in the common - envelope parameters @xmath53 . by comparing the binary parameters at birth with those after the common - envelope phase we subsequently calculate the two parameters @xmath71 and @xmath72 . the measured rate of mass loss for hu aqr of @xmath73 from the inner planetary orbit , which from the binary system itself would entail a mass - loss rate of @xmath74 , when we adopt the initial binary to have a semi - major axis of @xmath75 , which is bracketed by our derived range of @xmath76160 . this is consistent with a mass - loss rate of @xmath77 from numerical common - envelope studies @xcite . by adopting that the binary survives its common envelope at a separation between @xmath78 ( at which separation the secondary star will just fill it s roche - lobe to the white dwarf ) and @xmath79 ( for gravitational wave radiation to drive the binary into roche - lobe overflow within 10gyr ) , we derive the value of @xmath802.0 ( for @xmath75 we arrive at @xmath81 ) . this value is a bit small compared to numerous earlier studies , which tend to suggest @xmath82 . the alternative @xmath72-formalism for common - envelope ejection gives a value of @xmath831.80 ( for @xmath75 we arrive at @xmath84 ) , which is consistent with the determination of @xmath72 in 30 other cvs @xcite . the inner planet in hu aqr formed at @xmath85@xmath86 , with a best value of @xmath87 , which is consistent with the planets found to orbit other binaries , like kepler 16 @xcite and for kepler 34 and 35 @xcite , although these systems have lower primary mass and secondary mass stars . it seems unlikely that more planets were formed inside the orbit of the inner most planet , even though currently there is sufficient parameter space for many more stable planets ; in the zero - age binary there has not been much room for forming additional planets further in . it is however possible that additional planets formed further out and those , we predict , will have even higher eccentricity than those already found . * acknowledgements * it is a pleasure to thank edward p.j . van den heuvel , tom marsh , inti pelupessy , nathan de vries , arjen van elteren and the anonymous referee for comments on the manuscript and discussions . this work was supported by the netherlands research council nwo ( grants # 612.071.305 [ lgm ] , # 639.073.803 [ vici ] and # 614.061.608 [ amuse ] ) and by the netherlands research school for astronomy ( nova ) .

cataclysmic variables ( cvs ) are binaries in which a compact white dwarf accretes material from a low - mass companion star . the discovery of two planets in orbit around the cv hu aquarii opens unusual opportunities for understanding the formation and evolution of this system . in particular the orbital parameters of the planets constrains the past and enables us to reconstruct the evolution of the system through the common - envelope phase . during this dramatic event the entire hydrogen envelope of the primary star is ejected , passing the two planets on the way . the observed eccentricities and orbital separations of the planets in hu aqr enable us to limit the common - envelope parameter @xmath0 or @xmath1 and measure the rate at which the common envelope is ejected , which turns out to be copious . the mass in the common envelope is ejected from the binary system at a rate of @xmath2 . the reconstruction of the initial conditions for hu aqr indicates that the primary star had a mass of @xmath3 and a @xmath4 companion in a @xmath5160 ( best value @xmath6 ) binary . the two planets were born with an orbital separation of @xmath7 and @xmath8 respectively . after the common envelope , the primary star turns into a @xmath9 helium white dwarf , which subsequently accreted @xmath10 from its roche - lobe filling companion star , grinding it down to its current observed mass of @xmath11 . methods : numerical planets and satellites : dynamical evolution and stability planet star interactions planets and satellites : formation stars : individual : hu aquarius stars : binaries : evolution

arxiv

the @xmath0 ncsm / rgm was presented in @xcite as a promising technique that is able to treat both structure and reactions in light nuclear systems . this approach combines a microscopic cluster technique with the use of realistic interactions and a consistent @xmath0 description of the nucleon clusters . the method has been introduced in detail for two - body cluster bases and has been shown to work efficiently in different systems @xcite . however , there are many interesting systems that have a three - body cluster structure and therefore can not be successfully studied with a two - body cluster approach . the extension of the ncsm / rgm approach to properly describe three - body cluster states is essential for the study of nuclear systems that present such configuration . this type of systems appear , @xmath3 , in structure problems of two - nucleon halo nuclei such as @xmath1he and @xmath4li , resonant systems like @xmath5h or transfer reactions with three fragments in their final states like @xmath6h(@xmath6h,2n)@xmath2he or @xmath6he(@xmath6he,2p)@xmath2he . recently , we introduced three - body cluster configurations into the method and presented the first results for the @xmath1he ground state @xcite . here we present these results as well as first results for the continuum states of @xmath1he within a @xmath2he+n+n basis . the extension of the ncsm / rgm approach to properly describe three - cluster configurations requires to expand the many - body wave function over a basis @xmath7 of three - body cluster channel states built from the ncsm wave function of each of the three clusters , @xmath8 @xmath9^{(j^{\pi}t ) } \times \frac{\delta(x-\eta_{a_2-a_3})}{x\eta_{a_2-a_3 } } \frac{\delta(y-\eta_{a - a_{23}})}{y\eta_{a - a_{23}}}\ , , \label{eq:3bchannel } \end{aligned}\ ] ] where @xmath10 is the relative vector proportional to the displacement between the center of mass ( c.m . ) of the first cluster and that of the residual two fragments , and @xmath11 is the relative coordinate proportional to the distance between the centers of mass of cluster 2 and 3 . in eq . ( [ eq1 ] ) , @xmath12 are the relative motion wave functions and represent the unknowns of the problem and @xmath13 is the intercluster antisymmetrizer . projecting the microscopic @xmath14-body schrdinger equation onto the basis states @xmath15 , the many - body problem can be mapped onto the system of coupled - channel integral - differential equations @xmath16 g_{\nu}^{j^\pi t}(x , y ) = 0,\label{eq:3beq1 } \end{aligned}\ ] ] where @xmath17 is the total energy of the system in the c.m . frame and @xmath18 are integration kernels given respectively by the hamiltonian and overlap ( or norm ) matrix elements over the antisymmetrized basis states . finally , @xmath19 is the intrinsic @xmath14-body hamiltonian . in order to solve the schrdinger equations ( [ eq:3beq1 ] ) we orthogonalize them and transform to the hyperspherical harmonics ( hh ) basis to obtain a set of non - local integral - differential equations in the hyper - radial coordinate , @xmath20 which is finally solved using the microscopic r - matrix method on a lagrange mesh . the details of the procedure can be found in @xcite . at present , we have completed the development of the formalism for the treatment of three - cluster systems formed by two separate nucleons in relative motion with respect to a nucleus of mass number a@xmath21 . it is well known that @xmath1he is the lightest borromean nucleus @xcite , formed by an @xmath2he core and two halo neutrons . it is , therefore , an ideal first candidate to be studied within this approach . in the present calculations , we describe the @xmath2he core only by its g.s . wave function , ignoring its excited states . this is the only limitation in the model space used . we used similarity - renormalization - group ( srg ) @xcite evolved potentials obtained from the chiral n@xmath6lo nn interaction @xcite with @xmath22 = 1.5 @xmath23 . the set of equations ( [ rgmrho ] ) are solved for different channels using both bound and continuum asymptotic conditions . we find only one bound state , which appears in the @xmath24 channel and corresponds to the @xmath1he ground state . [ [ ground - state ] ] ground state + + + + + + + + + + + + [ tab : a ] lccc approach & & e@xmath25(@xmath2he ) & e@xmath25(@xmath1he ) + ncsm / rgm & ( @xmath26=12 ) & @xmath27 mev & @xmath28 mev + ncsm & ( @xmath26=12 ) & @xmath29 mev & @xmath30 mev + ncsm & ( extrapolated ) & @xmath31 mev & @xmath32 mev + the results for the g.s . energy of @xmath1he within a @xmath2he(g.s.)+n+n cluster basis and @xmath26 = 12 , @xmath33 = 14 mev harmonic oscillator model space are compared to ncsm calculations in table [ tab : a ] . at @xmath34 12 the binding energy calculations are close to convergence in both ncsm / rgm and ncsm approaches . the observed difference of approximately 1 mev is due to the excitations of the @xmath2he core , included only in the ncsm at present . therefore , it gives a measure of the polarization effects of the core . the inclusion of the excitations of the core will be achieved in a future work through the use of the no - core shell model with continuum approach ( ncsmc ) @xcite , which couples the present three - cluster wave functions with ncsm eigenstates of the six - body system . contrary to the ncsm , in the ncsm / rgm the @xmath2he(g.s.)+n+n wave functions present the appropriate asymptotic behavior . the main components of the radial part of the @xmath1he g.s . wave function @xmath35 can be seen in fig . ( [ fig:1 ] ) for different sizes of the model space demostrating large extension of the system . in the left part of the figure , the probability distribution of the main component of the wave function is shown , featuring two characteristic peaks which correspond to the di - neutron and cigar configurations . a thorough study of the converge of the results with respect to different parameters of the calculation was presented in @xcite , showing good convergence and stability . he+@xmath36+@xmath36 relative motion wave function for the @xmath37 ground state , @xmath38 and @xmath39 are the distances between the two neutrons and between the @xmath40 particle and center of mass of the two neutrons , respectively . in the right , the three main components of the radial part of the @xmath1he g.s . wave functions @xmath35 for @xmath26=6,8,10 , and 12 . , title="fig:",height=226 ] he+@xmath36+@xmath36 relative motion wave function for the @xmath37 ground state , @xmath38 and @xmath39 are the distances between the two neutrons and between the @xmath40 particle and center of mass of the two neutrons , respectively . in the right , the three main components of the radial part of the @xmath1he g.s . wave functions @xmath35 for @xmath26=6,8,10 , and 12 . , title="fig:",height=207 ] [ [ continuum - states ] ] continuum states + + + + + + + + + + + + + + + + the use of three - cluster dynamics is essential for describing @xmath1he states in the continuum . therefore , this formalism is ideal for such study . using continuum asymptotic conditions , we solved the set of equations ( [ rgmrho ] ) in order to obtain the low - energy phase shifts for the @xmath41 and @xmath42 channels in the continuum . in our preliminary results , we obtain the experimentally well - known @xmath43 resonance as well as a second low - lying @xmath44 resonance recently measured at ganil @xcite . a resonance is also found in the @xmath45 channel while no low - lying resonances are present in the @xmath46 or @xmath47 channels . in fig . [ fig:2 ] some of the preliminary phase shifts for different channels are shown . results for bigger model spaces and a study of their stability respect to the parameters in the formalism are presently being calculated and will be presented elsewhere . he for different @xmath48 channels.,height=226 ] in this work , we present an extension of the ncsm / rgm which includes three - body dynamics in the formalism . this new feature permits us to study a new range of systems that present three - body configurations . in particular , we presented results for both bound and continuum states of @xmath1he studied within a basis of @xmath2he+n+n . the obtained wave functions feature an appropriate asymptotic behavior , contrary to bound - state @xmath0 methods such as the ncsm . computing support for this work came from the llnl institutional computing grand challenge program and from an incite award on the titan supercomputer of the oak ridge leadership computing facility ( olcf ) at ornl . prepared in part by llnl under contract de - ac52 - 07na27344 . support from the u.s . doe / sc / np ( work proposal no . scw1158 ) and nserc grant no . 401945 - 2011 is acknowledged . triumf receives funding via a contribution through the canadian national research council . 13 quaglioni s , navrtil p ( 2008 ) ab initio many - body calculations of n-3h , n-(4)he , p-(3,4)he , and n-(10)be scattering . phys rev lett 101:092501 quaglioni s , navrtil p ( 2009 ) ab initio many - body calculations of nucleon - nucleus scattering . phys rev c 79:044606 navrtil p , quaglioni s ( 2011 ) ab initio many - body calculations of deuteron - he-4 scattering and li-6 states . phys rev c 83:044609 navrtil p , quaglioni s ( 2012 ) ab initio many - body calculations of the h-3(d , n)he-4 and he-3(d , p)he-4 fusion reactions . phys rev lett 108:042503 quaglioni s , romero - redondo c , navrtil p ( 2013 ) three - cluster dynamics within an ab initio framework . phys rev c 88:034320 tanihata i ( 1996 ) neutron halo nuclei . j phys g 22:157 - 198 tanihata i , hamagaki h , hashimoto o , shida y , yoshikawa n , sugimoto k , yamakawa o , kobayashi t , takahashi n ( 1985 ) measurements of interaction cross sections and nuclear radii in the light p - shell region . phys rev lett 55:2676 - 2679 bogner sk , furnstahl rj , perry rj ( 2007 ) similarity renormalization group for nucleon - nucleon interac- tions . phys rev c 75:061001 roth r , reinhardt s , hergert h ( 2008 ) unitary correlation operator method and similarity renormalization group : connections and differences . phys rev c 77:064003 entem dr , machleidt r ( 2003 ) accurate charge - dependent nucleon - nucleon potential at fourth order of chiral perturbation theory . phys rev c 68:041001 baroni s , navrtil p , quaglioni s ( 2013 ) ab initio description of the exotic unbound @xmath49he nucleus . phys rev lett 110:022505 baroni s , navrtil p , quaglioni s ( 2013 ) unified ab initio approach to bound and unbound states : no - core shell model with continuum and its application to @xmath49he . phys rev c 87:034326 mougeot x , lapoux v , mittig w , alamanos n , auger f , et al ( 2012 ) new excited states in the halo nucleus he-6 . phys lett b 718:441 - 446

we introduce an extension of the @xmath0 no - core shell model / resonating group method ( ncsm / rgm ) in order to describe three - body cluster states . we present results for the @xmath1he ground state within a @xmath2he+n+n cluster basis as well as first results for the phase shifts of different channels of the @xmath2he+n+n system which provide information about low - lying resonances of this nucleus . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

arxiv

the experimental evidence of the theoretically predicted skyrmions in non - centrosymmetric compounds with dzyloshinskii - moriya interaction has intrigued many scientists over the last years.@xcite recently , the preparation of thin films of b20 type mnsi on silicon substrates @xcite has offered promising prospects with regard to possible applications in future spintronic devices . on the one hand mnsi films offer a variety of interesting magnetic phases and on the other hand they are easy to integrate into devices due to the use of silicon as substrate material being well established in technology . the benefit of thin films compared to bulk material is the existence of the skyrmion phase in an extended region of the magnetic phase diagram due to the uniaxial anisotropy.@xcite this pioneers new opportunities for data storage devices . the drawback using mnsi films is the low magnetic ordering temperature , which is considerably below liquid nitrogen temperature . therefore , it is the aim to find compounds with similar spin order at higher temperatures . a suitable candidate is the b20 compound mnge ( bulk lattice constant of 4.795 ) with a magnetic ordering temperature @xmath0 of 170k.@xcite the magnetic ground state of mnge is a helical spin structure with a helix length between 3 nm at lowest temperatures and 6 nm near @xmath0.@xcite the helix axis is due to magnetic anisotropy pinned along @xmath1001@xmath2,@xcite but rotates into field direction in an applied field . recently , a large topological hall effect exceeding by 40 times that of mnsi was observed in mnge which was attributed to a skyrmion phase in analogy to mnsi.@xcite . further evidence for the existence of skyrmions was given by small angle neutron scattering experiments.@xcite unfortunately , the synthesis of mnge is considerably laborious , since it forms only under high pressure and temperatures between 600 and 1000@xmath3c.@xcite however , molecular beam epitaxy ( mbe ) allows for thin film growth under strong non - equilibrium conditions . nevertheless , there has been no successfull attempt to grow mnge on ge , since mn and ge tends to form mn@xmath4ge@xmath5.@xcite the use of si(111 ) as substrate offers the opportunity to prepare a seedlayer of mnsi , which realizes the b20 crystal structure for mnge growth . the lattice constant of mnge within the ( 111 ) plane matches that of si with a misfit of only 2@xmath6 , thus , compressively strained mnge films may be grown on si(111 ) substrates . in this paper we show a preparation method for mnge thin films on si substrates with the aid of a mnsi seedlayer . the structure and morphology of the films have been investigated by reflection high - energy electron diffraction ( rheed ) , atomic force microscopy ( afm ) and x - ray diffraction ( xrd ) . to determine the physical properties of the samples magnetization and magnetoresistance measurements have been performed . for the growth of mnge thin films p - doped si(111 ) substrates were used , which possess a resistivity between 1 and 10 @xmath7 cm at room temperature . prior to film deposition the substrates were heated to 1100@xmath3c under uhv conditions in order to remove the oxide layer and to achieve a clean and flat surface with 7@xmath87-reconstruction , which was verified by in - situ rheed investigations . the depostion of mn and ge directly on the si(111 ) surface does not produce b20 mnge films but results in a mn@xmath4ge@xmath5 layer . in order to establish the b20 crystal structure a 5mn layer was deposited onto the si surface and heated to 300@xmath3c subsequently . this procedure provides for the formation of a thin mnsi seedlayer with a thickness of 10 . in a second step , mnge is codeposited by simultanoeus evaporation of mn and ge from an effusion cell and an electron beam evaporator , respectively . during film growth with a rate of 0.15 / s the substrate is held at a temperature of 250@xmath3c . $ ] crystal direction and b ) line scans across the rheed streaks for the mnge film in comparison with the si substrate . the scans were taken parallel to the shadow edge.,scaledwidth=35.0% ] the mnge films have been investigated by in - situ rheed in order to determine their structure and morphology . the rheed pattern of a 135 mnge film observed along the @xmath9 $ ] direction of the si substrate indicates two - dimensional film growth [ fig . [ fig : rheed](a ) ] . the arrangement of the streaks is very similar to the pattern of mnsi thin films,@xcite and suggests that mnge sustains the b20 crystal structure provided by mnsi seedlayer . the uniformity of the intensity of the detected streaks implies a flat surface of a size comparable to the area contributing to the rheed pattern of around 100 nm in diameter . line scans across rheed patterns of a 135 mnge film [ fig . [ fig : rheed](b ) ] compared to the si substrate reveal a nearly pseudomorphic growth of the mnge layer . however , a small deviance of the mnge streaks from the corresponding si reflections indicates that the mnge lattice has at least partly relaxed from the compressive strain imposed by the substrate . + afm images of films with thicknesses of 45 , 90 and 135 give evidence that island growth of vollmer - weber type is the predominant growth mode [ fig . [ fig : afm ] ] . the thinnest film of 45 thickness [ fig . [ fig : afm](a ) ] consists of islands with a typical diameter of 100 nm separated by valleys of similar size . with increasing film thickness the islands are enlarged and gradually fill the space between them . for the 135 film only very thin valleys of a few nm can be observed [ fig . [ fig : afm](c ) ] , and the morphology has transformed into elongated islands with a length of up to 2@xmath10 m and a width of around 200 nm . $ ] and @xmath11 $ ] crystal directions . inset : intensity plot along the @xmath12 $ ] direction.,scaledwidth=40.0% ] x - ray diffraction measurements were performed using synchrotron radiation with @xmath13 at the swiss - norwegian beamline bm1a of the esrf ( grenoble , france ) with the pilatus@snbl diffractometer . the investigation of the 135 film confirms the b20 crystal structure of the mnge . in fig . [ fig : xray ] the ( 111 ) and ( 333 ) peaks of the si substrate and the mnge thin film are clearly resolved as single crystal peaks . the inset shows an integrated diffraction pattern along the [ 111 ] direction . from the position of the mnge(111 ) peak the lattice parameter of the mnge film of ( 4.750 @xmath14 0.004) is obtained , which is 1% smaller than the value for bulk mnge due to compressive strain . the magnetic characterization of the mnge films was carried out using a quantum design mpms-5s squid magnetometer . for films of different thickness the temperature dependence of the magnetic susceptibility has been measured in the range from 5k to 300k in an applied magnetic field of 10mt [ fig . [ fig : susceptibility ] ] . below 40k the susceptibility slightly increases due to the mnsi seedlayer , that orders magnetically in this temperature range . the measurements were normalized with respect to the saturation magnetization of the mnsi seedlayer , because this layer is the same for all three films . the susceptibility of mnge films exhibits an ordering temperature of @xmath15k indicated by a broad peak . regarding mnge bulk material , the susceptibility shows a qualitatively similar behavior with a lower @xmath16k.@xcite an enhancement of the ordering temperature has also been observed for mnsi thin films.@xcite in contrast to mnsi thin films , no thickness dependence of @xmath0 can be detected for films between 45 and 135 . possibly , the spin - spin correlation length is shorter than the value for mnsi films ( 7 monolayers ) @xcite , i.e. the thickness dependence may occur for mnge when the films are thinner than investigated in this work . both materials belong to the b20 compounds , which possess a helical spin structure , since their magnetic properties are governed by the interplay of ferromagnetic exchange and dzyaloshinskii - moriya interactions . nevertheless , the susceptibility of mnge shows a behavior which is typical for antiferromagnetic order , whereas for mnsi an increase to a constant value of magnetization towards low temperature is observed . the helix length in mnsi is very long ( 18 nm ) and , thus , the local magnetic structure is related to ferromagnetism . a small field easily deforms the soft helix and induces a net magnetization . in the case of mnge the helix is more rigid . therefore , at low temperature no net magnetic moment is induced by a field as small as 10mt . the helix wavelength is connected to the dzyaloshinskii constant @xmath17 via @xmath18 , where @xmath19 is the magnetic stiffness.@xcite since the helix in mnge is extremely short ( @xmath20nm)@xcite the dzyaloshinskii constant is expected to be large , and the magnetic structure is very close to an antiferromagnet . field dependent magnetization measurements at @xmath21k were carried out on the same three samples as in fig . [ fig : susceptibility ] . for all samples the magnetization increases in fields up to 1 t [ fig . [ fig : magnetization ] ] . the inset of fig . [ fig : magnetization ] shows a magnetization measurement on the 135 film in fields up to 5 t , which reveals that saturation is reached around 1 t . this is in agreement with measurements performed on bulk mnge at temperatures close to @xmath0.@xcite since the helix length is much shorter than the size of the mnge islands , the magnetization behavior is not expected to be different from the bulk . the magnetic moment per mn atom was calculated assuming that the complete amount of mn deposited during growth has reacted to mnge . however , since the magnetic moment is only half of the bulk value , some part of the deposited mn did not form mnge . furthermore , we observe an apparently larger magnetic moment for thicker films . this can be explained by the fact that especially in the beginning of mnge growth not every mn atom is incorporated into the mnge crystal . evidence for this is also given by magnetization measurements at 5k , where mainly the ordering of the mnsi seedlayer is observed , since mnge only gives a linear contribution in the considered field range . we observe a moment of the mnsi layer that is about twice as large as expected for 10 mnsi . thereby , we conclude that some part of the deposited mn has reacted with si from the substrate to form mnsi . + resistivity measurements were performed on the 135 film using the van - der - pauw method . the sample was found to be metallic , and the residual resistivity at 3k was determined as 83 @xmath22 cm . the field dependence of the resistivity was measured in magnetic fields up to 5 t for several temperatures . in fig . [ fig : mr ] three curves are depicted , which represent the magnetoresistivity defined by @xmath23 . the data were obtained at temperatures between 60k and 100k , where the sample is in a magnetically ordered state . the mr effect is negative for all temperatures and fields and exhibits no remarkable features in the investigated field range . for comparison the equivalent data for a 19 nm film mnsi is shown in the inset of fig . [ fig : mr ] . in the case of mnsi the critical magnetic field , where the spins align ferromagnetically , occurs around 1 t . at this field a clear kink accompanied by a change in curvature is observed . furthermore the size of the mr effect is considerably larger for mnsi . regarding mnge the absence of magnetic phase transitions in moderate magnetic fields and the smallness of the magnetoresistivity evidences that the helical structure is more rigid than in mnsi . as discussed in the previous paragraph this can be ascribed to a stronger dzyaloshinskii - moriya interaction . in this work we have proved that we succeeded in growing crystalline mnge as a thin film on a si(111 ) substrate . the film adopts the b20 structure from a thin seedlayer of mnsi prepared prior to mnge growth . morphological investigations using rheed and afm give evidence that the mnge thin films consist of islands with a flat surface , which enlarge during growth . the b20 structure was confirmed by xrd and the lattice parameter was determined to be 1% smaller than in bulk mnge due to compressive strain imposed by the si substrate . + although the magnetic properties of mnge thin films are found to be qualitatively similar to bulk , the ordering temperature is enhanced to 200k . in magnetoresistivity measurements no critical fields were observed up to 5 t . compared to mnsi , the helix in mnge is shorter and more rigid than in mnsi . therefore , the magnetic structure is related to antiferromagnetism rather than to ferromagnetism . we would like to thank dmitry chernyshov for his support with the x - ray measurements at the european synchrotron radiation facility . the afm measurements were performed at the institute of semiconductor technology in braunschweig . we thank alexander wagner for his help with the equipment . 99 s. mhlbauer , b. binz , f. jonietz , c. pfleiderer , a. rosch , a. neubauer , r. georgii , and p. bni , science * 323 * , 915 ( 2009 ) . a. neubauer , c. pfleiderer , b. binz , a. rosch , r. ritz , p. g. niklowitz , and p. bni , phys . * 102 * , 186602 ( 2009 ) . u. k. rler , a. n. bogdanov , and c. pfleiderer , nature * 442 * , 797 ( 2006 ) . j. engelke , t. reimann , l. hoffmann , s. gass , d. menzel , and s. sllow , j. phys . . jpn . * 81 * , 124709(2012 ) . e. a. karhu , u. k. rler , a. n. bogdanov , s. kahwaji , b. j. kirby , h. fritzsche , m. d. robertson , c. f. majkrzak , and t. l. monchesky , phys . b * 85 * , 094429 ( 2012 ) . m. n. wilson , e. a. karhu , a. s. quingley , u. k. rler , a. b. butenko , a. n. bogdanov , m. d. robertson , and t. l. monchesky , phys . b * 86 * , 144420 ( 2012 ) . a. b. butenko , a. a. leonov , u. k. rler , and a. n. bogdanov , phys . rev . b * 82 * , 052403 ( 2010 ) . y. li , n. kanazawa , x. z. yu , a. tsukazaki , m. kawasaki , m. ichikawa , x. f. jin , f. kagawa , and y. tokura , phys . * 110 * , 117202 ( 2013 ) . h. takizawa , t. sato , t. endo , and m. shimada , j. solid state chem . * 73 * , 40 ( 1988 ) . n. kanazawa , y. onose , t. arima , d. okuyama , k. ohoyama , s. wakimoto , k. kakurai , s. ishiwata , and y. tokura , phys . lett . * 106 * , 156603 ( 2011 ) . o. l. makarova , a. v. tsvyashchenko , g. andre , f. porcher , l. n. fomicheva , n. rey , and i. mirebeau , phys . rev . b. * 85 * , 205205 ( 2012 ) . n. kanazawa , j .- h . kim , d. s. inosov , j. s. white , n. egetenmeyer , j. l. gavilano , s. ishiwata , y. onose , t. arima , b. keimer , and y. tokura , phys . b * 86 * , 134425 ( 2012 ) . s. olive - mendez , a. spiesser , l. a. michez , v. le thanh , a. glachant , j. derrien , t. devillers , a. barski , and m. jamet , thin solid films * 517 * , 191 ( 2008 ) . r. gunnella , l. morresi , n. pinto , r. murri , l. ottaviano , m. passacantando , f. dorazio , and f. lucari , surf . sci . * 577 * , 22 ( 2005 ) . e. karhu , s. kahwaji , and t. l. monchesky , phys . rev . b * 82 * , 184417 ( 2010 ) .

mnge has been grown as a thin film on si(111 ) substrates by molecular beam epitaxy . a 10 layer of mnsi was used as seedlayer in order to establish the b20 crystal structure . films of a thickness between 45 and 135 have been prepared and structually characterized by rheed , afm and xrd . these techniques give evidence that mnge forms in the cubic b20 crystal structure as islands exhibiting a very smooth surface . the islands become larger with increasing film thickness . a magnetic characterization reveals that the ordering temperature of mnge thin films is enhanced compared to bulk material . the properties of the helical magnetic structure obtained from magnetization and magnetoresistivity measurements are compared with films of the related compound mnsi . the much larger dzyaloshinskii - moriya interaction in mnge results in a higher rigidness of the spin helix .

arxiv

recall a polynomial @xmath0 is _ postcritically finite _ if the forward orbits of its critical points are finite . then the filled julia set of @xmath1 contains a forward invariant , finite topological tree , called the _ hubbard tree _ @xcite . the _ core entropy _ of @xmath1 is the topological entropy of the restriction of @xmath1 to its hubbard tree . we shall restrict ourselves to quadratic polynomials . given @xmath2 , the external ray at angle @xmath3 determines a postcritically finite parameter @xmath4 in the mandelbrot set @xcite . we define @xmath5 to be the core entropy of @xmath6 . the main goal of this paper is to prove the following result : [ t : main ] the core entropy function @xmath7 extends to a continuous function from @xmath8 to @xmath9 . the theorem answers a question of w. thurston , who first introduced and explored the core entropy of polynomials . as thurston showed , the core entropy function can be defined purely combinatorially , but it displays a rich fractal structure ( figure [ f : core ] ) , which reflects the underlying geometry of the mandelbrot set . [ f : core ] the concept of core entropy generalizes to complex polynomials the entropy theory of real quadratic maps , whose monotonicity and continuity go back to milnor and thurston @xcite : indeed , the invariant real segment is replaced by the invariant tree , which captures all the essential dynamics . on the topological side , hubbard trees have been introduced to classify postcritically finite maps @xcite , @xcite , and their entropy provides a new tool to study the parameter space of polynomials : for instance , the restriction of @xmath5 to non - real veins of the mandelbrot set is also monotone @xcite , which implies that the lamination for the mandelbrot set can be reconstructed by looking at level sets of @xmath10 . note by comparison that the entropy of @xmath11 on its julia set is constant , independent of @xmath3 , hence it does not give information on the parameter . furthermore , the value @xmath5 equals , up to a constant factor , the hausdorff dimension of the set of biaccessible angles for @xmath11 ( @xcite , @xcite ) . for dendritic julia sets , the core entropy also equals the asymptotic stretch factor of @xmath11 as a rational map @xcite . a rational external angle @xmath3 determines a postcritically finite map @xmath11 , which has a finite hubbard tree @xmath12 . the simplest way to compute its entropy is to compute the markov transition matrix for the map @xmath11 acting on @xmath12 , and take ( the logarithm of ) its leading eigenvalue : however , this requires to know the topology of @xmath12 , which changes wildly after small perturbations of @xmath3 . in this paper , we shall by - pass this issue by leveraging an algorithm , devised by thurston , which considers instead a larger matrix , whose entries are _ pairs _ of postcritical angles ( see section [ s : algo ] ) ; no knowledge of the topology of the hubbard tree is required . in order to prove @xmath10 is continuous , we develop an infinite version of such algorithm , which is defined for _ any _ angle @xmath13 . in particular , instead of taking the leading eigenvalue of a transition matrix , we shall encode the possible transitions in a directed graph , which will now have countably many vertices . by taking the _ spectral determinant _ of such graph ( see section [ s : spectral ] for a definition ) , for each angle @xmath13 , we construct a power series @xmath14 which converges in the unit disk and such that its smallest zero " @xmath15 is related to the core entropy by @xmath16 we then produce an algorithm to compute each coefficient of the taylor expansion of @xmath14 , and show that essentially these coefficients vary continuously with @xmath3 : the result follows by rouch s theorem . as a corollary of our method , we shall prove that the entropy function is actually hlder continuous at angles @xmath3 such that @xmath17 ( using renormalization , it can be proven that @xmath10 is not hlder continuous where @xmath18 ) . on the way to our proof , we shall develop a few general combinatorial tools to deal with growth rates of countable graphs , which may be of independent interest . we define a countable graph @xmath19 with bounded ( outgoing ) degree to have _ bounded cycles _ if it has finitely many closed paths of any given length . in this case , we define the _ growth rate _ of @xmath19 as the growth rate with @xmath20 of the number of closed paths of length @xmath20 ( see definition [ d : growth ] ) ; as it turns out ( theorem [ t : rootofp ] ) , the inverse of the growth rate equals the smallest zero of the following function @xmath21 , constructed by counting the multi - cycles in the graph : @xmath22 where @xmath23 is the number of components of the multi - cycle @xmath24 , and @xmath25 is its length . ( for the definition of multi - cycle , see section [ s : graphs ] ; note that our graphs have finite outgoing degree but possibly infinite ingoing degree , hence the adjacency operator has infinite @xmath26-norm and the usual spectral theory ( see e.g. @xcite ) does not apply ) . we then define a general combinatorial object , called _ labeled wedge _ , which consists of pairs of integers which can be labeled either as being _ separated _ ( representing two elements of the postcritical set which lie on opposite sides of the critical point ) or _ non - separated _ ; to such object we associate an infinite graph , and prove that it has bounded cycles , hence one can apply the theory developed in the first part ( see theorem [ t : zero ] ) . finally , we shall apply these combinatorial techniques to the core entropy ( section [ s : core ] ) ; indeed , we associate to any external angle @xmath3 a labeled wedge @xmath27 , hence an infinite graph @xmath28 , and verify that : 1 . the growth rate of @xmath28 varies continuously as a function of @xmath3 ( theorem [ t : newmain ] ) ; 2 . the growth rate of @xmath28 coincides with the core entropy for rational angles ( theorem [ t : coincide ] ) . let us remark that , as a consequence of monotonicity along veins and theorem [ t : main ] , the continuous extension we define also coincides with the core entropy for parameters which are not necessarily postcritically finite , but for which the julia set is locally connected and the hubbard tree is topologically finite . the core entropy of polynomials has been introduced by w. thurston around 2011 ( even though there are earlier related results , e.g. @xcite , @xcite , @xcite ) but most of the theory is yet unpublished ( with the exception of section 6 in @xcite ) . several people are now collecting his writings and correspondence into a foundational paper @xcite . in particular , the validity of thurston s algorithm has been proven by tan l. , gao y. , and w. jung ( see @xcite , @xcite ) . continuity of the core entropy along principal veins in the mandelbrot set is proven by the author in @xcite , and along all veins by w. jung @xcite . note that the previous methods used for veins do not easily generalize , since the topology of the tree is constant along veins but not globally . after @xcite , alternative proofs of monotonicity and continuity of entropy for real maps are given in @xcite , @xcite ( the present proof independently yields continuity ) . biaccessible external angles and their dimension have been discussed in @xcite , @xcite , @xcite , @xcite , @xcite . the method we use to count closed paths in the graph bears many similarities with the theory of _ dynamical zeta functions _ @xcite , and several forms of the spectral determinant are used in thermodynamic formalism ( see e.g. @xcite , @xcite ) . moreover , the spectral determinant @xmath21 we use is an infinite version of the _ clique polynomial _ used in @xcite to study finite directed graphs with small entropy . the motivation for our combinatorial construction is thurston s algorithm to compute the core entropy for rational angles , which we shall now describe . let @xmath2 a rational number @xmath29 . then the external ray at angle @xmath3 in the mandelbrot set lands at a misiurewicz parameter , or at the root of a hyperbolic component : let @xmath1 denote the corresponding postcritically finite quadratic map . we shall call @xmath30 the critical point of @xmath1 , and for each @xmath31 , @xmath32 the @xmath33 iterate of the critical point . recall that the _ hubbard tree _ of @xmath1 is the union of the regulated arcs @xmath34 $ ] for all @xmath35 ( for more details , see @xcite ) . in the postcritically finite case , it is a finite topological tree , and it is forward - invariant under @xmath1 ( see figure [ f : tree ] ) . it is possible to compute the core entropy of @xmath1 by writing the markov transition matrix for the action of @xmath1 on the tree , and take the logarithm of its leading eigenvalue . however , given the external angle @xmath3 it is quite complicated to figure out the topology of the tree : the following algorithm by - passes this issue by looking at pairs of external angles . in order to explain the algorithm in more detail , let us remark that a rational angle @xmath3 is eventually periodic under the doubling map @xmath36 ; that is , there exist integers @xmath37 and @xmath38 such that the elements of the set @xmath39 are all distinct modulo @xmath40 , and @xmath41 . the elements of @xmath42 will be called _ postcritical angles _ ; the number @xmath43 is called the _ period _ of @xmath3 , and @xmath44 is the _ pre - period_. if @xmath45 , we shall call @xmath3 _ purely periodic_. denote @xmath46 the partition of the circle @xmath47 in the two intervals @xmath48 moreover , for each @xmath49 let us denote @xmath50 , which we see as a point in @xmath51 . let us now construct a matrix @xmath52 which will be used to compute the core entropy . denote @xmath53 the set of unordered pairs of ( distinct ) elements of @xmath42 : this is a finite set , any element of which will be denoted @xmath54 . we now define a linear map @xmath55 by defining it on its basis vectors in the following way : * if @xmath56 and @xmath57 belong to the same element of the partition @xmath46 , or at least one of them lies on the boundary , we shall say that the pair @xmath54 is _ non - separated _ , and define @xmath58 * if @xmath56 and @xmath57 belong to the interiors of two different elements of @xmath46 , then we say that @xmath54 is _ separated _ , and define @xmath59 ( in order to make sure the formulas are defined in all cases , we shall set @xmath60 whenever @xmath61 ) . by abuse of notation , we shall also denote @xmath52 the matrix representing the linear map @xmath52 in the standard basis . as suggested by thurston in his correspondence , the leading eigenvalue of @xmath52 gives the core entropy : [ t : algo ] let @xmath3 a rational angle.then the core entropy @xmath5 of the quadratic polynomial of external angle @xmath3 is related to the largest real eigenvalue @xmath62 of the matrix @xmath52 by the formula @xmath63 the explanation of this algorithm is the following . let @xmath64 be a complete graph whose vertices are labeled by elements of @xmath42 , and which we shall consider as a topological space . ( more concretely , we can take the unit disk and draw segments between all possible pairs of postcritical angles on the unit circle : the union of all such segments is a model for @xmath64 ( see figure [ f : chords_algo ] ) ) . let us denote by @xmath65 the hubbard tree of @xmath1 . we can define a continuous map @xmath66 which sends the edge with vertices @xmath56 , @xmath57 homeomorphically to the regulated arc @xmath34 $ ] in @xmath65 ( except in the case @xmath67 , where we map all the edge to a single point ) . finally , we can lift the dynamics @xmath68 to a map @xmath69 such that @xmath70 . in order to do so , note that : * if the pair of angles @xmath54 is non - separated , then @xmath1 maps the arc @xmath34 $ ] homeomorphically onto @xmath71 $ ] ; hence to define @xmath72 we have to lift @xmath1 so that it maps the edge @xmath73 $ ] homeomorphically onto @xmath74 $ ] ; * if instead @xmath54 is separated , then the critical point @xmath30 lies on the regulated arc @xmath34 $ ] in the hubbard tree ; thus , @xmath1 maps the arc @xmath75 = [ c_i , c_0 ] \cup [ c_0 , c_j]$ ] onto @xmath76 \cup [ c_1 , c_{j+1}]$ ] ; so we define @xmath72 by lifting @xmath1 and so that it maps @xmath73 $ ] continuously onto the union @xmath77 \cup [ x_1 , x_{j+1}]$ ] . the map @xmath72 is a markov map of a topological graph , hence its entropy is the logarithm of the leading eigenvalue of its transition matrix , which is by construction the matrix @xmath52 . in case @xmath3 is purely periodic , one can prove that the map @xmath78 is surjective and finite - to - one , and semiconjugates the dynamics @xmath72 on @xmath64 to the dynamics @xmath1 on @xmath65 , hence @xmath79 more care is needed in the pre - periodic case , since @xmath78 can collapse arcs to points ( for details , see @xcite , @xcite , or @xcite ) . in the following , by _ graph _ we mean a directed graph @xmath19 , i.e. a set @xmath80 of vertices ( which will be finite or countable ) and a set @xmath81 of edges , such that each edge @xmath82 has a well - defined _ source _ @xmath83 and a _ target _ @xmath84 ( thus , we allow edges between a vertex and itself , and multiple edges between two vertices ) . given a vertex @xmath85 , the set @xmath86 of its _ outgoing edges _ is the set of edges with source @xmath85 . the _ outgoing degree _ of @xmath85 is the cardinality of @xmath86 ; a graph is _ locally finite _ if the outgoing degree of all its vertices is finite , and has _ bounded outgoing degree _ if there is a uniform upper bound @xmath87 on the outgoing degree of all its vertices . note that we do _ not _ require that the ingoing degree is finite , and indeed in our application we will encounter graphs with vertices having countably many ingoing edges . we denote as @xmath88 the set of vertices of @xmath19 , and as @xmath89 its set of edges . moreover , we denote as @xmath90 the number of edges from vertex @xmath85 to vertex @xmath91 . a _ path _ in the graph based at a vertex @xmath85 is a sequence @xmath92 of edges such that @xmath93 , and @xmath94 for @xmath95 . the _ length _ of the path is the number @xmath20 of edges , and the set of vertices @xmath96 visited by the path is called its _ vertex - suppoert _ , or just _ support _ for simplicity . similarly , a _ closed path _ based at @xmath85 is a path @xmath92 such that @xmath97 . note that in this definition a closed path can intersect itself , i.e. two of the sources of the @xmath98 can be the same ; moreover , closed paths with different starting vertices will be considered to be different . on the other hand , a _ simple cycle _ is a closed path which does not self - intersect , modulo cyclical equivalence : that is , a simple cycle is a closed path @xmath92 such that @xmath99 for @xmath100 , and two such paths are considered the same simple cycle if the edges are cyclically permuted , i.e. @xmath92 and @xmath101 designate the same simple cycle . finally , a _ multi - cycle _ is the union of finitely many simple cycles with pairwise disjoint ( vertex-)supports . the length of a multi - cycle is the sum of the lengths of its components . given a countable graph with bounded outgoing degree , we define the _ adjacency operator _ @xmath102 on the space of summable sequences indexed by the vertex set @xmath80 . in fact , for each vertex @xmath103 we can consider the sequence @xmath104 which is @xmath40 at position @xmath49 and @xmath105 otherwise , and define for each @xmath106 the @xmath107 component of the vector @xmath108 to be @xmath109 equal to the number of edges from @xmath49 to @xmath110 . since the graph has bounded outgoing degree , the above definition can actually be extended to all @xmath111 , and the operator norm of @xmath112 induced by the @xmath113-norm is bounded above by the outgoing degree @xmath114 of the graph . note moreover that for each pair @xmath115 of vertices and each @xmath20 , the coefficient @xmath116 equals the number of paths of length @xmath20 from @xmath49 to @xmath110 . we say a countable graph @xmath19 has _ bounded cycles _ if it has bounded outgoing degree and for each positive integer @xmath20 , @xmath19 has at most finitely many simple cycles of length @xmath20 . note that , if @xmath19 has bounded cycles , then for each @xmath20 it has also a finite number of closed paths of length @xmath20 , since the support of any closed path of length @xmath20 is contained in the union of the supports of simple cycles with length @xmath117 . thus , for such graphs we shall denote @xmath118 the number of closed paths of length @xmath20 . note that in this case the trace @xmath119 is also well - defined for each @xmath20 , and equal to @xmath120 . [ d : growth ] if @xmath19 is a graph with bounded cycles , we define the _ growth rate _ @xmath121 as the exponential growth rate of the number of its closed paths : that is , @xmath122{c(\gamma , n)}.\ ] ] if @xmath19 is a finite graph with adjacency matrix @xmath112 , then is well - defined its characteristic polynomial @xmath123 , whose roots are the eigenvalues of @xmath112 . in the following we shall work with the related polynomial @xmath124 which we call the _ spectral determinant _ of @xmath19 . we shall now extend the theory to countable graphs with bounded cycles . it is known that the spectral determinant @xmath21 of a finite graph @xmath19 is related to its multi - cycles by the following formula ( see e.g. @xcite ) : @xmath125 where @xmath25 denotes the length of the multi - cycle @xmath24 , while @xmath23 is the number of connected components of @xmath24 . if @xmath19 is now a ( directed ) graph with countably many vertices and bounded cycles , then the number of multi - cycles of any given length is finite , hence the formula above is still defined as a formal power series . note that we include also the empty cycle , which has zero components and zero length , hence @xmath21 begins with the constant term @xmath40 . now , let @xmath126 denote the number of multi - cycles of length @xmath20 in @xmath19 , and let us define @xmath127{k(\gamma , n)}\ ] ] its growth rate . then the main result of this section is the following : [ t : rootofp ] suppose we have @xmath128 ; then the formula defines a holomorphic function @xmath129 in the unit disk @xmath130 , and moreover the function @xmath129 is non - zero in the disk @xmath131 ; if @xmath132 , we also have @xmath133 . the proof uses in a crucial way the following combinatorial statement . [ l:1/pt ] let @xmath19 be a countable graph with bounded cycles , and @xmath112 its adjacency operator . then we have the equalities of formal power series @xmath134 where @xmath21 is the spectral determinant . note that , since @xmath21 is a power series with constant term @xmath40 , then @xmath135 is indeed a well - defined power series . the second equality is just obtained by expanding the exponential function in power series . to prove the first equality , let us first suppose @xmath19 is a finite graph with @xmath20 vertices , and let @xmath136 be the eigenvalues of its adjacency matrix ( counted with algebraic multiplicity ) . then @xmath137 hence @xmath138 hence the claim follows since @xmath139 . now , let @xmath19 be infinite with bounded cycles , and let @xmath140 . note that both sides of the equation depend , modulo @xmath141 , only on multi - cycles of length @xmath142 , which by the bounded cycle condition are supported on a finite subgraph @xmath143 . thus by applying the previous proof to @xmath143 we obtain equality modulo @xmath141 , and since this holds for any @xmath144 the claim is proven . by the root test , the radius of convergence of @xmath145 is @xmath146 . thus , since the exponential function has infinite radius of convergence , the radius of convergence of @xmath147 is at least @xmath146 , and since @xmath148 where @xmath149 has positive coefficients , then the radius of convergence of @xmath147 is exactly equal to @xmath146 . hence , by lemma [ l:1/pt ] , the radius of convergence of @xmath135 around @xmath150 is also @xmath146 . on the other hand , since @xmath128 , then by the root test the power series @xmath21 converges inside the unit disk , and defines a holomorphic function ; thus , @xmath135 is meromorphic for @xmath151 , and holomorphic for @xmath152 , hence @xmath153 if @xmath152 . moreover , if @xmath154 , then the radius of convergence of @xmath135 equals the smallest modulus of one of its poles , hence @xmath146 is the smallest modulus of a zero of @xmath21 . finally , since @xmath135 has all its taylor coefficients real and nonnegative , then the smallest modulus of its poles must also be a pole , so @xmath133 . [ l : perroneig ] if @xmath19 is a finite graph , then its growth rate equals the largest real eigenvalue of its adjacency matrix . note that , by the perron - frobenius theorem , since the adjacency matrix is non - negative , it has at least one real eigenvalue whose modulus is at least as large as the modulus of any other eigenvalue . moreover , if @xmath62 is the largest real eigenvalue , then @xmath155 is the smallest root of the spectral determinant @xmath21 , hence the claim follows from theorem [ t : rootofp ] . let @xmath156 two ( locally finite ) graphs . a _ graph map _ from @xmath157 to @xmath158 is a map @xmath159 on the vertex sets and a map on edges @xmath160 which is compatible , in the sense that if the edge @xmath82 connects @xmath85 to @xmath91 in @xmath157 , then the edge @xmath161 connects @xmath162 to @xmath163 in @xmath158 . we shall usually denote such a map as @xmath164 . weak cover _ of graphs is a graph map @xmath164 such that : * the map @xmath159 between the vertex sets is surjective ; * the induced map @xmath165 between outgoing edges is a bijection for each @xmath166 . note that the map between outgoing edges is defined because @xmath167 is a graph map , and @xmath167 also induces a map from paths in @xmath157 to paths in @xmath158 . as a consequence of the definition of weak cover , you have the following _ unique path lifting _ property : let @xmath164 a weak cover of graphs . then given @xmath166 and @xmath168 , for every path @xmath24 in @xmath158 based at @xmath91 there is a unique path @xmath169 in @xmath157 based at @xmath85 such that @xmath170 . let @xmath166 and @xmath168 , and let @xmath171 be a path in @xmath158 based at @xmath91 . since the map @xmath172 is a bijection , there exists exactly one edge @xmath173 such that @xmath174 . by compatibility , the target @xmath175 of @xmath176 projects to the target of @xmath177 , hence we can apply the same reasoning and lift the second edge @xmath178 uniquely starting from @xmath175 , and so on . an immediate consequence of the property is the following [ l : easyineq ] each graph map @xmath164 induces a map @xmath179 for each @xmath180 and each @xmath181 . moreveor , if @xmath167 is a weak cover , then @xmath182 is injective . a general way to construct weak covers of graphs is the following . suppose we have an equivalence relation @xmath183 on the vertex set @xmath80 of a locally finite graph , and denote @xmath184 the set of equivalence classes of vertices . such an equivalence relation is called _ edge - compatible _ if whenever @xmath185 , for any vertex @xmath91 the total number of edges from @xmath186 to the members of the equivalence class of @xmath91 equals the total number of edges from @xmath187 to the members of the equivalence class of @xmath91 . when we have such an equivalence relation , we can define a quotient graph @xmath188 with vertex set @xmath184 . namely , we denote for each @xmath189 the respective equivalence classes as @xmath190 $ ] and @xmath191 $ ] , and define the number of edges from @xmath190 $ ] to @xmath191 $ ] in the quotient graph to be @xmath192 \to [ w ] ) : = \sum_{u \in [ w ] } \#(v \to u).\ ] ] by definition of edge - compatibility , the above sum does not depend on the representative @xmath85 chosen inside the class @xmath190 $ ] . moreover , it is easy to see that the quotient map @xmath193 is a weak cover of graphs . let us now relate the growth of a graph to the growth of its weak covers . [ l : growthcover ] let @xmath164 a weak cover of graphs with bounded cycles , and @xmath194 a finite set of vertices of @xmath157 . 1 . suppose that every closed path in @xmath157 passes through @xmath194 . then for each @xmath20 we have the estimate @xmath195 which implies @xmath196 2 . suppose that @xmath197 is a set of closed paths in @xmath158 such that each @xmath198 crosses at least one vertex @xmath91 with the property that : the set @xmath199 is non - empty , and any lift of @xmath24 from an element of @xmath200 ends in @xmath200 . then there exists @xmath201 , which depends on @xmath194 , such that for each @xmath20 we have @xmath202 \(1 ) let @xmath24 be a closed path in @xmath157 of length @xmath20 , based at @xmath85 . let now @xmath203 be the first vertex of @xmath24 which belongs to @xmath194 ( by hypothesis , there is one ) . if we call @xmath204 the cyclical permutation of @xmath24 based at @xmath203 , the projection of @xmath204 to @xmath158 now yields a closed path @xmath205 in @xmath158 which is based at @xmath206 . now note that for each such pair @xmath207 , there are at most @xmath208 possible choices for @xmath209 ; then , given @xmath203 there is a unique lift of @xmath205 from @xmath203 , and @xmath20 possible choices for the vertex @xmath85 on that lift , giving the estimate . \(2 ) recall first that in every directed graph we can define an equivalence relation by saying @xmath210 if there is a path from @xmath85 to @xmath91 and a path from @xmath91 to @xmath85 . the equivalence classes are known as _ strongly connected components _ , or @xmath211 for short . moreover , we can form a graph @xmath212 whose vertices are the strongly connected components and there is an edge from the s.c.c . @xmath213 to @xmath214 if in the original graph there is a path from an element of @xmath213 to an element of @xmath214 . note that by construction this graph has no cycles . moreover , let us consider a pair @xmath215 of distinct elements of @xmath194 ; then either there is no path from @xmath186 to @xmath187 , or we can pick some path from @xmath186 to @xmath187 , which will be denoted @xmath216 . then the set @xmath217 of such paths is finite , and let @xmath72 be the maximum length of an element of @xmath218 . let now @xmath24 be a closed path in @xmath158 based at @xmath219 . by hypothesis , @xmath24 passes through some @xmath91 such that @xmath220 , and each lift of @xmath24 from @xmath200 ends in @xmath200 . let us now consider the set @xmath221 of all s.c.c . of @xmath157 which intersect @xmath200 : since the graph @xmath212 constructed above has no cycles , there is a component @xmath222 which has the property that there is no path from @xmath223 to some other component of @xmath200 . thus , if we pick @xmath224 and lift @xmath24 from @xmath85 to a path @xmath169 in @xmath157 , the endpoint @xmath175 of @xmath169 must lie inside @xmath200 by hypothesis . thus , by the property of @xmath223 , the vertex @xmath175 lies in @xmath223 , hence there is a path @xmath225 from @xmath175 to @xmath85 in the previously chosen set @xmath218 , hence if we take @xmath226 , this is a closed path in @xmath157 of length between @xmath20 and @xmath227 . thus we have a map @xmath228 given by @xmath229 . now , given @xmath85 we can recover @xmath230 , and given @xmath205 we can recover @xmath24 , since @xmath24 is the path given by the first @xmath20 edges of @xmath231 starting at @xmath91 . finally , we have at most @xmath20 choices for the starting point @xmath219 on @xmath24 , hence the fibers of the above map have cardinality at most @xmath20 , proving the claim . an immediate corollary of the lemma is the following , when @xmath157 and @xmath158 are both finite . [ l : sameentro ] let @xmath156 be finite graphs , and @xmath164 a weak cover . then the growth rate of @xmath157 equals the growth rate of @xmath158 . apply the lemma with @xmath232 , and @xmath197 equal to the set of all closed paths in @xmath158 . let us consider the set @xmath233 of pairs of disjoint positive integers . the set @xmath234 will be sometimes called the _ wedge _ and , given an element @xmath235 , the coordinate @xmath49 will be called the _ height _ of @xmath85 , while the coordinate @xmath110 will be called the _ width _ of @xmath85 . the terminology becomes more clear by looking at figure [ f : wedge ] . @xmath236 we call a _ labeled wedge _ an assignment @xmath237 of a label @xmath238 ( which stands for _ non - separated _ ) or @xmath194 ( which stands for _ separated _ ) to each element of @xmath234 . now , to each labeled wedge @xmath239 we assign a graph @xmath240 in the following way . the vertex set of @xmath240 is the wedge @xmath234 , while the edges of @xmath240 are labeled by the set @xmath241 and determined according to the following rule . * if @xmath242 is non - separated , then @xmath243 has as its ( unique ) successor the vertex @xmath244 ; we say that the edge @xmath245 is an _ upward edge _ and we label it with @xmath246 ; * if @xmath247 is separated , then @xmath247 has two successors : * * first , we add the edge @xmath248 , which we call _ forward edge _ and label it with @xmath249 . * * second , we add the edge @xmath250 , which we call _ backward edge _ and label it with @xmath251 . in order to explain the names , note that following an upward or forward edge increases the width by @xmath40 , while following a backward edge ( weakly ) decreases it . moreover , following an upward edge increases the height by @xmath40 , while the targets of both backward and forward edges have height @xmath40 . @xmath252\txt{(2,4 ) } & ( 2,5 ) & \cdots \\ * + [ f]\txt{(1,2 ) } & ( 1,3 ) & * + [ f]\txt{(1,4 ) } & ( 1,5 ) & \cdots \\ } \ ] ] @xmath253^u & ( 3,5 ) & \cdots \\ & ( 2,3 ) \ar[ur]^u & * + [ f]\txt{(2,4 ) } \ar[dr]^f \ar@/_/[dl]_b & ( 2,5 ) & \cdots \\ * + [ f]\txt{(1,2 ) } \ar@(u , l)_b \ar[r]^f & ( 1,3 ) \ar[ur]_u & * + [ f]\txt{(1,4 ) } \ar[r]^f \ar@/^/[ll]^b & ( 1,5 ) & \cdots \\ } \ ] ] [ p : properties ] let @xmath240 the graph associated to the labeled wedge @xmath239 . then the following are true : 1 . each vertex of a closed path of length @xmath20 has height at most @xmath20 ; 2 . the support of each closed path of length @xmath20 intersects the set @xmath254 ; 3 . each vertex of a closed path of length @xmath20 has width at most @xmath255 ; 4 . for each @xmath256 , there exists at most one separated vertex in the @xmath257 diagonal @xmath258 which is contained in the support of at least one closed path ; 5 . there are at most @xmath259 multi - cycles of length @xmath20 . note that ( 1 ) , ( 2 ) , ( 3 ) are sharp , as seen by the simple cycle @xmath260^u & 25 \ar@/^/[r]^u & * + [ f]\txt{36 } \ar@/^/[ll]^b } \ ] ] \(1 ) let us first note that every closed path contains at least one backward edge , since the upward and forward edges always increase the height . moreover , the endpoint of a backward edge has always height @xmath40 , and each edge increases the height by at most @xmath40 , hence the height of a vertex along the closed path is at most @xmath20 . \(2 ) since the target of each backward edge is @xmath261 , where @xmath49 is the height of the source of the edge , which is at most @xmath20 by the previous point , then the target of each backward edge along the closed path belongs to the set @xmath262 . \(3 ) by the previous point , there is at least a vertex along the closed path with width at most @xmath263 . since every move increases the width by at most @xmath40 , then the largest possible width of a vertex along the path is @xmath264 . \(4 ) let @xmath265 be the separated vertex in @xmath266 with smallest height , if there is one . we claim that no vertex @xmath267 of @xmath266 with @xmath268 belongs to any closed path . note by looking at the rules that , if a vertex @xmath269 of height @xmath270 is the target of some edge , then it must be the target of an upward edge , more precisely an edge from the vertex @xmath271 immediately to the lower left of @xmath85 , which then must be non - separated . thus , since @xmath272 is separated , the vertex @xmath273 does not belong to any closed path ; the claim then follows by induction on @xmath274 , since , by the same reasoning , if @xmath275 belongs to some closed path , then also @xmath276 must belong to the same path . \(5 ) let @xmath277 the backward edges along a multi - cycle @xmath24 of length @xmath20 , and denote @xmath278 the source of @xmath98 , and @xmath279 the height of @xmath278 . note that the set @xmath280 determines @xmath24 , as you can start from @xmath186 , follow the backward edge , and then follow upward or forward edges until you either close the loop or encounter another @xmath278 , and then continue this way until you walk along all of @xmath24 . moreover , we know that for each @xmath278 there are at most @xmath255 possible choices , as each @xmath278 is separated and by ( 4 ) there is at most one for each diagonal @xmath266 , and by ( 3 ) it must lie on some @xmath266 with @xmath281 . we now claim that @xmath282 which is then sufficient to complete the proof . let us now prove the claim . by definition of multi - cycle , then the targets of the @xmath98 must be all distinct , and by the rule these targets are precisely @xmath283 with @xmath284 , hence all @xmath279 must be distinct . moreover , let us note that each @xmath98 must be preceded along the multi - cycle by a sequence @xmath285 of upward edges of length @xmath286 , and all such sequences for distinct @xmath49 must be disjoint . hence we have that all @xmath279 are distinct and their total sum is at most the length of the multi - cycle , i.e. @xmath20 . thus we have @xmath287 which proves the claim . [ t : zero ] let @xmath239 be a labeled wedge . then its associated graph @xmath19 has bounded cycles , and its spectral determinant @xmath21 defines a holomorphic function in the unit disk . moreover , the growth rate @xmath288 of the graph @xmath19 equals the inverse of the smallest real positive root of @xmath129 , in the following sense : @xmath289 for @xmath131 and , if @xmath132 , then @xmath133 . by construction , the outgoing degree of any vertex of @xmath19 is at most @xmath290 . moreover , by proposition [ p : properties ] ( 5 ) the graph has bounded cycles , and the growth rate of the number @xmath126 of multi - cycles is @xmath291 , since @xmath292{(2n)^{\sqrt{2n } } } = \limsup_{n } e^{\frac{\sqrt{2 } \log(2n)}{\sqrt{n } } } = 1.\ ] ] the claim then follows by theorem [ t : rootofp ] . we shall sometimes denote as @xmath293 the growth rate of the graph associated to the labeled wedge @xmath239 . we say that a sequence @xmath294 of labeled wedges converges to @xmath239 if for each finite set of vertices @xmath295 there exists @xmath238 such for each @xmath296 the labels of the elements of @xmath194 for @xmath297 and @xmath239 are the same . [ l : contwedge ] if a sequence of labeled wedges @xmath294 converges to @xmath239 , then the growth rate of @xmath297 converges to the growth rate of @xmath239 . let @xmath298 and @xmath21 denote respectively the spectral determinants of @xmath297 and @xmath239 . then for each @xmath256 , the coefficient of @xmath299 in @xmath298 converges to the coefficient of @xmath299 in @xmath21 , because by proposition [ p : properties ] ( 1 ) and ( 3 ) the support of any multi - cycle of length @xmath300 is contained in the finite subgraph @xmath301 . thus , since the modulus of the coefficient of @xmath299 is uniformly bounded above by @xmath302 , then @xmath303 uniformly on compact subsets of the unit disk . thus , by rouch s theorem , the smallest real positive zero of @xmath298 converges to the smallest real positive zero of @xmath21 , hence by theorem [ t : zero ] we have @xmath304 . given integers @xmath37 and @xmath38 we define the equivalence relation @xmath305 on @xmath306 by saying that @xmath307 if : * either @xmath308 and @xmath309 ; * or @xmath310 and @xmath311 . note that if @xmath45 the equivalence relation @xmath312 is simply the congruence modulo @xmath43 . a set of representatives for the equivalence classes of @xmath312 is the set @xmath313 . the equivalence relation induces an equivalence relation on the set @xmath314 of ordered pairs of integers by saying that @xmath315 if @xmath316 and @xmath317 . moreover , it also induces an equivalence relation on the set of _ unordered pairs _ of integers by saying that the unordered pair @xmath318 is equivalent to @xmath319 if either @xmath320 or @xmath321 . a labeled wedge is _ periodic _ of period @xmath43 and pre - period @xmath44 if the following two conditions hold : * any two pairs @xmath247 and @xmath322 such that @xmath323 have the same label ; * if @xmath307 , then the pair @xmath247 is non - separated . if @xmath324 , the labeled wedge will be called _ purely periodic_. a pair @xmath247 with @xmath325 will be called _ diagonal _ ; hence the second point in the definition can be rephrased as every diagonal pair is non - separated " . @xmath326 & \dots \\ & & ( 3,4 ) \ar[ur ] & ( 3 , 5 ) \ar[ur ] & \dots\\ & * + [ f]\txt{(2,3 ) } \ar[d ] \ar[dr ] & * + [ f]\txt{(2,4 ) } \ar[dl ] \ar[dr ] & * + [ f]\txt{(2 , 5 ) } \ar[dll ] \ar[dr ] & \dots \\ ( 1,2 ) \ar[ur ] & * + [ f]\txt{(1,3 ) } \ar[r ] \ar[l ] & * + [ f]\txt{(1 , 4 ) } \ar[r ] & * + [ f]\txt{(1 , 5 ) } \ar[r ] & \dots}\ ] ] @xmath327\txt{(2,3 ) } \ar@<0.5ex>[d ] \ar@<-0.5ex>[d ] \\ ( 1,2 ) \ar[ur ] & * + [ f]\txt{(1,3 ) } \ar[l ] \ar@(d , r ) } \ ] ] given a periodic wedge @xmath239 of period @xmath43 and pre - period @xmath44 , with associated ( infinite ) graph @xmath19 , we shall now construct a finite graph @xmath328 which captures the essential features of the infinite graph @xmath19 , in particular its growth rate . the set of vertices of @xmath328 is the set of @xmath312-equivalence classes of non - diagonal , unordered pairs of integers . a set of representatives of @xmath329 is the set @xmath330 the edges of @xmath328 are induced by the edges of @xmath19 , that is are determined by the following rules : if the unordered pair @xmath318 is non - separated , then @xmath318 has one outgoing edge , namely @xmath331 ; while if @xmath318 is separated , then @xmath318 has the two outgoing edges @xmath332 , and @xmath333 . the main result of this section is the following . [ p : finiteinfinite ] let @xmath239 be a periodic labeled edge , with associated ( infinite ) graph @xmath19 . then the growth rate of @xmath19 equals the growth rate of its finite model @xmath328 . in order to prove the proposition , we shall also introduce an intermediate finite graph , which we call the _ finite 2-cover _ of @xmath328 , and denote @xmath334 . the set of vertices of @xmath334 is the set of @xmath305-equivalence classes of non - diagonal , ordered pairs of integers , and the edges are induced by the edges of @xmath19 in the usual way . the reason to introduce the intermediate graph @xmath334 is that @xmath328 does not inherit the labeling of edges from @xmath19 , as backward and forward edges in @xmath19 may map to the same edge in @xmath328 , while @xmath334 naturally inherits the labels . let @xmath19 be the graph associated to a periodic labeled wedge . 1 . first , let us observe that no diagonal vertex is contained in the support of any closed path of @xmath19 : in fact , every diagonal vertex is non - separated , and its outgoing edge leads to another diagonal vertex with larger height , hence the path can never close up . thus , we can construct the subgraph @xmath335 by taking as vertices all pairs which are non - diagonal , and as edges all the edges of @xmath19 which do not have either as a source or target a diagonal pair . by what has been just said , the growth rate of @xmath19 and @xmath335 is the same , @xmath336 2 . since the maps @xmath337 are given by quotienting with respect to equivalence relations , they are both weak covers of graphs . thus , since both @xmath334 and @xmath328 are finite , by lemma [ l : sameentro ] , the growth rates of @xmath334 and @xmath328 are the same . we are now left with proving that the growth rate of @xmath335 is the same as the growth rate of @xmath334 . since the cardinality of the fiber of the projection @xmath338 is infinite , the statement is not immediate . note that the finite @xmath290-cover @xmath334 is a graph with labeled edges : indeed , if @xmath320 , then @xmath339 , and so on , thus the labeling of @xmath334 inherited from @xmath19 is well - defined , and the graph map @xmath338 preserves the labels . 4 . let us call _ backtracking _ a path in @xmath19 or @xmath334 such that at least one of its edges is labeled by @xmath251 (= backward ) , and _ non - backtracking _ otherwise . now let us note the following : 1 . every closed path in @xmath19 is backtracking ; in fact , following any edge which is upward or forward increases the height , thus a path in @xmath19 made entirely of @xmath246 and @xmath249 edges can not close up . 2 . every closed path in @xmath19 passes through the finite set @xmath340 in order to prove this , we first prove that any element in the support of any closed path in @xmath19 has height at most @xmath341 . in order to do so , let us fix a diagonal @xmath342 . by periodicity , either there is a separated pair @xmath247 in @xmath266 of height @xmath343 , or all elements of @xmath266 are non - separated ; in the latter case , no element of @xmath266 is part of any closed path , since any path based at an element of @xmath266 is non - backtracking . in the first case , let @xmath344 be the separated pair with smallest height in @xmath266 ; then , only the elements with height less than @xmath345 can be part of any closed path , and @xmath346 , so the first claim is proven . as a consequence , the target of any @xmath251-labeled edge which belongs to some closed path is of type @xmath347 , where @xmath348 is the height of the source , hence the target belongs to @xmath194 . by the same reasoning , if @xmath24 is any path in @xmath19 based at a vertex of height @xmath40 , then the target of any @xmath251-labeled edge along @xmath24 lies in @xmath194 . 3 . every backtracking closed path in @xmath334 has at least one @xmath251-labeled edge , hence the target of such @xmath251-labeled edge lies in the set @xmath349 , and every lift to @xmath19 starting from an element of @xmath194 must end in the target of a @xmath251-labeled edge in @xmath19 , hence must end in @xmath194 . 5 . finally , let us note that for each @xmath20 , the number of non - backtracking paths of length @xmath20 in @xmath334 is at most the cardinality of @xmath350 ; indeed , from each vertex there is at most one edge labeled @xmath246 or @xmath249 , thus for each vertex of @xmath334 there is at most one non - backtracking path of length @xmath20 based at it . let us now put together the previous statements . indeed , by ( 4)(a)-(b ) every closed path in @xmath19 passes through @xmath194 , hence we can apply lemma [ l : growthcover ] ( 1 ) and get that @xmath351 to prove the other inequality , let us note that by ( 4)(b)-(c ) we know that lemma [ l : growthcover ] ( 2 ) applies with @xmath197 the set of backtracking closed paths in @xmath334 . moreover , by point ( 5 ) above we have that the number of non - backtracking paths is bounded independently of @xmath20 : thus we can write @xmath352 @xmath353 from which follows @xmath354 as required . when dealing with purely periodic external angles , we shall also need the following lemma . [ l : twoperiodics ] let @xmath355 and @xmath356 be two labeled wedges which are purely periodic of period @xmath43 . suppose moreover that for every pair @xmath247 with @xmath357 the label of @xmath243 in @xmath355 equals the label in @xmath356 . then the finite models @xmath358 and @xmath359 are isomorphic graphs . as a consequence , the growth rates of @xmath355 and @xmath356 are equal . let @xmath360 an equivalence class of unordered pairs . if neither @xmath49 nor @xmath110 are divisible by @xmath43 , then the label of @xmath318 is the same in @xmath355 and @xmath356 , hence the outgoing edges from @xmath318 are the same in @xmath358 and @xmath359 . suppose on the other hand that @xmath361 ( hence , @xmath362 because the pair is non - diagonal ) . then , if the pair @xmath363 is non - separated , then its only outgoing edge goes to @xmath364 in @xmath365 . on the other hand , if @xmath363 is separated , then its two possible outgoing edges are @xmath366 and @xmath367 . however , the pair @xmath366 is diagonal , hence no vertex in the graph has such label . thus , independently of whether @xmath363 is separated , it has exactly one outgoing edge with target @xmath367 , proving the claim . we shall now apply the theory of labeled wedges to the core entropy . as we have seen in section [ s : algo ] , thurston s algorithm allows one to compute the core entropy for periodic angles @xmath3 ; in order to interpolate between periodic angles of different periods , we shall now define for _ any _ angle @xmath13 a labeled wedge @xmath27 , and thus an infinite graph @xmath368 as described in the previous sections . recall that for each @xmath31 we denote @xmath50 , which we see as a point in @xmath51 . for each pair @xmath247 which belongs to @xmath234 , we label @xmath247 as _ non - separated _ if @xmath369 and @xmath370 belong to the same element of the partition @xmath46 , or at least one of them lies on the boundary of the partition . if instead @xmath369 and @xmath370 belong to the interiors of two different intervals of @xmath46 , then we label the pair @xmath247 as _ separated_. the labeled wedge just constructed will be denoted @xmath27 , and its associated graph @xmath28 . the main quantity we will work with is the following : let @xmath3 have period @xmath43 and pre - period @xmath44 . by definition of the equivalence relation @xmath312 , we have @xmath325 if and only if @xmath373 , which proves the first condition in the definition of periodic wedge . moreover , if @xmath247 is a diagonal pair , then @xmath374 , hence the pair @xmath247 is non - separated , verifying the second condition . using the results of the previous section , we are now ready to prove that the logarithm of the growth rate of @xmath28 coincides with the core entropy for rational angles , proving the first part of theorem [ t : newmain ] . let @xmath3 a rational angle , @xmath28 the infinite graph associated to the labeled wedge @xmath27 , and @xmath376 the finite model of @xmath28 as described in section [ s : finitemodel ] by unraveling the definitions , the matrix @xmath52 constructed in section [ s : algo ] is exactly the adjacency matrix of the finite model @xmath376 . by proposition [ p : finiteinfinite ] , the growth rate of @xmath28 coincides with the growth rate of its finite model @xmath376 . moreover , by lemma [ l : perroneig ] , the growth rate of @xmath376 coincides with the largest real eigenvalue of its adjacency matrix , that is the largest real eigenvalue of @xmath52 . thus , by theorem [ t : algo ] its logarithm is the core entropy @xmath5 . as @xmath379 , the labeled wedge @xmath380 stabilizes . in fact , for each @xmath49 consider the position of @xmath50 with respect to the partition @xmath46 . if @xmath381 , then , for all @xmath382 in a neighborhood of @xmath3 , @xmath383 lies on the same side of the partition . otherwise , note that for any @xmath384 and @xmath385 , the functions @xmath386 and @xmath387 are both continuous and orientation preserving but have different derivative : thus , for all @xmath388 close enough to @xmath3 , the point @xmath383 lies on one side of the partition , and for @xmath389 close enough to @xmath3 lies on the other side . thus , the limits @xmath390_^+ : = _ ^+ _ , _ ^- : = _ ^- _ @xmath390 exist , and are both equal to @xmath380 if @xmath3 is not purely periodic , because then no @xmath369 with @xmath31 lies on the boundary of the partition . the claim then follows by lemma [ l : contwedge ] . this proves continuity of @xmath371 at all angles which are not purely periodic . we shall now deal with the purely periodic case , where for the moment we only know that the left - hand side and right - hand side limits of @xmath371 exist . [ l : diffpairs ] let @xmath3 be purely periodic of period @xmath43 . then @xmath27 , @xmath391 and @xmath392 are purely periodic of period @xmath43 , and differ only in the labelings of pairs @xmath247 with either @xmath361 or @xmath393 . let @xmath3 be purely periodic of period @xmath43 . note that @xmath369 lies on the boundary of the partition @xmath46 if and only if @xmath361 . thus , for all the pairs @xmath247 for which neither component is divisible by @xmath43 , the label of @xmath247 is continuous across @xmath3 , proving the second statement . now we shall show that if for all @xmath388 and close enough to @xmath3 one has @xmath394 on one side of the partition , then also @xmath395 is on the same side for each @xmath256 . this will prove that @xmath396 is purely periodic of period @xmath43 . the proof for @xmath397 is symmetric , and @xmath27 is purely periodic of period @xmath43 by lemma [ l : periodic ] . in order to prove the remaining claim , let us denote @xmath398 , @xmath399 ; since @xmath3 is purely periodic , there exists @xmath400 such that @xmath401 for all @xmath256 . now , note that for each @xmath256 the derivatives satisfy the inequality @xmath402 , thus for each @xmath256 there exists @xmath403 such that for each @xmath404 one has @xmath405 , hence the points @xmath406 all belong to the same side of the partition independently of @xmath300 , as required . [ p : bothsides ] if @xmath3 is purely periodic of period @xmath43 , then the ( infinite ) graphs @xmath28 , @xmath407 and @xmath408 , associated respectively to @xmath27 , @xmath391 and @xmath392 have the same growth rate , i.e. we have the equality @xmath409 by lemma [ l : diffpairs ] and lemma [ l : twoperiodics ] , the graphs @xmath28 , @xmath407 and @xmath408 have the same finite form , hence by proposition [ p : finiteinfinite ] they have the same growth rate , proving the claim . let @xmath2 , and denote by @xmath413 the smallest real zero of @xmath14 . the proof is a simple quantitative version of rouch s theorem . for simplicity , let us assume that @xmath3 is pre - periodic , and let @xmath414 be the minimum distance , in @xmath51 , between ( distinct ) elements of the set @xmath415 . given another angle @xmath416 , denote @xmath417 the smallest real zero of @xmath418 , and choose @xmath20 such that @xmath419 then , by looking at how @xmath369 moves with respect to the partition @xmath46 , we realize that , for @xmath420 , the pair @xmath247 is separated for @xmath3 if and only if it is separated for @xmath416 . by construction , the @xmath421 coefficient of @xmath14 depends only on multi - cycles of total length @xmath20 , and by proposition [ p : properties ] ( 3 ) all such multi - cycles live on the subgraph @xmath422 , hence the first @xmath20 coefficients of @xmath14 and @xmath418 coincide . thus we can write @xmath423 for some @xmath424 and @xmath425 , where we used that @xmath426 by proposition [ p : properties ] ( 5 ) . now , using @xmath427 we have @xmath428 where @xmath256 is the order of zero of @xmath46 at @xmath429 and @xmath430 is a holomorphic function , with @xmath431 . by combining , and , we get @xmath432 with @xmath433 , which implies the claim as @xmath434 . in the periodic case the proof the same , except one should argue separately for @xmath435 and @xmath436 , and use proposition [ p : bothsides ] . a. douady , _ topological entropy of unimodal maps : monotonicity for quadratic polynomials _ , in _ real and complex dynamical systems ( hillerd , 1993 ) _ , nato adv . sci . * 464 * , 6587 .

the core entropy of polynomials , recently introduced by w. thurston , is a dynamical invariant which can be defined purely in combinatorial terms , and provides a useful tool to study parameter spaces of polynomials . the theory of core entropy extends to complex polynomials the entropy theory for real unimodal maps : the real segment is replaced by an invariant tree , known as hubbard tree , which lives inside the filled julia set . we prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle , answering a question of thurston .

arxiv

the detection of x - ray , optical and radio afterglows from some well - localized gamma - ray bursts ( grbs ) definitely shows that at least most long grbs are of cosmological origin ( e.g. , costa et al . 1997 ; frail et al . 1997 ; galama et al . 1998 ; akerlof et al . 1999 ; zhu et al . 1999 ) . the so called fireball model is thus strongly favoured . however , we are still far from resolving the puzzle of grbs ( piran 1999 ; van paradijs , kouveliotou & wijers 2000 ) . a major problem is that we do not know whether grbs are due to highly collimated jets or isotropic fireballs , so that the energetics involved can not be determined definitely ( e.g. , pugliese , falcke & biermann 1999 ; kumar & piran 2000 ; dar & rjula 2000 ; wang & loeb 2001 ; rossi , lazzati & rees 2002 ) . this issue has been extensively discussed in the literature . generally speaking , three methods may help to determine the degree of beaming in grbs . first , based on analytic solutions , it has been proposed that optical afterglows from a jetted grb should be characterized by a break in the light curve during the relativistic phase , i.e. , at the time when the lorentz factor of the blastwave is @xmath0 , where @xmath1 is the half opening angle ( rhoads 1997 ; kulkarni et al . 1999 ; mszros & rees 1999 ) . some grbs such as 990123 , 990510 are regarded as good examples ( castro - tirado et al . 1999 ; wijers et al . 1999 ; harrison et al . 1999 ; halpern et al . 2000 ; pian et al . 2001 ; castro ceron et al . 2001 ; sagar et al . 2001 ) . however , detailed numerical studies show that the break is usually quite smooth ( panaitescu & mszros 1998 ; moderski , sikora & bulik 2000 ) , and huang et al . went further to suggest that the light curve break should in fact occur at the trans - relativistic phase ( huang et al . 2000a , b , c , d ) . additionally , many other factors can also result in light curve breaks , for example , the cooling of electrons ( sari , piran & narayan 1998 ) , a dense interstellar medium ( dai & lu 1999 ) , or a wind environment ( dai & lu 1998a ; chevalier & li 2000 ; panaitescu & kumar 2000 ) . all these facts combine together to make the first method not so conclusive . second , gruzinov ( 1999 ) argued that optical afterglows from a jet can be strongly polarized , in principle up to tens of percents . some positive observations have already been reported ( wijers et al . 1999 ; rol et al . 2000 ) . but polarization can be observed only under some particular conditions , i.e. , the co - moving magnetic fields parallel and perpendicular to the jet should have different strengths and we should observe at the right time from the right viewing angle ( gruzinov 1999 ; hjorth et al . 1999 ; mitra 2000 ) . the third method was first proposed by rhoads ( 1997 ) , who pointed out that due to relativistic beaming effects , @xmath2-ray radiation from the vast majority of jetted grbs can not be observed , but the corresponding late time afterglow emission is less beamed and can safely reach us . these afterglows are called orphan afterglows , which means they are not associated with any detectable grbs . the ratio of the orphan afterglow rate to the grb rate might allow measurement of the grb collimation angle . great expectations have been put on this method ( rhoads 1997 ; mszros , rees & wijers 1999 ; lamb 2000 ; paczyski 2000 ; djorgovski et al . 2001 ) . in fact , the absence of large numbers of orphan afterglows in many surveys has been regarded as evidence that the collimation can not be extreme ( rhoads 1997 ; perna & loeb 1998 ; greiner et al . 1999 ; grindlay 1999 ; rees 1999 ) . recently dalal , griest & pruet ( 2002 ) argued that measurement of the grb beaming angle using optical orphan searches is extremely difficult . the main reason is that when the afterglow emission from a jet begins to go into a much larger solid angle than the initial burst does , the optical flux density usually becomes very low . generally speaking , this problem can be overcome by improving the detection limit . in fact , an interesting result was recently reported by vanden berk et al . ( 2002 ) , who discovered a possible optical orphan at @xmath3 . they suggested that grbs should be collimated . in this article , we will point out another difficulty associated with the third method : there should be many `` failed gamma - ray bursts ( fgrbs ) '' , i.e. , baryon - contaminated fireballs with initial lorentz factor @xmath4 . fgrbs can not be observed in gamma - rays , but their long - lasting afterglows are detectable , thus they will also manifest themselves as orphan afterglows . our paper is organized as follows . in section 2 we explain the concept of fgrbs . section 3 describes the difficulty in distinguishing an fgrb orphan and a jetted grb orphan . some possible solutions that may help to overcome the difficulty are suggested . section 4 is a brief discussion . occurring in the deep universe , grbs are the most relativistic phenomena ever known . the standard fireball model ( mszros & rees 1992 ; dermer , bttcher & chiang 1999 ) requires that to successfully produce a grb , the initial lorentz factor of the blastwave should typically be @xmath5 1000 during the main burst phase ( piran 1999 ; lithwick & sari 2000 ) . generally speaking the requirement of ultra - relativistic motion is to avoid the so called `` compactness problem '' . a modest variation in the lorentz factor will result in a difference of the opacity of the high - energy @xmath2-ray photons by a factor of @xmath6 ( totani 1999 ) . additionally , assuming synchrotron radiation , the observed peak frequency is strongly dependent on @xmath2 , @xmath7 ( mszros , rees & wijers 1998 ) . thus a lorentz factor of @xmath8 makes the blastwave very inefficient in emitting @xmath2-ray photons . so , to successfully produce a grb , we need @xmath5 1000 . however , theoretically it is not easy to construct a model to generate such ultra - relativistic motions . currently there are mainly two kinds of progenitor models , the collapse of massive stars ( with mass @xmath9 ) , or the collision of two compact stars ( such as two neutron stars or a neutron star and a black hole ) . since a baryon - rich environment is involved in all these models , some researchers are afraid that the baryon - contamination problem may exist . but this problem maybe is not as serious as we previously expected . let us first take the collapsar model ( macfadyen & woosley 1999 ) as an example . we can imagine that the baryon mass and energy released in different collapsar events should vary greatly , then @xmath10 of the resultant fireballs may also vary in a relatively wide range . in most cases , @xmath10 should be very low ( i.e. , @xmath4 ) , but there still could be a few cases ( e.g. , one percent or even one in a thousand ) in which the fireball is relatively clean so that the blastwave can be successfully accelerated to @xmath5 1000 and produces a grb . since the collapsar rate is high enough in a typical galaxy , there should be no problem that such collapsars can meet the requirement of grb rate ( i.e. , @xmath11 @xmath12 event per typical galaxy per year , under isotropic assumption ) . cases are similar in the collisions of two compact stars . in short , we can not omit an important fact : if grbs are really due to isotropic fireballs , then there should be much more failed grbs ( i.e. , fireballs with lorentz factor much less than one hundred , but still much greater than unity ) . these fgrb fireballs can contain similar initial energy as normal grb fireballs , i.e. , @xmath13 @xmath14 ergs , but they are polluted by baryons with mass @xmath15 @xmath16 . radiation from these fgrbs should mainly be in x - ray bands in the initial bursting phase , not in @xmath2-ray bands . in fact , bepposax team has reported the discovery of several anomalous events named as fast x - ray transients , x - ray rich grbs , or even x - ray - grbs . they resemble usual grbs except that they are extremely x - ray rich ( frontera et al . 2000 ; kippen et al . 2001 ; gandolfi & piro 2001 ) . observational data on this kind of events are being accumulated rapidly . recent good examples include grbs 011030 , 011130 and 011211 ( gandolfi et al . 2001a , b ; ricker et al . 2001 ) . we propose that these events are probably just fgrbs . huang et al . ( 1998 ) and dai , huang & lu ( 1999 ) have pointed out that for afterglow behaviour , the parameter @xmath17 is decisive , while @xmath18 is only of minor importance , especially at late stages . so , fgrbs should also be associated with prominent afterglows . in figure 1 we compare the theoretical optical afterglows from fgrbs with those from isotropic grbs and jetted grbs . we can see that the light curve of an fgrb afterglow differs from that of a successful isotropic burst only slightly , i.e. , only notable at early stages . in our calculations , we have used the methods developed by huang et al.(1999a , b , 2000b , d ) , i.e. , for the dynamical evolution of isotropic fireballs we use @xmath19 where @xmath20 is the swept - up mass and @xmath21 is the radiation efficience . ( 1 ) has been proved to be proper in both ultra - relativistic phase and non - relativistic phase ( huang , dai & lu 1999a , b ) . for jetted ejecta , the following equation is added ( huang et al . 2000b , d ) , @xmath22 where @xmath23 is the blastwave radius , and the co - moving sound speed @xmath24 is given realistically by @xmath25 with @xmath26 the adiabatic index . in fact , in beamed grb models , there should also be many fgrbs , i.e. , beamed ejecta with @xmath27 . we call them beamed fgrbs . afterglow from beamed fgrbs has also been illustrated in figure 1 . in this article emphasises will be put on isotropic fgrbs , so by using `` fgrbs '' we will only mean isotropic fgrbs unless stated explicitly . both fgrbs and jetted but off - axis grbs can produce isolated fading objects , i.e. , orphan afterglows . theoretically , when orphan afterglows are really discovered observationally , it is still risky to conclude that grbs are beamed . we should study these orphans carefully to determine whether they come from fgrbs or jetted grbs . however , we will show below that it is not an easy task . usually the light curve of grb afterglows is plotted as @xmath28 vs. @xmath29 , where @xmath30 is the flux density at observing frequency @xmath31 and @xmath32 is observer s time measured from the burst trigger . in such plots , the behaviour of afterglows from isotropic grbs and jetted ones are possibly quite different . the former is generally characterized by a simple flat straight line with slope @xmath33 @xmath34 and the latter can be characterized by a break in the light curve or by a steep straight line with a slope sharper than @xmath35 ( figure 1 , also see huang et al . 2000a , b , c , d ) . but for orphan afterglow observations , the derivation of such a @xmath28 @xmath29 light curve is not direct : we do not know the trigger time so that the exact value of @xmath32 for each observed data point can not be determined . as the first step , the best that we can do is to produce a light curve with a linear time axis , which , however , is of little help for unveiling the nature of the orphan . figure 2 illustrates the matter . in this figure we plot @xmath28 vs. @xmath32 for the two kinds of orphans theoretically . the uncerntainty in trigger time means the observed light curve can be shifted along x axis , while the unknown distance results in a shift along y axis . we see that after some simple manipulations , the segment ab on the dashed curve ( i.e. , from @xmath36 d to @xmath37 d ) can be shifted to a place ( a@xmath38b@xmath38 ) that differs from the solid line only slightly . it hints that a linear light curve as long as @xmath39 days is still not enough . note that the solid and the dashed curves in figure 2 are only two examples . the variation of some intrinsic parameters , such as @xmath17 , @xmath40 , @xmath10 , @xmath41 , @xmath42 , @xmath43 , @xmath44 and @xmath45 as defined in the caption of figure 1 , can change the shape of the two curves notably , thus brings in much more difficulties . in figure 3 , we compare the theoretical @xmath28 @xmath29 light curves of optical afterglows from fgrbs and jetted but off - axis grbs directly . to investigate the influence of the uncertainty in trigger time , we also shift the light curve of fgrbs by @xmath46 d , @xmath47 d and @xmath48 d intentionally . from the dashed curves , we can see that the shape of the fgrb afterglow light curve is seriously affected by the uncertainty of the trigger time . but fortunately , these dashed curves still differ from the theoretical light curve of the jetted grb orphan markedly , i.e. , they are much flatter at very late stages . this means it is still possible for us to discriminate them . in figure 4 , similar results to figure 3 are given , but this time the light curve of the jetted grb orphan is shifted . again we see that the two kinds of orphans can be discriminated by their late time behaviour . figures 3 and 4 explain what we should do when an orphan afterglow is discovered . first , we have to assume a trigger time for it arbitrarily , so that the logarithmic light curve can be plotted . we then need to change the trigger time to many other values to see how the light curve is affected . in all our plots , we should pay special attention to the late time behaviour , which will be less affected by the uncertainty in the trigger time . if the slope tends to be @xmath49 @xmath34 , then the orphan afterglow may come from an fgrb event . but if the slope tends to be steeper than @xmath35 , then it is very likely from a jetted but off - axis grb . in fact , from figure 1 , we know that for all kinds of grbs , either successful or failed , the optical afterglow approximately follows a simple power - law decay at late stages ( i.e. , @xmath50 ) so that the light curve is a straight line . in such a relation , if we shift the time by @xmath51 , @xmath52 then the line would become curved . the slope at each point on the curve is @xmath53 for positive @xmath51 values the lines bend up - ward , while for negative values the lines bend down - ward . it hints that in plotting the orphan afterglow light curve , we could select the trigger time properly to get a straight line at late stages , then we can determine not only the late time slope , but also the true trigger time . in other words , we can use @xmath54 as the condition to determine the trigger time and to get the straight line at late stages . however , we must bear in mind that it is in fact not an easy task . first , to take the process we need to follow the orphan as long as possible , and the simple discovery of an orphan is obviously insufficient . note that currently optical afterglows from most well - localized grbs can be observed for only less than 100 days . it is quite unlikely that we can follow an orphan for a period longer than that . second , since the orphan is usually very faint , errors in the measured magnitudes will seriously prevent us from deriving the straight line . due to all these difficulties , a satisfactory light curve is usually hard to get for most orphans . we see that measurement of the grb beaming angle using orphan searches is not as simple as we originally expected . in fact , it is impractical to some extent . recently it was suggested by rhoads ( 2001 ) that grb afterglows can be effectly identified by snapshot observations made with three or more optical filters . the method has been successfully applied to grb 001011 by gorosabel et al . it is believed that this method is also helpful for orphan afterglow searches . however , please note that a jetted grb orphan and an fgrb one still can not be discriminated directly . we have shown that the derivation of a satisfactory light curve for an orphan afterglow is difficult . the major problem is that we do not know the trigger time . anyway , there are still some possible solutions that may help to determine the onset of an orphan afterglow . firstly , of course we should improve our detection limit so that the orphan afterglow could be followed as long as possible . the longer we observe , the more likely that we can get the true late - time light curve slope . secondly , we know that fgrbs usually manifest themselves as fast x - ray transients or x - ray - grbs . if an orphan can be identified to associate with such a transient , then it is most likely an fgrb one . in this case , the trigger time can be well determined . thirdly , maybe in some rare cases we are so lucky that the rising phase of the orphan could be observed . for a jetted grb orphan the maximum optical flux is usually reached within one or two days and for an fgrb orphan it is even within hours . then the uncertainty in trigger time is greatly reduced . additionally , a jetted grb orphan differs markedly from an fgrb one during the rising phase . the former can be brightened by more than one magnitude in several hours ( see figures 1 4 ) , while the brightening of the latter can hardly be observed . so , if an orphan afterglow with a short period of brightening is observed , then it is most likely a jetted grb orphan . of course , we should first be certain that it is not a supernova . fourthly , valuable clues may come from radio observations . in radio bands , the light curve should be highly variable at early stages due to interstellar medium scintillation , and it will become much smoother at late times . so the variability in radio light curves provides useful information on the trigger time . and fifthly , in the future maybe gravitational wave radiation or neutrino radiation associated with grbs could be detected due to progresses in technology , then the trigger time of an orphan could be determined directly and accurately . in fact , with the successful detection of gravitational waves or neutrino emission , our understanding on grb progenitors will surely be promoted greatly ( paolis et al . 2001 ) . sixthly , the redshift of the orphan afterglow can help us greatly in determining the isotropic energy involved , which itself is helpful for inferring the trigger time . seventhly , the microlensing effect may be of some help . since the size of the radiation zone of a jetted grb orphan is much smaller than that of an fgrb one , they should behave differently when microlensed . finally , although a successful detection of some orphan afterglows does not directly mean that grbs be collimated , the negative detection of any orphans can always place both a stringent lower limit on the beaming angle for grbs and a reasonable upper limit for the rate of fgrbs . to successfully produce a grb , the blastwave should be ultra - relativistic , with lorentz factor typically larger than 100 1000 . however , in almost all popular progenitor models , the environment is unavoidably baryon - rich . we believe that only in very rare cases can an ultra - relativistic blastwave successfully break out to give birth to a grb , and there should be much more failed grbs , i.e. , fireballs with lorentz factor much less than 100 but still much larger than unity . in fact , this possibility has also been mentioned by a number of authors , such as mszros & waxman ( 2001 ) . owing to the existence of fgrbs , there should be many orphan afterglows even if grbs are due to isotropic fireballs . then the simple discovery of orphan afterglows does not necessarily mean that grbs be highly collimated . to make use of information from orphan afterglow surveys correctly , we should first know how to discriminate a jetted grb orphan and an fgrb one . this can be done only by checking the detailed afterglow light curve . however , we have shown that the derivation of a satisfactory light curve for an orphan afterglow is difficult . the major problem is that we do not know the trigger time . in section 3.2 , some possible solutions to the problem are suggested . unfortunately many of these solutions are still quite impractical in the foreseeable future , which means measure of grb beaming angle using orphan afterglow searches is extremely difficult currently . however , special attention should be paid to the second solution . usually , fgrbs manifested themselves as fast x - ray transients during the main burst phase , while jetted but off - axis grbs went unattended completely . if the fast x - ray transients ( or x - ray - grbs ) observed by bepposax are really due to fgrbs , then afterglows should be detectable . we propose that this kind of events should be followed rapidly and extensively in all bands , just like what we are doing for grbs . if observed , afterglows from these anomalous events can be used to check our concept of fgrbs , and even to test the fireball model under quite different conditions ( i.e. , when @xmath4 ) . also , these fgrbs can provide valuable information for our understanding of grbs , especially on the progenitor models . note that beamed fgrbs can also give birth to fast x - ray transients if they are directed toward us , but afterglows from such a beamed fgrb and an isotropic fgrb can be discriminated easily from the light curves ( see figure 1 ) . it is very interesting to note that optical afterglows from two x - ray - grbs , 011130 and 011211 , have been observed ( garnavich , jha & kirshner 2001 ; grav et al . their redshifts were measured to be @xmath55 0.5 and 2.14 respectively ( jha et al . 2001 ; fruchter et al . 2001 ) , eliminates the possibility that they were ordinary classic grbs residing at extremely high redshifts ( @xmath56 ) . we propose that they should be fgrbs ( either isotropic or beamed ) or just jetted grb `` orphan '' . however , the observational data currently available are still quite poor so that we could not determine their nature definitely . as for other x - ray - grbs without a measured redshift , the possibility that they were at redshifts of @xmath57 can not be excluded . finally , the concept of fgrbs is based on the fact that most popular progenitor models for grbs are baryon - rich . but cases are quite different for another kind of progenitor models where strange stars are involved . strange stars , composed mainly of u , d , and s quarks , are compact objects which are quite similar to neutron stars observationally ( alcock , farhi & olinto 1986 ) . a typical strange star ( with mass @xmath58 ) can have a normal matter crust of less than @xmath59 ( alcock , farhi & olinto 1986 ) , or even as small as @xmath60 ( huang & lu 1997a , b ) . then baryon contamination can be directly avoided if grbs are due to the phase transition of neutron stars to strange stars ( cheng & dai 1996 ; dai & lu 1998b ) or collisions of binary strange stars . in these models , there should be very few fgrbs . we thank an anonymous referee for valuable comments and suggestions . yfh thanks l. j. gou and x. y. wang for helpful discussion . this research was supported by the special funds for major state basic research projects , the national natural science foundation of china ( grants 10003001 , 19825109 , and 19973003 ) , and the national 973 project ( nkbrsf g19990754 ) . note added after acceptance * ( this paragraph might not appear in the published version ) * : the optical orphan at z=0.385 reported by vanden berk et al . 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it is believed that orphan afterglow searches can help to measure the beaming angle in gamma - ray bursts ( grbs ) . great expectations have been put on this method . we point out that the method is in fact not as simple as we originally expected . due to the baryon - rich environment that is common to almost all popular progenitor models , there should be many failed gamma - ray bursts , i.e. , fireballs with lorentz factor much less than 100 1000 , but still much larger than unity . in fact , the number of failed gamma - ray bursts may even be much larger than that of successful bursts . owing to the existence of these failed gamma - ray bursts , there should be many orphan afterglows even if grbs are due to isotropic fireballs , then the simple discovery of orphan afterglows never means that grbs be collimated . unfortunately , to distinguish a failed - grb orphan and a jetted but off - axis grb orphan is not an easy task . the major problem is that the trigger time is unknown . some possible solutions to the problem are suggested . = -0.5 in stars : neutron ism : jets and outflows gamma - rays : bursts

arxiv

cosmic bursts of gamma - rays are some of the most elusive and mysterious events in the universe . typical burst lasts less than a minute and disappears thereafter . however , since the _ bepposax _ s first discovery on 1997 february 28 ( costa et al . 1997 ) it is known that gamma - ray bursts ( grbs ) also produce x - ray afterglows , which can be detected with modern x - ray observatories for several hours and sometimes several days after the burst . more than a few x - ray afterglows have been observed in last five years by _ bepposax , asca and chandra _ satellites . several afterglow observations were carried out with _ xmm - newton_. in this _ letter _ we report on two of the latter observations . grb 011211 was detected and localized in _ bepposax _ wfc1 on 2001 december 11 , 19:09:21 ut(gandolfi 2001 ) . the distinguishing features of grb 011211 were its long duration ( the longest event localized with _ bepposax _ ) and its faintness both in x- and @xmath0-rays ( frontera et al . 2002 ) . grav et al . ( 2001 ) discovered a new point source in optical r - band ( r@xmath719 ) at r.a.=11@xmath815@[email protected] , dec=@xmath1121@xmath1256@[email protected] ( j2000 , @xmath61@xmath14 ) identified as the afterglow of grb 011211 . fruchter et al . ( 2001 ) and gladders et al . ( 2001 ) measured the optical spectrum and found an absorption line system corresponding to a redshift of z=2.14 . later the optical transient was found to be superposed on an apparent host galaxy with r = [email protected] ( burud et al . 2001 ) . grb 001025 was detected by the _ rxte all - sky monitor _ on 2000 october 25 at about 03:10:05 ut ( smith et al . . lasted approximately 15 - 20 s and reached a peak 5 - 12 kev flux of @xmath154 crab . optical afterglow of grb 001025 has never been detected down to r=24.5 for any significantly variable object ( fynbo et al . the observation of grb 011211 afterglow was started on 2001 december 12 at 06:16:56 ut ( santos - lleo et al . a source was initially located so close to the edge of ccd#7 in epic - pn detector that some of the source photons were falling in the inter - ccd gap . therefore , the telescope was re - pointed the re - pointing slew started at 08:31:16ut and finished at 08:40:50ut . useful exposure of total observation is @xmath1527 ks . xmm - newton observed the location of grb 001025 from oct.27.003 to oct.27.46 ut , starting @xmath151.9 days after the burst . two x - ray sources were detected in the error box ( altieri et al . below we discuss the brighter source ( r.a.=8@xmath836@[email protected] , decl.=-13@xmath1204@[email protected] , j2000 ) . we analyzed data products from the two _ xmm - newton _ observations . in all observations the epic instruments ( turner et al . 2001 , strueder et al . 2001 ) were operated in the _ full window mode _ ( @xmath16 diameter fov ) . the medium optical blocking filter was used with mos detectors for grb 011211 observation . thin filter was used with pn detector for grb 011211 and with mos detectors in grb 001025 observation . no pn data were available for grb 001025 . we reduced epic data with the _ xmm - newton _ science analysis system ( sas v.5.2 ) . epic detectors data were fit together with common model parameters . the count rates were converted into energy fluxes using analytical fits to the spectra . we have used galactic absorption values provided by the heasarc `` nh '' tool ( dickey & lockman , 1990 ) . central part of mos1 detector ( converted to celestial coordinates ) is shown in fig . the bright source in the middle of the image corresponds to the position of grb 011211 optical afterglow . significant x - ray flux decline during the observation ( fig . 1b ) provides an additional evidence for an identification of the x - ray source with the grb . x - ray light curve can be fit with f@xmath17t@xmath18 ( @xmath19=1.5 - 1.7 ) or with f@xmath17e@xmath20 ( @[email protected] ks ) . average measured flux in 0.2 - 10 kev band was equal to 1.7@xmath2210@xmath23 erg / s/@xmath24 . extrapolating back to t=100 s since the initial detection of the burst we get an x - ray flux f@xmath25=5@xmath2210@xmath26 erg / s/@xmath24 or x - ray afterglow luminosity l@xmath25=2@xmath2210@xmath27 erg / s ( 0.6 - 30 kev ) in the rest frame at z=2.14 and a @xmath28=65 km s@xmath3mpc@xmath3 , @xmath29 , @xmath30 cosmology . spectrum of the grb 011211 afterglow can be described as a simple power - law with photon index @[email protected] modified by the galactic absorption only ( fig . no significant spectral evolution has been found in a sequence of 5-ks intervals within the observation ( fig . 2b ) . we built the spectrum of the brightest source in error box of grb 001025 ( fig . combined spectrum of two mos detectors can be readily approximated by a power - law with photon index @[email protected] and galactic absorption value n@xmath31=6@xmath2210@xmath32 @xmath2 . average flux from the source during the observation was 5.3@xmath2210@xmath33erg / s/@xmath24 . no significant flux variability was detected , but the spectrum is very typical for x - ray afterglows of grb , supporting the identification of this source with grb 001025 . it is interesting to consider an alternative identification of the x - ray source with a background quasar . an x - ray bright quasar 3c273 has a r magnitude of @xmath1512.5 ( odell et al . 1978 ) and the 0.5 - 10 kev x - ray flux of 23@xmath2210@xmath34 erg/@xmath24/s ( reeves & turner 2000 ) . neglecting the k - correction and simply scaling the r magnitude to the x - ray flux we find that if a 3c273-like quasar is a source of the detected x - ray flux of @xmath154.7@xmath2210@xmath33 erg/@xmath24/s ( 0.5 - 10 kev ) then its optical counterpart would have a r magnitude of @xmath1521.7 . as in fact an optical counterpart for grb 001025 had not been found down to r=24.5 ( fynbo et al . 2000 ) we consider unlikely that the x - ray source is an agn . we have detected power - law spectra with index @xmath152 from two grb afterglows . in case of grb 011211 an identification of the x - ray source with the grb is supported by the observation of optical transient and also by the decline of x - ray flux during the _ xmm - newton _ observation . for grb 001025 we do not have such supporting evidence , but the spectral shape itself together with the position of x - ray source inside ipn / rxte error box allows us to suggest that the source is indeed an afterglow of grb 001025 . power - law fit with slope @xmath152 is very typical for detected x - ray afterglows of grbs ( see e.g. harrison et al . 2001 , int zand et al . 2001 , antonelli et al . we did not detect any significant changes in the x - ray spectrum during the long observation of grb 011211 . afterglow of grb 001025 was observed significantly later after the burst , and measured x - ray flux was much lower , but the spectrum was almost identical to grb 011211 . conspicuously the x - ray spectra in all or most of the observed afterglows are generated by common physical process and do not depend much on the differences in the burst environments . overall spectral shape can be fit to popular model of synchrotron emission with possible inclusion of inverse compton scattering ( piran 1999 ; granot & sari 2002 ; sari & esin 2001 ) . no significant absorption above the galactic values has been detected in the x - ray spectra . high absorption would be naturally expected if the burst occurs in a high - density star - forming region . ramirez - ruiz , trentham & blain ( 2002 ) suggested that high absorption in nearby ( relative to the burst birthplace ) interstellar media may be the reason for a lack of optical detections in a significant fraction of grb . the absorption should be detectable in soft x - rays ( 0.5 - 2 kev ) . contrary to such expectations , we did not detect any significant absorption above the galactic value in both a grb with optical afterglow ( grb 011211 ) and without it ( grb 001025 ) . the lack of x - ray absorption in grb 011211 is consistent with the lack of detected reddenning in the spectrum of optical afterglow ( simon et al . 2001 ) . reeves et al . ( 2002a , hereafter rwo ) reported on the discovery of a blue - shifted line complex in the spectrum of grb 011211 . the lines have been detected only with pn detector and only during first 5 ks of the observation , with the source located close to the ccd chip boundary ( fig . we have extracted source and background spectra from the same regions as rwo ( reeves et al . . the spectrum can be satisfactory fit to power - law with @xmath5=2.4 and n@xmath31 = 8 @xmath22 10@xmath32 @xmath2 ( @xmath35=1.03 for 44 d.o.f . joint fit of pn , mos1 and mos2 data gives @xmath35=0.92 ( 72 d.o.f . ) for @[email protected] and an absorption fixed at the galactic value . hence we are able to get the perfect fit using a simple power - law model we do not see much value in adding extra lines to the model . addition of 5 lines at the energies specified by rwo gives @xmath35=0.93 ( 62 d.o.f . ) for joint fit of the pn and mos data . the analysis of pn data alone allowed us to reproduce line fluxes reported by rwo with somewhat lower significance ( fig . 4c and table 2 ) . we note that the pn data should be treated with special caution for the first 5 ks of the observation because of the unfortunate position of the source on the ccd . in their analysis rwo collected source data from two different pn chips and from the areas near the interchip boundary , which are the least suitable for fine spectroscopy . our extensive analysis showed that any alternative choice of extraction regions for the source and background leads to further reduction in the lines significance . we also found that background spectrum collected over the interchip edge is dominated by a strong feature at 0.7 kev , exactly at the energy of the most significant `` line '' reported by rwo ( see fig.4f ) . though the bulk of the events forming this line are eliminated by the event filtering we are still concerned that some of these bad events may be present in the spectrum of grb 011211 . our concern is amplified by the lack of any of the rwo `` lines '' in the pn spectrum after the reorientation ( fig.4d ) . spectral evolution of the source in sync with satellite revolutions looks quite suspicious unless one suggests that presense of the `` lines '' depends on the position of the source on the chip . such alternative hypothesis is further confirmed by a lack of `` lines '' in the mos data ( fig . it may therefore be concluded that there is no definitive evidence for the presence of the line complex at a redshift of 1.88 in the x - ray spectrum of grb 011211 . the existence of such complex is a possibility , but its statistical significance is greatly overestimated by rwo . our analysis suggests that the spectrum of grb 011211 is featureless and does not contain any significant line emission . we have used publicly available data obtained with _ xmm - newton _ satellite . we are grateful to the personnel of the _ xmm - newton _ science operations centre at vilspa , spain for satellite operations and expedited preparation of data products for scientific analysis . we are thankful to w.priedhorsky and l.titarchuk for encouraging advices , and to anonymous referee for her / his helpful comments . we acknowledge an interesting discussion of the draft version of our paper with j.osborne . altieri , b. , schartel , n. , santos , m. , tomas , l. , guainazzi , m. , piro , l. , parmar , a. 2000 , gcn circ . 869 antonelli , a. , piro , l. , vietry , m. , et al . 2000 , apj , 545 , l39 burud , i. , rhoads , j. , fruchter a. , & hjorth j. , on behalf of grace ( gamma - ray afterglow collaboration at eso ) , 2001 , gcn circ . 1213 costa , e. , frontera , f. , heise , j. , et al . 1997 , _ nature _ , 387 , 783 covino , s. , ghisellini , g. , saracco , p. , et al . , 2002 , gcn circ . 1214 dickey , j. m. , & lockman , f. j. 1990 , araa , 28 , 215 frontera , f. , amati , l. , guidorzi , c. , montanari , e. , costa , e. , feroci , m. , piro , l. , heise , j. , int zand , j.j.m . , 2002 , gcn circ . 1215 fruchter , a. , vreeswijk , p. , rhoads , j. , burud , i. , 2001 , gcn circ . 1200 fynbo , j. p. u. , moller , p. , milvang - jensen , b. , burud , i. , andersen , m. i. , pedersen , h. , jensen , b. l. , hjorth , j. , gorosabel , j. 2000 , gcn circ . 867 gandolfi , g. , on behalf of bepposax mission scientist , 2001 , gcn circ . 1189 gladders , m. , holland , s. , garnavich , p. m. , jha , s. , stanek , k. z. , bersier , d. , barrientos , l.f . , 2001 , 1209 granot , j. , sari , r. 2002 , apj , 568 , 820 grav , t. , hansen , m.w . , pedersen , h. , hjorth , j. , michelsen , r. , jensen , b.l . , andersen , m.i . , gorosabel , j. , fynbo , j.u . , 2001 , 1191 harrison , f.a . , yost , s.a . , sari , r. , berger , e. , galama , t. , et al . 2001 , apj , 559 , 123 int zand , j.j.m . , kuiper , l.m . , amati , l. , antonelli , l.a . , butler , r.c . , et al . 2001 , apj , 559 , 710 odell , s. l. , puschell , j. j. , stein , w. a. , owen , f. , porcas , r. w. , mufson , s. , moffett , t. j. , ulrich , m .- h . 1978 , apj , 224 , 22 piran , t. 1999 , phys.rep . , 314 , 575 ramirez - ruiz , e. , trentham , n. , & blain , a. w. 2002 , m.n.r.a.s . , 329 , 465 reeves , j. n. , & turner , m. j. l. 2000 , mnras , 316 , 234 reeves , j. n. , watson , d. , osborne , j. p. , pounds , k. a. , obrien , p. t. et al . , 2002a , _ nature _ , 416 , l512 ( rwo ) reeves , j. n. , watson , d. , osborne , j. p. , pounds , k. a. , obrien , p. t. 2002b , astro - ph/0206480 santos - lleo , m. , loiseau , n. , rodriguez , p. , altieri , b. , and schartel , n. , 2001 , gcn circ . 1192 sari , r. , esin , a.a . 2001 , apj , 548 , 787 simon , v. , hudec , r. , pizzichini , g. , masetti , n. , 2001 , gcn circ . 1211 smith , d. a. , levine , a. m. , remillard , r. , hurley , k. , cline , t. 2000 , gcn circ . 861 strueder , l. et al . , 2001 , a&a , l18 turner , m. et al . , 2001 , a&a , 365 , l27

we present the _ xmm - newton _ observations of x - ray afterglows of the @xmath0-ray bursts grb 011211 and grb 001025 . for grb 011211 _ xmm _ detected fading x - ray object with an average flux in 0.2 - 10 kev declining from @xmath1 erg @xmath2 s@xmath3 during the first 5 ks of 27-ks observation to @xmath4 erg @xmath2 s@xmath3 toward the end of the observation . the spectrum of the afterglow can be fit to a power law with @[email protected] modified for the galactic absorption . no significant evolution of spectral parameters has been detected during the observation . similar x - ray spectrum with @[email protected] has been observed by the xmm from the grb 001025 . the non - detection of any extra absorption in these spectra above the galactic value is an interesting fact and may impose restrictions to the favorable grb models involving burst origin in star - forming regions . finally we discuss soft x - ray lines from grb 011211 reported by reeves et al . and conclude that there is no definitive evidence for the presense of these lines in the spectrum . * # 1#1

arxiv

the layered organic material 3 ( ai3 ) , which has been studied since the 1980s,@xcite has recently attracted renewed interest because it reveals low - energy massless dirac fermions under hyrdostatic pressure ( @xmath2 gpa).@xcite compared to graphene , certainly the most popular material with low - energy dirac fermions@xcite or electronic states at the surface of three - dimensional topological insulators,@xcite ai3 is strikingly different in several respects . apart from the tilt of the dirac cones and the anisotropy in the fermi surface,@xcite its average fermi velocity is roughly one order of magnitude smaller than that in graphene . this , together with an experimentally identified low - temperature charge - ordered phase at ambient pressure,@xcite indicates the relevance of electronic correlations . indeed , because the effective coupling constant for coulomb - type electron - electron interactions is inversely proportional to the fermi velocity , it is expected to be ten times larger in ai3 than in graphene . the material ai3 thus opens the exciting prospective to study strongly - correlated dirac fermions that are beyond the scope of graphene electrons.@xcite . another specificity of ai3 is the presence of additional massive carriers in the vicinity of the fermi level , as recently pointed out in ab - initio band - structure calculations.@xcite however , the interplay between massless dirac fermions and massive carriers has , to the best of our knowledge , not yet been proven experimentally . finally , one should mention a topological merging of dirac points that is expected for high but experimentally accessible pressure.@xcite here , we present magneto - transport measurements of ai3 crystals under hydrostatic pressure larger than @xmath3 gpa where dirac carriers are present . we show not only the existence of high - mobility dirac carriers as reported elsewhere,@xcite but we prove also experimentally the presence of low - mobility massive holes , in agreement with recent band - structure calculations.@xcite the interplay between both carrier types at low energy is the main result of our studies . furthermore , we show that the measured mobilities for the two carrier types hint at scattering mechanisms due to strongly screened interaction potentials or other short - range scatterers . the remainder of the paper is organized as follows . in sec . [ sec:1 ] , we present the experimental set - up and the results of the magneto - transport measurements ( sec . [ sec:1.1 ] ) under hydrostatic pressure . the subsection [ sec:1.2 ] is devoted to a discussion of the temperature dependence of the carrier densities , in comparison with the model of ( a ) massless dirac fermions and ( b ) massive carriers . furthermore thermopower measurements are presented to corroborate the two - carrier scenario . the measured temperature dependence of the extracted carrier mobilities is exposed in sec . [ sec:1.3 ] , and a theoretical discussion of the experimental results , in terms of short - range ( such as screened coulomb ) scatterers may be found in sec . [ sec:2 ] . we present our conclusions and future perspectives in sec . [ sec:3 ] . the single crystals of ai3 used in our study have been synthesized by electro - crystallization . their typical size is @xmath4 mm@xmath5 ( @xmath6 plane ) x @xmath7 m ( @xmath8 direction ) . six @xmath9 nm thick gold contacts were deposited by joule evaporation on both sides of the sample , allowing for simultaneous longitudinal and transverse resistivity measurements . a picture of one of the three samples studied is shown in the inset of figure [ magneto ] . the resistivities were measured using a low - frequency ac lock - in technique . the magnetic field @xmath10 , oriented along the @xmath8 axis , was swept between @xmath11 and @xmath12 t at constant temperature between @xmath13 and @xmath3 k. to account for alignment mismatch of patterned contacts , the longitudinal ( transverse ) resistivity has been symmetrized ( anti - symmetrized ) with respect to the orientation of @xmath10 to obtain even [ @xmath14 and odd [ @xmath15 functions respectively . hydrostatic pressure was applied at room temperature in a nicral clamp cell using daphne 7373 silicone oil as the pressure transmitting medium . the pressure was determined , at room temperature , using a manganine resistance gauge located in the pressure cell close to the sample . the values given below take into account the pressure decrease during cooling . the analysis of our data is based on the study of the magneto - conductivity and is similar to the one presented in ref . for multi - carrier semiconductor systems . the magneto - conductivity is obtained from the measured resistivity tensor by means of @xmath16 $ ] . for a single carrier system , its analytical expression reads@xcite @xmath17 where @xmath18 , @xmath19 is the electron charge , @xmath20 the mobility , and @xmath21 is the carrier density . figure [ magneto ] displays a typical magneto - conductivity curve of ai3 under pressure , where two ` plateaus ' can be clearly seen . as conductivity in ai3 has a strong 2d character , conductivity is shown both as 3d conductivity ( @xmath22 ) and as 2d conductivity ( @xmath23 of each bedt - ttf plane ) according to @xmath24 . as conductivity is additive , in a two - carrier system , the contributions of each carrier type a and b can be added , @xmath25 the two `` plateaus '' , observed in fig . [ magneto ] , indicate the existence of two different carrier types ( @xmath26 or @xmath27 ) with significantly different mobilities . from this curve , we can extract the mobilities , @xmath28 , of each carrier type , their zero - field conductivities , @xmath29 , and their carrier densities , @xmath30 , by @xmath31 . regime at high fields . the left axis shows the square ( 2d ) conductivity of each bedt - ttf plane while the right axis shows the `` bulk '' ( 3d ) longitudinal conductivity ( see text ) . inset : photograph of one sample . ] figure [ sigmaxxt ] shows magneto - conductivity curves of ai3 at a fixed pressure for several temperatures . the previous analysis has been repeated for each of these magneto - conductivity curves to obtain the densities ( fig . [ density ] ) and mobilities ( fig . [ mobility ] ) for each carrier type as a function of temperature and for three different pressures , @xmath32 , @xmath33 and @xmath34 gpa . the strong temperature dependence of the carrier density is a signature that temperature is higher than @xmath35 for both a and b carriers even at the lowest measured temperature , @xmath36 k. this low fermi temperature hints at the absence of charge inhomogeneities that prevent the approach of the dirac point in graphene on si0@xmath37 substrates.@xcite the carrier density can be calculated from @xmath38 , where @xmath39 is the fermi - dirac distribution and @xmath40 is the density of states for massive ( @xmath41 ) and dirac ( @xmath42 ) carriers:@xcite gpa for different temperatures , from bottom to top : @xmath3 , @xmath43 , @xmath34 , @xmath44 , @xmath45 , @xmath46 , @xmath47 , @xmath48 , @xmath49 , @xmath50 and @xmath51 k. the left axis shows the square ( 2d ) conductivity of each bedt - ttf plane while the right axis shows the `` bulk '' ( 3d ) longitudinal conductivity . ] gpa , triangles @xmath33 gpa and squares @xmath34 gpa ; blue thin symbols for a carriers and red thick symbols for b carriers ) . the left axis shows the density for each bedt - ttf plane ( @xmath52 ) while the right axis shows the bulk density ( @xmath53 ) . the lines represent power - law fits of to the a and b carrier densities that yield exponents 0.9 and 2.2 , respectively . inset : band structure calculations at @xmath54 gpa where both dirac cones ( a ) and parabolic bands ( b ) cross the fermi level ( adapted from ref . ) . ] gpa , triangles : @xmath33 gpa and squares : @xmath34 gpa ; blue thin symbols for a carriers and red thick symbols for b carriers ) . the continuous lines represent power laws fits for the mobilities dependences with temperature , which gives exponents @xmath55 ( a carriers ) and @xmath56 ( b carriers ) . the low temperature dispersion of a carriers mobility is due to a decrease of the saturating mobility ( dotted line ) by increasing pressure . ] @xmath57 @xmath58 where @xmath59 and @xmath60 are valley and spin degeneracies and @xmath61 is the effective mass of massive carriers described by a schrdinger equation . in ai3 under pressure , two dirac cones but only one massive band exist at the fermi level.@xcite for large temperatures , @xmath62 , the carrier density depends linearly on temperature for massive carriers and quadratically for dirac carriers : @xmath63 @xmath64 figure [ density ] represents the measured temperature dependence of the mobilities and reveals a power - law behavior , @xmath65 . indeed one obtains an exponent of @xmath66 for the low - mobility carriers ( b ) , in good agreement with what [ eq . ( [ nmassif ] ) ] expected for massive carriers , whereas one finds @xmath67 for the high - mobility carriers ( a ) , as roughly expected for massless dirac particles [ eq . ( [ ndirac ] ) ] . besides , the nature of the carriers can be extracted from hall measurements . furthermore , we have performed thermopower measurements under pressure on a second sample ( figure [ thermopower ] ) . these data show a sign change for the seebeck coefficient ( s ) around 5k . thermopower is the voltage per unit of temperature produced by a thermal gradient . the carrier type determines the sign while the density and mobility of the carriers establish the amplitude . thus , a sign change of the thermopower indicates that the relevant carriers at low temperature have a different charge than those at high temperature , requiring a two - carrier scenario . in agreement with ref . , a carriers which dominate the low - field conduction are electrons . on the contrary , at large fields the conduction is dominated by holes ( b carriers ) . notice that our results are consistent with ab - initio calculations of the band structure of ai3 under a pressure of @xmath68 gpa ( inset of figure [ density])@xcite and do not depend on pressure ( within the range @xmath69 gpa ) . this supports the idea that massless and massive particles coexist in a broad pressure range . however , since @xmath70 in the whole temperature range under study , both dirac electrons and dirac holes are excited . thus there are indeed not two but three carrier types : dirac holes , dirac electrons and massive electrons . for @xmath62 , the electron and hole densities are actually identical ( semimetal with symmetric band structure ) : @xmath71 . the absence of a third ` plateau ' in the magneto - conductivity data allows us to consider that dirac electrons and holes have roughly the same mobilities : @xmath72 . therefore , the results obtained in figure [ density ] and [ mobility ] still hold when we consider two types of dirac carriers ( electrons and holes ) in addition to the massive holes . this analysis allows us to avoid using hall effect measurements for the determination of carrier densities . indeed , hall effect interpretation becomes challenging as dirac electron and hole contributions partially compensate , leading to the determination of only an ` effective ' dirac carrier density , and they are both mixed with massive carriers contribution . this problem is solved here by analyzing the magneto - conductivity where all carriers contributions are additive . the effective mass of the massive carriers has been extracted from eq . ( [ nmassif ] ) . the obtained value is quite small @xmath73 ( @xmath74 is the free electron mass ) . meanwhile , from eq . ( [ ndirac ] ) , @xmath75 m / s can be extracted , in agreement with previous theoretica@xcite and experimental estimates.@xcite in fig . [ density ] , no significant variation of this argument is observed upon sweeping pressure ( which should appear as a vertical shift of the @xmath76 line ) . this indicates that @xmath77 does not change with pressure . in principle , pressure should enhance hopping while reducing the unit cell volume . thus , an enhancement of the @xmath77 with pressure could be expected according to the approximate expression @xmath78 , where @xmath79 is the hopping integral . this expression can be simplified by means of harrison s law ( @xmath80 ) into @xmath81 . as pressure slightly modifies the lattice constant ( @xmath82gpa @xcite ) , @xmath77 is expected to vary by the same order of magnitude which is smaller than our current experimental uncertainty . this accounts for the apparent absence of pressure effects on the carrier density in the range @xmath83 gpa . in fig . [ mobility ] , the mobility of the dirac carriers ( a ) reaches @xmath84 cm@xmath5/vs at low temperatures ( @xmath86 k ) , a value comparable to already published values.@xcite it is quite high compared to typical graphene on 2 values ( @xmath87 to @xmath88 cm@xmath5/vs ) but similar to suspended graphene and graphene on bn mobilities at very low carrier density.@xcite on the other hand , the mobility for massive carriers is @xmath89 cm@xmath5/vs at @xmath86 k , which is two orders of magnitude smaller than for dirac carriers . the temperature dependence of the mobility follows power laws for both massive ( exponent @xmath56 ) and dirac carriers ( exponent @xmath55 ) . moreover , the dirac carrier mobility seems to saturate at @xmath90 . a similar saturation has been reported in others dirac systems.@xcite table [ tabcomp ] summarizes the main parameters of massive and dirac carriers in ai3 , in comparison with graphene on 2 . .dirac and massive carriers parameters in ai3 at high pressure , in comparison with graphene electrons . [ cols="<,<,<,<",options="header " , ] in order to better understand the difference in the mobility , we investigate the ratio @xmath91 , in terms of the scattering times @xmath92 and @xmath93 for the massless dirac and massive carriers , respectively . furthermore , @xmath94 is the density - dependent cyclotron mass of the dirac carriers , in terms of the fermi energy @xmath95 . the scattering times may be obtained from fermi s golden rule ( for @xmath96 ) @xmath97 in terms of the impurity density @xmath98 , the matrix element @xmath99 , and the density of states ( [ ddirac ] ) for dirac and ( [ dmassif ] ) for massive carriers . we consider implicitly that both carrier types are affected by the same impurities , and the matrix element is independent of @xmath100 for short - range impurity scattering . apart from atomic defects , screened coulomb - type impurities approximately fulfill this condition , as it may be seen within the thomas - fermi ( tf ) approximation . indeed , the screening length of the coulomb interaction is dominated by the thomas - fermi wave vector @xmath101 m@xmath102 of the massive carriers , for an effective bohr radius @xmath103 , whereas the thomas - fermi wave vector for massless dirac carriers @xmath104 m@xmath102 , for a density @xmath105 cm@xmath102 and a fine - structure constant @xmath107 . the thomas - fermi wave vector is thus roughly one order of magnitude larger than the fermi wave vector of the massive carriers , which is itself much larger than that of the dirac carriers . the screened coulomb potential for @xmath100-type carriers may therefore be approximated by its @xmath108 value , @xmath109 , which is thus the same for both carrier types , as mentioned above . here , @xmath110 is the permittivity of the dielectric environment and @xmath111 is the dielectric function calculated within the thomas - fermi approximation . in view of the above considerations , we thus obtain , for the mobility ratio in the limit @xmath112 @xmath113 which does neither depend on the form of the matrix element nor on the impurity density . one expects a ratio in the @xmath114 range , whereas the measured ratio is @xmath115 at @xmath116 k. notice that for @xmath62 , that is in the experimentally relevant regime here , one may replace the energy dependence in the density of states of the massless dirac carriers by a linear dependence in temperature , @xmath117 , such that one expects a linear temperature dependence of the mobility ratio ( [ eq : mu ] ) , in agreement with our experimental findings ( @xmath118 for @xmath119 , see fig . [ mobility ] ) . to conclude , we present an interpretation of magneto - transport in ai3 that indicates that both massive and dirac carriers are present even at high pressures . thermopower measurements performed on one of the three studied samples are also in agreement with this two carrier scenario . so far in the literature , the conduction in this system has been attributed solely to dirac carriers.@xcite moreover , this coexistence holds with little perturbation in the whole range of pressure under study . as dirac carriers have high mobility , they dominate the conduction at low magnetic field and high temperatures . on the contrary , for high magnetic fields and low temperatures , the massive holes drive the conduction properties . this crossover can be clearly seen from our magneto - conductivity curves and is responsible for their peculiar ` plateau ' shape . it should also be noted that a proper separation of massive carriers has to be done prior to using any expression that concerns solely dirac carriers . in order to confirm the picture of coexisting dirac and massive carriers , complementary studies , such as spectroscopic measurements , are highly desirable but beyond the scope of the present paper . k. s. novoselov , a. k. geim , s. v. morozov , d. jiang , m. i. katsnelson , i. v. gregorieva , s. v. dubonos , and a. a. firsov , nature * 438 * , 197 ( 2005 ) ; y. zhang , y .- w . tan , h. l. stormer , and p. kim , nature * 438 * 201 , ( 2005 ) . castro neto , f. guinea , n.m.r . peres , k.s . novoselov , and a.k . geim , rev . phys * 81 * , 109 ( 2009 ) ; m. o. goerbig rev . phys * 83 * , 1193 ( 2011 ) ; v. n. kotov , b. uchoa , v. m. pereira , f. guinea , and a. h. castro neto , rev . * 84 * , 1067 .

transport measurements were performed on the organic layered compound 3 under hydrostatic pressure . the carrier types , densities and mobilities are determined from the magneto - conductance of 3 . while evidence of high - mobility massless dirac carriers has already been given , we report here , their coexistence with low - mobility massive holes . this coexistence seems robust as it has been found up to the highest studied pressure . our results are in agreement with recent dft calculations of the band structure of this system under hydrostatic pressure . a comparison with graphene dirac carriers has also been done . 3 @xmath0 2 @xmath1

arxiv

fe - based superconductors ( febs ) represent a non - cuprate class of high-@xmath9 systems with the unconventional superconducting state . the origin of the latter is still debated . in general , febs can be divided into the two subclasses , pnictides and chalcogenides @xcite , with the square lattice of iron as the basic element , though with orthorhombic distortions in lightly doped materials . iron is surrounded by as or p situated in the tetrahedral positions within the first subclass and by se , te , or s within the second subclass . fermi surface ( fs ) is formed by fe @xmath10-orbitals and excluding the cases of extreme hole and electron dopings it consists of two hole sheets around the @xmath11 point and two electron sheets around the @xmath12 and @xmath13 points in the two - dimensional brillouin zone ( bz ) corresponding to one fe per unit cell ( the so - called 1-fe bz ) @xcite . in the 2-fe bz , electron pockets are centered at the @xmath14 point . nesting between these two groups of sheets leads to the enhanced antiferromagnetic fluctuations with the maximal scattering near the wave vector @xmath2 equal to @xmath12 or @xmath13 in the 1-fe bz or to @xmath15 in the 2-fe bz . different mechanisms of cooper pairs formation result in the distinct superconducting gap symmetry and structure @xcite . in particular , the rpa - sf ( random - phase approximation spin fluctuation ) approach gives the extended @xmath16-wave gap that changes sign between hole and electron fs sheets ( @xmath17 state ) as the main instability for the wide range of doping concentrations @xcite . on the other hand , orbital fluctuations promote the order parameter to have the sign - preserving @xmath6 symmetry @xcite . thus , probing the gap structure can help in elucidating the underlying mechanism . in this respect , inelastic neutron scattering ( ins ) is a useful tool since the measured dynamical spin susceptibility @xmath18 in the superconducting state carries information about the gap structure . there are been many reports of a well - defined peak in neutron spectra in 1111 , 122 , and 11 systems appearing only for @xmath19 at or around @xmath20 @xcite . the common explanation is that the peak is the spin resonance appearing due to the @xmath7 state . indeed , since @xmath2 connects fermi sheets with different signs of @xmath7 gaps , the resonance condition for the interband susceptibility is fulfilled and the spin resonance peak is formed at a frequency @xmath21 below @xmath22 with @xmath23 being the gap size @xcite . such simple explanation was indirectly questioned by the angle - resolved photoemission spectroscopy ( arpes ) results and recent measurements of gaps via andreev spectroscopy . latter clearly shows that there are at least two distinct gaps present in 11 , 122 , and 1111 systems @xcite and even three gaps in lifeas @xcite . larger gap ( @xmath0 ) is about 9mev and the smaller gap ( @xmath1 ) is about 4mev in baco122 materials . from arpes we know that electron fs sheets and the inner hole sheet are subject to opening the lager gap while the smaller gap is located at the outer hole fs @xcite . the very existence of the smaller gap rise the question what would be the spin resonance frequency in the system with two distinct gaps ? naive expectation is that the frequency shifts to the lower gap scale and @xmath24 . then the observed peak in ins in baco122 system at frequency @xmath25mev @xcite can not be the spin resonance since it is greater than @xmath26mev @xcite . thus the peak could be coming from the @xmath6 state @xcite , where it forms at frequencies _ above _ @xmath27 due to the redistribution of the spectral weight upon entering the superconducting state and a special form of scattering in the normal state . here we study this question in details and show that the naive expectation is wrong and that the true minimal energy scale is @xmath28 . latter is consistent with the maximal frequency of the observed peak in ins in baco122 and confirms that it is the true spin resonance evidencing the @xmath7 gap symmetry . the maximal energy scale is @xmath29 . whether the minimal or maximal energy scale will be realized depends on the relation between the exact band structure of a particular material and the wave vector of the spin resonance @xmath2 . to describe spin response in normal and superconducting states of febs , we use random phase approximation ( rpa ) with the local coulomb interactions ( hubbard and hund s exchange ) . in the multiband system , transverse dynamical spin susceptibility @xmath30 is the matrix in orbital ( or band ) indices . it can be obtained in the rpa from the bare electron - hole matrix bubble @xmath31 by summing up a series of ladder diagrams : @xmath32^{-1 } \hat\chi_{(0)+-}(\q,\omega ) , \label{eq : chi_s_sol}\end{aligned}\ ] ] where @xmath33 is the momentum , @xmath34 is the frequency , @xmath35 and @xmath36 are interaction and unit matrices in orbital ( or band ) space . exact form of @xmath35 and bare susceptibility @xmath31 depends on the underlying model . later we use two types of tight - binding models for the two - dimensional iron layer . first we study the four - band model of ref . with the following single - electron hamiltonian @xmath37 where @xmath38 is the annihilation operator of the @xmath10-electron with momentum @xmath39 , spin @xmath40 , and band index @xmath41 , @xmath42 are the on - site single - electron energies , @xmath43 is the electronic dispersion that yields hole pockets centered around the @xmath44 point , and @xmath45 is the dispersion that results in the electron pockets around the @xmath46 point of the 2-febz . parameters are the same as in ref . . in the superconducting state we assume either the @xmath6 state with @xmath47 or the @xmath7 state with @xmath48 . physical spin susceptibility @xmath49 obtained by calculating matrix elements @xmath50 via equation ( [ eq : chi_s_sol ] ) with the interaction matrix @xmath51 and with the bare spin susceptibility @xmath52 in the superconducting state ( see ref . for details ) . we assume here the effective hubbard interaction parameters to be @xmath53 and @xmath54 in order to stay in the paramagnetic phase @xcite . the model described above is simple enough to gain qualitative description of the spin response of superconductor with unequal gaps . but it lack for the orbital content of the bands that is important for the detailed structure of the susceptibility . that is why we also present results for the tight - binding model from ref . based on the fit to the dft band structure for lafeaso @xcite . the model includes all five iron @xmath10-orbitals ( @xmath55 , @xmath56 , @xmath57 , @xmath58 , @xmath59 ) enumerated by index @xmath60 and is given by @xmath61 d_{l \k \sigma}^\dagger d_{l ' \k \sigma } , \label{eq : h0}\ ] ] where @xmath62 is the annihilation operator of a particle with momentum @xmath39 , spin @xmath40 , and orbital index @xmath60 . later we use numerical values of hopping matrix elements @xmath63 and one - electron energies @xmath64 from ref . . this model for the undoped and moderately electron doped materials gives fs composed of two hole pockets , @xmath65 and @xmath66 , around the @xmath67 point and two electron pockets , @xmath68 and @xmath69 , centered around @xmath12 and @xmath13 points of the 1-fe bz . similar model for iron pnictides was proposed in ref . . the general two - particle on - site interaction would be represented by the hamiltonian @xcite : h_int & = & u _ f , m n_f m n_f m + u _ f , m < l n_f l n_f m + & & + j _ f , m < l _ , d_f l ^d_f m ^d_f l d_f m + & & + j _ f , m l d_f l ^d_f l ^d_f m d_f m . [ eq : hint ] where @xmath70 , @xmath71 is the number of particles operator at the site @xmath72 , @xmath73 and @xmath74 are the intra- and interorbital hubbard repulsion , @xmath75 is the hund s exchange , and @xmath76 is the so - called pair hopping . we choose the following values for the interaction parameters : @xmath77ev , @xmath78 , and make use of the spin - rotational invariance constraint @xmath79 and @xmath80 . green functions are diagonal in the band basis but not in the orbital basis . let us introduce creation and annihilation operators @xmath81 and @xmath82 of electrons with band index @xmath83 , in terms of which green functions are diagonal , @xmath84 . transformation from the orbital to the band basis is done via the matrix elements @xmath85 , @xmath86 , and for the transverse component of the bare spin susceptibility @xcite we have [ eq : chipmmu ] & & ^ll,mm_(0)+- ( , ) = -t _ , _ n , , , where @xmath87 and @xmath88 are matsubara frequencies , @xmath89 and @xmath90 are the normal and anomalous ( superconducting ) green s functions , respectively . components of the physical spin susceptibility @xmath91 are calculated using eq . ( [ eq : chi_s_sol ] ) with the interaction matrix @xmath92 from ref . . since calculation of the cooper pairing instability is not a topic of the present study , here we assume that the superconductivity is coming from some other theory and study either the @xmath6 state with @xmath93 or the @xmath7 state with @xmath94 , where @xmath83 is the band index . here we present results for susceptibilities at the wave vector @xmath95 as functions of frequency @xmath34 obtained via analytical continuation from matsubara frequencies ( @xmath96 with @xmath97 ) . imaginary part of bare and rpa spin susceptibilities in the four - band model ( [ eq : h04band ] ) are shown in fig . [ fig : imchi4band ] . first , we discuss result for equal gaps on electron ( @xmath98 , @xmath99 ) and hole ( @xmath100 , @xmath101 ) fss , @xmath102 . since @xmath103 describes particle - hole excitations and in the superconducting state all excitations are gapped below approximately @xmath104 ( at @xmath105 ) , then @xmath106 becomes finite only after that frequency . for the @xmath6 state , there is a gradual increase of the spin response for @xmath107 . for the @xmath7 state , @xmath2 connects fss with different signs of gaps , @xmath108 , and within rpa ( [ eq : chi_s_sol ] ) this results in the spin resonance peak divergence of @xmath109 at a frequency @xmath110 , see fig . [ fig : imchi4band ] , bottom panel . ( top ) and @xmath109 with @xmath111 in the 2-fe bz for the four - band model in the normal , @xmath6 and @xmath7 superconducting states . two cases of superconducting states are shown : equal @xmath23 s with @xmath102 , and unequal gaps with @xmath112 and @xmath113 . [ fig : imchi4band],title="fig : " ] ( top ) and @xmath109 with @xmath111 in the 2-fe bz for the four - band model in the normal , @xmath6 and @xmath7 superconducting states . two cases of superconducting states are shown : equal @xmath23 s with @xmath102 , and unequal gaps with @xmath112 and @xmath113 . [ fig : imchi4band],title="fig : " ] now let s consider the case of unequal gaps with a small gap scale on outer hole fs , @xmath113 , and a larger gap scale on all other fss , @xmath112 . as seen from fig . [ fig : imchi4band ] , top panel , for the @xmath7 state the discontinuous jump and , thus , @xmath114 , moved to lower frequencies . this new energy scale clearly tracked down in the @xmath6 state as the starting point of the susceptibility gradual increase . it is equal to @xmath115 , where @xmath0 and @xmath1 being the larger and smaller gap scales . consequently , the spin resonance peak in @xmath7 moved to lower frequencies , @xmath116 , see fig . [ fig : imchi4band ] , bottom panel . additional feature is the hump around the @xmath117 energy scale . note that the susceptibility in the @xmath6 state havent changed much compared to the equal gaps case . in the superconducting state , @xmath118 , as a function of momentum @xmath39 along the @xmath119 direction , i.e. @xmath120 . scattering wave vector @xmath2 entering the spin susceptibility is also shown . [ fig:5orbdelta ] ] with @xmath121 in the 1-fe bz for the five - orbital model in the normal , @xmath6 and @xmath7 superconducting states . two cases of superconducting states are shown : equal gaps with @xmath122 , and unequal gaps with @xmath123 and @xmath124 , where @xmath125 . latter case is shown in the inset , where gaps at the fs are plotted together with the wave vector @xmath2 . [ fig:5orbimchi ] ] to demonstrate where the new energy scale is coming from we turn our attention to the five - orbital model ( [ eq : h0 ] ) . its energy spectrum near the fermi level in the superconducting state , @xmath118 , is shown in fig . [ fig:5orbdelta ] . we consider here the case of unequal gaps with the smaller gap @xmath124 on the outer hole fs and larger gaps @xmath123 on inner hole and electron fss . to be consistent with the experimental data , we choose @xmath126 , see the inset in fig . [ fig:5orbimchi ] . naturally , the two energy scales , @xmath127 and @xmath128 , appear in the energy spectrum @xmath129 and they are connected with hole @xmath66 and electron @xmath130 bands , respectively . on the other hand , the susceptibility @xmath131 contains scattering _ between _ hole and electron bands with the wave vector @xmath2 . the energy gap that have to be overcome to excite electron - hole pair is the indirect gap with the scale @xmath4 . that is why spin excitations in the @xmath6 state start with the frequency proportional to the indirect gap @xmath132 , see fig . [ fig:5orbimchi ] . the same is true for the discontinuous jump in @xmath133 for the @xmath7 state it shifts to frequency @xmath134 . this , together with the corresponding @xmath135 singularity in @xmath136 , produce the spin resonance peak in rpa at frequency @xmath8 . such shift of resonance peak to lower frequencies compared to the equal gaps situation is seen in fig . [ fig:5orbimchi ] , where the spin response @xmath109 for the cases of equal and distinct gaps is shown . the changes in the band structure and/or doping level can result in the change of the indirect gap . in particular , since for the hole doping hole fss become larger the wave vector @xmath2 may connect states on the electron fs and on the _ inner _ hole fs . gaps on both these fss are determined by @xmath0 and thus the indirect gap will be equal to @xmath5 . this sets up a maximal energy scale for the spin resonance , i.e. @xmath29 . thus we conclude that depending on the relation between the wave vector @xmath2 and the exact fs geometry , the indirect gap in most febs can be either @xmath4 or @xmath5 . the peak in the dynamical spin susceptibility at the wave vector @xmath2 will be the true spin resonance if it appears below the indirect gap scale , @xmath8 . now we can compare energy scales extracted from arpes , andreev spectroscopy , and inelastic neutron scattering . latter gives peak frequency @xmath137mev in bafe@xmath138co@xmath139as@xmath140 with @xmath141k @xcite . for the same system , gap sizes extracted from arpes are @xmath142mev and @xmath143mev @xcite , and for a similar system with @xmath144k , @xmath145mev and @xmath146mev @xcite . gap sizes extracted from andreev spectroscopy are @xmath147mev and @xmath148mev in bafe@xmath149co@xmath150as@xmath140 with @xmath151k @xcite . evidently , @xmath152 and we can safely state that the peak in ins is the spin resonance . for the hole doped systems , peak frequency in ins is about @xmath153mev in ba@xmath154k@xmath155fe@xmath140as@xmath140 with @xmath156k @xcite . there is a slight discrepancy between gap sizes extracted from arpes and andreev spectra . former gives @xmath157mev and @xmath158mev in the same material with @xmath159k @xcite , thus @xmath152 . gap sizes from andereev spectroscopy are @xmath160mev and @xmath161mev in ba@xmath162k@xmath163fe@xmath140as@xmath140 with lower @xmath164k @xcite . in this case , @xmath165 but @xmath166 and we still can assume that the peak in ins is the spin resonance . however , in such a case definitive conclusion can be given only by the calculation of spin response for the particular experimental band structure . for more extensive review of available experimental data on @xmath167 and gap scales , see the supplemental material @xcite . on the separate note , we would like to mention that the appearance of a hump structure in the superconducting state at frequencies larger than the main peak frequency ( the so - called double resonance feature ) may be related to the @xmath128 energy scale , see fig . [ fig : imchi4band ] . such hump structure was observed in nafe@xmath168co@xmath169as @xcite and fete@xmath170se@xmath170 @xcite . somehow similar structure was found in polarized inelastic neutron studies of bafe@xmath171ni@xmath172as@xmath140 @xcite and ba(fe@xmath173co@xmath174)@xmath140as@xmath140 @xcite , but its origin may be related to the spin - orbit coupling @xcite rather than the simple @xmath128 energy scale . another explanation of the double resonance feature is related to the pre - existing magnon mode , i.e. the dispersive low - energy peak in underdoped materials is associated with the spin excitations of the magnetic order with the intensity enhanced below @xmath9 due to the suppression of the damping @xcite . we analysed the spin response of febs with two different superconducting gap scales , @xmath175 . spin resonance appears in the @xmath7 state below the indirect gap scale @xmath3 that is determined by the sum of gaps on two different fermi surface sheets connected by the scattering wave vector @xmath2 . in the @xmath6 state , spin excitations are absent below @xmath3 unless additional scattering mechanisms are assumed @xcite . for the fermi surface geometry characteristic to the most of febs materials , the indirect gap is either @xmath4 or @xmath5 . this gives the simple criterion to determine whether the experimentally observed peak in inelastic neutron scattering is the true spin resonance if the peak frequency @xmath21 is less than the indirect gap @xmath3 , then it is the spin resonance and , consequently , the superconducting state has the @xmath7 gap structure . comparison of energy scales extracted from ins , andreev spectroscopy , arpes and other techniques allowing to determine superconducting gaps , for most materials gives confidence that the observed feature in ins is the spin resonance peak . however , sometimes it is not always clear experimentally which gaps are connected by the wave vector @xmath2 . even without knowing this exactly , one can draw some conclusions . for example , if one of the gaps is @xmath0 , then there are three cases possible : ( 1 ) @xmath176 and the peak at @xmath21 is the spin resonance , ( 2 ) @xmath177 and the peak is definitely not a spin resonance , and ( 3 ) @xmath29 and the peak is most likely the spin resonance but the definitive conclusion can be drawn only from the calculation of the dynamical spin susceptibility for the particular experimental band structure . we would like to thank h. kontani , s.a . kuzmichev , t.e . kuzmicheva , v.m . pudalov , and i.s . sandalov for useful discussions . mmk is grateful to b. keimer and max - 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doped bafe@xmath178co@xmath179as@xmath140 system , nafe@xmath178co@xmath179as system , and fese , @xmath116 and , thus the peak in ins is the true spin resonance evidencing sign - changing gap . some hole doped ba@xmath178k@xmath179fe@xmath140as@xmath140 materials satisfy @xmath176 condition , and some satisfy @xmath180 condition . latter comes especially from newer tunneling @xcite and andreev reflection @xcite data reveling smaller gap values . the fact that @xmath180 is still consistent with the sign - changing gap , but as we mentioned before , the calculation of the spin response for the particular experimental band structure is required to make a final conclusion . 3 . the only case when @xmath181 is fete@xmath170se@xmath170 . according to our analysis , there should be no sign - changing gap structure . but before concluding this since this is the single case only , gap data coming from @xmath83sr @xcite should be double checked by independent techniques . interesting to note , that arpes in all cases gives gaps values larger than extracted from other techniques . natural question arise whether the arpes overestimates or all other methods underestimates superconducting gaps ? m. wang , m. yi , h.l . sun , p. valdivia , m.g . kim , z.j . xu , t. berlijn , a.d . christianson , s. chi , m. hashimoto , d.h . li , e. bourret - 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we study the spin resonance in superconducting state of iron - based materials within multiband models with two unequal gaps , @xmath0 and @xmath1 , on different fermi surface pockets . we show that due to the indirect nature of the gap entering the spin susceptibility at the nesting wave vector @xmath2 the total gap @xmath3 in the bare susceptibility is determined by the sum of gaps on two different fermi surface sheets connected by @xmath2 . for the fermi surface geometry characteristic to the most of iron pnictides and chalcogenides , the indirect gap is either @xmath4 or @xmath5 . in the @xmath6 state , spin excitations below @xmath3 are absent unless additional scattering mechanisms are assumed . the spin resonance appears in the @xmath7 superconducting state at frequency @xmath8 . comparison with available inelastic neutron scattering data confirms that what is seen is the true spin resonance and not a peak inherent to the @xmath6 state .

arxiv

the presence of impurities usually deeply modify the nature of the spectrum of a quantum system , and thus its coherence and transport properties . in the absence of interactions , if the impurity distribution is completely random , all states of the spectrum are exponentially localized in dimensions one ( 1d ) and two ( 2d ) , while a mobility edge exists in dimensions three ( 3d)@xcite . if the impurity positions are correlated , as for instance if it exists a minimum distance between the impurities @xcite , some delocalized states can appear in the spectrum . this was demonstrated in 1d in the context of the random dimer model ( rdm ) and of the dual random dimer model ( drdm ) @xcite . in 1d , the effects of correlated disorder was studied in different physical contexts ( see for instance @xcite ) . in 2d , the effect of correlations is almost unexplored , except for the case of a speckle potential @xcite , and for the case of pseudo-2d random dimer lattices with separable dimensions @xcite . correlations in speckle potentials may mimic the presence of a mobility edge @xcite , but in the thermodynamic limit all states are localized @xcite . random dimers introduce a set of delocalized states in pseudo-2d lattices @xcite as in 1d @xcite . from a statistical point of view , the main difference between these two models is the decay of the correlation function that is algebraic for the first and exponential for the second . this `` short - range '' feature of the random dimer model is at the basis of the delocalization mechanism . in interacting systems , the presence of disordered impurities gives rise to a remarkable richness of phenomena . for instance , the condensate and the superfluid fraction are modified by the presence of the disorder @xcite , and this can shift the onset of superfluidity @xcite , and , on lattice systems , can induce exotic phases such as the bose glass @xcite . in this work we study the effect of a short - range correlated disorder on a bose gas confined on a 2d square lattice . first we introduce a 2d generalization of the drdm ( 2d - drdm ) . in such a model , impurities can not be first neighbours and each impurity also modifies the hopping with its nearest neighbor sites . using a decimation and renormalization procedure @xcite , we show that , in the non - interacting regime , it exists a resonance energy at which the structured impurity is transparent and the states around this energy are delocalized . it is remarkable that this resonance energy does not depend on the system dimensionality and it is the same as the drdm in 1d @xcite . then , we consider the case of a weakly interacting bose gas confined on such a potential . within a gutzwiller approach , we show that the effect of the 2d - drdm is to drive the homogeneity of the ground state . the disorder induces a non - monotonic behavior of the condensate spatial delocalization and of the condensate fraction as a function of the disorder strength , and enhances both in correspondence of the resonance energy of 2d - drdm single - particle hamiltonian . we show that the dependence of such quantities on the interaction strength can be explained by including the effect of the healing length in the resonance condition discussion . the manuscript is organized as follows . in sec . [ sec : model ] , we introduce the 2d - drdm potential and we demonstrate its single - particle delocalization properties in the region of the spectrum around the resonance energy . the effect of such a potential on a weakly - interacting bose gas is studied in sec . [ sec : results ] , where we also introduce a suitable inverse participation ratio for our many - body system and study it for the case of the 2d - drdm potential and for an uncorrelated random disorder . moreover , we compute the density distribution and the condensate fraction as functions of the disorder strength . our concluding remarks in sec . [ sec : concl ] complete this work . we consider the tight - binding single - particle hamiltonian @xmath0 where @xmath1 are the on - site energies , @xmath2 the first neighbor hopping terms , @xmath3 the number of sites and @xmath4 denotes the sum over first neighbor sites . we focus on a 2d square lattice of linear dimension @xmath5 ( @xmath6 lattice sites ) , and compare the ordered lattice with @xmath7 and @xmath8 @xmath9 @xmath10 , as schematized in fig . [ fig1](a ) with a lattice where we introduce an impurity at the site 0 , @xmath11 that modifies the hopping terms involving this site , @xmath12 [ fig . [ fig1](b ) ] . schematic representation of ( a ) the unperturbed hamiltonian ; ( b ) the hamiltonian in the presence of a single impurity ; ( c ) the effective hamiltonian after decimation of the site 0 in the hamiltonian ( a ) ; ( d ) the effective hamiltonian after decimation of the site 0 in the hamiltonian ( b ) . ] with the aim of understanding the effect of the impurity , we consider the green s function @xmath13 projected on the subspace @xmath14 , including all sites except the site @xmath15 with coordinates @xmath16 . using a decimation and renormalization technique @xcite , it can be shown that @xmath17 with @xmath18 & \phantom{bla}{\rm site\;of\;the\;site\;}0\\[1 mm ] h_{aa}&\phantom{bla}{\rm elsewhere } \end{array}\right.\ ] ] where @xmath19 . the effective hamiltonian for the unperturbed case in fig . [ fig1 ] ( a ) , is schematically illustrated in fig . [ fig1 ] ( c ) ; whereas the effective hamiltonian for the case with a single impurity in fig . [ fig1 ] ( b ) , is illustrated in fig.[fig1 ] ( d ) . the subspace @xmath14 does not `` feel '' the presence of the impurity if @xmath20 ( @xmath21 ) remains the same in the absence or in the presence of the impurity , namely if @xmath22 the condition ( [ condition ] ) is satisfied if @xmath23 . if @xmath24 is an allowed energy of the system , namely if @xmath25 , at @xmath26 the impurity will not affect the eigenstate at this energy ( in the subspace @xmath14 ) . if we add other impurities in the system , as the one in fig . [ fig1 ] ( b ) , with the supplementary condition that on - site impurities can not occupy first neighbor sites ( fig . [ fig2 ] ) , one can repeat the same argument as above , properly redefining the subspace @xmath14 , and one obtains exactly the same condition ( [ condition ] ) imposing that _ all _ the @xmath27 impurities do not perturb the system ( the subspace @xmath14 ) . thus at @xmath26 , the impurities are transparent as in the 1d drdm @xcite . indeed , with this procedure , we are defining a 2d - drdm , where at each `` isolated '' impurity correspond a structure of 4 hopping terms forming a cross , as shown in fig . let us remark that this definition of the model provides the same condition ( [ condition ] ) independently from the dimensionality of the system @xcite . however our model is fully 2d and the hamiltonian can not be mapped onto two 1d drdm as opposed to ref . @xcite . schematic representation of the 2d drdm . ] with the aim of analyzing the localization properties of this model , we consider the inverse participation ratio ( ipr ) @xmath28 the symbol @xmath29 denotes the average over different disorder configurations , and @xmath30 the wavefunction on site @xmath31 and at energy @xmath32 . if @xmath33 is an eigenvalue of the system and @xmath34 is an extended state , then @xmath35 decreases as a function of @xmath5 . on the other side , if @xmath36 is a localized state , then @xmath35 does not depend on @xmath5 ( if @xmath5 is larger than the localization length ) . in fig . [ fig3 ] we show the behavior of @xmath37 and @xmath38 ( column left and right respectively ) , for the hamiltonian illustrated in fig . [ fig2 ] . [ cols="^,^ " , ] we consider three set of parameters , ( i ) @xmath39 and @xmath40 , ( ii ) @xmath41 and @xmath42 , and ( iii ) @xmath43 and @xmath44 that give the same resonance energy , @xmath45 . in all the three cases , the curves @xmath46 collapse around @xmath26 meaning that the states are delocalized in this energy region . moreover , due to the large strength of the disorder , the spectrum varies considerably for the cases ( ii ) and ( iii ) , and an energy gap appears in ( iii ) . the inverse participation ratio , eq . ( [ eq : ipr ] ) , in two dimensions has the following asymptotic behavior @xcite @xmath47 thus , the asymptotic behavior of the function @xmath48 is @xmath49 with @xmath50 for localized states , and @xmath51 for extended states . in fig . [ fig : esponenti ] we have analyzed the exponent @xmath52 as a function of the energy for the set of parameters ( iii ) . we observe a high - energy band of localized states that has been created by the disorder ; the original ( without noise ) band has been distorted , and the states at its boundaries are localized . the center of the band , around @xmath24 , is mainly constituted of extended states . the width of the feature around @xmath24 corresponds to the width of the resonance dip of the inverse participation ratio at this energy value ( fig . [ fig3 ] ) . ) as a function of the energy for @xmath43 and @xmath44 . the exponent has been obtained using calculations for lattice - linear dimensions @xmath53 averaged over 20 realizations , the error bars correspond to the standard deviation of the fit of the data to eq . ( [ eq : asympatty ] ) . the vertical dashed line indicates @xmath24 . ] these results confirm that our 2d extension of the drdm introduced by dunlap and collaborators in ref . @xcite for 1d systems introduces a set of delocalized states even at higher dimensions . we now consider the case of weakly interacting bosons in the presence of the potential defined in sec . [ sec : model ] . this system is described by the bose - hubbard hamiltonian in the grand canonical ensemble @xmath54 where @xmath55 is the creation operator defined at the lattice site @xmath31 , @xmath56 , @xmath57 the interparticle on - site interaction strength , and @xmath58 denotes the chemical potential fixing the average number of bosons . we use a gutzwiller approach to find the ground state wavefunction for a given set of parameters and average number of particles . the gutzwiller ansatz is given by the site product wavefunction in the occupation number representation @xmath59 where @xmath60 are the probability amplitudes of finding @xmath61 particles on site @xmath31 . the ansatz provides an interpolating approximation correctly describing both the bose - condensed and mott - insulating phases for low and high @xmath57 , respectively , in dimensions larger than one . in addition , the approximation becomes exact for all @xmath57 in the limit of infinite dimensions @xcite . we minimize the average energy given by hamiltonian ( [ eq : bhgw ] ) as a function of the set of amplitudes @xmath60 with the normalization and average number of particle constraint for at least 30 disorder realizations for each set of parameters . the minimization is done using standard conjugate - gradient and/or broyden - fisher techniques @xcite which provides reasonable performance for moderate lattice sizes . to quantify the extent of delocalization of the ground state @xmath62 in the interacting regime , we decompose it onto the localized basis @xmath63 , @xmath64 , representing the distribution of a homogeneous condensate with average density @xmath65 on the lattice @xcite . we define the many - body ground - state ipr @xmath66 with respect to this basis as @xmath67 @xmath66 measures the homogeneity of the ground state in the condensation regime : the smaller @xmath66 the more spatially delocalized is the condensate . in fig . [ fig : igs_l20 ] we show the behavior of @xmath66 as a function of @xmath68 , by fixing @xmath69 , @xmath70 and @xmath71 , for several values of @xmath72 . ( color online ) @xmath73 as a function of @xmath74 for @xmath69 , @xmath70 and @xmath71 particles per site . the different curves correspond to different values of @xmath72 as indicated in the figure . the filled symbols correspond to the 2d - drdm potential and the empty symbols correspond to the un - rand potential . ] we compare the case of @xmath75 of correlated impurities @xmath27 with the one of the same percentage of uncorrelated impurities , where there is no restriction for the position distribution of the on - site impurities @xmath68 and no correlations between them and the additional hopping @xmath72 ( un - rand ) . we note that due to the correlations present in the 2d - drdm the maximum percentage of allowed impurities is @xmath76 ( in this limit the system would be an ordered checkerboard ) . we can observe that , in the case of the 2d - drdm potential , @xmath66 has a minimum as a function of @xmath68 , whose position depends on the value of @xmath72 . this non - monotonic behavior is a signature of the resonance induced by the correlations of the disordered potential . indeed , it disappears for the case of the un - rand potential and for large values of @xmath72 ( strong disorder ) . the dip in the @xmath66 for the un - rand potential and weak disorder ( @xmath40 ) indicates that some drdm impurities may still statistically appear , in the absence of correlations . the effect of such impurities is not fully destroyed by the other defects only if the strength of the disorder is weak . in the perturbative regime for negligible interactions , one would expect that correlations modify the ground state if @xmath77 , @xmath78 being the ground - state energy per particle , which corresponds to @xmath79 in the weak disorder regime . this condition , that can be written @xmath80 , \label{cond - pat}\ ] ] determines the location of the minimum of @xmath66 at @xmath81 for @xmath40 , @xmath82 for @xmath42 , and @xmath83 for @xmath44 . however , in the limit of strong disorder , due to the interactions these values strongly differ from those shown in fig . [ fig : igs_l20 ] . in fact , we calculate @xmath66 for smaller values of @xmath65 and verify that the minimum location of @xmath66 depends on @xmath24 and that the shift observed is indeed an effect of the interactions . the results are illustrated in fig . [ fig : igs_l20_n ] , where we focus on the case @xmath42 . ( color online ) @xmath73 as a function of @xmath74 for @xmath69 , @xmath84 and @xmath42 . the different curves correspond to different values of the average density @xmath65 as indicated in the figure . all the curves correspond to the 2d - drdm potential . the vertical dashed line indicates the non - interacting resonance condition given in eq . ( [ cond - pat ] ) . ] by decreasing the value of @xmath65 , the minimum position @xmath85 of @xmath66 shifts from @xmath86 to about @xmath87 as expected by the perturbative argument . this shift can be understood as follows . the interactions introduce the so - called healing length @xmath88 @xcite that represents a coherence length over which the system feels the effect of an impurity , or in other words , the distance a site affects its neighborhood . for @xmath70 and @xmath65 from 20 to 5 , the value of @xmath89 ranges approximately from @xmath90 to 3 lattice spacing @xmath91 , which shows that , already for this @xmath57 value , the role of the interactions is important , effectively reducing the coherence length . to quantify this effect , we can partition the system into independent boxes of dimension @xmath92 , and use a mode - matching argument to determine their ground states : the condensate is more homogeneous if the lowest eigenvalue of each box is the same despite the presence of an impurity . ( color online)boxes of different sizes , in the presence and in the absence of an impurity . ] therefore , this mode - matching argument fixes the value of @xmath68 . for the case @xmath70 and @xmath71 , @xmath93 , and this gives @xmath94 while for @xmath95 , @xmath96 , and we expect to find @xmath97 , in good agreement with the results showed in fig . [ fig : igs_l20_n ] . namely , the larger is @xmath89 , the better we recover the non - interacting condition eq . ( [ cond - pat ] ) . this effect is summarized in table [ tab : uloco ] . ccc @xmath98 & fig . & @xmath68 + + @xmath91 & [ figboxes](a)&@xmath99 $ ] + @xmath100 & [ figboxes](b ) & @xmath101 $ ] + @xmath102 & [ figboxes](c ) & @xmath103 $ ] + we remark that this mode - matching condition is equivalent to match the resonance energy @xmath24 with the lowest eigenvalue of the unperturbed system of size @xmath92 . these simple arguments , allow us to understand the shift of @xmath68 as a function of the interaction energy @xmath104 and the role of the structured impurities in the presence of the interactions . we study the scaling behavior of @xmath73 with respect to @xmath5 . analogously to the case of the single - particle ipr @xmath105 [ see eq . ( [ eq : asympatty ] ) ] , we expect that @xmath106 with @xmath50 for a condensate localized on few sites , and @xmath51 for a homogeneous extended condensate . the behavior of @xmath73 for different values of @xmath5 is shown in fig . [ fig : igs_l ] . ( color online ) @xmath73 as a function of @xmath74 for @xmath42 , @xmath70 and @xmath71 particles per site . the different curves correspond to different values of @xmath5 as indicated in the figure . the filled symbols correspond to the 2d - drdm potential and the empty symbols correspond to the un - rand potential . ] we observe that the minima , corresponding to different system sizes , collapse all together , meaning that the ground state corresponds to a spatial homogeneous condensate in the parameter regime where the correlations are dominant . at lower values of @xmath68 , @xmath73 scales as @xmath107 , and larger values of @xmath68 , @xmath73 scales as @xmath108 , with @xmath109 and @xmath110 @xmath111 . this sort of `` super - delocalization '' , in the low @xmath68 region , is determined by the large value of @xmath72 that compensates , in the structured impurities , the effect of the site defect . indeed , we observe an analogous behavior for the un - rand potential . for such a potential , where the effect of @xmath72 is no more dominant , all the curves collapse together . thus we expect that in this region the effect of the uncorrelated impurities on the ground state density distribution does not depend on the system size . with the aim of characterizing the ground state configurations in the different regions , we show in figs . [ fig : denlatt2.1][fig : denlatt15.1 ] the spatial density distribution @xmath61 for @xmath69 , @xmath71 at @xmath112 ( fig . [ fig : denlatt2.1 ] ) , @xmath113 ( fig . [ fig : denlatt6.6 ] ) , and @xmath114 ( fig . [ fig : denlatt15.1 ] ) together with a pattern showing the locations of impurities . ( color online ) lattice density plots together with site and bond impurities locations for @xmath42 , @xmath115 and drdm disorder ( top ) and un - rand ( bottom).,title="fig : " ] + ( color online ) lattice density plots together with site and bond impurities locations for @xmath42 , @xmath115 and drdm disorder ( top ) and un - rand ( bottom).,title="fig : " ] ( color online ) same as fig . [ fig : denlatt6.6 ] for @xmath116.,title="fig : " ] + ( color online ) same as fig . [ fig : denlatt6.6 ] for @xmath116.,title="fig : " ] ( color online ) same as fig . [ fig : denlatt6.6 ] for @xmath117.,title="fig : " ] + ( color online ) same as fig . [ fig : denlatt6.6 ] for @xmath117.,title="fig : " ] the addition of a hopping term @xmath72 favors the delocalization of the density both for the 2d - drdm and un - rand disorders . however , in the case of the 2d - drdm , it is more beneficial as it tends to partially compensate the decrease in the density caused by the site impurity , reducing the decrease by means of the structured disorder . for small values of @xmath68 ( see fig . [ fig : denlatt2.1 ] ) , in the region where the effect of @xmath72 is dominant , the density in the impurity regions is even larger with respect to the density elsewhere . for large values of @xmath68 ( see fig . [ fig : denlatt15.1 ] ) , the effect of both types of disorder is similar as the change in the on - site energies dominates . this limit gives rise to a strongly depleted density at the impurity location plus a rather uniform background . the largest differences among the 2d - drdm and un - rand results are seen at the minimum of @xmath66 ( see fig.[fig : denlatt6.6 ] ) , where we can clearly observe a more homogeneous density spread over the lattice ( lower @xmath66 ) , and a consequently larger delocalization for the 2d - drdm than for the un - rand potential . the density behavior determines the condensate fraction @xmath118 , as shown in fig . [ fig : n0_l ] . ( color online ) condensate fraction @xmath119 as a function of @xmath74 for @xmath42 , @xmath70 and @xmath71 particles per site . the different curves correspond to different values of @xmath5 as indicated in the figure . the filled symbols correspond to the 2d - drdm potential and the empty symbols correspond to the un - rand potential . ] in correspondence of the minimum of the function @xmath66 , we observe that the condensate fraction @xmath119 does not depends of the system size , in the presence of the 2d - drdm potential . the resonance condition minimizes the fluctuations with respect the chosen homogeneous basis @xmath63 and fixes @xmath119 . at lower value of @xmath68 , we observe a `` super - delocalization '' ( @xmath73 scales as @xmath107 ) , and for both the 2d - drdm and the un - rand potentials , the large value of @xmath72 enhances the coherence and @xmath119 increases with system size . at larger values of @xmath68 , where @xmath73 scales as @xmath108 , the 2d - drdm impurities create holes in the system , and @xmath119 decreases with system size . for the case of the un - rand potential , one can observe a monotonic behavior of @xmath119 as a function of @xmath68 . as for the case of the 2d - drdm , the region where all the curves @xmath73 collapse together corresponds to a region where @xmath119 does not depend on the system size . the difference with the 2d - drdm is a larger decrease of @xmath119 in this region . for 2d - drdm , only one value of @xmath68 has this peculiarity , and the maximum position of the condensate fraction foregoes this point . let us remark that the minimum of @xmath120 corresponds to the minimum deviation with respect a homogeneous condensate , and , because of border effects , this target state is not necessarily the one that ensures a maximum value of @xmath119 in finite systems . the predicted condensate fraction enhancement for the drdm at low @xmath68 , being it very small , could be very difficult to be measured . however the non - diminishing of the coherence in a range of about @xmath121 should be observable , and could be directly compared with the result for the un - rand where the decrease of the coherence should be sizable . in summary , we introduce a correlated disorder model that is the natural extension of the drdm in 2d . we show that , in the non - interacting regime , such a disorder introduces some delocalized states if the resonance energy characterizing these structures belongs to the spectrum of the unperturbed system . in the presence of weak interactions , the 2d - drdm drives the density spatial fluctuations . by means of a mode - matching argument that includes the effect of the interactions , we show that the resonance energy is at the origin of these phenomena . a direct consequence is a non - monotonic behavior of the condensate fraction as a function of the disorder strength , and its enhancement for values close to the resonance condition . this work shows that short - range correlations in a disordered potential can modify and enhance the coherence of a many - body system in the weak interacting regime . such effects could be measured in the context of ultracold atoms with an accurate measurement of the density and coherence , via , for instance , a fringes contrast interference experiment . our results could also be extended to homogeneous systems provided one is able to engineer suitable impurities that are transparent for a given energy .

we introduce a two - dimensional short - range correlated disorder that is the natural generalization of the well - known one - dimensional dual random dimer model [ phys . rev . lett * 65 * , 88 ( 1990 ) ] . we demonstrate that , as in one dimension , this model induces a localization - delocalization transition in the single - particle spectrum . moreover we show that the effect of such a disorder on a weakly - interacting boson gas is to enhance the condensate spatial homogeneity and delocalisation , and to increase the condensate fraction around an effective resonance of the two - dimensional dual dimers . this study proves that short - range correlations of a disordered potential can enhance the quantum coherence of a weakly - interacting many - body system .

arxiv

in a graph @xmath14 , an independent set @xmath15 is a subset of the vertices of @xmath0 such that no two vertices in @xmath15 are adjacent . the independence number @xmath2 is the cardinality of a largest set of independent vertices and an independent set of size @xmath2 is called an @xmath16-set . the maximum independent set problem is to find an independent set with the largest number of vertices in a given graph . it is well - known that this problem is np - hard @xcite . therefore , many attempts are made to find upper and lower bounds , or exact values of @xmath2 for special classes of graphs . this paper is aimed toward studying this problem for the generalized petersen graphs . for each @xmath7 and @xmath8 @xmath17 , a generalized petersen graph @xmath18 , is defined by vertex set @xmath19 and edge set @xmath20 ; where @xmath21 and subscripts are reduced modulo @xmath7 . an induced subgraph on @xmath22-vertices is called the inner subgraph , and an induced subgraph on @xmath23-vertices is called the outer cycle . + in addition , we call two vertices @xmath24 and @xmath25 as twin of each other and the edge between them as a spoke . in @xcite , coxeter introduced this class of graphs . later watkins @xcite called these graphs `` generalized petersen graphs '' , @xmath18 , and conjectured that they admit a tait coloring , except @xmath26 . this conjecture later was proved in @xcite . since 1969 this class of graphs has been studied widely . recently vertex domination and minimum vertex cover of @xmath18 have been studied . for more details see for instance @xcite , @xcite , @xcite and @xcite . a set @xmath27 of vertices of a graph @xmath28 is called a vertex cover of @xmath0 if every edge of @xmath0 has at least one endpoint in @xmath27 . a vertex cover of a graph @xmath0 with the minimum possible cardinality is called a minimum vertex cover of @xmath0 and its size is denoted by @xmath29 . in @xcite and @xcite @xmath30 , has been studied . since for every simple graph @xmath0 , @xmath31 @xcite , their results imply the following results for @xmath3 , and @xmath9 : * @xmath32 * for all @xmath33 , @xmath34 . * for all @xmath35 , @xmath36 * if both @xmath7 and @xmath8 are odd , then @xmath37 . + also , if @xmath38 , then @xmath39 . * @xmath40 @xmath41 if and only if @xmath7 is even and @xmath8 is odd . * for all even @xmath8 , we have * * if @xmath42 then @xmath43 . * * if @xmath44 then @xmath45 . * for all odd @xmath7 , we have @xmath46 , where @xmath47 . recently , fox et al . proved the following results in @xcite : * for all @xmath48 , @xmath49 * for any integer @xmath50 , we have that @xmath51 * if @xmath52 are integers with @xmath7 odd and @xmath8 even , then @xmath40 @xmath53 , where @xmath47 . * if @xmath52 are even , then @xmath40 @xmath54 , where @xmath47 . notice that the problem of finding the size of a maximum independent set in the graph @xmath18 is trivial for even @xmath7 and odd @xmath8 , since @xmath18 is a bipartite graph . for odd @xmath7 and @xmath8 , @xmath18 is not bipartite but we can remove a set of @xmath55 edges from @xmath18 to obtain a bipartite graph . thus in this case we have @xmath56 . so , for odd @xmath8 , we have upper and lower bounds for @xmath40 that are at most @xmath55 away from @xmath7 . in contrast , for even @xmath8 , @xmath18 has a lot of odd cycles . in fact , the number of odd cycles in @xmath18 is at least as large as @xmath13 . this observation shows that for even @xmath8 , the graph @xmath18 is far from being a bipartite graph and as we see in continuation , we need more complicated arguments for finding lower and upper bounds for @xmath40 compared to the case that @xmath8 is an odd number . this paper is organized as follows . in section @xmath57 , we provide an upper bound for @xmath40 for even @xmath58 . in section @xmath59 , we present some lower bounds when @xmath8 is even . in both section @xmath57 and section @xmath59 we compare our bounds with the previously existing bounds . some exact values for @xmath40 are given in section @xmath60 by applying results presented in sections @xmath57 and section @xmath59 . finally , in section @xmath61 we prove behsaz - hatami - mahmoodian s conjecture for some cases by using known lower bounds . we checked the conjecture with our table for @xmath11 , and it had no inconsistency . in this section we present an upper bound for @xmath40 when @xmath62 is even and we will show that the presented upper bound is equal to @xmath40 in some cases . our upper bound is better than the upper bound given by behsaz et.al . in @xcite . + for @xmath63 , we call the set @xmath64 a @xmath65-segment and we denote it by @xmath66 . let @xmath67 $ ] be the subgraph of @xmath18 induced by @xmath66 . let @xmath68 be the set of all maximum independent sets of @xmath18 . for every @xmath69 we denote by @xmath70 the number of @xmath65-segments @xmath66 for which @xmath71 . + define @xmath72 . since @xmath68 is nonempty , @xmath73 is also nonempty . let @xmath74 [ note ] for any @xmath75 , @xmath76 if and only if @xmath77 . [ type ] for any @xmath69 , we say @xmath66 is of type @xmath78 with respect to @xmath79 if @xmath80 , of type @xmath57 with respect to @xmath79 if @xmath81 , and of type @xmath59 with respect to @xmath79 if @xmath82 . + let @xmath83 . for a given @xmath84 , we say @xmath66 is of special type @xmath57 with respect to @xmath79 , if @xmath85 and @xmath86 becomes an independent set for @xmath67 $ ] . since @xmath67 $ ] has a perfect matching of spokes @xmath87 , @xmath88 . so every @xmath66 is one of the above types . + note that @xmath89 . [ lemma type 1 ] if @xmath8 is an even number then @xmath90 ) = 2k$ ] and @xmath67 $ ] has a unique @xmath16-set shown in figure [ type 1 ] . + as a type @xmath78 segment.,scaledwidth=80.0% ] @xmath67 $ ] has a perfect matching @xmath91 . so @xmath90 ) \leq \frac{|v(g[i_t])|}{2 } = 2k.$ ] on the other hand , figure [ type 1 ] is an example of an independent set of @xmath67 $ ] of size @xmath65 . so @xmath90)= 2k$ ] . + to show that @xmath67 $ ] has a unique @xmath16-set , let @xmath79 be an @xmath16-set of @xmath92)$ ] . since @xmath90)= 2k$ ] , @xmath93 and @xmath79 must contain precisely one vertex from each edge @xmath94 where @xmath95 . notice that the set of @xmath23-vertices of @xmath92)$ ] induces a path of length @xmath65 . therefore @xmath96 the set of @xmath22-vertices of @xmath92)$ ] induces a matching of size @xmath8 . this means that @xmath97 these two observations show that any @xmath16-set @xmath79 of @xmath67 $ ] has @xmath8 vertices from @xmath23-vertices and @xmath8 other vertices from @xmath22-vertices of @xmath98).$ ] moreover , every such @xmath79 contains precisely one vertex from each edge @xmath99 where @xmath100 and @xmath101 where @xmath102 . now , consider two cases : + case 1 : @xmath103 . + in this case , @xmath104 and @xmath105 are forced not to be in @xmath79 . so @xmath106 is forced to be in @xmath79 . then @xmath107 and @xmath108 are forced not to be in @xmath79 and this forces @xmath109 and @xmath110 to be in @xmath79 . since @xmath110 is in @xmath79 , @xmath111 . therefore @xmath112 , so @xmath113 and thus @xmath114 so , we showed that if @xmath103 then @xmath115 too . now , if we repeat the same argument for @xmath116 instead of @xmath117 , we can deduce that @xmath118 and by a simple induction , it follows that @xmath119 for any @xmath120 . particularly , @xmath121 therefore @xmath122 this shows that @xmath123 . hence @xmath124 and @xmath125 . so @xmath126 but we already showed that @xmath109 is forced to be in @xmath79 . this contradiction shows that there is no type @xmath78 @xmath66 for which @xmath127 + case 2 : @xmath128 . + in this case , similar to the argument in case 1 , each vertex is either forced to be in @xmath79 or it is forced not to be in @xmath79 . so , there is a unique pattern for @xmath129 when @xmath130 . since the pattern shown in figure [ type 1 ] is an instance of an independent set of size @xmath65 for @xmath67 $ ] , it is the unique pattern for such an independent set . lemma [ lemma type 1 ] guarantees that there is a unique pattern for @xmath131 , if @xmath66 is of special type @xmath57 with respect to @xmath79 . [ lemma uv ] for every @xmath69 , if @xmath132 then @xmath133 , and @xmath134 also , if @xmath135 is a special type @xmath57 segment with respect to @xmath79 then @xmath136 and @xmath137 . if @xmath132 then @xmath80 . so by lemma [ lemma type 1 ] , there is a unique pattern for @xmath131 . based on this pattern , @xmath138 and @xmath139 . therefore @xmath136 and @xmath137 , since @xmath79 is an independent set of vertices of @xmath18 . a similar argument shows that @xmath140 and @xmath141 . the proof of the second part of the lemma is similar . [ corollary 1 ] if @xmath132 then @xmath142 notice that if @xmath143 then for any edge @xmath144)$ ] either @xmath145 or @xmath146 . since @xmath132 , lemma [ lemma uv ] implies that @xmath136 and @xmath147 . on the other hand , @xmath148)$ ] for @xmath149 . thus @xmath150 [ maintheorem ] @xmath151 for any even number @xmath62 and any integer @xmath152 . let @xmath153 . we consider two cases . + case @xmath78 : @xmath154 . + in this case @xmath155 . so @xmath156 for any @xmath157 . if we add all of these @xmath7 inequalities we get : + @xmath158 on the other hand @xmath159 , since every element of @xmath160 is contained in precisely @xmath65 of the sets @xmath66 . thus : @xmath161 case @xmath57 : @xmath162 + in this case @xmath163 . similar to the inequality [ e : 1 ] we have : + @xmath164 so , to prove the theorem , it suffices to show that there exists @xmath165 such that @xmath166 . + if we can show that for any @xmath167 , there exists an @xmath168 so that @xmath169 , then it follows that @xmath166 . + on the contrary , suppose that there exists @xmath170 in such a way that in the sequence @xmath171 before we see an element of @xmath172 , we see an element of @xmath173 . without loss of generality we can assume that @xmath174 by lemma [ lemma type 1 ] , @xmath175 is of the form depicted in figure [ type 11 ] . + as a type @xmath78 segment.,scaledwidth=80.0% ] since @xmath176 , by corollary [ corollary 1 ] , @xmath177 based on our assumption , @xmath178 in particular , @xmath179 . since @xmath180 , by lemma [ lemma uv ] we have @xmath181 . on the other hand , we know that @xmath160 must have one vertex from each edge @xmath182 where @xmath183 since @xmath184 , either @xmath185 or @xmath186 . but notice that @xmath187 is adjacent to @xmath188 which is in @xmath160 , for @xmath189 . thus , @xmath190 and @xmath185 must be in @xmath160 . this means that @xmath191 . now , define @xmath192 . one can easily see that @xmath193 . based on the choice of @xmath153 , @xmath194 . therefore , there must be an index @xmath195 so that @xmath196 . since @xmath160 and @xmath197 agree on every element except @xmath198 and @xmath199 , the only candidate for @xmath200 is @xmath201 . so @xmath202 and @xmath203 . moreover @xmath204 and @xmath205 . thus @xmath206 . by proposition [ note ] , @xmath207 . notice that if any of @xmath208 are of type @xmath209 with respect to @xmath197 , they are of type @xmath209 with respect to @xmath160 , as well . so , in the sequence @xmath208 , any type @xmath59 segment with respect to @xmath197 appears after an element of type @xmath78 with respect to @xmath197 . since @xmath202 by corollary [ corollary 1 ] , @xmath210 . then from our assumption @xmath211 . + this means that the same argument can be applied to @xmath197 and if we define @xmath212 , then @xmath213 . if we consecutively repeat this argument for @xmath214 where @xmath215 and @xmath216 , then we observe that @xmath217 and @xmath218 for @xmath219 , and none of @xmath220 are of type @xmath78 with respect to @xmath160 . also , @xmath221 for @xmath222 are of special type @xmath57 with respect to @xmath223 . since @xmath223 and @xmath160 agree on the @xmath224 , then @xmath221 for @xmath222 are of special type @xmath57 with respect to @xmath160 . + in other words , if @xmath225 belongs to @xmath226 and the next element of @xmath226 appears before the first element of @xmath227 in the sequence @xmath228 then all of @xmath229 are special type @xmath57 with respect to @xmath160 . in particular , @xmath230 is of special type @xmath57 with respect to @xmath160 . + as @xmath231 , by lemma [ lemma uv ] , @xmath232 and since @xmath233 and @xmath160 agree on the @xmath234 we conclude that @xmath235 . + now consider three cases : * @xmath236 : + since @xmath230 is of special type @xmath57 with respect to @xmath160 , by lemma [ lemma uv ] , we have @xmath237 . this is a contradiction as we assumed @xmath180 and therefore @xmath238 . * @xmath239 : + since @xmath180 , by lemma [ lemma uv ] , @xmath240 . also we know that @xmath241 . thus , @xmath242 is of type @xmath59 with respect to @xmath160 and none of @xmath243 are of type @xmath78 with respect to @xmath160 which is a contradiction . * @xmath244 : + @xmath225 is of type @xmath78 with respect to @xmath160 and for every @xmath245 , @xmath221 is of special type @xmath57 with respect to @xmath160 . in particular , @xmath230 is of special type @xmath57 with respect to @xmath160 , and therefore @xmath246 , ( see figure [ type 2 ] ) . on the other hand , @xmath247 as @xmath180 , and since @xmath248 , @xmath249 is adjacent to @xmath250 . this is a contradiction . so in all the cases , we get a contradiction which means , after any type @xmath78 segment @xmath251 , a type @xmath59 segment @xmath252 will appear before we see another type @xmath78 segment . this means that @xmath253 and the theorem follows , as we argued earlier . in this section we introduce some lower bounds for @xmath40 where @xmath8 is even and @xmath189 . + here we explain a construction for an independent set in @xmath18 for even numbers @xmath7 and @xmath8 . it happens that for every even @xmath254 , our lower bound is equal to the actual value , using a computer program for finding the independence number in @xmath18 . [ nk even ] if @xmath7 and @xmath8 are even and @xmath62 then : @xmath255 where @xmath200 is the remainder of @xmath7 modulo @xmath65 . we partition the vertices of @xmath18 into @xmath256 @xmath65-segments and one @xmath200-segment . since @xmath7 and @xmath8 are even numbers , @xmath200 is also an even number and it is straightforward to see that if we choose a subset of the form shown in figure [ type 2 ] , from each @xmath65-segment they form an independent set @xmath160 of size @xmath257 . then we try to extend this independent set by adding more vertices from the remaining @xmath200-segment . without loss of generality , we may assume that the @xmath200-segment consists of the vertices @xmath258 . consider two cases : * @xmath259 : + in this case the set @xmath260 is an independent set of size @xmath261 . * @xmath262 : + in this case the set : + @xmath263 is an independent set of size @xmath264 as a special type @xmath57 segment.,scaledwidth=80.0% ] in the next theorem we establish a lower bound for @xmath12 for odd @xmath7 and even @xmath8 . + [ n odd k even ] if @xmath7 is odd and @xmath62 is even then we have : @xmath265 where @xmath200 is the remainder of @xmath7 modulo @xmath65 . we construct an independent set for the graph @xmath18 . similar to the proof of theorem [ nk even ] , first we partition the vertices of the graph into @xmath266 @xmath65-segments and a remaining segment of size @xmath200 . without loss of generality , we can assume that the last @xmath65-segment starts from the first spoke and the remaining segment starts from the @xmath267-st spoke and finishes at the @xmath268-th spoke . we also label the @xmath65-segments with indices @xmath269 . from each of @xmath65-segments @xmath270 , we choose @xmath271 vertices as shown in figure [ type 2 ] . we also choose the following vertices from the last @xmath65-segment and the remaining @xmath200-segment : * @xmath272 : + @xmath273 * @xmath274 + @xmath275 * @xmath276 + @xmath277 one can easily check that in each case , the given set is an independent set of size specified in the theorem . notice that the upper bound given in theorem [ maintheorem ] and the lower bound in theorem [ nk even ] , and theorem [ n odd k even ] are very close to each other for every fixed even @xmath189 . more precisely , we have the following corollary : if @xmath278 is an even number then @xmath279 notice that our lower bounds are considerably better than the lower bounds obtained in @xcite and @xcite . in this section , we will find the exact value of @xmath40 for some pairs of @xmath52 . + [ k=4 ] if @xmath280 , then : + @xmath281 @xmath282 this result is straight consequence of theorems [ maintheorem ] , [ nk even ] , and [ n odd k even ] . notice that for @xmath283 and @xmath284 or @xmath285 , the upper bound and lower bound differ by @xmath78 . in fact , for @xmath286 the exact of @xmath3 is the same as our lower bound as we checked by computer . if @xmath278 is an even number and @xmath287 or @xmath288 then @xmath289 . this assertion is trivial consequence of theorems [ maintheorem],[nk even ] and [ n odd k even ] . in fact the upper bound and lower bounds we have for @xmath3 are identical in these cases . [ conj 1 ] ( @xcite ) . for all @xmath7 , @xmath8 we have @xmath290 . + notice that , since @xmath291 , this conjecture is equivalent to @xmath292 . the above conjecture is valid in the following cases : * @xmath293 . * @xmath7 is even and @xmath8 is odd . * @xmath52 are odd and @xmath294 . * @xmath52 are even . * @xmath7 is odd , @xmath8 is even and @xmath295 . * this case is a straight consequence of ( i ) , ( ii ) , ( iii ) , proposition [ k=4 ] and ( viii ) . * in this case @xmath18 is a bipartite graph and @xmath296 . * @xmath297 ( @xcite ) . for @xmath298 this lower bound is greater than @xmath299 . * let @xmath300 where @xmath301 and @xmath302 . we consider the following subcases : * * if @xmath259 and @xmath303 then by theorem [ nk even ] , @xmath304 for any @xmath305 . for @xmath306 conjecture holds based on the information provided in table @xmath78 . * * if @xmath307 and @xmath308 then by theorem [ nk even ] , @xmath309 for any @xmath310 . for @xmath311 , conjecture follows from @xmath312 . * * if @xmath262 and @xmath303 then by theorem [ nk even ] , @xmath313 for any @xmath314 . for @xmath315 conjecture holds based on the information provided in table @xmath78 . * * if @xmath316 and @xmath308 then by theorem [ nk even ] , @xmath317 for any @xmath318 . for @xmath319 , conjecture follows from @xmath312 . * similar to the previous part , let @xmath300 where @xmath301 and @xmath302 . we consider the following subcases : * * if @xmath320 and @xmath303 then @xmath321 which is isomorphic to @xmath322 . ( for more information about isomorphic generalized petersen graphs see @xcite ) . * * if @xmath320 and @xmath323 then by theorem [ n odd k even ] , @xmath324 for every @xmath325 . * * if @xmath326 then @xmath327 has to be larger than @xmath78 . in fact if @xmath326 and @xmath303 then @xmath328 . for @xmath326 and @xmath329 then by theorem [ n odd k even ] , @xmath330 for every @xmath331 . ( note that since @xmath7 is odd and @xmath8 is even , @xmath332 implies that @xmath333 ) . * * if @xmath334 then by theorem [ n odd k even ] , @xmath335 for @xmath336 . for @xmath337 then the assertion is concluded from part @xmath312 . if @xmath295 then @xmath338 , and behsaz - hatami - mahmoodian s conjecture holds . in this section we will prove that the independence number of generalized petersen graphs with fixed @xmath8 can be found in linear time , @xmath13 . this result is a special case of a deep theorem stating that the problem of finding the independence number of graphs with bounded treewidth can be solved in linear time of the number of vertices of the graph . in the continuation we will show that for every fixed @xmath8 and any integer @xmath9 the treewidth of @xmath18 is bounded . first , we need to formally define the concepts of tree decomposition and treewidth of a graph . let @xmath14 be a graph . a tree decomposition of @xmath0 is a pair @xmath339 , where @xmath340 is a family of subsets of @xmath341 , and @xmath342 is a tree whose nodes are the subsets @xmath343 , satisfying the following properties : * the union of all sets @xmath343 equals @xmath341 . * for every edge @xmath344 in the graph , there is a subset @xmath343 that contains both @xmath22 and @xmath345 . * if @xmath343 is on the path from @xmath346 to @xmath347 in @xmath342 then @xmath348 . in other words , for all vertices @xmath349 all nodes @xmath346 which contain @xmath22 induce a connected subtree of @xmath342 . the width of @xmath350 is defined to be the size of the largest @xmath346 minus one . the treewidth , @xmath351 , of the graph @xmath0 is defined to be the minimum width of all its tree decompositions . the treewidth will be taken as a measure of how much a graph resembles a tree . ( @xcite ) [ alghorithm ] the problem of finding a maximum independent set of a graph @xmath0 with bounded treewidth , @xmath352 can be solved in @xmath353 by dynamic programming techniques , where @xmath7 is the number of vertices of graph . for more details see for instance @xcite , @xcite , @xcite , and @xcite . for any fixed @xmath8 , the problem of finding independence number of the graphs @xmath18 can be solved by an algorithm with running time @xmath13 . by theorem [ alghorithm ] , we only need to show that for a given number @xmath8 , the treewidth of @xmath18 is bounded . consider the following tree decomposition of @xmath18 of width @xmath354 . without loss of generality we can only consider the case where @xmath355 . let @xmath342 be the path of order @xmath356 and define @xmath357 as follows : + @xmath358 + @xmath359 + @xmath360 + @xmath361 + @xmath362 and so on . + notice that in each step we remove two elements and add two other elements . therefore @xmath363 for all @xmath209 . one can easily see that @xmath350 is a tree decomposition for @xmath18 where @xmath364 . thus , @xmath365 and by theorem [ alghorithm ] , the proof is complete . table @xmath78 : independence number of @xmath366 . the authors would like to thank the referee for careful reading of this paper and very helpful comments . and the authors like to thank professor e. s. mahmoodian for suggesting the problem and very useful comments . we also thank nima aghdaei , and hadi moshaiedi for their computer program and algorithm to create presented table of @xmath3 . hans l. bodlaender . dynamic programming on graphs with bounded treewidth . in _ automata , languages and programming ( tampere , 1988 ) _ , volume 317 of _ lecture notes in comput . _ , pages 105118 . springer , berlin , 1988 .

determining the size of a maximum independent set of a graph @xmath0 , denoted by @xmath1 , is an np - hard problem . therefore many attempts are made to find upper and lower bounds , or exact values of @xmath2 for special classes of graphs . this paper is aimed toward studying this problem for the class of generalized petersen graphs . we find new upper and lower bounds and some exact values for @xmath3 . with a computer program we have obtained exact values for each @xmath4 . in @xcite it is conjectured that the size of the minimum vertex cover , @xmath5 , is less than or equal to @xmath6 , for all @xmath7 and @xmath8 with @xmath9 . we prove this conjecture for some cases . in particular , we show that if @xmath10 , the conjecture is valid . we checked the conjecture with our table for @xmath11 and it had no inconsistency . finally , we show that for every fixed @xmath8 , @xmath12 can be computed using an algorithm with running time @xmath13 . * keywords * : generalized petersen graphs , independent set , tree decomposition

arxiv

quantum key distribution ( qkd ) is often said to be unconditionally secure @xcite . more precisely , qkd can be proven to be secure against any eavesdropping _ given _ that the users ( alice and bob ) devices satisfy some requirements , which often include mathematical characterization of users devices as well as the assumption that there is no side - channel . this means that no one can break mathematical model of qkd , however in practice , it is very difficult for practical devices to meet the requirements , leading to the breakage of the security of practical qkd systems . actually , some attacks on qkd have been proposed and demonstrated successfully against practical qkd systems @xcite . to combat the practical attacks , some counter - measures @xcite , including device independent security proof idea @xcite , have been proposed . the device independent security proof is very interesting from the theoretical viewpoint , however it can not apply to practical qkd systems where loopholes in testing bell s inequality @xcite can not be closed . as for the experimental counter - measures , battle - testing of the practical detection unit has attracted many researchers attention @xcite since the most successful practical attack so far is to exploit the imperfections of the detectors . recently , a very simple and very promising idea , which is called a measurement device independent qkd ( mdiqkd ) has been proposed by lo , curty , and qi @xcite . in this scheme , neither alice nor bob performs any measurement , but they only send out quantum signals to a measurement unit ( mu ) . mu is a willing participant of the protocol , and mu can be a network administrator or a relay . however , mu can be untrusted and completely under the control of the eavesdropper ( eve ) . after alice and bob send out signals , they wait for mu s announcement of whether she has obtained the successful detection , and proceed to the standard post - processing of their sifted data , such as error rate estimation , error correction , and privacy amplification . the basic idea of mdiqkd is based on a reversed epr - based qkd protocol @xcite , which is equivalent to epr - based qkd @xcite in the sense of the security , and mdiqkd is remarkable because it removes _ all _ the potential loopholes of the detectors without sacrificing the performance of standard qkd since alice and bob do not detect any quantum signals from eve . moreover , it is shown in @xcite that mdiqkd with infinite number of decoy states and polarization encoding can cover about twice the distance of standard decoyed qkd , which is comparable to epr - based qkd . the only assumption needed in mdiqkd is that the preparation of the quantum signal sources by alice and bob is ( almost ) perfect and carefully characterized . we remark that the characterization of the signal source should be easier than that of the detection unit since the characterization of the detection unit involves the estimation of the response of the devices to unknown input signals sent from eve . with mdiqkd in our hand , we do not need to worry about imperfections of mu any more , and we should focus our attention more to the imperfections of signal sources . one of the important imperfections of the sources is the basis - dependent flaw that stems from the discrepancy of the density matrices corresponding to the two bases in bb84 states . the security of standard bb84 with basis - dependent flaw has been analyzed in @xcite which show that the basis - dependent flaw decreases the achievable distance . thus , in order to investigate the practicality of mdiqkd , we need to generalize the above works to investigate the security of mdiqkd under the imperfection . another problem in mdiqkd is that the first proposal is based on polarization encoding @xcite , however , in some situations where birefringence effect in optical fiber is highly time - dependent , we need to consider mdiqkd with phase encoding rather than polarization encoding . in this paper , we study the above issues simultaneously . we first propose two schemes of the phase encoding mdiqkd , one employs phase locking of two separate laser sources and the other one uses the conversion of phase encoding into polarization encoding . then , we prove the unconditional security of these schemes with basis - dependent flaw by generalizing the quantum coin idea @xcite . based on the security proof , we simulate the key generation rate with realistic parameters , especially we employ a simple model to evaluate the basis - dependent flaw due to the imperfection of the phase modulators . our simulation results imply that the first scheme covers shorter distances and may require less accuracy of the state preparation , while the second scheme can cover much longer distances when we can prepare the state very precisely . we note that in this paper we consider the most general type of attacks allowed by quantum mechanics and establish unconditional security for our protocols . this paper is organized as follows . in sec . [ sec : protocol ] , we give a generic description of mdiqkd protocol , and we propose our schemes in sec . [ sec : phase encoding scheme i ] and sec . [ sec : phase encoding scheme ii ] . then , we prove the unconditional security of our schemes in sec . [ sec : proof ] , and we present some simulation results of the key generation rate based on realistic parameters in sec . [ sec : simulation ] . finally , we summarize this paper in sec . [ sec : summary ] . in this section , we introduce mdiqkd protocol whose description is generic for all the schemes that we will introduce in the following sections . the mdiqkd protocol runs as follows . step ( 1 ) : each of alice and bob prepares a signal pulse and a reference pulse , and each of alice and bob applies phase modulation to the signal pulse , which is randomly chosen from @xmath0 , @xmath1 , @xmath2 , and @xmath3 . here , @xmath4 ( @xmath5 ) defines @xmath6 ( @xmath7)-basis . alice and bob send both pulses through quantum channels to eve who possesses mu . step ( 2 ) : mu performs some measurement , and announces whether the measurement outcome is successful or not . it also broadcasts whether the successful event is the detection of type-0 or type-1 ( the two types of the successful outcomes correspond to two specific bell states @xcite ) . step ( 3 ) : if the measurement outcome is successful , then alice and bob keep their data . otherwise , they discard the data . when the outcome is successful , alice and bob broadcast their bases and they keep the data only when the bases match , which we call sifted key . depending on the type of the successful event and the basis that they used , bob may or may not perform bit - flip on his sifted key . step ( 4 ) : alice and bob repeat ( 1)-(3 ) many times until they have large enough number of the sifted key . step ( 5 ) : they sacrifice a portion of the data as the test bits to estimate the bit error rate and the phase error rate on the remaining data ( code bits ) . step ( 6 ) : if the estimated bit error and phase error rates are too high , then they abort the protocol , otherwise they proceed . step ( 7 ) : alice and bob agree over a public channel on an error correcting code and on a hash function depending on the bit and phase error rate on the code bits . after performing error correction and privacy amplification , they share the key . the role of the mu in eve is to establish a quantum correlation , i.e. , a bell state , between alice and bob to generate the key . if it can establish the strong correlation , then alice and bob can generate the key , and if it can not , then it only results in a high bit error rate to be detected by alice and bob and they abort the protocol . as we will see later , since alice and bob can judge whether they can generate a key or not by only checking the experimental data as well as information on the fidelity between the density matrices in @xmath6- basis and @xmath7-basis , it does not matter who performs the measurement nor what kind of measurement is actually done as long as mu broadcasts whether the measurement outcome was successful together with the information of whether the successful outcome is type-0 or type-1 . in the security proof , we assume that mu is totally under the control of eve . in practice , however , we should choose an appropriate measurement that establishes the strong correlation under the normal operation , i.e. , the situation without eve who induces the channel losses and noises . in the following sections , we will propose two phase encoding mdiqkd schemes . in this section , we propose an experimental setup for mdiqkd with phase encoding scheme , which is depicted in fig . [ setup1 ] . this scheme will be proven to be unconditionally secure , i.e. , secure against the most general type of attacks allowed by quantum mechanics . in this setup , we assume that the intensity of alice s signal ( reference ) pulse matches with that of bob s signal ( reference ) pulse when they enter mu . in order to lock the relative phase , we use strong pulses as the reference pulses . in pl unit in the figure , the relative phase between the two strong pulses is measured in two polarization modes separately . the measurement result is denoted by @xmath8 ( here , the arrow represents two entries that correspond to the two relative phases ) . depending on this information @xmath8 , appropriate phase modulations for two polarization modes are applied to incoming signal pulse from alice . then , alice s and bob s signal pulses are input into the 50/50 beam splitter which is followed by two single - photon threshold detectors . the successful event of type-0 ( type-1 ) in step ( 2 ) is defined as the event where only d0 ( d1 ) clicks . in the case of type-1 successful detection event , bob applies bit flip to his sifted key ( we define the phase relationship of bs in such a way that d1 never clicks when the phases of the two input signal coherent pulses are the same ) . . then , the phase shift of @xmath8 for each polarization mode is applied to one of the signal pulses , and they will be detected by d0 and d1 after the interference at the 50:50 beam splitter bs . [ setup1 ] ] roughly speaking , our scheme performs double bb84 @xcite , i.e. , each of alice and bob is sending signals in the bb84 states , without phase randomization @xcite . differences between our scheme and the polarization encoding mdiqkd scheme include that alice and bob do not need to share the reference frame for the polarization mode , since mu performs the feed - forward control of the polarization , and our scheme intrinsically possesses the basis - dependent flaw . to see how this particular setup establishes the quantum correlation under the normal operation , it is convenient to consider an entanglement distribution scheme @xcite , which is mathematically equivalent to the actual protocol . for the simplicity of the discussion , we assume the perfect phase locking for the moment and we only consider the case where both of alice and bob use @xmath6-basis . we skip the discussion for @xmath7-basis , however it holds in a similar manner @xcite . in this case , the actual protocol is equivalently described as follows . first , alice prepares two systems in the following state , which is a purification of the @xmath6-basis density matrix , @xmath9 and sends the second system to mu through the quantum channel . here , @xmath10 and @xmath11 represent coherent states that alice prepares in the actual protocol ( @xmath12 represents the mean photon number or inetensity ) , @xmath13 and @xmath14 are eigenstate of the computational basis ( @xmath6 basis ) , which is related with @xmath7-basis eigenstate through @xmath15 and @xmath16 . for the later convenience , we also define @xmath17-basis states as @xmath18 and @xmath19 . moreover , the subscript of @xmath20 in @xmath21 represents that alice is to measure her qubit along @xmath6-basis , the subscript of @xmath22 in @xmath12 refers to the party who prepares the system , and the superscript @xmath23 represents the relative phase of the superposition . similarly , bob also prepares two systems in a similar state @xmath24 , sends the second system to mu , and performs @xmath6-basis measurement . note that @xmath6-basis measurement by alice and bob can be delayed after eve s announcement of the successful event without losing any generalities in the security analysis , and we assume this delay in what follows . in order to see the joint state of the qubit pair after the announcement , note that the beam splitter converts the joint state @xmath25 into the following state @xmath26 @xmath27 here , for the simplicity of the discussion , we assume that there is no channel losses , we define @xmath28 , and @xmath29 represents the vacuum state . moreover , the subscripts @xmath30 and @xmath31 represent the output ports of the beam splitter . if detector d0 ( d1 ) detects photons and the other detector d1 ( d0 ) detects the vacuum state , i.e. , type-0 ( type-1 ) event , it is shown in the appendix a that the joint probability of having type-0 ( type-1 ) successful event and alice and bob share the maximally entangled state @xmath32 ( @xmath33 ) is @xmath34 . we note that since @xmath35 , alice and bob do not always share this state , and with a joint probability of @xmath36 , they have type-0 ( type-1 ) successful event and share the maximally entangled state with the phase error , i.e. , the bit error in @xmath7-basis , as @xmath37 ( @xmath38 ) . note that the bit - flip operation in type-1 successful detection can be equivalently performed by @xmath2 rotation around @xmath17-basis before bob performs @xmath6 basis measurement . in other words , @xmath2 rotation around @xmath17-basis before @xmath6-basis measurement does not change the statistics of the @xmath6-basis measurement followed by the bit - flip . thanks to this property , we can conclude that alice and bob share @xmath32 with probability of @xmath39 and @xmath37 with probability of @xmath40 after the rotation . this means that even if alice and bob are given the successful detection event , they can not be sure whether they share @xmath37 or @xmath32 , however , if they choose a small enough @xmath41 , then the phase error rate ( the rate of the state @xmath37 in the qubit pairs remaining after the successful events or equivalently , the rate of @xmath7-basis bit error among all the shared qubit pairs ) becomes small and they can generate a pure state @xmath32 by phase error correction , which is equivalently done by privacy amplification in the actual protocol @xcite . we note that the above discussion is valid only for the case without noises and losses , and we will prove the security against the most general attack in sec . [ sec : proof ] without relying on the argument given in this section . we remark that in the phase encoding scheme i , it is important that alice and bob know quite well about the four states that they prepare . this may be accomplished by using state tomography with homodyne measurement involving the use of the strong reference pulse @xcite . [ scheme ii ] in this section , we propose the second experimental setup for mdiqkd with phase encoding scheme . like scheme i , this scheme will also be proven to be unconditionally secure . in this scheme , the coherent pulses that alice and bob send out are exactly the same as those in the standard phase encoding bb84 , i.e. , @xmath42 where subscripts @xmath43 and @xmath44 respectively denote the signal pulse and the reference pulse , @xmath45 is a completely random phase , @xmath46 is randomly chosen from @xmath47 to encode the information . after entering the mu , each pulse pair is converted from a phase coding signal to a polarization coding signal by a phase - to - polarization converter ( see details below ) . we note that thanks to the phase randomization by @xmath45 , the joint state of the signal pulse and the reference pulse is a classical mixture of photon number states . in fig . [ setup2 ] , we show the schematics of the converter . this converter performs the phase - to - polarization conversion : @xmath48 to @xmath49 , where @xmath50 is a projector that projects the joint system of the signal and reference pulses to a two - dimensional single - photon subspace spanned by @xmath51 where @xmath0 and @xmath52 represent the photon number , and @xmath53 ( @xmath54 ) represents the horizontal ( vertical ) polarization state of a single - photon . to see how it works , let us follow the time evolution of the input state . at the polarization beam splitter ( pbs in fig . [ setup2 ] ) , the signal and reference pulses first split into two polarization modes , h and v , and we throw away the pulses being routed to v mode . then , in h mode , the signal pulse and the reference pulse are routed to different paths by using an optical switch , and we apply @xmath2-rotation only to one of the paths to convert h to v. at this point , we essentially have @xmath55 , where the subscripts of `` @xmath56 '' and `` @xmath57 '' respectively denote the upper path and the lower path . finally , these spatial modes @xmath56 and @xmath57 are combined together by using a polarization beam splitter so that we have @xmath49 in the output port depicted as `` out '' . in practice , since the birefringence of the quantum channel can be highly time dependent and the polarization state of the input pulses to mu may randomly change with time , i.e. , the input polarization state is a completely mixed state , we can not deterministically distill a pure polarization state , and thus the conversion efficiency can never be perfect . in other words , one may consider the same conversion of the v mode just after the first polarization beam splitter , however it is impossible to combine the resulting polarization pulses from v mode and the one from h mode into a single mode . we assume that mu has two converters , one is for the conversion of alice s pulse and the other one is for bob s pulse , and the two output ports `` out '' are connected to exactly the same bell measurement unit @xcite in the polarization encoding mdiqkd scheme in fig . [ bell m ] @xcite . this bell measurement unit consists of a 50:50 beam splitter , two polarization beam splitters , and four single - photon detectors , which only distinguishes perfectly two out of the four bell states of @xmath38 and @xmath33 . the polarization beam splitters discriminate between @xmath58 and @xmath59 ( note that we choose @xmath60 and @xmath61 modes rather than h and v modes since our computational basis is @xmath60 and @xmath61 ) . suppose that a single - photon enters both from alice and bob . in this case , the click of d0 + and d0- or d1 + and d1- means the detection of @xmath38 , and the click of d0 + and d1- or d0- and d1 + means the detection of @xmath33 ( see fig . [ bell m ] ) . in this scheme , since the use of coherent light induces non - zero bit error rate in @xmath7-basis ( @xmath62-basis ) , we consider to generate the key from @xmath63-basis and we use the data in @xmath7-basis only to estimate the bit error rate in this basis conditioned on that both of alice and bob emit a single - photon , which determines the amount of privacy amplification . by considering a single - photon polarization input both from alice and bob , one can see that bob should not apply the bit flip only when alice and bob use @xmath7-basis and @xmath64 is detected in mu , and bob should apply the bit flip in all the other successful events to share the same bit value . accordingly , the bit error in @xmath6-basis is given by the successful detection event conditioned on that alice and bob s polarization are identical . as for @xmath7-basis , the bit error is @xmath64 detection given the orthogonal polarizations or @xmath65 detection given the identical polarization . assuming completely random input polarization state , our converter successfully converts the single - photon pulse with a probability of @xmath66 . note in the normal experiment that the birefringence effect between alice and the converter and the one between bob and the converter are random and independent , however it only leads to fluctuating coincidence rate of alice s and bob s signals at the bell measurement , but does not affect the qber . moreover , the fluctuation increases the single - photon loss inserted into the bell measurement . especially , the events that the output of the converter for alice is the vacuum and the one for bob is a single - photon , and vice versa would increase compared to the case where we have no birefringence effect . however , this is not a problem since the bell measurement does not output the conclusive events in these cases unless the dark counting occurs . thus , the random and independent polarization fluctuation in the normal experiment is not a problem , and we will simply assume in our simulation in sec . [ simulationii ] that this fluctuation can be modeled just by @xmath66 loss . we emphasize that we do not rely on these assumptions at all when we prove the security , and our security proof applies to any channels and mus . for the better performance and also for the simplicity of analysis , we assume the use of infinite number of decoy states @xcite to estimate the fraction of the probability of successful event conditioned on that both of alice and bob emit a single - photon . one of the differences in our analysis from the work in @xcite is that we will take into account the imperfection of alice s and bob s source , i.e. , the decay of the fidelity between two density matrices in two bases . we also remark that since the h and v modes are defined locally in mu , alice and bob do not need to share the reference frame for the polarization mode , which is one of the qualitative differences from polarization encoding miqkd scheme @xcite . this section is devoted to the unconditional security proof , i.e. , the security proof against the most general attacks , of our schemes . since both of our schemes are based on bb84 and the basis - dependent flaw in both protocols can be treated in the same manner , we can prove the security in a unified manner . if the states sent by alice and bob were basis independent , i.e. , the density matrices of @xmath6-basis and @xmath7-basis were the same , then the security proof of the original bb84 @xcite could directly apply ( also see @xcite for a bit more detailed discussion of this proof ) , however they are basis dependent in our case . fortunately , security proof of standard bb84 with basis - dependent flaw has already been shown to be secure @xcite , and we generalize this idea to our case where we have basis - dependent flaw from both of alice and bob . in order to do so , we consider a virtual protocol @xcite that alice and bob get together and the basis choices by alice and bob are made via measurement processes on the so - called quantum coin . in this virtual protocol of the phase encoding scheme i , alice and bob prepare joint systems in the state @xcite @xmath67 since just replacing the state , for instance @xmath68 where @xmath52 and @xmath0 in the ket respectively represents the single - photon and the vacuum , is enough to apply the following proof to the phase encoding scheme ii , we discuss only the security of the phase encoding scheme i in what follows . in eq . ( [ coin - joint - state ] ) , the first system denoted by @xmath69 is given to eve just after the preparation , and it informs eve of whether the bases to be used by alice and bob match or not . the second system , denoted by @xmath70 , is a copy of the first system and this system is given to bob who measures this system with @xmath71 basis to know whether alice s and bob s bases match or not . if his measurement outcome is @xmath72 ( @xmath73 ) , then he uses the same ( the other ) basis to be used by alice ( note that no classical communication is needed in order for bob to know alice s basis since alice and bob get together ) . the third system , which is denoted by @xmath22 and we call `` quantum coin '' , is possessed and to be measured by alice along @xmath74 basis to determine her basis choice , and the measurement outcome will be sent to eve after eve broadcasts the measurement outcome at mu . moreover , all the second systems of @xmath75 , @xmath76 , @xmath24 , and @xmath77 are sent to eve . note in this formalism that the information , including classical information and quantum information , available to eve is the same as those in the actual protocol , and the generated key is also the same as the one of the actual protocol since the statistics of alice s and bob s raw data is exactly the same as the one of the actual protocol . thus , we are allowed to work on this virtual protocol for the security proof . the first system given to eve in eq . ( [ coin - joint - state ] ) allows her to know which coherent pulses contain data in the sifted key and she can post - select only the relevant pulses . thus , without the loss of any generalities of the security proof , we can concentrate only on the post - selected version of the state in eq . ( [ coin - joint - state ] ) as @xmath78 the most important quantity in the proof is the phase error rate in the code bits . the definition of the phase error rate is the rate of bit errors along @xmath7-basis in the sifted key if they had chosen @xmath7-basis as the measurement basis when both of them have sent pulses in @xmath6-basis . if alice and bob have a good estimation of this rate as well as the bit error rate in the sifted key ( the bit error rate in @xmath6-basis given alice and bob have chosen @xmath6-basis for the state preparation ) , they can perform hashing in @xmath7-basis and @xmath6-basis simultaneously @xcite to distill pairs of qubits in the state whose fidelity with respect to the product state of the maximally entangled state @xmath32 is close to @xmath52 . according to the discussion on the universal composability @xcite , the key distilled via @xmath6-basis measurement on such a state is composably secure and moreover exactly the same key can be generated only by classical means , i.e. , error correction and privacy amplification @xcite . thus , we are left only with the phase error estimation . for the simplicity of the discussion , we assume the large number of successful events @xmath79 so that we neglect all the statistical fluctuations and we are allowed to work on a probability rather than the relative frequency . the quantity we have to estimate is the bit error along @xmath7-basis , denoted by @xmath80 , given alice and bob send @xmath81 state , which is different from the experimentally available bit error rate along @xmath7-basis given alice and bob send @xmath82 state . intuitively , if the basis - dependent flaw is very small , @xmath80 and @xmath83 should be very close since the states are almost indistinguishable . to make this intuition rigorous , we briefly review the idea by @xcite which applies bloch sphere bound @xcite to the quantum coin . suppose that we randomly choose @xmath17-basis or @xmath6-basis as the measurement basis for each quantum coin . let @xmath84 and @xmath85 be fraction that those quantum coins result in @xmath52 in @xmath17-basis and @xmath6-basis measurement , respectively . what bloch sphere bound , i.e. , eq . ( 13 ) or eq . ( 14 ) in @xcite or eq . ( a1 ) in @xcite , tells us in our case is that no matter how the correlations among the quantum coins are and no matter what the state for the quantum coins is , thanks to the randomly chosen bases , the following inequality holds with probability exponentially close to @xmath52 in @xmath79 , @xmath86 by applying this bound separately to the quantum coins that are conditional on having phase errors and to those that are conditional on having no phase error , and furthermore by combining those inequalities using bayes s rule , we have @xmath87 here , @xmath88 is equivalent to the probability that the measurement outcome of the quantum coin along @xmath6-basis is @xmath14 given the successful event in mu . note that this probability can be enhanced by eve who chooses carefully the pulses , and eve could attribute all the loss events to the quantum coins being in the state @xmath13 . thus , we have an upper bound of @xmath88 in the worst case scenario as @xmath89 and @xmath90 where @xmath91 is the frequency of the successful event . note that we have not used the explicit form of @xmath92 and @xmath93 , where @xmath94 , in the derivation of eqs . ( [ phase error bound ] ) , ( [ fdelta ] ) , and ( [ delta ] ) , and the important point is that the state @xmath92 and @xmath93 are the purification of alice s and bob s density matrices for both bases . since there always exists purification states of @xmath95 and @xmath96 , which are respectively denoted by @xmath97 and @xmath98 , such that @xmath99 , @xmath100 can be rewritten by @xmath101/2\ , , \label{fiddelta}\end{aligned}\ ] ] where @xmath102 represents alice s density matrix of @xmath6 basis and all the other density matrices are defined by the same manner . our expression of @xmath100 has the product of two fidelities , while the standard bb84 with basis - dependent flaw in @xcite has only one fidelity ( the fidelity between alice s density matrices in @xmath6 and @xmath7 bases ) . the two products may lead to poor performance of our schemes compared to that of standard qkd in terms of the achievable distances , however our schemes have the huge advantage over the standard qkd that there is no side - channel in the detectors . finally , the key generation rate @xmath103 , given @xmath6-basis , in the asymptotic limit of large @xmath79 is given by @xmath104 where @xmath105 is the bit error rate in @xmath6-basis , @xmath106 is the inefficiency of the error correcting code , and @xmath107 . we can trivially obtain the key generation rate for @xmath7-basis just by interchanging @xmath6-basis in all the discussions above to @xmath7-basis . we remark in our security proof that we have assumed nothing about what kind of measurement mu conducts but that it announces whether it detects the successful event and the type of the event ( this announcement allows us to calculate @xmath108 and the error rates ) . thus , mu can be assumed to be totally under the control of eve . in the following subsections , we show some examples of the key generation rate of each of our schemes assuming typical experimental parameters taken from gobby - yuan - shields ( gys ) experiment @xcite unless otherwise stated . moreover , we assume that the imperfect phase modulation is the main source of the decay of the fidelity between the density matrices in two bases , and we evaluate the effect of this imperfection on the key generation rate . in the phase encoding scheme i , the important quantity for the security @xmath100 can be expressed as @xmath109\ , . \label{delta11}\end{aligned}\ ] ] note that this quantity is dependent on the intensity of alice s and bob s sources . as we have mentioned in sec . iii , this quantity may be estimated relatively easily via tomography involving homodyne measurement . ) of @xmath110 and @xmath111 . dashed line : ( a ) mu is at bob s side , i.e. , @xmath112 . solid line : ( b ) mu is just in the middle between alice and bob . the lines achieving the longer distances correspond to @xmath110 of @xmath113 . see also the main text for the explanation . [ fig : key33 ] ] ) that outputs fig . [ fig : key33 ] as a function of the distance between alice and bob . [ fig : intensity33 ] ] to simulate the resulting key generation rate , we assume that the bit error stems from the dark counting as well as alignment errors due to imperfect phase locking or imperfect optical components . the alignment error is assumed to be proportional to the probability of having a correct click caused only by the optical detection not by the dark counting . moreover , we make assumptions that all the detectors have the same characteristics for the simplicity of the analysis , and alice and bob choose the intensities of the signal lights in such a way that the intensities of the incoming pulses to mu are the same . finally , we assume the quantum inefficiency of the detectors to be part of the losses in the quantum channels . with all the assumptions , we may express the resulting experimental parameters as @xmath114(1-p_{\rm dark})\nonumber\\ & + & ( 1-p_{\rm dark})e^{-2\alpha_{\rm in}}p_{\rm dark}\nonumber\\ \gamma_{\rm suc}&=&\gamma_{\rm suc}^{(x)}+\gamma_{\rm suc}^{(y)}\nonumber\\ \delta_x&=&\delta_y=\big[e_{\rm ali}(1-p_{\rm dark})^2(1-e^{-2\alpha_{\rm in}})\nonumber\\ & + & ( 1-p_{\rm dark})e^{-2\alpha_{\rm in}}p_{\rm dark}\big]/\gamma_{\rm suc}^{(x)}\nonumber\\ \alpha_{\rm in}&\equiv&\alpha_{a}\eta_{a}=\alpha_{b}\eta_{b}\nonumber\\ \eta_{a}&=&\eta_{{\rm det } , a}10^{-\xi_{a } l_{a}/10}\nonumber\\ \eta_{b}&=&\eta_{{\rm det } , b}10^{-\xi_{b } l_{b}/10}\ , . \label{ex data i}\end{aligned}\ ] ] here , @xmath115 is the dark count rate of the detector , @xmath113 is the alignment error rate , @xmath116 is alice s ( bob s ) overall transmission rate , @xmath117 ( @xmath118 ) is the quantum efficiency of alice s ( bob s ) detector , @xmath119 is alice s ( bob s ) channel transmission rate , and @xmath120 ( @xmath121 ) is the distance between alice ( bob ) and mu . the first term and the second term in @xmath122 or @xmath123 respectively represent the alignment error , which is assumed to be proportional to the probability of having correct bit value due to the detection of the light , and errors due to dark counting ( one detector clicks due to the dark counting while the other one does not ) . we take the following parameters from gys experiment @xcite : @xmath124 , @xmath125 , @xmath126 ( db / km ) , @xmath127 , and @xmath128 , and we simulate the key generation rate as a function of the distance between alice and bob in fig . [ fig : key33 ] . in the figure , we consider two settings : ( a ) mu is at bob s side , i.e. , @xmath112 ( b ) mu is just in the middle between alice and bob . the reason why we consider these setting is that the basis - dependent flaw is dependent on intensities that alice and bob employ , and it is not trivial where we should place mu for the better performance . since mdiqkd polarization encoding scheme without basis - dependent flaw achieves almost twice the distance of bb84 @xcite , we may expect that the setting ( b ) could achieve almost twice the distance of bb84 without phase randomization that achieves about 13 ( km ) @xcite with the same experimental parameters . the simulation result , however , does not follow this intuition since we have the basis - dependent flaw not only from alice s side but also from bob s side . thus , the advantage that we obtain from putting mu between alice and bob is overwhelmed by the double basis - dependent flaw . in each setting , we have optimized the intensity of the coherent pulses @xmath12 for each distance ( see fig . [ fig : intensity33 ] ) . in order to explain why the optimal @xmath129 is so small , note that scheme i intrinsically suffers from the basis - dependent flaw due to eq . ( [ delta11 ] ) . this means that if we use relatively large @xmath129 , then we can not generate the key due to the flaw . actually , when we set @xmath130 , which is a typical order of the amplitude for decoy bb84 , one can see that the upper bound of the phase error rate is @xmath131 even in the zero distance , i.e. , @xmath132 , and we have no chance to generate the key with this amplitude . thus , alice and bob have to reduce the intensities in order to suppress the basis - dependent flaw . also , as the distance gets larger and the losses get increased , alice and bob have to use weaker pulses since larger losses can be exploited by eve to enhance the basis - dependent flaw according to eq . ( [ fdelta ] ) , and they can reduce the intensities until it reaches the cut - off value where the detection of the weak pulses is overwhelmed by the dark counts . in the above simulation , we have assumed that alice and bob can prepare states very accurately , however in reality , they can only prepare approximate states due to the imperfection of the sources . this imperfection gives more basis - dependent flaw , and in order to estimate the effect of this imperfection , we assume that the fidelity between the two actually prepared density matrices in two bases is approximated by the fidelity between the following density matrices ( see appendix b for the detail ) @xmath133 and @xmath134 where we assume an imperfect phase modulator whose degree of the phase modulation error is proportional to the target phase modulation value , and @xmath135 represents the imperfection of the phase modulation that is related with the extinction ratio @xmath136 as @xmath137 in this equation , we assume that the non - zero extinction ratio is only due to the imperfection of the phase modulators . since imperfect phase modulation results in the same effect as the alignment errors , i.e. , the pulses are routed to a wrong output port , we assume that the alignment error rate is increased with this imperfection . thus , in the simulation accommodating the imperfection of the phase modulation , we replace @xmath113 with @xmath138 . here , we have used a pessimistic assumption that the effect of the phase modulation becomes @xmath139-times higher than before since each of alice and bob has one phase modulator and mu has two phase modulators for the phase shift of two polarization modes ( note from eq . ( [ mimperfect phase modulator ] ) that @xmath136 is approximately proportional to @xmath140 , thus 4 times degradation in terms of the accuracy of the phase modulation results in @xmath139-times degradation in terms of the extinction ratio ) . we also remark that in practice , it is more likely that the phase encoding errors are independent , in which case a factor of 4 will suffice and the key rate will actually be higher than what is presented in our paper . on the other hand , we have to use the following @xmath100 when we consider the security : @xmath141/2\,.\end{aligned}\ ] ] in figs . [ fig : keyimpferfectpmi ] and [ fig : intensityimpferfectpmi ] , we plot the key generation rate and the corresponding optimal alice s mean photon numbers ( @xmath12 ) as a function of the distance between alice and bob . in the figures , we define @xmath142 that satisfies @xmath143 as @xmath144 , where @xmath145 is the typical order of @xmath136 in some experiments @xcite . we have confirmed that we can not generate the key when @xmath145 . however , we can see in the figures that if the accuracy of the phase modulation is increased three times or five times , i.e. , @xmath146 and @xmath147 , then we can generate the key . like the case in fig . [ fig : intensity33 ] , the small optimal mean photon number can be intuitively understood by the arguments that we have already made in this section . in order to investigate the feasibility of the phase encoding scheme i with the current technologies , we replace @xmath125 , @xmath127 , and @xmath128 with @xmath148 , @xmath149 @xcite , and @xmath150 @xcite . we see in fig . [ fig : keyimpferfectpminew ] that the key generation is possible over much longer distances with those parameters assuming the precise control of the intensities of the laser source . we also show the corresponding optimal mean photon number @xmath129 in fig . [ fig : intensityimpferfectpminew ] . we note that thanks to the higher quantum efficiency , the success probability becomes higher , following that alice and bob can use larger mean photon number @xmath129 compared to those in figs . [ fig : intensityimpferfectpmi ] and [ fig : intensityimpferfectpminew ] . ) of @xmath110 and imperfect phase modulators . @xmath151 represents the typical amount of the phase modulation error , and we plot the key rate for smaller imperfection of @xmath152 and @xmath153 . dashed line : mu is at bob s side , i.e. , @xmath112 . solid line : mu is just in the middle between alice and bob.[fig : keyimpferfectpmi ] ] ) that outputs fig . [ fig : keyimpferfectpmi ] as a function of the distance between alice and bob . [ fig : intensityimpferfectpmi ] ] with @xmath148 , @xmath149 @xcite , and @xmath151 . dashed line : mu is at bob s side , i.e. , @xmath112 . solid line : mu is just in the middle between alice and bob.[fig : keyimpferfectpminew ] ] ) that outputs fig . [ fig : keyimpferfectpminew ] as a function of the distance between alice and bob . [ fig : intensityimpferfectpminew ] ] in the phase encoding scheme ii , note that we can generate the key only from the successful detection event in mu given both of alice and bob send out a single - photon since if either or both of alice and bob emit more than one photon , then eve can employ the so - called photon number splitting attack @xcite . thus , the important quantities to estimate are @xmath154 , @xmath155 , @xmath105 , @xmath156 , which respectively represents gain in @xmath6-basis given both of alice and bob emit a single - photon , the phase error rate given alice and bob emit a single - photon , overall bit error rate in @xmath6-basis , and overall gain in @xmath6-basis . to estimate these quantities stemming from the simultaneous single - photon emission , we assume the use of infinite number of decoy states for the simplicity of analysis @xcite . another important quantity in our study is the fidelity @xmath157 ( @xmath158 ) between alice s ( bob s ) @xmath6-basis and @xmath7-basis density matrices of only single - photon component , _ not _ whole optical modes . if this fidelity is given , then we have @xmath159 for the simplicity of the discussion , we consider the case of @xmath160 in our simulation . the estimation of the fidelity only in the single - photon part is very important , however to the best of our knowledge we do not know any experiment directly measuring this quantity . this measurement may require photon number resolving detectors and very accurate interferometers . thus , we again assume that the degradation of the fidelity is only due to the imperfect phase modulation given by eq . ( [ mimperfect phase modulator ] ) , and we presume that the fidelity of the two density matrices between the two bases is approximated by the fidelity between the following density matrices ( see appendix b for the detail ) @xmath161\nonumber\\ \rho_{y}^{(1)}&=&\frac{1}{2}\big[{\hat p}\left(\frac{{\left| 0_z \right\rangle}+i e^{i|\delta|/2}{\left| 1_z \right\rangle}}{\sqrt{2}}\right)\nonumber\\ & + & { \hat p}\left(\frac{{\left| 0_z \right\rangle}-i e^{-i|\delta|/2}{\left| 1_z \right\rangle}}{\sqrt{2}}\right)\big]\,.\end{aligned}\ ] ] with these parameters , we can express the key generation rate given alice and bob use @xmath6-basis as @xcite @xmath162-f(\delta_{x})q_{x}h(\delta_{x})\,,\end{aligned}\ ] ] where @xmath163 is the @xmath164 version of @xmath80 in eq . ( [ key rate ] ) . to simulate the resulting key generation rate , the bit errors are assumed to stem from multi - photon component , the dark counting , and the misalignment that is assumed to be proportional to the probability of obtaining the correct bit values only due to the detection by optical pulses . like before , we also assume that all the detectors have the same characteristics , alice and bob choose the intensities of the signal lights in such a way that the intensities of the incoming pulses to mu are the same , and all the quantum inefficiencies of the detectors can be attributed to part of the losses in the quantum channel . finally , alice s and bob s coherent light sources are assumed to be phase randomized , and the imperfect phase modulation is represented by the increase of the alignment error rate . with these assumptions , we may have the following resulting experimental parameters @xmath165\nonumber\\ & + & w^{(2,1)}+w^{(2,0)}\nonumber\\ \delta_{x}^{(1,1)}&=&\big\{4\alpha_{a}\alpha_{b}\eta_{a}\eta_{b}e^{-2(\alpha_{a}+\alpha_{b})}p_{\rm dark}(1-p_{\rm dark})^2/2\nonumber\\ & + & 2(e_{\rm ali}+4\eta_{\rm ex})\alpha_{a}\alpha_{b}\eta_{a}\eta_{b}e^{-2(\alpha_{a}+\alpha_{b})}(1-p_{\rm dark})^2\nonumber\\ & + & ( w^{(2,1)}+w^{(2,0)})/2\big\}/q^{(1,1)}_{x}\nonumber\\ q^{(1,1)}_{y}&=&q^{(1,1)}_{x}\nonumber\\ \delta^{(1,1)}_{y}&=&\delta^{(1,1)}_{x}\nonumber\\ w^{(2,1)}&\equiv&8\alpha_{a}\alpha_{b}e^{-2(\alpha_{a}+\alpha_{b})}\big[\eta_{a}(1-\eta_{b})+(1-\eta_{a})\eta_{b}\big]\nonumber\\ & \times&p_{\rm dark}(1-p_{\rm dark})^2\nonumber\\ w^{(2,0)}&\equiv&16\alpha_{a}\alpha_{b}(1-\eta_{a})(1-\eta_{b})e^{-2(\alpha_{a}+\alpha_{b})}\nonumber\\ & \times&p_{\rm dark}^2(1-p_{\rm dark})^2\nonumber\\ q_{x}&=&2\left[1-(1-p_{\rm dark})e^{-\alpha_{\rm in}}\right]^2(1-p_{\rm dark})^2e^{-2\alpha_{\rm in}}+v \nonumber\\ \delta_{x}&=&v+(e_{\rm ali}+4\eta_{\rm ex})2\left(1-e^{-\alpha_{\rm in}}\right)^2 \nonumber\\ & \times&(1-p_{\rm dark})^2e^{-2\alpha_{\rm in}}\nonumber\\ v&\equiv&\frac{p_{\rm dark}(1-p_{\rm dark})}{2\pi}\nonumber\\ & \times&\int_{0}^{2\pi}d\theta\big[1-(1-p_{\rm dark})e^{-\alpha_{\rm in}|1+e^{i\theta}|^2}\big ] \nonumber\\ & \times&\big[(1-p_{\rm dark})e^{-\alpha_{\rm in}|1-e^{i\theta}|^2}\big]\nonumber\\ & + & \frac{p_{\rm dark}(1-p_{\rm dark})}{2\pi}\nonumber\\ & \times&\int_{0}^{2\pi}d\theta\big[1-(1-p_{\rm dark})e^{-\alpha_{\rm in}|1-e^{i\theta}|^2}\big ] \nonumber\\ & \times&\big[(1-p_{\rm dark})e^{-\alpha_{\rm in}|1+e^{i\theta}|^2}\big]\nonumber\\ \alpha_{\rm in}&\equiv&\alpha_{a}\eta_{a}=\alpha_{b}\eta_{b}\nonumber\\ \eta_{a}&=&\eta_{{\rm det } , a}10^{-\xi_{a } l_{a}/10}/2\nonumber\\ \eta_{b}&=&\eta_{{\rm det } , b}10^{-\xi_{b } l_{b}/10}/2\ , \label{exp - data - ii}\end{aligned}\ ] ] note that @xmath12 ( @xmath166 ) represents each of the intensity of alice s ( bob s ) signal light and the reference light , _ not _ the total intensity of them , and @xmath167 and @xmath168 are divided by @xmath169 since the conversion efficiency of our converter is @xmath66 . @xmath170 in @xmath171 again comes from the pessimistic assumption that each of alice s and bob s phase modulator is imperfect , and @xmath172 ( @xmath173 ) represents the probability of the event where both of alice and bob emit a single - photon and only one ( zero ) photon is detected but the successful detection event is obtained due to the dark counting . on the other hand , the quantity that quantifies the basis - dependent flaw @xmath88 in the present case is upper bounded by @xmath174\nonumber\\ q^{(1,1)}&\equiv&(q^{(1,1)}_{x}+q^{(1,1)}_{y})/2\end{aligned}\ ] ] where @xmath175 is the probability that mu receives a single - photon both from alice and bob simultaneously conditioned on that each of alice and bob sends out a single - photon . we remark that @xmath100 in this scheme is only dependent on the accuracy of the phase modulation . this is different from scheme i where the manipulation of the intensities of the pulses can affect the basis - dependent flaw . in the simulation , we again assume gys experimental parameters and we consider two settings : ( a ) mu is at bob s side and ( b ) mu is just in the middle between alice and bob . note that @xmath100 is independent of @xmath12 and @xmath166 in the phase encoding scheme ii case . in fig . [ f1 ] , we plot the key generation rates of ( a ) and ( b ) for @xmath176 , @xmath177 , @xmath178 , @xmath179 ( recall from eq . ( [ mimperfect phase modulator ] ) that @xmath180 that corresponds to the typical extinction ratio of @xmath181 ) , which respectively correspond to @xmath182 , @xmath183 , @xmath184 , and @xmath185 , and the achievable distances of ( a ) and ( b ) increase with the improvement of the accuracy , i.e. , with the decrease of @xmath135 . we have confirmed that no key can be distilled in ( a ) and ( b ) when @xmath186 . the figure shows that the achievable distance drops significantly with the degradation of the accuracy of the phase modulator , and the main reason of this fast decay is that @xmath88 is approximated by @xmath187 and this dominator decreases exponentially with the increase of the distance . we also plot the corresponding optimal @xmath129 in fig . notice that the mean photon number increases in some regime in some cases of ( a ) , and recall that this increase does not change @xmath100 . if we increased the intensity in scheme i with the distance , then we would have more basis - dependent flaw , resulting in shortening of the achievable distance . this may be an intuitive reason why we see no such increase in figs . [ fig : intensity33 ] , [ fig : intensityimpferfectpmi ] , and 9 . like in the phase encoding scheme i , we investigate the feasibility of the phase encoding scheme ii with the current technologies by replacing @xmath125 , @xmath127 , and @xmath128 with @xmath148 , @xmath149 @xcite , and @xmath150 @xcite . with this upgrade , we have confirmed the impossibility of the key generation , however if we double the quantum efficiency of the detector or equivalently , if we assume the polarization encoding so that the factor of @xmath131 , which is introduced by the phase - to - polarization converter , is removed both from @xmath167 and @xmath168 in eq . ( [ exp - data - ii ] ) , then we can generate the key , which is shown in fig . [ f1new ] ( also see fig . finally , we note that our simulation is essentially the same as the polarization coding since the fact that we use phase encoding is only reflected by the dominator of 2 in @xmath188 and @xmath189 in eq . ( [ exp - data - ii ] ) . thus , the behavior of the key generation rate against the degradation of the state preparation is the same also in polarization based mdiqkd . also note that even in the standard bb84 , @xmath88 decays exponentially with increasing distance . thus , we conclude that very precise state preparation is very crucial in the security of not only mdiqkd but also in standard qkd . we also note that our estimation of the fidelity might be too pessimistic since we have assumed that the degradation of the extinction ratio is only due to imperfect phase modulation . in reality , the imperfection of mach - zehnder interferometer and other imperfections should contribute to the degradation , and the fidelity should be closer to @xmath52 than the one based on our model . . solid line : ( b ) mu is just in the middle between alice and bob . we plot the key generation rates of each case when @xmath176 , @xmath177 , @xmath178 , @xmath179 where @xmath135 is proportional to the amount of the phase modulation error , and for each case of ( a ) and ( b ) the key generation rates monotonously increase with the decrease of @xmath135 . , i.e. , with the improvement of the phase modulation . the key rates of ( a ) and ( b ) when @xmath190 are almost superposed . see also the main text for the explanation . [ f1 ] ] ) that outputs fig . the bold lines correspond to ( a ) . see also the main text for the explanation . [ f2 ] ] , @xmath191 , and @xmath151 . note that we double @xmath192 compared to the one of @xcite , or we effectively consider the polarization encoding @xcite . dashed line : ( a ) mu is at bob s side , i.e. , @xmath112 . solid line : ( b ) mu is just in the middle between alice and bob . the key rates are almost superposed . see also the main text for the explanation . [ f1new ] ] ) that outputs fig . [ f1new ] . the bold lines correspond to ( a ) . see also the main text for the explanation . [ f2new ] ] in summary , we have proposed two phase encoding mdiqkd schemes . the first scheme is based on the phase locking technique and the other one is based on the conversion of the pulses in the standard phase encoding bb84 to polarization modes . we proved the security of the first scheme , which intrinsically possesses basis - dependent flaw , as well as the second scheme with the assumption of the basis - dependent flaw in the single - photon part of the pulses . based on the security proof , we also evaluate the effect of imperfect state preparation , and especially we focus our attention to the imperfect phase modulation . while the first scheme can cover relatively short distances of the key generation , this scheme has an advantage that the basis - dependent flaw can be controlled by the intensities of the pulses . thanks to this property , we have confirmed based on a simple model that 3 or 5 times of the improvement in the accuracy of the phase modulation is enough to generate the key . moreover , we have confirmed that the key generation is possible even without these improvements if we implement this scheme by using the up - to - date technologies and the control of intensities of the laser source is precise . on the other hand , it is not so clear to us how accurate we can lock the phase of two spatially separated laser sources , which is important for the performance of scheme i. our result still implies that scheme i can tolerate up to some extent of the imperfect phase locking errors , which should be basically the same as the misalignment errors , but further analysis of the accuracy from the experimental viewpoint is necessary . we leave this problem for the future studies . -basis when @xmath176 , @xmath193 , @xmath194 where @xmath135 is the amount of the phase modulation error . [ sf1 ] ] ) that outputs fig . [ sf2 ] ] the second scheme can cover much longer distances when the fidelity of the _ single - photon components _ of @xmath7-basis and @xmath6-basis density matrices is perfect or extremely close to perfect . when we consider the slight degradations of the fidelity , however , we found that the achievable distances drop significantly . this suggests that we need a photon source with a very high fidelity , and very accurate estimation of the fidelity of the single - photon subspace is also indispensable . in our estimation of the imperfection of the phase modulation , we simply assume that the degradation of the extinction ratio is only due to imperfect phase modulation , which might be too pessimistic , and the imperfection of mach - zehnder interferometer and other imperfections contribute to the degradation . thus , the actual fidelity between the density matrices of the single - photon part in two bases might be very close to 1 , which should be experimentally confirmed for the secure communication . we note that the use of the passive device to prepare the state @xcite may be a promising way for the very accurate state preparation . we remark that the accurate preparation of the state is very important not only in mdiqkd but also in standard qkd where eve can enhance the imbalance of the quantum coin exponentially with the increase of the distance . to see this point , we respectively plot in fig . [ sf1 ] and fig . [ sf2 ] the key generation rate of standard bb84 with infinite decoy states in @xmath6-basis and its optimal mean photon number assuming @xmath148 , @xmath195 @xcite , @xmath150 , @xmath124 , and @xmath126 . again , @xmath196 is the typical value of the phase modulation error , and we see in the figure that the degradation of the phase modulator in terms of the accuracy significantly decreases the achievable distance of secure key generation . one notices that standard decoy bb84 is more robust against the degradation since the probability that the measurement outcome of the quantum coin along @xmath6-basis is @xmath14 given the successful detection of the signal by bob is written as @xmath197 rather than @xmath198 . on the other hand , one has to remember that we trust the operation of bob s detectors in this simulation , which may not hold in practice . finally , we neglect the effect of the fluctuation of the intensity and the center frequency of the laser light in our study , which we will analyze in the future works . in summary , our work highlights the importance of very accurate preparation of the states to avoid basis - dependent flaws . we thank x. ma , m. curty , k. azuma , t. yamamoto , r. namiki , t. honjo , h. takesue , y. tokunaga , and especially g. kato for enlightening discussions . part of this research was conducted when k. t and c - h . f. f visited the university of toronto , and they express their sincere gratitude for all the supports and hospitalities that they received during their visit . this research is in part supported by the project `` secure photonic network technology '' as part of `` the project uqcc '' by the national institute of information and communications technology ( nict ) of japan , in part by the japan society for the promotion of science ( jsps ) through its funding program for world - leading innovative r@xmath199d on science and technology ( first program ) " , in part by rgc grant no . 700709p of the hksar government , and also in part by nserc , canada research chair program , canadian institute for advanced research ( cifar ) and quantumworks . in this appendix , we give a detailed calculation about how scheme i works when there is no channel losses and noises . in order to calculate the joint probability that alice and bob obtain type-0 successful event , where only the detector d0 clicks , and they share the maximally entangled state @xmath32 , we introduce a projector @xmath200 that corresponds to type-0 successful event . here , @xmath201 represents the non - vacuum state . the state after alice and bob have the type-0 successful event @xmath202 ( see eq . ( [ normal schi ] ) for the definition of @xmath26 ) can be expressed by @xmath203 here , @xmath204 is an identity operator on @xmath205 and @xmath206 , @xmath207 and @xmath208 are complex numbers , and @xmath209 and @xmath210 are orthonormal bases , which are related with each other through @xmath211 by a direct calculation , one can show that latexmath:[\ ] ] here , @xmath240 , @xmath241 , and @xmath242 , and @xmath243 is defined by @xmath244 where @xmath245 is a purification of @xmath246 , which is the state that alice actually prepares for the bit value @xmath247 in basis @xmath248 , and @xmath249 is alice s qubit system . one can choose any purification for @xmath245 , and in particular it should be chosen in such a way that it maximizes the inner product in eq . ( c2 ) or ( c3 ) . one can similarly define @xmath250 , and @xmath46 is introduced via considering a joint state involving the quantum coin as @xmath251 due to this change , the figures for the key generation rate have to be revised . as the examples of revised figures , we show the revised version of figs . 8 , 9 , 12 , and 13 , which are the most important figures for our main conclusions to hold . notice that there are only minor changes in figs . 8 and 9 and the changes in figs . 12 and 13 are relatively big . however , the big changes do not affect the validity of the main conclusions in our paper , which is the importance of the state preparation in mdiqkd and the fact that our schemes can generate the key with the practical channel mode that we have assumed . for the derivation of eq . ( [ 0 ] ) , we invoke koashi s proof @xcite . to apply koashi s proof , it is important to ensure that i ) one of the two parties holds a virtual * qubit * ( rather than a higher dimensional system ) and ii ) the fictitious measurements performed on the virtual qubit have to form * conjugate * observables . therefore , it is not valid to consider fidelity alone ( which allows arbitrary purifications that may not satisfy the conjugate observables requirement ) . fortunately , it turns out to be easy to modify our equation to satisfy the above two requirements . since the difference between eq . ( c2 ) and eq . ( c3 ) comes from whether we consider alice s virtual qubit or bob s virtual qubit , we focus only on eq . ( c2 ) and the same argument holds for eq . ( c3 ) . in koashi s proof , the security is guaranteed via two alternative tasks , ( i ) agreement on x ( key distillation basis ) and ( ii ) alice s or bob s preparation of an eigenstate of y , the conjugate basis of x , with use of an extra communication channel . the problem with the original ( i.e. uncorrected ) version of eq . ( 9 ) is the following . if we use the uncorrected version of eq . ( 9 ) in our paper , then the use of the fidelity means that the real part in eq . ( c2 ) is equivalent to @xmath252 with the maximization over _ all possible _ local unitary operators @xmath253 . in this case , if alice performs a measurement along x basis , then it violates the correspondence between her sending state @xmath246 and her qubit state @xmath254 in general , and thus , in the uncorrected version of eq . ( 9 ) in our paper , the argument based on the fidelity does not guarantee the security of the protocol . in contrast , with the corrected version of eq . ( 9 ) in our paper , since the maximization over @xmath46 and @xmath255 in eq . ( c2 ) preserves the relationship between alice s sending state and her qubit state as well as the conjugate relationship between x and y , we can apply koashi s proof for the security argument of the protocol . b. qi , c .- h . f. fung , h .- k . lo and x. ma , quant . 73 - 82 ( 2007 ) , y. zhao , c .- h . f. fung , b. qi , c. chen and h .- k . lo , phys . a 78 , 042333 ( 2008 ) , c .- h . f. fung , b. qi , k. tamaki and h .- k . lo , phys . a 75 , 032314 ( 2007 ) , f. xu , b. qi and h .- k . lo , new j. phys . 12 , 113026 ( 2010 ) . l. lydersen , c. wiechers , c. wittmann , d. elser , j. skaar and v. makarov , nature photonics 4 , pp . 686 - 689 ( 2010 ) , z. l. yuan , j. f. dynes and a. j. shields , nature photonics 4 , pp . 800 - 801 ( 2010 ) , l. lydersen , c. wiechers , c. wittmann , d. elser , j. skaar and v. makarov , nature photonics 4 , 801 ( 2010 ) , i. gerhardt , q. liu , a. lamas - linares , j. skaar , c. kurtsiefer and v. makarov , nature comm . 2 , 349 ( 2011 ) , l. lydersen , m. k. akhlaghi , a. h. majedi , j. skaar and v. makarov , arxiv : 1106.2396 . f. fung , k. tamaki , b. qi , h .- k . lo and x. ma , quant . inf . comp . 9 , 131 ( 2009 ) , l. lydersen , j. skaar , quant . inf . comp . * 10 * , 0060 ( 2010 ) , . mary , l. lydersen , j. skaar , phys . a * 82 * , 032337 ( 2010 ) . d. mayers and a. c .- c . yao , in proceedings of the 39th annual symposium on foundations of computer science ( focs98 ) , ( ieee computer society , washington , dc , 1998 ) , p. 503 , a. acin , n. brunner , n. gisin , s. massar , s. pironio , and v. scarani , phys . lett . * 98 * , 230501 ( 2007 ) . the definition of the four bell state is as follows . @xmath257=\frac{1}{\sqrt{2}}[{\left| 0_x \right\rangle}_{a1}{\left| 0_x \right\rangle}_{b1}-{\left| 1_x \right\rangle}_{a1}{\left| 1_x \right\rangle}_{b1}]$ ] , @xmath258=\frac{1}{\sqrt{2}}[{\left| 0_x \right\rangle}_{a1}{\left| 1_x \right\rangle}_{b1}+{\left| 1_x \right\rangle}_{a1}{\left| 0_x \right\rangle}_{b1}]$ ] , @xmath259=\frac{1}{\sqrt{2}}[{\left| 0_x \right\rangle}_{a1}{\left| 0_x \right\rangle}_{b1}+{\left| 1_x \right\rangle}_{a1}{\left| 1_x \right\rangle}_{b1}]$ ] , and @xmath260 . one of the most simplest proofs is shor - preskill s proof @xcite . the intuition of this proof is as follows . note that if alice and bob share some pairs of @xmath32 , ( i.e. , alice has one half of each pair and bob has the other half ) , then they can generate a secure key by performing @xmath6-basis measurement . the reason of the security is that this state is a pure state , which means that this state has no correlations with the third system including eve s system . due to the intervention by eve , alice and bob do not share this pure state in general , but instead they share noisy pairs . the basic idea of the proof is to consider the distillation of @xmath32 from the noisy pairs . for the distillation , note that @xmath32 is only one qubit pair state that has no bit errors in @xmath6-basis ( we call this error as the bit error ) and has no bit errors in @xmath7-basis ( we call this error as the phase error ) . it is known that if alice and bob employ the so - called css code ( calderbank - shor - steane code ) @xcite , then the noisy pairs are projected to a classical mixture of the four bell states , i.e. , @xmath32 , @xmath37 ( @xmath32 with the phase error ) , @xmath33 ( @xmath32 with the bit error ) , and @xmath38 ( @xmath32 with both the phase and bit errors ) . moreover , if alice and bob choose a correct css code , which can be achieved by random sampling procedure , then css code can detect the position of the erroneous pair with high probability . thus , by performing bit and phase flip operation depending on the detected error positions , alice and bob can distill some qubit pairs that are very close in fidelity to the product state of @xmath32 . in general , implementation of the above scheme requires a quantum computer . fortunately , shor - preskill showed that the bit error detection and bit flip operation can be done classically , and the phase error detection and phase flip operation need not be done , but exactly the same key can be obtained by the privacy amplification , so that we do not need to possess a quantum computer for the key distillation . r. renner , and r. koenig , proc . of tcc 2005 , lncs , springer , vol . * 3378 * ( 2005 ) , m. ben - or , and dominic mayers , arxiv : quant - ph/0409062 , m. ben - or , michal horodecki , d. w. leung , d. mayers , j. oppenheim , theory of cryptography : second theory of cryptography conference , tcc 2005 , j.kilian ( ed . ) springer verlag 2005 , vol . * 3378 * of lecture notes in computer science , pp . 386 - 406 .

in this paper , we study the unconditional security of the so - called measurement device independent quantum key distribution ( mdiqkd ) with the basis - dependent flaw in the context of phase encoding schemes . we propose two schemes for the phase encoding , the first one employs a phase locking technique with the use of non - phase - randomized coherent pulses , and the second one uses conversion of standard bb84 phase encoding pulses into polarization modes . we prove the unconditional security of these schemes and we also simulate the key generation rate based on simple device models that accommodate imperfections . our simulation results show the feasibility of these schemes with current technologies and highlight the importance of the state preparation with good fidelity between the density matrices in the two bases . since the basis - dependent flaw is a problem not only for mdiqkd but also for standard qkd , our work highlights the importance of an accurate signal source in practical qkd systems . + * note : we include the erratum of this paper in appendix c. the correction does not affect the validity of the main conclusions reported in the paper , which is the importance of the state preparation in mdiqkd and the fact that our schemes can generate the key with the practical channel mode that we have assumed . *

arxiv

massive stars play a fundamental role in driving the energy flow and material cycles that influence the physical and chemical evolution of galaxies . despite receiving much attention , their formation process remains enigmatic . observationally , the large distances to the nearest examples and the clustered mode of formation make it difficult to isolate individual protostars for study . it is still not certain , for instance , whether massive stars form via accretion ( similar to low mass stars ) or through mergers of intermediate mass stars . advances in instrumentation , have enabled ( sub ) arcsecond resolution imaging at wavelengths less affected by the large column densities of material that obscure the regions at shorter wavelengths . recent observations exploiting these capabilities have uncovered the environment surrounding _ individual _ massive protostellar systems . from analysis of @xmath42.3 @xmath0 m co bandhead emission , @xcite have inferred keplerian disks very closely surrounding ( within a few au ) four massive young stellar objects , while interferometric , mm - continuum observations , find the mass - function of protostellar dust clumps lies close to a salpeter value down to clump radii of 2000au @xcite . these high resolution observations point toward an accretion formation scenario for massive stars . further discrimination between the two competing models is possible by examining the properties , in particular the young stellar populations , of hot molecular cores . the mid - infrared ( mir ) window ( 7 - 25 @xmath0 m ) offers a powerful view of these regions . the large column densities of material process the stellar light to infrared wavelengths , and diffraction limited observations are readily obtained . recent observations indicate that class ii methanol masers exclusively trace regions of massive star formation @xcite and are generally either not associated or offset from uchii regions @xcite . @xcite ( hereafter m05 ) have carried out multi - wavelength ( mm to mir ) observations toward five star forming complexes traced by methanol maser emission to determine their large scale properties . they found that maser sites with weak ( @xmath510mjy ) radio continuum flux are associated with massive ( @xmath650m@xmath7 ) , luminous ( @xmath610@xmath8l@xmath7 ) and deeply embedded ( a@xmath940 mag ) cores characterising protoclusters of young massive ( proto)stars in an earlier evolutionary stage than uchii regions . the spatial resolution of the observations ( @xmath68@xmath2 ) was , however , too low to resolve the sources inside the clumps . details of the regions from observations in the literature are described in m05 . we have since observed three of the m05 regions at high spatial resolution to uncover the embedded sources inside the cores at mir wavelengths . the data were obtained with michelle . ] on the 8-m , gemini north telescope in queue mode , on the 18@xmath10 , 22@xmath11 and 30@xmath10 of march 2003 . each pointing centre was imaged with four n band silicate filters ( centred on 7.9 , 8.8 , 11.6 and 12.5 @xmath0 m ) and the qa filter ( centred on 18.5 @xmath0 m ) with 300 seconds on - source integration time . g173.49 and g188.95 were observed twice on separate nights and g192.60 observed once . the n and q band observations were scheduled separately due to the more stringent weather requirements at q band . the standard chop - nod technique was used with a chop throw of 15@xmath2 and chop direction selected from msx images of the region , to minimise off - field contamination . the spatial resolution calculated from standard star observations was @xmath4 0.36@xmath2 at 10 @xmath0 m and @xmath4 0.57@xmath2 at 18.5 @xmath0 m . the 32@xmath2x24@xmath2 field of view fully covered the dust emission observed by m05 in each region . particular care was taken to determine the telescope pointing position but absolute positions were determined by comparing the mir data to sensitive , high resolution , cm continuum , vla images of the 3 regions ( minier et al . in prep ) . similar spatial distribution and morphology of the multiple components allowed good registration between the images . the astrometric uncertainty in the vla images is @xmath41@xmath2 . flux calibration was performed using standard stars within 0.3 airmass of the science targets . there was no overall trend in the calibration factor as a result of changes in airmass throughout the observations . the standard deviation in the flux of standards throughout the observations was found to be 7.4 , 3.1 , 4.4 , 2.4 and 9% for the four n - band and 18.5 @xmath0 m filters respectively . the statistical error in the photometry was dominated by fluctuations in the sky background . upper flux limits were calculated from the standard deviation of the sky background for each filter and a 3@xmath12 upper detection limit is used in table 1 . similarly , a 3@xmath12 error value is quoted for the fluxes in table 1 ( typical values for the n and q band filters were 0.005 and 0.03 jy respectively ) . the flux densities for the standard stars were taken from values derived on the gemini south instrument , t - recs which shares a common filter set with michelle . regions confused with many bright sources were deconvolved using the lucy - richardson algorithm with 20 iterations . this was necessary to resolve source structure and extract individual source fluxes . the instrumental psf was obtained for each filter using a bright , non - saturated standard star . the results were reliable and repeatable near the brighter sources when using different stars for the psf and observations of the objects taken over different nights . as a further check , the standard stars were used to deconvolve other standards and reproduced point sources down to 1% of the peak value after 20 iterations , so only sources greater than 3% of the peak value were included in the final images . the resulting deconvolutions are shown in fig 1 . [ tab_sources ] [ cols="^,^,^,^,^,^,^,^,^,^,^,^ " , ] as the large scale clump dust and gas morphology appears simple and centrally peaked ( see m05 ) , we make the reasonable assumption that the protocluster centres coincide with the central peak of dust emission . the spatial distribution of the point sources within the protocluster is similar between the clumps with close point sources toward the cluster centre . the methanol masers are found closest to the brightest mir point source ( within the assumed 1@xmath2 pointing error from image registration ) . these sources have temperatures sufficient to evaporate methanol ice from the dust grains into the gas phase ( @xmath690k ) as well as sufficient luminosity of ir photons to pump the masing transition conditions models suggest are required for such emission @xcite . it is known that more massive stars favour cluster centres ( e.g. @xcite ) , but it is unclear whether they form there or migrate in from outside . we have used the simple - harmonic model of ballistic motion developed by @xcite to consider the motion of sources within the cores . using the measured column density and radius from m05 ( listed in table 2 ) , the time required for migration from the edge to the centre is @xmath4 @xmath13 years . this is comparable to the predicted hmc lifetime of 10@xmath14 years @xcite so we can not rule out the possibility of migration within the clumps . any sources having migrated to the centre in this way would have acquired a velocity of @xmath4 2 kms@xmath15 with respect to the clump . massive stars in clusters are observed to have a high companion star fraction @xcite . in the m16 cluster , @xcite observed massive stars ( earlier than b3 ) with visual companions separated by 1000 - 3000au . if multiple systems are bound from birth , it is likely some of the sources we have observed will belong to multiple systems , even though the companions may lie below the detection limit . however , all three regions show two or more point sources at close angular separation ( see insets of figure 1 ) corresponding to linear separations of 1700 to 6000au . we can not determine whether these stars are physically bound or simply close due to projection effects but we can calculate the instrumental sensitivity required to confirm or deny the association . assuming they are physically bound in a keplerian orbit , the maximum proper motions ( projection angle = 0@xmath16 ) of @xmath4 0.1 mas / year are too small to be detected on short temporal baselines . the maximum velocity difference ( projection angle = 90@xmath16 ) @xmath4 2 kms@xmath15 is achievable by high spectral resolution observations of any line features . the mass distribution of stars is generally well described as a power law through the initial mass function ( imf ) . given the mass of gas available to form stars , we may estimate the likelihood that a cluster will end up with the most massive stars that are observed in it . the fraction of gas that forms stars is given by the star formation efficiency ( sfe ) and is observationally found to be less than 50% in any cloud and to be @xmath17 33% for nearby embedded clusters @xcite . for a cluster whose total stellar mass is 120 ( 50 , 320 ) m@xmath7 ( equivalent to the gas mass determined for the three cores ) , @xcite estimate that the mean maximum mass that a star may have in it is 10 ( 5 , 20 ) m@xmath7 . this is comparable to the largest observed mass in two out of the three cases . however , we also observed several other stars in each cluster so can estimate the probability of generating stars of equal or greater mass than the remaining mass distribution . we did this by running monte - carlo simulations to populate 10@xmath14 clusters using @xcite , @xcite and @xcite imfs until the available gas mass was exhausted . we only considered clusters which contained a star of at least equal mass to the most massive observed . the simulations show that even using the salpeter form of the imf ( most biased toward forming high - mass stars ) and allowing 50% of the gas to form stars , it is difficult to generate the observed mass distributions ( probabilities @xmath18 10@xmath19 , 10@xmath20 , 10@xmath15 for the three cores respectively ) . by itself , this may not be significant for a single cluster . however , since the probability is low for all three sources studied , it is unlikely that the mass distribution of the most massive stars can be produced by sampling a standard form of the imf from the reservoir of gas available for star formation . this conclusion would not hold if there was a substantial stellar mass already in the cluster that remains unseen , or if much of the original gas mass had already been dispersed from the core due to star formation . the former requires a sfe close to unity and given the relatively quiescent state of the cores , the latter seems unlikely . a larger sample of young , massive protoclusters is required to draw general conclusions . however , in all three hot molecular cores traced by methanol maser emission we have found : * multiple , mir sources which can be separated into three morphological types : unresolved point sources ( p ) ; unresolved point source with weak surrounding extended emission ( pe ) and extended sources ( e ) . * the point sources lie at close angular separations . future high spatial and spectral resolution observations may be able to determine whether or not they are physically bound . * the methanol masers are found closest to the brightest mir point source ( within the assumed 1@xmath2 pointing accuracy ) . * cooler , extended sources dominate the luminosity . * the time scale for a source at the core edge to migrate to the centre is comparable to the hot molecular core lifetime , so it is not possible to rule out large protostellar motions within the core . * from the derived gas mass of the core and mass estimates for the sources , monte carlo simulations show that it is difficult to generate the observed distributions for the most massive cluster members from the gas in the core using a standard form of the imf . this conclusion would not hold , however , if most of the original gas has already formed stars , or has been dispersed such that the original core mass is much greater than now observed . s.l . would like to thank alistair glass , scott fisher , tony wong and melvin hoare for helpful discussion of the data and scientific input . we thank the anonymous referee for the thorough response and insightful comments . this work was made possible by funding from the australian research council and unsw . the gemini observatory is operated by the association of universities for research in astronomy , inc . , under a cooperative agreement with the nsf on behalf of the gemini partnership : nsf ( usa ) , pparc ( uk ) , nrc ( canada ) , conicyt ( chile ) , arc ( australia ) , cnpq ( brazil ) and conicet ( argentina ) . b. t. , 1989 , interstellar extinction in the infrared . infrared spectroscopy in astronomy , proceedings of the 22nd eslab symposium held in salamanca , spain , 7 - 9 december , 1988 . edited by b.h . kaldeich . esa sp-290 . european space agency , 1989 . , p.93 , pp 93+ g. , bouvier j. , eislffel j. , simon t. , 2001 , in asp conf . ser . 243 : from darkness to light : origin and evolution of young stellar clusters statistical properties of visual binaries as tracers of the formation and early evolution of young stellar clusters .

we present high resolution , mid - infrared images toward three hot molecular cores signposted by methanol maser emission ; g173.49 + 2.42 ( s231 , s233ir ) , g188.95 + 0.89 ( s252 , afgl-5180 ) and g192.60 - 0.05 ( s255ir ) . each of the cores was targeted with michelle on gemini north using 5 filters from 7.9 to 18.5 @xmath0 m . we find each contains both large regions of extended emission and multiple , luminous point sources which , from their extremely red colours ( @xmath1 ) , appear to be embedded young stellar objects . the closest angular separations of the point sources in the three regions are 0.79 , 1.00 and 3.33@xmath2 corresponding to linear separations of 1,700 , 1,800 and 6,000au respectively . the methanol maser emission is found closest to the brightest mir point source ( within the assumed 1@xmath2 pointing accuracy ) . mass and luminosity estimates for the sources range from 3 - 22 m@xmath3 and 50 - 40,000 l@xmath3 . assuming the mir sources are embedded objects and the observed gas mass provides the bulk of the reservoir from which the stars formed , it is difficult to generate the observed distributions for the most massive cluster members from the gas in the cores using a standard form of the imf . masers stars : formation techniques : high angular resolution stars : early type stars : mass function infrared : stars .

arxiv

modified theories of gravity have attained much attention after the discovery of expanding accelerated universe . the basic ingredient responsible for this tremendous change in cosmic history is some mysterious type force having repulsive nature dubbed as dark energy . the enigmatic nature of this energy has motivated many researchers to unveil its hidden characteristics which are still not known . modified gravity approach is considered as the promising and optimistic scenario among several other proposals that have been presented to explore the salient features of dark energy . these modified theories are established by adding or replacing curvature invariants and their corresponding generic functions in the einstein - hilbert action . lovelock theory of gravity is the direct generalization of general relativity ( gr ) in @xmath3-dimensions which coincides with gr in @xmath4-dimensions @xcite . the ricci scalar @xmath5 is known as first lovelock scalar while gauss - bonnet ( gb ) invariant is the second lovelock scalar yielding einstein - gauss - bonnet gravity in @xmath6-dimensions @xcite . the gb invariant is a linear combination with an interesting feature that it is free from spin-2 ghost instabilities defined as @xcite @xmath7 where @xmath8 and @xmath9 are the ricci and riemann tensors , respectively . this quadratic curvature invariant is a topological term in 4-dimensions which possesses trivial contribution in the field equations . to discuss the dynamics of gb invariant in 4-dimensions , there are two interesting scenarios either to couple @xmath1 with scalar field or to add generic function @xmath10 in the einstein - hilbert action . the first scheme naturally appears in the effective action in string theory which investigates singularity - free cosmological solutions @xcite . the second approach known as @xmath10 gravity is introduced as an alternative for dark energy which successfully discusses the late - time cosmological evolution @xcite . this modified theory of gravity is endowed with a quite rich cosmological structure as well as consistent with solar system constraints @xcite . the current cosmic accelerated expansion has also been discussed in modified theories of gravity involving the curvature - matter coupling . harko et al . @xcite established @xmath11 gravity to study the curvature - matter coupling . recently , we introduced the curvature - matter coupling in @xmath10 gravity named as @xmath0 theory of gravity @xcite . this coupling yields non - zero covariant divergence of the energy - momentum tensor and an extra force appears due to which massive test particles follow non - geodesic trajectories while geodesic lines of geometry are followed by the dust particles . shamir and ahmad @xcite constructed some cosmologically viable models in @xmath0 gravity using noether symmetry approach . it is mentioned here that cosmic expansion can be obtained from geometric as well as matter components in such coupling . the reconstruction as well as stability of cosmic evolutionary models in modified theories of gravity are the captivating issues in cosmology . in reconstruction technique , any known cosmic solution is used in the modified field equations to find the corresponding function which reproduces the given evolutionary cosmic history . in stability analysis , the isotropic and homogeneous perturbations are usually considered in which hubble parameter as well as energy density are perturbed to examine the background stability as time evolves @xcite . nojiri et al . @xcite formulated the reconstruction scheme to reproduce some cosmological models in @xmath12 gravity . elizalde et al . @xcite applied the same scenario for @xmath13cdm cosmology ( @xmath13 denotes cosmological constant while cdm stands for cold dark matter ) in @xmath14 gravity as well as in modified gb theories of gravity . the stability of power - law solutions are also discussed in modified gravity theories @xcite . sez - gmez @xcite explored the cosmological solutions in @xmath12 horava - lifshitz gravity and analyzed their stability against first order perturbations around frw universe . myrzakulov and his collaborators @xcite discussed the cosmological models and found that @xmath10 gravity could successfully explain the cosmic evolutionary history . jamil et al . @xcite reconstructed the cosmological models in @xmath11 gravity and found that numerical analysis for hubble parameter is in good agreement with observational data for redshift parameter @xmath15 . the stability of de sitter , power - law solutions as well as @xmath13cdm are analyzed in the context of @xmath14 gravity @xcite . salako et al . @xcite studied the cosmological reconstruction , stability as well as thermodynamics including first and second laws for @xmath13cdm model in generalized teleparallel theory of gravity . sharif and zubair @xcite demonstrated that @xmath11 gravity can reproduce @xmath13cdm model , phantom or non - phantom eras , de sitter universe and power - law cosmic history . they also analyzed the stability of reconstructed de sitter as well as power - law solutions . in this paper , we reconstruct various cosmological models including de sitter universe , power - law solutions , phantom or non - phantom eras and @xmath13cdm model in @xmath0 theory . we also analyze the stability against linear homogeneous perturbations for de sitter as well as power - law solutions . the paper has the following format . in section * 2 * , we formulate the modified field equations while section * 3 * is devoted to reconstruct some known cosmological solutions in this gravity . section * 4 * analyzes the stability of specific solutions against linear perturbations around frw universe model . the results are summarized in the last section . the action for @xmath0 gravity is defined as @xcite @xmath16 where @xmath17 and @xmath18 represent coupling constant , determinant of the metric tensor ( @xmath19 ) and lagrangian associated with matter distribution , respectively . varying eq.([1 ] ) with respect to @xmath19 , we obtain the field equations @xmath20f_{\mathcal{g}}(\mathcal{g},t ) \\\nonumber&-&[2rg_{\alpha\beta } \box-4r_{\alpha\beta}\box-2r\nabla_{\alpha}\nabla_{\beta } + 4r^{\mu}_{\beta}\nabla_{\alpha}\nabla_{\mu } + 4r^{\mu}_{\alpha}\nabla_{\beta}\nabla_{\mu}\\\label{2}&- & 4g_{\alpha\beta}r^{\mu\nu } \nabla_{\mu}\nabla_{\nu}+4r_{\alpha\mu\beta\nu } \nabla^{\mu}\nabla^{\nu}]f_{\mathcal{g}}(\mathcal{g},t)=0,\end{aligned}\ ] ] where @xmath21 ( @xmath22 denotes a covariant derivative ) and @xmath23 is the energy - momentum tensor . the expressions for @xmath23 and @xmath24 are @xcite @xmath25 where we have assumed that @xmath18 depends only on @xmath19 rather than its derivatives . the non - zero divergence of @xmath23 is given by @xmath26.\end{aligned}\ ] ] the above equations indicate that the complete dynamics of @xmath0 gravity is based on the suitable choice of @xmath18 . the energy - momentum tensor for perfect fluid is @xmath27 where @xmath28 and @xmath29 represent the four velocity , energy density and pressure of matter distribution , respectively . in this case , the expression for @xmath24 becomes @xmath30 where @xmath31 . the line element for frw universe model is given by @xmath32 where @xmath33 is the scale factor . using eqs.([5])-([7 ] ) in ( [ 2 ] ) , we obtain the corresponding field equation as follows @xmath34 where @xmath35 and dot represents derivative with respect to time . the non - zero continuity equation ( [ 4 ] ) takes the form @xmath36.\ ] ] the standard conservation law holds if right hand side of this equation vanishes . for equation of state @xmath37 ( @xmath38 is the equation of state parameter ) , eq.([9 ] ) yields @xmath39 with additional constraint @xmath40 we rewrite the above equations in terms of new variable @xmath41 known as e - folding instead of @xmath42 which is also related with redshift parameter @xmath43 as @xcite @xmath44 using the above definition of @xmath41 , eqs.([8 ] ) and ( [ 9 ] ) become @xmath45,\end{aligned}\ ] ] where @xmath46 and prime denotes derivative with respect to @xmath41 . the simplest choice of @xmath0 model is @xmath47 which possesses no direct non - minimally coupling between curvature and matter . for this particular model , the field equation ( [ 13 ] ) splits into a set of two ordinary differential equations as @xmath48 @xmath49 where @xmath50 and @xmath51 . the field equations for perfect fluid matter distribution in @xmath10 gravity is recovered if @xmath52 vanishes while gr is achieved for @xmath53 . in this section , we reproduce different cosmological scenarios including de sitter universe , power - law solutions , phantom / non - phantom eras and @xmath13cdm model in @xmath0 gravity . the de sitter cosmic evolution is interesting and well - known as it elegantly describes current expansion of the universe . the scale factor of this evolutionary model grows exponentially with constant hubble parameter @xmath54 , defined as @xcite @xmath55 where @xmath56 is an integration constant . equation ( [ 10 ] ) gives energy density of the form @xmath57 where @xmath58 and @xmath59 is a constant . using eqs.([1d ] ) and ( [ 2d ] ) in ( [ 8 ] ) , we obtain @xmath60 where @xmath61 is the gb invariant at @xmath54 . the solution of the above differential equation is @xmath62 where @xmath63 s @xmath64 are integration constants . since we have used the continuity equation ( [ 10 ] ) in eq.([3d ] ) , so we must constrain its solution . using the above equation with eq.([11 ] ) , we obtain the following functions @xmath65 where @xmath66 s @xmath67 are constants in terms of @xmath38 and @xmath68 given in appendix * a*. for the model ( [ 15 ] ) , we have @xmath69 where the first equation corresponds to de sitter universe in the absence of matter contents in @xmath10 gravity @xcite . using the constraint ( [ 11 ] ) , the second equation becomes @xmath70 the solution of eqs.([7d ] ) and ( [ 8d ] ) leads to @xmath71 where @xmath72 s are constants of integration . equations ( [ 4d ] ) and ( [ 9d ] ) indicate that de sitter expansion can also be described in @xmath0 gravity . power - law solutions have significant importance to discuss different evolutionary phases of the universe in modified theory . these solutions describe the decelerated as well as accelerated cosmic eras which are characterized by the scale factor as @xcite @xmath73 the cosmic decelerated phase is observed for @xmath74 including the radiation @xmath75 as well as dust @xmath76 dominated eras while @xmath77 covers the accelerated phase of the universe . for this scale factor , the gb invariant takes the form @xmath78 using eqs.([10 ] ) , ( [ 1p ] ) and ( [ 2p ] ) , the field equation becomes @xmath79 whose solution is given by @xmath80 where @xmath81 s are integration constants and @xmath82,\\\nonumber\gamma_{2}&=&\left[\frac{3}{4}\lambda \tilde{c}_{2}(1+\omega)\{2\tilde{c}_{2}\lambda(1+\omega)+2(\lambda-1 ) -8\}+\frac{1}{4}(\lambda-1)(\lambda-7)+4\right.\\\nonumber&+&\left . 8\tilde{c}_{2}(\lambda-1)\left(\frac{1+\omega}{1 - 3\omega}\right ) \right]^{\frac{1}{2}},\quad\gamma_{3}=-\frac{1}{2}\left(\frac{1 - 3\omega } { 1+\omega}\right),\quad\gamma_{4}=\frac{2\kappa^2}{\omega-3},\\\nonumber \gamma_{5}&=&\left(\frac{18\lambda^{3}(1 - 3\omega)^{\frac{3\lambda ( 1+\omega)-2}{3\lambda(1+\omega)}}}{3\lambda(1 - 3\omega)+4}\right ) \rho_{0}^{\frac{-2}{3\lambda(1+\omega)}},\quad\gamma_{6}=\frac{2 } { 3\lambda(1+\omega)}.\end{aligned}\ ] ] inserting eq.([4p ] ) in ( [ 11 ] ) , we obtain @xmath83 where @xmath84 s and @xmath85 s @xmath86 are given in appendix * a*. now we find the expression of @xmath0 for the choice of model ( [ 15 ] ) . the differential equation ( [ 3p ] ) yields two ordinary differential equations in variables @xmath1 and @xmath2 . the first is the second order cauchy - euler s equation related to curvature given by @xmath87 whose solution is given by @xmath88 where @xmath89 s are constants of integration which is consistent with power - law solutions in @xmath10 gravity @xcite . the second equation is obtained using the additional constraint ( [ 11 ] ) as @xmath90 the solution of this equation corresponds to matter distribution given by @xmath91}\left(\frac{t}{\rho_{0}(1 - 3\omega)}\right ) ^{\frac{2}{3\lambda(1+\omega)}},\end{aligned}\ ] ] where @xmath92 s @xmath93 are integration constants . consequently , @xmath0 model becomes @xmath94}\\\label{7p}&\times & \left(\frac{t}{\rho_{0}(1 - 3\omega)}\right)^{\frac{2}{3\lambda(1+\omega)}}.\end{aligned}\ ] ] thus , the power - law solutions are reconstructed which may be helpful to explore the expansion history of the universe in this modified theory of gravity . here , we reconstruct @xmath0 model which can explain the system including both phantom and non - phantom eras . in the einstein gravity , the hubble parameter describing the phantom as well as non - phantom matter distribution is given by @xcite @xmath95 where @xmath96 and @xmath97 represent the model parameter , energy densities of phantom and non - phantom matter fluids , respectively . when the scale factor is large , the first term on right hand side dominates which corresponds to the phantom era of the universe with @xmath98 . the non - phantom era in the early universe is observed for @xmath99 when the scale factor is small and the second term dominates . we rewrite @xmath100 in terms of a new function @xmath101 as @xmath102 so that eq.([1q ] ) becomes @xmath103 where @xmath104 and @xmath105 . the gb invariant takes the form @xmath106 inserting eq.([2q ] ) in ( [ 3q ] ) , we obtain a quadratic equation in @xmath107 whose solution is given by @xmath108 for the sake of simplicity , we consider @xmath109 so that it reduces to @xmath110 using eqs.([2q ] ) and ( [ 4q ] ) in ( [ 8 ] ) , we have @xmath111 which is a complicated partial differential equation whose analytical solution can not be found . to find the reconstructed @xmath0 model , we consider its particular form ( [ 15 ] ) which provides the following set of differential equations @xmath112 where we have used the additional constraint in the second equation . solving these equations , it follows that @xmath113-\frac{2\kappa^{2}t}{1 - 3\omega}.\end{aligned}\ ] ] where @xmath114 s are constants of integration . thus , phantom and non - phantom cosmic history can be discussed in @xmath0 gravity . now we apply the usual reconstruction technique to reproduce the @xmath13cdm cosmology in this gravity . in gr , the @xmath13cdm cosmological evolution is discussed by adding @xmath13 in the einstein - hilbert action whereas we reconstruct such evolution in the absence of @xmath13 in the action ( [ 1 ] ) . the hubble parameter for @xmath13cdm model is given by @xcite @xmath115 the first term indicates the contribution of cdm while the second term corresponds to @xmath13 . the hubble parameter in terms of gb invariant takes the form @xmath116 equation ( [ 1h ] ) gives @xmath117 using eqs.([2h ] ) and ( [ 3h ] ) in ( [ 13 ] ) , we obtain @xmath118 this equation can not be solved analytically hence we solve it for a specific @xmath0 model ( [ 15 ] ) with dust fluid . the corresponding equations become @xmath119 using constraint ( [ 11 ] ) , the solution of eq.([4h ] ) is given by @xmath120 where @xmath121 s are constants of integration while solution of eq.([5h ] ) can not be found . let us consider the case when @xmath122 with the assumption @xmath123 which reduces eq.([5h ] ) to @xmath124 whose solution yields @xmath125 where @xmath126 s are integration constants . consequently , @xmath0 takes the form @xmath127 provided that @xmath128 . here @xmath0 gravity can not explain @xmath13cdm cosmological evolution rather it corresponds to cdm evolution . in this section , we analyze stability of some cosmological evolutionary solutions about linear homogeneous perturbations in this modified gravity . we construct the perturbed field as well as continuity equations using isotropic and homogeneous universe model for both general and particular cases including de sitter and power - law solutions . we assume a general solution @xmath129 which satisfies the basic field equations for frw universe model in @xmath0 gravity . in terms of the above solution , the expressions for @xmath130 and @xmath131 are @xmath132 for any particular @xmath0 model that can regenerate the above solution ( [ 1 t ] ) , the following equation of motion as well as non - zero divergence of the energy - momentum tensor must be satisfied @xmath133,\end{aligned}\ ] ] where superscript @xmath134 denotes that the function and its corresponding derivatives are calculated at @xmath135 and @xmath136 . if the conservation law holds , we get energy density in terms of @xmath137 as @xmath138 the first order perturbations in hubble parameter and energy density are defined as @xmath139 where @xmath140 and @xmath141 are the perturbation parameters . in order to analyze first order perturbations about the solution ( [ 1 t ] ) , we apply the series expansion on the function @xmath0 as @xmath142 where @xmath143 involves the terms proportional to quadratic or higher powers of @xmath1 and @xmath2 while only the linear terms are considered . using eqs.([6 t ] ) and ( [ 7 t ] ) in ( [ 8 ] ) , we obtain the following perturbed field equation @xmath144 where @xmath145 s @xmath146 are given in appendix * a*. inserting these perturbations in eq.([9 ] ) , the perturbed continuity equation is @xmath147 where @xmath148 s are provided in appendix * a*. if the conversation law holds in this modified gravity , eq.([9 t ] ) reduces to @xmath149 the perturbed equations ( [ 8 t ] ) and ( [ 9 t ] ) are helpful to analyze the stability of any specific frw cosmological evolutionary model in @xmath0 gravity . for the particular model ( [ 15 ] ) , these perturbed equations reduce to @xmath150 where the coefficients of @xmath151 and their derivatives are expressed in appendix * a*. in the following subsections , we investigate the stability of de sitter and power - law solutions . consider the de sitter solution @xmath152 , the perturbed equation ( [ 8 t ] ) takes the form @xmath153 f_{\mathcal{g}t}^{0}-3456\rho_{*}h_{0}^{8}(1 - 3\omega)(1+\omega ) f_{\mathcal{gg}t}^{0}\right)\delta+12\rho_{*}h_{0}^{3}\\\nonumber & \times&(1 - 3\omega)f_{\mathcal{g}t}^{0}\dot{\delta}_{m}+\left(\kappa^{2 } \rho_{*}+\frac{1}{2}\rho_{*}(3-\omega)f_{t}^{0}+\rho_{*}^{2}(1 - 3\omega ) ( 1+\omega)f_{tt}^{0}\right.\\\nonumber&-&\left.12\rho _ { * } h_{0}^{2}(1 - 3\omega)[h_{0}^{2}+3(1+\omega)h_{0}^{2}]f_{\mathcal{g}t}^{0 } -36\rho_{*}^{2}h_{0}^{4}(1 - 3\omega)^{2}(1+\omega ) \right.\\\label{13t}&\times&\left.f_{\mathcal{g}tt}^{0}\right)\delta_{m}=0,\end{aligned}\ ] ] where the superscript @xmath154 represents that the function and its corresponding derivatives are evaluated at @xmath155 and @xmath156 . we consider the conserved perturbed equation for stability analysis since the de sitter solutions are constructed using the constraint ( [ 11 ] ) in the previous section . the numerical technique is used to solve eqs.([10 t ] ) and ( [ 13 t ] ) for the model ( [ 5d ] ) . the evolution of @xmath140 and @xmath141 are shown in figure * 1*. we consider @xmath157 and @xmath158 throughout the stability analysis of de sitter universe models whereas integration constants are @xmath159 and @xmath160 . figure * 1 * shows smooth behavior of @xmath140 ( left ) and @xmath141 ( right ) which do not decay in late times indicating that de sitter model ( [ 5d ] ) is unstable . the stability analysis of model ( [ 6d ] ) with same integration constants is shown in figure * 2*. in the left panel , it is observed that small oscillations are produced about @xmath161 while it decays in late times , thus the model ( [ 6d ] ) shows stable behavior against perturbations . for model ( [ 9d ] ) , eq.([13 t ] ) becomes @xmath162 figure * 3 * represents the behavior of @xmath140 and @xmath141 for model ( [ 9d ] ) with integration constants @xmath163 and @xmath164 . it is shown that oscillations in perturbation parameters are produced initially as shown in figure * 3*. this oscillating behavior is clearly observed in figure * 4 * which decays in future for both @xmath140 as well as @xmath141 and hence the solution becomes stable . here we investigate the stability of power - law solutions . these solutions describe the accelerated as well as decelerated cosmological evolutionary phases in the background of frw universe . we first consider the reconstructed power - law solution ( [ 5p ] ) and numerically solve eqs.([8 t ] ) and ( [ 10 t ] ) . for this model , we choose integration constants @xmath165 and @xmath166 figure * 5 * shows the oscillating behavior of perturbed parameters @xmath167 for the cosmic accelerated era with @xmath168 and @xmath169 . the perturbations around the power - law solutions decay in future leading to stable results . the radiation ( @xmath170 and @xmath171 ) as well as matter ( @xmath172 and @xmath173 ) dominated eras can not be discussed for the model ( [ 5p ] ) because singular as well as complex terms appear which lead to non - physical case . secondly , we consider the model ( [ 6p ] ) and analyze its behavior against linear perturbations . figure * 6 * shows the fluctuating behavior of considered perturbations in the cosmic accelerated phase with @xmath168 and @xmath169 . here , we choose @xmath174 and @xmath166 it is observed that the oscillating behavior disappears in future while both perturbation parameters will not decay in late times leading to unstable cosmological solutions . the considered model can not explain the cosmological evolution corresponding to matter and radiation dominated eras like previous model ( [ 5p ] ) . lastly , we explore the stability of model ( [ 7p ] ) with integration constants @xmath175 and @xmath176 . figure * 7 * represents the evolution of ( @xmath177 ) versus time for @xmath178 with @xmath168 . the left panel shows that the oscillations of @xmath140 decay in late times while fluctuations of @xmath141 remain present in future . since a complete perturbation against any cosmological solution includes the matter perturbations therefore , the solutions are unstable . in this paper , we have employed the reconstruction scheme to @xmath0 gravity in the background of isotropic and homogeneous universe model to reproduce some important cosmological models . the basic aspect of this modified gravity is the coupling between curvature and matter components which yields non - zero divergence of the energy - momentum tensor . we have imposed additional constraint to obtain the standard conservation equation which has been used to explain the cosmic evolution in this gravity . the de sitter and power - law solutions have been reconstructed for general as well as particular cases which are of great interest and have significant importance in cosmology . we have also reconstructed the @xmath0 model which can explain cosmic history of the phantom as well as non - phantom phases of the universe . similar reconstruction technique is carried out for @xmath13cdm model . in this case , we have found that the considered gravity fails to reproduce @xmath13cdm cosmology for both cases . for the specific form of function , this result is consistent with @xmath10 gravity in the absence of matter @xcite . on physical grounds , the stability analysis of different forms of generic function leads to classify the modified theories of gravity . we have applied the first order perturbations to hubble parameter and energy density to analyze the stability of models which reproduce de sitter and power - law cosmic history . we have perturbed the field equation as well as conservation law whose numerical solutions provide the stable / unstable results . * for the de sitter universe , the evolution of perturbation has been plotted against time as shown in figures * 1 * -*4*. these indicate that models ( [ 6d ] ) and ( [ 9d ] ) are stable against linear perturbations . * for the power - law universe , the stability analysis is given in figures * 5 * -*7*. it is found that @xmath0 gravity fails to reproduce matter and radiation dominated eras while stable results are obtained for accelerated phase of the universe for model ( [ 5p ] ) . we conclude that the cosmological reconstruction and stability analysis might restrict @xmath0 gravity in the background of frw universe . the expressions for @xmath179 s in eqs.([5d ] ) and ( [ 6d ] ) are @xmath180\\\nonumber & \times&[1+\omega-36c_{1}h_{0}^{4}(1 - 3\omega)]^{-2},\\\nonumber \xi_{2}&=&18c_{1}h_{0}^{4}[(1 - 11\omega)(1-\omega^{2})-8c_{1 } h_{0}^{4}\{2(2 + 59\omega^{2})-11\omega(5 - 3\omega^{2})\}]\\\nonumber & \times&[(1+\omega)(1 - 24c_{1}h_{0}^{4})\{1+\omega-6c_{1}h_{0}^{4 } ( 5 - 4\omega-33\omega^{2})\}]^{-1},\\\nonumber\xi_{3}&=&-[18c_{1 } h_{0}^{4}(1 - 32c_{1}h_{0}^{4})-3\omega\{1 - 6c_{1}h_{0}^{4}(3 - 352c_{1 } h_{0}^{4})\}\\\nonumber&-&2\omega^{2}\{1 - 9c_{1}h_{0}^{4 } ( 7 - 1248c_{1}h_{0}^{4})\}+\omega^{3}\{1 - 54c_{1}h_{0}^{4}(7 - 480c_{1 } h_{0}^{4})\}]\\\nonumber&\times&[(1 - 3\omega)(1 - 24c_{1}h_{0}^{4 } ) \{1+\omega-6c_{1}h_{0}^{4}(5 - 4\omega-33\omega^{2})\}]^{-1}.\end{aligned}\ ] ] the expressions for @xmath84 s and @xmath85 s in eqs.([5p ] ) and ( [ 6p ] ) are @xmath181 where @xmath182,\\\nonumber\gamma_{8}&=&\frac{\tilde{c}_{2 } } { 6\lambda}\left[6\tilde{c}_{2}\lambda(1+\omega)^{2}-3\lambda ( 1 + 5\omega+2\omega^2)+2(\gamma_{1}-\gamma_{2})\right].\end{aligned}\ ] ] the values of @xmath145 s in eq.([8 t ] ) are given as follows @xmath183f_{\mathcal{g}t}^{*}-864(1+\omega)(1 - 3\omega)\rho _ { * } h_{*}^{6}(4h_{*}^{2}\\\nonumber&+&\dot{h}_{*})f_{\mathcal{gg}t}^ { * } , \\\nonumber\chi_{4}&=&12(1 - 3\omega)\rho_{*}h_{*}^{3}f_{\mathcal{g}t}^ { * } , \\\nonumber\chi_{5}&=&\kappa^{2}\rho_{*}-\frac{1}{2}(\omega-3 ) \rho_{*}f_{t}^{*}+(1 - 3\omega)(1+\omega)\rho_{*}^{2}f_{tt}^{*}-12 ( 1 - 3\omega)\rho_{*}h_{*}^{2}\\\nonumber&\times&[(4 + 3\omega)h_{*}^{2 } + \dot{h}_{*}]f_{\mathcal{g}t}^{*}+288(1 - 3\omega)\rho_{*}h_{*}^{4 } ( 4h_{*}^{2}\dot{h}_{*}+2\dot{h}_{*}^{2}+h_{*}\ddot{h } _ { * } ) \\\nonumber&\times&f_{\mathcal{gg}t}^{*}-36(1+\omega ) ( 1 - 3\omega)^{2}\rho_{*}^{2}h_{*}^{4}f_{\mathcal{g}tt}^{*}.\end{aligned}\ ] ] the expressions for @xmath148 s are @xmath184f_{\mathcal{g}t}^ { * } -72\rho_{*}^{2}h_{*}^{2}(1 - 3\omega)(1+\omega)^{2}(4h_{*}^{2 } \\\nonumber&+&\dot{h}_{*})f_{\mathcal{g}tt}^{*}+576\rho_{*}h_{*}^{3 } ( 1+\omega)(4h_{*}^{2}+\dot{h}_{*})(4h_{*}^{2}\dot{h}_{*}+2 \dot{h}_{*}^{2}\\\nonumber&+&4h_{*}\ddot{h}_{*})f_{\mathcal{gg}t}^ { * } , \\\nonumber\upsilon_{2}&=&-12\rho_{*}h_{*}^{2}(1+\omega)\left[3h_{*}^{2 } ( 1-\omega)-4(2h_{*}^{2}+3\dot{h}_{*})\right]f_{\mathcal{g}t}^ { * } + 576\rho_{*}h_{*}^{2}\\\nonumber&\times&(1+\omega)(4h_{*}^{2 } \dot{h}_{*}+2\dot{h}_{*}^{2}+h_{*}\dot{h}_{*})f_{\mathcal{gg}t}^ { * } -72\rho_{*}^{2}h_{*}^{4}(1 - 3\omega)\\\nonumber&\times&(1+\omega^{2 } ) f_{\mathcal{g}tt}^{*},\\\nonumber\upsilon_{3}&=&24\rho_{*}h_{*}^{3 } ( 1+\omega)f_{\mathcal{g}t}^{*},\\\nonumber\upsilon_{4}&=&-\frac{3}{2 } \rho_{*}h_{*}(1+\omega)\left[(1-\omega)f_{t}^{*}+2\rho_{*}^{2 } ( 1+\omega)(1 - 3\omega)^{2}f_{ttt}^{*}\right]-\frac{15}{2 } \rho_{*}^{2}h_{*}\\\nonumber&\times&(1 - 3\omega)(1+\omega)^{2}f_{tt}^ { * } + 24\rho_{*}h_{*}(1+\omega)(4h_{*}^{2}\dot{h}_{*}+2\dot{h}_{*}^{2}+h _ { * } \ddot{h}_{*})\\\nonumber&\times&\left[f_{\mathcal{g}t}^{*}+\rho _ { * } ( 1 - 3\omega)f_{tt\mathcal{g}}^{*}\right],\\\nonumber\upsilon_{5 } & = & \rho_{*}\left(\kappa^2+\frac{1}{2}(3-\omega)f_{t}^{*}\right ) + ( 1+\omega)(1 - 3\omega)\rho_{*}^{2}f_{tt}^{*}.\end{aligned}\ ] ] for model ( [ 15 ] ) , the coefficients of @xmath151 have the following expressions @xmath185-\frac{15}{2 } \rho_{*}^{2}h_{*}\\\nonumber&\times&(1 - 3\omega)(1+\omega)^{2 } \mathcal{f}_{tt}^{*},\\\nonumber\hat{\upsilon}_{5}&=&\rho _ { * } \left(\kappa^2+\frac{1}{2}(3-\omega)\mathcal{f}_{t}^{*}\right ) + ( 1+\omega)(1 - 3\omega)\rho_{*}^{2}\mathcal{f}_{tt}^{*}.\end{aligned}\ ] ]

the aim of this paper is to reconstruct and analyze the stability of some cosmological models against linear perturbations in @xmath0 gravity ( @xmath1 and @xmath2 represent the gauss - bonnet invariant and trace of the energy - momentum tensor , respectively ) . we formulate the field equations for both general as well as particular cases in the context of isotropic and homogeneous universe model . we reproduce the cosmic evolution corresponding to de sitter universe , power - law solutions and phantom / non - phantom eras in this theory using reconstruction technique . finally , we study stability analysis of de sitter as well as power - law solutions through linear perturbations . * keywords : * reconstruction ; stability analysis ; modified gravity . + * pacs : * 04.50.kd ; 98.80.-k .

arxiv

majorana fermions ( mfs ) are real solutions of the dirac equation and which are their own antiparticles @xmath0 @xcite . although proposed originally as a model for neutrinos , mfs have recently been predicted to occur as quasi - particle bound states in engineered condensed matter systems @xcite . this exotic particle obeys non - abelian statistics , which is one of important factors to realize subsequent potential applications in decoherence - free quantum computation @xcite and quantum information processing @xcite . over the recent few years , the possibility for hosting mfs in exotic solid state systems focused on topological superconductors @xcite . currently , various realistic platforms including topological insulators @xcite , semiconductor nanowires ( snws ) @xcite , and atomic chains @xcite have been proposed to support majorana states based on the superconducting proximity effect . although various schemes have been presented , observing the unique majorana signatures experimentally is still a challenging task to conquer . mfs are their own antiparticles , and they are predicted to appear in tunneling spectroscopy experiments , in which majoranas manifest themselves as characteristic zero - bias peaks ( zbps ) @xcite . the theoretical predictions of zbps have been observed experimentally in snws which are interpreted as the signatures of mfs mourikv , dasa , dengmt , churchillhoh , finckadk . remarkably , nadj - perge et al . @xcite recently designed a chain of magnetic fe atoms deposited on the surface of an s - wave superconducting pb with strong spin - orbit interactions , and reported the striking observation of a zbp at the end of the atomic chains with stm , which provides evidence for majorana zero modes . however , these above experimental results can not serve as definitive evidences to prove the existence of mfs in condensed matter systems , and it is also a major challenge in these experiments to uniquely distinguish majoranas from conventional fermionic subgap states . the first reason is that the zero - bias conductance peaks can also appear in terms of the other mechanisms @xcite , such as the zero - bias anomaly due to kondo resonance @xcite and the disorder or band bending in the snw @xcite . the second one is that andreev bound states in a magnetic field can also exhibit a zero - energy crossing as a function of exchange interaction or zeeman energy @xcite , and therefore give rise to similar conductance features . as far as we know , most of the experimental evidences for majorana bound states largely relies on measurements of the tunneling conductance at present , and the observation of majorana signature based on electrical methods still remains a subject of debate . identifying mfs only through tunnel spectroscopy is somewhat problematic . therefore , to obtain definitive signatures of mfs , alternative setups or proposals for detecting mfs are necessary . here , we will propose an alternative all - optical scheme to detect mfs . benefitting from recent advances in nanotechnology and nanofabrication , nanostructures such as quantum dots ( qds ) and nanomechanical resonators ( nrs ) have been obtained significant progress in modern nanoscience and nanotechnology . qd , as a simple stationary atom with well optical property @xcite , lays the foundation for numerous possible applications urbaszekb . due to high natural frequencies and large quality factors of nrs @xcite , if qds coupled to nrs @xcite to form hybrid systems , the coherent optical properties will be enhanced remarkably , which will be an alternative ultrasensitive detection means . although probing mfs with qds @xcite have been proposed , we notice that all the schemes are still based on electrical means . in the present work , we propose an optical measurement scheme to detect the existence of mfs in iron chains on the superconducting pb surface @xcite via a coupled hybrid qd - nr system with optical pump - probe scheme @xcite . compared with electrical detection means where the qds are coupled to mfs via the tunneling @xcite , in our optical scheme , there is no direct contact between mfs and the hybrid qd - nr system , which can effectively avoid introducing other signals disturbing the detecting of mfs . the interaction between mfs in iron chains and qd in hybrid qd - nr system is mainly due to the dipole - dipole interaction , and the distance between the two systems can be adjusted by several tens of nanometers , therefore the tunneling between the qd and mfs can be neglected safely . in addition , the qd is considered as a two - level system rather than a single resonant level with spin - singlet state , and once mfs appear in the end of iron chains and couple to the qd , the majorana signature will be carried out via the coherent optical spectrum of the qd . the change in the coherent optical spectrum as a possible signature for mfs is another potential evidence in the iron chains . this optical scheme will provide another method for the detection of mfs , which is very different from the zero - bias peak in the tunneling experiments mourikv , dasa , dengmt , churchillhoh , finckadk , nadj - perges2 . furthermore , in order to investigate the role of the nr in the hybrid system , we further introduce the exciton resonance spectrum to detect mfs . the results shows that the vibration of the nr acting as a phonon cavity will enhance the exciton resonance spectrum significantly and make mfs more sensitive to be detectable . the technique proposed here provide a new platform for applications ranging from robust manipulation of mfs and mfs based quantum information processing . figure 1(b ) shows the schematic setup that will be studied in this work , where iron ( fe ) chains on the superconducting pb(110 ) surface nadj - perges2 , and we employ a two - level qd with optical pump - probe technology to detect mfs . the fe chain is ferromagnetically ordered nadj - perges2 with a large magnetic moment , which takes the role of the magnetic field in the nanowire experiments @xcite . different from the proposal of mourik et al . @xcite , this `` magnetic field '' is mostly localized on the fe chain , with small leakage outside , and superconductivity is not destroyed along the chain . in this setup , the energy scale of the exchange coupling of the fe atoms is typically much larger than that of the rashba spin - orbit coupling and the superconducting pairing . figure 1(c ) displays that a qd is implanted in the nr to form a coupled hybrid qd - nr system . the whole system includes two kinds of couplings which are qd - mf coupling and qd - nr coupling as shown in fig . 1(a ) . in the following , we will discuss the two kinds of coupling in detail , respectively . in the hybrid qd - nr system , the qd is modeled as a two - level system consisting of the ground state @xmath1 and the single exciton state @xmath2 at low temperatures zrennera , stuflers , and the hamiltonian of the qd can be described as @xmath3 with the exciton frequency @xmath4 , where @xmath5 and @xmath6 are the pseudospin operator describing the two - level exciton with the commutation relation @xmath7 = \pm s^{\pm } $ ] and @xmath8 = 2s^{z}$ ] . for the nr , the thickness of the beam is much smaller than its width , the lowest - energy resonance corresponds to the fundamental flexural mode that will constitute the resonator mode @xcite which can be characterized by a quantum harmonic oscillator with hamiltonian @xmath9 , where @xmath10 is the resonator frequency and @xmath11 is the annihilation operator of the resonator mode . since the flexion induces extensions and compressions in the structure @xcite , this longitudinal strain will modify the energy of the electronic states of qd through deformation potential coupling . then the coupling between the resonator mode and the qd is described by @xmath12 , where @xmath13 is the coupling strength between the resonator mode and qd wilson - raei . thus we obtain the hamiltonian of the coupled hybrid qd - nr system@xmath14 for the qd - mf coupling , as each mf is its own antiparticle , we introduce an operator @xmath15 with @xmath16 and @xmath17 to describe mfs . supposed that the qd couples to the nearby mf @xmath18 in the end of iron chains , then the hamiltonian is written by liude , flensbergk , leijnsem , caoys , lij @xmath19to detect mfs , it is helpful to switch the majorana representation to the regular fermion one via the exact transformation @xmath20 and @xmath21 @xmath22 , where @xmath23 and @xmath24 are the fermion annihilation and creation operators obeying the anti - commutative relation @xmath25 . accordingly , in the rotating wave approximation @xcite , the above hamiltonian can be rewritten as@xmath26where the first term gives the energy of mf with frequency @xmath27 and @xmath28 with the iron chains length ( @xmath29 ) and the pb superconducting coherent length ( @xmath30 ) . if the iron chains length ( @xmath29 ) is large enough , @xmath31 will approach zero . in the following , we will discuss the two conditions of @xmath32 and @xmath33 , and define the two conditions as coupled mfs ( @xmath34 ) and uncoupled mfs ( @xmath33 ) , respectively . the second term describes the coupling between the nearby mf and the qd with the coupling strength @xmath35 , where the coupling strength is related to the distance between the hybrid qd - nr system and the iron chains . it should be also noted that the term of non - conservation for energy , i.e. @xmath36 , is generally neglected . we have made the numerical calculations ( not shown in the following figures ) and shown that the effect of this term is too small to be considered in our theoretical treatment . currently , the optical pump - probe technique has become a popular topic , which affords an effective way to investigate the light - matter interaction . the optical pump - probe technology includes a strong pump laser and a weak probe laser @xcite . in the optical pump - probe technology , the strong pump laser is used to stimulate the system to generate coherent optical effect , while the weak laser plays the role of probe laser . therefore , the linear and nonlinear optical effects can be observed via the probe absorption spectrum based on the optical pump - probe scheme . xu et al . have obtained coherent optical spectroscopy of semiconductor qd when driven simultaneously by two optical fields @xcite . their results open the way for the demonstration of numerous quantum level - based applications , such as qd lasers , optical modulators , and quantum logic devices . in terms of this scheme , we apply the pump - probe scheme to the qd of the hybrid qd - nr system simultaneously . when the optical pump - probe technology is applied on the qd , the majorana signature will be carried out via the coherent optical spectrum . the hamiltonian of the exciton of the qd coupled to the two fields is given by @xcite @xmath37 , where @xmath38 is the dipole moment of the exciton , and @xmath39 is the slowly varying envelope of the field . therefore , we obtain the whole hamiltonian of the hybrid system as @xmath40 . in a rotating frame at the pump field frequency @xmath41 , we obtain the total hamiltonian of the system as @xmath42where @xmath43 is the detuning of the exciton frequency and the pump frequency , @xmath44 is the rabi frequency of the pump field , and @xmath45 is the detuning of the probe field and the pump field . @xmath46 is the detuning of the mf frequency and the pump frequency . actually , we have neglected the regular fermion like normal electrons in the nanowire that interact with the qd in the above discussion . to describe the interaction between the normal electrons and the exciton in qd , a tight binding hamiltonian of the whole iron chains is introduced chenhj . according to the heisenberg equation of motion and introducing the corresponding damping and noise terms , the quantum langevin equations of the whole system are derived as@xmath47@xmath48s^{-}+2(\beta f - i\omega _ { pu})s^{z}-\frac{2i\mu e_{pr}}{\hbar } e^{-i\delta t}s^{z}+\hat{% \tau}(t)\text{,}\]]@xmath49@xmath50where @xmath51 ( @xmath52 ) is the exciton spontaneous emission rate ( dephasing rate ) , @xmath53 is the position operator , @xmath54 is the decay rate of the nr , and @xmath55 is the decay rate of the mf . @xmath56 is the @xmath57-correlated langevin noise operator , which has zero mean @xmath58 and obeys the correlation function @xmath59 . the resonator mode is affected by a brownian stochastic force with zero mean value , and @xmath60 has the correlation function@xmath61,\]]where @xmath62 and @xmath63 are the boltzmann constant and the temperature of the reservoir of the coupled system . mfs have the same correlation relation as the resonator mode as @xmath64.\]]in eq.(9 ) and eq.(10 ) , both the nr and majorana mode will be affected by a thermal bath of brownian and non - markovian processes @xcite . in the low temperature , the quantum effects of both the majorana and nr mode are only observed in the case of @xmath65 and @xmath66 . due to the weak coupling to the thermal bath , the brownian noise operator can be modeled as markovian processes . in addition , both the qd - mfs coupling and qd - nr mode coupling in the hybrid system are stronger than the coupling to the reservoir that influences the two kinds coupling . in this case , owing to the second order approximation gardinercw , we can obtain the form of the reservoir that affects both the nr mode and majorana mode as eq.(9 ) and eq.(10 ) . to go beyond weak coupling , the heisenberg operator can be rewritten as the sum of its steady - state mean value and a small fluctuation with zero mean value@xmath67since the driving fields are weak , but classical coherent fields , we will identify all operators with their expectation values , and drop the quantum and thermal noise terms . simultaneously , inserting these operators into the langevin equations eqs.(5)-(8 ) and neglecting the nonlinear term , we can obtain two equation sets about the steady - state mean value and the small fluctuation . the steady - state equation set consisting of @xmath68 , @xmath69 and @xmath70 is related to the population inversion ( @xmath71 ) of the exciton which is determined by@xmath72 + 4w_{0}\gamma _ { 2}\omega _ { pu}^{2}(\delta _ { m}^{2}+\kappa _ { m}^{2}/4)=0.\end{gathered}\]]for the equation set of small fluctuation , we make the ansatz @xcite @xmath73 ( @xmath74 ) . solving the equation set and working to the lowest order in @xmath75 but to all orders in @xmath76 , we can obtain the linear susceptibility as @xmath77 , where @xmath78 is given by@xmath79\gamma _ { 2}}{\pi { % _ { 2}\pi _ { 4}^{\ast } -\lambda _ { 1}\lambda _ { 2}\pi _ { 1}\pi _ { 3}^{\ast } } } , \]]@xmath68 , @xmath70 and @xmath69 can be derived from the steady - state equations , and @xmath80 , @xmath81 , @xmath82 , @xmath83/(\gamma _ { 1}-i\delta ) $ ] , @xmath84/(\gamma _ { 1}-i\delta ) $ ] , @xmath85 , @xmath86 , @xmath87 , @xmath88 , @xmath89 ( @xmath90 indicates the conjugate of @xmath91 ) . the imaginary and real parts of @xmath92 indicate absorption and dissipation , respectively . in addition , the average population of the exciton states can be obtained as@xmath93}{\pi { _ { 2}\pi _ { 4}^{\ast } -\lambda _ { 1}\lambda _ { 2}\pi _ { 1}\pi _ { 3}^{\ast } } } , \]]which is benefited for readout the exciton states of qd . for illustration of the numerical results , we choose the realistic hybrid systems of the coupled qd - nr system @xcite and the iron chains on the superconducting pb surface @xcite . for an inas qd in the coupled qd - nr system , we use parameters @xcite : the exciton relaxation rate @xmath94 ghz , the exciton dephasing rate @xmath95 ghz . the physical parameters of gaas nr are @xmath96 , @xmath97 , @xmath98 ghz , @xmath99 kg , @xmath100 , where @xmath101 , @xmath97 , and @xmath102 are the resonator frequency , the effective mass , and quality factor of the nr , respectively . the decay rate of the nr is @xmath103 @xmath104 khz , and the coupling strength between the qd and nr is @xmath105 . for mfs , there are no experimental values for the lifetime of the mfs and the coupling strength between the exciton and mfs in the recent literature . however , according to a few recent experimental reports @xcite , it is reasonable to assume that the lifetime of the mfs is @xmath106 mhz . since the coupling strength between the qd and nearby mfs is dependent on their distance , we also expect the coupling strength @xmath107 ghz via adjusting the distance between the hybrid qd - nr system and the iron chains . figure 2(a ) shows the coherent optical properties of the qd as functions of probe - exciton detuning @xmath108 at the detuning of the exciton frequency and the pump frequency @xmath109 , i.e. , the absorption ( @xmath110 ) and dissipation ( @xmath111 ) properties of the qd without considering any coupling ( @xmath112 ) , which indicates the normal absorption and dissipation of the qd , respectively . turning on the qd - nr coupling ( @xmath105 ) and without considering the qd - mf coupling ( @xmath113 ) , two sharp peaks will appear in both the absorption and dissipation spectra as shown in fig . 2(b ) . from the curves , we find that the two sharp peaks at both sides of the spectra just correspond to the vibrational frequency of the nr . the physical origin of this result is due to mechanically induced coherent population oscillation , which makes quantum interference between the resonator and the beat of the two optical fields via the qd when the probe - pump detuning is equal to the nr frequency @xcite . this reveals that if fixing the pump field on - resonance with the exciton and scan through the frequency spectrum , the two sharp peaks can obtain immediately in the coherent optical spectra , which also indicates a scheme to measure the frequency of the nr . this phenomenon stems from the quantum interference between the vibration nr and the beat of the two optical fields via the exciton when probe - pump detuning @xmath114 is adjusted equal to the frequency of the nr . therefore , the qd - nr coupling play a key role in the hybrid system , and if we ignore the coupling ( @xmath115 ) , the above phenomenon will disappear completely as shown in fig . compared with fig.2(b ) , in fig.2(c ) , we consider the qd coupled with the nearby mf @xmath18 without taking the qd - nr coupling into account , i.e. the condition of @xmath115 and @xmath107 ghz . as the mfs appear in the ends of iron chains and coupled to the qd , both the probe absorption ( the blue curve ) and dissipation ( the green curve ) spectra will present an remarkable signature of mfs under @xmath116 ghz . the physical origin of this result is due to the qd - mf coherent interaction and we can interpret this physical phenomenon with dressed state between the exciton and mfs . the qd coupled to the nearby mf will induce the upper level of the state @xmath117 to split into @xmath118 and @xmath119 ( @xmath120 denotes the number states of the mfs ) . the left peak in the coherent optical spectra signifies the transition from @xmath1 to @xmath121 while the right peak is due to the transition of @xmath122 to @xmath119 chenhj . to determine this signature is the true mfs rather than the normal electrons that couple with the qd , we have used a tight binding hamiltonian to describe the electrons in whole iron chains , the numerical results indicate the signals in the absorption and dissipation spectra are the true mfs signature @xcite . if we consider both the two kinds coupling , i.e. the qd - nr coupling ( @xmath105 ) and qd - mfs coupling ( @xmath107 ghz ) as shown in fig . 2(d ) , not only the two sharp peaks locate at the nr frequency induced by its vibration , i.e. two peaks are at @xmath123 ghz ( @xmath124ghz ) , there is also mfs signal appear at @xmath125 ghz ( @xmath116 ghz ) induced by the qd - mf coupling . in fig . 2(c ) , we only consider the situation of @xmath34 . in fact , if the iron chains length @xmath29 is much larger than the pb superconducting coherent length @xmath30 , @xmath31 will approach zero . therefore , it is necessary to consider the conditions of @xmath126 and @xmath33 , and we define them as coupled mfs mode ( @xmath32 ) and uncoupled mfs mode ( @xmath33 ) , respectively . figure 3(a ) and figure 3(b ) show the absorption and dissipation spectra as a function of detuning @xmath127 with qd - mf coupling constants @xmath128 ghz under @xmath34 and @xmath33 , respectively . compared with the coupled mfs mode , the uncoupled qd - mf hamiltonian will reduce to @xmath129 which is analogous j - c hamiltonian of standard model under @xmath33 , and the probe absorption spectrum ( the blue curve ) shows a symmetric splitting as the qd - mf coupling strength @xmath107 ghz which is different from of coupled mfs mode presenting unsymmetric splitting due to a detuning @xmath130 ghz . therefore , our results reveal that the signals in the coherent optical spectra is a real signature of mf , and the optical detection scheme can work at both the coupled majorana edge states and the uncoupled majorana edge states . in fig . 3(c ) , we further make a comparison of the probe absorption spectrum under the coupled mfs mode ( @xmath34 ) and uncoupled mfs mode ( @xmath131 ) . it is obvious that the probe absorption spectrum display the analogous phenomenon of electromagnetically induced transparency ( eit ) @xcite under both the two conditions . the dip in the probe absorption spectrum goes to zero at @xmath132 and @xmath133 ghz with @xmath33 and @xmath34 , respectively , which means the input probe field is transmitted to the coupled system without absorption . such a phenomenon is attributed to the destructive quantum interference effect between the majorana modes and the beat of the two optical fields via the qd . if the beat frequency of two lasers @xmath57 is close to the resonance frequency of mfs , the majorana mode starts to oscillate coherently , which results in stokes - like ( @xmath134 ) and anti - stokes - like ( @xmath135 ) scattering of light from the qd . the stokes - like scattering is strongly suppressed because it is highly off - resonant with the exciton frequency . however , the anti - stokes - like field can interfere with the near - resonant probe field and thus modify the probe field spectrum . here the majorana modes play a vital role in this coupled system , and we can refer the above phenomenon as majorana modes induced transparency , which is analogous with eit in atomic systems @xcite . on the other hand , we can propose a means to determine the qd - mf coupling strength @xmath35 via measuring the distance of the two peaks with increasing the qd - mf coupling strength in the probe absorption spectrum . figure 3(d ) indicates the peak - splitting width as a function of the qd - mf coupling strength @xmath35 under the condition of the coupled mfs mode ( @xmath136 ) and the uncoupled mfs mode ( @xmath33 ) which follows a nearly linear relationship . it is obvious that the two lines ( the uncoupled mfs and the coupled mfs mode ) have a slight deviation . however , the deviation becomes slighter with increasing coupling strength . therefore , it is essential to enhance the coupling strength for a clear peak splitting via adjusting the distance between the qd and the nearby mfs . in this case the coupling strength can obtain immediately by directly measuring the distance of the two peaks in the probe absorption spectrum . as shown in fig . 2(d ) , there are not only two sharp peaks locate at the nr frequency induced by its vibration but also the mfs signal appear at @xmath137 induced by the qd - mf coupling in the probe absorption spectrum ( the blue curve ) under the two kinds coupling . in fig . 4(a ) , we further consider switching the detuning @xmath116 ghz to @xmath138 ghz at small exciton - pump detuning @xmath139 ghz . it is obvious that the resonance amplification process ( 1 ) and the resonance absorption process ( 2 ) in the probe absorption spectrum without considering the qd - mf coupling ( the blue curve , @xmath113 ) will accordingly transform into the the resonance absorption process ( 3 ) and the resonance amplification process ( 4 ) due to the qd - mf coupling ( the green curve , @xmath140 ghz ) . return to fig . 1(a ) , there are two kinds of coupling which are qd - nr coupling and qd - mf coupling in the hybrid system . for the qd - nr system , the two sharp peaks in the probe absorption corresponding to the resonance amplification ( 1 ) and absorption process ( 2 ) can be elaborated with dressed states @xmath141 , @xmath142 , @xmath143 , @xmath144 ( @xmath145 denotes the number state of the resonance mode ) , and the two sharp peaks indicate the transition between the dressed states @xcite . however , once mfs appear in the ends of iron chains and coupled to the qd , the ground state @xmath146 and the exciton state @xmath2 of the qd will also modify by the number states of the mfs @xmath120 and induce the majorana dressed states @xmath147 , @xmath148 , @xmath118 , @xmath149 . with increasing the qd - mf coupling , the majorana dressed states will affect the amplification ( 1 ) and absorption process ( 2 ) significantly , and even realize the inversion between the absorption ( 3 ) and amplification ( 4 ) process due to the qd - mf coherent interaction ( the green curve ) . to illustrate the advantage of the nr in the hybrid system , we introduce the exciton resonance spectrum to investigate the role of nr in the coupled qd - nr , which is benefited for readout the exciton states of qd . in fig . 4(b ) , we adjust the detuning @xmath116 ghz to @xmath150 ghz , therefore , the location of the two sideband peaks induced by the qd - mf coupling coincides with the two sharp peaks induced by the vibration of nr , thus the nr is resonant with the coupled qd - mf system and makes the coherent interaction of qd - mf more strong . figure 4(b ) shows the exciton resonance spectrum of the probe field as a function of the probe detuning @xmath151 with the detuning @xmath139 ghz under the coupled mfs mode @xmath34 . the black and red curves correspond to @xmath115 and @xmath152 for the qd - mf coupling @xmath153 ghz , respectively . it is obvious that the role of nr is to narrow and to increase the exciton resonance spectrum . in this case , the nr behaves as a phonon cavity will enhance the sensitivity for detecting mfs . we have proposed an all - optical means to detect the existence of mfs in iron chains on the superconducting pb surface with a hybrid qd - nr system . the signals in the coherent optical spectra indicate the possible majorana signature , which provides another supplement for detecting mfs . due to the vibration of nr , the exciton resonance spectrum becomes much more significant and then enhances the detection sensitivity of mfs . in addition , the qd - mf coupling in our system is a little feeble , while ref . [ 35 ] presents a strong qd - mf coupling and the coupling strength can reach kilohertz , which is beneficial for mfs detection . on the other hand , if we consider embedding a metal nanoparticle - quantum dot ( mnp - qd ) complex chenhj , lijj in the nr , the surface plasmon induced by the mnp will enhance the coherent optical property of qd , which may be robust for probing mfs . the concept proposed here , based on the quantum interference between the nr and the beat of the two optical fields , is the first all - optical means to probe mfs . this coupled system will provide a platform for applications in all - optically controlled topological quantum computing based on mfs . the authors gratefully acknowledge support from the national natural science foundation of china ( no.11574206 , no.10974133 , no.11274230 , no.61272153 , no.61572035 , no.51502005 , and no.11404005 ) , the key foundation for young talents in college of anhui province ( no . 2013sqrl026zd ) , and the foundation for phd in anhui university of science and technology . i. yeo , p. l. de assis , a. gloppe , e. dupont - ferrier , p. verlot , n. s. malik , e. dupuy , j. claudon , j. m. grard , a. auffves , g. nogues , s. seidelin , j. p. poizat , o. arcizet , and m. richard , nat . nanotechnol . * 9 * , 106 ( 2014 ) . fig.1 sketch of the proposed setup for optically detecting majorana fermions ( mfs ) . ( a ) the energy - level diagram of a qd coupled to mfs and nr , which includes two kinds coupling , i.e. the qd - mf coupling ( the dotted frame ) and the qd - nr coupling ( the dashed frame ) . ( b ) the iron chains on the superconducting pb surface , and a pair of mfs appear in the ends of the iron chains . the nearby mf is coupled to ( c ) a qd embedded in a nanomechanical resonator ( nr ) with optical pump - probe technology . fig.2 the absorption ( the blue curve ) and dispersion ( the green curve ) spectra of probe field as a function of the probe detuning @xmath127 under different conditions . ( a ) without considering any coupling , i.e. , @xmath115 and @xmath113 . ( b ) the qd - nr coupling strength is @xmath105 and @xmath113 . ( c ) the qd - mf coupling strength is @xmath107 ghz and @xmath115 . ( d ) considering both the qd - nr coupling and qd - mf coupling , i.e. , @xmath105 and @xmath154 ghz . the parameters used are @xmath94 ghz , @xmath95 ghz , @xmath155 khz , @xmath124 ghz , @xmath156 mhz , @xmath157(ghz)@xmath158 , @xmath116 ghz , and @xmath109 . fig.3 ( a ) and ( b ) show the probe absorption ( the blue curve ) and dispersion ( the green curve ) spectra with qd - mf coupling strengths @xmath107 ghz under @xmath34 and @xmath33 , respectively . ( c ) the probe absorption spectrum under @xmath34 ( the green curve ) and @xmath131 ( the blue curve ) , respectively . ( d ) the linear relationship between the distance of peak splitting and the coupling strength of qd - mf @xmath159 . the other parameters used are the same as in fig.2 . fig.4 ( a ) the probe absorption spectrum as a function of the probe detuning @xmath160 with considering ( the blue curve , @xmath153 ghz ) and without considering ( the green curve , @xmath113 ) the qd - mf coupling under the qd - nr coupling strength @xmath105 . ( b ) the exciton resonance spectrum as a function of @xmath127 with @xmath115 and @xmath105 at the qd - mf coupling strength @xmath153 ghz . @xmath150 ghz , @xmath139 ghz , @xmath161(ghz)@xmath158 , the other parameters used are the same as fig.2 . under different conditions . ( a ) without considering any coupling , i.e. , @xmath115 and @xmath162 . ( b ) the qd - nr coupling strength is @xmath105 and @xmath163 . ( c ) the qd - mf coupling strength is @xmath164 ghz and @xmath115 . ( d ) considering both the qd - nr coupling and qd - mf coupling , i.e. , @xmath105 and @xmath164 ghz . the parameters used are @xmath165 ghz , @xmath166 ghz , @xmath167 khz , @xmath168 ghz , @xmath169 mhz , @xmath170(ghz)@xmath158 , @xmath116 ghz , and @xmath109.,width=453 ] ghz under @xmath171 and @xmath172 , respectively . ( c ) the probe absorption spectrum under @xmath173 ( the green curve ) and @xmath174 ( the blue curve ) , respectively . ( d ) the linear relationship between the distance of peak splitting and the coupling strength of qd - mf @xmath175 . the other parameters used are the same as in fig.2.,width=453 ] with considering ( the blue curve , @xmath176 ghz ) and without considering ( the green curve , @xmath162 ) the qd - mf coupling under the qd - nr coupling strength @xmath105 . ( b ) the exciton resonance spectrum as a function of @xmath127 with @xmath115 and @xmath152 at the qd - mf coupling strength @xmath177 ghz . @xmath138 ghz , @xmath139 ghz , @xmath178(ghz)@xmath158 , the other parameters used are the same as fig.2.,width=453 ]

motivated by a recent experiment [ nadj - perge et al . , science 346 , 602 ( 2014 ) ] providing evidence for majorana zero modes in iron chains on the superconducting pb surface , in the present work , we theoretically propose an all - optical scheme to detect majorana fermions , which is very different from the current tunneling measurement based on electrical means . the optical detection proposal consists of a quantum dot embedded in a nanomechanical resonator with optical pump - probe technology . with the optical means , the signal in the coherent optical spectrum presents a distinct signature for the existence of majorana fermions in the end of iron chains . further , the vibration of the nanomechanical resonator behaving as a phonon cavity will enhance the exciton resonance spectrum , which makes the majorana fermions more sensitive to be detectable . this optical scheme affords a potential supplement for detection of majorana fermions and supports to use majorana fermions in fe chains as qubits for potential applications in quantum computing devices .

arxiv

first evidence of the violation of time reversal symmetry has been found in the kaon system @xcite . despite strong efforts no other signal of violation of time reversal symmetry has been found to date . however , by now , studying time reversal symmetry has become a corner stone of the search for physics beyond the standard model of elementary particles @xcite . some alternatives or extensions of the standard model are due to dynamical symmetry breaking , multi higgs models , spontaneous symmetry breaking , grand unified theories ( e.g. so(10 ) ) , extended gauge groups ( leading e.g. to right - handed bosons @xmath3 in left - right symmetric models ) , super symmetric ( susy ) theories , etc . , each implying specific ways of @xmath4 violation . for a recent review of models relevant in the context of @xmath4 violation see e.g. @xcite , and refs . therein . these theories `` beyond '' the standard model are formulated in terms of quarks and leptons whereas nuclear low energy tests of @xmath4 involve hadronic degrees of freedom ( mesons and nucleons ) @xcite . to extract hadronic degrees of freedom from observables one may introduce effective @xmath1odd nucleon nucleon potentials @xcite , or more specific @xmath1odd mesonic exchange potentials @xcite . as in the context of @xmath0-violation see e.g. @xcite , these potentials have been proven quite useful to treat the nuclear structure part involved and to extract effective @xmath1odd hadronic coupling constants @xcite . in turn they allow to compare the sensitivity of different experiments , which has been done recently in ref . however , in order to compare upper bounds on a more fundamental level of @xmath1odd interactions , it is necessary to relate hadronic degrees of freedom to quark degrees of freedom in some way . this step is hampered by the absence of a complete solution of quantum chromo dynamics ( qcd ) at the energies considered here . in many cases a rough estimate in the context of time reversal violation may be sufficient , and , in the simplest case , factors arising from hadronic structure may be neglected . in the context of @xmath0odd time reversal violation e.g. concepts such as pcac and current algebra @xcite have been utilized to improve the evaluation of hadronic structure effects . in the @xmath0even case , which is considered here , this approach is not applicable ( no goldstone bosons involved here ) . however , it may be useful to utilize quark models specifically designed for and quite successful in describing the low energy sector . in fact , experimental precision tests still continue to make progress and so theorists face a renewed challenge to translate these experimental constrains to a more fundamental interaction level . the purpose of the present paper is to give estimates on hadronic matrix elements that arise when relating quark operators to the effective hadronic parameterizations of the @xmath0even @xmath1odd interaction . these are the charge @xmath2 type exchange and the axial vector type exchange nucleon nucleon interaction @xcite . they will shortly be outlined in the next section . the ansatz to calculate @xmath5 matrix elements from the quark structure is described in section iii . the last section gives the result for different types of quark models and a conclusion . for completeness , note that in general also @xmath1-odd and @xmath0-odd interactions are possible , and in fact most of the simple extensions of the standard model mentioned above give rise to such type of @xmath1violation . parameterized as one boson exchanges they lead e.g. to effective pion exchange potentials that are essentially long range , see @xcite . limits on @xmath0odd @xmath1odd interactions are rather strongly bound by electric dipole moment measurements , in particular by that of the neutron @xcite . in contrast bounds on @xmath0even @xmath1odd interactions are rather weak . note , also that despite theoretical considerations @xcite new experiments testing generic @xmath1odd @xmath0even observables have been suggested ; for the present status see e.g. refs . due to the moderate energies involved in nuclear physics tests of time reversal symmetry , hadronic degrees of freedom are useful and may be reasonable to analyze and to compare different types of experiments . for a recent discussion see ref . @xcite . in the following only @xmath1-odd and @xmath0-_even _ interactions will be considered . they may be parameterized in terms of effective one boson exchange potentials . due to the behavior under @xmath6 , @xmath0 , and @xmath1 symmetry transformations , see e.g. @xcite , two basic contributions are possible then : a charged @xmath2 type exchange @xcite and an axial vector exchange @xcite . the effective @xmath2 type @xmath1odd interaction is @xmath6odd due to the phase appearing in the isospin sector and is only possible for charged @xmath2 exchange . it has been suggested by simonius and wyler , who used the tensor part to parameterize the interaction @xcite , @xmath7 there is some question of whether to choose an `` anomalous '' coupling @xcite , viz . @xmath8 . the numerical value of @xmath9 is usually taken to be @xmath10 close to the strong interaction case @xcite . we shall see in the following that it is not unreasonable to introduce such a factor since in may be related to `` nucleonic structure effects '' , which are not of @xmath1 violating origin ( similar to nuclear structure effects that are also treated separately ) . combining the @xmath1odd vertex with the appropriate @xmath1even vertex leads to the following effective @xmath1odd @xmath0even one boson exchange @xmath5 interaction , @xmath11 where @xmath12 , and @xmath13 , and @xmath14 is the strong coupling constant , as e.g. provided by the bonn potential @xcite . the axial vector type interaction has been suggested by @xcite . unlike the @xmath2type interaction the isospin dependence is not restricted and may be isoscalar , vector , and/or tensor type . the effective lagrangian for the @xmath15 coupling for example is given by @xmath16 combined with the appropriate @xmath1even vertex this leads to an effective axial vector type exchange @xmath5 potential @xcite , @xmath17 the bounds on the @xmath1odd coupling strengths arising from various experiments have been discussed in ref . a more recent bound not included there is from an improved analysis @xcite of the @xmath18fe @xmath19decay experiment @xcite . bounds are in the order of 10% if derived from generic @xmath1odd @xmath0even observables and slightly more than an order of magnitude smaller , if related to the electric dipole moments @xcite . to complete this section , note that although possible , two boson exchanges have not been considered up to now . we now turn to effective @xmath1-odd @xmath0-even quark operators . the simplest operator that leads to an effective @xmath1odd @xmath0even vector type vertex @xmath20 analogous to the @xmath2type interaction eq . ( [ eqn : vnn ] ) is @xmath21 here @xmath22 , @xmath23 denote flavored quark fields . again the flavor dependence is responsible for @xmath6 , viz . @xmath1violation due to the phase dependence . a tensor term has not been introduced for simplicity . on the basis of eq . ( [ quarkv ] ) such a term will arise in a natural way in the effective hadronic @xmath24 @xmath1-odd lagrangian through the quark structure effects as will be explained below . the second generic quark operator utilizing axial vector bilinear operators is given by @xcite , @xmath25 with the on shell equivalent @xmath26 in order to recover eqs . ( [ eqn : vnn ] ) and ( [ eqn : ann ] ) we utilize the constituent quark model . this model has been rather successful and valuable in reproducing gross features of low energy phenomenae , such as mass spectra , form factors , coupling constants , magnetic moments etc . , see e.g. @xcite . to relate quark operators to effective hadronic operators we utilize the fock space representation of hadrons in terms of constituent quarks , viz . @xmath27 since there is no low energy solution of qcd , the evaluation of the matrix elements of the l.h.s of eq . ( [ fock ] ) needs further consideration . in general , the same problem arises in the context of strong interactions . an extensive overview of the different approaches to tackle the problem in this case has been given by ref . @xcite . here we follow the ideas first formulated in ref . @xcite , and extensively studied for different quark models in @xcite . the resulting strong interaction potential is a generic hybrid model connecting quark degrees of freedom with effective meson nucleon degrees of freedom . the basic idea is summarized in the following . suppose the two nucleons overlap , and two quarks are sufficiently close together . this situation is depicted in figure [ fig : qqnn]a ) . then , to begin with , the matrix elements may be evaluated without introducing any mesonic fields . in terms of the constituent quark model @xmath28 excitations are neglected ( or partially parameterized in the constituent quark mass ) . only at larger distances of the nucleons , mesons are essential and may appear as @xmath28 correlations on the nonperturbative qcd vacuum @xcite that might be the physical vacuum of the low energy regime @xcite . however , the appearance of mesons is disconnected from the problem of @xmath1odd force . therefore , in the following we assume that the hadronization mechanism is the _ same for both @xmath1odd and the usual @xmath1even strong interaction _ and investigate the relative strength of the @xmath1odd matrix elements to the @xmath1even matrix elements . this is done in the framework of the virginia potential that assumes a quark pairing mechanism to generate effective meson nucleon coupling constants @xcite . to illustrate the quality of this ansatz table [ tab : virginia ] shows the resulting coupling constants using different quark models compared to the values of a recent version of the bonn potential @xcite . in this framework we utilize the factorization approximation to evaluate the matrix element of eq . ( [ fock ] ) , see figure [ fig : qqnn]b ) @xmath29 to demonstrate the calculation , we use the simple constituent quark model . this model is supplemented by an explicit lower component , which already occurs implicitly in the dirac magnetic moments @xcite and in the two body pair current through electromagnetic gauge invariance @xcite . this way a treatment of relativistic effects has been introduced , see e.g. @xcite and ref . therein . the integration of the internal degrees of freedom reads , @xmath30\ ] ] with @xmath31 the three quark wave internal function . in the rest system the space part of the wave function is given by @xcite @xmath32\ ] ] where numerical values may be chosen as @xmath33 , and @xmath34 . the coordinates are normalized lovelace coordinates , viz . @xmath2 for the pair and @xmath35 for the odd quark . integration is done in the breit system . the symbol @xmath36 denotes either one of the vertices in ( [ quark ] ) . evaluation for the different type of operators leads to the following expressions ( using the isospin formalism for @xmath37 quarks ) @xmath38 the factors arising are related to the quark structure of nucleons . note , that in eq . ( [ axial ] ) one recognizes the well known coupling of the axial vector current ( for @xmath39 ) , viz . @xmath40 of the nonrelativistic constituent quark model , besides factors arising from relativistic corrections due to the lower dirac component . the latter reduce the value of @xmath41 close to the experimental one @xcite . in eq . ( [ tensor ] ) a tensor coupling appears , which belongs to the @xmath2 type @xmath1odd exchange . indeed due to the quark structure factors its relative strength is larger than the first term of the r.h.s . of eq . ( [ tensor ] ) , and therefore preferable in an ansatz of a @xmath2type @xmath1odd force as done by simonius and wyler @xcite . the factor in front the tensor term may be interpreted as `` anomalous '' coupling . it appears in analogy to the electromagnetic interaction , where the pauli term of the electromagnetic photon nucleon interaction can be recovered from a pure dirac coupling on quark level . this has been explained and shown in ref . the resulting relation between quark and hadronic @xmath1odd coupling strength on the basis of the constituent quark model is , for @xmath2 type exchange @xmath42 and for axial type of exchange @xmath43 here the expression for isoscalar and isovector are the same . similar results may be obtained using different types of quark models . in the context of the virginia potential those studied are the mit bag model and a relativistic model with linear confining potential , see ref . these are used here in the same way as demonstrated for the constituent quark model in the previous section . the values for the quark structure effects evaluated using typical quark model parameters of low energy phenomenology are given in table [ tab : values ] . in fact , due to the symmetries inherent in the quark pairing mechanism ( viz . the virginia potential ) it is possible to arrive at the following relations between the coupling constants , viz . @xmath44 this equation shows that the factors appearing in the @xmath45 type exchange may be related to the anomalous coupling @xmath46 . so , inclusion of @xmath9 might give a bound closer to the more basic quark degrees of freedom . in conclusion , provided the hadronization process does not substantially differ for @xmath1odd and @xmath1even interactions , the factors arising reflect the _ nucleon _ structure effects . the origin of the structure factors are due to the spin , isospin structure and the different mass scales ( i.e. @xmath47 vs. @xmath48 ) . these have also been essential in deriving the relative strength of the strong coupling constants as given in table [ tab : virginia ] . the author gratefully acknowledges support by the national institute for nuclear theory at the university of washington , seattle , during his stay on the int program `` physics beyond the standard model at low and intermediate energies '' . this work has been supported by deutsche forschungsgemeinschaft be 1092/4 - 1 . p. herczeg , nucl . phys . * 75 * ( 1966 ) 655 r. bryan , a. gersten , phys . * 26 * ( 1971 ) 1000 , * 27 * ( 1971 ) 1102(e ) e.c.g . sudarshan , proc . a305 * , ( 1968 ) 319 m. simonius , phys . lett . * 58b * ( 1975 ) 147 m. simonius , d. wyler , nucl a286 * ( 1977 ) 182 w.c . haxton , e.m . henley , phys . * 51 * ( 1983 ) 1937 p. herczeg , in _ tests of time reversal invariance in neutron physics _ , eds , n.r . roberson et al . , ( world scientific publishing , singapore , 1987 ) p. 24 . e.g. adelberger , w.c . haxton ann . nucl . part . * 35 * 501 ( 1985 ) . i.s . towner and a.c . hayes , phys . rev . * c 49 * , 2391 ( 1994 ) . gudkov , x .- he , and b.h.j . mckellar , phys . * c47 * , 2365 ( 1993 ) . m. beyer , nucl . a493 * 335 ( 1989 ) . m. beyer , phys . * c48 * 906 ( 1993 ) . w.c . haxton and a. hoering , nucl . phys . * a 560 * , 469 ( 1993 ) . haxton , a. hring , m.j . musolf , phys . d50 3422 ( 1994 ) . crewther , p. di vecchia , g. veneziano , e. witten , phys . lett . * 88b * 123 ( 1979 ) . altarev et al . , phys . lett . * b276 * 242 ( 1992 ) . smith , phys . b234 * 191 ( 1990 ) . pendlebury , nucl . phys . * a546 * 359c ( 1992 ) . montanet et al . ( particle data group ) , phys.rev . * d50 * 1173 ( 1994 ) . p. herczeg , private communication : p. herzceg , j. kambor , m. simonius , and d. wyler to be published . j. engel , p. frampton , and r. springer , preprint nucl - th/9505026 ( 1995 ) . c. itzykson and j .- b . zuber , _ quantum field theory _ ( mcgraw hill international , singapore , 1980 ) . p. herczeg , private communication . ressell , j. engel , p. vogel , preprint map-189 , nucl - th/9512013 ( 1995 ) . cheung , h.e . henrikson , f. boehm , phys . rev . * c16 * , 2381 ( 1977 ) . khriplovich , nucl . phys . * b352 * 382 ( 1991 ) . see e.g. f.e . close , _ an introduction to quarks and partons _ ( academic press , 1979 ) ; a. le yaouanc , l. oliver , o. pene , j .- c . raynal , _ hadron transitions in the quark model _ ( gordon and breach , 1988 ) ; r.k . bhaduri , _ models of the nucleon : from quarks to soliton _ ( addison - wesley , 1988 , lecture notes and supplements in physics , 22 ) . f. myhrer , rev . 60 629 ( 1988 ) h.j . weber , z. phys . * a297 * ( 1980 ) 261 , * a301 * , 141 ( 1981 ) , phys . rev . * c26 * , 2333 ( 1982 ) m. bozoian and h.j . weber , phys . * c28 * 811 ( 1983 ) m. beyer , h.j . weber , phys . * 146b * ( 1984 ) 383 ; phys . rev . * c35 * ( 1987 ) 14 m.a . shifman , a.i . vainshtein , and v.i . zakharov , nucl . * b147 * , 385 and 448 ( 1980 ) for a recent discussion of constituent quarks see : d. diakonov , gatchina preprint , nucl - th/960323 . r. machleidt , k. holinde and ch . elster , phys . rep . * 149 * , 1 ( 1987 ) n. isgur and g. karl , phys * d18 * , 4187 ( 1978 ) ; * d 19 * , 2653 ( 1979 ) . m. beyer , d. drechsel , and m.m . giannini , phys . 122 b * , 1 ( 1983 ) h.j . weber , m. weyrauch phys . * c32 * , 1342 ( 1985 ) m. beyer and s.k . singh , phys . * b 160 * , 26 ( 1985 ) ; z. phys . * c31 * , 421 ( 1986 ) @xmath49 & 3.8 & [ -0.5 ] & 9.85 & [ -0.49]&10.6 & [ 0.0]\\ \rho(763 ) & ( 1^{--},1 ^ -)&0.7 & [ 2.2 ] & 0.42&[3.2 ] & 1.10 & [ 1.53 ] & 0.41 & [ 6.1]\\ f_1(1285 ) & ( 1^{++},0^+)&0.5 & [ -1.5 ] & 0.22 & & 0.59 & [ 0 ] & & \\ a_1(119 ) & ( 1^{++},1 ^ -)&1.0 & [ -1.5 ] & 0.6 & & 1.64 & [ 0 ] & & \\ \hline\hline \end{array}\ ] ]

tests of time reversal symmetry at low and medium energies may be analyzed in the framework of effective hadronic interactions . here , we consider the quark structure of hadrons to make a connection to the more fundamental degrees of freedom . it turns out that for @xmath0even @xmath1odd interactions hadronic matrix elements evaluated in terms of quark models give rise to factors of 2 to 5 . also , it is possible to relate the strength of the anomalous part of the effective @xmath2 type @xmath1odd @xmath0even tensor coupling to quark structure effects .

arxiv

type ia supernovae ( sne ia ) , characterized by no @xmath1 but strong si lines in the spectra at the maximum brightness , are brighter than most of sne classified into the other types and exhibit uniform light curves . thus they are used as a standard candle to measure distances to remote galaxies . a plausible explosion model for sne ia is the accreting white dwarf ( wd ) model , in which a white dwarf in a binary system accretes material from the companion star , increases its mass , usually up to the chandrasekhar mass limit ( @xmath2 ) , and then explodes ( e.g. , * ? ? ? . there have been significant progresses in the accreting wd model since @xcite introduced the stellar wind from the wd while it accretes materials from the companion . their model succeeded in sustaining a stable mass transfer in the progenitor systems of sne ia . according to their model , there are two main evolutionary paths leading to sne ia , the super soft x - ray source ( sss ) channel and the symbiotic channel @xcite . accordingly their model predicts which companion stars lead to sne ia . the above evolutionary scenario for sne ia has not been confirmed by observations , which will require the identification of the companion star that should remain in the vicinity of the explosion site . @xcite argued that their group identified a g2iv star as the companion of tycho brahe s supernova remnant ( snr ) by measuring the velocities and distances of stars in the vicinity of the center of the snr . they concluded that this g2iv star was moving much faster than the other neighbor stars and that the distance to this star seemed consistent with the distance to tycho s snr . although @xcite has argued that the observed velocity of tycho g might correspond to the velocity of stars belonging to the thick disk population , the expected stellar mass in thick disk stars within the cone with 2.87 arcsec radius ( which corresponds to the angular distance from the center of tycho s snr to the tycho g star ) , at 3 kpc from earth , is only 2 @xmath3 , which makes the thick disk star alternative very unlikely . for this estimate we use the density in the vicinity of tycho s snr @xcite . although the coincidence of the kinematic characteristics of tycho g with its being at the position and distance of the snr appears significant , confirmation by other means would nonetheless be very useful . in this paper we propose a direct method to prove that the companion star is located inside the sn ejecta . a hint was dropped by observations for a star called s - m star discovered by @xcite near a type ia snr 1006 . @xcite proved that the s - m star was not the companion of this supernova by investigating features of fe ii absorption lines in the ultraviolet ( uv ) spectrum . very broad wings were observed in both blue and red sides of the absorption lines . the line width was a few thousand km / s much larger than the thermal velocity of stellar atmosphere , which is thought to be @xmath410 km / s . the broad wings are likely to be formed by fe ii in the ejecta of snr 1006 ; photons in the blue wing are absorbed by the matter ejected toward us , and those in the red wing away from us . thus it was proved that the s - m star is located behind the snr 1006 ejecta . when a star is inside the ejecta of sne ia , photons emitted from the star are absorbed only by the ejecta moving toward us . hence the broad wing must be present only in the blue side . the absorption line with only blue wing enables us to identify companion stars of sne ia . @xcite used the uv range to observe the s - m star with the faint object spectrograph on the hubble space telescope . however , in addition to difficulties in uv observations from the ground , companion stars on the evolutionary paths suggested by the above mentioned scenario @xcite may not be bright in the uv range . then we will focus on absorption lines in the visible range . furthermore the corresponding transitions need to be from the ground state because most fe ions in the ejecta are expected to be in the ground state . thus only fe i can produce such absorption lines in sne ia ejecta . in this paper , we estimate the amount of fe i in the ejecta of tycho s snr by taking account of collisional processes in non - equilibrium and ionizations by photons emitted from the shocked ejecta , and calculate spectra of a star located at the center of a snr and discuss whether we can identify the feature of fe i absorption lines in the spectrum of the companion star . [ cols="^,^,^,^,^,^ " , ] [ tbl : ew ] even when we find a star that exhibits the absorption feature discussed in this paper , there is a chance that a star other than the companion star happens to be inside the ejecta . a star existing in the vicinity of a sn ia gets a fraction of the explosion energy and is accelerated . if the companion star similar to tycho g star with the mass of @xmath5 1.2 @xmath6 and the radius of @xmath7 2 @xmath8 is located at the distance of @xmath9 5.5 @xmath8 from the progenitor , the size of the roche lobe is comparable to the size of the companion . thus the star in this situation will get the maximum velocity after the explosion . then the orbital velocity before the explosion is @xmath10 km / s . in a 3-d hydrodynamical simulation of sne ia by @xcite , they obtain a plausible kick velocity @xmath11 km / s . therefore , including the orbital velocity , the velocity of the companion star becomes @xmath12 km / s . if the companion star has been moving away from the explosion site at that speed , the companion star is now at the distance of @xmath130.08 pc from the center . the stellar mass density in the neighborhood of tycho s snr estimated from a model of the galaxy @xcite is less than 0.03 @xmath14 . thus the expected stellar mass inside this volume is only @xmath15 . therefore if we find a star showing absorption lines with broad blue wings in the spectrum , it is likely that the star is the companion star . as a consequence , we have demonstrated that there exhibit very deep absorption lines with unique shapes in the spectrum of the companion star located at the center of a young snr such as tycho s snr . there are , however , a few factors that might reduce or even erase these distinct absorption features . first , the number of fe i in the freely expanding ejecta is very sensitive to the number of ionizing photons emitted from the shocked ejecta . an increase in the number of ionizing photons by a few factors might decrease the number of fe i by a few orders of magnitude or more . since the main source of ionizing photons is o in the outer ejecta , the distributions of o and density in the outer ejecta need to be known precisely . unfortunately , these regions in w7 have some problems to reproduce the observed optical spectra ( * ? ? ? * and references therein ) . due to this uncertainty in the explosion model , it is not conclusive if the companion star of tycho s sn will exhibit unique fe i absorption features discussed in this paper . nevertheless , it is true that every sn ia has a period during which the companion star has the distinct absorption features discussed above because the ejecta become cool enough to have plenty of fe i for a time after the optical brightening . second , it is assumed in our calculations that ions in the shocked region are ionized to fe@xmath16 , si@xmath17 , o@xmath18 , c@xmath19 immediately after the shock passes following the procedure taken by @xcite . since ions in lower ionization stages are a strong source of ionizing photons , ionization may be more advanced in real young snrs . arnaud , m. rothenflug , r. 1985 , a&as , 60 , 425 branch , d. 1998 , , 36 , 17 colella , p. , woodward , p. 1984 , jcoph , 54 , 174 decourchelle , a. , et al . 2001 , a&as , 365 , l218 dehnen , w. , binney , j. , 1998 , mnras , 294 , 429 engstrom , l. , et al . 1992 , jphb , 25 , 2459 fuhrmann , k. 2005 , mnras , 359 , l35 hachisu , i. , kato , m. , nomoto , k. 1996 , apj , 470 , l97 hachisu , i. , kato , m. , nomoto , k. , umeda , h. 1999 , apj , 519 , 314 hachisu , i. , kato , m. , nomoto , k. 1999 , apj , 522 , 487 hamilton , a. , sarazin , c. 1984 , apj , 287 , 282 hamilton , a. , fesen , r. 1988 , apj , 327 , 178 itoh , h. , masai , k. , nomoto , k. 1988 , apj , 334 , 279 kato , t. 1976 , apjs , 30 , 397 reinecke , m. , hillebrandt , w. , & niemeyer , j. c. 2002 , , 391 , 1167 lotz , w. 1967 , apjs , 14 , 207l marietta , e. , burrows , a. , fryxell , b. 2000 , apjs , 128 , 615 mihalas , d. , 1978 , stellar atmospheres . nomoto , k. 1982 , , 253 , 798 nomoto , k , thielemann , f. , yokoi , k. 1984 , apj , 286 , 644 ruiz - lapuente , p. , et al . 2004 , nature , 431 , 1069 schweizer , f. middleditch , j. 1980 , apj , 241 , 1039 sorokina , e. i. , et al . 2004 , astl , 30 , 737 spitzer , l. 1978 , physical processes in the interstellar medium . tunklev , m. , et al . 1997 , phys , 55 , 707 verner , d. , et al . 1996 , apj , 465 , 487 vink , j. , et al . 2003 , apjl , 587 , l31 wang , l. , baade , d. , hflich , p. , wheeler , j. c. , kawabata , k. , & nomoto , k. 2004 , , 604 , l53 woosley , s. e. , & weaver , t. a. 1986 , lecture notes in physics , berlin springer verlag , 255 , 91 wu , c. , et al . 1993 , apj , 416 , 247

we propose a method to identify the companion stars of type ia supernovae ( sne ia ) in young supernova remnants ( snrs ) by recognizing distinct features of absorption lines due to fe i appearing in the spectrum . if a sufficient amount of fe i remains in the ejecta , fe i atoms moving toward us absorb photons by transitions from the ground state to imprint broad absorption lines exclusively with the blue - shifted components in the spectrum of the companion star . to investigate the time evolution of column depth of fe i in the ejecta , we have performed hydrodynamical calculations for snrs expanding into the uniform ambient media , taking into account collisional ionizations , excitations , and photo - ionizations of heavy elements . as a result , it is found that the companion star in tycho s snr will exhibit observable features in absorption lines due to fe i at @xmath0 nm and 385.9911 nm if a carbon deflagration sn model @xcite is taken . however , these features may disappear by taking another model that emits a few times more intense ionizing photons from the shocked outer layers . to further explore the ionization states in the freely expanding ejecta , we need a reliable model to describe the structure of the outer layers .

arxiv

quantum entanglement is a feature of quantum mechanics that has captured much recent interest due to its essential role in quantum information processing @xcite . it may be characterized and manipulated independently of its physical realization , and it obeys a set of conservation laws ; as such , it is regarded and treated much like a physical resource . it proves useful in making quantitative predictions to quantify entanglement.when one has complete information about a bipartite system subsystems @xmath0 and @xmath1the state of the system is pure and there exists a well established measure of entanglement the _ entropy of entanglement _ , evaluated as the von neumann entropy of the reduced density matrix , @xmath2 with @xmath3 . this measure is unity for the bell states and is conserved under local operations and classical communication . unfortunately , however , quantum systems in nature interact with their environment ; states of practical concern are therefore mixed , in which case the quantification of entanglement becomes less clear . given an ensemble of pure states , @xmath4 with probabilities @xmath5 , a natural generalization of @xmath6 is its weighted average @xmath7 . a difficulty arises , though , when one considers that a given density operator may be decomposed in infinitely many ways , leading to infinitely many values for this average entanglement . the density operator for an equal mixture of bell states @xmath8 , for example , is identical to that for a mixture of @xmath9 and @xmath10 , yet by the above measure the two decompositions have entanglement one and zero , respectively . various measures have been proposed to circumvent this problem , most of which evaluate a lower bound . one such measure , the _ entanglement of formation _ , @xmath11 @xcite , is defined as the minimal amount of entanglement required to form the density operator @xmath12 , while the _ entanglement of distillation _ , @xmath13 @xcite , is the guaranteed amount of entanglement that can be extracted from @xmath12 . these measures satisfy the requirements for a physical entanglement measure set out by horodecki _ et al_. @xcite . they give the value zero for @xmath14 , which might be thought somewhat counterintuitive , since this state can be viewed as representing a sequence of random `` choices '' between two bell states , both of which are maximally entangled . this is unavoidable , however , because assigning @xmath15 a non - zero value of entanglement would imply that entanglement can be generated by local operations . the problem is fundamental , steming from the inherent uncertainty surrounding a mixed state : the state provides an incomplete description of the physical system , and in view of the lack of knowledge a definitive measure of entanglement can not be given . an interacting system and environment inevitably become entangled . the problem of bipartite entanglement for an open system is therefore one of tripartite entanglement for the system and environment . complicating the situation , the state of the environment is complex and unknown . conventionally , the partial trace with respect to the environment is taken , yielding a mixed state for the bipartite system . if one wishes for a more complete characterization of the entanglement than provided by the above measures , somehow the inherent uncertainty of the mixed state description must be removed . to this end , nha and carmichael @xcite recently introduced a measure of entanglement for open systems based upon quantum trajectory unravelings of the open system dynamics @xcite . central to their approach is a consideration of the way in which information about the system is read , by making measurements , from the environment . the evolution of the system conditioned on the measurement record is followed , and the entanglement measure is then contextual dependent upon the kind of measurements made . suppose , for example , that at some time @xmath16 the system and environment are in the entangled state @xmath17 a partial trace with respect to @xmath18 yields a mixed state for @xmath19 . if , on the other hand , an observer makes a measurement on the environment with respect to the basis @xmath20 , obtaining the `` result '' @xmath21 , the reduced state of the system and environment is @xmath22 with conditional system state @xmath23 where @xmath24 is the probability of the particular measurement result . thus , the system and environment are disentangled , so the system state is pure and its bipartite entanglement is defined by the von neumann entropy , eq . ( [ eq : von - neumann ] ) . nha and carmichael @xcite apply this idea to the continuous measurement limit , where @xmath25 executes a conditional evolution over time . in this paper we follow the lead of nha and carmichael , also carvalho _ et al . _ @xcite , not to compute their entanglement measure _ per se _ , but to examine the entanglement _ dynamics _ of a cascaded qubit system coupled through the oneway exchange of photons . the system considered has been shown to produce unconditional entangled states generally a superposition of bell states as the steady - state solution to a master equation @xcite . for a special choice of parameters ( resonance ) , a maximally entangled bell state is achieved @xmath26 except that the approach to the steady state takes place over an infinite amount of time . here we analyze the conditional evolution of the qubit system to illuminate the dynamical creation of entanglement in the general case , and to explain , in particular , the infinitely slow approach to steady - state in the special case . we demonstrate that in the special case the conditional dynamics exhibit a distinct bimodality , where the approach to the bell state is only one of two possibilities for the asymptotic evolution : the second we call an _ entangled - state cycle _ , where the qubits execute a sustained stochastic switching between two bell states . though involving just two qubits and elementary quantum transitions , the situation is similar to that of a bimodal system in classical statistical physics in the limit of a vanishing transition rate between attractors . the physical model of the cascaded qubit system is presented in sec . [ subsec2_1 ] and the quantum trajectory unraveling of its conditional dynamics in sec . [ subsec3_1 ] . in sec . [ subsec4_1 ] we analyze the quantum trajectory equations to demonstrate bimodality and the existence of entangled - state cycles . finally , a discussion and conclusions are presented in sec . [ subsec5_1 ] . in this section we briefly outline the physical model for the cascaded qubit system to be analyzed . a more detailed description , together with the techniques and assumptions used to derive the model master equation presented here , is available in @xcite . the system considered consists of two high - finesse optical cavities , each containing a single tightly - confined atom , the cavities arranged in a cascaded configuration with unidirectional coupling from cavity 1 to cavity 2 ( fig . [ fig : fig1 ] ) . for simplicity , we consider the cavity modes to be identical , with resonance frequency @xmath27 and field decay rate @xmath28 . inefficiencies and losses in the coupling between the cavities are modeled by a real parameter @xmath29 , @xmath30 , with perfect coupling corresponding to @xmath31 . the atoms are assumed to have five relevant electronic levels , of which two ground states , @xmath32 and @xmath33 , represent an effective two - state system , or qubit . and @xmath33 , to three excited states , @xmath34 , @xmath35 , and @xmath36.,scaledwidth=40.0% ] for each atom , the cavity field in combination with auxiliary laser fields ( incident from the side of the cavity ) drives two separate resonant raman transitions between states @xmath32 and @xmath33 . an additional laser field coupled to the @xmath37 transition provides a tunable light shift of the energy of state @xmath32 . all fields are assumed far detuned from the atomic excited states , so these states may be adiabatically eliminated and atomic spontaneous emission ignored . under the further assumption that the cavity field decay rate is much larger than the transition rates between @xmath32 and @xmath33 , the cavity fields may also be adiabatically eliminated to yield a master equation for the reduced two - atom density matrix @xmath12 , @xmath38}+{\left[\hat r_2,\rho\hat r_1^\dag\right]}\right)},\end{aligned}\ ] ] with @xmath39 where @xmath40 , and @xmath41 and @xmath42 are the rates of @xmath43 and @xmath44 transitions , respectively . by virtue of the cavity output , the system is an open system and solutions to master equation ( [ eq : me ] ) generally describe mixed states . under appropriate conditions , however , the system evolves to a pure and entangled steady state . if the coupling between cavities is perfect ( @xmath31 ) and the parameters of the subsystems are the same ( @xmath45 , @xmath46 ) then the steady state is the pure state @xmath47 where we use the abbreviated notation @xmath48 and @xmath49 . then when @xmath50 , which we shall refer to as the _ resonance _ condition , the steady state is a maximally - entangled bell state . this may seem to be ideal , but a problem arises when we consider the eigenvalues of the operator @xmath51 . specifically , the characteristic time for the system to reach steady state , @xmath52 , where @xmath53 denotes the eigenvalue of @xmath51 with smallest ( in magnitude ) non - zero real part , approaches infinity as the resonance condition is approached . this is shown by the plot in fig . [ fig : fig2 ] . thus the master equation itself , in particular its steady state , offers limited insight into the behavior of the system at resonance . we wish to learn more about this special case ; in particular , how does the entanglement develop dynamically . also , if additional information is factored into the description , by making measurements on the environment , can we better characterize the long term behavior , or possibly find perfect entanglement after a finite time ? we demonstrate that quantum trajectory theory can provide answers to these questions . the relaxation time @xmath54 plotted as a function of @xmath55 . note the singularity at resonance , @xmath56.,scaledwidth=40.0% ] as with any open system , the first step in unraveling the master equation is to identify the points of coupling to the environment . the first is obvious the output from cavity 2 . to measure this output , let us assume the existence of an ideal photon detector in the path of the output from cavity 2 ; we call it _ detector 1_. the second point of coupling to the environment is more subtle . our model does not assume the inter - cavity coupling to be perfect ; only a fraction @xmath29 of the output photon flux from cavity 1 makes it into cavity 2 . physically , this loss may be caused , for example , by non - ideal transmissivity of the faraday isolators or by absorption in the cavity mirrors . these imperfections cause photons to be scattered into the environment in some uncontrollable fashion . formally , though , this is equivalent to assuming that the apparatus is ideal , except that there exists a beamsplitter between the cavities , as drawn schematically in fig . [ fig : fig3 ] . we therefore further assume the existence of a second photon detector to collect photons reflected by this beamsplitter ; we call it _ detector 2_. we now proceed to develop the quantum trajectory formalism for the cascaded qubit system . in this approach the system is described by a pure state which is dependent on ( conditioned on ) the counting histories , or records , of detectors 1 and 2 . firstly , we rewrite the master equation in a form suitable for translation into the quantum trajectory language . we reexpress eq . ( [ eq : me ] ) in the form @xmath57 with @xmath58-\frac{1}{2}\sum_{i=1,2 } \left(\hat c_i^{\dag}\hat c_i \rho+\rho\hat c_i^{\dag}\hat c_i\right),\\ \mathcal{s}\rho&=\sum_{i=1,2}\hat c_i \rho\hat c_i^{\dag},\end{aligned}\ ] ] where @xmath59 then , within quantum trajectory theory , the evolution of the system is described by a pure state @xmath60 which evolves under the non - hermitian effective hamiltonian @xmath61 the continuous evolution interrupted at random times by quantum jumps , @xmath62 , where the jumps occur with probability @xmath63 in time interval @xmath64 . physically , the jump operators @xmath65 and @xmath66 account for the reduction of the state of the system , given a photon count is recorded by detector 1 or detector 2 , respectively . thus , within the quantum trajectory description of the coupled cavity system , we consider an experiment in which ideal detectors are employed , such that every scattered photon is detected and recorded . given the history of detector ` clicks ' , one has complete information about the system state , in the sense that that state is always pure ; hence , although the solution to the master equation is generally mixed , one is able to characterize the entanglement in an unambiguous ( conditional ) fashion @xcite . consider the special case where the coupling between the cavities is optimal ( @xmath31 ) . in this case there is only one output from the system , that from cavity 2 , recorded by detector 1 . standard numerical algorithms @xcite have been used to simulate typical quantum trajectories for various values of @xmath55 . specifically , we consider the evolution of the conditional expectation of the operator product @xmath67 , where @xmath68 is the pauli operator diagonal in the @xmath69representation , @xmath70 this expectation has a number of convenient properties ; for example , the steady - state value @xmath71 regardless of the value of @xmath55 , which makes it easy to compare rates of convergence to the steady state for different system parameters . -0.5 cm figure [ fig : fig4 ] contrasts the solution to the master equation and a single quantum trajectory . the solution to the master equation exhibits a completely smooth evolution that tends asymptotically towards the steady state . the quantum trajectory , on the other hand , undergoes a sequence of switches between two extreme values of @xmath72 , which occur at each photon detection . provided the parameters are chosen away from resonance , the photon detections eventually stop and the trajectory settles into the steady state ( [ eq : ss ] ) , with @xmath73 ; the steady state is clearly a dark state . at resonance , however , the photon detections may continue indefinitely . physically , this seems plausible , since it simply implies that the atoms continue to switch between states @xmath32 and @xmath33 , scattering one photon with each transition . at resonance , apparently , a unique equilibrium dark state can not be established . the cyclic behavior that replaces it is completely invisible if we consider only the ensemble average a vivid demonstration of how single quantum trajectories can provide additional insight into the evolution of an open quantum system . the oscillatory behavior featured in fig . [ fig : fig4 ] hints at a simple cyclic process . in fact , it is simple enough that we can understand why it occurs without resorting to numerics . in this section we formulate a graphical description of individual trajectories . figure [ fig : fig4 ] demonstrates that the conditional expectation @xmath72 is conserved during the periods of evolution between quantum jumps . the positively and negatively correlated subspaces @xmath74 are coupled only through quantum jumps . noting that @xmath75 are each @xmath76-dimensional ( assuming real amplitudes without loss of generality ) , we manage to break up a @xmath77-dimensional space into two @xmath76-dimensional planes , linked to one another by the quantum jumps . we refer to this representation as the _ cascaded system phase space_. trajectories within it can be viewed as lines moving continuosly within either plane and jumping discontinuously between the planes . we use phase space portraits within @xmath78 and @xmath79 to characterize the behavior of the system , where for the sake of simplicity , and without loss of generality , we are assuming @xmath80 and @xmath81 to be real . we define @xmath82 and scale time by setting @xmath83 . the master equation then takes the form ( @xmath31 ) @xmath84 where @xmath85 the resonance condition is now @xmath86 . it is useful to convert to a matrix notation , such that a pure state @xmath60 of the system is represented by a @xmath77-vector , @xmath87 and system operators are written as @xmath88 matrices , e.g. , @xmath89 and @xmath90 the evolution of @xmath60under @xmath91 is written as a linear differential equation in four variables , @xmath92\ ! { |\phi\rangle}\nonumber\\ & = \left(\begin{array}{cccc } -2&0&0&2r\\ 0&-(1+r^2)&2r^2&0\\ 0&2&-(1+r^2)&0\\ 2r&0&0&-2r^2\\\end{array}\right ) \label{eqn : basicde}\!{|\phi\rangle}.\end{aligned}\ ] ] as noted above , this evolution is constrained within either @xmath78 or @xmath79 . thus we can write @xmath60 as a vector sum of two orthogonal components @xmath93 and @xmath94 , @xmath95 , to obtain the decoupled dynamics @xmath96 eigenvectors of the two dynamical matrices correspond to states of the system that are preserved under the evolution between quantum jumps . note , however , that it does not necessarily follow that such a state is a steady state of the quantum trajectory evolution as a whole ; it must eventually experience a quantum jump if its norm decays i.e . , the corresponding eigenvalue is not zero . recall from quantum trajectory theory that the probability for a state not to jump prior to time @xmath16 is given by its norm @xcite . for the systems of equations given above we find the following ( unnormalised ) eigenstates and eigenvalues : 1 . @xmath97 , @xmath98 ; this is the steady state of the system for @xmath99 . 2 . @xmath100 , @xmath101 ; this state in @xmath78 is orthogonal to @xmath102 and must eventually jump to a state in @xmath79 . 3 . @xmath103 , @xmath104 ; this state in @xmath79 must eventually jump to a state in @xmath78 unless @xmath86 ; in the latter case it plays no role once an entangled - state cycle is initiated ( see below ) . 4 . @xmath105 , @xmath106 ; this state in @xmath79 must eventually jump to a state in @xmath78 . in the special case of resonance , @xmath86 , there are two independent steady states , @xmath102 and @xmath107 , which helps to explain the failure of the master equation evolution to approach a unique steady state . it also suggests a fundamental feature of the indefinite switching , the cyclic behavior , revealed by individual quantum trajectories : during such an _ entangled - state cycle _ , the system state must remain orthogonal to @xmath102 and @xmath107 . we verify this shortly , after examining the trajectory evolution away from resonance , where the steady state @xmath102 is always reached for perfect inter - cavity coupling . typical quantum trajectories for @xmath108 are shown in figs . [ fig : fig5 ] and [ fig : fig6 ] , where the @xmath78 and @xmath79 subspaces are drawn as circular planes . normalized states are located on the circumferences of the circles . the bell states @xmath109 lie at intersections of the circumference with the dotted lines as shown . between quantum jumps , under the influence of the non - hermitian hamiltonian @xmath91 , the norm of the state decays and the point representing it within the phase space moves to the interior of one of the circles . quantum jumps cause a switch from @xmath78 to @xmath79 or vice - versa . they are represented by the lines connecting the two planes , where for illustrative purposes , the system state is renormalized after each quantum jump ; thus jumps terminate at points on the circumference of the circles . we restrict ourselves to separable initial states located in one or other of the two subspaces ; for example , the states @xmath110 and @xmath111 , respectively , are considered in figs . [ fig : fig5 ] and [ fig : fig6 ] . the action of the jump operator @xmath65 on states located in @xmath78 ( with renormalization ) is @xmath112 while the action of @xmath65 on states in @xmath79 is @xmath113 thus , when a quantum jump occurs , any state within @xmath78 collapses onto the bell state @xmath114 in @xmath79 , while any state within @xmath79 collapses onto the state @xmath115 in @xmath78 . consider an initial normalized state in @xmath78 , @xmath116 , for some ( real ) @xmath117 . given that @xmath102 is a steady state of the evolution between quantum jumps , the probability of an eventual quantum jump to @xmath79 is @xmath118 while with probability @xmath119 the system evolves to the steady state @xmath102 without any photon emissions . if a jump from @xmath79 to @xmath120 has just occurred , then by the same argument one shows that the probability of a future quantum jump to @xmath79 is @xmath121 , or , alternatively , the probability of reaching the steady state after such a jump is @xmath122 ^ 2 $ ] . consider now an initial state in @xmath79 , @xmath123 , for some ( real ) @xmath124 . owing to the instability of both @xmath107 and @xmath125 for @xmath99 , an eventual quantum jump is guaranteed ; thus , @xmath126 armed with this information , we move to an explanation of the quantum trajectories displayed in figs . [ fig : fig5 ] and [ fig : fig6 ] . in fig . [ fig : fig5 ] we plot three typical phase - space trajectories for @xmath108 and @xmath127 . [ fig : fig5](a ) illustrates the case where the system evolves directly to the steady state @xmath102 . the probability of this event is @xmath128 , so it is the most likely occurrence for the chosen parameters . if a first quantum jump does occur , then typical trajectories are shown in figs . [ fig : fig5](b ) and ( c ) . following the jump to @xmath129 in @xmath79 , a second jump returning the state to @xmath78 is guaranteed . for @xmath108 , this leaves the system in the state @xmath130 , from which the probability of a further cycle of jumps is @xmath131 . thus , after a first quantum jump cycle , it is most likely that further cycles will follow , as seen in figs . [ fig : fig5](b ) and ( c ) , where in both cases a total of five cycles ( ten photon detections ) occur before the system finally reaches the steady state . in fig . [ fig : fig6 ] we plot three typical phase - space trajectories for @xmath108 and @xmath132 . in this case , at least one quantum jump is certain to occur , following which the probability of further jumps is @xmath131 , as above . so for this initial condition , the most likely outcome is a sequence of quantum jump cycles following a first guaranteed photon detection . in fig . [ fig : fig6 ] ( a ) only the first detection occurs , while in figs . [ fig : fig6](b ) and ( c ) this detection is followed by a sequence of cycles before the steady state is eventually achieved . the case @xmath86 is of particular interest . the normalized eigenstates of the evolution between quantum jumps are the bell states @xmath133 , @xmath134 , @xmath135 , and @xmath136 . the eigenvalues are @xmath137 and @xmath138 . the action of the jump operator @xmath65 on states within @xmath78 simplifies to @xmath139 and its action on states within @xmath79 to @xmath140 for @xmath86 , photon detections , if they occur , are associated with collapses onto one of two maximally - entangled bell states . for initial states @xmath141 and @xmath142 in @xmath78 and @xmath79 , respectively , the system evolves continuously , without the emission of any photons , to @xmath133 and @xmath143 , with probabilities @xmath144 and @xmath145 in @xmath79 or @xmath146 in @xmath78 . in this case , as both terminal states are unstable under the between - jump evolution , a second detection and quantum jump must follow . according to eqs . ( [ eq : plusjump ] ) and ( [ eq : minusjump ] ) this simply exchanges @xmath147 for @xmath146 and vice - versa . hence , a perpetual switching between bell states @xmath147 and @xmath146 occurs . we designate this behavior an _ entangled - state cycle_. thus , at resonance we find a distinctly bimodal behavior . the system either evolves into a maximally - entangled bell state without emitting photons , or an entangled - state cycle is initiated under which the system switches indefinitely between orthogonal bell states while emitting a continual stream of photons . as an aside , such behavior can be regarded as a quantum measurement that distinguishes the bell states @xmath148 from @xmath149 . the two alternative outcomes of the quantum trajectory evolution are illustrated in figs . [ fig : fig7 ] and [ fig : fig8 ] for the initial states @xmath150 in @xmath78 and @xmath132 in @xmath79 , respectively . with this choice of initial states there are equal probabilities for reaching the steady states , @xmath151 [ fig . [ fig : fig7](a ) ] and @xmath152 [ fig . [ fig : fig8](a ) ] , and for commencing an entangled - state cycle [ figs . [ fig : fig7](b ) and [ fig : fig8](b ) ] . note that once an entangled - state cycle is initiated , the trajectory remains in a plane orthogonal to the lines defining @xmath151 and @xmath152 ; the cycle continues indefinitely our original model allowed for the possibility of imperfect intercavity coupling , through the parameter @xmath29 and the jump operator @xmath66 which describe the effects of photon loss in propagation between the two cavities . focusing on the resonant case ( @xmath86 ) , we now consider the situation in which @xmath153 . typical trajectories for @xmath154 are shown in figs . [ fig : fig9](a ) and [ fig : fig10](a ) , with the two photon count records shown in frames ( b ) and ( c ) of the figures . remarkably , entangled - state cycles persist , but now the system settles into one or other of two distinct cycles , involving either the symmetric or antisymmetric bell states . to understand the behavior , consider the forms of the operators involved ; in particular , for @xmath86 , we have effective hamiltonian @xmath155 and jump operators @xmath156 and @xmath157 significantly , these operators commute with one another , @xmath158=[\hat c_1,\hat h_{\rm eff}]=[\hat c_2,\hat h_{\rm eff}]=0.\end{aligned}\ ] ] their operation upon the bell states is given by @xmath159 and @xmath160 thus , the bell states are eigenstates of @xmath91 , and the jump operators interchange bell states in @xmath78 and @xmath79 : each jump operator converts the symmetric ( antisymmetric ) bell state in @xmath78 to the symmetric ( antisymmetric ) bell state in @xmath79 and vice - versa . now , let us consider a particular quantum trajectory for which a total of @xmath161 jumps occur , separated by the time intervals @xmath162 . for an initial state @xmath163 , the ( unnormalized ) state at the conclusion of the @xmath161 jumps is written as @xmath164 where each @xmath165 is either @xmath65 or @xmath66 . since all operators in the string acting on @xmath166 commute , this expression can be rewritten in a variety of forms , two of which prove to be especially useful in explaining the distinct behaviors illustrated by figs . [ fig : fig9 ] and [ fig : fig10 ] . in the first case , we may write @xmath167 passing all @xmath168 ocurrences of @xmath65 to the right and all @xmath169 occurrences of @xmath66 to the left ( @xmath170 ) ; in the second we write @xmath171 where all jump operators are passed to the left . the arbitrary ( pure ) initial state can be expressed as a superposition of bell states , @xmath172 where @xmath173 , @xmath174 , @xmath175 , and @xmath176 are expansion coefficients , generally complex . substituting this expansion into eqs . ( [ phit1 ] ) and ( [ phit2 ] ) , and using eqs . ( [ heff1])([c2psi])assuming for simplicity that @xmath168 and @xmath169 are even the two forms for the state @xmath177 are @xmath178 where @xmath179 observe now that the ratio of the eigenvalues satisfies @xmath180 it follows that @xmath181 and @xmath182 allow us to predict quite distinct asymptotic behaviors for the system state . for sufficiently large @xmath168 , the contribution to @xmath181 from the symmetric bell states is negligible compared with the contribution from the antisymmetric bell states , in which case , using eqs . ( [ eq : cycle1 ] ) and ( [ eq : cycle1prime ] ) , @xmath183 the system is locked into a cycle between the two antisymmetric bell states , the situation illustrated in fig . [ fig : fig9 ] ( for @xmath184 , @xmath185 ) . in contrast , for sufficiently large @xmath16 , the contribution to @xmath182 from the antisymmetric bell states is negligible compared with that from the symmetric bell states , and using eqs . ( [ eq : cycle2 ] ) and ( [ eq : cycle2prime ] ) , @xmath186 the system is locked into a cycle between the two symmetric bell states , as shown in fig . [ fig : fig10 ] . which of the two cycles is chosen in a particular realization of the photon counting record is random , as is the time taken to settle into the cycle . effectively , the decision is the outcome of a competition between the periods of evolution between quantum jumps and the jumps themselves specifically , those associated with photon counts at detector 1 . considering eqs . ( [ eq : cycle1prime ] ) and ( [ eq : ratio ] ) , we see that every count at detector 1 results in an increased probability to find the system in one of the antisymmetric bell states . on the other hand , from eqs . ( [ eq : cycle2prime ] ) and ( [ eq : ratio ] ) , the periods of evolution between counts have the reverse effect they increase the probability for the system to be found in a symmetric bell state . the critical factor that decides which tendency wins is the number of photon counts occuring at detector 1 over a given ( substantial ) interval of time . if there are many , as in fig . [ fig : fig9](b ) , the entangled - state cycle between antisymmetric bell states wins out ; if there are few , fig . [ fig : fig10](b ) , the cycle between symmetric bell states occurs . the same decision mechanism is observed in other examples @xcite . note that counts at detector 2 are not involved not directly at least . they do figure indirectly as a mechanism reducing the average number of counts at detector 1 ; indeed , they are the ultimate source of the asymmetry reflected in the ratio @xmath187 . as the system approaches a particular cycle the quantum trajectory evolution tends to reinforce the establishment of the cycle . close to the antisymmetric cycle , the evolution between jumps is dominantly governed by @xmath188 and is therefore relatively fast . this leads to frequent photon counts at detector 1 [ fig . [ fig : fig9](b ) ] . close to the symmetric cycle , the between - jump evolution is dominantly governed by @xmath189 , hence is relatively slow . photon counts at detector 1 become much less frequent [ fig . [ fig : fig10](b ) ] . from the dramatic difference in count rates at detector 1 for the two cycles , it is clear that one can determine which entanglement cycle the system evolves to for a particular realization . however , without knowledge of the record of photon counts at detector 2 , which by definition we do not have , one can not know where on the cycle the system is , i.e. , whether the state is in @xmath78 or @xmath79 . thus , the ensemble average state of the system is mixed , described by one of the density operators @xmath190 consider a thought experiment where the cascaded qubit system , set to resonance , evolves freely and its entire output is collected and stored inside a black box . at some time the lasers driving the raman transitions are turned off , so the evolution ceases . the box and qubits are separated and moved to causally disconnected regions of space time . let alice and bob be standard observers of the qubits , and give eve jurisdiction over the box . we can now ask , how much entanglement exists between the qubits of alice and bob ? while this is simply a roundabout way of asking how entanglement evolves , it helps elucidate some of the key concepts behind the quantum trajectory measure of entanglement . conventional entanglement measures are based upon an analysis of the density matrix at this time . they throw away the box and look at the system of qubits alone they disregard eve and view the system from the perspective of alice and bob . yet in general every interaction between two objects entangles them , and as the qubit system and box interacted in the past , their states are intertwined . neither possess an independent reality , and neither , considered alone , can be completely described . eve s box contains information , which , if discarded , adds entropy to the qubit system of alice and bob . this entropy is the source of ambiguity in the quantification of entanglement . from this point of view , as noted in the introduction , the problem of bi - partite entanglement in an open system relates to that of tri - partite entanglement in a closed one . to completely characterize the entanglement of the present example , in addition to the entanglement between alice and bob , we must consider their entanglement with eve . a quantum description of the box is impractical , but it is feasible to extract classical information about what it contains , through measurement . quantum trajectories facilitate this , and allow us not to discard the box completely . in turn , the system state retains its purity , conditional on the classical information extracted from the box . with this extra information , we can extract more entanglement from the cascaded qubit system . working from the master equation for the cascaded system @xcite , previously it was assumed that the system evolved gradually into a pure state , whereby entanglement was generated . the behaviour at resonance , however , was unclear , since there the master equation had two zero eigenvalues and no well - defined steady state . by considering the conditional evolution we have shown that , at resonance , asymptotically the system is either in the bell state @xmath151 or oscillating ( stochastically switching ) between two bell states , @xmath146 and @xmath147 . from the density matrix point of view , the latter is an equal mixture of bell states and would yield no entanglement under any mixed state measure ; physically , alice and bob , without collaboration from eve , can not extract any entanglement from their qubits . suppose , however , that eve opens her box to count the number of photons inside . seeing whether the count is even or odd , she is able to deduce exactly which bell state alice and bob s system is in . thus , her measurement unravels the density operator , creating entanglement , despite the fact that the measurement is not causally connected to alice and bob s qubits . it is tempting to say that the entanglement was always there , as a matter of fact , until one realizes that there are many other ways in which eve could choose to measure her state , each producing a different unravelling of the qubit system and yielding a different value of entanglement . the entanglement facilitated by eve s measurements is _ contextual _ in this sense . this thought experiment demonstrates why any attempt to quantify the entanglement of an open system from the density operator alone can not be considered complete . the density operator should not be treated as a fundamental object , as it does not provide a complete description of the physical state . we have presented a simple example where oscillations between maximally entangled states are hidden within a separable density operator . the fact that the density operator contains entropy , implies that information about its entanglement with an external system was discarded at some time . in studying such a mixed state , there is benefit from considering , not only the mixed state itself , but the process through which it was generated , and the access this potentially gives to a conditional dynamics . the results of this paper could be extended by employing quantum trajectories in a broader sense . in cases where the results of environmental interactions can not be measured , such as coupling loss , wiseman and vaccaro @xcite have shown that only certain unravelings can be physically realized . a conceivable measure of entanglement would take the minimum of all physically realizable unravelings . alternatively , one might take the maximum of all physically realizable unravellings , which would measure the maximum distillable entanglement when local measurements on the environment are taken into account . h. j. carmichael , p. kochan , and l. tian , `` coherent states and open quantum systems : a comment on the stern - gerlach experiment and schrdinger cats , '' in _ coherent states : past , present , and future _ , eds . d. h. feng , j. r. klauder , and m. r. strayer ( world scientific , singapore , 1994 ) , pp . 75 - 91 .

a system of cascaded qubits interacting via the oneway exchange of photons is studied . while for general operating conditions the system evolves to a superposition of bell states ( a dark state ) in the long - time limit , under a particular _ resonance _ condition no steady state is reached within a finite time . we analyze the conditional quantum evolution ( quantum trajectories ) to characterize the asymptotic behavior under this resonance condition . a distinct bimodality is observed : for perfect qubit coupling , the system either evolves to a maximally entangled bell state without emitting photons ( the dark state ) , or executes a sustained entangled - state cycle random switching between a pair of bell states while emitting a continuous photon stream ; for imperfect coupling , two entangled - state cycles coexist , between which a random selection is made from one quantum trajectory to another .

arxiv

in the unification scheme of agn the difference between type 1 and type 2 agn is explained by angle - dependent circumnuclear obscuration of the accretion disk and broad - line region @xcite . this obscuring dusty medium commonly referred to as `` dust torus '' is optically and geometrically thick and probably extends from sub - parsec scales outward to several 10s of parsecs , or beyond for high luminosity objects . the dust in the torus absorbs the incident uv / optical radiation an re - emits the received energy in the infrared . observations have shown that type 1 agn show significantly more emission in the near - ir than type 2 agn for the same given intrinsic luminosity ( e.g. * ? ? ? this is consistent with the picture where the face - on view onto the torus in type 1 agn exposes the innermost hot dust to the observer . on the other hand in type 2 agn the torus is seen edge - on so that internal obscuration blocks the line - of - sight to the hot dust . owing to this effect , it is expected that for a given agn luminosity the infrared emission of type 1 agn is generally stronger than from type 2s . in the light of attempts at forming isotropic agn samples based on ir fluxes it seems important to know exactly how strong of a bias towards type 1s over type 2s may occur when invoking flux limits . moreover , probing the wavelength dependence of this anisotropy in the infrared has some constraining power on our understanding of how the torus obscures the agn . it may be possible to distinguish torus models where the dust is smoothly distributed from those where the dust is arranged in clouds ( e.g. * ? ? ? * ) : if the dust is smoothly distributed within the torus , a large degree of anisotropy is expected . if , however , the dust is arranged in clouds the anisotropy is expected to be smaller . a problem commonly encountered when studying agn samples in the local universe is a significant contribution of the host galaxy to the ir . this is related to the typical lower luminosity seyfert galaxies which dominate the nearby agn population . one way around this problem is the use of high - spatial resolution observations , as possible with the largest ground - based telescopes or interferometers , which are able to resolve out the host and isolate the agn emission ( for details see * ? ? ? * ) . however , it is difficult to set up representative samples owing to the observational limitations . another possibility is the use of high luminosity objects typically at higher redshift where the agn outshines the host galaxy by a large factor in the optical and near - ir . if pah features are absent , the agn most likely dominates the mid - ir wavelength region as well ( in our sample : host @xmath4 for wavelengths @xmath5 17@xmath2 ) . in this paper we aim at quantifying the wavelength dependence of the anisotropy of the agn emission in the infrared from @xmath1 . for that we use a ( nearly ) isotropically selected and complete sample of quasars and radio galaxies with hidden quasars at @xmath0 as recently presented in @xcite and described in sect . [ sec : sample ] . here we improve the analysis by using host - galaxy subtraction for the radio galaxies and use clumpy torus models for interpretation . in sect . [ sec : results ] we show the average seds of each of the subsamples which are representing obscured ( type 2 ) and unobscured ( type 1 ) agn . we further analyze the origin of the anisotropy by fitting extinction models and clumpy torus models to the observations in sect . [ sec : analysis ] . in sect . [ sec : obsaniso ] we discuss our results by comparing them to previous ir anisotropy estimates in literature . the results are summarized in sect . [ sec : summary ] . the object sample for this paper comprises all 3crr @xcite radio galaxies and quasars with @xmath6 this lobe dominated sample presents a well matched set of radio galaxies and quasars in terms of their intrinsic luminosity ( @xmath7erg / s for the quasars and @xmath8erg / s for the radio galaxies ; errors indicate standard deviation of the sample ) . the data used here have been presented previously in @xcite and @xcite and we refer the reader to these papers for further details on the source selection , data reduction , and building of the average seds . to summarize briefly , we obtained mid - ir photometry in all six filters from 3.6@xmath2 to 24@xmath2 and spectroscopy from 19@xmath2 to 38@xmath2 utilizing all three instruments onboard the _ spitzer _ space telescope @xcite . after the data reduction in a standard manner , the individual source seds were interpolated onto a common rest frame wavelength grid . the quasars and radio galaxies were then averaged into a mean sed for each class of objects . the individual seds ( including observed and interpolated photometry ) as well as the the average seds are presented in ( * ? ? ? * their figs.1 & 2 ) . for this paper , additional corrections have been applied to the radio galaxy data before the averaging process outlined above : at the shortest wavelengths considered here ( @xmath9@xmath10 m , rest frame ) the radio galaxy seds show contributions from the host galaxy . since we want to isolate the emission coming from the active nucleus , we have to correct for the stellar emission in these cases . this correction was performed by fitting the observed irac photometry with a combination of a moderately old ( 5 - 10gyr ) elliptical galaxy sed to represent the stellar emission ( taken from the grasil webpage ; @xcite ) and a hot black body for the agn dust whose temperature we allowed to vary ( see e.g. * ? ? ? * ; * ? ? ? the resulting black body temperatures in the radio galaxies range from 600 to 970k , with a median value of 860k . the fraction of host galaxy light contributing to the flux measured at the observed frame wavelengths 3.6 , 4.5 , 5.8 , and 8.0@xmath10 m was found to be 0.9 , 0.6 , 0.3 , and 0.1 , respectively . despite a negligible contribution from the host galaxy , we performed similar fits to the quasar sample as well to obtain characteristics of the hot dust emission and compare it to the radio galaxies . for the quasars we find notably hotter temperatures in the range from 8801250k ( median 1020k ) . this is consistent with the idea that the hottest dust is obscured in radio galaxies , while directly seen in quasars . in the radio galaxies we then subtracted the estimated host galaxy contribution from the observed irac photometry using the scaled template sed . for the irs and mips measurements the host galaxy contributions were considered negligible at restframe wavelength @xmath11 and no corrections have been applied . we refrain from further corrections related to possible starformation . as pointed out in @xcite neither the individual seds nor the averages showed any pah features ( see also fig . [ fig : aver_sed ] ) , which indicates that any contribution from starformation to the mid - ir is probably negligible . the radio galaxy 3c368 has a galactic m - star superimposed close to the position of the radio galaxy nucleus ( e.g. * ? ? ? * ; * ? ? ? both sources are partly blended even at the shortest irac wavelengths which makes the correction for the host galaxy emission in this source quite uncertain . consequently , we removed this source from the sample considered here . this leaves us with 11 quasars and 8 radio galaxies from which the average seds have been calculated . in fig . [ fig : aver_sed ] we show the average sed of @xmath13 quasars ( red ) and radio galaxies ( blue ) respectively . the error bars reflect the mean absolute deviation while the shaded areas show the range of the respective subsample at each wavelength point of the interpolated data ( see sect . [ sec : sample ] ) . for each of the object types we calculated a spline fit through the mean data points in order to guide the eye . it is obvious that the radio galaxies are systematically lower in infrared emission than the quasars . the discrepancy is largest in the near - ir and flattens out towards longer wavelengths . @xcite showed similar average seds . here we used additional host galaxy subtraction , which isolates the agn light much better ( see sect . [ sec : sample ] ) . this is most obvious in the near - ir part of the radio galaxies shortward of 5@xmath2 . the sed keeps on falling toward shorter wavelengths consistent with the wien tail from hot dust emission , instead of making an upward turn ( see * ? ? ? since both types have the same radio luminosities due to our selection , this difference between quasars (= type 1 agn ) and radio galaxies (= type 2 agn ) is a generic property of the sample . it either reflects a difference in line - of - sight extinction ( e.g. by cold dust in the host galaxy ) , or traces the anisotropy in re - emission of the agn - heated dust . @xcite tested the former possibility and found a surprisingly good match of the difference between radio galaxies and quasars in the mid - ir by a single extinction law . this , however , breaks down in the near - ir . in this paper we will test if , instead , a single absorber and emitter (= the dust torus ) may be responsible for the radio galaxy / quasar anisotropy ( see sect . [ sec : modcomp ] ) . to quantify the wavelength dependence of the anisotropy we plot the quasar / radio galaxy ratio in fig . [ fig : ratio ] . also shown is the mean absolute deviation of the ratio calculated by propagating the standard deviations of each subsample . from 2 to 8@xmath2 the emission ratio gradually decreases from 20 to 2 . in the silicate feature this ratio increases again up to about 3 and flattens out towards 15@xmath2 at a value of @xmath14 . our sample comprises a range of sed shapes , meaning that a range of ratios is observed . most of this sample range comes from the radio galaxies which show much less uniformity than the quasars ( see fig . [ sec : sample ] ) . we illustrated the range of ratios covered by the radio galaxies in fig . [ fig : range ] where we plot the wavelength dependence of the ratio of each radio galaxy using the average quasar sed and normalize it to 15@xmath2 . at around 10@xmath2 the radio galaxies show the silicate feature in absorption while quasars display a weak silicate emission feature . we used the spline fits described above ( see fig . [ fig : aver_sed ] ) to locate the centers of the silicate features following the method outlined by @xcite . the silicate absorption feature center is found to be at @xmath15 while the peak of the silicate emission feature was measured at @xmath16 . this difference in central wavelength is similar to the ones observed in local galaxies . as recent high spatial resolution studies of seyferts suggest , the `` shift '' in wavelength is not a pure radiative transfer effect due to the location or distribution of the dust , but implies a change of dust chemistry within the torus @xcite . as demonstrated by @xcite the silicate emission feature may be located at around 10.5@xmath2 if a fraction of the hot silicate dust consists of porous silicate grains . we note that the transition from quasars to radio galaxies is not as smooth as one may expect . in spite of some overlap in the range of seds of both types in the near - ir part in fig . [ fig : aver_sed ] , quasars generally show infrared emission characteristics expected for a type 1 agn ( blue ir color ; silicate emission feature ) and radio galaxies have ir seds with type 2 emission properties ( redder sed ; silicate absorption feature ) . in this section we analyze the scenarios that can lead to the observed anisotropy in radio galaxies and quasars . there are two possibilities : ( 1 ) extinction by a cold dust screen and ( 2 ) absorption and emission in a warm dusty medium . the former possibility may be associated with cold dust in the host galaxy while the latter one is equivalent to the dust torus in the agn unification scheme ( which we may call `` intrinsic anisotropy '' ) . we will first discuss the plausibility and consequences of a cold dust screen on the seds . if the ir anisotropy is dominated by cold host galaxy dust , then we have a situation where the ir emission originates from the torus while the absorption is coming from a different component ( i.e. dust in the host ) . this means that we require an additional component outside the agn to model the data ( e.g. as used for a significant minority of high-@xmath17 type 2 qsos in @xcite ) . however , the objects suffering extinction ( here : radio galaxies ) would be offset from quasars by only a single extinction law . this has been tested and ruled out by @xcite for the same set of objects as presented here . in fig . [ fig : ratio ] we show the anisotropy ratio between quasars and radio galaxies ( black circles with error bars ) . we overplot a standard ism extinction curve , resembling cold screen extinction , scaled to the observed 15@xmath2 anisotropy ( dark - blue dotted line ; using a mixture of 53% silicates and 47% graphite , based on updated dust opacity cuves by @xcite , and a grain size distribution according to @xcite ) . the extinction curve significantly overpredicts the anistropy in the silicate feature with better agreement in the near - ir . for reference we also plot the extinction curve based on ( * light - blue dotted line ; pixie dust ) , as used in @xcite , which results in better agreement within the silicate feature but significant offsets in the near - ir . in fact , @xcite pointed out that a good correspondence of quasars and radio galaxies can be achieved only if the quasar sed is attenuated by at least _ two _ instead of one extinction components which is reminiscent of radiative transfer (= absorption _ and _ emission ) within the torus rather than a cold screen . moreover , we found a mean anisotropy of about 1.4 at 15@xmath2 ( see sect . [ sec : results ] ) . dust opacity curves typically have opacity ratios @xmath18 . in order to obtain the observed anisotropy , the dust screen would have to have @xmath19 . while such optical depth values are in reach for galactic dust lanes ( e.g. extinction towards our own galactic center ) , it requires very edge - on views onto disk galaxies since the scale height of galactic disks is small . on statistical grounds this possibility may be viable for a minority of all radio galaxies , but it is unlikely that the whole population is dominated by host extinction . we note that the same line of argument can be made using the near - ir anisotropy leading to even higher @xmath20 and illustrating the need for more then just one cold extinction screen in this scenario . we use our clumpy torus models _ cat3d _ @xcite to test if the observed anisotropy ratio can be explained by the intrinsic anisotropy as predicted in the unification scheme . generally smooth dust torus models predict stronger anisotropy than clumpy models @xcite . @xcite argue that the observed small anisotropy in the mid - ir / x - ray correlation ( see sect . [ sec : obsaniso ] ) is qualitatively in agreement with a clumpy torus . here we aim at being more quantitative and show consistency of the observed anisotropy with torus orientation . in fig . [ fig : ratio ] we overplot the observed ratio with predicted ratios of a clumpy torus model ( red dashed line ) . as mean torus inclination in quasars we assumed 39@xmath21 while the mean radio galaxy inclination is set to 75@xmath21 . this corresponds to a mean opening angle of the torus of 60@xmath21 or a type 1/type 2 ratio of 1:1 , which is consistent with the number statistics of our complete and isotropic sample @xcite . [ fig : ratio ] shows that the model is following the continuum anisotropy curve quite well . there is , however , a slight deviation within the silicate feature where in the center of the feature the model curve predicts slightly lower anisotropy than shown by the observed curve . this may be indicative of additional , off - torus obscuration in some objects ( e.g. from the host galaxy ) , which effectively deepens silicate absorption features in radio galaxies ( the denominator of the plot in fig . [ fig : ratio ] ) and , to a lesser degree , changes the spectral slope . if this happens in individual objects , the average curve and scatter will tend to be slightly more anisotropic . some local examples of host obscuration are centaurus a , ngc 5506 , or the nucleus of the circinus galaxy where host - galactic dust lanes are projected onto the nucleus producing deep silicate features . in fact the silicate absorption feature in the average radio galaxy sed seems to be deeper than in typical local seyfert 2 galaxies without host obscuration @xcite . note that this requires a statistical alignment of cold host dust with that of the torus ( see sect . [ sec : extinct ] ) . the model used for reproducing the ratio uses a dust cloud distribution with radial power law @xmath22 and 5 clouds along the line - of - sight in equatorial direction ( for details see * ? ? ? * ) . in comparison to models for seds and mid - ir interferometry of local seyfert galaxies , these parameters suggest only a slightly more centrally condensed and transparent torus @xcite , while the half - opening angle may be wider ( 60@xmath21 instead of 45@xmath21 ) . in fact a range of torus model parameters satisfies the observed type 1/type 2 anisotropy spectrum within the error bars of the sample ( e.g. various steeper and shallower dust distributions ) . from bayesian inference analysis we found that the torus model parameters are generally poorly constrained ( broad posterior distributions for individual parameters ) . a weakness is certainly that modeling the flux ratio is not very constraining for model parameters since it does only take relative fluxes into account , while absolute fluxes ( e.g. actual silicate strength of type 1s or type 2s ) are not included . on the other hand , what the modeling shows is that the observed small ratios at long wavelengths and the change of anisotropy from the near- to the mid - ir are in agreement with expectations from clumpy torus models without fine - tuning parameters . in summary , the torus model seems to reproduce the observed anisotropy reasonably well over most of the wavelength range , while single extinction laws result in much worse fits . the model parameters used in the clumpy torus model fit are reasonable in comparison to fits to local agn . it demonstrates plausibility of the scenario that the anisotropy is related to torus orientation . this strongly suggests that the observed anisotropy is a measure for the intrinsic anisotropy of luminous type 1 and type 2 agn at @xmath12 . in sect . [ sec : analysis ] we argued that the observed ir anisotropy is probably reflecting the `` intrinsic anisotropy '' as caused by the dust torus . the isotropic and complete selection of the sample helps to minimize any biasing effects on the anisotropy . on the other hand , optically- and x - ray - selected samples often suffer from missing some of the most obscured objects . moreover , since the ratio is @xmath23 at 15@xmath2 , flux - limited mid - ir selected samples are potentially biased towards type 1 agn as well . thus , even using 12@xmath2-selected agn samples will slightly suffer from the assumption of isotropy . one popular way of comparing type 1s and type 2s is the correlation between x - ray and mid - ir luminosity . in case the x - luminosity is emitted isotropically ( and traces the dust - heating emission ) , and the mid - ir emission is radiated very anisotropically , this correlation is expected to be different for type 1 and type 2 agn . however , using high spatial resolution observations @xcite showed that local seyfert 1 and seyfert 2 galaxies , up to column densities of few @xmath24 @xmath25 , essentially follow the same correlation . a conservative estimate suggests that at 12@xmath2 the difference between both types is smaller than a factor of 3 . despite including some `` mildly '' compton - thick objects in this study , the most obscured objects are still missing due to the lack of intrinsic x - ray data . this essentially makes anisotropy estimates from the mid - ir / x - ray - correlation a lower limit on the `` true '' anisotropy . nevertheless this result is fully consistent with our finding for powerful @xmath12 agn . at 12@xmath2 we find a type 1/type 2 ratio of @xmath26 . assuming that x - ray selection misses the highest - inclination objects our result would predict that the anisotropy in the mid - ir / x - ray correlation is @xmath27 , at least for powerful agn as presented here . @xcite used a sample of local seyfert galaxies and compared the average 535@xmath2 sed of type 1s and type 2s , scaled to their respective 8.4ghz emission . however since the sources have been selected according to a flux limit at 12@xmath2 , the sample can not be considered isotropic . @xcite report generally higher fluxes for type seyfert 1 agn as compared to seyfert 2s . the anisotropy decreases from a factor of about 8 to @xmath32.5 from 5 to 8@xmath2 . this is significantly larger than what we find for our isotropically selected radio - loud sample . at longer wavelengths the seyfert 1/seyfert 2 ratio of @xcite flattens to about a factor of 23 ( with no convergence to unity as expected at long wavelengths ) which is , again , larger than what we found . the discrepancy between our results and @xcite may be either due to ( 1 ) the different selection criteria chosen , ( 2 ) a difference between low and high luminosity agn , or ( 3 ) a difference between radio - quiet and radio - loud agn . while the data in @xcite has not been corrected for host galaxy , the contribution by starformation should not be significant given the lack of pah features in the average spectrum . if a host - correction were applied , it would predominantly affect type 2 agn , thus making the anisotropy even larger . furthermore , if the mid - ir selection had any effect , then the sample would miss out agn at highest obscuration , so that the real type 1/type 2 anisotropy would again be larger . in conclusion , possible selection effects and host contamination in the @xcite study would tend to result in an underestimated anisotropy , making the difference to our findings even stronger . it is well possible that the difference in anisotropy between our high - luminosity radio - loud sample and the low - luminosity radio - quiet sample is real . this would imply that either luminosity or radio power are the drivers for the observed characteristics . radio jets are highly collimated so that any influence of the jet can be expected perpendicular to the torus plane and just in a very small solid angle . it is , therefore , more reasonable to assume that the higher agn luminosity would cause the lower anisotropy than the jet . the classical receding torus picture changes the opening angle of the torus for higher luminosity ( e.g. * ? ? ? * ; * ? ? ? this mainly affects the relative number of type 1s and type 2s in a sample . the relatively high fraction of about 50% unobscured agn in our sample would support this scenario . however , to change the anisotropy between both types , it would be necessary to also change the obscuration properties ( i.e. type 2s must on average look more like type 1s ) . such a scenario of `` radiation - limited obscuration '' has been proposed by @xcite and supported by observations of @xcite . in this case about the same effect is expected for radio - quiet and radio - loud agn . we use the sample of quasars and radio galaxies at @xmath12 recently presented in @xcite . since the sample was selected isotropically , it should cover all torus inclination angles ( weighted by solid angle ) . for these objects an infrared 117@xmath2 restframe sed has been constructed . average seds were calculated for the quasar (= type 1 agn ) and radio galaxy (= type 2 agn ) samples , respectively , in order to study the intrinsic anisotropy of the ir emission of the dust torus . it is shown that the ratio between type 1 and type 2 agn in our parameter space of very luminous radio galaxies , the value gradually decreases from 20 to 2 at wavelengths 2 to 8@xmath2 . within the 10@xmath2 silicate feature the ratio raises slightly . at longer wavelength the mid - ir emission becomes more isotropic . the intrinsic ratio between our type 1 and type 2 agn is @xmath14 at 15@xmath2 . when using ir - selected flux - limited samples this anisotropy has to be taken into account . by analyzing the silicate feature in the sample averages we find the well - established `` shift '' of the central peak of the silicate emission feature with respect to the center of the absorption feature . the resulting central wavelengths at @xmath15 for the absorption and @xmath16 for the emission feature are in agreement with previous reports ( e.g. * ? ? ? * ; * ? ? ? we discussed our results in the frame of other anisotropy estimators . our findings are consistent with upper limits derived from the x - ray / mid - ir correlation of local seyfert galaxies @xcite . some discrepancy exists with respect to a similar study of @xcite for nearby seyferts . if real it would imply that nearby , radio - quiet lower - luminosity agn show a higher degree of anisotropy in the ir than higher luminosity , radio - loud sources . this may be explained by a receding torus model with luminosity - dependent obscuration . we also show that the overall relatively small degree of anisotropy is consistent with the torus being clumpy rather than smooth . our clumpy torus model reproduces the observed type 1/type 2 ratio reasonably well . we would like to thank our referee prof . andy lawrence for helpful and constructive comments which significantly improved the paper , as well as poshak gandhi who also commented on this manuscript . the paper was made possible by deutsche forschungsgemeinschaft ( dfg ) in the framework of a research fellowship ( `` auslandsstipendium '' ) for sh . mh is supported by the nordrhein - westflische akademie der wissenschaften und der knste . this work is based on observations made with the spitzer space telescope , which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . antonucci , r. 1993 , ara&a , 31 , 473 best , p. n. , longair , m. s. , & roettgering , j. h. a. 1997 , mnras , 292 , 758 buchanan , c. l. , gallimore , j. f. , odea , c. p. , baum , s. a. , axon , d. j. , et al . 2006 , aj , 132 , 401 chiar , j. e. , & tielens , a. g. g. m. 2006 , apj , 637 , 774 de breuck , c. , seymour , n. , stern , d. , willner , s. p. , eisenhardt , p. r. m. , et al . 2010 , apj , 725 , 36 draine , b. t. 2003 , apj , 598 , 1026 gandhi , p. , horst , h. , smette , a. , hnig , s. , comastri , a. , et al . 2009 , a&a , 502 , 457 haas , m. , willner , s. p. , heymann , f. , ashby , m. l. n. , fazio , g. , et al . 2008 , apj , 688 , 122 hammer , f. , proust , d. , & le fevre , o. 1991 , apj , 374 , 91 hnig , s. f. , & beckert , t. 2007 , mnras , 380 , 1172 hnig , s. f. , kishimoto , m. , gandhi , p. , smette , a. , asmus , d. , duschl , w. , polletta , m. , weigelt , g. 2010 , a&a , 515 , 23 hnig , s. f. , & kishimoto , m. 2010 , a&a , 523 , 27 laing , r. a. , riley , j. m. , & longair , m. s. 1983 , mnras , 204 , 151 lawrence , a. 1991 , mnras , 252 , 586 leipski , c. , haas , m. , meusinger , h. , siebenmorgen , r. , chini , r. , et al . 2005 , a&a , 440 , l5 leipski , c. , haas , m. , siebenmorgen , r. , meusinger , h. , albrecht , m. , et al . 2007 , 473 , 121 leipski , c. , haas , m. , willner , s. p. , ashby , m. l. n. , wilkes , b. j. , et al . 2010 , apj , 717 , 766 levenson , n. a. , radomski , j. t. , packham , c. , mason , r. e. , schaefer , j. j. , telesco , c. m. 2009 , apjl , 703 , 390 mathis , j. s. , rumpl , w. , & nordsieck , k. h. 1977 , apj , 217 , 425 polletta , m. , weedman , d. w. , hnig s. f. , lonsdale , c. j. , smith , h. e. , houck , j. r. 2008 , apj , 675 , 960 schartmann , m. , meisenheimer , k. , camenzind , m. , wolf , s. , tristram , k. r. w. , & henning , t. 2008 , a&a , 482 , 67 seymour , n. , stern , d. , de breuck , c. , vernet , j. , rettura , a. , et al . 2007 , apjs , 171 , 363 shi , y. , rieke , g. h. , hines , d. c. , gorjian , v. , werner , m. w. , et al . 2006 , apj , 653 , 127 silva , l. , granato , g.l . , bressan , a. , danese , l. , 1998 , apj , 509 , 103 simpson , c. 2005 , mnras , 360 , 565 sirocky , m. m. , levenson , n. a. , elitzur , m. , spoon , h. w. w. , armus , l. 2008 , apj , 678 , 729 smith , h. a. , li , a. , li , m. p. , khler , m. , ashby , m. l. n. , et al . 2010 , apj , 716 , 490 sturm , e. , schweitzer , m. , lutz , d. , contursi , a. , genzel , r. , et al . 2005 , apj , 629 , l21 treister , e. , virani , s. , gawiser , e. , urry , c. m. , lira , p. , et al . 2009 , apj , 693 , 1713 urry , c. m. , & padovani , p. 1995 , pasp , 107 , 803 werner , m. w. , et al . 2004 , apjs , 154 , 1

we use restframe near- and mid - ir data of an isotropically selected sample of quasars and radio galaxies at @xmath0 , which have been published previously , to study the wavelength - dependent anisotropy of the ir emission . for that we build average seds of the quasar subsample (= type 1 agn ) and radio galaxies (= type 2 agn ) from @xmath1 and plot the ratio of both average samples . from 2 to 8@xmath2 restframe wavelength the ratio gradually decreases from 20 to 2 with values around 3 in the 10@xmath2 silicate feature . longward of 12@xmath2 the ratio decreases further and shows some high degree of isotropy at 15@xmath2 ( ratio @xmath31.4 ) . the results are consistent with upper limits derived from the x - ray / mid - ir correlation of local seyfert galaxies . we find that the anisotropy in our high - luminosity radio - loud sample is smaller than in radio - quiet lower - luminosity agn which may be interpreted in the framework of a receding torus model with luminosity - dependent obscuration properties . it is also shown that the relatively small degree of anisotropy is consistent with clumpy torus models .

arxiv

continuous - variable quantum - key distribution ( cvqkd ) , as an unconditionally secure communication scheme between two legitimate parties alice and bob , has achieved advanced improvements in theoretical analysis and experimental implementation in recent years @xcite . practical implementation systems , such as fiber - based gaussian - modulated @xcite and discrete - modulated @xcite coherent - state protocol qkd systems over tens of kilometers , have been demonstrated in a few groups . the unconditional security of such systems with prepare - and - measure ( pm ) implementation has been confirmed by the security analysis of the equivalent entanglement - based ( eb ) scheme @xcite . however , the traditional security analysis of the eb scheme of cvqkd just includes the signal beam and not the local oscillator ( lo ) , which is an auxiliary light beam used as a reference to define the phase of the signal state and is necessary for balanced homodyne detection . this will leave some security loopholes for eve because lo is also unfortunately within eve s manipulating domain . the necessity of monitoring lo intensity for the security proofs in discrete qkd protocols embedded in continuous variables has been discussed @xcite . moreover , in @xcite , the excess noise caused by imperfect subtraction of balanced homodyne detector ( bhd ) in the presence of lo intensity fluctuations has been noted and quantified with a formulation . however , in the practical implementation of cvqkd , shot noise scaling with lo power measured before keys distribution is still assumed to keep constant if the fluctuations of lo intensity are small . and in this circumstance , pulses with large fluctuation are just discarded as shown in @xcite . unfortunately , this will give eve some advantages in exploiting the fluctuation of lo intensity . in this paper , we first describe bob s measurements under this fluctuation of lo intensity , and propose an attacking scheme exploiting this fluctuation . we consider the security of practical cvqkd implementation under this attack and calculate the secret key rate with and without bob monitoring the lo for reverse and direct reconciliation protocol . and then , we give a qualitative analysis about the effect of this lo intensity fluctuation on the secret key rate alice and bob hold . we find that the fluctuation of lo could compromise the secret keys severely if bob does not scale his measurements with the instantaneous lo intensity values . finally , we briefly discuss the accurate monitoring of lo intensity to confirm the security of the practical implementation of cvqkd . generally , in practical systems of cvqkd , the local oscillator intensity is always monitored by splitting a small part with a beam splitter , and pulses with large lo intensity fluctuation are discarded too . however , even with such monitoring , we do not yet clearly understand how fluctuation , in particular small fluctuation , affects the secret key rate . to confirm that the secret key rate obtained by alice and bob is unconditionally secure , in what follows , we will analyze the effects of this fluctuation on the secret key rate only , and do not consider the imperfect measurement of bhd due to incomplete subtraction of it in the presence of lo intensity fluctuations , which has been discussed in @xcite . ideally , with a strong lo , a perfect pulsed bhd measuring a weak signal whose encodings are @xmath0 will output the results@xcite , @xmath1 where _ k _ is a proportional constant of bhd , @xmath2 is the amplitude of lo , @xmath3 is the relative phase between the signal and lo except for the signal s initial modulation phase . so scaling with lo power or shot noise , the results can be recast as @xmath4 with @xmath3 in eq . ( [ eq : x0 ] ) is 0 or @xmath5 . here the quadratures @xmath6 and @xmath7 are defined as @xmath8 and @xmath9 , where @xmath10 is the quadrature of the vacuum state . however , in a practical system , the lo intensity fluctuates in time during key distribution . with a proportional coefficient @xmath11 , practical lo intensity can be described as @xmath12 , where @xmath2 is the initial amplitude of lo used by normalization and its value is calibrated before key distribution by alice and bob . if we do not monitor lo or quantify its fluctuation @xcite , especially just let the outputs of bhd scale with the initial intensity or power of lo , the outputs then read @xmath13 unfortunately , this fluctuation will open a loophole for eve , as we will see in the following sections . in conventional security analysis , like the eb scheme equivalent to the usual pm implementation depicted in fig . [ fig:1](a ) , lo is not taken into consideration and its intensity is assumed to keep unchanged . however , in practical implementation , eve could intercept not only the signal beam but also the lo , and she can replace the quantum channel between alice and bob with her own perfect quantum channel as shown in figs . [ fig:1](b ) and [ fig:1](c ) . in so doing , eve s attack can be partially hidden by reducing the intensity of lo with a variable attenuator simulating the fluctuation without changing lo s phase , and such an attack can be called a lo intensity attack ( loia ) . in the following analysis , we will see that , in the parameter - estimation procedure between alice and bob , channel excess noise introduced by eve can be reduced arbitrarily , even to its being null , just by tuning the lo transmission . consequently , alice and bob would underestimate eve s intercepted information and eve could get partial secret keys that alice and bob hold without being found under this attack . figure [ fig:1](b ) describes the loia , which consists of attacking the signal beam with a general gaussian collective attack @xcite and attacking the lo beam with an intensity attenuation by a non - changing phase attenuator * a * , such as a beam splitter whose transmission is variable . this signal - beam gaussian collective attack consists of three steps : eve interacts her ancilla modes with the signal mode by a unitary operation * u * for each pulse and stores them in her quantum memory , then she makes an optimal collective measurement after alice and bob s classical communication . figure [ fig:1](c ) is one practical loia with * u * being a beam - splitter transformation . its signal attack is also called an entangling cloner attack , which was presented first by grosshan @xcite and improved by weedbrook @xcite . in appendix [ sec : security ] , we will demonstrate that with this entangling cloner , eve can get the same amount of information as that shown in fig . [ fig:1](b ) . we analyze a practical cvqkd system with homodyne protocol to demonstrate the effect of loia on the secret key rate , and for simplicity we do not give the results of the heterodyne protocol , which is analogous to the homodyne protocol . in the usual pm implementation , alice prepares a series of coherent states centered on @xmath0 with each pulse , and then she sends them to bob through a quantum channel which might be intercepted by eve . here , @xmath14 and @xmath15 , respectively , satisfy a gaussian distribution independently with the same variance @xmath16 and zero mean . this initial mode prepared by alice can be described as @xmath17 , and @xmath18 is the quadrature variable . here @xmath9 describes the quadrature of the vacuum mode . note that we denote an operator with a hat , while without a hat the same variable corresponds to the classical variable after measurement . so the overall variance of the initial mode prepared by alice is @xmath19 . when this mode comes to bob , bob will get a mode @xmath20 , @xmath21 where @xmath22 describes eve s mode introduced through the quantum channel whose quadrature variance is @xmath23 . bob randomly selects a quadrature to measure , and if eve attenuates the lo intensity during the key distribution , but bob s outputs still scale with the initial lo intensity , just as in eq . ( [ eq : x0 ] ) , he will get the measurement @xmath24 where @xmath25 is @xmath26 ( or equivalently @xmath27 ) . however , if bob monitors lo and also scales with the instantaneous intensity value of lo with each pulse , he will get @xmath25 without any loss of course . note that , for computation simplicity , hereafter we assume the variable transmission rate @xmath28 ( or attenuation rate @xmath29 ) of each pulse of lo is the same without loss of generality . thus the variance of bob s measurements and conditional variance on alice s encodings with and without monitoring ( in what follows , without monitoring specially indicates that bob s measurement is obtained just by scaling with the initial lo intensity instead of monitoring instantaneous values , and vice versa ) can be given by @xmath30\;,\label{eq : vbm}\\ v_{b|a}&=t+(1-t)n\ ; , \\ v_{b|a}^w&=\eta\left[t+(1-t)n\right]\;,\end{aligned}\ ] ] where the superscript @xmath31 indicates without monitoring " and all variances are in shot - noise units ; the conditional variance is defined as @xcite @xmath32 hence , the covariance matrix of alice s and bob s modes can be obtained as @xmath33\mathbb{i } \end{pmatrix}\label{eq : rab},\end{split } \\ & \gamma_{ab}^w=\begin{pmatrix } v\mathbb{i } & \sqrt{\eta t(v^2 - 1)}\sigma_z\\ \sqrt{\eta t(v^2 - 1)}\sigma_z & \eta[tv+(1-t)n]\mathbb{i } \end{pmatrix},\end{aligned}\ ] ] where @xmath34 is the pauli matrix and @xmath35 is a unit matrix . from eqs . ( [ eq : vb ] ) and ( [ eq : vbm ] ) we can derive that the channel transmission and excess noise are @xmath36 , @xmath37 with monitoring , and @xmath38 , @xmath39 without monitoring . hence , by attenuating the lo intensity as fig . [ fig:1 ] shows , to make @xmath40 , eve could arbitrarily reduce @xmath41 to zero , thus she will get the largest amount of information permitted by physics . in the following numerical simulation we always make @xmath42 , namely , @xmath43 . thus , the covariance matrix @xmath44 and eve s introducing noise @xmath23 [ in an entangling cloner it is eve s epr state s variance as fig . [ fig:1](c ) shows ] should be selected to be @xmath45 to estimate the secret key rate , without loss of generality , we first analyze the reverse reconciliation then consider the direct reconciliation . from alice and bob s points of view , the secret key rate for reverse reconciliation with monitoring or not are given , respectively , by @xmath46 where the mutual information between alice and bob with and without monitoring are the same , and that is @xmath47 this is because bob s measurements in these two cases are just different with a coefficient @xmath28 , and they correspond with each other one by one , so they are equivalent according to the data - processing theorem @xcite . however , the mutual information between eve and bob given by the holevo bound @xcite in these two cases is not identical . as the previous analysis showed , in bob s point of view , channel transmission and the excess noise estimation are different . but from eve s point of view , they are identical according to the data - processing theorem because she estimates bob s measurements in these two cases just by multiplying a coefficient @xmath28 . we ll calculate the real information intercepted by eve first . it can be given by @xmath48 where @xmath49 is the von neumann entropy @xcite . for a gaussian state @xmath50 , this entropy can be calculated by the symplectic eigenvalues of the covariance matrix @xmath51 characterizing @xmath50 @xcite . to calculate eve s information , first eve s system @xmath52 can purify @xmath53 permitted by quantum physics , so that @xmath54 . second , after bob s projective measurement , the system @xmath55 is pure , so that @xmath56 . designating @xmath57 , @xmath58 , and @xmath59 , the symplectic eigenvalues of @xmath60 are given by @xmath61 where @xmath62 and @xmath63 . similarly , the entropy @xmath64 is determined by the symplectic eigenvalue @xmath65 of the covariance matrix @xmath66 @xcite , namely , @xmath67 where @xmath68 and @xmath69 stands for the moore - penrose inverse of a matrix . then @xmath70 , and the holevo bound reads @xmath71 where @xmath72 . however , eve s information estimated by bob without monitoring is given by @xmath73 by substituting eqs . ( [ eq : xbe ] ) and ( [ eq : xbem ] ) into eqs . ( [ eq : krr ] ) and ( [ eq : krrm ] ) , respectively , the secret key rate with and without bob s monitoring can be obtained . however , the secret key rate in eq . ( [ eq : krrm ] ) without monitoring is unsecured in evidence . eve s interception of partial information from @xmath74 is not detected , in other words , alice and bob underestimate eve s information without realizing it . actually , the real or unconditionally secure secret key rate @xmath74 , which we called a truly secret key rate , should be available by replacing @xmath75 in eq . ( [ eq : krrm ] ) with eq . ( [ eq : xbe ] ) . note that it is identical with the monitoring secret key rate in eq . ( [ eq : krr ] ) due to eq . ( [ eq : iab ] ) . we investigate the secret key rate @xmath74 bob measured without monitoring and the true one or equivalently monitoring one @xmath76 for reverse reconciliation under eve attacking the intensity of lo during key distribution . as fig . [ fig:2 ] shows , with various values of transmission of lo that can be controlled by eve , the truly secret key rate alice and bob actually share decreases rapidly over long distances or small channel transmissions . ( color online ) reverse reconciliation pseudosecret key rate and the truly secret one vs channel transmission @xmath36 under loia . solid lines are secret key rate estimated by bob without monitoring lo intensity and dashed lines are the truly secret ones . colored lines correspond to the lo transmissions @xmath28 as labeled . here alice s modulation variance @xmath77 . ] additionally , because the mutual information between alice and bob with and without monitoring is identical as eq.([eq : iab ] ) shows , subtracting eq . ( [ eq : krr ] ) from eq . ( [ eq : krrm ] ) we can estimate eve s intercepted information @xmath78 which is plotted in fig . [ fig:3 ] . we find that eve could get partial or full secret keys which alice and bob hold by controlling the different transmissions of lo . taking a 20-km transmission distance as an example , surprisingly , just with lo intensity fluctuation or attenuating rate 0.08 , eve is able to obtain the full secret keys for reverse reconciliation without bob s monitoring the lo . we now calculate the secret key rate for direct reconciliation , which is a little more complicated , and we investigate the effect of lo intensity attack by eve on cvqkd . the secret key rate estimated by bob with and without lo monitoring is given , respectively , by @xmath79 note that we have already calculated @xmath80 and @xmath81 in eq.([eq : iab ] ) and they are identical for direct and reverse reconciliation . for eve we have @xmath82 where @xmath54 has been already computed in the previous section , and @xmath83 using the fact that after alice s projective measurement on modes @xmath84 and @xmath85 obtaining @xmath86 in the eb scheme shown in fig . [ fig:1](a ) , the system @xmath87 is pure . to calculate @xmath88 , we have to compute the symplectic eigenvalues of covariance matrix @xmath89 , which is obtained by @xmath90 where @xmath91 and @xmath92 can be read in the decomposition of the matrix @xmath93 which is available by elementary transformation of the matrix [ see fig . [ fig:1](a ) ] @xcite @xmath94 where @xmath95 is a unit matrix . it is obtained by applying a homodyne detection on mode a after mixing @xmath84 and @xmath85 with a balanced beam - splitter transformation @xmath96 . the matrix @xmath97 and @xmath98 actually is @xmath60 in eq . ( [ eq : rab ] ) , @xmath99 is a unit matrix . so we can get @xmath100 and the symplectic eigenvalues of it @xmath101 where @xmath102 and @xmath103 . the holevo bound then reads @xmath104 substituting eqs . ( [ eq : xae ] ) and ( [ eq : xaem ] ) into eqs . ( [ eq : kdr ] ) and ( [ eq : kdrm ] ) , respectively , the secret key rates in these two cases are obtained . in fig . [ fig:4 ] , we plotted them for channel transmission @xmath36 with various values of @xmath28 and find that the difference between the pseudosecret key rate with bob not monitoring lo and the truly secret one is still increasing with the channel transmission @xmath36 becoming smaller . for eve , when bob does not monitor lo , she will get the partial or total secret key rate @xmath105 without being found by subtracting eq . ( [ eq : kdr ] ) from eq . ( [ eq : kdrm ] ) when she reduces the intensity of lo . in fig . [ fig:5 ] , we plotted the pseudosecret key rate for direct reconciliation and the mutual information overestimated by alice and bob . we find that for short distance communication ( less 15 km or 3 db limit ) , a small fluctuation of lo intensity could still hide eve s attack partially or totally . note that in the above estimation we assume each pulse s transmission rate @xmath28 ( or attenuation rate @xmath29 ) is identical . however , when @xmath28 is different for each pulse ( eve simulates the fluctuation of lo to hide her dramatic attack on lo ) , eve still could intercept as much as or even more secret key rates than above for reverse and direct reconciliation , as long as the largest value of @xmath28 among all pulses ( or approximately most pulse transmission rates ) is smaller than the above constant value . our analysis shows that reverse reconciliation is more sensitive than direct reconciliation about the fluctuation of lo intensity , and , even with a small attenuation of lo intensity , eve can get full secret keys but not be found . this is consistent with the fact that channel excess noise has a more severe impact on reverse reconciliation than on direct reconciliation . of course , when the intensity of lo fluctuates above the initial calibrated value ( i.e. , @xmath106 ) , eve could not get any secret keys , but alice and bob would overestimate eve s intercepted information due to the overestimation of channel excess noise . however , when lo fluctuates around the initial calibrated value , how to quantify eve s information is still an open question , because the distribution of the fluctuation of lo ( or @xmath28 ) is not a normal distribution and unclear for alice and bob due to eve s arbitrary manipulation . but in this circumstance , eve still could intercept partial secret keys if she increases the channel excess noise of one part of the signal pulses when she controls @xmath107 and decreases it for the other part when controlling @xmath106 , i.e. , making the overall estimated excess noise by alice and bob lower than the real one . remarkably , lo intensity fluctuation opens a loophole for eve to attack the practical system , especially in the case of communication with low channel transmission or over long distance . consequently , in the practical implementation of cvqkd , we must monitor the lo fluctuation carefully and in particular scale the measurements with instantaneous intensity values of lo . alternatively , we can also scale with the lowest intensity value of lo if the fluctuations are very small , but it will estimate the secret key rate pessimistically thus leading to the reduction of the efficiency of the key distribution . however , we can not use the average intensity value of lo to normalize the measurements as most current implementations do , because it still could overestimate the secret key rate for alice and bob . additionally , for reverse reconciliation communication over long distance , very small fluctuation of lo might compromise the secret key rate completely , which presents a big challenge for accurately monitoring lo intensity . finally , we point out that in this paper we do not consider the imperfections of bhd such as detection efficiency , electronic noise , and incomplete subtraction , which may make lo intensity fluctuation have a more severe impact on estimating the secret key rate for alice and bob . in conclusion , we have analyzed the effect of lo intensity fluctuation on the secret key rate estimation of alice and bob for reverse and direct reconciliation . incredibly , bob s estimation of the secret key rate will be compromised severely without monitoring lo or if his measurements do not scale with lo instantaneous intensity values even with monitoring but just discard large fluctuation pulses like in @xcite . furthermore , we have shown that eve could hide her attack partially by reducing the intensity of lo and even could steal the total secret keys alice and bob share without being found by a small attenuation of lo intensity , especially for reverse reconciliation . finally , we have also briefly discussed the monitoring of lo and pointed out that it would be a challenge for highly accurate monitoring . this work is supported by the national natural science foundation of china , grant no . is supported by the program for new century excellent talents . c.m . is supported by the hunan provincial innovation foundation for postgraduates . acknowledge support from nudt under grant no . kxk130201 . in this appendix , we calculate the holevo bound obtained by eve for direct and reverse reconciliation using weedbrook s entangling cloner model @xcite , and then give the secret key rate shared by alice and bob under loia . we begin the analysis by calculating the von neumann entropy of eve s intercepting state first . as fig . [ fig:1](c ) shows , the entangling cloner consists of eve replacing the gaussian quantum channel between alice and bob with a beam splitter of transmission @xmath36 and an epr pair of variance @xmath23 . half of the epr pair mode @xmath108 is mixed with alice s mode in the beam splitter and is sent to bob to match the noise of the real channel by tuning n. the other half mode @xmath109 is kept by eve to reduce the uncertainty on one output of the beam splitter , the mode @xmath110 , which can be read as @xmath111 where @xmath112 is the quadrature of mode @xmath108 . thus , the variance of mode @xmath110 is given by @xmath113 and the conditional variance @xmath114 can be calculated as , using eq . ( [ eq : cvar ] ) , @xmath115 hence , eve s covariance matrix can be obtained as @xmath116 where @xmath117 and the notation @xmath118 stands for a matrix with the arguments on the diagonal elements and zeros everywhere else . the symplectic eigenvalues of this covariance matrix are given by @xmath119 where @xmath120 , and @xmath121 . hence , the von neumann entropy of eve s state is given by @xmath122 for the direct reconciliation protocol of cvqkd , the holevo bound between eve and alice is given by eq . ( [ eq : xae0 ] ) , where @xmath123 has been calculated by eq . ( [ eq : se ] ) . @xmath124 can be obtained by the conditional covariance matrix @xmath125 and its symplectic eigenvalues are given by @xmath126 where @xmath127 , and @xmath128 . thus , the conditional entropy is @xmath129 substituting eqs . ( [ eq : se ] ) and ( [ eq : sea ] ) into eq . ( [ eq : xae0 ] ) , we can get the mutual information between alice and eve , @xmath130 under loia , bob s estimation of the holevo bound without monitoring lo intensity then reads , using eq . ( [ eq : xae1 ] ) , @xmath131 with eqs . ( [ eq : xae1 ] ) and ( [ eq : xaem1 ] ) , the secret key rates in eqs . ( [ eq : kdr ] ) and ( [ eq : kdrm ] ) then can be calculated respectively , and the calculation numerically demonstrates that they are perfectly consistent with the fig . [ fig:4 ] . the calculation of the holevo bound between eve and bob for reverse reconciliation is a bit more complicated . using eq . ( [ eq : xbe0 ] ) , we only need to calculate the conditional entropy @xmath132 , which is determined by the symplectic eigenvalues @xmath133 of the covariance matrix @xmath134 , @xmath135 where @xmath136 and @xmath137 , @xmath138 . then , @xmath134 can be recast as @xmath139 where + @xmath140 , @xmath141 , + and + @xmath142 . + hence , its symplectic eigenvalues are given by @xmath143 where @xmath144 , @xmath145 , and @xmath146 is the determinant of a matrix . so , we get the conditional entropy @xmath147 and then the holevo bound @xmath148 consequently , without monitoring lo intensity , alice and bob will give eve the holevo bound @xmath149 substituting eqs . ( [ eq : xbe1 ] ) and ( [ eq : xbem1 ] ) into eqs . ( [ eq : krr ] ) and ( [ eq : krrm ] ) , respectively , the secret key rates with and without bob s monitoring can be obtained , and for channel transmission @xmath36 with various values of @xmath28 , they are numerically demonstrated to be perfectly consistent with fig . [ fig:2 ] , too . hence , it also indirectly confirms that either for direct or reverse reconciliation , the entangling cloner could reach the holevo bound against the optimal gaussian collective attack . in this paper , lo intensity fluctuation indicates the deviation of each pulse s intensity from the initial calibrated value during the key distribution . it does not mean the quantum fluctuation of each pulse itself , because lo is a strong classical beam whose quantum fluctuation is very small relative to itself and can be neglected .

we consider the security of practical continuous - variable quantum key distribution implementation with the local oscillator ( lo ) fluctuating in time , which opens a loophole for eve to intercept the secret key . we show that eve can simulate this fluctuation to hide her gaussian collective attack by reducing the intensity of the lo . numerical simulations demonstrate that , if bob does not monitor the lo intensity and does not scale his measurements with the instantaneous intensity values of lo , the secret key rate will be compromised severely .

arxiv

the goal of the development of the code was to have a simple and efficient tool for the computation of adiabatic oscillation frequencies and eigenfunctions for general stellar models , emphasizing also the accuracy of the results . not surprisingly , given the long development period , the simplicity is now less evident . however , the code offers considerable flexibility in the choice of integration method as well as ability to determine all frequencies of a given model , in a given range of degree and frequency . the choice of variables describing the equilibrium model and oscillations was to a large extent inspired by @xcite . as discussed in section [ sec : eqmodel ] the equilibrium model is defined in terms of a minimal set of dimensionless variables , as well as by mass and radius of the model . fairly extensive documentation of the code , on which the present paper in part is based , is provided with the distribution packagejcd / adipack.n ] . @xcite provided an extensive review of adiabatic stellar oscillations , emphasizing applications to helioseismology , and discussed many aspects and tests of the aarhus package , whereas @xcite carried out careful tests and comparisons of results on polytropic models ; this includes extensive tables of frequencies which can be used for comparison with other codes . the equilibrium model is defined in terms of the following dimensionless variables : @xmath0 here @xmath1 is distance to the centre , @xmath2 is the mass interior to @xmath1 , @xmath3 is the photospheric radius of the model and @xmath4 is its mass ; also , @xmath5 is the gravitational constant , @xmath6 is pressure , @xmath7 is density , and @xmath8 , the derivative being at constant specific entropy . in addition , the model file defines @xmath4 and @xmath3 , as well as central pressure and density , in dimensional units , and scaled second derivatives of @xmath6 and @xmath7 at the centre ( required from the expansions in the central boundary condition ) ; finally , for models with vanishing surface pressure , assuming a polytropic relation between @xmath6 and @xmath7 in the near - surface region , the polytropic index is specified . the following relations between the variables defined here and more `` physical '' variables are often useful : @xmath9 we may also express the characteristic frequencies for adiabatic oscillations in terms of these variables . thus if @xmath10 is the buoyancy frequency , @xmath11 is the lamb frequency at degree @xmath12 and @xmath13 is the acoustical cut - off frequency for an isothermal atmosphere , we have @xmath14 where @xmath15 is the adiabatic sound speed , and @xmath16 is the pressure scale height , @xmath17 being the gravitational acceleration . finally it may be noted that the squared sound speed is given by @xmath18 these equations also define the dimensionless characteristic frequencies @xmath19 , @xmath20 and @xmath21 as well as the dimensionless sound speed @xmath22 , which are often useful . as is well known the displacement vector of nonradial ( spheroidal ) modes can be written in terms of polar coordinates @xmath23 as @xmath24 \exp ( - { { \rm i}}\omega t ) \right\ } \ ; . \nonumber\end{aligned}\ ] ] here @xmath25 is a spherical harmonic of degree @xmath12 and azimuthal order @xmath2 , @xmath26 being co - latitude and @xmath27 longitude ; @xmath28 is an associated legendre function , and @xmath29 is a suitable normalization constant . also , @xmath30 , @xmath31 , and @xmath32 are unit vectors in the @xmath1 , @xmath26 , and @xmath27 directions . finally , @xmath33 is time and @xmath34 is the angular frequency of the mode . similarly , e.g. , the eulerian perturbation to pressure may be written @xmath35 \ ; . \label{eq : e2.2}\ ] ] as the oscillations are adiabatic ( and only conservative boundary conditions are considered ) @xmath34 is real , and the amplitude functions @xmath36 , @xmath37 , @xmath38 , etc . can be chosen to be real . the equations of adiabatic stellar oscillations , in the nonradial case , are expressed in terms of the following variables : , @xmath39 results from the earlier use of an unconventional sign convention for @xmath40 ; now , as usual , @xmath40 is defined such that the perturbed poisson equation has the form @xmath41 , where @xmath42 is the eulerian density perturbation . ] @xmath43 here @xmath40 is the perturbation to the gravitational potential . also , we introduce the dimensionless frequency @xmath44 by @xmath45 corresponding to eqs [ eq : buoy ] [ eq : cutoff ] . these quantities satisfy the following equations : @xmath46 y_1 + ( a - 1 ) y_2 + \eta a y_3 \ ; , \\ \label{eq : ea.3 } x { { { \rm d}}y_3 \over { { \rm d}}x } & = & y_3 + y_4 \ ; , \\ \label{eq : ea.4 } x { { { \rm d}}y_4 \over { { \rm d}}x } & = & - a u y_1 - u { v_g \over \eta } y_2 \\ & & + [ l ( l + 1 ) + u(a - 2 ) + u v_g ] y_3 + 2(1 - u ) y_4 \ ; . \nonumber\end{aligned}\ ] ] here @xmath47 , and the notation is otherwise as defined in eq . [ eq : fivea ] . in the @xcite approximation , where the perturbation to the gravitational potential is neglected , the terms in @xmath48 are neglected in eqs [ eq : ea.1 ] and [ eq : ea.2 ] and eqs [ eq : ea.3 ] and [ eq : ea.4 ] are not used . the dependent variables @xmath49 in the nonradial case have been chosen in such a way that for @xmath50 they all vary as @xmath51 for @xmath52 . for large @xmath12 a considerable ( and fundamentally unnecessary ) computational effort would be needed to represent this variation sufficiently accurately with , e.g. , a finite difference technique , if these variables were to be used in the numerical integration . instead i introduce a new set of dependent variables by @xmath53 these variables are then @xmath54 in @xmath55 near the centre . they are used in the region where the variation in the @xmath56 is dominated by the @xmath51 behaviour , for @xmath57 , say , where @xmath58 is determined on the basis of the asymptotic properties of the solution . this transformation permits calculating modes of arbitrarily high degree in a complete model . for radial oscillations only @xmath59 and @xmath60 are used , where @xmath59 is defined as above , and @xmath61 here the equations become @xmath62 y_1 + a y_2 \ ; . \label{eq : ea.6}\end{aligned}\ ] ] the equations are solved on the interval @xmath63 $ ] in @xmath55 . here , in the most common case involving a complete stellar model @xmath64 , where @xmath65 is a suitably small number such that the series expansion around @xmath66 is sufficiently accurate ; however , the code can also deal with envelope models with arbitrary @xmath67 , typically imposing @xmath68 at the bottom of the envelope . the outermost point is defined by @xmath69 where @xmath70 is the surface radius , including the atmosphere ; thus , typically , @xmath71 . the centre of the star , @xmath72 , is obviously a singular point of the equations . as discussed , e.g. , by @xcite boundary conditions at this point are obtained from a series expansion , in the present code to second significant order . in the general case this defines two conditions at the innermost non - zero point in the model . for radial oscillations , or nonradial oscillations in the cowling approximation , one condition is obtained . the surface in a realistic model is typically defined at a suitable point in the stellar atmosphere , with non - zero pressure and density . here the simple condition of vanishing lagrangian pressure perturbation is implemented and sometimes used . however , more commonly a condition between pressure perturbation and displacement is established by matching continuously to the solution in an isothermal atmosphere extending continuously from the uppermost point in the model . a very similar condition was presented by @xcite . in addition , in the full nonradial case a condition is obtained from the continuous match of @xmath40 and its derivative to the vacuum solution outside the star . in full polytropic models , or other models with vanishing surface pressure , the surface is also a singular point . in this case a boundary condition at the outermost non - singular point is obtained from a series expansion , assuming a near - surface polytropic behaviour ( see * ? ? ? * for details ) . the code also has the option of considering truncated ( e.g. , envelope ) models although at the moment only in the cowling approximation or for radial oscillations . in this case the innermost boundary condition is typically the vanishing of the radial displacement @xmath73 although other options are available . the numerical problem can be formulated generally as that of solving @xmath74 with the boundary conditions @xmath75 @xmath76 here the order @xmath77 of the system is 4 for the full nonradial case , and 2 for radial oscillations or nonradial oscillations in the cowling approximation . this system only allows non - trivial solutions for selected values of @xmath78 which is thus an eigenvalue of the problem . the programme permits solving these equations with two basically different techniques , each with some variants . the first is a shooting method , where solutions satisfying the boundary conditions are integrated separately from the inner and outer boundary , and the eigenvalue is found by matching these solutions at a suitable inner fitting point @xmath79 . the second technique is to solve the equations together with a normalization condition and all boundary conditions using a relaxation technique ; the eigenvalue is then found by requiring continuity of one of the eigenfunctions at an interior matching point . for simplicity i do not distinguish between @xmath80 and @xmath56 ( cf . section [ sec : eq ] ) in this section . it is implicitly understood that the dependent variable ( which is denoted @xmath56 ) is @xmath80 for @xmath57 and @xmath56 for @xmath81 . the numerical treatment of the transition between @xmath80 and @xmath56 has required a little care in the coding . it is convenient here to distinguish between @xmath77 = 2 and @xmath77 = 4 . for @xmath77 = 2 the differential eqs [ eq : e3.1 ] have a unique ( apart from normalization ) solution @xmath82 satisfying the inner boundary conditions [ eq : e3.2 ] , and a unique solution @xmath83 satisfying the outer boundary conditions [ eq : e3.3 ] . these are obtained by numerical integration of the equations . the final solution can then be represented as @xmath84 . the eigenvalue is obtained by requiring that the solutions agree at a suitable matching point @xmath85 , say . thus @xmath86 these equations clearly have a non - trivial solution @xmath87 only when their determinant vanishes , i.e. , when @xmath88 equation [ eq : e3.5 ] is therefore the eigenvalue equation . for @xmath77 = 4 there are two linearly independent solutions satisfying the inner boundary conditions , and two linearly independent solutions satisfying the outer boundary conditions . the former set @xmath89 is chosen by setting @xmath90 and the latter set @xmath91 is chosen by setting @xmath92 the inner and outer boundary conditions are such that , given @xmath59 and @xmath48 , @xmath60 and @xmath39 may be calculated from them ; thus eqs [ eq : e3.6 ] and [ eq : e3.7 ] completely specify the solutions , which are obtained by integrating from the inner or outer boundary . the final solution can then be represented as @xmath93 at the fitting point @xmath79 continuity of the solution requires that @xmath94 this set of equations only has a non - trivial solution if @xmath95 where , e.g. , @xmath96 . thus eq . [ eq : e3.9 ] is the eigenvalue equation in this case . clearly @xmath97 as defined in either eq . [ eq : e3.5 ] or eq . [ eq : e3.9 ] is a smooth function of @xmath78 , and the eigenfrequencies are found as the zeros of this function . this is done in the programme using a standard secant technique . however , the programme also has the option for scanning through a given interval in @xmath78 to look for changes of sign of @xmath97 , possibly iterating for the eigenfrequency at each change of sign . thus it is possible to search a given region of the spectrum completely automatically . the programme allows the use of two different techniques for solving the differential equations . one is the standard second - order centred difference technique , where the differential equations are replaced by the difference equations @xmath98 , \quad i = 1 , \ldots , i \ ; . \label{eq : e3.11}\ ] ] here i have introduced a mesh @xmath99 in @xmath55 , where @xmath10 is the total number of mesh points ; @xmath100 , and @xmath101 . these equations allow the solution at @xmath102 to be determined from the solution at @xmath103 . the second technique was proposed by @xcite ; here on each mesh interval @xmath104 we consider the equations @xmath105 with constant coefficients , where @xmath106 . these equations may be solved analytically on the mesh intervals , and the complete solution is obtained by continuous matching at the mesh points . this technique clearly permits the computation of solutions varying arbitrarily rapidly , i.e. , the calculation of modes of arbitrarily high order . on the other hand solving eqs [ eq : e3.12 ] involves finding the eigenvalues and eigenvectors of the coefficient matrix , and therefore becomes very complex and time consuming for higher - order systems . thus in practice it has only been implemented for systems of order 2 , i.e. , radial oscillations or nonradial oscillations in the cowling approximation . if one of the boundary conditions is dropped , the difference equations , with the remaining boundary condition and a normalization condition , constitute a set of linear equations for the @xmath107 which can be solved for any value of @xmath44 ; this set may be solved efficiently by forward elimination and backsubstitution ( e.g. , * ? ? ? * ) , with a procedure very similar to the so - called henyey technique ( e.g. , * ? ? * see also christensen - dalsgaard 2007 ) used in stellar modelling . the eigenvalue is then found by requiring that the remaining boundary condition , which effectively takes the role of @xmath108 , be satisfied . however , as both boundaries , at least in a complete model , are either singular or very nearly singular , the removal of one of the boundary conditions tends to produce solutions that are somewhat ill - behaved , in particular for modes of high degree . this in turn is reflected in the behaviour of @xmath97 as a function of @xmath44 . this problem is avoided in a variant of the relaxation technique where the difference equations are solved separately for @xmath109 and @xmath110 , by introducing a double point @xmath111 in the mesh . the solution is furthermore required to satisfy the boundary conditions [ eq : e3.2 ] and [ eq : e3.3 ] , a suitable normalization condition ( e.g. @xmath112 ) , and continuity of all but one of the variables at @xmath85 , e.g. , @xmath113 ( when @xmath77 = 2 clearly only the first continuity condition is used ) we then set @xmath114 and the eigenvalues are found as the zeros of @xmath97 , regarded as a function of @xmath78 . with this definition , @xmath97 may have singularities with discontinuous sign changes that are not associated with an eigenvalue , and hence a little care is required in the search for eigenvalues . however , close to an eigenvalue @xmath97 is generally well - behaved , and the secant iteration may be used without problems . as implemented here the shooting technique is considerably faster than the relaxation technique , and so should be used whenever possible ( notice that both techniques may use the difference eqs [ eq : e3.11 ] and so they are numerically equivalent , in regions of the spectrum where they both work ) . for _ second - order systems _ the shooting technique can probably always be used ; the integrations of the inner and outer solutions should cause no problems , and the matching determinant in eq . [ eq : e3.5 ] is well - behaved . for _ fourth - order systems _ , however , this needs not be the case . for modes where the perturbation to the gravitational potential has little effect on the solution , the two solutions @xmath115 and @xmath116 , and similarly the two solutions @xmath117 and @xmath118 , are almost linearly dependent , and so the matching determinant nearly vanishes for any value of @xmath78 . this is therefore the situation where the relaxation technique may be used with advantage . this applies , in particular , to the calculation of modes of moderate and high degree which are essential to helioseismology . to make full use of the increasingly accurate observed frequencies the computed frequencies should clearly at the very least match the observational accuracy , for a given model . only in this way do the frequencies provide a faithful representation of the properties of the model , in comparisons with the observations . however , since the numerical errors in the computed frequencies are typically highly systematic , they may affect the asteroseismic inferences even if they are smaller than the random errors in the observations , and hence more stringent requirements should be imposed on the computations . also , the fact that solar - like oscillations , and several other types of asteroseismically interesting modes , tend to be of high radial order complicates reaching the required precision . the numerical techniques discussed so far are generally of second order . this yields insufficient precision in the evaluation of the eigenfrequencies , unless a very dense mesh is used in the computation ( see also * ? ? ? the code may apply two techniques to improve the precision . one technique ( cf . * ) uses the fact that the frequency approximately satisfies a variational principle @xcite . the variational expression may formally be written as @xmath119 where @xmath120 and @xmath121 are integrals over the equilibrium model depending on the eigenfunction , here represented by @xmath122 . the variational property implies that any error @xmath123 in @xmath122 induces an error in @xmath124 that is @xmath125 . thus by substituting the computed eigenfunction into the variational expression a more precise determination of @xmath78 should result . this has indeed been confirmed @xcite . the second technique uses explicitly that the difference scheme [ eq : e3.11 ] , which is used by one version of the shooting technique , and the relaxation technique , is of second order . consequently the truncation errors in the eigenfrequency and eigenfunction scale as @xmath126 . if @xmath127 and @xmath128 are the eigenfrequencies obtained from solutions with @xmath129 and @xmath10 meshpoints , the leading - order error term therefore cancels in @xmath130 \ ; . \label{eq : e3.18}\ ] ] this procedure , known as _ richardson extrapolation _ , was used by @xcite . it provides an estimate of the eigenfrequency that is substantially more accurate than @xmath131 , although of course at some added computational expense . indeed , since the error in the representation [ eq : e3.11 ] depends only on even powers of @xmath132 , the leading term of the error in @xmath133 is @xmath134 . even with these techniques the precision of the computed frequencies may be inadequate if the mesh used in stellar - evolution calculations is used also for the computation of the oscillations . the number of meshpoints is typically relatively modest and the distribution may not reflect the requirement to resolve properly the eigenfunctions of the modes . @xcite discussed techniques to redistribute the mesh in a way that takes into account the asymptotic behaviour of the eigenfunctions ; a code to do so , based on four - point lagrangian interpolation , is included in the adipls distribution package . on the other hand , for computing low - order modes ( as are typically relevant for , say , @xmath135 scuti or @xmath136 cephei stars ) , the original mesh of the evolution calculation may be adequate . it is difficult to provide general recommendations concerning the required number of points or the need for redistribution , since this depends strongly on the types of modes and the properties of the stellar model . it is recommended to carry out experiments varying the number and distribution of points to obtain estimates of the intrinsic precision of the computation ( e.g. , * ? ? ? * ; * ? ? ? in the latter case , considering simple polytropic models , it was found that 4801 points yielded a relative precision substantially better than @xmath137 for high - order p modes , when richardson extrapolation was used . in the discussion of the frequency calculation it is important to distinguish between _ precision _ and _ accuracy _ , the latter obviously referring to the extent to which the computed frequencies represent what might be considered the ` true ' frequencies of the model . in particular , the manipulations required to derive eq . [ eq : varprinc ] and to demonstrate its variational property depend on the equation of hydrostatic support being satisfied . if this is not the case , as might well happen in an insufficiently careful stellar model calculation , the value determined from the variational principle may be quite precise , in the sense of numerically stable , but still unacceptably far from the correct value . indeed , a comparison between @xmath138 and @xmath133 provides some measure of the reliability of the computed frequencies ( e.g. * ? ? ? the programme finds the order of the mode according to the definition developed by @xcite and @xcite , based on earlier work by @xcite . specifically , the order is defined by @xmath139 here the sum is over the zeros @xmath140 in @xmath59 ( excluding the centre ) , and @xmath141 is the sign function , @xmath142 if @xmath143 and @xmath144 if @xmath145 . for a complete model that includes the centre @xmath146 for radial oscillations and @xmath147 for nonradial oscillations . thus the lowest - order radial oscillation has order @xmath148 . although this is contrary to the commonly used convention of assigning order 0 to the fundamental radial oscillation , the convention used here is in fact the more reasonable , in the sense that it ensures that @xmath149 is invariant under a continuous variation of @xmath12 from 0 to 1 . with this definition @xmath150 for p modes , @xmath151 for f modes , and @xmath152 for g modes , at least in simple models . it has been found that this procedure has serious problems for dipolar modes in centrally condensed models ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the eigenfunctions @xmath59 are shifted such that nodes disappear or otherwise provide spurious results when eq . [ eq : e4.1 ] is used to determine the mode order . a procedure that does not suffer from this difficulty has recently been developed by @xcite ; i discuss it further in section [ sec : develop ] . a powerful measure of the characteristics of a mode is provided by the _ normalized inertia_. the code calculates this as @xmath153 \rho r^2 { { \rm d}}r \over m [ \xi_r ( r_{\rm phot } ) ^2 + l(l+1 ) { \xi_{\rm h}}(r_{\rm phot } ) ^2 ] } \nonumber \\ & = & { \int_{x_1}^{{x_{\rm s } } } \left [ y_1 ^ 2 + y_2 ^ 2 / l ( l + 1 ) \right ] q u { { \rm d}}x / x \over 4 \pi [ y_1 ( x_{\rm phot } ) ^2 + y_2 ( x_{\rm phot } ) ^2/l(l+1 ) ] } \ ; .\end{aligned}\ ] ] ( for radial modes the terms in @xmath60 are not included . ) here @xmath154 and @xmath155 are the distance of the innermost mesh point from the centre and the surface radius , respectively , and @xmath156 is the fractional photospheric radius . the normalization at the photosphere is to some extent arbitrary , of course , but reflects the fact that many radial - velocity observations use lines formed relatively deep in the atmosphere . a more common definition of the inertia is @xmath157 where @xmath158 is the so - called _ mode mass_. the code has the option to output the eigenfunctions , in the form of @xmath159 . in addition ( or instead ) the displacement eigenfunctions can be output in a form indicating the region where the mode predominantly resides , in an energetical sense , as @xmath160 ( for radial modes only @xmath161 is found ) . these are defined in such a way that @xmath162 { { \rm d}}x / x \over 4 \pi [ y_1 ( x_{\rm phot } ) ^2 + y_2 ( x_{\rm phot } ) ^2/l(l+1 ) ] } \ ; . \label{eq : e4.4}\ ] ] the form provided by the @xmath163 is also convenient , e.g. , for computing rotational splittings @xmath164 ( e.g. , * ? ? ? * ) , where @xmath165 is the frequency of a mode of radial order @xmath149 , degree @xmath12 and azimuthal order @xmath2 . for slow rotation the splittings are obtained from first - order perturbation analysis as @xmath166 characterized by _ kernels _ @xmath167 , where in general the angular velocity @xmath168 depends on both @xmath1 and @xmath26 . the code has built in the option to compute kernels for first - order rotational splitting in the special case where @xmath168 depends only on @xmath1 . several revisions of the code have been implemented in preliminary form or are under development . a substantial improvement in the numerical solution of the oscillation equations , particularly for high - order modes , is the installation of a fourth - order integration scheme , based on the algorithm of @xcite . this is essentially operational but has so far not been carefully tested . comparisons with the results of the variational expression and the use of richardson extrapolation , of the same formal order , will be particularly interesting . as discussed by @xcite the use of @xmath169 ( or , as here , @xmath170 ) as one of the integration variables has the disadvantage that the quantity @xmath171 enters into the oscillation equations . in models with a density discontinuity , such as results if the model has a growing convective core and diffusion is neglected , @xmath171 has a delta - function singularity at the point of the discontinuity . in the adipls calculations this is dealt with by replacing the discontinuity by a very steep and well - resolved slope . however , it would obviously be an advantage to avoid this problem altogether . this can be achieved by using instead the lagrangian pressure perturbation @xmath172 as one of the variables . implementing this option would be a relatively straightforward modification to the code and is under consideration . the proper classification of dipolar modes of low order in centrally condensed models has been a long - standing problem in the theory of stellar pulsations , as discussed in section [ sec : results ] . such a scheme must provide a unique order for each mode , such that the order is invariant under continuous changes of the equilibrium model , e.g. , as a result of stellar evolution . as a major breakthrough , takata in a series of papers has elucidated important properties of these modes and defined a new classification scheme satisfying this requirement @xcite . a preliminary version of this scheme has been implemented and tested ; however , the latest and most convenient form of the takata classification still needs to be installed . a version of the code has been established which computes the first - order rotational splitting for a given rotation profile @xmath173 , in addition to setting up the corresponding kernels . this is being extended by k. burke , sheffield , to cover also second - order effects of rotation , based on the formalism of @xcite . an important motivation for this is the integration , discussed by @xcite , of the pulsation calculation with the astec evolution code to allow full calculation of oscillation frequencies for a model of specified parameters ( mass , age , initial rotation rate , etc . ) as the result of a single subroutine call . i am very grateful to w. dziembowski and d. o. gough for illuminating discussions of the properties of stellar oscillations , and to a. moya and m. j. p. f. g. monteiro for organizing the comparisons of stellar oscillation and model calculations within the esta collaboration . i thank the referee for useful comments which , i hope , have helped improving the presentation . this project is being supported by the danish natural science research council and by the european helio- and asteroseismology network ( helas ) , a major international collaboration funded by the european commission s sixth framework programme . christensen - dalsgaard , j. , berthomieu , g. : theory of solar oscillations . in : cox , a. n. , livingston , w. c. , matthews , m. ( eds ) , _ solar interior and atmosphere _ , p. 401 space science series , university of arizona press ( 1991 ) moya , a. , christensen - dalsgaard , j. , charpinet , s. , lebreton , y. , miglio , a. , montalbn , j. , monteiro , m. j. p. f. g. , provost , j. , roxburgh , i. , scuflaire , r. , surez , j. c. , suran , m. : inter - comparison of the g- , f- and p - modes calculated using different oscillation codes for a given stellar model . apss , this volume ( 2007 )

development of the aarhus adiabatic pulsation code started around 1978 . although the main features have been stable for more than a decade , development of the code is continuing , concerning numerical properties and output . the code has been provided as a generally available package and has seen substantial use at a number of installations . further development of the package , including bringing the documentation closer to being up to date , is planned as part of the helas coordination action .

arxiv

an important manifestation of the activity inside the disks of gas - rich galaxies is their highly structured hi distribution , marked by cavities , shells and supershells . first discovered in the milky way ( heiles 1979 , 1984 ) , such features are now known to exist in a number of spiral galaxies ( e.g. lehnert & heckman 1996 , irwin & seaquist 1990 , puche et al . 1992 , brinks & bajaja 1986 ) . exceptionally huge hi arcs and loops extending across several kiloparsecs have been identified with greater clarity in the hi images of a number of edge - on spirals , such as ngc 5775 ( irwin 1994 ) , ngc 4631 ( rand & van der hulst 1993 ) , ngc 3044 ( lee & irwin 1997 , hereafter li97 ) and ngc 3556 ( m 108 , king & irwin 1997 , hereafter ki97 ) . these have been interpreted as expanding supershells because of a loop - like or circular appearance in projection and either a persistence over a wide velocity range or , in a few cases , as some evidence for expansion in position - velocity space . two main classes of explanations for the supershells posit the source of their kinetic energy to be , respectively , internal and external to the parent galaxy . the internal source model involves starbursts , driving stellar winds ( or superwinds ) and subsequent supernova explosions ( e.g. lehnert & heckman 1996 ) . the chimney model " ( norman & ikeuchi 1989 ) , for example , attempts to explain disk - halo features and other halo gas via processes related to underlying star formation . the association between extraplanar h@xmath1 filaments and star forming regions in the disk of ngc 891 and other correlations between halo emission and in - disk tracers of star formation ( dahlem et al . 1995 ; rand 1997 ) argue in favour of such models . if the presence of hi supershells is found to correlate with the existence of other halo gas , as might be expected in the chimney model , then stellar winds and supernovae are expected to be responsible for the hi supershells as well . the main difficulty with the starburst model for hi supershells lies in the required input energies for the largest shells . using standard assumptions that the expanding supershells are in the post - sedov phase following an ` instantaneous ' injection of energy ( cf . chevalier 1974 ) , hi supershells often require energy input from staggering numbers of spatially correlated supernova events . this was realized early on for our own galaxy ( heiles 1979 , 1984 ) . for external edge - on galaxies , since we are selectively observing only the largest shells , the energy deficit problem is exacerbated . in some cases , hundreds of thousands of clustered supernovae are required ( e.g. ki97 , li97 ) , a conclusion which is not changed significantly if the energy is injected continuously over the lifetime of the shells . other evidence against star formation processes creating the hi shells is also emerging . rhode et al . ( 1999 ) find no optical evidence for recent star formation in the numerous lower energy hi holes of holmberg ii and note that x - ray and fuv emission are also absent . they conclude that supernovae have not played a part in the formation of the hi shells . efremov et al . ( 1998 ) outline numerous other examples in which there appears to be no relation between hi shells and star formation . they , as well as loeb & perna ( 1998 ) , propose that the hi shells are produced , instead , by gamma ray bursts . the alternative external source hypothesis invokes infall of massive gas clouds on to the galactic plane , as a result of gravitational interaction with neighbouring galaxies ( see tenorio - tagle & bodenheimer 1988 ) . this resolves the energy problem since input energy is then a function of the mass and velocity of the infalling cloud . evidence in favour of this hypothesis comes from observations of high velocity clouds ( hvcs ) around our own milky way and the signatures of interaction in m 101 ( van der hulst & sancisi 1988 ) and ngc 4631 ( rand & stone 1996 ) . it does , however , require that the galaxy be in some way interacting with a companion or , at least , that sufficiently massive clouds be in the vicinity . recent observations are revealing galaxies which are apparently isolated , yet harbour extremely large hi supershells . two striking examples are the nearby , sb(s)cd galaxy , ngc 3556 ( ki97 ) and the sbc galaxy , ngc 3044 ( li97 ) . both of these galaxies exhibit radio continuum halos extending to @xmath2 kpc from the galactic plane and have a number of supershells requiring energies up to a few @xmath3 10@xmath4 ergs . these supershells are too large and energetic to have been produced by conventional clustered supernovae . at the same time , there appears to be no evidence for interaction or nearby companions , either . we propose here a new explanation for hi supershells . that is , that they have been formed by radio jets which plow through the interstellar medium ( ism ) , accreting ism gas and sometimes inflating bubbles . this allows for an internal energy source for the hi shells , provides a natural explanation for any spatial symmetries seen in the hi features , and also resolves the energy problem . in sect . 2 , we provide arguments in favour of jet inflated hi bubbles , sect . 3 presents the model , and sect . 4 discusses the implications of this scenario . seyferts are one class of disk galaxy for which several examples of the nucleus ejecting a radio jet pair have been found ( e.g. ulvestad & wilson 1984a , 1984b , kukula et al . 1995 , aoki et al . likewise , several cases of jets occurring in normal spiral galaxies have been reported ( e.g. hummel et al . 1983 ) . prominent examples include ngc 3079 ( de bruyn 1977 ) , ngc 5548 ( ulvestad et al . 1999 ) and circinus ( elmouttie et al . 1998 ) . the total energy output from such nuclear activity can approach @xmath5 erg , assuming that the nuclear activity lasts @xmath0 years , and arises from accretion onto a supermassive black hole of @xmath6 at 10% of the eddington limit with the canonical 10% efficiency . in normal " spiral galaxies , the central mass may be lower ; for example , the bipolar outflow from the sab(rs)cd galaxy , m 101 may be produced by a central @xmath7 10@xmath8 m@xmath9 black hole ( moody et al . while jets are not always observed directly , the growing number of supermassive black holes inferred to be present in the nuclei of normal disk galaxies ( kormendy & richstone 1995 , van der marel 1999 ) , including late - type disks ( ho et al . 1997 ) , the milky way itself ( e.g. genzel et al . 1997 ) and gas - rich , large low - surface - brightness galaxies ( schombert 1998 ) , lend weight to the idea that many such galaxies may have undergone phases of nuclear activity accompanied with a bi - directional ejection of relativistic plasma jets , before entering a dormant nuclear phase . recent studies have also revealed that the ejection of jets can take place at small angles to the large - scale galactic disk ( e.g. nagar & wilson 1999 ; kinney et al 2000 ) , plausibly leading to clear signatures of their dynamical interaction with the ism . a classic case of such an interaction is the bow - shaped morphology of the radio lobe in the well known seyfert galaxy , ngc 1068 , which strongly suggests that the kiloparsec - scale radio jets are ploughing through the disk medium ( wilson & ulvestad 1987 , axon et al . a dynamical interaction of the jets with the disk is also evident in the case of the spiral galaxy m 51 which is at the low end of nuclear activity scale . here , the jets seem to have created a pair of ` radio lobes ' on opposite sides of the nucleus ; whereas one of the lobes is bow - shaped , the other one is actually resolved into a ring which is also detected in h@xmath1 emission ( ford et al . another example is ngc 4258 whose vlbi jet has been ejected close to the galaxy disk ( see cecil et al . 1995 , herrnstein et al . 1997 , cecil et al . large scale ( 14 kpc length ) anomalous arms " are also seen _ within _ the galactic disk and have been interpreted as manifestations of a larger - scale jet ( e.g. martin et al . 1989 , but see cox & downes 1996 ) . in m 101 , the bipolar outflow is also roughly confined to the disk , with the outflow not expected to extend beyond a height of 400 pc ( moody et al . additional support for the jet - disk interaction hypothesis in these disk galaxies comes from the detection of shock - excited optical emission lines associated with the radio lobes , as discussed in some of the references cited above . it is further interesting to note that radio bubbles and shells of synchrotron plasma have been discovered within the lobes of a few radio galaxies . two spectacular examples are 3c310 ( van breugel & fomalont , 1984 ) and hercules a ( dreher & feigelson 1984 ) . in hercules a , in addition , peculiar dark shells , about 3 kpc in size , have been found straddling the nucleus along the radio outflow axes ( baum et al . for the dark shells , baum et al . prefer an explanation in terms of expanding bubbles of hot gas ejected from the active nucleus along the radio axis , in agreement with plasmon - like ejection from the core , though they do not rule out other possibilities . these cases provide a clue that the radio bubbles / shells may have been ` puffed ' up due to localized instabilities in the radio jets , occurring at distances of a few kiloparsecs from the active nucleus ( e.g. morrison & sadun 1996 ) . should such instabilities arise also in the jets ejected within disk galaxies ( sect . 2.1 ) , it is quite plausible that the resulting increase in pressure at those locations would inflate bubbles and shells out of the ambient ism material which is hi rich , in the case of spirals . such features , puffed out of the disk during the brief phase of jet instability , would remain visible past the radiative lifetime of the synchrotron plasma against the radiative and expansion losses , which is typically of order @xmath10 years . in ngc 3556 , the two most prominent supershells are located symmetrically near the opposite extremities of the galaxy . ki97 have carried out a detailed mapping of the hi and radio continuum emission from this nearly edge - on galaxy located 11.6 mpc away . the superimposition of 2 channels of hi emission , symmetrically placed in velocity about systemic , are shown in fig . 1 a. the most obvious features are two loops of hi , producing extensions to the east and west along the major axis . fig . 1b shows the hi emission from these velocities superimposed on the radio continuum map , the latter taken from entirely independent observations . the abrupt density drop - off in the radio continuum map occurs roughly where the optical galaxy ends ( fig . 1a inset ) . the salient features of this system most relevant to the present study ( see fig . 1 ; ki97 ) are : \(a ) two symmetrically placed giant hi loops are seen originating at a projected galactocentric radius of about 12 kpc on the east and west major axis . they appear to originate right at the opposite edges of the optical disk and extend , not perpendicular to the disk plane , but rather parallel to it and outwards . \(b ) the giant hi loops or supershells are slightly bent into a z - symmetry . \(c ) these hi loops are most obvious in velocity channels which are equally spaced about the systemic velocity of the galaxy ( though they extend over a larger velocity range , see ki97 ) . \(d ) from their velocity dependent morphology , each of the loops shows evidence for shell expansion , but with only half of the shell present . in both cases , the receding side ( with respect to galaxy rotation ) is open . \(e ) other smaller hi loops and extensions do exist at smaller galactocentric radii . \(f ) the parameters of these two hi supershells are as follows : the eastern supershell has a mass , m = 5.9 @xmath3 10@xmath11 m@xmath9 , a radius , r = 3.2 kpc , an expansion velocity , @xmath12 = 51.7 km s@xmath13 , a kinematic age , @xmath14 = 6.0 @xmath3 10@xmath11 yr , a kinetic energy , e@xmath15 = 1.6 @xmath3 10@xmath16 ergs , and an implied input energy ( assuming instantaneous input from supernovae ) of e = 2.6 @xmath3 10@xmath4 ergs . the parameters of the western supershell are : m = 2.3 @xmath3 10@xmath11 m@xmath9 , r = 1.85 kpc , @xmath12 = 41.4 km s@xmath13 , @xmath14 = 4.4 @xmath3 10@xmath11 yr , e@xmath15 = 3.9 @xmath3 10@xmath17 ergs , and e = 3.4 @xmath3 10@xmath18 ergs . \(g ) the energy needed to create the eastern supershell alone would require a star cluster populated by @xmath19 ob stars , if supernovae and stellar winds are the drivers . \(h ) the brighter and more spectacular eastern supershell has associated hi gas which extends as far as 30 kpc beyond the eastern edge of the optical disk and possibly much farther ( see fig . 1a inset ) . the huge eastern hi extension appears to open up from one side of the eastern shell and extends fairly straight outwards in the radial , rather than the vertical direction as if formed in a short period of time in comparison to galactic rotation . \(i ) the galaxy contains a large nonthermal radio halo with a scale height of @xmath2 kpc above the plane ( see also de bruyn & hummel 1979 , bloemen et al . 1993 ) . \(j ) radio continuum emission appears to be associated with the hi loops , but is not spatially coincident with them . the radio continuum emission extends farther out than the hi supershells , as if funnelling " along or through the hi features ( ki97 ; fig . \(k ) ngc 3556 has no obvious interacting companions . the brightest galaxy in the vicinity is the 16th magnitude , mcg+09 - 19 - 001 , 25@xmath20 in size , undisturbed in appearance and @xmath7 12@xmath21 to the east of ngc 3556 . this is most likely a background source . ngc 5775 , an interacting galaxy , shows symmetrically placed hi features ( lee et al . 2000 ) in the sense that hi extensions occur on opposite sides of the major axis in which there appears to be an underlying disturbance ; these features occur at similar galactocentric radii , in projection ( see also fig . 3 of irwin 1994 which shows 3 of the extensions ) . ngc 2613 ( chaves & irwin 2000 ) also shows six hi extensions which occur in pairs symmetrically on either side of the major axis ; two of the pairs occur near the ends of the optical disk . another potentially very interesting example is m 31 . blitz et al . ( 1999 ) show that two massive hi clouds exist symmetrically placed on opposite sides of this galaxy . they bear a remarkable resemblance to extragalactic radio jets . the two clouds are redshifted by @xmath22 km s@xmath13 with respect to the systemic velocity of m 31 . conceivably , this could be explained by ram pressure sweeping as m 31 falls through the igm towards the milky way . as mentioned above , the galaxy ngc 3556 puts to a severe test both of the standard explanations for the supershell phenomenon , namely : ( i ) localized starburst ( generating intense stellar winds , followed by multiple supernova explosions ) , and ( ii ) infall of massive gas clouds on to the galactic disk . the major difficulties faced by these models , as highlighted in ki97 , are : \(a ) * localized starbursts : * the energy needed to create the eastern supershell alone would require a star cluster populated by @xmath23 ob stars . even for the recently discovered ` super star clusters ( ssc ) ' in some starburst galaxies , not more than @xmath24 ob stars are implied ( meurer et al . 1995 ) . there is no indication of an ob association even remotely approaching the level of sscs at the locations of the supershells in ngc 3556 . since the kinematic ages of the supershells are @xmath7 5 @xmath3 10@xmath11 yr , the starburst would have to have occurred within this time frame . yet starburst durations and ob association ages are typically also of this order , so some evidence of the starburst might be expected to have survived . no such evidence is presently found , however . ngc 3556 does not show a markedly high supernova rate globally ( irwin et al . 1999 ) and independent low and high resolution radio continuum observations show no evidence for a source of energy at the bases of the supershells ( ki97 , irwin et al . 2000 ) . \(b ) * infalling clouds : * several arguments have been advanced against this possibility ( ki97 ) . firstly , this galaxy appears isolated , making the source of the putative clouds puzzlesome . secondly , to impart sufficient kinetic energy for the creation of the eastern supershell , an infalling cloud of mass @xmath25 would be required . a cloud this massive would easily have been detected on the sensitive hi maps , yet no such cloud or clouds are seen . ( an exception would be if the clouds had very narrow line widths , low fractal dimensions and were optically thick , in which case they could have been missed due to low filling factors . ) thirdly , if an encounter with intergalactic clouds has occurred in the past , it would be necessary to postulate that two very massive clouds just happened to fall in parallel to the major axis at opposite ends of the galaxy , roughly at the same time , a scenario which seems implausible . fourthly , we could instead suggest that a rain " of hi clouds is infalling ( since other high latitude hi structures are also seen in this galaxy ) , including two which fell in along the major axis at either end . however , in order for the rain to be no longer visible , the infall would have to be completed over a timescale which is less than or equal to the age of the supershells , i.e. a few 10@xmath11 years . this is unlikely since infall timescales are of order a dynamical timescale which is typically a few 10@xmath26 yr . while these arguments have been applied to ngc 3556 alone , the difficulty with internal energy from starbursting applies to a number of galaxies , as outlined in sect . 1 . infalling clouds are certainly plausible , but run into difficulties for isolated galaxies such as ngc 3556 and ngc 3044 . barring new information coming to light on these galaxies , we therefore argue that a new model should be considered . as mentioned earlier , the duality of the expanding hi supershells in ngc 3556 , together with their locations at the opposite edges of the disk marked by steep density gradients , their z - symmetric deviations from spherical symmetry , as well as their anomalously large energy requirements ( sect . 2.3.1 ) , together lead us to argue that the origin of these two supershells is linked to the jet phenomenon . in sect . 2 we have noted several manifestations of jet instabilities giving rise to bubble - shaped features , or shells , in the form of optical nebulosities or nonthermal radio emission in normal spirals and radio galaxies . in the present context of hi supershells , we examine the possibility of the supershell being inflated due to localised instabilities in the radio jets as they plough through the ism of the disk of ngc 3556 . the present discussion should only be viewed as a feasibility check based on energy considerations , and not as a detailed quantitative model for the supershells . even in the case of ngc 3556 , the standard mechanisms for the supershells ( sect . 1 ) may well have contributed at some level . according to our proposal , the two radio jets ejected close to the plane of the disk , during the active phase of the nucleus , undergo a rapid _ flaring _ as they encounter the region of large scale density decline near the outskirts of the galactic disk . consequently , both synchrotron plasma and the entrained thermal plasma deposited by the jets in these two regions get heated and the resulting high - pressure bubble of hot plasma undergoes a rapid expansion , sweeping the gas - rich ambient medium into the shape of the hi supershells . an idealization to such a situation is the flaring of a jet crossing an ism / igm interface , which was first discussed in the context of radio galaxies analytically by gopal - krishna & wiita ( 1987 ) , followed by two - dimensional ( norman et al . 1988 , wiita et al . 1990 , zhang et al . 1999 ) and three - dimensional ( loken et al . 1995 , hooda & wiita 1996 , 1998 ) numerical simulations . the simulations by loken et al . showed that intermediate power radio jets associated with wide - angle - tail ( wat ) sources undergo a rapid flaring upon crossing the ism / igm density discontinuity where a moderately supersonic jet becomes subsonic . as shown by the numerical simulations , a density drop of a factor of just a few can cause such a flaring ( e.g. hooda & wiita 1996 ) . observational evidence for the jet flaring comes from wide - angle - tail ( wat ) radio sources which are associated with the dominant members of rich clusters of galaxies ( e.g. odonoghue et al . further evidence is provided , e.g. , by the recent radio / optical / x - ray study of the wat radio galaxy 3c465 in the abell cluster a2634 ( sakelliou & merrifield 1999 ) . particularly instructive for the present study is the case of the active disk galaxy iras 04210 + 0400 ; the two @xmath27 kpc long radio jets associated with this disk galaxy are seen to flare up near the opposite optical boundaries of the galaxy ( holloway et al 1996 ) . it may be noted that during their passage through the inner parts of the galactic disk , the jets are likely to encounter large ism density fluctuations . however , these are less prone to disrupt the jets because of the higher jet thrust there , combined with the expected small spatial scale of the density fluctuations compared with the jets cross - section . since the postulated radio jets in ngc 3556 are currently too weak for detection , a point which we re - address below , we shall adopt here the jet parameters inferred for another massive disk galaxy , ngc 4258 ( sect . 2.1 ) , and apply them to known conditions in ngc 3556 . the jet velocity , as measured from the emission line gas , is _ at least _ v@xmath28 = 2000 km s@xmath13 which , together with a density of @xmath29 = 0.02 @xmath30 g @xmath31 , where @xmath30 is the mass of the proton , correspond to a kinetic luminosity of l@xmath28 @xmath7 10@xmath32 erg s@xmath13 ( falcke & biermann 1999 ) . assuming that the ram pressure in the jet dominates over the internal thermal pressure , then across the shock front we require @xmath33 , where @xmath34 is the ambient gas density , and @xmath35 is the shock velocity . taking @xmath34 = 0.26 @xmath30 g @xmath31 at the positions of the supershells ( ki97 ) , we obtain @xmath35 = 435 km s@xmath13 . this is slightly higher than the value of @xmath7 300 km s@xmath13 estimated for ngc 4258 . the shock velocity puts a minimum timescale on the duration of the jet in this model , since there must be sufficient time for the jet to propagate out to the locations of the supershells at 12 kpc galactocentric radius . thus , the minimum duration of the jets is 2.7 @xmath3 10@xmath11 yr for the parameters in this illustration . the mechanical power of the jet would then be @xmath36 = @xmath37 = 5 @xmath3 10@xmath38 erg s@xmath13 , where the effective jet radius , r has a typical value of 1 kpc ( e.g. cecil et al 1995 ) . this is similar to the jet power estimated for the well known spiral galaxy m 51 , which is at the low end of the scale of nuclear activity ( cf . ford et al 1985 ) . in ngc 3556 , the integrated mechanical luminosity of the jet over its minimum lifetime is then @xmath7 4 @xmath3 10@xmath4 erg . this lower limit is adequate to account for the observed kinetic energy associated with the larger eastern supershell ( sect . here we have assumed that the efficiency for converting the input energy into the kinetic energy of the shell is of order 1% as usually taken for the multiple supernova model . for a higher efficiency , the required jet power can be lower . the expansion of a typical bubble in the ism should then proceed similarly to previously modelled scenarios , the difference being the source of input energy . since the input energy is much higher than conventional supernovae , the bubbles are more likely to achieve blowout , providing a natural conduit through which cosmic rays ( including those supplied by the jets ) can escape into the halo . thus , the presence of supershells is expected to correlate with the existence of a nonthermal radio halo . the locations of the anchor points of the two supershells in ngc 3556 suggest that the postulated blasting of each supershell would have occurred at a galactocentric distance which is just past the peak of the hi rotation curve ( see fig . 6 of ki97 ) . the radially outward expansion of each supershell would then expose it to the velocity shear in the medium , exerting a side - ways wind pressure counter to the galactic rotation . due to this , the half of the supershell towards the direction of the galactic rotation would be compressed and hence brightened , whilst the other half would be dragged out due to the velocity shear in the external medium and , consequently , dimmed . such a deformation of the two supershells from spherical symmetry , in course of their expansion , would give rise to a z - symmetry , which is consistent with the observations ( sect . 2.3.1 ; ki97 ) . the cooling time for the x - ray emitting sheaths around the jets of ngc 4258 , assuming unity filling factor , is only 4 @xmath3 10@xmath8 yr ( cecil et al . 1995 ) and therefore , once the jets have turned off , such a signature of their existence would disappear rapidly . the lifetime of the synchrotron emitting particles in the jets is longer , typically from 10@xmath8 to 10@xmath11 yr . for ngc 4258 , it has been estimated to be between 1 to 5 @xmath3 10@xmath11 yr ( martin et al . 1989 ) . however , shorter timescales are possible and will depend on local magnetic field strength and spectral index . for instance , the magnetic field in the north - east radio jet in ngc 1068 , which extends to @xmath7 450 pc from the nucleus , is @xmath7 5 @xmath3 10@xmath39 gauss and the spectral index is -1 ( wilson & ulvestad 1987 ) . this gives a synchrotron lifetime of only 1.5 @xmath3 10@xmath40 yr . the particle lifetime is expected to be shorter in shocks where the magnetic field is compressed . thus , we expect the radio jet in ngc 3556 to fade over a timescale less than the ages of the supershells . similar arguments apply to the radio core which is expected to decay faster than the jets , once the nuclear activity has switched off , given the likelihood of a stronger magnetic field and flatter spectral index in the core . even if some radio emission from a core were to persist after the activity ceases , it may be below the rms noise level of the available maps . chary & becklin ( 1997 ) , for example , estimate the radio luminosity of the core of ngc 4258 , which is known to have a supermassive central object of mass 3.6 @xmath3 10@xmath11 m@xmath9 and a vlbi jet , to be l@xmath41 = 100 l@xmath9 . the 2@xmath42 noise level of the radio maps of irwin et al . ( 2000 ) corresponds to a radio power of 6.8 @xmath3 10@xmath43 w hz@xmath13 . integrating over 10@xmath44 hz , this yields a radio luminosity of 180 l@xmath9 . thus , even if ngc 3556 harbours a radio core of the same luminosity as ngc 4258 , it would not have been detected in the observations mentioned above . in this study we have sketched a scenario whereby radio jets ejected during an active nuclear phase in the life of a large spiral galaxy could inflate out of the disk exceptionally large shells , i.e. supershells of neutral hydrogen gas . when applied to the scd galaxy , ngc 3556 , this model can account for each of the several features enumerated in sect . 2.3.1 . in the context of the hi supershells seen in this galaxy , this scheme appears to fare distinctly better than either of the two standard models for supershells which invoke either a starburst induced superwind or an infall of external gas clouds on to the galaxy disk . thus , the jet - ism interaction scenario for disk galaxies could effectively supplement the other two mechanisms already proposed for the supershell phenomenon , with a greater relevance whenever the shells are extraordinarily large as well as energetic and occur at large galactocentric distances . the energy requirement is no longer a major issue in the present model . even a modest energy input from a 10@xmath11 m@xmath9 compact object at the galactic nucleus , such as that observed in ngc 4258 , is more than adequate to supply the input energy required for the supershells of ngc 3556 . this model can therefore account for hi supershells which are difficult to form via conventional processes involving massive star formation , or those occurring in galaxies lacking potentially interacting companions . larger supershells ( or parts thereof , given the probable development of instabilities ) may also be predicted , since known central masses and/or accretion rates may well surpass those considered here . furthermore , the present model can account for the symmetric hi features in galaxies . for instance , it is more likely that bubbles will be inflated by the propagating jets at galactocentric radii where the ism density has declined to sufficiently low values for blow - out . at the same time , symmetry is not a generic outcome our model , since local shocks , dictated by the interaction of the jet with local density perturbations , can be important in determining where the bubbles will be inflated . for example , known bubbles in the lobes of radio galaxies ( sect . 2.2 ) are not always seen to be equidistant on either side of the nucleus . in our sample illustration ( sect . 3.2 ) , the jets must exist for long enough to reach large galactocentric radii ( 3 @xmath3 10@xmath11 yr ) but need to inject energy over only a fraction of the lifetime of the supershells ( 2 @xmath3 10@xmath11 yr , or even less , if the efficiency exceeds 1% ) . the jets presumably switch off thereafter , with the synchrotron emitting particles decaying in 10@xmath8 - 10@xmath11 yr . it is difficult to predict whether jets and hi supershells should co - exist in spirals . nuclear activity in galaxies is commonly believed to be a transient phenomenon , lasting for @xmath45 years ( e.g. eilek 1996 , scheuer 1995 ; soltan 1982 ; haehnelt & rees 1993 ; ho et al 1997 ) . if agn activity occurs on timescales longer than a few @xmath3 10@xmath11 yr , this would suggest that there might be disk galaxies in which both phenomena should be observed at the same time . however , such nuclear activity timescales are likely to be more representative of radio galaxies ; agn activity in spirals is probably shorter lived . an important next step is to consider what jet parameters are required to reach large , or at least kpc - 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taking a clue from the pair of hi supershells found in the scd galaxy ngc 3556 ( m 108 ) , we propose a new mechanism for the origin of hi supershells in gas - rich massive galaxies . in this scenario , the two supershells were inflated out of the neutral hydrogen disk due to the localised flaring of a pair of radio lobes formed by the jets ejected from the nucleus during an active phase about @xmath0 years ago , but have faded away by now . it is shown that the salient features of this supershell pair , such as their symmetrical locations about the galactic centre , the anomalously large energy requirements , the large galacto - centric distances , as well as the z - symmetric hemispherical shapes , find a more natural explanation in terms of this scenario , as compared to the standard models which postulate either a massive starburst , or the infall of external gas clouds . other possible implications of this hypothesis are briefly discussed .

arxiv

the first color - magnitude diagrams ( cmd ) obtained by baade for the dwarf spheroidal ( dsph ) companions of the milky way , and in particular for the draco system ( baade & swope 1961 ) , showed all of the features present in the cmd s of globular clusters . this , together with the presence of rr lyrae stars ( baade & hubble 1939 ; baade & swope 1961 ) led to the interpretation that dsph galaxies are essentially pure population ii systems . but baade ( 1963 ) noted that there are a number of characteristics in the stellar populations of dsph galaxies that differentiate them from globular clusters , including extreme red horizontal branches and the distinct characteristics of the variable stars . when carbon stars were discovered in dsph galaxies , these differences were recognized to be due to the presence of an intermediate - age population ( cannon , niss & norgaard nielsen 1980 ; aaronson , olszewski & hodge 1983 ; mould & aaronson 1983 ) . in the past few years this intermediate - age population has been shown beautifully in the cmds of a number of dsph galaxies ( carina : mould & aaronson 1983 ; mighell 1990 ; smecker - hane , stetson & hesser 1996 ; hurley - keller , mateo & nemec 1998 ; fornax : stetson , hesser & smecker - hane 1998 ; leo i : lee et al . 1993 , l93 hereinafter ; this paper ) . other dsph show only a dominant old stellar population in their cmds ( ursa minor : olszewski & aaronson 1985 ; martnez - delgado & aparicio 1999 ; draco : carney & seitzer 1986 ; stetson , vandenbergh & mcclure 1985 ; grillmair et al . 1998 ; sextans : mateo et al . 1991 ) . an old stellar population , traced by a horizontal - branch ( hb ) , has been clearly observed in all the dsph galaxies satellites of the milky way , except leo i , regardless of their subsequent star formation histories ( sfh ) . in this respect , as noted by l93 , leo i is a peculiar galaxy , showing a well populated red - clump ( rc ) but no evident hb . this suggests that the first substantial amount of star formation may have been somehow delayed in this galaxy compared with the other dsph . leo i is also singular in that its large galactocentric radial velocity ( 177@xmath53 km @xmath6 , zaritsky et al . 1989 ) suggests that it may not be bound to the milky way , as the other dsph galaxies seem to be ( fich & tremaine 1991 ) . byrd et al . ( 1994 ) suggest that both leo i and the magellanic clouds seem to have left the neighborhood of the andromeda galaxy about 10 gyr ago . it is interesting that the magellanic clouds also seem to have only a small fraction of old stellar population . leo i presents an enigmatic system with unique characteristics among local group galaxies . from its morphology and from its similarity to other dsph in terms of its lack of detectable quantities of hi ( knapp , kerr & bowers 1978 , see section [ leoi_prev ] ) it would be considered a dsph galaxy . but it also lacks a conspicuous old population and it has a much larger fraction of intermediate - age population than its dsph counterparts , and even , a non - negligible population of young ( @xmath7 1 gyr old ) stars . in this paper , we present new _ hst _ f555w ( @xmath1 ) and f814w ( @xmath2 ) observations of leo i. in section [ leoi_prev ] , the previous work on leo i is briefly reviewed . in section [ obs ] , we present the observations and data reduction . in section [ phot ] we discuss the photometry of the galaxy , reduced independently using both allframe and dophot programs , and calibrated using the ground - based photometry of l93 . in section [ cmd ] we present the cmd of leo i , and discuss the stellar populations and the metallicity of the galaxy . in section [ discus ] we summarize the conclusions of this paper . in a companion paper , ( gallart et al . 1998 , paper ii ) we will quantitatively derive the sfh of leo i through the comparison of the observed cmd with a set of synthetic cmds . leo i ( ddo 74 ) , together with leo ii , was discovered by harrington & wilson ( 1950 ) during the course of the first palomar sky survey . the distances to these galaxies were estimated to be @xmath8 200 kpc , considerably more distant than the other dsph companions of the milky way . it has been observed in hi by knapp et al . ( 1978 ) using the nrao 91-m telescope , but not detected . they set a limit for its hi mass of @xmath9 in the central 10(@xmath8 780 pc ) of the galaxy . recently , bowen et al . ( 1997 ) used spectra of three qso / agn to set a limit on the hi column density within 24 kpc in the halo of leo i to be @xmath10 . they find no evidence of dense flows of gas in or out of leo i , and no evidence for tidally disrupted gas . the large distance to leo i and the proximity on the sky of the bright star regulus have made photometric studies difficult . as a consequence , the first cmds of leo i were obtained much later than for the other nearby dsphs ( fox & pritchet 1987 ; reid & mould 1991 ; demers , irwin & gambu 1994 ; l93 ) . from the earliest observations of the stellar populations of leo i there have been indications of a large quantity of intermediate - age stars . hodge & wright ( 1978 ) observed an unusually large number of anomalous cepheids , and carbon stars were found by aaronson et al . ( 1983 ) and azzopardi , lequeux & westerlund ( 1985 , 1986 ) . a prominent rc , indicative in a low z system of an intermediate - age stellar population , is seen both in the @xmath11 $ ] cmd of demers et al . ( 1994 ) and in the @xmath12 $ ] cmd of l93 . the last cmd is particularly deep , reaching @xmath13 ( @xmath14 ) , and suggests the presence of a large number of intermediate age , main sequence stars . there is no evidence for a prominent hb in any of the published cmd s . l93 estimated the distance of leo i to be @xmath15 based on the position of the tip of the red giant branch ( rgb ) ; we will adopt this value in this paper . they also estimated a metallicity of [ fe / h ] = [email protected] dex from the mean color of the rgb . previous estimates of the metallicity ( aaronson & mould 1985 ; suntzeff , aaronson & olszewski 1986 ; fox & pritchet 1987 ; reid & mould 1991 ) using a number of different methods range from [ fe / h]=1.0 to 1.9 dex . with the new _ hst _ data presented in this paper , the information on the age structure from the turnoffs will help to further constrain the metallicity . we present wfpc2 _ hst _ @xmath1 ( f555w ) and @xmath2 ( f814w ) data in one 2.6@xmath16 2.6 field in leo i obtained in march 5 , 1994 . the wfpc2 has four internal cameras : the planetary camera ( pc ) and three wide field ( wf ) cameras . they image onto a loral 800@xmath16800 ccd , which gives an scale of 0.046 pixel@xmath17 for the pc camera and 0.10 pixel@xmath17 for the wf cameras . at the time of the observations the camera was still operating at the higher temperature of 77.0 @xmath18c . figure [ carta ] shows the location of the wfpc2 field superimposed on digitized sky survey image of leo i. the position of the ground - based image of l93 is also shown . the position was chosen so that the pc field was situated in the central , more crowded part of the galaxy . three deep exposures in both f555w ( @xmath1 ) and f814w ( @xmath2 ) filters ( 1900 sec . and each , respectively ) were taken . to ensure that the brightest stars were not saturated , one shallow exposure in each filter ( 350 sec . in f555w and 300 sec in f814w ) was also obtained . figure [ mosaic ] shows the @xmath1 and @xmath2 deep ( 5700 sec . and 4800 sec . respectively ) wf chip2 images of leo i. all observations were preprocessed through the standard stsci pipeline , as described by holtzmann et al . in addition , the treatment of the vignetted edges , bad columns and pixels , and correction of the effects of the geometric distortion produced by the wfpc2 cameras , were performed as described by silbermann et al . photometry of the stars in leo i was measured independently using the set of daophot ii / allframe programs developed by stetson ( 1987 , 1994 ) , and also with a modified version of dophot ( schechter , mateo & saha 1993 ) . we compare the results obtained with each of these programs below . allframe photometry was performed in the 8 individual frames and the photometry list in each band was obtained by averaging the magnitudes of the corresponding individual frames . in summary , the process is as follows : a candidate star list was obtained from the median of all the images of each field using three daophot ii / allstar detection passes . this list was fed to allframe , which was run on all eight individual frames simultaneously . we have used the psfs obtained from the public domain _ hst _ wfpc2 observations of the globular clusters pal 4 and ngc 2419 ( hill et al . the stars in the different frames of each band were matched and retained if they were found in at least three frames for each of @xmath1 and @xmath2 . the magnitude of each star in each band was set to the error - weighted average of the magnitudes for each star in the different frames . the magnitudes of the brightest stars were measured from the short exposure frames . a last match between the stars retained in each band was made to obtain the @xmath19 photometry table . dophot photometry was obtained with a modified version of the code to account for the _ hst _ psf ( saha et al . dophot reductions were made on average @xmath1 and @xmath2 images combined in a manner similar to that described by saha et al . ( 1994 ) in order to remove the effects of cosmic rays . photometry of the brightest stars was measured from the @xmath1 and @xmath2 short exposure frames . the dophot and allframe calibrated photometries ( see section [ transjohn ] ) show a reasonably good agreement . there is a scatter of 2 - 3% for even the brightest stars in both @xmath1 and @xmath2 . no systematic differences can be seen in the @xmath1 photometry . in the @xmath2 photometry there is good systematic agreement among the brightest stars , but a small tendency for the dophot magnitudes to become brighter compared to the allframe magnitudes with increasing @xmath2 magnitude . this latter effect is about 0.02 magnitudes at the level of the rc , and increases to about 0.04 - 0.05 mag by @xmath2 = 26 . we can not decide from these data which program is ` correct ' . however , the systematic differences are sufficiently small compared to the random scatter that our final conclusions are identical regardless of which reduction program is used . in the following we will use the star list obtained with daophot / allframe . our final photometry table contains a total of 31200 stars found in the four wfpc2 chips after removing stars with excessively large photometric errors compared to other stars of similar brightness . the retained stars have @xmath20 , chi@xmath21 and @xmath22sharp@xmath23 . for our final photometry in the johnson - cousins system we will rely ultimately in the photometry obtained by l93 . before this last step though , we transformed the profile fitting photometry using the prescription of holtzmann et al . ( 1995 ) . in this section , we will describe both steps and discuss the differences between the _ hst_-based photometry and the ground based photometry . the allframe photometry has been transformed to standard magnitudes in the johnson - cousins system using the prescriptions of holtzmann et al . ( 1995 ) and hill et al . ( 1998 ) as adopted for the _ hst _ @xmath24 key project data . psf magnitudes have been transformed to instrumental magnitudes at an aperture of radius 0.5 " ( consistent with holtzmann et al . 1995 and hill et al . 1998 ) by deriving the value for the aperture correction for each frame using daogrow ( stetson 1990 ) . the johnson - cousins magnitudes obtained in this way were compared with ground - based magnitudes for the same field obtained by l93 by matching a number of bright ( @xmath25 ) , well measured stars in the _ hst _ ( @xmath26 , @xmath27 ) and ground - based photometry ( @xmath28 , @xmath29 ) . the zero - points between both data sets have been determined as the median of the distribution of ( @xmath30 ) and ( @xmath31 ) . in table [ zeros ] the values for the median of ( @xmath30 ) , ( @xmath31 ) and its dispersion @xmath32 are listed for each chip ( no obvious color terms are observed , as expected , since both photometry sets have been transformed to a standard system taking into account the color terms where needed of the corresponding telescope - instrument system ) . @xmath33 is the number of stars used to calculate the transformation . although the value of the median zero point varies from chip to chip , it is in the sense of making the corrected @xmath1 magnitudes brighter by @xmath34 mag than @xmath26 and the corrected @xmath2 magnitudes fainter than @xmath27 by about the same amount . therefore , the final @xmath35 colors are one tenth of a magnitude bluer in the corrected photometry . lcccc chip & filter & median & @xmath32 & n + chip 1 & f555w & -0.037 & 0.100 & 17 + chip 2 & f555w & -0.110 & 0.103 & 59 + chip 3 & f555w & -0.080 & 0.063 & 43 + chip 4 & f555w & -0.059 & 0.067 & 57 + chip 1 & f814w & 0.035 & 0.075 & 17 + chip 2 & f814w & 0.013 & 0.064 & 53 + chip 3 & f814w & 0.076 & 0.042 & 43 + chip 4 & f814w & 0.080 & 0.041 & 53 + note that the cte effect , which may be important in the case of observations made at the temperature of @xmath36 c , could contribute to the dispersion on the zero - point nevertheless , if the differences @xmath37 , @xmath38 are plotted for different row intervals , no clear trend is seen , which indicates that the error introduced by the cte effect is not of concern in this case . the fact that the background of our images is considerable ( about 70 @xmath39 ) can be the reason for the cte effect not being noticeable . we adopt the l93 calibration because it was based on observations of a large number of standards from graham ( 1981 ) and landolt ( 1983 ) and because there was very good agreement between independent calibrations performed on two different observing runs and between calibrations on four nights of one of the runs . in addition , the holtzmann et al . ( 1995 ) zero points were derived for data taken with the wide field camera ccd s operating at a lower temperature compared to the present data set . in figure [ 4cmd ] we present four @xmath12 $ ] cmds for leo i based on the four wfpc2 chips . leo i possesses a rather steep and blue rgb , indicative of a low metallicity . given this low metallicity , its very well - defined rc , at @xmath40 21.5 , is characteristic of an intermediate - age stellar population . the main sequence ( ms ) , reaching up to within 1 mag in brightness of the rc , unambiguously shows that a considerable number of stars with ages between @xmath8 1 gyr and 5 gyr are present in the galaxy , confirming the suggestion by l93 that the faintest stars in their photometry might be from a relatively young ( @xmath8 3 gyr ) intermediate - age population . our cmd , extending about 2 magnitudes deeper than the l93 photometry and reaching the position expected for the turnoffs of an old population , shows that a rather broad range in ages is present in leo i. a number of yellow stars , slightly brighter and bluer than the rc , are probably evolved counterparts of the brightest stars in the ms . finally , the lack of discontinuities in the turnoffs / subgiant region indicate a continuous star formation activity ( with possible changes of the star formation rate intensity ) during the galaxy s lifetime . we describe each of these features in more detail in section [ compaiso ] , and discuss their characteristics by comparing them with theoretical isochrones and taking into account the errors discussed in section [ photerrors ] . we will quantitatively study the sfh of leo i in paper ii by comparing the distribution of stars in the observed cmd with a set of model cmds computed using the stellar evolutionary theory as well as a realistic simulation of the observational effects in the photometry ( see gallart et al . 1996b , c and aparicio , gallart & bertelli 1997 a , b for different applications of this method to the study of the sfh in several lg dwarf irregular galaxies ) . before proceeding with an interpretation of the features present in the cmd , it is important to assess the photometric errors . to investigate the total errors present in the photometry , artificial star tests have been performed in a similar way as described in aparicio & gallart ( 1994 ) and gallart , aparicio & vlchez ( 1996a ) . for details on the tests run for the leo i data , see paper ii . in short , a large number of artificial stars of known magnitudes and colors were injected into the original frames , and the photometry was redone again following exactly the same procedure used to obtain the photometry for the original frames . the injected and recovered magnitudes of the artificial stars , together with the information of the artificial stars that have been lost , provides us with the true total errors . in figure [ errors ] , artificial stars representing a number of small intervals of magnitude and color have been superimposed as white spots on the observed cmd of leo i. enlarged symbols ( @xmath16 , @xmath41 , @xmath42 ) show the recovered magnitudes for the same artificial stars . the spread in magnitude and color shows the error interval in each of the selected positions . this information will help us in the interpretation of the different features present in the cmd ( section [ compaiso ] ) . a more quantitative description of these errors and a discussion of the characteristics of the error distribution will be presented in appendix a of paper ii . we will adopt , here and in paper ii the distance obtained by l93 @xmath43 from the position of the tip of the rgb . since the ground based observations of l93 cover a larger area than the _ hst _ observations presented in this paper , and therefore sample the tip of the rgb better , they are more suitable to derive the position of the tip . on another hand , since we derive the callibration of our photometry from theirs , we do nt expect any difference in the position of the tip in our data . the adopted distance provides a good agreement between the position of the different features in the cmd and the corresponding theoretical position ( figures [ leoi_isopa ] and [ leoi_isoya ] ) , and its uncertainty does not affect the ( mostly qualitative ) conclusions of this paper . in figure [ leoi_isopa ] , isochrones of 16 gyr ( @xmath44 ) , 3 and 1 gyr ( @xmath45 ) from the padova library ( bertelli et al . 1994 ) have been superimposed upon the global cmd of leo i. in figure [ leoi_isoya ] , isochrones of the same ages and metallicities from the yale library ( demarque et al . 1996 ) are shown . in both cases ( except for the padova 1 gyr old , z=0.001 isochrone ) , only the evolution through the rgb tip has been displayed ( these are the only phases available in the yale isochrones ) . in figure [ leoi_isoclump ] , the hb agb phase for 16 gyr ( @xmath44 ) and the full isochrones for 1 gyr , 600 and 400 myr ( @xmath45 ) from the padova library are shown . a comparison of the yale and padova isochrones in figures [ leoi_isopa ] and [ leoi_isoya ] shows some differences between them , particularly regarding the shape of the rgb ( the rgb of the padova isochrones are in general _ steeper _ and redder , at the base of the rgb , and bluer near the tip of the rgb , than the yale isochrones for the same age and z ) and the position of the subgiant branches of age @xmath46gyr ( which is brighter in the padova isochrones ) . in spite of these differences , the general characteristics deduced for the stellar populations of leo i do not critically depend on the set chosen . however , based on these comparisons , we can gain some insight into current discrepancies between two sets of evolutionary models widely used , and therefore into the main uncertainties of stellar evolution theory that we will need to take into account when analyzing the observations using synthetic cmds ( paper ii ) . in the following , we will discuss the main features of the leo i cmd using the isochrones in figures [ leoi_isopa ] to [ leoi_isoz ] . this will allow us to reach a qualitative understanding of the stellar populations of leo i , as a starting point of the more quantitative approach presented in paper ii . the broad range in magnitude in the ms turnoff region of leo i cmd is a clear indication of a large range in the age of the stars populating leo i. the fainter envelope of the subgiants coincides well with the position expected for a @xmath8 1015 gyr old population whereas the brightest blue stars on the main sequence ( ms ) may be as young as 1 gyr old , and possibly younger . figure [ leoi_isoclump ] shows that the blue stars brighter than the 1 gyr isochrone are well matched by the ms turnoffs of stars a few hundred myr old . one may argue that a number of these stars may be affected by observational errors that , as we see in figure [ errors ] , tend to make stars brighter . they could also be unresolved binaries comprised of two blue stars . nevertheless , it is very unlikely that the brightest blue stars are stars @xmath8 1 gyr old affected by one of these situations , since one has to take into account that : a ) a 1 gyr old binary could be only as bright as @xmath47 in the extreme case of two identical stars , and b ) none of the blue artificial stars at @xmath48 ( which are around 1 gyr old ) got shifted the necessary amount to account for the stars at @xmath49 and only about 4% of them have been shifted a maximum of 0.5 mag . we conclude , therefore , that some star formation has likely been going on in the galaxy from 1 gyr to a few hundreds myr ago . the presence of the bright yellow stars ( see subsection [ yellow ] below ) , also supports this conclusion . concerning the age of the older population of leo i , the present analysis of the data using isochrones alone does not allow us to be much more precise than the range given above ( 1015 gyr ) , although we favour the hypothesis that there may be stars older than 10 gyr in leo i. in the old age range , the isochrones are very close to one another in the cmd and therefore the age resolution is not high . in addition , at the corresponding magnitude , the observational errors are quite large . nevertheless , the characteristics of the errors as shown in figure [ errors ] make it unlikely that the faintest stars in the turnoff region are put there due to large errors because i ) a significant migration to fainter magnitudes of the stars in the @xmath810 gyr turnoff area is not expected and , ii ) because of the approximate symmetric error distribution , errors affecting intermediate - age stars in their turnoff region are not likely to produce the well defined shape consistent with a 16 gyr isochrone ( see figure [ leoi_isopa ] ) . finally , the fact that there are not obvious discontinuities in the turnoff / subgiant region suggests that the star formation in leo i has proceeded in a more or less continuous way , with possible changes in intensity but no big time gaps between successive bursts , through the life of the galaxy . these possible changes will be quantified , using synthetic cmds , in paper ii . core he - burning stars produce two different features in the cmd , the hb and the rc , depending on age and metallicity . very old , very low metallicity stars distribute along the hb during the core he - burning stage . the rc is produced when the core he - burners are not so old , or more metal - rich , or both , although other factors may also play a role ( see lee 1993 ) . rc area in leo i differs from those of the other dsph galaxies in the following two important ways . first , the lack of a conspicuous hb may indicate , given the low metallicity of the stars in the galaxy , that leo i has only a small fraction of very old stars . there are a number of stars at @xmath50 , @xmath51 that could be stars on the hb of an old , metal poor population , but their position is also that of the post turn - off @xmath8 1 gyr old stars ( see figure [ leoi_isoclump ] ) . the relatively large number of these stars and the discontinuity that can be appreciated between them and the rest of the stars in the herszprung - gap supports the hypothesis that hb stars may make a contribution . this possible contribution will be quantified in paper ii . second , the leo i rc is very densely populated and is much more extended in luminosity than the rc of single - age populations , with a width of as much as @xmath52 1 mag . the intermediate - age lmc populous clusters with a well populated rc ( see e.g. bomans , vallenari & de boer 1995 ) have @xmath53 values about a factor of two smaller . the rcs of the other dsph galaxies with an intermediate - age population ( fornax : stetson et al . 1998 ; carina : hurley keller , mateo & nemec 1998 ) are also much less extended in luminosity . the leo i rc is more like that observed in the cmds of the general field of the lmc ( vallenari et al . 1996 ; zaritzky , harris & thompson 1997 ) . a rc extended in luminosity is indicative of an extended sfh with a large intermediate age component . the older stars in the core he burning phase lie in the lower part of the observed rc , younger rc stars are brighter ( bertelli et al . 1994 , their figure 12 ; see also caputo , castellani & deglinnocenti 1995 ) . the brightest rc stars may be @xmath8 1 gyr old stars ( which start the core he - burning phase in non - degenerate conditions ) in their blue loop phase . the stars scattered above the rc , ( as well as the brightest yellow stars , see subsection [ yellow ] ) , could be a few hundred myr old in the same evolutionary phase ( see figure 1 in aparicio et al . 1996 ; gallart 1998 ) . the rc morphology depends on the fraction of stars of different ages , and will complement the quantitative information about the sfh from the distribution of sub - giant and ms stars ( paper ii ) . there are a number of bright , yellow stars in the cmd ( at @xmath54 mag and @xmath55 mag ) . l93 indicate that a significant fraction of these stars show signs of variability , and two of the stars in their sample were identified by hodge & wright ( 1978 ) to be anomalous cepheids ) estimated for them implies that they should be relatively young stars , or mass transfer binaries . since the young age hypothesis appeared incompatible with the idea of dsph galaxies being basically population ii systems , it was suggested that anomalous cepheids could be products of mass - transfer binary systems . nevertheless , we know today that most dsph galaxies have a substantial amount of intermediate - age population , consistent with anomalous cepheids being relatively young stars that , according to various authors ( gingold 1976 , 1985 ; hirshfeld 1980 ; bono et al . 1997 ) , after undergoing the he - flash , would evolve towards high enough effective temperatures to cross the instability strip before ascending the agb . ] . some of them also show signs of variability in our _ hst _ data . in figure [ leoi_isoclump ] however , it is shown that these stars have the magnitudes and colors expected for blue loop stars of few hundred myr . this supports our previous conclusion that the brightest stars in the ms have ages similar to these . given their position in the cmd , it is interesting to ask whether some of the variables found by hodge & wright ( 1978 ) in leo i could be classical cepheids instead of anomalous cepheids . from the bertelli et al . ( 1994 ) isochrones , we can obtain the mass and luminosity of a 500 myr blue - loop star , which would be a representative star in this position of the cmd . such a star would have a mass , @xmath56 , and a luminosity , l@xmath8 350 l@xmath57 . from eq . 8 of chiosi et al . ( 1992 ) we calculate that the period that corresponds to a classical cepheid of this mass and metallicity is 1.2 days , which is compatible with the periods found by hodge & wright ( 1978 ) , that range between 0.8 and 2.4 days . we suggest that some of these variable stars may be similar to the short period cepheids in the smc ( smith et al . 1992 ) , i.e. classical cepheids in the lower extreme of mass , luminosity and period . if this is confirmed , it would be of considerable interest in terms of understanding the relationship between the different types of cepheid variables . a new wide field survey for variable stars , more accurate and extended to a fainter magnitude limit ( both to search for cepheids and rr lyrae stars ) would be of particular interest in the case of leo i. the rgb of leo i is relatively blue , characteristic of a system with low metallicity . assuming that the stars are predominantly old , with a small dispersion in age , l93 obtained a mean metallicity [ fe / h]=2.02 @xmath5 0.10 dex and a metallicity dispersion of @xmath58 } < -1.8 $ ] dex . this estimate was based on the color and intrinsic dispersion in color of the rgb at @xmath59 using a calibration based on the rgb colors of galactic globular clusters ( da costa & armandroff 1990 ; lee , freedman & madore 1993b ) . for a younger mean age of about 3.5 gyr , they estimate a slightly higher metallicity of [ fe / h]=1.9 , based on the difference in color between a 15 and a 3.5 gyr old population according to the revised yale isochrones ( green et al . other photometric measurements give a range in metallicity of [ fe / h]= 1.85 to 1.0 dex ( see l93 and references therein ) . the metallicity derived from moderate resolution spectra of two giant stars by suntzeff ( 1992 , unpublished ) is [ fe / h]@xmath60 dex . since leo i is clearly a highly composite stellar population with a large spread in age , the contribution to the width of the rgb from such an age range may no longer be negligible compared with the dispersion in metallicity . therefore , an independent estimate of the age range from the ms turnoffs is relevant in the determination of the range in metallicity . in the following , we will discuss possible limits on the metallicity dispersion of leo i through the comparison of the rgb with the isochrones shown in figures [ leoi_isopa ] through [ leoi_isoz ] . as we noted in the introduction of section [ cmd ] , there are some differences between the padova and the yale isochrones , but their positions coincide in the zone about 1 magnitude above the rc . we will use only this position in the comparisons discussed below . we will first check whether the whole width of the rgb can be accounted for by the dispersion in age . in subsection [ ms ] above , we have shown that the ages of the stars in leo i range from 1015 gyr to less than 1 gyr . in figure [ leoi_isoz ] we have superimposed padova isochrones of z=0.0004 and ages 10 , 1 and 0.5 gyr on the leo i cmd . this shows that the full width of the rgb above the rc can be accounted for by the dispersion in age alone . a similar result is obtained for a metallicity slightly lower or higher . this provides a lower limit for the metallicity range , which could be negligible . the agb of the 0.5 gyr isochrone appears to be too blue compared with the stars in the corresponding area of the cmd . however , these agbs are expected to be poorly populated because a ) stars are short lived in this phase and b ) the fraction of stars younger than 1 gyr is small , if any . second , we will discuss the possible range in z at different ages from a ) the position of the rgb , taking into account the fact that isochrones of the same age are redder when they are more metal rich and isochrones of the same metallicity are redder when they are older and b ) that the extension of the blue - loops depends on metallicity : \a ) for stars of a given age , the lower limit of z is given by the blue edge of the rgb area we are considering : isochrones of any age and z=0.0001 have colors in the rgb above the rc within the observed range . therefore , by means of the present comparison only , we can not rule out the possibility that there may be stars in the galaxy with a range of ages and z as low as z=0.0001 . the oldest stars of this metallicity would be at the blue edge of the rgb , and would be redder as they are younger . the upper limit for the metallicity of stars of a given age is given by the red edge of the rgb : for old stars , the red edge of the observed rgb implies an upper limit of z @xmath7 0.0004 ( see figure [ leoi_isoz ] ) , since more metal rich stars would have colors redder than observed . for intermediate - age stars up to @xmath8 3 gyr old we infer an upper limit of z=0.001 , and for ages @xmath8 3 - 1 gyr old an upper limit of z=0.004 . \b ) we can use the position of the bright yellow stars to constrain z : the fact that there are a few stars in blueward extended blue loops implies that their metallicity is as low as [email protected] or even lower ( figure [ leoi_isoclump ] ) , because higher metallicity stars do nt produce blueward extended blue - loops at the observed magnitude . this does not exclude the possibility that a fraction of young stars have metallicity up to z=0.004 . these upper limits are compatible with z slowly increasing with time from z@xmath8 0 to [email protected] , on the scale of the padova isochrones . in summary , we conclude that the width of the leo i rgb can be accounted for the dispersion of the age of its stellar population and , therefore , the metallicity dispersion could be negligible . alternatively , considering the variation in color of the isochrones depending on both age and metallicity , we set a maximum range of metallicity of @xmath620.0010.004 : a lower limit of z=0.0001 is valid for any age , and the upper limit varies from z=0.0004 to z=0.004 , increasing with time . these upper limits are quite broad ; they will be better constrained , and some information on the chemical enrichment law gained , from the analysis of the cmd using synthetic cmds in paper ii . from the new _ hst _ data and the analysis presented in this paper , we conclude the following about the stellar populations of leo i : \1 ) the broad ms turnoff / subgiant region and the wide range in luminosity of the rc show that star formation in leo i has extended from at least @xmath8 1015 gyr ago to less than 1 gyr ago . a lack of obvious discontinuities in the ms turnoff / subgiant region suggests that star formation proceeded in a more or less continuous way in the central part of the galaxy , with possible intensity variations over time , but no big time gaps between successive bursts , through the life of the galaxy . \2 ) a conspicuous hb is not seen in the cmd . given the low metallicity of the galaxy , this reasonably implies that the fraction of stars older than @xmath8 10 gyr is small , and indicates that the beginning of a substantial amount of star formation may have been delayed in leo i in comparison to the other dsph galaxies . it is unclear from the analysis presented in this paper whether leo i contains any stars as old as the milky way globular clusters . \3 ) there are a number of bright , yellow stars in the same area of the cmd where anomalous cepheids have been found in leo i. these stars also have the color and magnitude expected for the blue - loops of low metallicity , few hundred myr old stars . we argue that some of these stars may be classical cepheids in the lower extreme of mass , luminosity and period . 4)the evidence that the stars in leo i have a range in age complicates the determination of limits to the metallicity range based on the width of the rgb . in one extreme , if the width of the leo i rgb is atributted to the dispersion of the age of its stellar population alone , the metallicity dispersion could be negligible . alternatively , considering the variation in color of the isochrones depending on both age and metallicity , we set a maximum range of metallicity of @xmath620.0010.004 : a lower limit of z=0.0001 is valid for any age , and the ( broad ) upper limit varies from z=0.0004 to z=0.004 , increasing with time . in summary , leo i has unique characteristics among local group galaxies . due to its morphology and its lack of detectable quantities of hi , it can be classified as a dsph galaxy . but it appears to have the youngest stellar population among them , both because it is the only dsph lacking a conspicuous old population , and because it seems to have a larger fraction of intermediate - age and young population than other dsph . the star formation seems to have proceeded until almost the present time , without evidence of intense , distinct bursts of star formation . important questions about leo i still remain . an analysis of the data using synthetic cmds will give quantitative information about the strength of the star formation at different epochs . further observations are needed to characterize the variable - star population in leo i , and in particular , to search for rr lyrae variable stars . this will address the issue of the existence or not of a very old stellar population in leo i. it would be interesting to check for variations of the star formation across the galaxy and to determine whether the hb is also missing in the outer parts of leo i. answering these questions is important not only to understand the formation and evolution of leo i , but also in relation to general questions about the epoch of galaxy formation and the evolution of galaxies of different morphological types . the determination of the strength of the star formation in leo i at different epochs is important to assess whether it is possible that during intervals of high star formation activity , leo i would have been as bright as the faint blue galaxies observed at intermediate redshift . in addition , the duration of such a major event of star formation may be important in explaining the number counts of faint blue galaxies . we want to thank allan sandage for many very useful discussions and a careful reading of the manuscript . we thank also nancy b. silbermann , shoko sakai and rebecca bernstein for their help through the various stages of the _ hst _ data reduction . support for this work was provided by nasa grant go-5350 - 03 - 93a from the space telescope science institute , which is operated by the association of universities for research in astronomy inc . under nasa contract nasa526555 . c.g . also acknowledges financial support from a small research grant from nasa administered by the aas and a theodore dunham jr . grant for research in astronomy . thanks the carnegie observatories for their hospitality . is supported by the ministry of education and culture of the kingdom of spain , by the university of la laguna and by the iac ( grant pb3/94 ) . m.g.l is supported by the academic research fund of ministry of education , republic of korea , bsri-97 - 5411 . the digitized sky surveys were produced at the space telescope science institute under u.s . government grant nag w-2166 . the images of these surveys are based on photographic data obtained using the oschin schmidt telescope on palomar mountain and the uk schmidt telescope . smecker - hane , t. a. , stetson , p. b. , hesser , j. e. & van den bergh , d. a. 1996 in _ from stars to galaxies : the impact of stellar physics on galaxy evolution _ ( ed . c. leitherer , u. fritze - van alvensleben & j. huchra ) . asp conf series , vol 98 , p. 328 .

we present deep @xmath0 f555w ( @xmath1 ) and f814w ( @xmath2 ) observations of a central field in the local group dwarf spheroidal ( dsph ) galaxy leo i. the resulting color - magnitude diagram ( cmd ) reaches @xmath3 and reveals the oldest @xmath4 gyr old turnoffs . nevertheless , a horizontal branch is not obvious in the cmd . given the low metallicity of the galaxy , this likely indicates that the first substantial star formation in the galaxy may have been somehow delayed in leo i in comparison with the other dsph satellites of the milky way . the subgiant region is well and uniformly populated from the oldest turnoffs up to the 1 gyr old turnoff , indicating that star formation has proceeded in a continuous way , with possible variations in intensity but no big gaps between successive bursts , over the galaxy s lifetime . the structure of the red - clump of core he - burning stars is consistent with the large amount of intermediate age population inferred from the main sequence and the subgiant region . in spite of the lack of gas in leo i , the cmd clearly shows star formation continuing until 1 gyr ago and possibly until a few hundred myrs ago in the central part of the galaxy . subject headings : galaxies : individual ( leo i ) ; galaxies : evolution ; galaxies : stellar content ; galaxies : photometry ; stars : hertzsprung - russell ( hr - diagram ) .

arxiv

this article is motivated by the following result of berestycki , et al . given in @xcite for the laplacian perturbed by a divergence - free drift in dimensions @xmath10 . let @xmath11 be a bounded @xmath12 regular open set and let @xmath13 be a bounded @xmath7-dimensional vector field such that @xmath14 on @xmath2 in the sense of distributions ( distr . ) , i.e. @xmath15 for @xmath5 , let ( @xmath16 ) be the principal eigen - pair corresponding to the dirichlet problem for the operator @xmath17 . theorem 0.3 of @xcite asserts that @xmath18 remains bounded as @xmath9 , if and only if the equation @xmath19 has a solution @xmath20 ( called a first integral of @xmath6 ) , such that @xmath21 and @xmath22 . the result can be interpreted intuitively in the following way : functions @xmath20 satisfying are constant along the flow of the vector field @xmath23 ( see section [ sec : ub ] ) , and the existence of ( non - trivial ) first integrals allows for flow lines that are contained in @xmath2 . on the other hand , if no such @xmath20 exist , then the flow leaves @xmath2 with speed proportional to @xmath24 . adding the laplacian @xmath25 to @xmath26 , or equivalently the brownian motion to the flow , results in a stochastic process whose trajectories gradually depart from the integral curves of @xmath6 , but the general picture is similar : if nontrivial first integrals exist , then the trajectories may remain in @xmath2 with positive probability during a finite time interval , even as @xmath9 . in this case we are lead to a nontrivial limiting transition mechanism between the flow lines . the result described in the foregoing enjoys many extensions and has proved quite useful in various applications describing the influence of a fluid flow on a diffusion , see for example @xcite . in the context of a compact , connected riemannian manifold a sufficient and necessary condition for @xmath18 to remain bounded , as @xmath9 , expressed in terms of the eigenspaces of the advection operator @xmath27 , has been given in ( * ? ? ? * theorem 1 ) . the purpose of the present paper is to verify that a similar property of the principal eigenvalue holds when the classical laplacian is replaced by the fractional laplacian @xmath4 with @xmath28 . we consider @xmath29 defined as the set of all the nonzero first integrals in the sobolev space @xmath30 equipped with the norm coming from the dirichlet form @xmath31 of @xmath4 ( see below ) . the sobolev norm condition on the first integrals reflects smoothing properties of the green function of the fractional laplacian , while is related to the flow defined by @xmath6 . the main difficulty in our development stems from roughness of general elements of @xmath32 and non - locality of @xmath4 , which prevent us from a direct application of the differential calculus in the way it has been done in @xcite . instead , we use conditioning suggested by a paper of bogdan and dyda @xcite , approximation techniques for flows given by diperna and lions in @xcite , and the properties of the green function and heat kernel of gradient perturbations of @xmath4 obtained by bogdan , jakubowski in @xcite and chen , et al . in @xcite for @xmath3 and bounded @xmath1-regular open sets @xmath2 . these properties allow to define and study , via the classical krein - rutman theorem and compactness arguments , the principal eigen - pair @xmath33 for @xmath34 and @xmath3 . our main result can be stated as follows . [ main - thm ] suppose that @xmath35 is a bounded domain with @xmath1-regular boundary that is simply connected , i.e. @xmath36 - the complement of @xmath37 - is connected . if @xmath28 , and @xmath38 is of zero divergence , then @xmath39 and the infimum is attained . here we use the convention that @xmath40 , hence @xmath41 if and only if the zero function is the only first integral . equality results from the following lower and upper bounds of @xmath18 , @xmath42 @xmath43 the bounds are proved in sections [ sec : lb ] and [ sec : ub ] , correspondingly . in section [ proof - coro ] we explain that the minimum on the right hand side of is attained , and we finish the proof of the theorem . comparing our approach with the arguments used in the case of local operators , cf . @xcite , we note that the use of the green function seems more robust whenever we lack sufficient differentiability of functions appearing in variational formulas . recall that in the present case we need to deal with @xmath30 , which limits the applicability of the arguments based on the usual differentiation rules of the classical calculus , e.g. the leibnitz formula or the chain rule . we consider the use of the green function as one of the major features of our approach . in addition , the non - locality of the quadratic forms forces a substantial modifications of several other arguments , e.g. those involving conditioning of nonlocal operators and quadratic forms in the proof of the upper bound in section [ sec : ub ] . finally , we stress the fact that the dirichlet fractional laplacian on a bounded domain @xmath2 _ is not _ a fractional power of the dirichlet laplacian on @xmath2 , e.g. the eigenfunctions of these operators have a different power - type decay at the boundary , see @xcite in this connection . as a preparation for the proof , we recall in section [ sec : p ] the estimates of @xcite for the green function and transition density of @xmath44 for the dirichlet problem on @xmath2 . these functions are defined using hunt s formula , which in principle requires the drift @xmath45 to be defined on the entire @xmath46 . we show however , in corollary [ cor010212 ] , that they are determined by the restriction of the drift to the domain @xmath2 . in section [ sec3 ] we prove that the corresponding green s and transition operators are compact , see lemmas [ lem : gdc1 ] and [ lem : gdc1 kb ] . this result is used to define the principal eigen - pair of @xmath44 , via the krein - rutman theorem . in theorem [ thm011307 ] of section [ sec4 ] we prove that the domains of @xmath4 and @xmath44 in @xmath47 coincide . in section [ sec : pt11 ] we employ the bilinear form of @xmath44 to estimate the principal eigenvalue . the technical assumption @xmath48 is only needed in sections [ sec : ub ] to characterize the first integrals of @xmath6 by means of the theory of flows developed by diperna and lions in @xcite for sobolev - regular vector fields . we start with a brief description of the setting and recapitulation of some of the results of @xcite . further details and references may be found in those papers ( see also @xcite and the references therein ) . in what follows , @xmath49 is the euclidean space of dimension @xmath50 , scalar product @xmath51 , norm @xmath52 and lebesgue measure @xmath53 . all sets , measures and functions in @xmath49 considered throughout this paper will be borel . we denote by @xmath54 the ball of center @xmath55 and radius @xmath56 . we will consider nonempty , bounded open set @xmath35 , whose boundary is of class @xmath1 . the latter means that @xmath56 exists such that for every @xmath57 there are balls @xmath58 and @xmath59 , which are tangent at @xmath60 ( the _ inner _ and _ outer _ tangent ball , respectively ) . we will refer to such sets @xmath2 as to @xmath1 domains , without requiring connectivity . for an alternative analytic description and localization of @xmath1 domains we refer to ( * ? ? ? * lemma 1 ) . we note that each connected component of @xmath2 contains a ball of radius @xmath61 , and the same is true for @xmath62 . therefore @xmath2 and @xmath62 have a finite number of components , which will play a role in a later discussion of extensions of the vector field to a neighborhood of @xmath63 . the distance of a given @xmath55 to @xmath62 will be denoted by @xmath64 _ constants _ mean positive numbers , that do not depend on the considered arguments of the functions being compared . accordingly , notation @xmath65 means that there is a constant @xmath66 such that @xmath67 for all @xmath68 . as usual , @xmath69 and @xmath70 . we will employ the function space @xmath47 , consisting of all square integrable real valued functions , with the usual scalar product @xmath71 generally , given @xmath72 , the norms in @xmath73 shall be denoted by @xmath74 . customarily , @xmath75 denotes the space of smooth functions on @xmath49 with compact support in @xmath2 . also , @xmath76 denotes the closure of @xmath75 in the norm @xmath77 , where @xmath78 in the last equality we have used plancherel theorem . the fourier transform of @xmath79 is given by @xmath80 the discussion in section [ sec : iso ] and section [ sec : kill ] is valid for any @xmath82 . let @xmath83 the coefficient is chosen in such a way that @xmath84\nu(y)dy=|\xi|^\alpha\,,\quad \xi\in \rd\,.\ ] ] we define the _ fractional laplacian _ as the @xmath85-closure of the operator @xmath86\nu(y)dy\ , , \quad x\in \rd,\,\phi\in c^\infty_c(\rd).\ ] ] its fourier symbol is given by @xmath87 , cf . the fractional laplacian is the generator of the semigroup of the isotropic @xmath81-stable lvy process @xmath88 on @xmath49 . here @xmath89 and @xmath90 are the law and expectation for the process starting at @xmath91 . these are defined on the borel @xmath92-algebra of the canonical cdlg path space @xmath93 via transition probability densities as follows . we let @xmath94 be the canonical process , i.e. @xmath95 and define time - homogeneous transition density @xmath96 where @xmath97 according to ( [ eq : trf ] ) and the lvy - khinchine formula , @xmath98 is a probabilistic convolution semigroup of functions with the lvy measure @xmath99 , see e.g. @xcite . from ( [ eq : dpt ] ) we have @xmath100 it is well - known ( @xcite ) that @xmath101 , hence @xmath102 let @xmath103 be the _ time of the first exit _ of the ( canonical ) process from @xmath2 . for each @xmath104 , the measure @xmath105\ ] ] is absolutely continuous with respect to the lebesgue measure on @xmath2 . its density @xmath106 is continuous in @xmath107 and satisfies g. hunt s formula ( see @xcite , @xcite ) , @xmath108.\ ] ] in addition , the kernel is symmetric ( see @xcite for discussion and references ) : @xmath109 and defines a strongly continuous semigroup on @xmath47 , @xmath110 we shall denote @xmath111 . the _ green function _ of @xmath4 for @xmath2 is defined as @xmath112 and the respective _ green operator _ is @xmath113 = \int_{\rd}g_d(x , y)f(y)dy.\end{aligned}\ ] ] we have ( @xcite ) , @xmath114 and @xmath115 see ( * ? ? ? * lemma 5.3 ) , where the fractional laplacian operator appearing above is defined in the sense of distributions theory . see formula below for another statement . thus the generator of @xmath116 is @xmath117 with zero ( dirichlet ) exterior conditions , i.e. with the domain equal to the range of @xmath118 , @xmath119 the following estimate has been proved by kulczycki @xcite and chen and song @xcite ( see also ( * ? ? ? * theorem 21 ) ) , @xmath120^\alpha}\label{greenestimatestj}\\ & & \approx\ ; |x - y|^{\alpha - d}\left(\frac{\delta_d(x)^{\alpha/2}\delta_d(y)^{\alpha/2}}{|x - y|^\alpha } \land 1\right)\nonumber \ , , \qquad x , y \in d\,.\nonumber\end{aligned}\ ] ] in particular , @xmath121 from it also follows that @xmath122 from ( * ? ? ? * corollary 3.3 ) we have the following gradient estimate , @xmath123 we define @xmath124 as the subspace of @xmath125 made of those elements for which @xmath126 is given as follows ( cf @xcite ) : @xmath127 ^ 2\nu(y - x)dxdy,\label{011109b}\end{aligned}\ ] ] the above formula can be also used to define a bilinear form @xmath128 on @xmath129 by polarization . we also have @xmath130 therefore @xmath131 we define @xmath32 as the closure of @xmath75 in the norm @xmath132 . by theorems 4.4.2 , a.2.10 and formula ( 4.3.1 ) of @xcite , we have ( cf . ) ) , @xmath133 if @xmath134 and @xmath135 belongs to the domain of the fractional laplacian , then @xmath136 throughout the remainder of the paper we always assume that @xmath137 , and @xmath138 is a bounded vector field . for @xmath139 and @xmath140 we let @xmath141 and for each @xmath142 @xmath143 let @xmath144 it follows from ( * ? ? ? * theorem 2 and example 2 ) that series converges uniformly on compact subsets of @xmath145 . from the results of @xcite , we know that @xmath146 is a transition probability density function , i.e. it is non - negative and @xmath147 in addition , @xmath148 is continuous on @xmath145 , and @xmath149 where @xmath150 if @xmath151 . in fact , this holds under much weaker , kato - type condition on @xmath6 , see ( * ? ? ? * theorems 1 and 2 ) . we denote by @xmath152 and @xmath153 the law and expectation on @xmath154 for the ( canonical ) markov process starting at @xmath68 and defined by the transition probability density @xmath155 , @xmath152 may also be defined by solving stochastic differential equation @xmath156 . such equations have been studied in dimension @xmath157 in @xcite under the assumptions of boundedness and continuity of the vector field ; also for @xmath158 . we refer the reader to ( * ? ? ? * formula ( 13 ) ) , for a closer description of a connection to and . we may also define the perturbation series for @xmath159 . in what follows , objects pertaining to @xmath160 will be marked with the superscript hash ( @xmath161 ) , e.g. @xmath162 if @xmath163 on @xmath2 in the sense of distributions theory , and @xmath164 , then @xmath165 if also @xmath166 and we substitute @xmath167 for @xmath79 in , then we obtain @xmath168 the last equality extends to arbitrary @xmath169 . [ lem : duality ] if @xmath163 on @xmath49 , then we have @xmath170 and @xmath171 for all @xmath139 , @xmath172 , and @xmath173 . let @xmath174 , @xmath172 . by ( see also ( * ? ? ? * lemma 4 ) ) , @xmath175 from we conclude that @xmath176 . for a general @xmath177 , by we have @xmath178 where @xmath179 , @xmath180 , @xmath181,\ ] ] and @xmath182,\ ] ] with the convention that @xmath183 , cf . . using formula @xmath184 times in space , and then integrating in time we see that @xmath185 which yields the identities stated in the lemma . a strengthening of proposition [ lem : duality ] will be given in corollary [ cor010908 ] below . if @xmath163 on @xmath49 , then and proposition [ lem : duality ] yield @xmath186 we recall that @xmath2 is a bounded @xmath1 domain in @xmath49 . hunt s formula may be used to define the transition probability density of the ( first ) perturbed and ( then ) killed process @xcite . thus , for @xmath139 , @xmath187 we let @xmath188\,.\ ] ] we have @xmath189=\int_d \tp_d(t , x , y)f(y)dy,\quad t>0,\,x\in d,\,f\in l^\infty(d),\ ] ] and @xmath190 where @xmath191 } c_t<+\infty$ ] for any @xmath192 . by ( * ? * formula ( 40 ) ) , there exist constants @xmath193 such that @xmath194 we define the green function of @xmath2 for @xmath195 : @xmath196 clearly @xmath197 is nonnegative . by blumenthal s 0 - 1 law , @xmath198 for all @xmath139 ( and thus @xmath199 ) if @xmath200 or @xmath201 , see ( [ ptxy_comp ] ) . it follows from and ( see ( * ? ? ? * lemma 7 ) ) , that @xmath202 the main result of @xcite asserts that @xmath203 and @xmath204 is continuous for @xmath205 ( for estimates of @xmath206 see @xcite ) . we consider the integral operators @xmath207 and @xmath208 in light of and , the above operators are bounded on every @xmath73 , @xmath209 . the analogous operators @xmath210 and @xmath211 , defined for @xmath159 , turn out to be mutually adjoint on @xmath47 , as follows from corollary [ cor010908 ] below . from , and we obtain that @xmath212 is a semigroup of contractions in every @xmath73 , @xmath213 $ ] . it is strongly continuous for @xmath209 . let @xmath195 be the @xmath47 generator of the semigroup , with the domain @xmath214 . we have @xmath215 it has been shown in @xcite that the following crucial recursive formula holds @xmath216 and for all @xmath217 and @xmath218 we have @xmath219 later on we shall also consider the operator @xmath220 corresponding to the vector field @xmath221 , where @xmath5 and we let @xmath222 . clearly , if @xmath24 is fixed , then there is no loss of generality to focus on @xmath223 . [ sec4 ] the following pointwise version of is proved in ( * ? ? ? * lemma 12 ) , @xmath224 with @xmath225 we define @xmath226 after a series of auxiliary estimates , we will prove @xmath227 to be compact on @xmath47 . [ lem011107 ] there exist @xmath228 such that @xmath229 using ( [ eq : gradestimgreen ] ) we obtain that @xmath230}{\delta_d(x)|x - y|},\ ] ] which we bound from above , thanks to ( [ greenestimatestj ] ) , by @xmath231\delta_d(y)^{\alpha/2}}{[\delta_d(x)\vee |x - y| \vee \delta_d(y)]^{\alpha}}\ , \nonumber\\ & \le c_1\,\delta_d(x)^{\alpha/2 - 1}|x - y|^{\alpha-1-d } \dfrac{\delta_d(y)^{\alpha/2}}{[\delta_d(x)\vee |x - y| \vee \delta_d(y)]^{\alpha-1}}\ , , \end{aligned}\ ] ] and this yields . [ lem051307 ] if @xmath232 , then there is @xmath233 such that @xmath234 we only need to examine points close to @xmath235 . given such a point we consider the integral over its neighborhood @xmath236 . the neighborhood can be chosen in such a way that , after a bi - lipschitz change of variables ( see ( * ? ? ? * formula ( 75 ) ) , ( * ? ? ? * formula ( 11 ) ) ) , we can reduce our consideration to the case when @xmath237 $ ] , @xmath238 $ ] and @xmath239 for @xmath240 . the respective integral over @xmath236 is then estimated by @xmath241 the result is valid for all bounded open _ lipschitz _ sets ( @xcite ) in all dimensions @xmath242 . lemma [ lem051307 ] yields the following . [ cordk ] let @xmath243 and @xmath244 . suppose also that @xmath245 denote @xmath246 then , @xmath247 constants @xmath248 depend only on @xmath249 the above result shall be used to establish that operator @xmath250 ( thus also @xmath227 ) is @xmath251 bounded via schur s test , see ( * ? ? ? * theorem 5.2 ) . note that we have @xmath252 , for some constant @xmath253 , provided that @xmath254 . likewise , @xmath255 , provided that @xmath256 . summarizing , we will require the following conditions : @xmath257 [ lemary ] conditions ( [ c1])-([c3 ] ) hold , if @xmath258 and @xmath259 where @xmath260 . we have @xmath261 , etc . [ prop021307 ] operator @xmath227 is compact on @xmath47 . we note that @xmath262 , where @xmath263 is given by and @xmath253 is some constant . let @xmath264 be as in lemma [ lemary ] . by corollary [ cordk ] and the discussion preceding lemma [ lemary ] , there are constants @xmath265 such that @xmath266 using schur s test for boundedness of integral operators on @xmath251 spaces ( see ( * ? ? ? * theorem 5.2 ) ) we obtain @xmath267 for @xmath268 , @xmath269 we let @xmath270 , @xmath271 , and @xmath272 each operator @xmath273 is hilbert - schmidt , hence compact ( ( * ? ? ? * theorem 4 , p. 247 ) ) , due to the fact that @xmath274 we have @xmath275 , unless @xmath276 . the latter may only happen if @xmath277 satisfy @xmath278 , with @xmath228 ( cf ) , or equivalently when @xmath279 , and then there are constants @xmath280 such that @xmath281 this estimate for @xmath282 actually holds for all @xmath283 , hence yields @xmath284 . since @xmath227 is a norm limit of compact operators , it is compact , too . in view of proposition[prop021307 ] we may regard the gradient operator @xmath26 as a small perturbation of @xmath4 when @xmath3 . in fact @xmath26 is relatively compact with respect to @xmath4 with dirichlet conditions in the sense of ( * ? ? ? * iv.1.3 ) . [ thm011307 ] @xmath285 . by virtue of , we can write @xmath286 by , @xmath118 is injective , therefore so is @xmath287 . in particular , @xmath157 is not an eigenvalue of @xmath227 . since @xmath227 is compact , by the riesz - schauder theory ( * ? ? ? * theorem x.5.1 , p. 283 ) , @xmath287 is invertible . thus , @xmath288 for future reference we remark that @xmath289 is invertible , too . [ cor011307 ] @xmath290 . by ( * lemma 10 ) , @xmath291 for any bounded function @xmath135 . invoking the argument used in proposition [ prop021307 ] , we conclude that there is a number @xmath292 independent of @xmath135 for which @xmath293 by approximation , and extend to all @xmath294 . furthermore , if @xmath295 and @xmath296 in @xmath47 , then @xmath297 and @xmath298 on @xmath235 , hence @xmath299 ( see ( * ? ? ? * theorem 5.37 ) ) . therefore , @xmath300 . the following result justifies our notation @xmath301 . [ prop011009 ] if @xmath302 , then @xmath303 . let @xmath304 and @xmath305 . applying @xmath306 to @xmath307 and using , we obtain @xmath308 which , thanks to , concludes the proof . we shall also observe the following localization principle for our perturbation problem . [ cor010212 ] operators @xmath306 and @xmath309 do not depend on the values of @xmath6 on @xmath62 . if @xmath310 and @xmath311 are two ( bounded ) vector fields on @xmath49 equal to @xmath6 on @xmath2 , and @xmath312 and @xmath313 are the corresponding green s operators , then , by , @xmath314 which identifies @xmath315 as operators on @xmath47 , hence also as functions defined in section [ sec : deftg ] . a similar conclusion for @xmath316 follows from the fact that @xmath317 is the generator of the semigroup . we will identify the adjoint operator of @xmath227 on @xmath47 . [ lemhs ] @xmath318 . if @xmath319 , then by corollary [ cor011307 ] and , @xmath320 functions @xmath321 form a dense set in @xmath47 , which ends the proof . when defining @xmath322 , via , we may encounter the situation when @xmath6 is given only on @xmath2 . we may extend the field outside @xmath2 by letting e.g. @xmath323 on @xmath62 . of course such an extension needs not satisfy @xmath324 , even though the condition may hold on @xmath2 . however , we still have a local analogue of proposition [ lem : duality ] . recall that the transition probability densities and green function corresponding to the vector field @xmath160 are marked with a hash ( @xmath161 ) . [ cor010908 ] if @xmath325 on @xmath2 , then for all @xmath139 and @xmath283 , @xmath326 and @xmath327 we shall first prove , or , equivalently , that @xmath328 where @xmath329 is adjoint to @xmath330 on @xmath47 . from applied to @xmath331 we have , @xmath332 applying the operator @xmath118 from the left to both sides of the equality we obtain @xmath333 by lemma [ lemhs ] , @xmath334 taking adjoints of both sides of we also obtain , @xmath335 we have already noted in the proof of theorem [ thm011307 ] that @xmath289 is a linear automorphism of @xmath47 . therefore and give . furthermore , let @xmath336 denote the adjoint of @xmath195 on @xmath47 . we have @xmath337 where the last equality follows from ( * ? ? ? * lemma xii.1.6 ) . let @xmath338 be the semigroup adjoint to @xmath339 . by ( * ? * corollary 4.3.7 ) , the generator of @xmath338 is @xmath340 . since @xmath341 , the semigroups are equal . the corresponding kernels are defined pointwise , therefore they satisfy . [ lem : gdc1 ] operator @xmath342 is hilbert - schmidt on @xmath47 for each @xmath139 . let @xmath343 be an orthonormal base in @xmath47 . by and plancherel s identity , we bound the hilbert - schmidt norm as follows . using and @xmath344 [ lem : gdc1 kb ] @xmath197 is compact on @xmath47 . let @xmath268 , and @xmath345 . the integral operator @xmath346 on @xmath47 corresponding to the kernel @xmath347 is compact . indeed , it has a finite hilbert - schmidt norm : @xmath348 ^ 2dxdy\le n^2|d|^2<\infty.\ ] ] the norm of @xmath349 on @xmath47 may be directly estimated as follows , @xmath350 see , e.g. , theorem 3 , p. 176 of @xcite . we let @xmath351 . the functions @xmath352 ( and @xmath353 ) are uniformly integrable on @xmath2 by . since @xmath197 is approximated in the norm topology by compact operators , it is compact . in the special case when @xmath354 , @xmath355 equals @xmath356 , a symmetric contraction semigroup on @xmath47 , whence the green operator @xmath118 is symmetric , compact and positive definite . the spectral theorem yields the following . [ cor011206 ] @xmath357 is symmetric and compact for every @xmath358 . by and , @xmath197 is also irreducible . krein - rutman theorem ( see @xcite ) implies that there exists a unique nonnegative @xmath359 and a number @xmath360 such that @xmath361 and @xmath362 we shall call @xmath363 the _ principal eigenpair _ corresponding to @xmath195 . from we have @xmath364 and @xmath365 . furthermore , @xmath366 the formula yields extra regularity of @xmath367 , as follows . [ lem011106 ] if holds for some @xmath368 and @xmath369 , then @xmath370 and there is @xmath371 such that latexmath:[\[\label{011106 } starting from , for an arbitrary integer @xmath142 we obtain @xmath373 where @xmath374 and @xmath375 to estimate @xmath376 , we use basic properties of the bessel potentials , which can be found in ( * ? ? ? * ch.ii.4 ) . recall that for @xmath377 the bessel potential kernel @xmath378 is the unique , extended - continuous , probability density function on @xmath49 , whose fourier transform is @xmath379 thus , @xmath380 for all @xmath381 , see ( 4.7 ) of ibid . if @xmath382 , then by ( * ? ? ? * ( 4.2 ) ) , @xmath383 is locally comparable with @xmath384 . by there is a constant @xmath253 such that @xmath385 hence , @xmath386 , which is bounded if @xmath387 , see ( * ? ? ? * ( 4.2 ) ) again . considering such @xmath184 we conclude that @xmath367 is bounded . the boundary decay of @xmath367 follows from , and . the continuity of @xmath367 is a consequence of the continuity of @xmath204 for @xmath388 , and the uniform integrability of the kernel , which stems from . we say that @xmath20 is a _ first integral _ of @xmath6 if @xmath389 cf . . we write @xmath390 if @xmath391 and @xmath20 is not equal to @xmath392 a.e . recall that for @xmath5 , the operator @xmath393 is considered with the dirichlet exterior condition on @xmath2 , i.e. it acts on @xmath394 , see theorem [ thm011307 ] . the green operator and krein - rutman eigen - pair of @xmath44 shall be denoted by @xmath395 and @xmath33 , respectively . we also recall that @xmath396 and @xmath397 the proof of shall be obtained by demonstration of lower and upper bounds for @xmath18 ( as @xmath222 ) . [ prop010906 ] if @xmath398 , then @xmath399 . _ let @xmath398 . according to proposition [ prop011009 ] , @xmath79 belongs to @xmath400 and @xmath401 . in addition , @xmath402 . taking the scalar product of both sides of the equality against @xmath79 and using we get the result , because the second term vanishes . according to proposition [ prop010906 ] , @xmath403 suppose @xmath404 , as @xmath405 , but @xmath406 stay bounded . by and corollary [ cor011206 ] , the sequence @xmath407 is pre - compact in @xmath47 . suppose that @xmath20 is a weak limit of @xmath408 in @xmath30 , thus a strong limit in @xmath47 . we have @xmath409 , and for @xmath410 , @xmath411 dividing both sides by @xmath412 and passing to the limit , we obtain that @xmath413 thus @xmath414 . fatou s lemma and yield @xmath415 , therefore follows . the proof of uses `` conditioning '' of truncations of @xmath416 by the principal eigenfunction inspired by @xcite and @xcite ( see below ) . here @xmath20 is a first integral in @xmath32 . an important part of the procedure is to prove that the truncation of @xmath416 is also a first integral in @xmath32 . this is true if @xmath417 : suppose that @xmath418 and @xmath419 are bounded and @xmath420 . let @xmath163 and that @xmath421 a.e . then , a.e . we have @xmath422 thus @xmath423 is also a first integral of @xmath6 . however , the a.e . differentiability of @xmath424 is not guaranteed for @xmath425 . in fact , for @xmath424 , the condition @xmath426 is understood in the sense of distributions theory , cf , and the calculations in may only serve as a motivation . to build up tools for a rigorous proof of the distributional version of , in section [ sec5.2.1 ] for a sobolev - regular , divergence - free vector field @xmath138 , we consider the incompressible flow @xmath427 of mappings on @xmath46 constructed by diperna and lions in ( * ? ? ? * theorem iii.1 ) ( the integral curves of @xmath6 ) . the flow is used in lemma [ lem012707 ] to characterize the first integrals in @xmath428 as those locally invariant under the flow . then , we are able to conclude that the composition of a first integral in @xmath428 with a lipschitz function is also a first integral in @xmath428 , see corollary [ prop011007 ] , and , finally , use the conditioning of @xmath416 . unless stated otherwise , in this section we consider general @xmath138 such that @xmath429 and @xmath163 a.e . on @xmath46 . according to ( * ? ? ? * theorem iii.1 ) , there exists a unique a.e . defined jointly borelian family of mappings @xmath430 ( flow generated by @xmath6 ) with the following properties : first , for a.e . @xmath55 , the function @xmath431 is continuous and @xmath432 so that , in particular , @xmath433 second , for all @xmath434 and borel measurable sets @xmath435 , @xmath436 where @xmath437 is the @xmath7-dimensional lebesgue measure , and , third , for all @xmath438 , @xmath439 by and , if @xmath440 are nonnegative , then @xmath441 we let @xmath213 $ ] , @xmath442 , and define @xmath443 note that , @xmath444 , @xmath445 , defines a group of isometries on @xmath446 . if @xmath447 , then @xmath448 for a.e . @xmath55 and all @xmath434 , because @xmath163 implies @xmath449 . then for all @xmath434 , @xmath450 dxds.\end{aligned}\ ] ] let @xmath451 , @xmath452 and @xmath453 ( a mollifier ) . define @xmath454 the function is an approximate solution of the transport equation @xmath455 in the following sense : if @xmath456 @xmath457 $ ] , @xmath458 and @xmath459 , then for all @xmath434 and finite @xmath460 , @xmath461 this follows from ( * ? ? ? * theorem ii.1 ) . we shall use @xmath462 to characterize the first integrals @xmath463 as those elements of @xmath32 , which are constant along the flow @xmath464 . a word of explanation may be helpful . suppose that @xmath465 , @xmath163 a.e . , and @xmath20 is an @xmath466-integrable first integral , i.e. @xmath467 , as distributions on space - time @xmath468 , see ( * ? ? ? * ( 13 ) ) . by ( * ? ? ? * corollary ii.1 ) , there is a unique solution to the transport equation with the initial condition @xmath469 . the equation is understood in the sense of distributions on space - time , too . by ( * ? ? ? * theorem iii.1 ) , the solution has the form @xmath470 . however , since @xmath20 is a first integral , the mapping @xmath471 defines another solution . thus , by uniqueness , @xmath472 a.e . in our case this argument needs to be slightly modified since the first integral is defined only on @xmath2 and not on the entire @xmath49 . in particular , the identity @xmath472 is bound to hold only for small times @xmath473 . [ lem012707 ] let @xmath474 , @xmath475 , and @xmath476 . then @xmath414 if and only for every @xmath477 there is @xmath478 such that @xmath479 suppose that @xmath480 satisfies with @xmath478 and let @xmath481 . choose @xmath477 so that @xmath482 for all @xmath68 in the support of @xmath483 . using and , for all @xmath484 we obtain , @xmath485 applying , we rewrite the rightmost side of to obtain , @xmath486 as a result , @xmath487 letting @xmath488 we see that @xmath489 the limiting passage is justified by boundedness of @xmath6 , integrability of @xmath20 ( cf . the discussion preceding ) and dominated convergence theorem . since @xmath481 is arbitrary , we conclude that @xmath390 . conversely , let us assume for some @xmath476 . by sobolev embedding theorem ( * ? ? * theorem v.1 , p. 119 ) , we have that @xmath490 , where @xmath491 . let @xmath474 , @xmath492 on @xmath2 and @xmath493 on @xmath62 . note that @xmath494 . let @xmath495 and @xmath473 be such that @xmath496 by and , both @xmath497 and @xmath498 are supported in @xmath2 . using and we have @xmath499 therefore , @xmath500 since @xmath501 , the remainder @xmath502 satisfies . by hlder inequality , the right hand side of tends to @xmath392 , as @xmath503 . this proves for @xmath484 . as an immediate consequence of lemma [ lem012707 ] we obtain the following . [ prop011007 ] if @xmath79 is lipschitz on @xmath504 , @xmath420 and @xmath505 , then either @xmath506 or @xmath507 . by ( * theorem 1.4.2 ( v ) ) , @xmath508 . we apply @xmath79 to , and use lemma [ lem012707 ] . in particular , we may consider truncations of @xmath20 at the level @xmath268 , @xmath509 [ cor - x1 ] if @xmath268 and @xmath510 , then @xmath511 . we shall first prove the upper bound under the assumptions that @xmath6 is bounded and of zero divergence on the whole of @xmath49 , and @xmath512 . [ prop021007 ] suppose that @xmath513 , @xmath495 , @xmath514 and @xmath20 is bounded . then , @xmath515 we denote @xmath516 . by ( * theorem 1.4.2 ( ii ) , ( iv ) ) , @xmath517 . also , @xmath518 . considering , we observe that the right hand side of equals @xmath519 . by proposition [ prop011009 ] , the left hand side of is @xmath520 however , the second term on the right hand side vanishes , because @xmath521dz=0,\ ] ] and @xmath522 , for a sufficiently large @xmath523 , is a first integral by virtue of corollary [ cor - x1 ] . thus follows from . the following elementary identity holds for functions @xmath524 , @xmath525 ^ 2+u^2(x)\frac{v(y)-v(x)}{v(x)}+u^2(y)\frac{v(x)-v(y)}{v(y)}\\ & & = v(x)v(y)\left[\frac{u(x)}{v(x)}-\frac{u(y)}{v(y)}\right]^2\nonumber.\end{aligned}\ ] ] let @xmath526 , @xmath527 , and @xmath528 . by , @xmath529 ^ 2\ge \left\{\frac{w^2_n(x)}{\phi_a(x)+\eps}-\frac{w^2_n(y)}{\phi_a(y)+\eps}\right\}\left[\phi_a(x)-\phi_a(y)\right].\ ] ] multiplying both sides by @xmath530 and integrating over @xmath531 , we obtain @xmath532\nu(x - y)dxdy.\ ] ] using , we see that @xmath533 letting @xmath534 , we conclude that @xmath535 . letting @xmath536 and using ( * ? ? ? * part ( iii ) of theorem 1.4.2 ) , we obtain @xmath537 . this proves . [ sec5.2.3 ] [ sec6 ] suppose that @xmath538 , with @xmath474 , and @xmath539 on @xmath2 . here , as usual , @xmath2 is a bounded domain with the @xmath1 class boundary . by the discussion in section [ sec : gener ] , @xmath540 has finitely many , say , @xmath541 connected components . denote them by @xmath542 , and assume that @xmath543 belongs to the compactification of @xmath544 . we start with the following extension result . [ prop013110 ] there exist @xmath545 and @xmath546 such that @xmath547 @xmath548 has compact support and is smooth outside of @xmath549 . according to the results of section 4 of @xcite we can find @xmath550 , supported in @xmath551 and such that @xmath552 for @xmath553 , numbers @xmath554 , and points @xmath555 such that @xmath556 here @xmath557 . we denote by @xmath558 the right hand side of the above equality . the field @xmath559 is incompressible on @xmath560 . we choose @xmath545 so that @xmath561 for @xmath562 . let @xmath563 equal to @xmath157 on @xmath564 and vanish outside of @xmath565 . the field @xmath566 has the desired properties . we consider the flow @xmath567 of measurable mappings generated by @xmath548 , which satisfies , and , see theorem iii.2 of @xcite . condition needs to be modified as follows : there exists @xmath228 , such that @xmath568 let @xmath569 be a mollifier , let @xmath570 and let @xmath571 be the flow generated by @xmath572 . it has been shown in section 3 of @xcite that @xmath573}\int_{b(0,n)}|\phi(x(t , x))-\phi(x_\eps(t , x))|\wedge 2^ndx=0\label{eq:2.31.10}\ ] ] for all @xmath574 and measurable functions @xmath575 . we shall need the following modification of lemma [ lem012707 ] . [ lem012707a ] if @xmath576 then , the conclusion of lemma [ lem012707 ] holds for the flow @xmath567 generated by @xmath548 . suppose that @xmath577 . since @xmath578 on @xmath2 , we can repeat the proof of the respective part of lemma [ lem012707 ] . namely , keeping the notation from that lemma , for every @xmath481 we have @xmath579 such that latexmath:[\[\label{022807a } \int_{d } w(x)u_0(x(-t , x))dx=\int_{d } w(x)u_0(x)dx,\quad @xmath579 can be so adjusted that @xmath581 this part can not be guaranteed directly from the definition of the flow , as the extended field @xmath548 needs not be divergence - free . equality holds however , when @xmath582 is replaced by @xmath583 for a sufficiently small @xmath495 . indeed , by the liouville theorem , the jacobian @xmath584 of @xmath585 satisfies @xmath586 since @xmath587 in an open neighborhood of @xmath37 we conclude that @xmath588 on @xmath2 for ( sufficiently small ) @xmath589 . since @xmath20 and @xmath483 are supported in @xmath2 , @xmath590 letting @xmath503 and using we obtain . the rest of the proof follows that of lemma [ lem012707 ] . having established lemma [ lem012707a ] we proceed with the proofs of corollaries [ prop011007 ] and [ cor - x1 ] and proposition [ prop021007 ] with no alterations . these results yield . we consider the principal eigen - pair , say @xmath591 , of @xmath4 for the unit ball in @xmath46 . by rotation invariance of @xmath4 and uniqueness , @xmath592 is a smooth radial function in the ball . for @xmath553 we take radially symmetric functions @xmath593 , such that @xmath594 . let @xmath595 , @xmath596 . the vector field @xmath557 is of zero divergence and tangent to the spheres @xmath597 for all @xmath56 . indeed , since @xmath598 , @xmath599 as a result @xmath600 for any @xmath601 smooth radially symmetric function @xmath602 . thus , we conclude that @xmath592 is the principal eigenfunction of @xmath44 for every @xmath24 . we have @xmath603 , and @xmath604 attains its infimum , @xmath605 , at @xmath606 , see section [ sec : lb ] . in passing we note that the considered limiting eigenproblems are essentially different for different values of @xmath81 , in accordance with the fact that the `` escape rate '' @xmath607 of the isotropic @xmath81-stable lvy processes from @xmath2 depends on @xmath81 . we refer the interested reader to @xcite , @xcite , @xcite for more information on the eigenproblem of @xmath4 , see also @xcite . to complete the proof of theorem [ main - thm ] we only need to explain the attainability of infimum appearing on the right hand side of . this is done in the following . [ lem1 ] if @xmath608 , then @xmath609 attains its infimum on the set . if @xmath610 is the infimum , then we can choose functions @xmath611 in the set given in the statement of the lemma , such that @xmath612 and , by choosing a subsequence , that @xmath611 weakly converge to @xmath613 . since @xmath614 is precompact in @xmath47 , we may further assume that @xmath615 in @xmath47 and a.e . this implies that @xmath616 , @xmath617 is a first integral and , by and fatou s lemma , that @xmath618 . from the definition of @xmath619 , we conclude that equality actually occurs . krzysztof bogdan thanks the department of mathematics at stanford university for hospitality during his work on the paper , lenya ryzhik for discussions on incompressible flows and michael frazier for a discussion on schur s test .

we study the principal dirichlet eigenvalue of the operator @xmath0 , on a bounded @xmath1 regular domain @xmath2 . here @xmath3 , @xmath4 is the fractional laplacian , @xmath5 , and @xmath6 is a bounded @xmath7-dimensional divergence - free vector field in the sobolev space @xmath8 . we prove that the eigenvalue remains bounded , as @xmath9 , if and only if @xmath6 has non - trivial first integrals in the domain of the quadratic form of @xmath4 for the dirichlet condition .

arxiv

inferring macroscopic properties of physical systems from their microscopic description is an ongoing work in many disciplines of physics , like condensed matter , ultra cold atoms or quantum chromo dynamics . the most drastic changes in the macroscopic properties of a physical system occur at phase transitions , which often involve a symmetry breaking process . the theory of such phase transitions was formulated by landau as a phenomenological model @xcite and later devised from microscopic principles using the renormalization group @xcite . one can identify phases by knowledge of an order parameter which is zero in the disordered phase and nonzero in the ordered phase . whereas in many known models the order parameter can be determined by symmetry considerations of the underlying hamiltonian , there are states of matter where such a parameter can only be defined in a complicated non - local way @xcite . these systems include topological states like topological insulators , quantum spin hall states @xcite or quantum spin liquids @xcite . therefore , we need to develop new methods to identify parameters capable of describing phase transitions . such methods might be borrowed from machine learning . since the 1990s this field has undergone major changes with the development of more powerful computers and artificial neural networks . it has been shown that such neural networks can approximate every function under mild assumptions @xcite . they quickly found applications in image classification , speech recognition , natural language understanding and predicting from high - dimensional data . furthermore , they began to outperform other algorithms on these tasks @xcite . in the last years physicists started to employ machine learning techniques . most of the tasks were tackled by supervised learning algorithms or with the help of reinforcement learning @xcite . supervised learning means one is given labeled training data from which the algorithm learns to assign labels to data points . after successful training it can then predict the labels of previously unseen data with high accuracy . in addition to supervised learning , there are unsupervised learning algorithms which can find structure in unlabeled data . they can also classify data into clusters , which are however unlabelled . it is already possible to employ unsupervised learning techniques to reproduce monte - carlo - sampled states of the ising model @xcite . phase transitions were found in an unsupervised manner using principal component analysis @xcite . we employ more powerful machine learning algorithms and transition to methods that can handle nonlinear data . a first nonlinear extension is kernel principal component analysis @xcite . the first versions of autoencoders have been around for decades @xcite and were primarily used for dimensional reduction of data before feeding it to a machine learning algorithm . they are created from an encoding artificial neural network , which outputs a latent representation of the input data , and a decoding neural network that tries to accurately reconstruct the input data from its latent representation . very shallow versions of autoencoders can reproduce the results of principal component analysis @xcite . in 2013 , variational autoencoders have been developed as one of the most successful unsupervised learning algorithms @xcite . in contrast to traditional autoencoders , variational autoencoders impose restrictions on the distribution of latent variables . they have shown promising results in encoding and reconstructing data in the field of computer vision . in this work we use unsupervised learning to determine phase transitions without any information about the microscopic theory or the order parameter . we transition from principal component analysis to variational autoencoders , and finally test how the latter handles different physical models . our algorithms are able to find a low dimensional latent representation of the physical system which coincides with the correct order parameter . the decoder network reconstructs the encoded configuration from its latent representation . we find that the reconstruction is more accurate in the ordered phase , which suggests the use of the reconstruction error as a universal identifier for phase transitions . whereas for physicists this work is a promising way to find order parameters of systems where they are hard to identify , computer scientists and machine learning researchers might find an interpretation of the latent parameters . the ising model is one of the most - studied and well - understood models in physics . whereas the one - dimensional ising model does not possess a phase transition , the two - dimensional model does . the hamiltonian of the ising model on the square lattice with vanishing external magnetic @xmath0 field reads @xmath1 with uniform interaction strength @xmath2 and discrete spins @xmath3 on each site @xmath4 . the notation @xmath5 indicates a summation over nearest neighbors . a spin configuration @xmath6 is a fixed assignment of a spin to each lattice site , @xmath7 denotes the set of all possible configurations @xmath8 . we set the boltzmann constant @xmath9 and the interaction strength @xmath10 for the ferromagnetic case and @xmath11 for the antiferromagnetic case . a spin configuration @xmath8 can be expressed in matrix form as @xmath12 lars onsager solved the two dimensional ising model in 1944 @xcite . he showed that the critical temperature is @xmath13 . for the purpose of this work , we assume a square lattice with length @xmath14 such that @xmath15 , and periodic boundary conditions . we sample the ising model using a monte - carlo algorithm @xcite at temperatures @xmath16 $ ] to generate @xmath17 samples in the ferromagnetic case and @xmath18 samples in the antiferromagnetic case . the ising model obeys a discrete @xmath19-symmetry , which is spontaneously broken below @xmath20 . the magnetization of a spin sample is defined as @xmath21 the partition function @xmath22 allows us to define the corresponding order parameter . it is the expectation value of the absolute value of the magnetization at fixed temperature @xmath23 similarly , with the help of the matrix @xmath24 , we define the order parameter , as the expectation value of the staggered magnetization . the latter is calculated from an element - wise product with a matrix form of the spin configurations @xmath25 the mermin - wagner - hohenberg theorem @xcite prohibits continuous phase transitions in @xmath26 dimensions at finite temperature when all interactions are sufficiently short - ranged . hence , we choose the xy model in three dimensions as a model to probe the ability of a variational autoencoder to classify phases of models with continuous symmetries . the hamiltonian of the xy model reads @xmath27 with spins on the one - sphere @xmath28 . employing @xmath10 , the transition temperature of this model is @xmath29 @xcite using a cubic lattice with @xmath30 , such that @xmath31 , we perform monte - carlo simulations to create 10000 independent sample spin configurations in the temperature range of @xmath32 $ ] . the order parameter is defined analogously to the ising model magnetization , but with the @xmath33-norm of a magnetization consisting of two components . _ principal component analysis _ @xcite is an orthogonal linear transformation of the data to an ordered set of variables , sorted by their variance . the first variable , which has the largest variance , is called the first principal component , and so on . the linear function @xmath34 , which maps a collection of spin samples @xmath35 to its first principal component , is defined as @xmath36 \ , \label{eq : pca}\end{aligned}\ ] ] where @xmath37 is the vector of mean values of each spin averaged over the whole dataset . further principal components are obtained by subtracting the already calculated principal components and repeating . _ kernel principal component analysis _ @xcite projects the data into a kernel space in which the principal component analysis is then performed . in this work the nonlinearity is induced by a radial basis functions kernel . _ traditional neural network - based autoencoders _ consist of two artificial neural networks stacked on top of each other . the encoder network is responsible for encoding the input data into some latent variables . the decoder network is used to decode these parameters in order to return an accurate recreation of the input data , shown in . the parameters of this algorithm are trained by performing gradient descent updates in order to minimize the reconstruction loss ( reconstruction error ) between input data and output data . _ variational autoencoders _ are a modern version of autoencoders which impose additional constraints on the encoded representations , see latent variables in . these constraints transform the autoencoder to an algorithm that learns a latent variable model for its input data . whereas the neural networks of traditional autoencoders learn an arbitrary function to encode and decode the input data , variational autoencoders learn the parameters of a probability distribution modeling the data . after learning the probability distribution , one can sample parameters from it and then let the encoder network generate samples closely resembling the training data . to achieve this , variational autoencoders employ the assumption that one can sample the input data from a unit gaussian distribution of latent parameters . the weights of the model are trained by simultaneously optimizing two loss functions , a reconstruction loss and the kullback - leibler divergence between the learned latent distribution and a prior unit gaussian . in this work we use autoencoders and variational autoencoders @xcite with one fully connected hidden layer in the encoder as well as one fully connected hidden layer in the decoder , each consisting of 256 neurons . the number of latent variables is chosen to match the model from which we sample the input data . the activation functions of the intermediate layers are rectified linear units . the activation function of the final layer is a _ sigmoid _ in order to predict probabilities of spin @xmath38 or @xmath39 in the ising model , or _ tanh _ for predicting continuous values of spin components in the xy model . we do not employ any @xmath40 or dropout regularization . however , we tune the relative weight of the two loss functions of the variational autoencoder to fit the problem at hand . the kullback - leibler divergence of the variational autoencoder can be regarded as reguarization of the traditional autoencoder . in our autoencoder the reconstruction loss is the cross - entropy loss between the input and output probability of discrete spins , as in the ising model . the reconstruction loss is the mean - squared - error between the input and the output data of continuous spin variables in the xy model . to understand why a variational autoencoder can be a suitable choice for the task of classifying phases , we recall what happens during training . the weights of the autoencoder learn two things : on the one hand , they learn to encode the similarities of all samples to allow for an efficient reconstruction . on the other hand , they learn a latent distribution of the parameters which encode the most information possible to distinguish between different input samples . let us translate these considerations to the physics of phase transitions . if all the training samples are in the unordered phase , the autoencoder learns the common structure of all samples . the autoencoder fails to learn any random entropy fluctuations , which are averaged out over all data points . however , in the ordered phase there exists a common order in samples belonging into the same phase . this common order translates to a nonzero latent parameter , which encodes correlations on each input sample . it turns out that in our cases this parameter is the order parameter corresponding to the broken symmetry . it is not necessary to find a perfect linear transformation between the order parameter and the latent parameter as it is the case in . a one - to - one correspondence is sufficient , such that one is able to define a function that maps these parameters onto each other and captures all discontinuities of the derivatives of the order parameter . we point out similarities between principal component analysis and autoencoders . although both methods seem very different , they both share common characteristics . principal component analysis is a dimensionality reduction method which finds the linear projections of the data that maximizes the variance . reconstructing the input data from its principal components minimizes the mean squared reconstruction error . although the principal components do not need to follow a gaussian distribution , principal components have the highest mutual agreement with the data if it emerges from a gaussian prior . moreover , a single layer autoencoder with linear activation functions closely resembles principal component analysis @xcite . principal component analysis is much easier to apply and in general uses less parameters than autoencoders . however , it scales very badly to a large dataset . autoencoders based on convolutional layers can reduce the number of parameters . in extreme cases this number can be even less than the parameters of principal component analysis . furthermore , such autoencoders can promote locality of features in the data . + + + + + pixels , from the latent parameter . the brightness indicates the probability of the spin to be up ( white : @xmath41 , black : @xmath42 ) . the first row is a reconstruction of sample configurations from the ferromagnetic ising model . the second row corresponds to the antiferromagnetic ising model . the third row is the prediction from the af latent parameter , where each second spin is multiplied by @xmath43 , to show that the second row indeed predicts an antiferromagnetic state.,title="fig:",scaledwidth=50.0% ] + + the four different algorithms can be applied to the ising model to determine the role of the first principal components or the latent parameters . shows a clear correlation between these parameters and the magnetization for all four methods . however , the traditional autoencoder is inaccurate ; this fact leads us to enhancing traditional autoencoders to variational autoencoders . the principal component methods show the most accurate results , slightly better than the variational autoencoder . this is to be expected , since the former are modeled by fewer parameters . in the following results section , we concentrate on the variational autoencoder as the most advanced algorithm for unsupervised learning . to begin with , we choose the number of latent parameters in the variational autoencoder to be one . after training for 50 epochs and a saturation of the training loss , we visualize the results in . on the left , we see a close linear correlation between the latent parameter and the magnetization . in the middle we see a histogram of encoded spin configurations into their latent parameter . the model learned to classify the configurations into three clusters . having identified the latent parameter to be a close approximation to the magnetization @xmath44 allows us to interpret the properties of the clusters . the right and left clusters in the middle image correspond to an average magnetization of @xmath45 , while the middle cluster corresponds to the magnetization @xmath46 . employing a different viewpoint , from we conclude that the parameter which holds the most information on how to distinguish ising spin samples is the order parameter . on the right panel , the average of the magnetization , the latent parameter and the reconstruction loss are shown as a function of the temperature . a sudden change in the magnetization at @xmath47 defines the phase transition between paramegnetism and ferromagnetism . even without knowing this order parameter , we can now use the results of the autoencoder to infer the position of the phase transition . as an approximate order parameter , the average absolute value of latent parameter also shows a steep change at @xmath20 . the averaged reconstruction loss also changes drastically at @xmath20 during a phase transition . while the latent parameter is different for each physical model , the reconstruction loss can be used as a universal parameter to identify phase transitions . to summarize , without any knowledge of the ising model and its order parameter , but sample configurations , we can find a good estimation for its order parameter and the occurrence of a phase transition . it is a priori not clear how to determine the number of latent neurons in the creation of the neural network of the autoencoder . due to the lack of theoretical groundwork , we find the optimal number by experimenting . if we expand the number of latent dimensions by one , see , the results of our analysis only change slightly . the second parameter contains a lot less information compared to the first , since it stays very close to zero . hence , for the ising model , one parameter is sufficient to store most of the information of the latent representation . while the ferromagnetic ising model serves as an ideal starting ground , in the next step we are interested in models where different sites in the samples contribute in a different manner to the order parameter . we do this in order to show that our model is even sensitive to structure on the smallest scales . for the magnetization in the ferromagnetic ising model , all spins contribute with the same weight . in contrast , in the antiferromagnetic ising model , neighboring spins contribute with opposite weight to the order parameter . again the variational autoencoder manages to capture the traditional order parameter . the staggered magnetization is strongly correlated with the latent parameter , see . the three clusters in the latent representation make it possible to interpret different phases . furthermore , we see that all three averaged quantities - the magnetization , the latent parameter and the reconstruction loss - can serve as indicators of a phase transition . demonstrates the reconstruction from the latent parameter . in the first row we see the reconstruction from samples of the ferromagnetic ising model , the latent parameter encodes the whole spin order in the ordered phase . reconstructions from the antiferromagnetic ising model are shown in the second and third row . since the reconstructions clearly show an antiferromagnetic phase , we infer that the autoencoder encodes the spin samples even to the most microscopic level . in the xy model we examine the capabilities of a variational autoencoder to encode models with continuous symmetries . in models like the ising model , where discrete symmetries are present , the autoencoder only needs to learn a discrete set , which is often finite , of possible representations of the symmetry broken phase . if a continuous symmetry is broken , there are infinitely many possibilities of how the ordered phase can be realized . hence , in this section we test the ability of the autoencoder to embed all these different realizations into latent variables . the variational autoencoder handles this model equally well as the ising model . we find that two latent parameters model the phase transition best . the latent representation in the middle of shows the distribution of various states around a central cluster . the radial symmetry in this distribution leads to the assumption that a sensible order parameter is constructed from the @xmath33-norm of the latent parameter vector . in , one sees the correlation between the magnetization and the absolute value of the latent parameter vector . averaging the samples for the same temperature hints to the facts that the latent parameter and the reconstruction loss can serve as an indicator for the phase transition . we have shown that it is possible to observe phase transitions using unsupervised learning . we compared different unsupervised learning algorithms ranging from principal component analysis to variational autoencoders and thereby motivated the need for the upgrade of the traditional autoencoder to a variational autoencoder . the weights and latent parameters of the variational autoencoder approach are able to store information about microscopic and macroscopic properties of the underlying systems . the most distinguished latent parameters coincide with the known order parameters . furthermore , we have established the reconstruction loss as a new universal indicator for phase transitions . we have expanded the toolbox of unsupervised learning algorithms in physics by powerful methods , most notably the variational autoencoder , which can handle nonlinear features in the data and scale very well to huge datasets . using these techniques , we expect to predict unseen phases or uncover unknown order parameters , e.g. in quantum spin liquids . we hope to develop deep convolutional autoencoders which have a reduced number of parameters compared to fully connected autoencoders and can also promote locality in feature selection . furthermore , since there exists a connection between deep neural networks and renormalization group @xcite , it may be helpful to employ deep convolutional autoencoders to further expose this connection . _ acknowledgments _ we would like to thank timo milbich , bjrn ommer , michael scherer , manuel scherzer and christof wetterich for useful discussions . we thank shirin nkongolo for proofreading the manuscript . s.w . acknowledges support by the heidelberg graduate school of fundamental physics . stefano curtarolo , dane morgan , kristin persson , john rodgers , and gerbrand ceder . predicting crystal structures with data mining of quantum calculations . _ physical review letters _ , 910 ( 13 ) , doi : 10.1103/physrevlett.91.135503 . matthias rupp , alexandre tkatchenko , klaus - robert mller , and o. anatole von lilienfeld . fast and accurate modeling of molecular atomization energies with machine learning . _ physical review letters _ , 1080 ( 5 ) , jan 2012 . doi : 10.1103/physrevlett.108.058301 . zhenwei li , james r. kermode , and alessandro de vita . molecular dynamics with on - the - fly machine learning of quantum - mechanical forces . _ physical review letters _ , 1140 ( 9 ) , mar 2015 . doi : 10.1103/physrevlett.114.096405 . erin ledell , prabhat , dmitry yu . zubarev , brian austin , and william a. lester . classification of nodal pockets in many - electron wave functions via machine learning . _ journal of mathematical chemistry _ , 500 ( 7):0 20432050 , may 2012 . doi : 10.1007/s10910 - 012 - 0019 - 5 . g. pilania , j. e. gubernatis , and t. lookman . structure classification and melting temperature prediction in octet ab solids via machine learning . _ physical review b _ , 910 ( 21 ) , jun 2015 . doi : 10.1103/physrevb.91.214302 . yousef saad , da gao , thanh ngo , scotty bobbitt , james r. chelikowsky , and wanda andreoni . data mining for materials : computational experiments withabcompounds . _ physical review b _ , 850 ( 10 ) , mar 2012 . doi : 10.1103/physrevb.85.104104 . o. s. ovchinnikov , s. jesse , p. bintacchit , s. trolier - mckinstry , and s. v. kalinin . disorder identification in hysteresis data : recognition analysis of the random - bondrandom - field ising model . _ physical review letters _ , 1030 ( 15 ) , oct 2009 . doi : 10.1103/physrevlett.103.157203 . louis - franois arsenault , alejandro lopez - bezanilla , o. anatole von lilienfeld , and andrew j. millis . machine learning for many - body physics : the case of the anderson impurity model . _ physical review b _ , 900 ( 15 ) , oct 2014 . doi : 10.1103/physrevb.90.155136 . john c. snyder , matthias rupp , katja hansen , klaus - robert mller , and kieron burke . finding density functionals with machine learning . _ physical review letters _ , 1080 ( 25 ) , jun 2012 . doi : 10.1103/physrevlett.108.253002 . geoffroy hautier , christopher c. fischer , anubhav jain , tim mueller , and gerbrand ceder . finding nature s missing ternary oxide compounds using machine learning and density functional theory . _ chemistry of materials _ , 220 ( 12):0 37623767 , jun 2010 . doi : 10.1021/cm100795d . pierre baldi and kurt hornik . neural networks and principal component analysis : learning from examples without local minima . _ neural networks _ , 20 ( 1):0 5358 , jan 1989 . doi : 10.1016/0893 - 6080(89)90014 - 2 . n. d. mermin and h. wagner . absence of ferromagnetism or antiferromagnetism in one- or two - dimensional isotropic heisenberg models . _ physical review letters _ , 170 ( 22):0 11331136 , nov 1966 . doi : 10.1103/physrevlett.17.1133 . aloysius p. gottlob and martin hasenbusch . critical behaviour of the 3d xy - model : a monte carlo study . _ physica a : statistical mechanics and its applications _ , 2010 ( 4):0 593613 , dec 1993 . doi : 10.1016/0378 - 4371(93)90131-m .

we employ unsupervised machine learning techniques to learn latent parameters which best describe states of the two - dimensional ising model and the three - dimensional xy model . these methods range from principal component analysis to artificial neural network based variational autoencoders . the states are sampled using a monte - carlo simulation above and below the critical temperature . we find that the predicted latent parameters correspond to the known order parameters . the latent representation of the states of the models in question are clustered , which makes it possible to identify phases without prior knowledge of their existence or the underlying hamiltonian . furthermore , we find that the reconstruction loss function can be used as a universal identifier for phase transitions .

arxiv

main laws of quantum mechanics forbids the perfect cloning of the quantum states , see corresponding discussion for the pure states in @xcite , @xcite , and for the mixed states in @xcite . but it is possible to carry out an approximate copying of the quantum states @xcite . quantum cloning machines ( qcm ) depend on the conditions accepted at its designing . they can produce identical copies of the initial state ( symmetric qcm ) , nonidentical copies ( non - symmetric qcm ) , the quality of the copying can be either identical for all states ( universal qcm ) or depend on the state ( state - dependente qcm ) . detailed discussion of the different variants of qcm and theirs possible applications in quantum cryptography and quantum informatics can be found in @xcite , @xcite . one possible application of the qcm is an eavesdropping of the quantum channel . the aim of such eavesdropping defines the main properties of the designing qcm . one can design qcm which copies only part of the quantum state , for instance . such qcm can be useful if eavesdropper , usually called eve , intends to catch part of the transmitted quantum information only . some classical analogue of this situation can be classical eavesdropping of the key words in the transmitted classical information . at quantum cloning we can choose the different parts of the quantum signal in which we are interested . in this paper we intend to discuss some `` partial '' qcm , which copies one constituent of the two - partite states . our approach gives the possibility to consider qcm for a mixed states too . it is well known fact , that any mixed state can be considered as a reduction of a pure state , which is called `` purification '' of the mixed state @xcite . so , cloning of the mixed state can be considered as a `` partial '' cloning of the `` purification '' of the mixed state . some difference between the `` partial '' cloning machine and the cloning machine for the mixed states is connected with the corresponding difference of the sets of the initial states , see details below . note , that the main attention in the present literature was devoted to the cloning of the pure states @xcite , @xcite . we consider two - partite qubit states , qubits are elements of two - dimensional hilbert space @xmath0 . in order to construct qcm we need in tensor product of three such spaces on the ancilla space : @xmath1 , here different components are marked by indexes . the first and third qubit components constitute a quantum state which carries information in the quantum channel , and the state of first component is interesting for eve . the second component is a blank state , where we will copy the first component , the last component is necessary for the realization of the qcm . let quantum channel carries the quantum state @xmath2 @xmath3 where normalization condition holds , @xmath4 here and below @xmath5 are base vectors in @xmath6 . we suppose , that eve s goal is a copying of the first component of this state . after tracing one can obtain : @xmath7 @xmath8 so , eve has to realize the cloning to produce the pair of states ( in the first and second components respectively ) closest to @xmath9 we consider here symmetric qcm , so , we suppose , that states in the first and second components have to coincide . then produced state must be symmetric with regard to permutation of the first and second components . let us introduce the orthonormal basis in the subspace of @xmath10 symmetric regarding this permutation : @xmath11 let s assume , that the second component be in state @xmath12 initially . description of the qcm is , in essence , the definition of the corresponding unitary operator @xmath13 . following to @xcite , @xcite , we set @xmath14 @xmath15 where @xmath16 are some vectors , belonging to @xmath17 . symmetry of qcm is provided by the fact , that right - hand part of this relation contains linear combinations of vectors @xmath18 only . taking into account ( [ init ] ) , we obtain : @xmath19 @xmath20 @xmath21 generally speaking , the choice of the unitary operator @xmath13 is very broad and corresponding analysis is quite complex even for the lowest dimensions , so usually one admits some additional restrictions . we suppose as in @xcite , that following conditions ( which guarantee the unitarity of @xmath13 ) are fulfilled : @xmath22 @xmath23 let @xmath24 so as @xmath25 , @xmath26 . in this case qcm produces the next state from @xmath27 : @xmath28 @xmath29 @xmath30 reducing this state on the first component , we obtain : @xmath31 @xmath32 it is necessary to compare the initial state and state which is produced by the qcm , in other words , we have to choose the measure of the closeness of these states . there are different measures , specifically , fidelity . it is defined for the mixed states as @xmath33 ^2 $ ] , this value is not very suitable for the analytical considerations . we use here more convenient measure : @xmath34 ^2 = w(\zeta , \nu , \psi ) , \ ] ] where @xmath35 + \ ] ] @xmath36 @xmath37 .\ ] ] this value estimates the difference between initial and final states with fixed parameters @xmath38 . for the determination of the qcm parameters we average this value respect to the set of all initial states . we use here the next parametrization of the initial state @xmath39 : @xmath40 @xmath41 where @xmath42 here the first component has zero phase due to the corresponding freedom of the choice . for the averaging we need in corresponding measure . supposing that all states @xmath43 are equiprobable , we choose as such a measure @xmath44 @xmath45 simple calculations lead to the conclusion , that @xmath46 takes its minimal value at @xmath47 , @xmath48 . this value @xmath49 implies , that vectors @xmath50 and @xmath51 are parallel . the values of the fidelity @xmath33 ^2 $ ] , calculated at @xmath52 , for the states on the `` real '' part of the bloch sphere , @xmath53 @xmath54 , @xmath55 , are plotted in fig.1 . evidently , that this qcm is state - dependent one , because the quality of the cloning depends on the quantum state . the construction described above can be used for the cloning of the mixed states . note , that final state produced by qcm depend on the reduction of the initial state @xmath56 ( [ red ] ) only . it means , that one can reverse our considerations and take the mixed state @xmath56 as initial one . then pure state @xmath43 defined by relation ( [ init ] ) belongs to the space of the larger dimension and it is a `` purification '' of the state @xmath56 . `` purification '' of the given mixed state @xmath56 can be realized by different methods , as it follows from ( [ init ] ) , but this nonuniqueness has not influence in the results . qcm constructed in accordance with relations ( [ ccm1 ] ) , ( [ ccm2 ] ) produces states ( [ out12 ] ) , ( [ out ] ) , which depend on the parameters of the initial state @xmath56 only . but the set of the initial states is changing , one has to use another parametrization for this set . as such parametrization one can take a bloch sphere , see @xcite . namely , density matrix @xmath56 can be described as @xmath57 where @xmath58 in the spherical coordinates we have : @xmath59 @xmath60 and @xmath61 in order to obtain the parameters of the qcm one has to average the value @xmath62 on the bloch sphere . we suppose that all states in the bloch sphere are equiprobable , so averaging is reduced to the integral @xmath63 note , that if eve has a priori information about transmitted quantum information she has to choose corresponding weight multiplier . we are searching in such values @xmath64 which correspond to the minimal value @xmath65 . as a result we obtain @xmath66 . the plot of fidelity for these parameters differs in a small way from the preceding one , so we omit it . we have discussed here a `` partial '' quantum cloning , when only one component of the two - partite pure state is cloning . such cloning can be considered as a variant of the eavesdropping of the quantum channel . the choice of the parameters of the qcm was realized with help of the some natural criterion . namely , we seek in parameters corresponding to the minimum of the integral average of the `` distance '' between the initial state and output state . note , that fidelity of the initial and output states for the most part of the bloch sphere exceeds value 5/6 which corresponds to the universal qcm for the pure states @xcite . this fact has two reasons . firstly , described qcm copies only part of the two - partite state . second , this qcm is state - dependent one , and this non - universality raises the quality of the cloning . moreover , here was discussed the cloning of the mixed qubit states . in order to consider such device we use the `` purification '' of the mixed state and then apply `` partial '' cloning machine . let s emphasize , that `` purification '' of the mixed state is not unique , but the output state produced by our qcm does not depend on this nonuniqueness . parameters of the qcm was sought by minimization of the integral average of the distance between the initial and output states . note , that this averaging differs from the used above because the sets of the states are different . in this case the fidelity of the initial and output states on the real part of the bloch sphere exceeds the value 5/6 too . evidently , that choice of the parameters @xmath67 for the qcm is defined by the strategy of the eavesdropping . w have discussed here only one possible qcm for the mixed states . evidently , that there are many other variants for the qcm , may be , without restrictions like ( [ ccm1 ] ) , ( [ ccm2 ] ) , ( [ con1 ] ) , asymmetric qcm etc . w.k.wooters , w.h.zurek_. nature , 299 , 802 ( 1982 ) . _ d.dieks_. phys . lett.a , 54 , 1844 ( 1982 ) . _ h.barnum , c.m.caves , c.a.fuchs , r.jozsa , b.schumacher_. phys.rev.lett . , 76 , 2818 ( 1996 ) ( arxiv quant - ph/9511010 ) . _ v.buzek , m.hillery._ phys.rev.a , 54 , 1844 ( 1996 ) . _ v.scarani , s.ibilsdur , n.gisin._ rev.mod.phys . , v.77 , 1225 - 1256 ( 2005 ) . _ n.j.cerf , j.fiurasek_. optical quantum cloning . in : progress in optics , v.49 , ed . e.wolf , elsevier , 2006 . _ j. preskill . _ lecture notes for physics : quantum information and computation , 1998 . _ s.adhikari , a.k.pati , i.chakrabarty , b.s.choudhury._ arxiv : quant - ph/0705.0631 , 2007 .

we discuss the `` partial '' quantum cloning of the pure two - partite states , when the `` part '' of initial state related to the one qubit is copied only . the same approach gives the possibility to design the quantum copying machine for the mixed qubit states . * `` partial '' quantum cloning and quantum cloning of the mixed states * * a.ya.kazakov * + laboratory of quantum information , + st .- petersburg state university of aerospace instrumentation , 67 b.morskaya str . , st .- petersburg , 190000 russia pacs : 03.67 -a , 03.67 dd

arxiv

the nature of dark matter remains one of the outstanding questions of modern astrophysics . the success of the cold dark matter cosmological model ( albeit with `` dark energy '' now required : @xmath3cdm ) argues strongly for a major component of the dark matter being in the form of an elementary particle . however , the inventory of baryons which we can observe locally falls far short of the total inferred from observations of the cosmic microwave background fluctuations @xcite , leaving open the possibility that there may be a significant baryonic component of dark matter . furthermore , although @xmath3cdm is very successful in describing the growth of structure in the universe on large scales , we still lack a direct detection of any of the candidate dark matter particles . lacking this decisive piece of observational evidence , some authors have proposed models which include a large component of baryonic dark matter . in particular there have been many papers dealing with the possibility that cold , self - gravitating molecular clouds constitute a major component of the dark matter @xcite . a variety of different forms , including isolated , clustered , and fractal , have been considered for the clouds , but all proposals involve dense gas of high column - density , in contrast to the diffuse gas in the interstellar medium which is easily detected in emission and/or absorption . one of the fundamental predictions of a model featuring dense gas clouds is the gamma - ray emission resulting from cosmic - ray interactions within the clouds @xcite . because of the potentially large total mass of gas involved , this process may yield a diffuse flux in the galactic plane comparable to the flux from known sources for photon energies around 1 gev @xcite . considering the high quality data on diffuse emission acquired by the egret detector aboard the compton gamma ray observatory @xcite , it is worth considering this source of gamma - ray emission in detail as it is possible to use these data to constrain the dark matter models ( see @xcite ; @xcite ) . most previous investigations of this problem have neglected the self - shielding and cascade phenomena which can be important at high column densities @xcite , and have employed emissivities appropriate to the low - density limit . these effects alter the emergent gamma - ray spectrum , and we note that this could be relevant to the observed excess galactic flux above 1 gev @xcite . we have noted elsewhere @xcite that massive ( @xmath4 ) aggregates of dense gas clouds could potentially account for many of the unidentified discrete sources detected by egret @xcite . here we present detailed calculations of the gamma - ray spectra arising from cosmic - ray interactions with dense gas clouds . we have used a monte carlo simulation code , geant4 , developed for simulating interaction events in detectors used in high - energy particle physics . not surprisingly , we find that the predicted spectra differ substantially between high and low column - density clouds , and we discuss the interpretation of our results in the context of the observed galactic gamma - ray emission . our calculations are undertaken for cold , dense molecular gas in clouds of radius @xmath5 cm , similar to those proposed by @xcite to explain the extreme scattering events @xcite during which compact extragalactic radio sources are magnified and demagnified as a plasma `` lens '' moves across the line of sight ( see @xcite for a criticism of this model ) . however , the results of our calculations depend primarily on the column - density of the individual clouds , @xmath6 , under consideration , and their fractional contribution to the galaxy s dark matter halo , and our results can be taken as representative of other models which are characterised by similar values of these quantities . previous calculations of gamma - ray spectra from cosmic - ray irradiation assumed single interactions of protons with the interstellar medium ( @xcite and references therein ) . in order to investigate cosmic - ray interactions with dense gas , where cascade processes and particle transport are important , we have used a monte carlo code , geant4 , to derive gamma - ray production spectra . this code is a general purpose monte carlo code of particle interactions and is widely used for simulation of high - energy particle detectors in accelerator experiments . cross - sections and interactions of various hadronic processes , i.e. , fission , capture , and elastic scattering , as well as inelastic final state production , are parametrized and extrapolated in high and low particle energy limits , respectively . the @xmath7 production in this code , which is important because of the @xmath8 decay that dominates the emissivity of the gas at high energies , has been tested against accelerator data @xcite . initially we experienced one slight difficulty in applying geant4 to our physical circumstance : the low - energy hadron interaction code , called gheisha , did not conserve energy very accurately ( geant4 bug reports no . 171 and 389 ) . a `` patch '' was available for gheisha ( geant ver . 4.4.1 ) , but this patch appeared to introduce further problems of its own in the energy deposition distribution ( geant4 bug report no . these difficulties have been overcome by the geant team , and we are not aware of any such problems in the latest release ( geant ver . 4.5.1 ) . our calculations assume a spherical cloud of molecular hydrogen of uniform density and temperature ( 10 k ) . the radius of the sphere was assumed to be @xmath9 au . protons and electrons are injected randomly at a surface point of the cloud and particles subsequently emanating from this surface are counted as products . the adopted spectra of cosmic - ray protons and electrons were taken from @xcite ( here we use the `` median '' flux ; note that the units on his equation ( 3 ) should read @xmath10s@xmath11sr@xmath11gev@xmath11 ) , and @xcite , respectively . in figs . [ fig : f1 ] and [ fig : f2 ] we plot these spectra together with observational data . ( note that the adopted spectra are those for cosmic rays in the galaxy , whereas the measured points are subject to the modulating influence of the magnetic field of the solar wind ) . the simulated range of kinetic energy of cosmic rays is from 10 mev to 10 tev . we divided this energy range into four and superposed the resulting spectra with appropriate weight factors in order to increase the simulation statistics at higher energies , considering the rapidly falling spectrum of cosmic rays . the density of molecular hydrogen , @xmath12 , was varied from @xmath13 to @xmath14 g @xmath15 in factors of 10 . this corresponds to the column density , @xmath16 , of @xmath17 g @xmath10 , respectively , where @xmath18 is the incident angle of a cosmic ray into a cloud and @xmath19 for random injection . figure [ fig : f3 ] shows the resulting gamma - ray spectra obtained in our geant4 simulation for proton injection into a cloud of column density @xmath20 . ( the quantity plotted is @xmath21 , with @xmath22 being the emissivity as given in 3 . ) the dashed , dotted and dot - dashed lines show the spectral components classified by the parent processes producing gamma - rays , i.e. , @xmath7 decay , bremsstrahlung and positron - electron annihilation , respectively . the latter two components are necessarily omitted in calculations which assume single interactions only ( i.e. the thin material limit ) . the error bars are calculated from monte carlo statistics . although the @xmath7 decay component shows a broad peak at @xmath23 mev ( note that the quantity plotted is the emissivity multiplied by @xmath24 ) and dominates above about 200 mev , the electron bremsstrahlung component broadens the emissivity peak . the bremsstrahlung and annihilation contributions are negligible in the limit of small column - density ; their fractional contribution to the emission is greatest for clouds which are just thick enough ( @xmath25 ) to attenuate the bulk of the incident proton power . this is as expected considering that the interaction mean free path of gev protons is @xmath26 g @xmath10 and is in accordance with the result of @xcite who treated a similar problem by solving one - dimensional transport equations . the resulting gamma - ray emissivities for clouds of various column densities are shown in figure [ fig : f4 ] ( proton injection ) and figure [ fig : f5 ] ( electron injection ) . here the emissivities are defined for irradiation by cosmic - rays of all species ( see 2.3 , equation ( 3 ) ) ; to take account of the contribution of heavier nuclei than helium , the emissivity due to proton irradiation ( figure 4 ) has been multiplied by a nuclear enhancement factor @xcite of 1.52 @xcite . note that for high densities the monte carlo statistics are rather poor , since the yield itself is low . figure [ fig : f4 ] includes a comparison of our calculated gamma - ray production functions with that of @xcite ( corresponding to the `` thin material '' limit ) . the results are consistent with those of @xcite for column densities less than about @xmath27 g @xmath10 , except in the energy range @xmath28 mev where the effect of the maximum energy assumed in the monte carlo simulation is evident . we note the very low values of the emissivity at energies @xmath29 mev , for column densities @xmath30 . a slightly steeper spectrum in the @xmath31@xmath32 mev region comes from our omission of the contribution of heavy nuclei , which were taken into account in @xcite . a somewhat surprising feature of these curves is that the power - law index above 1 gev is almost the same as the input cosmic - ray proton flux for column densities less than about 1000 g @xmath10 ( for higher column densities the statistics of the simulations are not good enough to decide whether this result still holds ) . this is already indicated by @xcite , but is contrary to the expectation of @xcite who suggested a spectral change above 1 gev at the point where self - shielding becomes important . for some purposes the energy - integrated emissivities are of more interest than their differential counterparts , so we present these quantities in figures [ fig : f6 ] and [ fig : f7 ] , for cosmic ray protons and electrons , respectively . at low column densities , where cascades and self - shielding are unimportant , there is very little variation of the integrated emissivity with cloud column density , and the thin material limit can be adopted for columns less than @xmath33 for protons ( @xmath34 for electrons ) . above this point , however , the emissivity drops rapidly with increasing cloud column - density . in order to gauge the sensitivity of these calculations to the assumed cosmic - ray spectra , we have computed our results for two different incident cosmic - ray proton spectra , and three different incident cosmic - ray electron spectra , as shown in figures 6 and 7 , respectively . the differences are seen to be small in comparison with the variation as a function of cloud column density , but at a fixed column - density the systematic uncertainties associated with the input cosmic - ray spectra are nevertheless significant . @llll@ + + @xmath6 & @xmath35 & @xmath36&@xmath37 + + @xmath38 & @xmath39 & @xmath40 & @xmath41 + @xmath42 & @xmath43 & @xmath44 & @xmath45 + @xmath46 & @xmath47 & @xmath48 & @xmath49 + @xmath50 & @xmath51 & @xmath52 & @xmath53 + @xmath54 & @xmath55 & @xmath56 & @xmath57 + @xmath58 & @xmath59 & @xmath60 & @xmath61 + @xmath62 & @xmath63 & @xmath64 & @xmath65 + @xmath66 & @xmath67 & @xmath68 & @xmath69 + @lllll@ + + @xmath70&10&100&1000&10000 + + @xmath6 & @xmath71 & @xmath71&@xmath71 & @xmath71 + @xmath38&@xmath72&@xmath73&@xmath74&@xmath75 + @xmath42&@xmath76&@xmath77&@xmath78&@xmath79 + @xmath46&@xmath80&@xmath81&@xmath82&@xmath83 + @xmath50&@xmath84&@xmath85&@xmath86&@xmath87 + @xmath54&@xmath88&@xmath89&@xmath90&@xmath91 + @xmath58&@xmath92&@xmath93&@xmath94&@xmath95 + @xmath62&@xmath96&@xmath97&@xmath98&@xmath99 + @xmath66&@xmath100&@xmath101&@xmath102&@xmath103 + using the gamma - ray production spectra obtained in the previous section , we have calculated the diffuse gamma - ray emission from the galaxy as follows . the predicted gamma - ray spectrum for each case is @xmath104 where @xmath105 is the spectrum returned by the simulation in units of photons / mev / primary , for an individual cloud , appropriate to the incident cosmic - ray spectrum . the quantity @xmath106 is the intensity of cosmic rays at a distance @xmath107 along the line of sight , in units of primaries @xmath10 s@xmath11 sr@xmath11 , and @xmath108 is the mean density in gas clouds of column density @xmath6 . the galactic variation of the spectrum @xmath109 is not well constrained by existing data , and consequently we adopt the simplifying assumption that the shape of the cosmic - ray spectra ( both electrons and protons ) is the same everywhere in the galaxy , with variations only in the normalisation . with this assumption it is convenient to recast the calculation as @xmath110 ( @xmath111 ) , where the emissivity is @xmath112 ( @xmath113 ) with @xmath114 the cosmic - ray mean intensity in the solar neighbourhood and @xmath115 is the weighted column density ( @xmath116 ) of the cloud population along the line - of - sight under consideration . this formulation is convenient because the emissivity , @xmath22 , describes the properties of the gas clouds themselves and is independent of the galactic variations in mean dark matter density and cosmic - ray density ; conversely the quantity @xmath117 characterises these properties of the galaxy , and is independent of the properties of the gas clouds themselves . the emissivity shown in figures 4 and 5 is the quantity @xmath21 , whereas figures 6 and 7 show @xmath118 . for the inner galactic disk , where we are interested in @xmath119 , we need to average over the whole solid angle , @xmath120 , under consideration : @xmath121 . in order to calculate @xmath117 we need to adopt models for both the galactic cosmic - ray distribution and the galactic distribution of the clouds . the quantity @xmath108 , the density in cold , dense gas clouds , is only weakly constrained by direct observation , because the hypothetical clouds constitute a form of _ dark _ matter . we therefore proceed by adopting a conventional dark matter density distribution for the galaxy , namely a cored isothermal sphere , as our model cloud density distribution , with a fiducial normalisation which is equivalent to the assumption that all of the dark matter is in the form of dense gas clouds . this corresponds to the model @xmath122 in terms of cylindrical coordinates @xmath123 , with @xmath124 . we have adopted a core radius of @xmath125 kpc based on the preferred model of walker ( 1999 ) . ( this choice corresponds to walker s preferred value of cloud column density @xmath126 . ) walker s model exhibits a core radius which is a function of cloud column density , but we have fixed the core radius at 6.2 kpc for all of our computations . this choice permits more straightforward consideration of the observational constraints because @xmath117 is independent of @xmath6 in this case . it then remains to specify the cosmic - ray energy - density as a function of position in the galaxy . @xcite ( hereafter wlg92 ) constructed numerical models of cosmic - ray propagation in the galaxy ; they did not give any analytic forms for their model cosmic - ray distributions , but an appropriate analytic approximation can be deduced from the results which they obtained . they found that the cosmic - ray radial distribution reflects , in large part , the radial dependence of cosmic - ray sources , with a modest smoothing effect introduced by diffusion . we have therefore adopted wlg92 s preferred model ( their model 3 ) for the radial distribution of sources as our model for the radial distribution of cosmic rays . the various spectra of cosmic - ray isotope ratios considered by wlg92 favour models in which the diffusion boundaries are in the range @xmath127 kpc above and below the plane of the galaxy . we adopt the midpoint of this range . wlg92 do not give a simple functional form for the vertical variation of cosmic - ray density within this zone , so we have simply assumed an exponential model : @xmath128 . we know that in wlg92 s models the cosmic - ray density is fixed at zero at the diffusion boundaries , and consequently the scale height of the exponential should be approximately half the distance to the diffusion boundary , i.e. @xmath129 kpc . these considerations lead us to the model cosmic - ray mean intensity distribution @xmath130 : @xmath131,\ ] ] in terms of cylindrical coordinates @xmath132 . here @xmath133 kpc is the radius of the solar circle , while @xmath134 kpc , @xmath135 kpc and @xmath136 . this distribution has the character of a disk with a central hole . these models for @xmath108 and @xmath137 allow us to compute the quantity @xmath117 , as per equation 4 , and the resulting variation over the sky is plotted in figure 8 . for reference we give the values of @xmath138 evaluated at the cardinal points , as follows : @xmath139 , @xmath140 , @xmath141 , and @xmath142 . in order to compare with the egret results of @xcite , we have also evaluated the average of @xmath117 over the inner galactic disk : @xmath143 . several physical processes contribute to the observed diffuse gamma - ray intensity . in order of decreasing fractional contribution these are thought to be pion production and bremsstrahlung from cosmic - ray interactions with _ diffuse _ galactic gas , inverse compton emission from cosmic - ray electrons interacting with ambient photons , and an isotropic background , which is presumably extragalactic and due to many faint , discrete sources . the sum of these contributions offers a good model for the diffuse emission which is actually observed in the 100 mev1 gev band @xcite ; however , the @xmath144 gev emission is poorly modeled @xcite with the prediction @xcite amounting to only @xmath145 of the observed intensity . some authors @xcite have used the low - energy ( @xmath146 gev ) data to argue that the agreement between model and data allows no room for any significant unmodeled emission and that these data therefore place tight constraints on any baryonic contribution to the dark matter halo of the galaxy . this line of argument is clearly suspect because of the failure of the same model to account for the high - energy ( @xmath144 gev ) data . however , even if we ignore the high - energy data the calculations presented in 2.3 demonstrate that the gamma - ray constraints on high column density gas clouds in the dark halo are much weaker than those on low column density gas because the emissivity of the latter is much greater ; consequently , we revisit the published constraints in the following section ( 3.2 ) , returning to the question of the high - energy data in 3.3 . @xcite argued that the known ( diffuse ) gas accounts for essentially all of the gamma - ray intensity in the @xmath147 mev band observed by the cos b satellite @xcite , and he suggested that any contribution from baryonic clouds should amount to no more than @xmath148 ( @xmath147 mev ) at high galactic latitudes . toward the galactic poles we found ( 3 ) @xmath149 , implying that the average emissivity of the material in the dark halo should be @xmath150 for @xmath151 . from table 1 ( or figures 6 and 7 ) we see that this requirement is not met for the models which are in the thin material limit but that any model with @xmath152 is acceptable . in other words the constraint imposed by @xcite permits the galaxy s dark halo to be entirely baryonic , provided the individual components have a column - density @xmath152 . @xcite pointed out that obervations at low galactic latitudes can provide tighter constraints on any baryonic dark halo , because of the higher intensity expected when looking through the cosmic - ray disk edge - on . this point is manifest in the large values of @xmath153 , which we computed in 2.3 ( see also figure 8) . @xcite argued that a suitable constraint on any baryonic component of the dark halo is that it should not contribute more than the uniform background intensity component observed in any given field . the strongest constraint then comes from observations of the ophiuchus region @xcite , for which the requirement imposed by @xcite is an intensity contribution from the dark matter halo of @xmath154 in the band 300 mev @xmath155 500 $ mev . for this line of sight our calculation yields @xmath156 , and in the thin material limit , for which we find @xmath157 ( 300 mev @xmath155 500 $ mev ; see table 1 ) , this corresponds to a predicted intensity of @xmath158 in the 300500 mev band , implying that at most 8% of the dark halo can be resident in low column density clouds . kpc and a spherical halo ) , a difference which is accounted for by their choice of a smaller core radius for the dark halo . @xcite chose a core radius of 3.5 kpc , which yields @xmath159 , whereas we have employed a core radius of 6.2 kpc ( see 2.3 ) . ] however , our calculations extend to clouds of higher column - densities , and we find that for @xmath160 the limit relaxes to the point where all of the dark halo is permitted to be in the form of high column density clouds . the constraints discussed in the previous subsection make reference only to the low - energy ( @xmath146 gev ) gamma - ray data . as noted earlier in this section , the observed intensity of the inner galactic disk at high photon energies ( @xmath161 gev ) is substantially greater than expected @xcite . much effort has been expended on explaining this discrepancy , with most of the attention given to models in which the galactic cosmic - ray electron and/or proton spectra differ from their locally measured values @xcite . however , none of these models offers a satisfactory explanation of the observed mean gamma - ray spectrum of the galactic disk , and consequently a successful match to the low energy data should not be taken to mean that the emission model is correct . in turn this suggests that the constraints formulated by @xcite and @xcite , on the basis of the low - energy data , may be too restrictive . the question then arises as to what constraints the gamma - ray data do in fact place on unmodeled emission , given the current state of understanding of the observed emission . the galactic diffuse emission model used in @xcite is based on @xcite , and contains the following contributions : @xmath162 where @xmath163 is the gamma - ray intensity contributed by cosmic - ray interactions with diffuse atomic hydrogen , and similarly for the ionised and molecular components of the interstellar medium ( hii and @xmath164 , respectively ) . the emissivity for these components @xcite is , of course , computed in the thin material limit , and each component therefore has the same spectral shape , differing only in normalisation . here @xmath165 is the gamma - ray flux by inverse compton emission ( cosmic - ray electrons up scattering low - energy photons ) , and @xmath166 is the isotropic extragalactic background flux . although the atomic and ionised hydrogen components can be observed directly via their line emission , and are thus well constrained , this is not true for the molecular component . the molecular hydrogen column is assumed to be proportional to the co emission line strength , as measured by the columbia co survey @xcite , for example , but the constant of proportionality ( usually denoted by @xmath167 ) is unknown and one is forced to determine its value by fitting to the gamma - ray data . although the uncertainty in the best - fit determination of @xmath168 ( averaged over the whole sky ) is small , the systematic uncertainties are acknowledged to be much larger , `` at least 10%15% '' @xcite . in turn , this estimate of the uncertainty is small in comparison with the differences among the various values of @xmath167 which have been reported in the literature ( see the review by @xcite ) and the likely range of variation in @xmath167 within the galaxy @xcite . although these are substantial uncertainties , molecular hydrogen contributes only a fraction of the total observed gamma - ray intensity roughly 20% of the local surface density of gas in the galactic disk is thought to be in molecular form @xcite so the implied fractional uncertainty in the total galactic emission is perhaps as small as 3% . moreover , uncertainty in @xmath167 affects only the normalisation and not the spectral shape of the predicted emission from diffuse molecular hydrogen , so the observed high - energy excess can not be explained in this way even if @xmath167 could assume an arbitrarily large value . the other main contributions to uncertainty in the diffuse model are associated with ( 1 ) unmodeled spatial variations in the cosmic - ray spectral energy density and ( 2 ) unmodeled spatial variations in the low - energy photon spectral energy density ; the former affects all of the galactic contributions to @xmath169 , whereas the latter affects only the inverse compton component . these uncertainties affect both the normalisation and the spectral shape of the predicted gamma - ray emission ; however , the uncertainties are difficult to quantify . for our purposes it is not actually necessary to quantify the uncertainties on the model input parameters ; it suffices to use the discrepancy between model and data as a measure of the uncertainty in our understanding of the observed emission . in turn this measure determines the constraints which we can apply to any putative unmodeled emission , such as the contribution from dense gas which we are concerned with here . at photon energies @xmath170 the fractional discrepancy is roughly 60% @xcite , in the sense that the observed emission is 1.6 times larger than the model , and we henceforth adopt @xmath171% of the total observed intensity as our estimate of the unmodeled emission . although this estimate is derived from data at high energies , the effects of the various contributing processes are all very widely spread , and _ the estimate therefore applies independent of photon energy . _ the constraints appropriate to high / low galactic latitudes can now be re - evaluated . at high galactic latitudes the observed intensity is @xmath172 for @xmath173 @xcite , implying that any unmodeled emission should be @xmath174 in this band . this result is actually slightly stricter than the criterion used by @xcite and thus leads us to tighten our high - latitude constraints , relative to those quoted in 3.2 : the observed high - latitude gamma - ray intensity constrains the amount of low column - density gas to @xmath175% of the total density of the galactic dark halo , with this fraction rising to 100% for gas clouds of column density @xmath176 . at low galactic latitudes we can make use of the mean intensity of the inner galactic disk , which has been accurately determined by @xcite . for example at 1 gev the mean intensity ( @xmath177 , @xmath178 ) is @xmath179 , and our calculation of @xmath180 for this region yields ( 2.3 ) @xmath181 , implying that the emissivity of the galactic dark halo material must be , on average , @xmath182 . by comparison , the actual emissivity of low column - density gas is computed to be ( 2.3 , table 2 ) @xmath183 , implying that @xmath184% of the galaxy s dark halo may be comprised of low column density gas . for higher column densities the emissivity falls , and table 2 shows that for @xmath185 the emissivity is only @xmath186 . the gamma - ray data on the inner galactic disk thus indicate all of the galaxy s dark halo to be made of dense clouds of column - density @xmath152 . the constraint we have just given is based on the mean spectrum of the inner galactic disk , in contrast to those given by @xcite who employed limits based on the angular structure of the observed gamma - ray intensity . specifically , @xcite required that the putative contribution of emission from a baryonic component of the galaxy s dark halo be less than that of the isotropic component of the observed intensity ; this procedure seems to us to be less reliable than the procedure we have employed , for two reasons . first , even if the dark halo were spherically symmetric the emission from any baryonic component would not be , both because our point of observation is quite distant from the centre of the galaxy and because the resulting gamma - ray emission is strongly dependent on the galactic cosmic - ray distribution , which in turn is strongly concentrated in the disk of the galaxy . indeed the cosmic - ray distribution appears to correlate with the distribution of interstellar matter @xcite , thus complicating the interpretation of the observed gamma - ray intensity distribution . in particular this coupling leads to gamma - ray emission from a baryonic dark halo being correlated with the diffuse gas column density , even if the dense gas is uncorrelated with the diffuse gas . second , on any given line of sight , such as the ophiuchus field considered by @xcite , a highly structured dark matter halo might exhibit , by chance , a low dark matter column density . the chances of this are good if , as in the case of @xcite , the field is specifically chosen to have a low `` background '' intensity . the gamma - ray spectra arising from cosmic - ray interactions with gas clouds of various column - densities have been calculated using a monte carlo event simulator , geant4 . our calculations reproduce the analytic result in the low column - density limit , where only single particle interactions need to be considered , but exhibit significant differences for clouds of column - density @xmath187 where the emissivity declines substantially for photon energies @xmath188 . the low emissivity of dense gas means that the baryonic content of the galaxy s dark halo is not so tightly constrained by the gamma - ray data as had previously been thought . for @xmath176 we find that the existing gamma - ray data , taken in isolation , do not exclude purely baryonic models for the galactic dark halo . aharonian , f. a. , & atoyan , a.m. 2000 , 362 , 937 asakimori , k. et al . 1998 , , 502 , 278 barwick , s. w. et al . 1998 , , 498 , 779 berezhko , e.g. , & vlk , h.j . 2000 , , 540 , 923 bertsch , d.l . 1993 , , 416 , 587 bloemen h. 1989 , , 27 , 469 boezio , m. et al . 1999 , , 518 , 457 boezio , m. et al . 2000 , , 532 , 653 bsching , i. , pohl , m. , & schlickeiser , r. 2001 , 377 , 1056 casadei , d. , & bindi , v. astro - ph/0302307 cavallo , g. , & gould , r. j. 1971 , nuovo cimento , 2b , 77 combes , f. & pfenniger , d. 1997 , , 327 , 453 dame , t. m. et al . 1987 , , 322 , 706 dannheim , d. `` test of @xmath7-production in geant4 '' , cern summer student report , 15 september 1999 ( unpublished : available at cern website ) de paolis , f. et al . 1995 , , 74 , 14 dermer , c.d . 1986 , , 157 , 223 fiedler , r. et al . 1994 , , 430 , 581 fukugita m. 2004 , in iau symp . 220 , in press ( astro - ph/0312517 ) gaisser , t. k. & shaefer , r. k. 1992 , , 394 , 174 gerhard , o. & silk , j. 1996 , , 472 , 34 gilmore , g. 1994 , , 92 , 539 golden , r. l. et al . 1984 , , 287 , 622 golden , r. l. et al . 1994 , , 436 , 769 hartman et al . 1999 , , 123 , 79 henriksen r. n. , & widrow l. m. 1995 , , 441 , 70 hunter , s.d . , digel , s.w . , de geus , e.j . , & kanbach g. 1994 apj 436 , 216 hunter , s.d . 1997 , , 481 , 205 ivanenko , i. p. et al . 1993 , in proc . of the 23rd international cosmic ray conference , calgary , 2 , 17 kalberla , p.m.w . , shchekinov , yu.a . & dettmar , r .- j . 1999 , 350 , l9 kawamura , y. et al . 1989 , , 40 , 729 kniffen d.a . 1996 , suppl . 120 , 615 kobayashi , t. et al . 1999 , proc . . cosmic - ray conf . ( salt lake city ) , 3 , 61 mckee , c.f . 2001 asp conf ser 243 , 155 mori , masaki 1997 , , 478 , 225 moskalenko , i.v . & strong , a.w . 1998 , 493 , 694 pfenniger , d. , combes , f. , & martinet , l. 1994 , , 285 , 79 ryan , m. j. et al . 1972 , , 28 , 985 salati , p. , chardonnet , p. , luo , x. , silk , j. , & taillet , r. 1996 , 313 , 1 sanuki , t. et al . , 2000 , , 545 , 1135 sciama , d.w . 2000a , , 312 , 33 sciama , d.w . 2000b , , 319 , 1001 seo , e. s. et al . 1991 , , 378 , 763 shikaze , y. 2001 , ph.d . thesis , university of tokyo skibo , j.g . and ramaty , r. 1993 , suppl . , 97 , 145 smith , l. h. et al . 1973 , , 180 , 987 stephens , s.a . & badhwar , g.d . 1981 , , 76 , 213 strong , a.w . , moskalenko , i.v . & reimer , o. 2000 , , 537 , 763 tang , k .- k . , , 1984 , 278 , 881 torii , s. et al . , , 2001 , 559 , 973 umebayashi , t. & nakano , t. , , 1981 , 33 , 617 walker , m. & wardle , m. , 1998 , , 498 , l125 walker , m. 1999 , , 308 , 551 walker , m. , mori , m. & ohishi , m. 2003 , , 589 , 810 wardle , m. & walker , m. 1999 , , 527 , l109 webber , w. r. et al . 1987 , in proc . of the 20th international cosmic ray conference , moscow , 1 , 325 webber , w.r . , lee , m. & gupta , m. 1992 390 , 96 ( wlg92 ) wouterloot , j.g.a . , brand , j. , burton , w.b . , & kwee , k.k . 1990 230 , 21

gamma - ray spectra from cosmic - ray proton and electron interactions with dense gas clouds have been calculated using a monte carlo event simulation code , geant4 . such clouds are postulated as a possible form of baryonic dark matter in the universe . the simulation fully tracks the cascade and transport processes which are important in a dense medium , and the resulting gamma - ray spectra are computed as a function of cloud column - density . these calculations are used for predicting the galactic diffuse gamma - ray spectrum which may be contributed by baryonic dark matter ; the results are compared with data from the egret instrument , and used to constrain the fraction of galactic dark matter which may be in the form of dense gas clouds . in agreement with previous authors , we find useful constraints on the fraction of galactic dark matter which may be in the form of low column - density clouds ( @xmath0 ) . however , this fraction rises steeply in the region @xmath1 , and for @xmath2 we find that baryonic dark matter models are virtually unconstrained by the existing gamma - ray data .

arxiv

colloidal suspensions present the possibility to develop novel materials via self - assembly . of particular interest are colloidal crystals , whose optical properties can generate iridescent colours , and provide a means by which photonic crystals may be produced@xcite , while further applications range from lasers @xcite to display devices @xcite , with recent advances demonstrating tunable colours through control of lattice spacing with an external field @xcite . further to the practical importance of colloidal crystals , their well - defined thermodynamic temperature allows colloidal dispersions to be viewed as mesoscopic ` model atoms ' @xcite . recently , the ability to tune the colloid - colloid interactions has led to the observation of a wide variety of structures @xcite . of particular interest here , to first order ac electric fields can induce dipolar interactions between the colloidal particles , leading to anisotropic interparticle potentials and exotic crystal structures , some of which are not observed in atomic and molecular systems @xcite , while external control of the colloid - colloid interactions allows direct observation of phase transitions @xcite . furthermore , direct microscopic observation at the single - particle level allows an unprecedented level of detail to be accessed @xcite , opening the possibility of tackling long - standing problems in condensed matter , such as freezing @xcite . the introduction of a rotating ac field opens up even more possibilities . in this case , the dipolar interactions lead to an attraction in the plane of rotation and to repulsions above and below . studies with a rotating magnetic field on granular matter indeed produced disc like patterns consistent with expectations @xcite . unlike granular matter , since colloidal dispersions exhibit brownian motion , thermodynamic equilibrium structures ( ie crystals ) , may be obtained @xcite . in previous work snoswell _ @xcite showed that lattice spacing within quasi-2d colloidal crystals could be controlled _ in - situ _ , by means of coplanar rotating electric field . the interparticle dipolar interactions in the plane of the electric field may be treated to first order as a circularly symmetric attraction , due to the time averaging effect of a rapidly rotating field ( 1000hz ) on relatively large particles on the micron lengthscale , where the diffusive timescale is of the order of seconds @xcite . in considering the interactions between the particles , the asymmetry between the colloids ( 10nm-1@xmath0 m ) and smaller molecular and ionic species must be addressed . a number of coarse - graining schemes have been developed where the smaller components are formally integrated out @xcite . this generates a one - component picture , where only the effective colloid - colloid interactions are considered , and the complexity of the description is vastly reduced . the equilibrium behaviour of the colloids in the original multi - component system may then be faithfully reproduced by appeal to liquid state theory @xcite and computer simulation @xcite . central to the success of this one - component approach is the use of a suitable effective colloid - colloid interaction @xmath1 . in this study , we use a simple numerical treatement in which we can predict the lattice spacing in the quasi 2d crystal from the electric field strength . we consider a model system of charged colloids , in a rotating electric field @xcite . by exploiting the knowledge both of the electrostatic repulsions and dipolar attractions , we present a direct , quantitative comparison of a tunable interaction and a material property of the crystalline ` rafts ' formed . we combine experimental measurements of the crystal lattice constant @xmath2 as a function of field strength @xmath3 with monte - carlo simulations according to a screened coulomb repulsion plus dipolar attraction where the only fitting parameter is the debye screening length . in the simulations , we use pairwise interactions , in other words we assume that at the higher densities at which the crystalline rafts are formed , the system is still accurately described by interactions calculated for two particles in isolation . we note that deviations from this assumption of pairwise additivity have been measured both in the case of strongly charged colloids @xcite and in the case of _ repulsive _ dipolar interactions @xcite . we further compare simulation results with the minimum of the effective potential , which we take as a measure of the lattice constant of the crystalline rafts , which we also determine from experimental data . this paper is organised into six sections . in section [ theory ] we present expressions for the effective interactions between the colloids , summing the attractions and repulsions to provide an effective one - component description of the system . section [ experimental ] describes our experimental metholodogy . section [ simulation ] outlines the monte - carlo simulation technique employed . the comparison of simulation and experimental results is presented in section [ results ] and in section [ tunability ] we extrapolate our findings to maximise the tunability of the crystal lattice constant , which may be useful for applications . we conclude our findings in section [ conclusions ] . in the following we will consider a system consisting of two particles in a surrounding medium . we shall assume that these particles are charged , leading to a repulsive interaction , and that the rotating ac electric field induces a dipole moment in the two particles and thus induces an attractive interaction . to describe this system , we start from the derjaguin , landau , verwey and overbeek ( dlvo ) approach @xcite , which consists of attractive van der waals interactions at short range , and long - ranged repulsive electrostatic interactions . the van der waals interactions are very short - ranged , and are neglected , as electrostatic repulsions inhibit the close approach at which van der waals interactions become important . we shall therefore assume that the only relevant attractions result from the long - ranged dipolar interactions induced by the rotating electric field . in the linear poisson - boltzmann regime , the electrostatic repulsions may be expressed as a hard core yukawa , or screened coulomb interaction @xcite , @xmath4 where @xmath5 where @xmath6 is boltzmann s constant , @xmath7 is temperature and @xmath8 is the colloid diameter . the potential at contact , @xmath9 is given by @xmath10 where @xmath11 is the colloid charge , @xmath12 is the bjerrum length and @xmath13 is the inverse debye screening length where @xmath14 is the total number density of monovalent ions . now the regime of linear poisson - boltzmann theory in which equation ( [ eqyuk ] ) holds corresponds to relatively weak charging . although this is not the case here , a potential of the yukawa form is recovered at larger separations , if a smaller , renormalized charge is considered @xcite . we tabulate measurements of the @xmath15-potential of dilute suspensions in table [ tablekappasigma ] . these values suggest that we expect a renormalised charge , in the conditions under which the colloids form the crystalline rafts , rather high colloid concentration , hence high counter ion concentration @xcite . therefore , noting that the debye length is much smaller than the colloid radius , we follow bocquet _ et . _ @xcite and take the following expression for the renormalised charge @xmath16 : @xmath17 which we substitute for @xmath11 in equation ( [ eqepsilonyuk ] ) . this expression gives good agreement with measurements of the effective colloid charge for particles with a comparable @xmath15-potential @xcite . we recall that many - body effects can lead to a density - dependence in the effective colloid - colloid interactions @xcite . however , we shall neglect these effects in the present work . the attractive potential between the colloids resulting from the rotating ac electric field is treated to first order as a dipolar interaction : @xmath18 where @xmath19 is the induced dipole moment of each particle , @xmath20 is the dielectric constant and @xmath21 is the permittivity of free space . the dipole moment may be calculated from the strength of the electric field @xmath22 where @xmath23 is the clausius - mosotti factor and takes values between 1 and -1/2 depending upon the origin of the dipole moment . we note that in this case of an alternating field , the root - mean - square of the time dependent field is taken . substituting @xmath24 leads us to a potential at contact @xmath25 which has a cubic dependence on the colloid diameter , in the case of all other contributions being unchanged . in considering the value of the clausius - mosotti factor , two regimes are relevant , corresponding to applied electric fields of high and low frequency , in which the dipolar interactions result from dielectric permittivity differences between the particles and the solvent and differences in conductivity respectively @xcite . the crossover frequency @xmath26 between these regimes is given by @xmath27 where @xmath28 are conductivities and the indices @xmath19 and @xmath29 refer to the colloidal particles and the medium respectively . following @xcite the conductivity of the particles @xmath30 is taken to be the sum of the bulk conductivity @xmath31 and the conductivity on the surface @xmath32 . @xmath33 in principle equation ( [ eqconductivity ] ) holds for particles larger than 1 @xmath0 m . in the case of much smaller particles , one should take into account that the contribution of the diffuse layer to the conductivity of the particles . noting that the smallest particles used in the experiments are 757 nm in diameter and that the conductivity of the de - ionised water is relatively low , neglect the double layer contribution and thus use equation ( [ eqconductivity ] ) . since the largest particles studied are 2070 nm in diameter and the frequency is 1 khz and the bulk ionic strength is @xmath34 mmol as determined from conductivity measurements , we work in the low frequency regime where the contributions from the conductivities dominate @xcite . the clausius - mosotti factor therefore takes the following form : @xmath35 due to the surface conductivity , @xmath30 is much larger than the conductivity of the medium @xmath36 . the value of @xmath37 is therefore close to one . we now combine the contributions from the electrostatic repulsions and the dipolar attractions , yielding the expression . @xmath38 @xmath39 has the form indicated in fig . we see a minimum in the potential , which one might expect to provide a first approximation to the lattice constant in the 2d colloidal crystal . in sections [ results ] and [ tunability ] we shall compare this minimum to our simulation results . colloidal crystals underwent self - assembly as a result of an applied electric field . we used anionic , sulphate stabilised polystyrene latex particles , either synthesised using a standard technique , surfactant free emulsion polymerisation @xcite in the case of @xmath40757 and 945 nm , or particles produced by the same method but purchased from microparticles gmbh for @xmath401390 and 2070 nm . particle electrophoretic mobility was measured using a brookhaven zetaplus light scattering instrument . particle sizes were determined by scanning electron microscopy , either in - house using a jeol jsm-6330f , in the case of @xmath40757 and 945 nm or by microparticles gmbh for @xmath401390 and 2070 nm are are listed in table [ tablekappasigma ] . experiments were performed with dilute aqueous suspensions ( 0.5 - 1.5 wt% ) . schematics of the experimental set up are shown in fig . [ figschematic ] all glass surfaces were chemically washed with 0.1 m koh and washed with copious quantities of milliq water . particles were deionised by direct contact with ion exchange beads before being made up to the desired electrolyte concentration with kcl . very low salt conditions are required as even moderate ( mmol ) salt concentrations lead to electrohydrodynamic pattern formation due to ion flow @xcite . for further details of the experimental set up and procedure the reader is referred to snoswell _ et . _ @xcite . we consider our experiment as a 2d system . however , occasionally we noticed some overlap of the crystalline rafts . we do not include these results in our analysis . in order to treat the system in 2d , one might expect the gravitational length @xmath41 , where @xmath29 is the bouyant mass of the particle and @xmath42 is the acceleration due to gravity , to be much less than some characteristic length such as the particle diameter . due to the fairly small density mismatch between polystyrene and water , in fact even for the largest particles @xmath43 nm , @xmath44 . however , the dipolar interactions between the particles are attractive in - plane , but strongly repulsive in the vertical direction . this promotes the formation of ` rafts ' and ` sheets ' , and these large assemblies of many particles have very small gravitational lengths . thus we argue that the system behaves in a quasi 2d manner . the lattice parameter @xmath2 was taken as the average of typically ten crystalline ` rafts ' , measured across the ` raft ' , of around ten lattice spacings . the response to changing the electric field strength was determined to be less than 100 ms , we waited 5s after changing the field strength before acquiring data . we use standard monte - carlo ( mc ) simulations in the nvt ensemble @xcite . the particles interact via equation ( [ equ ] ) in two dimensions . to mimic the experiments , we initialise the system in a random configuration at a relatively low concentration , corresponding to an area fraction @xmath45 . we confirmed that different area fractions gave indistinguishable results , however smaller area fractions often led to longer equilibration times . the attractive interactions cause the particles to approach one another , and form crystallites which then coalesce to form a crystalline ` rafts ' which contain of order @xmath46 particles , in a qualitatively similar way to the experimental system . each simulation was equilibrated for typically 30000 mc moves per particle , followed by 3000 production moves per particle . we recall that a potential of the form @xmath47 converges in 2d . the lattice constant for this 2d hexagonal crystal was taken as @xmath48 where @xmath49 is the pair correlation function , and @xmath50 the minimum between the first and second peaks of @xmath49 . we found that when we used @xmath51 , that typically one crystalline domain was formed , around the centre of the simulation box . we therefore do not use periodic boundary conditions , and consider the lattice spacing of the single crystal formed in the simulation . one parameter is the number of particles typically present in each crystalline domain . this is estimated to be around @xmath52@xmath53 from experimental data ( see fig . [ figpix ] ) . this value is governed by the overall concentration of the colloidal suspension . we have considered the effects of varying @xmath54 as shown in fig . [ figfinitesize ] . the results of a considerable range of @xmath54 resulted in a variation of a around one percent in @xmath2 for the @xmath55 nm system for a field @xmath3=20 @xmath56 . each simulation was repeated four times except @xmath57 . we found a slight trend to tighter binding for larger @xmath54 , however , as shown in fig . [ figfinitesize ] , this effect is rather small . we henceforth use one simulation per state point unless otherwise stated . the slight scatter in the simulation data is perhaps indicative the existence of different magic numbers for these crystalline rafts @xcite . these may be thought of as 2d hexagonal clusters , whose ` magic numbers ' , ie low energy states , are expected to include @xmath58 , @xmath59 , @xmath60 ... close to a ` magic number ' the binding may be expected to be relatively tighter . a more detailed exploration of this phenomenon lies beyond the scope of this work . overall , the variation in @xmath2 as a function of @xmath54 is smaller than the experimental scatter , nonetheless we take both @xmath61 and @xmath62 values for @xmath2 when comparing with experimental data . the slightly larger value for @xmath63 in fig . [ figfinitesize ] is attributed to the small size of the crystalline ` raft ' , such that the surface particles which are more widely spaced make a greater contribution . the domain size is of order 100 particles , thus , for larger @xmath54 , defects and grain boundaries may lead to a smaller contribution to the measurement of @xmath2 . this also applies to experimental data . for fitting , all parameters are known , except the debye length @xmath64 which we take as a free parameter for each particle size and salt concentration , although we note that the effective colloid charge is itself a function of the debye length , equation ( [ eqzeff ] ) . it has previously been shown that comparison of simulation and experimental data can yield a reasonable _ local _ measure of the debye length @xcite . furthermore , in the region of the sample in which the electric field is applied , the colloid volume fraction is much higher , and due to the colloidal counter - ions the ionic strength may increase , leading to a reduction in the debye length with respect to the bulk . however , it is the debye length in this region which is relevant for the effective colloid - colloid interactions . the bulk debye length that may be determined , for example from conductivity measurements , may therefore be taken only as an upper bound . system parameters for different particle sizes are tabulated in table [ tablekappasigma ] , and the debye length is plotted in fig . [ figkappasigma](a ) and the contact potential of the yukawa interaction [ equation ( [ eqepsilonyuk ] ) ] in fig . [ figkappasigma](b ) are plotted as a function of particle diameter . clearly , our mc simulations may be expected to provide a reasonable treatment of the crystalline rafts in ( quasi ) equilibrium , rather than to describe the formation process . we therefore restrict our analysis to a simple characterisation of the crystal rafts , and primarily consider the lattice spacing . we furthermore note both here and in previous work @xcite that considerable variation in shapes of crystalline rafts was observed , but that this had little impact upon the measurement of the lattice spacing @xmath2 . likewise , at the level of this work , we neglect possible local variations in @xmath2 due to the proximity of an interface . in any case we note that for display applications , all lattice spacings contribute to the diffraction . a few words on equilibration are in order . this applies both to the experimental system , and to the simulations . neither system is strictly in equilibrium , in that case we might expect a rather regularly - shaped raft such as a hexagon . however , the insensitivity of the lattice parameter either as a function of @xmath54 ( fig . [ figfinitesize ] ) , and the very close agreement between statistically independent simulation runs lead us to conclude that our approach is sufficient to compare lattice parameters between experiment and simulation . the effect of changing the electric field strength is readily demonstrated in fig . [ figpix ] , which shows optical microscope images of the region of the sample in which the field is applied . two field strengths are shown , 29 kvm@xmath65 ( a ) and 80 kvm@xmath65 ( b ) for @xmath55 nm and we see a correspondingly tighter lattice in the case of the higher field strength . similar behaviour was observed in @xcite . figure [ figpix](c ) illustrates simulated coordinates for @xmath66@xmath67 @xmath68 , @xmath69 , @xmath70@xmath71 @xmath68 , which corresponds to @xmath3=29.0 vm@xmath65for the @xmath55 nm system ( see table [ tablekappasigma ] ) . a more quantitative comparison between experiment and simulation is shown in fig . [ figg ] , which plots the radial distribution function @xmath49 . we see that the structure appears to be crystalline , with reasonably long - ranged correlations , although , as is clear from fig . [ figpix ] , the system is too small to test for truly long - ranged order . we thus note that it is difficult to distinguish the true crystal from a hexatic phase in this system . however , our motivation is to model the experimental system , with @xmath72 and thus we argue that identification of the hexatic phase lies beyond the scope of this work . having illustrated the general behaviour of the system , we now turn our attention to the fitting of the experimental data , with the model described in section ii . the experimental data was fitted to the theory by taking the debye - length as the fitting parameter . the main results are shown in figs . [ figsmallexpmc ] and [ figbigexpmc ] for @xmath73 and 945 nm , and @xmath74 and 2070 nm systems respectively . these concern the lattice parameter @xmath2 as a function of applied field strength @xmath3 . in general , the simulation is able to capture the behaviour of the system in a reasonably quantitative manner . furthermore , simply plotting the minimum in the potential @xmath75 given by equation ( [ equ ] ) provides a first approximation to the lattice constant @xmath2 . figure @xmath76(a ) shows the results for @xmath73 nm particles . due to the small particle size , and the inverse cubic dependence of the strength of the dipolar attractions on the particle diameter , as characterised by @xmath77 [ equation ( [ eqepsilondip ] ) ] the attraction for a given field strength is comparitively small , so we find relatively larger lattice parameters for this system . at low field strengths ( @xmath78 @xmath56 ) we see an increase of the lattice constant for both the experimental data and the simulation , compared to the minimum in the potential . apparently , as the crystal approaches melting , fluctuations become more pronounced , leading to an increase in @xmath2 , which is not captured in equation ( [ equ ] ) , or perhaps a molten surface layer increases the apparent value of @xmath2 . we leave this intriguing question for further investigations . meanwhile , at high field strengths , we find some deviation between simulation , experiment and equation ( [ equ ] ) . apparently , due to the long ranged nature of the interaction , second nearest neighbours experience an attraction of sufficient strength that the crystal is compressed by its own cohesion , leading to a smaller lattice constant . in the case of the @xmath73 nm particles , we find that for small field strengths , ( @xmath79 @xmath56 ) , the system does not form crystallites , rather it remains as a colloidal liquid . we determined this both by experimental observation and with simulations . in the latter case , we identified crystallisation with a splitting in the second peak of the pair correlation function @xmath49 . the kink around @xmath80 @xmath56 in fig . [ figsmallexpmc ] is likely related an artifact of measuring multiple rafts , so defects and grain boundaries contribute to the value of @xmath2 . at weak field strengths the rafts tend to be smaller , thus increasing this effect . at larger particle sizes , ( fig . [ figbigexpmc ] ) , we again see reasonable agreement between experiment and simulation , albeit with some deviation between simulation and equation ( [ equ ] ) at higher field strengths , as the simulation predicts a smaller lattice constant than that observed in the experiments . however , for the @xmath74 nm system in particular , in fact equation ( [ equ ] ) provides a more accurate description of the experimental data . apparently some of our assumptions in the simulations , perhaps that of pairwise additivity , begin to break down at high field strengths . according to equation ( [ equ ] ) , the potential at the minimum of the attractive well is some 250 @xmath68 for an applied field of 50 kvm@xmath65 for @xmath74 nm . thus we conclude that for moderate interaction strengths , equation ( [ equ ] ) provides a reasonable description of the system . we now consider the dependence of the system upon the colloid diameter @xmath8 . firstly , we see from fig . [ figkappasigma](a ) that the absolute value of the ( fitted ) debye length @xmath64 does not change significantly for all the four particle sizes studied . thus , the reduced inverse debye length @xmath81 is linear in @xmath8 . the contact potential @xmath82 has an approximately inverse cubic dependence on @xmath8 for @xmath81@xmath83 , equation ( [ eqepsilonyuk ] ) , but also depends upon the ( effective ) charge @xmath16 . plotting @xmath82 as a function of @xmath8 in fig . [ figkappasigma](b ) we find an approximate power law dependence with an exponent of @xmath84 . as noted above , the prefactor of the dipolar attraction , @xmath77 [ equation ( [ eqepsilondip ] ) ] , has a @xmath85 dependence . although the electrostatic interactions are non - negligible , we nonetheless expect from equation ( [ eqepsilondip ] ) that upon decreasing @xmath8 we find a relatively larger lattice constant for a given field strength , and that larger field strengths are required to provide sufficient interactions that the crystalline rafts form . this we indeed find , as shown in figs . [ figsmallexpmc ] and [ figbigexpmc ] , and also in the next section . now we have noted that this system has optical display applications , due to the possibility of externally tuning the lattice parameter @xmath2 . these applications may be most usefully realised when the possibility to tune the system is maximised , ie when the range of @xmath2 is maximised . we are therefore motivated to consider a range of particle sizes , and calculate the range of tunability of @xmath2 . now @xmath8 sets a lower bound to @xmath2 while the upper bound is set by the melting transition which can be determined by monte - carlo simulation . hitherto , we have used the debye length @xmath64 as a fitting parameter . however , as fig . [ figkappasigma](a ) shows , there is relatively little change in the absolute value of the debye length for the range of colloid diameters investigated . we therefore fix @xmath86 nm and vary the colloid size , and apply the same methodology as that outlined in sections [ theory ] and [ simulation ] , in calculating the response for @xmath87 nm colloids to an external electric field . while an accurate determination of the melting transition is complicated by the small system size , melting is approximately determined by the splitting of the second peak in the radial distribution function @xmath49 @xcite . we decrease the electric field to identify the weakest field at which the second peak in @xmath49 exhibits clear splitting [ fig . [ figtuning](a ) , inset ] , thus yielding @xmath88 , the lattice spacing just prior to melting . the melting value of the lattice parameter @xmath89 is taken as @xmath90 where @xmath91 is the amount by which the field strength is varied between simulations , typically @xmath92 @xmath56 . the error in @xmath89 is then taken as @xmath2@xmath93 . in terms of the particle diameter , @xmath89 grows considerably at small sizes , as the relatively larger debye length leads to a minimum in the effective potential at relatively larger distances . however , we recall that our analysis suggests a roughly @xmath94 dependence of @xmath77 the strength of the dipolar attraction . this suggests that we might expect a stronger electric field to be required for relatively small colloids , and indeed we find that a higher field is required to provide sufficient cohesive energy to hold the colloids together in the crystal - like ` rafts ' , so at 100 @xmath56 , a value we take as a reasonably accessible maximum field strength , the lattice spacing is still around @xmath95@xmath8 for @xmath96 @xmath97 , where @xmath98 is the lattice spacing corresponding to a field strength of 100 @xmath56 . conversely , at larger particle sizes , the debye length is relatively small , so we find , even at melting , that the lattice constant only approaches around @xmath99@xmath8 . we note that , at high field strengths , the assumptions of section [ theory ] will ultimately break down . the results presented in figs . [ figsmallexpmc ] and [ figbigexpmc ] show that @xmath100 @xmath56 is reasonable , at least for @xmath101 nm , supporting our conclusion of a maximum in tuneability around @xmath99@xmath8 . we show the values of the well depth of equation ( [ equ ] ) at melting for the different experimental system in table [ tablekappasigma ] , which indicates a trend towards deeper wells for larger particles , i.e. shorter ranges of the repulsive interaction relative to the particle size . larger field strengths may be expected to yield greater tunability for the small particle sizes . since the overall minimum lattice spacing is @xmath102 , in principle smaller particles have greater tunability , however experimental observations reveal intense fluid turbulance disrupts crystal formation at higher field strengths . this is caused by large field gradients at the electrode edges that induce strong dielectrophoretic forces . in addition , higher field strengths can cause bubble formation by electrolysis and electrochemical degradation of the electrodes . the result is thus that intermediate particle sizes have the highest degree of tuneability , for a given maximum field strength . here we have considered 100 @xmath56 , which leads to a maximum tuneability of @xmath103 in the range 1000 nm @xmath1041500 nm . 2d colloidal crystalline ` rafts ' with externally controllable properties have been modeled with monte - carlo simulations , and the resulting lattice constant , both experimental and simulated , has been compared to the minimum in the effective interaction potential . in treating the effective colloidal interactions , we find that a straightforward , pairwise approach provides a reasonable description in describing the lattice parameter . the minimum in the effective interaction formed from combining the electrostatic repulsions and dipolar attractions is a useful means to approximate the lattice parameter , although close to melting fluctuations lead to a larger value in the lattice parameter than this minimum . conversely , at high field strengths , our treatment appears to be less accurate . apparently , higher order terms which we have not accounted for may become important , such as non - linear poisson - boltzmann contributions to the electrostatic interactions , leading to a deviations from the yukawa form [ equation ( [ eqyuk ] ) ] @xcite or limitations in our pairwise treatment of the dipolar attractions @xcite . other possibilities include inaccuracies in our assumptions for the effective colloid charge [ equation ( [ eqzeff ] ) ] , limitations of the 2d behaviour of the experimental system . perhaps due to a cancellation of errors , at high field strengths , the minimum in the effective interaction can sometimes provide a more accurate value of the lattice parameter than the simulations . another important assumption lies in our derivation on the electrostatic interactions . while their yukawa - like form is well - accepted @xcite , the values the effective colloid charge comes from equation ( [ eqzeff ] ) . we note that the debye length we determine seems rather constant across the four particle sizes considered , [ fig.[figkappasigma ] ( a ) ] , which suggests that out approach is at least consistent . however challenging it may be , a more quantitative measurement of the local ion concentration in the vicinity of the crystalline rafts is desirable and will be considered in the future . furthermore , there may be some variation in the value of @xmath16 . in fact this should affect all state points in a similar manner ( for each particle size ) , and therefore we anticipate a similar outcome , but that a different value might be arrived at in the fitting of the debye length , however , owing to the relatively short - ranged nature of the interaction , the effect of a different value for the effective charge is unlikely to impinge significantly the debye length and should have a negligible effect on our main results . we have also assumed that a given system is described by _ _ one single__debye length . in fact , in the counterion - dominated regime , the debye length is in fact a function of colloid concentration @xcite , an effect we have neglected . this might lead to a tightening at higher colloid concentration ( i.e. high field strength ) , which may be expected to lead to an increase in deviation with the experimental results ( figs . [ figsmallexpmc ] and [ figbigexpmc ] ) . nevertheless , varying the debye length as a function of local colloid concentration would be worth considering in the future . we have extended our approach a range of colloid size , to optimise the lattice tuneability for display applications . tunable 2d crystal rafts can behave as tunable diffraction gratings capable of filtering white light into visible colours@xcite . similar tunable diffraction has been proposed for display devices @xcite.the range of colours obtainable for a given geometry is governed by the lattice tunability . fig 9b demonstrates that lattice tunability in our system is maximised for particles of approximately 1 to 1.5 microns . only higher field strengths applied to smaller particles can increase the lattice tunability . as already indicated , higher field strengths are not practical in the current experimental system , however in future experiments metal coated colloids would exhibit much stronger dipolar interactions , enabling a reduction in field strength for a given attraction . regardless , our methodology illustrates using the well - developed machinery of effective colloidal interactions , as a means to model potentially complex interactions , useful for engineering purposes . * acknowledgments * we thank jeremy baumberg , martin cryan , daan frenkel and junpei yamanaka for helpful discussions . cpr acknowledges the royal society for funding and hajime tanaka for kind provision of lab space and computer time . des thanks kodak european research centre , cambridge and epsrc ( ep / do33047/1 ) for funding . ) .fitted parameters of the debye screening length used in the mc simulations for systems with different colloid diameters . errors in the diameter are standard deviations in the data obtained found from electron microscopy measurements , while those in @xmath105 , @xmath81 , and the ionic strength are estimated from the fitting of @xmath2 from mc data.@xmath15 potential measurements were made in a 0.01 mmol kcl solution [ tablekappasigma ] [ cols="^,^,^,^,^,^ " , ]

we compare the behaviour of a new 2d aqueous colloidal model system with a simple numerical treatment . to first order the attractive interaction between the colloids induced by an in - plane rotating ac electric field is dipolar , while the charge stablisation leads to a shorter ranged , yukawa - like repulsion . in the crystal - like ` rafts ' formed at sufficient field strengths , we find quantitative agreement between experiment and monte carlo simulation , except in the case of strongly interacting systems , where the well depth of the effective potential exceeds 250 times the thermal energy . the ` lattice constant ' of the crystal - like raft is located approximately at the minimum of the effective potential , resulting from the sum of the yukawa and dipolar interactions . the experimental system has display applications , owing to the possibility of tuning the lattice spacing with the external electric field . limitations in the applied field strength and relative range of the electrostatic interactions of the particles results in a reduction of tunable lattice spacing for small and large particles respectively . the optimal particle size for maximising the lattice spacing tunability was found to be around 1000 nm .

arxiv

soft x ray transients ( sxrts ) , when in outburst , show properties similar to those of persistent low mass x ray binaries containing a neutron star ( lmxrbs ; white et al . 1984 ; tanaka & shibazaki 1996 ; campana et al . 1998 ) . the large variations in the accretion rate that are characteristic of sxrts allow the investigation of a variety of regimes for the neutron stars in these systems which are inaccessible to persistent lmxrbs . while it is clear that , when in outbursts , sxrts are powered by accretion , the origin of the low luminosity x ray emission that has been detected in the quiescent state of several sxrts is still unclear . an interesting possibility is that a millisecond radio pulsar ( msp ) turns on in the quiescent state of sxrts ( stella et al . this would provide a missing link " between persistent lmxrbs and recycled msps . aql x-1 is the most active sxrt known : more than 30 x ray and/or optical outbursts have been detected so far . the companion star has been identified with the k1v variable star v1333 aql and an orbital period of 19 hr has been measured ( chevalier and ilovaisky 1991 ) . the outbursts of aql x-1 are generally characterised by a fast rise ( 510 d ) followed by a slow decay , with an @xmath4folding time of 3070 d ( see tanaka & shibazaki 1996 and campana et al . 1998 and references therein ) . type i x ray bursts were discovered during the declining phase of an outburst ( koyama et al . 1981 ) , testifying to the presence of a neutron star . ray luminosities are in the @xmath5 range ( for the @xmath6 kpc distance inferred from its optical counterpart ; thorstensen et al . 1978 ) . close to the outburst maximum the x ray spectrum is soft with an equivalent bremsstrahlung temperature of @xmath7 kev . sporadic observations of aql x-1 during the early part of the outburst decay ( czerny et al . 1987 ; tanaka & shibazaki 1996 ; verbunt et al . 1994 ) showed that when the source luminosity drops below @xmath8 the spectrum changes to a power law with a photon index of @xmath2 , extending up to energies of @xmath9 kev ( harmon et al . 1996 ) . rosat pspc observations revealed aql x-1 in quiescence on three occasions at a level of @xmath1 ( 0.42.4 kev ; verbunt et al . 1994 ) . in this lower energy band the spectrum is considerably softer and consistent with a black body temperature of @xmath10 kev . an outburst from aql x-1 reaching a peak luminosity of @xmath11 ( 210 kev ) was discovered and monitored starting from mid - february , 1997 with the rossixte all sky monitor ( asm ; levine et al . six observations were carried out with the bepposax narrow field instruments ( nfis ) starting from march 8@xmath12 , 1997 ( see table 1 ) , with the aim of studying the final stages of the outburst decay . 1a shows the light curve of the aql x-1 outburst as observed by the rossixte asm and bepposax mecs . the first part of the outburst can be fit by a gaussian with sigma @xmath13 d. this is not uncommon in sxrts ( e.g. in the case of 4u 160852 ; lochner & roussel - dupr 1994 ) . the flux decay rate changed dramatically around mjd 50512 ( march 5@xmath12 , 1997 ) . at the time of the first bepposax observation ( which started on march 8@xmath12 , 1997 ) the source luminosity was decreasing very rapidly , fading by about 30% in 11 hr , from a maximum level of @xmath14 . the second observation took place on march 12@xmath12 , 1997 when the source , a factor of @xmath15 fainter on average , reduced its flux by about 25% in 12 hr . in the subsequent four observations the source luminosity attained a constant level of @xmath16 , consistent with previous measurements of the quiescent luminosity of aql x-1 ( verbunt et al . 1994 ) . the sharp decrease after mjd 50512 is well described by an exponential decay with an @xmath4folding time @xmath17 d. the bepposax lecs , mecs and pds spectra during the fast decay phase , as well as those obtained by summing up all the observations pertaining to quiescence , can be fit with a model consisting of a black body plus a power law ( see table 2 and fig . the soft black body component remained nearly constant in temperature ( @xmath18 kev ) , but its radius decreased by a factor of @xmath19 from the decay phase to quiescence . the equivalent radius in quiescence ( @xmath20 km ) is consistent with the rosat results ( verbunt et al . the power law component changed substantially from the decay phase to quiescence : during the decay the photon index was @xmath21 , while in quiescence it hardened to @xmath22 . the two values are different with @xmath23 confidence ( see table 1 ) . the ratio of the 0.510 kev luminosities in the power law and black body components decreased by a factor of five between the first bepposax observation and quiescence . [ cols="^,^,^,^,^,^ " , ] @xmath25 spectra from the lecs and mecs ( and pds for the first observation ) detectors have been considered . the spectra corresponding to the quiescent state have been summed up , in order to increase the statistics and an upper limit from the summed pds data was also used to better constrain the power law slope . @xmath26 unabsorbed x ray luminosity in the energy range 0.510 kev . in the case of the first observation the pds data were included in the fit ( the unabsorbed 0.5100 kev luminosity amounts to @xmath27 ) . the bepposax observations enabled us to follow for the first time the evolution of a sxrt outburst down to quiescence . the sharp flux decay leading to the quiescent state of aql x-1 is reminiscent of the final evolution of dwarf novae outbursts ( e.g. ponman et al . 1995 ; osaki 1996 ) , although there are obvious differences with respect to the x ray luminosities and spectra involved in the two cases , likely resulting from the different efficiencies in the gravitational energy release between white dwarfs and neutron stars . models of low mass x ray transient outbursts hosting an old neutron star or a black hole are largely built in analogy with dwarf novae outbursts . in particular , van paradijs ( 1996 ) showed that the different range of time - averaged mass accretion rates over which the dwarf nova and low mass x ray transient outbursts were observed to take place is well explained by the higher level of disk irradiation caused by the higher accretion efficiency of neutron stars and black holes . however , the outburst evolution of low mass x ray transients presents important differences . in particular , the steepening in the x ray flux decrease of aql x-1 has no clear parallel in low mass x ray transients containing black hole candidates ( bhcs ) . the best sampled light curves of these sources show an exponential - like decay ( sometimes with a superposed secondary outburst ) with an @xmath4folding time of @xmath29 d and extending up to four decades in flux , with no indication of a sudden steepening ( chen et al . in addition , bhc transients display a larger luminosity range between outburst peak and quiescence than neutron star sxrts ( garcia et al . 1998 and references therein ) . being the mass donor stars and the binary parameters quite similar in the two cases , it appears natural to attribute these differences to the different nature of the underlying object : neutron stars possess a surface and , likely , a magnetosphere , while bhcs do not . when in outburst accretion down to the neutron star surface takes place in sxrts , as testified by the similarity of their properties with those of persistent lmxrbs , especially the occurrence of type i bursts and the x ray spectra and luminosities . the mass inflow rate during the outburst decay decreases , causing the expansion of the magnetospheric radius , @xmath30 . thus , accretion onto the neutron star surface can continue as long as the centrifugal drag exerted by the corotating magnetosphere on the accreting material is weaker than gravity ( illarionov & sunyaev 1975 ; stella et al . this occurs above a limiting luminosity @xmath31 , where @xmath32 is the gravitational constant ; @xmath33 , @xmath34 , @xmath35 g and @xmath36 ms are the neutron star mass , radius , magnetic field and spin period , respectively ( here and in the following we assume @xmath37 and @xmath38 cm ) . as @xmath30 reaches the corotation radius , @xmath39 , accretion onto the surface is inhibited and a lower accretion luminosity ( @xmath40 ) of @xmath41 is released . after this luminosity gap the source enters the propeller regime . if the mass inflow rate decreases further , the expansion of @xmath30 can continue up to the light cylinder radius , @xmath42 , providing a lower limit to the accretion luminosity that can be emitted in the propeller regime @xmath43 . below @xmath44 the radio pulsar mechanism may turn on and the pulsar relativistic wind interacts with the incoming matter pushing it outwards . matter inflowing through the roche lobe is stopped by the radio pulsar radiation pressure , giving rise to a shock front ( illarionov & sunyaev 1975 ; shaham & tavani 1991 ) . clearly these regimes have no equivalent in the case of black hole accretion . during the february - march 1997 outburst of aql x-1 , rossixte observations led to the discovery of a nearly coherent modulation at @xmath45 hz ( @xmath46 ms ) during a type i x ray burst . a single qpo peak , with a centroid frequency ranging from @xmath47 to 830 hz , was also observed at two different flux levels , when the persistent luminosity was @xmath48 and @xmath49 ( zhang et al . 1998 ) . in the presence of a single qpo peak , the magnetospheric and sonic point beat frequency models ( alpar & shaham 1985 ; miller et al . 1997 ) interpretation is ambiguous in that the qpo peak could represent either the keplerian frequency at the inner disk boundary or the beat frequency . moreover , the burst periodicity at @xmath45 hz may represent the neutron star spin frequency , @xmath50 , or half its value ( zhang et al . 1998 ) . in either cases , the possibility that accretion onto the neutron star surface takes place even in the quiescent state of aql x-1 faces serious difficulties : for a quiescent luminosity of order @xmath51 a magnetic field of only @xmath52 g would be required , in order to fulfill the condition @xmath53 . for such a low magnetic field , aql x-1 and , by inference , lmxrbs with khz qpos can hardly be the progenitors of recycled msps . more crucially , the marked steepening in the outburst decay that takes place below @xmath54 , is accompanied by a marked spectral hardening , resulting from a sudden decrease of the flux in the black body spectral component . this is clearly suggestive of a transition taking place deep in the gravitational well of the neutron star , where most of the x rays are produced . the most appealing mechanism is a transition to the propeller regime , where most of the inflowing matter is stopped at the magnetospheric boundary ( zhang , yu & zhang 1998 ) . in fig . 1a , the luminosity at mjd 50512 is identified with @xmath55 and the lower horizontal lines indicate the luminosity interval during which aql x-1 is likely in the propeller regime . additional information on the neutron star magnetic field ( and spin ) can be inferred as follows . the observed ratio of the luminosity , @xmath56 , when the qpo at @xmath57 hz were detected and the luminosity @xmath58 when the centrifugal barrier closes is @xmath59 . at @xmath58 the keplerian frequency of matter at the magnetospheric boundary is , by definition , equal to the spin frequency , i.e. @xmath60 or @xmath61 hz for aql x-1 . based on beat - frequency models , at @xmath56 the keplerian frequency at the inner disk boundary can be either @xmath62 hz or @xmath63 hz , depending on whether the single khz qpos observed corresponds to the keplerian or the beat frequency . in the magnetospheric beat - frequency models , simple theory predicts that the keplerian frequency at the magnetospheric boundary is @xmath64 ; in the radiation pressure - dominated regime relevant to the case at hand , the ghosh and lamb ( 1992 ) model predicts instead @xmath65 . therefore we expect @xmath66 and @xmath67 , in the two models , respectively . such a low ratio clearly favors the interpretation in which @xmath62 hz and @xmath68 hz . in the sonic point beat - frequency model ( miller et al . 1997 ) , the innermost disk radius is well within the magnetosphere , implying that the keplerian frequency at the magnetospheric boundary is @xmath69 . in this case all possible combinations of @xmath50 and @xmath70 are allowed . by using the observed @xmath58 , a neutron star magnetic field of @xmath71 g ( depending on the adopted model of the disk - magnetosphere interaction ) is obtained in the case @xmath68 hz and @xmath72 g in the case @xmath73 hz . once in the propeller regime , the accretion efficiency decreases further as the magnetosphere expands for decreasing mass inflow rates ( @xmath74 ) . the @xmath75 d exponential - like luminosity decline observed with bepposax is considerably faster than the propeller accretion luminosity extrapolated from the first part of the outburst ( e.g. the gaussian profile shown by the dashed line in fig . we note here that the spectral transition accompanying the onset of the centrifugal barrier may also modify the irradiation properties of the accretion disk , contributing to x ray luminosity turn off . alternatively , an active contribution of the `` propeller '' mechanism or the neutron star spin - down energy dissipated into the inflowing matter can not be excluded . it is unlikely that the quiescent luminosity of aql x-1 is powered by magnetospheric accretion in the propeller regime . as shown in fig . 1a , the quiescent x ray luminosity is probably lower than the minimum magnetospheric accretion luminosity @xmath76 allowed in the propeller phase ( this remains true for @xmath77 g if @xmath68 hz , and for @xmath78 g if @xmath73 hz ) . moreover the bepposax x ray spectrum shows a pronounced decrease in the power law to black body flux ratio together with a flattening of the power law component between the fast decay phase and quiescence , suggesting that a transition to shock emission from the interaction of a radio pulsar wind with the matter outflowing from the companion star has taken place . note that an x ray spectrum with a slope of @xmath79 has been observed from the radio pulsar psr b125963 immersed in the wind of its be star companion . models of this interaction predict that a power law x ray spectrum with a slope around @xmath79 should be produced for a wide range of parameters ( tavani & arons 1997 ) . the additional soft x ray component observed during the outburst decay ( see table 2 ) might be emitted at the polar caps as a result of the residual neutron star accretion in the propeller phase . note that the equivalent black body radius decreases for decreasing x ray luminosities , just as it would be expected if the magnetospheric boundary expanded . alternatively , the black body - like spectral component observed in quiescence could be due to the streaming of energetic particles that hit the polar caps , in close analogy to the soft x ray component observed , in msps , at the weaker level of @xmath80 ( becker & trmper 1997 ) . assuming a magnetic field in the range derived in section 3.1 ( i.e. @xmath81 g for @xmath82 hz and @xmath83 g for @xmath84 hz ) , we can consistently explain the @xmath1 quiescent x ray luminosity as powered by a radio pulsar enshrouded by matter outflowing from the companion star , if the conversion efficiency of spin - down luminosity to x ray is @xmath85% . this is consistent with modeling and observations of enshrouded pulsars ( tavani 1991 ; verbunt et al . 1996 ) . there are chances of observing a msp ( a simple scaling from msps implies a signal at 400 mhz of @xmath86 mjy ; see kulkarni et al . 1990 ) , even though the emission would probably be sporadic , like in the case of the pulsar psr 174424a due to the large amount of circumstellar matter ( see lyne et al . 1991 ; shaham & tavani 1991 ) . in summary aql x-1 appears to provide the first example of an old fast rotating neutron stars undergoing a transition to the propeller regime at first , followed by a transition to the radio pulsar regime , as the transient x ray emission approaches its quiescent level . therefore , aql x-1 ( and possibly sxrts in general ) likely represents the long - sought `` missing link '' between lmxrbs and recycled msps .

we report on the march - april 1997 bepposax observations of aql x-1 , the first to monitor the evolution of the spectral and time variability properties of a neutron star soft x ray transient from the outburst decay to quiescence . we observed a fast x ray flux decay , which brought the source luminosity from @xmath0 to @xmath1 in less than 10 days . the x ray spectrum showed a power law high energy tail with photon index @xmath2 which hardened to @xmath3 as the source reached quiescence . these observations , together with the detection by rossixte of a periodicity of a few milliseconds during an x ray burst , likely indicate that the rapid flux decay is caused by the onset of the propeller effect arising from the very fast rotation of the neutron star magnetosphere . the x ray luminosity and hard spectrum that characterise the quiescent emission can be consistently interpreted as shock emission by a turned - on rotation - powered pulsar .

arxiv

on 9th october 2006 , @xcite reported k. itagaki s discovery of a possible supernova ( sn ) in ugc 4904 . although the sn was discovered after the peak , an upper limit of the @xmath6 magnitude ( @xmath7 ) was obtained at @xmath820 days before the discovery @xcite . interestingly , @xcite also reported that an optical transient had appeared in 2004 close to the position of sn 2006jc . the transient was as faint as @xmath9 and its duration was as short as @xmath10 days . since the event was faint and short - lived , they speculated that the transient was a luminous blue variable ( lbv)-like event . the spatial coincidence between the lbv - like event and sn 2006jc is confirmed by @xcite . because of such an intriguing association with the lbv - like event , many groups performed follow - up observations of sn 2006jc in various wavebands : x - ray , ultra violet ( uv ) , optical , infrared ( ir ) , and radio . spectroscopic observations showed many broad features and strong narrow emission lines . according to the he detection , sn 2006jc was classified as type ib @xcite . however , strange spectral features and their evolutions were reported . a bright blue continuum was prominent in the optical spectrum at early epochs @xcite . such a bright blue continuum had also been observed in type ii sn 1988z @xcite , but the origin of this feature is still unclear . as the blue continuum declined , the red wing brightened and the optical spectra showed `` u''-like shapes @xcite . this is a distinguishing feature of sn 2006jc in contrast to the spectra of usual sne that have a peak in optical bands . photometric observations in optical and ir bands were performed continuously . the optical light curve ( lc ) showed a rapid decline from 50 days after the discovery , as in the case of sn 1999cq @xcite . at the same epoch , near infrared ( nir ) emissions brightened @xcite . the nir brightness increased from @xmath11 days to @xmath12 days after the discovery and then declined @xcite . the epoch of the nir brightening corresponds to that of the development of the red wing in the optical spectra @xcite . the nir brightening , as well as the fact that the redder side of the he emission profile declined faster than the bluer side , has been interpreted as an evidence of an ongoing dust formation @xcite . additionally , on 29th april 2007 ( 200 days after the discovery ) , the _ akari _ satellite performed nir and mid - infrared ( mir ) photometric and spectroscopic observations @xcite and the _ magnum _ telescope obtained the nir photometries @xcite . they report the formation of amorphous carbon dust : another piece of evidences of the dust formation . x - ray and uv emissions have also been observed by the _ swift _ and _ chandra _ satellites @xcite . x - ray observations were performed at seven epochs and showed a brightening from @xmath13 days to @xmath14 days after the discovery @xcite . the x - ray detection suggests an interaction between the sn ejecta and the circumstellar matter ( csm ) . on the contrary , the radio emission was not detected by very large array ( vla ) @xcite . we present a sn explosion model of a wolf - rayet star that explains the bolometric and x - ray lcs . hydrodynamics , nucleosynthesis , and lc synthesis calculations are performed assuming the spherical symmetry . in this study , we assume the explosion date of sn 2006jc to be 15 days before the discovery ( @xmath15 ) and the energy source of the light to be the @xmath4ni-@xmath4co decay . the paper is organized as follows : in [ sec : bol ] , we describe how we derive the bolometric lc from observations in the various wavebands , in [ sec : presn ] , we briefly discuss the presupernova evolutionary properties of the progenitor star ; in [ sec : hyd ] , hydrodynamical and nucleosynthesis calculations are described ; in [ sec : lc ] , lc synthesis calculations are presented ; in [ sec : csm ] , we calculate the x - ray emission due to the ejecta - csm interaction ; in [ sec : conclude ] and [ sec : discuss ] , conclusions and discussion are presented . cc 20 & 370 + 21 & 340 + 24 & 250 + 27 & 180 + 28 & 170 + 33 & 110 + 36 & 87 + 38 & 75 + 39 & 70 + 40 & 66 + 42 & 58 + 44 & 53 + 47 & 44 + 49 & 40 + 53 & 36 + 58 & 28 + 60 & 27 + 62 & 25 + 64 & 23 + 65 & 22 + 70 & 15 + 77 & 6.3 + 79 & 4.8 + 81 & 4.0 + 89 & 2.2 + 92 & 2.1 + 103 & 1.0 + 119 & 0.36 + 138 & 0.23 + 195 & 0.15 the bolometric luminosities of sne are usually estimated from the integration over the optical and nir emission because the usual sne radiate dominantly in the optical and nir bands ( e.g. , @xcite ) . however , the spectra of sn 2006jc show the bright red and blue wings @xcite , which implies that the emissions in uv and ir bands considerably contribute to the bolometric luminosity . we construct the bolometric luminosity with the integration of the uv , optical , and ir photometries that are obtained with the _ hct _ @xcite , _ azt-24 _ @xcite , _ magnum _ @xcite , and _ subaru _ telescopes @xcite and the _ swift _ @xcite and _ akari _ satellites @xcite . since the uv fluxes are available only at @xmath16 days @xcite , the uv luminosity is estimated from the optical luminosity at the other epoch . available observations are shown in figure [ fig : lcobsall ] . details of optical observations will be presented in the forthcoming papers ( e.g. , @xcite ) . we adopt a distance of 25.8mpc corresponding to a distance modulus of 32.05 @xcite and a reddening of @xmath17 @xcite . the optical lcs were obtained with the _ hct _ and _ subaru _ telescopes @xcite . we integrate the optical fluxes with a cubic spline interpolation from @xmath18 hz to @xmath19 hz . the optical luminosities ( @xmath20 ) are summarized in table [ tab : uvopt ] and the lc is shown in figure [ fig : lcobs ] . the optical lc declines monotonically after the discovery . the decline suddenly becomes rapid at @xmath21 days and the optical luminosity finally goes down to @xmath22ergs s@xmath23 at @xmath24 days . the x - ray lc obtained with the _ swift _ and _ chandra _ satellites @xcite shows that the x - ray luminosities , @xmath25 , are much fainter than the optical luminosities @xcite . thus , the x - ray contribution to the bolometric luminosities is negligible . however , the uv luminosity , @xmath26 , is comparable to the optical luminosity at @xmath16 days ( @xmath27ergs s@xmath23 as estimated from the uvot observations , @xcite ) . the uv luminosity is @xmath28 of the optical luminosity , i.e. , the total flux is @xmath29 times brighter than the optical flux ( fig . [ fig : lcobs ] ) . since the uv flux declined as the optical flux @xcite , we assume that @xmath30 at every epoch . although the blue wing declines with time and @xmath26 might be over - estimated at @xmath31 days @xcite , the bolometric luminosity ( @xmath32 ) should be reliable because the ir contribution dominates in the bolometric luminosity at such late epochs ( [ sec : irest ] ) . ccccc 49 & 3.9 & 1580 & 2.9 & 9.0 + 57 & 12 & 1330 & 5.1 & 12 + 67 & 16 & 1340 & 6.6 & 15 + 70 & 19 & 1330 & 7.8 & 17 + 72 & 23 & 1300 & 8.9 & 19 + 77 & 27 & 1310 & 11 & 23 + 79 & 18 & 1400 & 9.1 & 21 + 127 & 46 & 1050 & 7.6 & 12 + 132 & 52 & 1010 & 7.2 & 11 + 154 & 17 & 1150 & 4.2 & 7.0 + 157 & 32 & 1010 & 4.5 & 6.6 + 159 & 75 & 900 & 6.7 & 8.7 + 160 & 26 & 1050 & 4.4 & 6.8 + 167 & 35 & 990 & 4.5 & 6.4 + 168 & 54 & 940 & 5.5 & 7.5 + 169 & 99 & 880 & 7.6 & 9.7 + 170 & 48 & 930 & 4.8 & 6.3 + 171 & 45 & 950 & 4.9 & 6.7 + 172 & 44 & 940 & 4.6 & 6.3 + 192 & 48 & 900 & 4.0 & 5.3 + 195 & 45 & 870 & 3.3 & 4.2 + 197 & 56 & 860 & 3.8 & 4.8 + 202 & 5.6 & 1190 & 1.5 & 2.7 + 215 & 28 & 870 & 2.1 & 2.7 the ir spectroscopy and photometries are obtained with the _ azt-24 _ and _ magnum _ telescopes ( nir photometries , @xcite ) and the _ akari _ satellite ( nir spectroscopy and mir photometries , @xcite ) . as indicated by the red wing in the optical spectra , the ir emission considerably contributes to the bolometric luminosity of sn 2006jc . the mir observation is available at @xmath33 days @xcite . the ir luminosity integrated over @xmath34 hz is estimated from the nir and mir observations as @xmath35 ergs s@xmath23 . @xcite concluded that the ir emission is originated from amorphous carbon grains with two temperatures of @xmath36k and 320k . the large difference between the two temperatures would imply that the origin of the hot carbon dust with @xmath36k is different from that of the warm carbon dust with @xmath37k . the hot carbon dust is suggested to be newly formed in the sn ejecta and heated by the @xmath4ni-@xmath4co decay by a dust formation calculation @xcite . on the other hand , the origin of the emission from the warm carbon dust is suggested to be a sn light echo of the csm carbon dust ( @xcite ; see also @xcite ) . therefore , we assume that the optical emission from sn 2006jc is absorbed and simultaneously re - emitted by the hot carbon dust and thus the luminosity emitted from the hot carbon dust should be included in the bolometric luminosity of sn 2006jc . according to the estimated temperatures and masses of the hot and warm carbon grains @xcite , the luminosities contributed by the hot and warm carbon grains are @xmath38 ergs s@xmath23 and @xmath39 ergs s@xmath23 , respectively . and @xmath40 stems from that the h - band luminosity is slightly brighter than the luminosity emitted from the hot carbon dust @xcite . ] for the epochs when the ir photometries at @xmath41 are unavailable , we estimate the contribution of the ir emission by fitting the jhk - band photometries with amorphous carbon emission . from the kirchhoff s law , the thermal radiation from a spherical dust grain x with a uniform radius @xmath42 and temperature @xmath43 is given by @xmath44 , where @xmath45 is the absorption efficiency of the grain . for the optically thin case , the observed emission from dust grains @xmath46 is written as @xmath47 where @xmath48 and @xmath6 denote the total number of the dust particles and the distance from the observer , respectively @xcite . in the followings , we convolve the @xmath49-independent coefficients as an emission coefficient @xmath50 . applying the absorption efficiency for the amorphous carbon grain with @xmath51 , we derive the temperature of the hot carbon dust , @xmath52 , and @xmath53 to reproduce the jhk - band photometries . to justify the above estimate , we compare the estimate with the actual mir observation at @xmath33 days @xcite . the fitting gives the temperature @xmath54 k and the emission coefficient @xmath55 for the hk - band photometries at @xmath33 days . the luminosity integrated over @xmath56 is @xmath57 ergs s@xmath23 . the temperature and luminosity are roughly consistent with those of the hot carbon dust . the agreement indicates that the fitting gives a good estimate of the ir emission due to the hot carbon dust . we note that the estimate can not account for the emission from the warm carbon dust . table [ tab : nirbb ] summarizes the emission coefficient , temperature , estimated luminosity at @xmath58 hz , and luminosity emitted below @xmath59 hz . the dust temperature roughly declines from @xmath60k at @xmath61 days to @xmath62k at @xmath33 days . this is consistent with a picture that the hot carbon dust was formed in the sn ejecta and cooled down gradually @xcite . the ir lc is shown in figure [ fig : lcobs ] . the estimated luminosity at @xmath58 hz evolves as the jhk lcs , and thus the ir lc brightens at @xmath63 days and declines at @xmath64 days . since there is no nir data at @xmath65 days , the bolometric lc can not be estimated at this epoch . the bolometric luminosity is derived from the summation of @xmath26 , @xmath20 , and @xmath66 and summarized in table [ tab : bol ] , where @xmath67 is applied . cc 49 & 81 + 51 & 81 + 53 & 75 + 58 & 64 + 60 & 61 + 62 & 59 + 65 & 55 + 66 & 54 + 70 & 45 + 77 & 33 + 79 & 29 + 119 & 14 + 138 & 10 + 195 & 4.7 the presupernova model has been extracted from a set of models already presented by @xcite and computed with the latest release of the stellar evolutionary code franec ( 5.050218 ) . since all the features of this code have been already presented , we will address here only the main points . the interaction between convection and local nuclear burning has been taken into account by coupling together and solving simultaneously the set of equations governing the chemical evolution due to the nuclear reactions and those describing the convective mixing . more specifically , the convective mixing has been treated by means of a diffusion equation where the diffusion coefficient is computed by the use of the mixing - length theory . the nuclear network is the same as that adopted in @xcite , but the nuclear cross sections have been updated whenever possible ( see table 1 in @xcite ) . a moderate amount of overshooting of 0.2 @xmath68 has been included into the calculation only on the top of the convective core during core h burning . mass loss has been taken into account following the prescriptions of @xcite for the blue supergiant phase ( @xmath69k ) , @xcite for the red supergiant phase ( @xmath70k ) , @xcite for the wnl wolf - rayet phase and @xcite during the wne / wco wolf - rayet phases . we adopt the following correspondence of the models to the various wr phases according to the surface abundances , as suggested by @xcite : wnl ( @xmath71 ) , wne ( @xmath72 and @xmath73 ) , wnc ( @xmath74 ) and wco ( @xmath75 ) . ( hereafter , c / n and c / o denote the number ratios . ) the x - ray emission , as well as the early bright blue continuum and the narrow lines , clearly indicates an interaction between the sn ejecta and the csm , i.e. , the existence of a dense csm . furthermore , the ir spectral energy distribution may be explained by the formation of amorphous carbon grains in the sn ejecta and the csm ( @xcite , see also @xcite ) . since the c - rich environment ( i.e. , @xmath76 ) is required to form carbon dust ( e.g. , @xcite ) , the ir observations suggest that the sn ejecta and csm contain a c - rich layer . this suggests that the progenitor star of sn 2006jc is a wco wolf - rayet star with a c - rich envelope and csm ( figs . [ fig : modpresn]a-[fig : modpresn]d ) . lr + @xmath77 [ myr ] & 4.64 + @xmath78 [ @xmath79 & 25.80 + @xmath80 [ @xmath79 & 35.40 + @xmath81 [ myr ] & 4.16 + @xmath82 [ @xmath79 & 10.01 + + @xmath83 [ myr ] & 0.46 + @xmath84 [ @xmath79 & 12.56 + @xmath80 [ @xmath79 & 7.04 + @xmath85 [ @xmath79 & 18.80 + @xmath86 & 0.28 + @xmath87 [ myr ] & 0.07 + @xmath88 [ myr ] ( @xmath89 ) & 0.11 ( 0.77 ) + @xmath90 [ myr ] ( @xmath89 ) & 0.054 ( 0.44 ) + @xmath91 [ myr ] ( @xmath89 ) & 0.21 ( 0.31 ) + + @xmath92 [ yr ] & 1.25(+4 ) + @xmath93 [ @xmath79 & 16.52 + @xmath94 [ @xmath79 & 4.83 + @xmath95 [ @xmath79 & 1.50 + @xmath96 [ @xmath79 & 6.88 + @xmath97 [ cm ] & 3.08(+10 ) + @xmath98 ( int .- ext . ) [ @xmath79 & 5.262 - 6.648 + @xmath99 ( int .- ext . ) [ @xmath79 & 2.736 - 4.097 + @xmath100 [ yr ] & 1.10(+5 ) + @xmath101 [ yr ] & 5.43(+4 ) + @xmath102 [ yr ] & 2.21(+5 ) + @xmath103 [ yr ] & 3.86(+5 ) inspection of all the presupernova models available in @xcite indicates that only massive models , i.e. , @xmath104 , fulfill the requirements from the ir observation and become wco stars . moreover , these are the only stars in which the chemical compositions of the mantle and csm are dominated mainly by c with a smaller amount of o ( fig . [ fig : modpresn]cd ) . in stars with initial masses smaller than @xmath105 , the mass of the he convective core increases or remains constant during the core he burning phase . at core he exhaustion , a sharp discontinuity of he abundance is produced at the outer edge of the co core . then , the co core begins to contract to ignite the next nuclear fuel while he burning shifts to a shell inducing a formation of a convective zone . the he convective shell forms beyond the he discontinuity at the outer edge of the co core . hence its chemical composition is dominated by he [ @xmath46(he)@xmath106 . because of the short lifetime of the advanced burning stages , only a small amount of he is burned inside the shell before the presupernova stage ( figs . [ fig : modpresn]ab ) . such a behavior is typical for stars in which the he core mass remains roughly constant during core he burning ( e.g. , @xcite ) . in stars with initial masses greater than @xmath105 , on the contrary , the mass loss is efficient enough ( @xmath107 ) to uncover the he core and they reduce progressively their mass during the core he burning phase . the star enters the wne wolf - rayet stage and its subsequent evolution is governed by the actual size of the he core . in particular , as the he core progressively reduces due to the mass loss , the star tends to behave as an initially - lower mass star , i.e. , essentially reduces its central temperature . this induces the he convective core to shrink progressively in mass as well , leaving a layer with a variable chemical composition that reflects the central abundances at various stages during core he burning . when the stellar mass is reduced below the maximum extension of the he convective core , the products of core he burning appear on the surface and the star becomes a wco wolf - rayet star . at core he exhaustion , he burning shifts to a shell inducing the formation of the convective shell . the convective shell forms in the region with variable chemical composition . as a consequence , at variance with what happens in stars with @xmath108 , in these stars , the chemical composition of the convective shell becomes a mixture of the central he burning products . hence it is mainly composed of c , o and he ( figs . [ fig : modpresn]cd ) . since all the models above @xmath109 have a similar presupernova structure , we selected a @xmath109 star as representative of a typical star becoming a wco wolf - rayet star . the mass at the presupernova stage ( @xmath96 ) is @xmath110 because of the strong mass loss . we underline that the @xmath111-@xmath96 relation is highly uncertain because it strongly depends on many details of the stellar evolution ( e.g. , the mass loss , overshooting , rotation , and metallicity , see @xcite ) . for this reason , for the purpose of this study , we mainly focus on a wco progenitor with @xmath112 @xmath113 , without paying much emphasis on @xmath111 . a detailed discussion of the presupernova evolution during all the nuclear burning stages is beyond the purpose of this paper . hence we report here in table [ tab : presn ] some key properties during the h , he , and advanced burning stages . in particular , for the h burning stage we report the following quantities : the h burning lifetime ( @xmath77 ) , the maximum extension of the convective core ( @xmath78 ) , the total mass ( @xmath80 ) at core h exhaustion , the time spent as an o - type star ( @xmath81 ) and the he core mass ( @xmath82 ) at h exhaustion . here , we assume that the temperature of the o - type stars is @xmath114 . for the he burning phase we report the following quantities : the he burning lifetime ( @xmath83 ) , the maximum size of the he convective core ( @xmath84 ) , @xmath80 at core he exhaustion , the maximum depth of the convective envelope ( @xmath85 ) , the central @xmath115 mass fraction at core he exhaustion [ @xmath86 ] , the time spent at the red side ( @xmath117 ) of the hr diagram ( @xmath87 ) , and the wnl , wne , and wco lifetimes ( @xmath88 , @xmath90 , and @xmath91 , respectively ) - in parenthesis the central he mass fraction [ @xmath89 ] when the star enters the wnl , wne and wco phases . for the advanced burning stage we report the following key quantities : the time until the explosion ( @xmath92 ) , the maximum size of the he core [ @xmath93 ] , the maximum size of the co core [ @xmath94 ] , the masses of the iron core ( @xmath95 ) and the star ( @xmath96 ) and the radius of the star ( @xmath97 ) at the presupernova stage , the final extension in mass of the he convective shell [ @xmath121 and of the convective c shell [ @xmath122 , and the total lifetimes during the wnl [ @xmath100 ] , wne [ @xmath101 ] , wco [ @xmath102 ] , and wr [ @xmath103 , where @xmath126 phases . figures [ fig : hr40 ] and [ fig:40pre2 ] show the evolutionary path in the hr diagram and the temperature and density profiles at the presupernova stage . the sn explosion and explosive nucleosynthesis are calculated for the progenitor star with @xmath110 . we apply various explosion energies ( @xmath127 ) for the sn explosion calculations ( e.g. , @xcite ) . the hydrodynamical calculation is performed by means of a spherical lagrangian hydrodynamics code with a piecewise parabolic method ( ppm , @xcite ) including nuclear energy production from the @xmath128-network . the equation of state takes account of the gas , radiation , @xmath129-@xmath130 pair @xcite , coulomb interactions between ions and electrons , and phase transition @xcite . after the hydrodynamical calculations , nucleosynthesis is calculated as a post - processing with a reaction network that includes 280 isotopes up to @xmath131br ( see table 1 in @xcite ) . since the explosion mechanism of a core - collapse sn for a massive star with an iron core is still an unsolved problem ( e.g. , @xcite ) , we initiate the sn explosion as a thermal bomb . although there are various ways to simulate the explosion ( e.g. , a kinetic piston , @xcite ) , it is suggested that the explosive nucleosynthesis does not depend sensitively on the way how the explosion energy is deposited @xcite . we set an inner reflective boundary at @xmath132 and @xmath133 km within the iron core and elevate temperatures at the inner boundary . in the spherical symmetry case , for any given progenitor model , hydrodynamics and nucleosynthesis are determined by the explosion energy . during the sn explosion , a shock propagates outward inducing local compression and heating , triggering explosive nucleosynthesis . behind the shock front the matter is accelerated and starts moving outward . however , if the progenitor has a deep gravitational potential and the explosion energy is low , the inner layers begin to fall back due to the gravitational attraction . a more compact star and a lower explosion energy leads a larger amount of fallback . the fallback has a deep implication on the sn nucleosynthesis because it decreases the matter ejection , especially , of the inner core ( e.g. , @xmath4ni ) . figure [ fig : vrho ] shows density structures at 100 s after the explosions when homologously expanding structures are reached ( @xmath134 ) . we find that the fallback takes place for the model with @xmath135 but not for the models with @xmath136 , 10 , and 20 . figure [ fig : escape ] shows a comparison between the escape velocity and the ejecta velocity for the model with @xmath135 and demonstrates that the matter below @xmath137 will fall back . on the other hand , in the models with @xmath136 , 10 , and 20 , the matter above the inner boundary will be ejected . the abundance distributions after the explosions are shown in figures [ fig : abn]a-[fig : abn]d . in every model , @xmath138ni is synthesized in the innermost layer ( @xmath139 ) due to the low electron fraction . thus , we can estimate the maximum amounts of synthesized @xmath4ni for given energies . the @xmath4ni - rich layer extending to @xmath140 [ where @xmath141 and @xmath142si - rich layer extending to @xmath143 [ where @xmath144 expand farther in the models with higher @xmath145 because the temperature achieved is higher in the outer layer for higher @xmath145 . @xmath146 and @xmath147 for each model are summarized in table [ tab : abn ] . ccccc 1 & 1.8 & 2.1 & & + 5 & 2.1 & 2.5 & 0.5 & 1.8 + 10 & 2.3 & 2.7 & 0.6 & 2.0 + 20 & 2.5 & 3.0 & 0.7 & 2.3 @xmath4ni is synthesized at @xmath148 . since @xmath149 in the model with @xmath135 , the model is likely not to eject @xmath4ni . on the other hand , the models with @xmath136 , 10 , and 20 can eject all synthesized @xmath4ni because the fallback does not occur . the total amounts of synthesized @xmath4ni for the models with @xmath136 , 10 , and 20 are summarized in table [ tab : abn ] . the energy source of the lc of sn 2006jc is still under debate . the possible sources include the @xmath4ni-@xmath4co decay like type i sne and the ejecta - csm interaction like type iin sne . however , both scenarios have the following problems . in the case of the @xmath4ni-@xmath4co decay , the @xmath150-ray photon and positron emitted from the @xmath4ni-@xmath4co decay are absorbed by the sn ejecta and the absorbed energy is thermalized . thus , the spectra would show a blackbody - like continuum as normal type i sne do . however , the spectra of sn 2006jc do not resemble those of normal type i sne but show a bright blue continuum in early epochs @xcite . in the case of the ejecta - csm interaction , the kinetic energy is transformed to an x - ray emission via bremsstrahlung radiation , and then converted to uv , optical , and ir emissions . thus , it is difficult to explain that the x - ray luminosity is much fainter than the optical luminosity unless the optical depth for the x - ray emission is much higher than that for the optical emission . another problem with the ejecta - csm interaction model is that the x - ray lc is not synchronized with the bolometric lc . in addition , the lc powered by the ejecta - csm interaction usually has a long - term plateau ( e.g. , sn 1997cy , @xcite ) . thus , we assume that the lc is powered by the @xmath4ni-@xmath4co decay . the bolometric lc of sn 2006jc is constructed from the uv , optical , and ir observations as described in [ sec : bol ] . the estimated peak bolometric magnitude of sn 2006jc is @xmath151 , being as bright as sn 2006aj ( e.g. , @xcite ) . thus , it is speculated that the ejected amount of @xmath4ni [ @xmath152 is similar to sn 2006aj , i.e. , @xmath153 @xcite . according to [ sec : hyd ] , the models with @xmath136 , 10 , and 20 can eject a large enough amount of @xmath4ni , while the @xmath4ni production of the model with @xmath135 is too small . the spherical explosion models with @xmath136 , 10 , and 20 yield too much @xmath154 because of no fallback . however , no fallback is a consequence of the assumption of the spherical symmetry . the fallback takes place in an aspherical explosion even with a high explosion energy and thus the aspherical explosion may well decrease @xmath154 and increase the central remnant mass @xmath155 ( @xcite ) . therefore , assuming that aspherical fallback takes place in the high - energy models with @xmath136 , 10 , and 20 , we estimate the amount of fallback to yield @xmath153 and then the ejected masses for the models as @xmath156 . as a result , the sets of @xmath155 , @xmath157 and @xmath145 are derived to be ( @xmath158 , @xmath159 , @xmath160 ) @xmath161 ( 1.8 , 5.1 , 5 ) , ( 2.0 , 4.9 , 10 ) , and ( 2.3 , 4.6 , 20 ) . applying the homologous density structures of the models ( fig . [ fig : vrho ] ) , we synthesize bolometric lcs for the models with @xmath136 , 10 , and 20 using the lte radiation hydrodynamics code and the gray @xmath150-ray transfer code @xcite . in the radiative transfer calculation , the electron scattering is calculated for the ionization states solved by the saha equation and the rosseland mean opacity is approximated with an empirical relation to the electron - scattering opacity @xcite . the peak width ( @xmath162 ) of the sn lc depends on the ejected mass @xmath157 , explosion energy @xmath145 , opacity @xmath163 , density structure , and @xmath4nidistribution , as @xmath164 @xcite , where @xmath165 represents the effects of the density structure and the @xmath4ni distribution . here , we assume for sake of simplicity a uniform mixing of @xmath4ni in the sn ejecta . also , the density structures after the sn explosions with various @xmath145 are analogous . thus , the dependence on @xmath165 is negligible and we investigate the lc properties depending on @xmath163 , @xmath157 and @xmath145 . the synthetic lcs obtained for the models with ( @xmath159 , @xmath160 ) @xmath161 ( 5.1 , 5 ) , ( 4.9 , 10 ) , and ( 4.6 , 20 ) are shown in figure [ fig : lc ] . figure [ fig : lc ] also shows the multicolor and bolometric lcs of sn 2006jc . the period of sn 2006jc is divided into four epochs depending on the available observations : ( 1 ) uv and optical photometries at @xmath166 days , ( 2 ) optical and nir photometries at @xmath63 days , ( 3 ) optical photometry at @xmath65 days , and ( 4 ) optical , nir , and mir photometries and nir spectroscopy at @xmath64 days . \(1 ) at @xmath166 days , the ir contributions to the bolometric luminosity may well be small because the ir contribution is only @xmath167 at @xmath168 days . thus , the peak bolometric luminosity derived from the uv and optical fluxes is reliable ( [ sec : optest ] ) . if the bolometric lc peaked at the discovery , the peak luminosity is reproduced by the @xmath4ni-@xmath4co decay of @xmath169 . the rapid decline after the peak prefers such high - energy models as ( @xmath159 , @xmath160 ) = ( 4.9 , 10 ) and ( 4.6 , 20 ) . \(2 ) at @xmath63 days , the ir contribution to the bolometric luminosity increases from @xmath167 at @xmath61 days to @xmath170 at @xmath171 days . the contribution of @xmath172 to @xmath173 changes from @xmath174 at @xmath61 days to @xmath175 at @xmath171 days . at this epoch , the optical and ir emissions contribute to the bolometric luminosities . combining the ir brightening and the optical decline , the bolometric lc including @xmath26 ( @xmath176 ) declines slowly . such a slow decline is consistent with the models of ( @xmath159 , @xmath160 ) = ( 4.9 , 10 ) and ( 4.6 , 20 ) . \(3 ) at @xmath65 days , nir photometries are not available . the decline of the optical luminosity at this epoch is more rapid than at @xmath177 days . such a rapid decline of the optical lc can not be reproduced by the @xmath4ni-@xmath4co decay . however , the bolometric lc may well decline more slowly than the optical lc because the ir emission dominates in the bolometric luminosity . \(4 ) at @xmath64 days , nir photometries are available continuously and optical photometries are available at @xmath178 days @xcite . the contribution of the optical emission to the bolometric luminosities is negligible ( @xmath179 at @xmath178 days ) . at this epoch , the contribution of @xmath172 to the total luminosity increases from @xmath180 at @xmath181 days to @xmath28 at @xmath33 days . the estimated ir luminosity is consistent with the luminosity emitted from the hot carbon dust at @xmath33 days ( [ sec : irest ] ) . since the dust temperature decreases with time , the ratio of the mir luminosities to the nir luminosities becomes larger with time . therefore , the amorphous carbon emission model reasonably estimates the ir luminosity due to the hot carbon dust at @xmath182 days . the model with @xmath183 , @xmath184 , and @xmath169 reproduces well the lc decline at @xmath64 days and the ir luminosities due to the hot carbon dust at @xmath33 days days was presented by @xcite . they estimated the total luminosity and the dust temperature as @xmath185ergs s@xmath23 and @xmath186 k , respectively . our model with @xmath183 , @xmath184 , and @xmath169 predicted the luminosity of @xmath187ergs s@xmath23and the dust temperature of @xmath188 k @xcite at @xmath189 days , which are lower than the observations . this suggests that the observed ir emissions at @xmath190 days may originate not only from the newly - formed dust in the sn ejecta heated by the @xmath4ni-@xmath4co decay but also from the light echo of the csm dust . ] . therefore , we conclude that the hypernova - like sn explosion model with @xmath183 , @xmath184 , and @xmath169 is the most preferable model among the exploded models of a wco wolf - rayet star with @xmath110 . x - rays from sn 2006jc were detected by the _ swift _ and _ chandra _ satellites @xcite . the x - ray detection indicates that the expanding sn ejecta collides with the csm . we calculate x - ray emission from the ejecta - csm interaction for the sn model with ( @xmath159 , @xmath160 ) @xmath161 ( 4.9 , 10 ) , and estimate the csm density structure on the basis of a comparison with the observed x - ray lc ( e.g. , @xcite ) . the observed x - ray luminosities estimated with the distance of @xmath191 mpc in @xcite are scaled using @xmath192 mpc . we adopt a csm density profile characterized by a power - law of @xmath193 and assume that the interaction starts at a distance @xmath194 cm . the parameters @xmath195 , @xmath196 , and @xmath197 are determined so that the ejecta - csm interaction reproduces the observed x - ray lc . the interaction generates reverse and forward shock waves in the sn ejecta and csm , respectively . both regions are heated by the shock waves and emit x - rays . in such a compact star , because the density in the shocked sn ejecta is higher than that in the shocked csm , the emitted x - rays from the shocked sn ejecta are more luminous than those from the shocked csm ( fig . [ fig : lx ] ) . figure [ fig : lx ] shows the synthesized x - ray lc for @xmath198 g @xmath199 and @xmath200 for @xmath201 cm and @xmath202 for @xmath203 cm , i.e. , for a flat ( inside ) and steep ( outside ) csm density profile of @xmath204 g @xmath199 for @xmath201 cm and @xmath205 g @xmath199 for @xmath203 cm . the total mass of the csm is @xmath206 to reproduce the peak of the observed x - ray lc and the subsequent decline . the density , velocity , and temperature structures and their evolutions are shown in figures [ fig : csm]abc . the velocity of the reverse shock is @xmath207 km s@xmath23 . the reverse shock reaches @xmath208 from the outer edge of the sn ejecta at @xmath209 days and heats up the swept - up sn ejecta . the temperature behind the reverse shock is higher than @xmath210 k where dust can not newly form and the dust formed in the sn ejecta is destroyed @xcite . our calculation does not show the formation of a cooling shell . this is because the csm interaction is so weak to emit x - ray of @xmath211 ergs s@xmath23 . if the bolometric luminosity is powered by the csm interaction , i.e. , if the csm interaction emits as high luminosity as @xmath212 ergs s@xmath23 , the cooling shell might form and thus the dust formation might be possible . further detailed studies , however , are required to confirm the dust formation behind the reverse shock . such a flat density profile of the inner csm implies that the stellar wind was not steady because the steady wind should form the csm of @xmath213 . this circumstellar environment might have been formed by a variable mass - loss rate @xmath214 and/or a variable wind velocity @xmath215 . for example , assuming that the stellar wind blew with a constant @xmath216 km s@xmath23 for two years before the explosion , the mass - loss rate must have changed from @xmath217 to @xmath218 @xmath219 yr@xmath23 in two years . such a drastic change of the mass - loss rate and/or the wind velocity is consistent with the fact that the progenitor of sn 2006jc was surrounded by the matter ejected by the lbv - like event two years before the explosion . we present a theoretical model for sn 2006jc whose properties are summarized as follows . \(1 ) * wco progenitor and dust formation * : the progenitor is a wco wolf - rayet star whose total mass has been reduced from @xmath220 to as small as @xmath110 . the wco star model has a thick c - rich envelope and csm . this is consistent with the formation of amorphous carbon grains in the sn ejecta and the csm suggested by _ akari _ observations @xcite . @xcite have calculated dust formation in the wco star explosion model and shown that carbon dust is formed in the c - rich layer at @xmath168 days . this is much earlier than the dust formation after @xmath221 yr in type ii sne ( sne ii ) , because of the much smaller @xmath157 in the wco star than sne ii . according to the models in @xcite , the stars with @xmath104 typically become wco stars to form a thick c - rich layer and csm . this limiting mass , however , is still uncertain and strongly depends on many details of the stellar evolution . the early dust formation in the sn ejecta and the csm suggests that the progenitor of sn 2006jc is a massive star becoming a wco wolf - rayet star . \(2 ) * explosion and bolometric light curve * : the multicolor lcs of sn 2006jc show peculiar evolutions , e.g. , a rapid decline of the optical lc and brightening of the ir lc . these can be interpreted as an ongoing dust formation . assuming the absorbed optical light is re - emitted in the ir band , the bolometric lc is constructed as a summation of @xmath26 ( @xmath176 ) , @xmath20 , and @xmath66 . by calculating the hydrodynamics , nucleosynthesis , and the bolometric lc for the sn explosion with the various explosion energies , @xmath135 , 5 , 10 , and 20 , we find that the hypernova - like sn explosion model with @xmath183 , @xmath184 , and @xmath169 best reproduces the bolometric lc of sn 2006jc with the radioactive decays . also , the temperature evolution of the carbon dust heated by the @xmath4ni-@xmath4co decay reasonably well explains the ir observations for @xmath222 days @xcite . \(3 ) * csm interaction and x - ray light curve * : applying the model with @xmath183 and @xmath184 , we calculate the ejecta - csm interaction and the resultant x - ray lc . we derive the csm density structure to reproduce the x - ray lc of sn 2006jc as @xmath204 g @xmath199 for @xmath201 cm and @xmath205 g @xmath199 for @xmath203 cm . the flat density distribution in the inner csm indicates a drastic change of the mass - loss rate and/or the wind velocity that is consistent with the lbv - like event two years before the explosion . * lbv connection * : our model does not take into account the lbv - like event that occurred two years before the explosion . the first reason is that the mechanism of the outburst is still unclear . the second reason is that , at least in the framework of the current understanding of standard stellar evolution , the envelope of a massive star practically freezes out after core he exhaustion ( i.e. , about 10000 years before the explosion ) due to the more rapid evolution of the core than the envelope . in addition , it is interesting to note that , and this is a confirmation of the theoretical expectation , there is no observational evidence that any wolf - rayet star has ever undergone such a luminous outburst @xcite . hence , there is no specific reason to associate the occurrence of a lbv - like outburst to the presupernova evolution . future studies on the mechanism of the outburst are required to firmly conclude the origin of the lbv - like outburst . it would be possible that a possible binary companion star could undergo the lbv - like outburst . * fallback * : according to our hydrodynamics and nucleosynthesis calculations , in the spherically symmetric models with @xmath223 , the fallback does not take place and thus the amount of synthesized @xmath4ni is much larger than @xmath169 which is required to power the lc of sn 2006jc ( [ sec:56ni ] ) . in the aspherical explosions , however , the fallback takes place even for @xmath223 . in this paper , we assume the fallback even for the models with @xmath223 and derive the amount of fallback to yield the appropriate amount of @xmath4ni . to justify the above assumption , we calculate an aspherical explosion induced by a jet with an opening angle of @xmath224 and an energy deposition rate of @xmath225 @xcite . the jet - induced model realizes an explosion with @xmath226 , @xmath227 , and @xmath228 that is consistent with the adopted model . we note that an aspherical radiative transfer calculation is required to confirm that the jet - induced explosion model can reproduce the lc of sn 2006jc . * light curve models * : the model with @xmath183 , @xmath184 , and @xmath169 is not an unique model to reproduce the bolometric lc of sn 2006jc . in the case of usual sne , the velocities of the absorption lines can disentangle the degeneracy of @xmath157 and @xmath145 by means of the comparison with the photospheric velocities ( e.g. , @xcite ) . however , the spectra of sn 2006jc are dominated by he emission lines and the nature is unclear . thus we can not fully resolve the degeneracy . since the lc shape is proportional to @xmath229 , the model with a larger @xmath157 requires a higher @xmath145 . an explosion of the progenitor star with a larger @xmath96 reproduces the lc of sn 2006jc with a higher @xmath145 and the x - ray lc with a lower csm density . on the other hand , an explosion of the progenitor star with a smaller @xmath96 reproduces the lc with a lower @xmath145 , e.g. , an explosion with @xmath230 and @xmath135 can explain the lc shape of sn 2006jc . however , such low @xmath145 explosions suppress explosive nucleosynthesis and enhance the fallback . as a result , the @xmath4niproduction is reduced for a small @xmath96 . if the lc of sn 2006jc is powered by the @xmath4ni-@xmath4co decay , the bright peak indicates a larger amount of @xmath4ni production ( @xmath231 ) than a normal sn [ @xmath232 , e.g. , sn 1987a , @xcite ] . therefore , sn 2006jc is likely a more energetic explosion than a normal sn with @xmath233 . * dust formation * : we assume that the energy source of the lc of sn 2006jc is the @xmath4ni-@xmath4co decay . this consistently explains the formation of carbon dust at the early epoch ( @xmath168days ) and the dust temperature at @xmath24days @xcite . in this scenario , however , the origin of the bright blue continuum remains an unsolved problem ( e.g. , @xcite ) . such a spectrum might be explained by the ejecta - csm interaction . in this scenario , however , the fine tunings are required to reproduce the bolometric lc ; most of the x - rays are absorbed and converted to the optical luminosity , which only a small fraction of the x - rays are emitted with changing the fraction from @xmath234 at @xmath235 days to @xmath236 at @xmath237 days . moreover , the formation of carbon dust with two temperatures would not be explained . since both scenarios are inconclusive so far , further investigations may give important implications on the emission mechanism of sn 2006jc . we would like to thank a. arkharov , n. efimova , a. di paola , c. corsi , e. di carlo , and m. dolci for providing us the data of the jhk - band photometries . we also would like to thank s. immler for the preprint on the swift observations . thanks the support from the 21st century coe program ( quest ) of jsps ( japan society for the promotion of science ) for his stay in the university of tokyo and the hospitality of the department of astronomy . , d.k.s . , and t.p.p . are supported by the jsps - insa ( indian national science academy ) exchange programme . m.t . are supported through the jsps research fellowship for young scientists . this work has been supported in part by world premier international research center initiative ( wpi initiative ) , mext , japan , and by the grant - in - aid for scientific research of the jsps ( 10041110 , 10304014 , 11740120 , 12640233 , 14047206 , 14253001 , 14540223 , 16740106 , 18104003 , 18540231 , 20540226 ) and mext ( 19047004 , 20040004 , 20041005 , 07ce2002 ) . holland , s. , et al . 2007 , `` supernova 1987a : 20 years after : supernovae and gamma - ray bursters '' , ed . s. immler , k.w . weiler , and r. mccray ( aip : new york ) , in press . the poster paper is available at http://astrophysics.gsfc.nasa.gov/conferences/supernova1987a/holland_poster.pdf .

we present a theoretical model for type ib supernova ( sn ) 2006jc . we calculate the evolution of the progenitor star , hydrodynamics and nucleosynthesis of the sn explosion , and the sn bolometric light curve ( lc ) . the synthetic bolometric lc is compared with the observed bolometric lc constructed by integrating the uv , optical , near - infrared ( nir ) , and mid - infrared ( mir ) fluxes . the progenitor is assumed to be as massive as @xmath0 on the zero - age main - sequence . the star undergoes extensive mass loss to reduce its mass down to as small as @xmath1 , thus becoming a wco wolf - rayet star . the wco star model has a thick carbon - rich layer , in which amorphous carbon grains can be formed . this could explain the nir brightening and the dust feature seen in the mir spectrum . we suggest that the progenitor of sn 2006jc is a wco wolf - rayet star having undergone strong mass loss and such massive stars are the important sites of dust formation . we derive the parameters of the explosion model in order to reproduce the bolometric lc of sn 2006jc by the radioactive decays : the ejecta mass @xmath2 , hypernova - like explosion energy @xmath3 ergs , and ejected @xmath4ni mass @xmath5 . we also calculate the circumstellar interaction and find that a csm with a flat density structure is required to reproduce the x - ray lc of sn 2006jc . this suggests a drastic change of the mass - loss rate and/or the wind velocity that is consistent with the past luminous blue variable ( lbv)-like event .

arxiv

the micro - quasar grs 1915 + 105 is an enigmatic black hole binary ( bhb ) exhibiting enormous variability which have been classified in more than 14 different variability classes @xcite . it is believed that the extreme variability and rapid state changes observed in grs 1915 + 105 are due to a very high accretion rate , which is close to , or at times higher than , the eddington accretion rate @xcite . it is also known for exhibiting large superluminal radio flares and steady radio emission which are always associated with specific x - ray variability classes @xcite . such an extreme and correlated multi - wavelength variability makes grs 1915 + 105 a unique bhb . in this context , igr j170913624 , a new x - ray transient source believed to be a bhb , generated considerable interest recently . it was detected by integral / ibis in 2003 @xcite . it has exhibited repeated outbursts with periods of two to four years in 1994 , 1996 , 2001 , 2003 , 2007 , and 2011 @xcite . the recent 2011 outburst of igr j170913624 was unusually long and the source was found to be active even after one year @xcite . during this outburst , igr j170913624 revealed its highly variable nature and showed variability patterns so far observed only in grs 1915 + 105 . the most prominent of these patterns was the ` heartbeat ' pattern , similar to the @xmath0-class in grs 1915 + 105 . @xcite documented the first six months of rxte observations and showed that not only @xmath0-class but many other variability patterns similar to @xmath1- , @xmath5- , @xmath6- , @xmath7- , @xmath8- , and @xmath9- classes have been observed during this outburst of igr j170913624 . @xcite also detected a high frequency quasi - periodic oscillation ( hfqpo ) in this source with a frequency of 66 hz , which is almost identical to the frequency of hfqpo in grs 1915 + 105 . despite striking morphological similarities , the most perplexing difference between the two sources lies in their observed intensities . while grs 1915 + 105 is one of the brightest x - ray sources with a typical brightness of @xmath100.5 2 crab , igr j170913624 is about 20 times fainter . in the present scenario , mass , distance , and inclination for this source are rather poorly constrained , with reports so far suggesting a mass range of <3 m@xmath4 @xcite to @xmath1015 m@xmath4 @xcite and a distance range of @xmath1011 kpc @xcite to @xmath1020 kpc @xcite . nevertheless , the apparent faintness of igr j170913624 is difficult to explain even after assuming the smallest possible mass of 3 @xmath11 for a black hole @xcite and the largest possible distance of @xmath1025 kpc for a galactic source . here , we attempt to investigate the possible reasons for this apparent faintness of igr j170913624 by simultaneously fitting spectra at different phases . the main idea is that the system parameters can not change over the phase of the oscillations . therefore , a simultaneous fitting of spectra at different phases , with system parameters tied across phases , may put a better constraint on them . this , along with a proposal that the ` heartbeats ' can be used as a ` standard candle ' , leads to a primary conclusion that the faintness of igr j170913624 is due to its low or negative spin . we have used data from long simultaneous observations of igr j170913624 made on 2011 march 27 with rxte ( obsid : 96420 - 01 - 05 - 000 , total exposure @xmath1021 ks ) and xmm - newton ( obsid : 0677980201 , total exposure @xmath1039 ks ) with net simultaneous exposure of @xmath1015 ks . the data reduction for the rxte / pca observation was carried out with heasoft version 6.8 following standard analysis procedure for good xenon data . we extracted 1 s light curve from pcu2 data . it showed the typical @xmath1-class oscillations with periods ranging from 30 to 50 s ( figure 1 ) . it contained a total of 385 bursts . we carried out ` phase - resolved ' spectroscopy for these bursts in the energy range of 3.035.0 kev for rxte / pca and 0.712.0 kev for xmm / pn data as described below . the peak time for each burst was identified in a semiautomatic manner using an idl script and the peak - to - peak interval between consecutive bursts was divided into 64 phases of equal length . the start and stop times of each phase , recorded in rxte mission time for 385 bursts , were used for extracting spectra for each phase . total counts for all 64 spectra and their corresponding exposure times were then used to generate the ` phase - folded ' light curve ( figure 2 ) . the 64 phase bins were grouped into five phases as shown in figure 2 and the spectra extracted for these five phases were used for simultaneous spectral fitting . the grouping was carried out mainly by the visual inspection of the folded rxte / pca lightcurve . the xmm observation was carried out in the _ fast timing _ mode of epic - mos and the _ burst _ mode of epic - pn and we followed the standard analysis procedures for these modes using _ sas v11.0.0 _ and the latest calibration files . we used data from xmm - pn only because mos2 data could not be checked for possible pileup ( generation of pattern plot always resulted in error ) whereas mos1 data are not useful in timing mode because of a dead pixel in the ccd . for pn data , the observed and the expected pattern behavior differed below 0.7 kev and hence the energy range for rest of the analysis was restricted to 0.712.0 kev . start and stop times of the 64 phases of all bursts from rxte mission were converted into xmm mission time using the _ xtime _ tool , available at heasarc , which were used to build gti files using sas task _ gtibuild_. these gti files were used for extracting the 64 phase spectra using the task _ evselect_. the ` phase - folded ' light curve was generated using the total counts and the exposure times , as described earlier . the subtle features were averaged out as a consequence of the quasi - periodicity and co - adding , but the overall profile of the ` phase - folded ' light curve followed a typical burst cycle . further , it was seen that the oscillations were more pronounced in the xmm light curve indicating that the accretion disk radiation was primarily participating in the oscillations and not the comptonized emission from the corona which dominates at higher energies . source spectra from the five grouped phases were extracted using rawx columns between 32 and 42 for single and double pixel events ( pattern @xmath124 ) . sas tools _ rmfgen _ and _ arfgen _ were used to generate redistribution matrices and ancillary files , respectively , and the same files were used for the spectra of all phases . the background spectrum was extracted from rawx columns between 5 and 7 after confirming that the region was not contaminated significantly with source photons in the selected energy range . a single background spectrum was used for the five phases . all spectra were rebinned using sastask _ specgroup _ to have a minimum of 25 counts per channel . once we extracted the spectra for the five phases for both pca and pn , the 10 spectra were fitted simultaneously with various parameters tied as follows . the 10 spectra were loaded into xspec as 10 data groups . for a given phase , all parameters for pca and pn spectra were tied together except the normalization constant which was frozen at 1.0 for pn whereas for pca it was kept free but tied across the five phases . for a particular spectral model , we tied all parameters representing system property , such as mass , distance , inclination , spin or combination of these , across the five phases . the parameters describing the accretion process such as inner disk temperature or accretion rate , inner disk radius , were fitted independently for each phase . wide - band x - ray spectrum of a black hole binary generally consists of two dominant components : a multi - color disk and a high - energy tail arising from compton scattering in an optically thin region surrounding the disk . we used diskpn @xcite as a simplified disk model , primarily because its parameters are cleanly separated in accretion - process - dependent parameters ( disk temperature and inner disk radius ) and system parameters ( normalization ) . for a general relativistic description of the multi - temperature disk spectrum , we used kerrbb @xcite , which is widely used for its accurate modeling of disk spectrum and to investigate black hole spin the high - energy tail of the spectrum is typically modeled as powerlaw to approximate the comptonized component . however , @xcite proposed a more physical model simpl to empirically describe the comptonized component . here , we have used simpl along with one of the two disk models ( diskpn and kerrbb ) to fit the spectra in the five phases simultaneously , as described below . for the first part of the analysis , we fitted rxte - pca and xmm - pn spectra for the five phases simultaneously with const*phabs*(simpl@xmath13diskpn ) . the parameter const was used to account for possible calibration uncertainties between the two instruments . normalizations of diskpn were tied across the five phases , whereas rest of the parameters were allowed to vary independently . though the interstellar absorption can be considered as a part of system parameters , we allowed it to vary to account for any phase - dependent absorption intrinsic to the source . we , however , found that the n@xmath14 values for the five phases were not significantly different and the fitted values of n@xmath14 were in agreement with the values reported by @xcite and @xcite . table 1 provides the results of spectral fits . it can be seen that the inner disk temperature is highest for phase 1 corresponding to the peak of the bursts , implying higher accretion rate as expected . we verified that neither the fe - line nor the reflection component was required to fit the data . a best fit was obtained with @xmath15 value of 1709.0 for 1030 degrees of freedom . however , for the present work , more important is the best - fit value of diskpn normalization , @xmath16 , as it can provide some constraints on mass ( @xmath17 ) , distance ( @xmath18 ) , and inclination ( @xmath2 ) . the best fit value of diskpn normalization was found to be @xmath19 . assuming a minimum mass of @xmath103 m@xmath4 and a maximum distance of @xmath1025 kpc for a galactic black hole candidate along with a standard value for color correction factor , @xmath20 , the best - fit value of @xmath21 resulted in a lower limit on inclination angle of 76@xmath3 . considering the 90% upper confidence limit of @xmath22 for the normalization , the lowest possible inclination angle was @xmath1053@xmath3 . this lower limit comes only from simultaneous spectral fitting and it is not dependent on any additional information , and hence it can be considered as fairly robust . since spectral fitting could not constrain lower limit of n@xmath23 , it was not possible to obtain an upper limit on @xmath2 with spectral fitting . however , simultaneous spectral analysis with diskpn to model accretion disk spectra suggests that igr j170913624 is a high inclination binary . this is consistent with the finding of @xcite and @xcite . , scattering fraction ( f@xmath24 ) , n@xmath14 , and @xmath25 for all combinations of @xmath18 , @xmath17 , @xmath2 and @xmath26 . the parameters are more or less similar for any combination of system parameters and hence distinction between them is not made . error bars ( 90% confidence ) are shown with gray and only a subset of data are shown here.,width=453,height=453 ] in the second part , we fitted the spectra using the model cons*phabs*(simpl@xmath13kerrbb ) . apart from mass , distance , inclination , and spin as independent fit parameters , kerrbb has parameters governing the second - order effects such as ` spectral hardening factor ' , ` returning radiation ' and ` limb darkening ' . we , however , froze these parameters to their default values . typically kerrbb is used for disk - dominated spectra with luminosity < 0.3 @xmath27 , however , @xcite have shown that simpl@xmath13kerrbb can be used to accurately describe spectra with higher luminosities as well . in this case , we tied all kerrbb parameters except n@xmath14 and mass accretion rate across the five phases . further , the normalization was frozen to 1.0 . since all system parameters can not be fitted simultaneously , we selected specific values for mass , inclination , and spin and fitted for mass accretion rate and distance . in order to systematically investigate the parameter space , we varied the black hole mass from 3 @xmath11 to 20 @xmath11 , inclination angle from @xmath28 to @xmath29 , and spin from 0 to 0.9 . results of this analysis have been shown in figures 3 and 4 . figure 3 shows variation of interstellar absorption , n@xmath14 , two parameters of simpl power - law index , @xmath30 and the scattering fraction , @xmath31 , and fit statistic , @xmath32 for the five phases . since these parameters are more or less the same for any combination of system parameters ( mass , distance , inclination , and spin ) , these are shown without any distinction . again it was found that neither the fe - line nor the reflection component is required for fitting data with the f - test chance improvement probability being > 88% for all combinations of mass , distance , and inclination . for a combination of mass @xmath17 , inclination @xmath2 ( > 50@xmath3 ) , and spin @xmath26 , the fitted values of distance @xmath18 ( top panel ) and mass accretion rate @xmath33 ( bottom panel ) have been shown in figure [ mdplot ] . only maximum ( phase 1 ) and minimum ( phase 3 ) mass accretion rates have been shown in figure [ mdplot ] for the sake of clarity . error bars ( 90% confidence limit ) are smaller than the symbols for most of the points in plot . this indicates that the distance and the accretion rate are very well constrained for a given combination of system parameters . figure [ mdplot ] can be used to put some significant limits on the spin of the black hole with some independent constraints on either black hole mass or distance . one such independent constraint comes from the total luminosity argument during ` heartbeat ' oscillations . @xcite studied such oscillations in grs 1915 + 105 and suggested that the radiation pressure instability augmented by the local eddington limit within inner accretion disk as the origin of such variability pattern . however , this mechanism requires high accretion rate . @xcite showed that during the peak of ` heartbeat ' , the bolometric disk luminosity is typically as high as 80%90 % of the eddington luminosity . given the similarity between the variability patterns in grs 1915 + 105 and igr j170913624 , it is natural to assume a similar mechanism operating in both the sources . in this way , the observed flux during the peak of the ` heartbeat ' oscillations can be used to estimate the distance for a given mass or in other words ` heartbeat ' can be used as a standard candle in order to constrain the parameters . we found the best - fit value of the peak bolometric flux as 4.10 @xmath34 erg s@xmath35@xmath36 . figure 1 shows that the peak flux in individual bursts is not the same and there is a burst - to - burst variation in the peak flux values . therefore , we have assumed the range of peak flux to be ( 3.0 5.0 ) @xmath34 erg s@xmath35 @xmath36 and a more conservative range of peak luminosity to be 60%-90% of eddington luminosity . thus the obtained possible @xmath37 range is shown as a gray area in the top left corner of figure [ mdplot ] . it can be seen that the points within this region correspond to inclination angle < 60@xmath3 and spin < 0.2 . however , the lower limit on inclination was found to be @xmath1076@xmath3 from diskpn normalization . this presents a tantalizing possibility of black hole spin to be retrograde indicating a black hole spin in opposite direction to the accretion disk . further , we have found a lower limit of @xmath1053@xmath3 on inclination angle from 90% upper confidence limit of diskpn normalization . the points corresponding to @xmath3850@xmath3 require spin to be < 0.2 . thus we exclude the possibility of high spin and it appears that the main reason for the observed faintness of igr j170913624 is very low ( or even negative ) spin of the black hole , in contrast to grs 1915 + 105 , which is known to have a very high spin @xcite . it can be seen that even for the exotic type of black hole with mass <3 @xmath11 @xcite , this inference on black hole spin remains valid . further , in order to verify the effect of other frozen parameters of kerrbb , such as returning radiation , limb darkening , and inner boundary stress , we have also enabled all of them ( one by one as well as all together ) and found that the lines corresponding to a particular combination of spin and inclination angle move away from the shaded region . we verified that the same is valid when powerlaw is used instead of simpl . thus for any combination of these parameters or model for the high - energy tail , the black hole spin is required to be either very low or negative . the 66 hz hfqpo of igr j170913624 detected by @xcite may also be considered as an independent constraint on black hole mass , if it is assumed to be related to mass . in this case , the black hole mass in igr j170913624 has to be @xmath39 . for such a high black hole mass , figure 4(a ) does not provide any constraint on the distance , however , both the inclination angle and the spin are required to be > 70@xmath3 and > 0.7 , respectively . for these high values of inclination angle and spin , figure 4(b ) shows that the required accretion rate is only a small fraction of the eddington accretion rate . thus , if we assume that the 66 hz qpo is related to the black hole mass , the basic accretion process giving rise to the apparent similar variability of grs 1915 + 105 and igr j170913624 has to be different . however , it is more likely , as also suggested by @xcite , that the 66 hz hfqpo is not directly related to the black hole mass . in that case , the previous argument for a low black hole mass based on the apparent flux and similar accretion process between igr j170913624 and grs 1915 + 105 holds , and the inference of low or negative spin remains valid . overall this work indicates that the black hole spin may not play a significant role in generating the observed extreme variability and such a behavior is generated mainly by a high accretion rate . p3.5 cm c c c c c & phase 1 & phase 2 & phase 3 & phase 4 & phase 5 + @xmath40 ( @xmath4110@xmath42@xmath36 ) & 0.93@xmath43 & 0.84@xmath44 & 0.81@xmath45 & 0.81@xmath46 & 0.88@xmath46 + t@xmath47 ( kev ) & 1.24@xmath48 & 1.11@xmath46 & 1.02@xmath46 & 1.05@xmath46 & 1.11@xmath46 + r@xmath49 ( r@xmath50 ) & 24.8@xmath51 & 24.9@xmath52 & 28.2@xmath53 & 26.3@xmath54 & 27.3@xmath55 + @xmath30 & 3.57@xmath56 & 2.66@xmath57 & 2.56@xmath58 & 2.52@xmath59 & 2.70@xmath58 + scattered fraction & 0.47@xmath60 & 0.27@xmath44 & 0.36@xmath44 & 0.37@xmath46 & 0.35@xmath44 + disk flux@xmath61 & 2.10 & 1.23 & 1.01 & 1.06 & 1.47 + total flux@xmath61 & 2.54 & 1.53 & 1.42 & 1.49 & 1.92 + @xmath62 norm@xmath63 & + + + + + + grs 1915 + 105 was so far the only bhb exhibiting extreme variability and spectral changes over timescales as short as a few tens of seconds . observations of similar variability in another bhb igr j170913624 establishes that such extreme variability of grs 1915 + 105 is not due to some specific coincidences unique to one particular bhb , but is a more generic phenomenon . here , we presented results of simultaneous fitting of 0.735.0 kev spectra during different phases of the ` heartbeat ' oscillations in igr j170913624 , which indicate that the most likely difference between grs 1915 + 105 and igr j170913624 is in the spins of the respective black holes . while the black hole in grs 1915 + 105 is known to be rotating with high spin , the black hole in igr j170913624 is found to have a very low spin . in fact , for inclination @xmath1070@xmath3 , which is favored by the data , the black hole could very well have a retrograde spin . in this case , igr j170913624 would be the first known astrophysical source having a retrograde spin . even though theoretically possible , such a scenario would be very challenging to explain from the point of view of evolution of such a system . this research has made use of data obtained from high energy astrophysics science archive research center ( heasarc ) , provided by nasa s goddard space flight center . we thank a. r. rao for useful discussions . we also thank the anonymous referee for very useful comments .

igr j170913624 is a transient x - ray source and is believed to be a galactic black hole candidate . recently , it has received a considerable attention due to the detection of peculiar variability patterns known as ` heartbeats ' , which are quasi - periodic mini - outbursts repeated over timescales ranging between 5 and 70 s. so far , such variability patterns have been observed only in grs 1915 + 105 and these are classified as @xmath0- and @xmath1-variability classes . here , we present the results of ` phase - resolved ' spectroscopy of the ` heartbeat ' oscillations of igr j170913624 using data from simultaneous observations made by rxte and xmm - newton . we find that the 0.735 kev spectra can be fitted with a ` canonical ' model for black hole sources consisting of only two components a multi - temperature disk black body and a power law ( or its equivalent ) . we attempt to constrain the system parameters of the source by simultaneously fitting spectra during different phases of the burst profile while tying the system parameters across the phases . the results indicate that the source is a high inclination binary ( @xmath2>53@xmath3 ) . further , the observed low flux from the source can be explained only if the black hole spin is very low , along with constraints on the black hole mass ( < 5 m@xmath4 ) and the distance ( > 20 kpc ) . for higher inclination angles , which is favored by the data , the black hole spin is required to be negative . thus low or retrograde spin could be the reason for the low luminosity of the source .

arxiv

the extinction curve describes how the extinction changes with the wavelength . extinction is due to the presence of dust grains in the interstellar medium and its characteristics are different in a diffuse interstellar medium as compared to a dense interstellar medium . thus , the knowledge of extinction curve is necessary to deredden magnitudes and colors of astronomical objects and to understand the physical properties of dust grains . * hereafter ccm ) derived a mean extinction law ( for @xmath2 ) that depends on only one parameter @xmath3 . they considered the sample used in the ultraviolet ( uv ) extinction study of @xcite based on _ international ultraviolet explorer _ extinction curves of 45 reddened milky way ob stars . ccm searched for the corresponding optical and near - infrared ( ubvrijhkl ) photometry from the literature . finally they used the intrinsic colors of @xcite for the appropriate spectral types to determine the extinction . they obtained the following one - parameter family of curves that represents the uv to infrared ( ir ) extinction law in terms of @xmath0 : @xmath4 where @xmath5 , and @xmath6 and @xmath7 are the wavelength - dependent coefficients . equation ( [ ccmlaw ] ) is very powerful because it allows one to determine the extinction in some spectral region based on the extinction in a different spectral region , given that one knows @xmath0 . the @xmath0 parameter ranges from about 2.0 to about 5.5 ( with a typical value of 3.1 ) when one goes from diffuse to dense interstellar medium . in this formalism @xmath0 therefore characterizes the dust properties in the region that produces the extinction . many authors used this parameter to study extinction , e.g. : @xcite searched for a relation between @xmath0 and other parameters that characterize the extinction curves ; @xcite studied the role of @xmath0 as the main regulatory agent of the penetration of radiation inside dark clouds ; @xcite discussed different methods to deredden the data and obtained a new estimate of the extinction law in terms of @xmath0 . * hereafter gs ) applied a @xmath8 minimization to compute the @xmath0 values for a sample of stars with uv extinction data using the linear relation ( [ ccmlaw ] ) . a similar method with weights was used by @xcite to determine @xmath0 and @xmath1 toward a sample of stars with known color excesses in ubvrijhkl . here we extend the method used by gs in order to obtain improved @xmath0 values for the lines of sight toward a sample of stars with known extinction data in the uv . the structure of this paper is the following . in [ th.cons . ] we discuss the theoretical basis of our @xmath0 derivation . in [ data ] we describe our data sources . we present the results and assess the consistency between different samples and theoretical approaches in [ results ] . finally in [ conclusion ] we discuss our results and comment on the future work . the interstellar dust grains span a wide range of sizes from a few angstroms to a few micrometers . in general , they reduce the intensity of the transmitted beam by two physical processes : absorption and scattering . the extinction is the result of these two processes . + the apparent magnitude @xmath9 of each star as a function of wavelength may be written as @xmath10 @xmath11 where @xmath12 , @xmath13 and @xmath14 represent absolute magnitude , distance and total extinction , respectively , and subscripts `` red '' and `` comp '' denote reddened and comparison stars , respectively . the extinction as a function of @xmath15 may be obtained by comparing corresponding stars paired according to spectral properties . in principle , the comparison star should be of the same spectral classification as the reddened star , but with a negligible extinction . if the reddened star and the comparison star have the same spectral classification it also means that they have very similar intrinsic spectral energy distributions . thus we have @xmath16 . we also assume that @xmath17 . the magnitude difference obtained from equation ( [ mred ] ) and ( [ mcomp ] ) is therefore : @xmath18 the quantity @xmath19 is a constant term and may be eliminated by normalizing with respect to extinction difference in two standard wavelengths @xmath20 and @xmath21 : @xmath22 generally , the extinction curves are normalized with respect to the b and v passbands in the @xcite system : @xmath23 where @xmath24 , @xmath25 is the observed color and @xmath26 is the intrinsic color ( by construction equal to the color of the comparison star ) . it is possible to obtain the absolute extinction by using the total to selective extinction ratio : @xmath27 then : @xmath28 ccm , for computational reasons , divided the complete extinction curve ( equation [ ccmlaw ] ) into three wavelengths regions and fitted the extinction law as a function of @xmath29 : + infrared ( @xmath30 ) , + optical / nir ( @xmath31 ) , + ultraviolet and far - ultraviolet ( @xmath32 ) . + for every wavelength , the coefficients @xmath6 and @xmath7 from equation ( [ ccmlaw ] ) are fixed and given by an appropriate expression . observations from _ international ultraviolet explorer _ cover a range from @xmath33 ( @xmath15[@xmath34 = 1549 , 1799 , 2200 , 2493 , and 3294 ) , so we use the equations for the coefficients for the last two regions listed above . for @xmath31 and @xmath35 we have : @xmath36 for @xmath37 : @xmath38+f_a(x);\\ & b(x)&=-3.090 + 1.825x+1.206/[(x-4.62)^2 + 0.263]+f_b(x).\end{aligned}\ ] ] where : @xmath39 @xmath40 @xcite compute @xmath0 values using equations ( [ ccmlaw ] ) and ( [ eps ] ) and by minimizing the quantity : @xmath41\}^2 \label{chi}\ ] ] the right side of equation ( [ chi ] ) is a second order polynomial ( parabola ) of @xmath0 with the minimum : @xmath42\right\}}{\sum_{i=1}^{n_{\rm bands}}(a(x_i)-1)^2 } \label{rvgs}\ ] ] + this formula is right when the errors in @xmath43 are identical for all bands . our data ( see [ data ] ) have errors that differ from band to band , so we use an improved @xmath8 , weighted by the observational errors . ducati et al . ( 2003 ) suggested the following @xmath8 for independent minimization with respect to @xmath0 and @xmath1 : @xmath44 ^ 2\ ] ] where @xmath45 are the weights associated with each band . we use a related but different approach which stems from the fact that in addition to uv bands we also use @xmath46 as our input data . we normalize our color excesses with @xmath46 to form @xmath47 . since @xmath48 , we minimize the following @xmath8 with respect to @xmath0 only : @xmath49 \ } ^2 ~e^2(b - v ) \label{chiweighted}\ ] ] setting @xmath50 and minimizing equation ( [ chiweighted ] ) with respect to @xmath0 we find : @xmath51/ \sigma_i^2 \}}{\sum_{i=1}^{n_{\rm bands } } \ { ( a(x_i)-1)^2/ \sigma_i^2 \ } } \label{rvweighted}\ ] ] where : @xmath52 \equiv \left(\frac{\partial \epsilon(\lambda_i - v)}{\partial e(\lambda_i - v ) } \sigma[e(\lambda_i - v)]\right)^2 + \left ( \frac{\partial \epsilon(\lambda_i - v)}{\partial e(b - v ) } \sigma[e(b - v)]\right)^2 \nonumber\\ & = & \left(\frac{e(\lambda_i - v)}{e(b - v ) } \right)^2 \left [ \left ( \frac{\sigma[e(\lambda_i - v)]}{e(\lambda_i - v)}\right)^2 + \left(\frac{\sigma[e(b - v)]}{e(b - v ) } \right)^2 \right ] \label{sigma}\end{aligned}\ ] ] and @xmath53 & \equiv & \sigma^2[(m_{\lambda_i}-m_v ) - ( m_{\lambda_i}-m_v)_0 ] \nonumber\\ & = & \sigma^2[m_{\lambda_i}]+\sigma^2[m_v]+\sigma^2_{i,{\rm mismatch } } \label{sigmae}\end{aligned}\ ] ] the error terms on the right side of equation ( [ sigmae ] ) are described in table [ table1 ] . in equation ( [ sigma ] ) we assumed for simplicity that the errors in @xmath54 and @xmath46 are independent . however , the values of @xmath47 and their errors for different bands are not independent . to get a good idea about the errors in @xmath0 we compute them in two ways . first , we calculate the maximum error in @xmath0 , which is the straight sum of errors coming from different sources : @xmath55 then we obtain the error in quadrature which would properly describe total uncertainty if the errors from different sources were uncorrelated : @xmath56 } \nonumber\\ & = & \frac{1}{\sum_{i=1}^{n_{\rm bands } } [ ( a(x_i)-1)^2/\sigma_i^2 ] } \cdot \sqrt { \sum_{j=1}^{n_{\rm bands } } \left ( \frac{a(x_j)-1}{\sigma_j } \right)^2 } \label{quaerr}\end{aligned}\ ] ] neither description ( [ errabs ] ) nor ( [ quaerr ] ) is strictly correct : the real error in @xmath0 lies likely between these two estimates . by definition : @xmath57 where the second equality is a consequence of equation ( [ rvweighted ] ) . therefore the maximum error in @xmath1 is given by } \frac{1}{\sigma_i^2 } = c$ ] , where @xmath58 is a constant . from equation ( [ sigma ] ) we conclude that this condition is in conflict with @xmath59=\sigma [ e(b - v)]$ ] assumed by gs . if we ignore this conflict and force both conditions , then : @xmath60 \cdot \sigma [ e(b - v)]\ ] ] the results reported in table 1 of gs suggest that they used the above formula rather than their equation ( 8) . ] : @xmath61 + \left | \frac{\partial a_v } { \partial e(b - v ) } \right| \sigma [ e(b - v ) ] \nonumber\\ & = & \frac{1}{\sum_{i=1}^{n_{\rm bands } } ( a(x_i)-1)^2/\sigma_i^2 } \left [ \sum_{j=1}^{n_{\rm bands } } \left | \frac{(a(x_j)-1)}{\sigma_j^2}\right| \sigma[e(\lambda_j - v ) ] \right . \nonumber\\ & + & \left . \left| \sum_{i=1}^{n_{\rm bands}}\frac{(a(x_i)-1)(-b(x_i))}{\sigma_i^2 } \right| \sigma[e(b - v ) ] \right ] \label{averrmax}\end{aligned}\ ] ] the error in quadrature is given by : @xmath62 + \left ( \frac{\partial a_v}{\partial e(b - v ) } \right ) ^2 \sigma^2[e(b - v ) ] } \nonumber\\ & = & \frac{1}{\sum_{i=1}^{n_{\rm bands } } ( a(x_i)-1)^2/\sigma_i^2 } \left[\left(\sum_{j=1}^{n_{\rm bands } } \frac{(a(x_j)-1)}{\sigma_j^2 } \right)^2\sigma^2[e(\lambda_j - v)]\right . \nonumber\\ & + & \left.\left(\sum_{i=1}^{n_{\rm bands}}\frac{(a(x_i)-1)(-b(x_i))}{\sigma_i^2 } \right)^2\sigma^2[e(b - v ) ] \right]^{1/2 } \label{sigmaquadav}\end{aligned}\ ] ] we use the data taken from the @xcite catalog of ultraviolet color excesses @xmath63 for stars of spectral types b7 and earlier . the uv measurements are taken from _ astronomical netherlands satellite _ ( ans ) data @xcite and consist of observations in five uv channels with central wavelengths : @xmath64 1549 , 1799 , 2200 , 2493 , and 3294@xmath65 . the sources of the data used to obtain @xmath66 as given by equation ( [ extinction ] ) and their errors are listed in table [ table1 ] . we also consider another type of error : a ` mismatch error ' , which is caused by the fact that the reddened star and the comparison star may have slightly different colors . @xcite give in their section 2c , table 1b ultraviolet color excess errors which include errors associated with spectral type misclassification ( mismatch error ) and errors in the intrinsic colors . we adopt their values for this total additional source of error , and we record them under the name of mismatch error . figure [ figure1 ] shows the histograms of the @xmath54 errors for the five ultraviolet bands which we obtain using equation ( [ sigmae ] ) . the errors are completely dominated by the mismatch errors which results in a few spikes observed in each panel . the errors adopted by gs and marked by vertical lines are shown for comparison . from the @xcite catalog we exclude some lines of sight using the same method of selection as gs . it means that we exclude the lines of sight that have @xmath67 , and the ones with @xmath68 . this selection results in 923 lines of sight considered previously by gs . in addition , we also exclude those stars that do not have spectral type classification , because for them we are not able to assign the mismatch errors . this last cut reduces the number of lines of sight we consider to 782 . figure [ figure2 ] shows the sky positions of the stars in our sample . the sample contains stars of spectral type b7 and earlier and this is the reason for which almost all the stars lie in the galactic plane at low latitudes . by using the method described in [ th.cons . ] we compute for our sample the @xmath0 and @xmath1 values listed in table [ table2 ] . here we present only the first 20 objects from our sample . the complete table is available in electronic form and on the world wide web . in the first column we list the names of stars , in the second and third the galactic coordinates , then the @xmath46 values taken from @xcite catalog ; in the remaining columns we list the @xmath0 and @xmath1 values with their errors obtained using formulae ( [ rvweighted])([sigmaquadav ] ) . + our determination of @xmath0 values with their errors is made using the gs method improved through the consistent treatment of observational errors . figure [ figure3 ] shows the good agreement between the @xmath0 values obtained with gs unweighted method and the weighted method applied here . in our case , the @xmath0 values are not so different between the two methods because our adopted errors in @xmath69 are of the same order in the five uv wavelengths . however , it s important to notice that typical errors in @xmath0 are very different between the two methods ( mostly due to the mismatch errors considered here ) . + figure [ figure4 ] shows the same points as in figure [ figure2 ] , but now different colors mark different values of @xmath0 with @xmath0 increasing from red to blue . as expected most lines of sight have @xmath0 of about 3.1 . this may be also seen in figure [ figure5 ] which shows @xmath0 values as a function of galactic coordinates . the circular red points are the mean values of @xmath0 for the data binned every @xmath70 and every @xmath71 for the galactic longitude and latitude , respectively . these mean values do not differ a lot one from another but some sky anisotropy is also quite apparent . figure [ figure6 ] shows the histogram of the @xmath0 values derived here . the weighted mean of @xmath0 values is @xmath72 . we consider systematic errors in extinction curve determination that can result from using biased @xmath46 values . to this aim we use the wegner s ( 2002 ) calibration of @xmath46 to estimate the effect of the calibration change on the value of @xmath0 . usually , the extinction curve is expressed in terms of a color excesses to @xmath46 ratio . since @xmath0 value depends on this ratio ( see equation [ eps ] or [ rvweighted ] ) , the adopted @xmath46 calibration will influence it . since we do not know which set of @xmath46 values is more appropriate [ @xcite or @xcite ] , the difference in obtained @xmath0 values will be a good indicator of a possible systematic error in @xmath0 . @xcite made a catalog of interstellar extinction curves of ob stars . he used the uv data from @xcite , but differently from @xcite who used the data sources described in table [ table1 ] , he took the visual magnitudes and spectral classification of o and b stars from the simbad database . the maximum error in @xmath46 adopted by @xcite is 0.04 mag ; the error in @xmath73 and @xmath74 is 0.01 , and he obtains the intrinsic colors using the ` artificial ' standard method by @xcite , who found a linear relation between @xmath75 and @xmath76 and used the coefficients of this relation also to compute the linear relation between the intrinsic colors . this method improves the accuracy of the intrinsic colors based on ans photometry . there are 190 stars that @xcite has in common with our sample . for these stars we compute the @xmath0 values using formula ( [ rvweighted ] ) weighted by the observational errors given by @xcite . figure [ figure7 ] shows the difference in the @xmath0 values given by different calibrations values is : @xmath77 . ] . the main effect on @xmath0 comes from the fact that the @xmath46 values from two calibrations differs on average by [email protected] in the sense that @xcite color excesses are smaller than the ones used in our primary determination . + table [ table3 ] reports results for the lines of sight in common between wegner s ( 2002 ) sample and our sample . the complete table is given in the electronic form on the world wide web . the first column lists stellar designations , the second and third report the galactic coordinates ; in the fourth column we list the @xmath46 values taken by @xcite ; the fifth column gives our @xmath0 values and the sixth their maximum errors ; the seventh contains the @xmath46 values used by @xcite , the eighth provides the @xmath0 values and the ninth their errors computed with our method and using the wegner s ( 2002 ) ultraviolet data . using ultraviolet color excesses we find @xmath0 and @xmath1 values and their errors for a sample of 782 lines of sight . we extend the analysis of @xcite by considering various sources of statistical and systematic errors . in a treatment related to the one by ducati et al . ( 2003 ) , we introduce the weights associated with the errors in each uv band to our @xmath8 minimization procedure . we explicitly give all the formulae we use to compute the @xmath0 and @xmath1 values and their errors . we compute the maximum errors and the errors in quadrature for @xmath0 and @xmath1 taking into account mismatch errors that affect the color excesses to the largest extent . we present the sky distribution of @xmath0 values and show their behavior as a function of galactic coordinates . finally , we emphasize how @xmath0 values change with different calibrations of @xmath46 . since @xmath0 value may characterize entire extinction curves , extending our study into wavelength regions beyond ultraviolet will provide a check on the universality of ccm law in various parts of the spectrum . we discuss this issue in the forthcoming paper . we are grateful to gregory rudnick for his very careful reading of the original version of this manuscript and a number of helpful suggestions . we thank paola mazzei and guido barbaro for their comments . ag acknowledges the financial support from earastargal fellowship at max - planck - institute for astrophysics , where this work has been completed . 99 cardelli , j. a. , clayton , g. c. , & mathis j. s. 1989 , , 345 , 245 cecchi - pestellini , c. , cervetto , c. , aiello , s. , & barsella , b. 1995 , , 43 , 1319 2003 , , 588 , 344 , m. p. 1970 , a&a , 4 , 234 , e. l. & massa , d. 1990 , , 72 , 163 , e. l. 1999 , , 111 , 63 gnaciski , p. & sikorski , j. 1999 , , 49 , 577 , p. & greenberg , j. m. 1993 , a&a , 274 , 439 johnson , h.l . 1966 , , 4 , 193 , d. m. & savage , b. d. 1981 , , 248 , 545 , b. 1978 , a&as , 34 , 1 , j. , krelowski , j. , & wegner , w. 1993 , , 273 , 575 , b. d. , massa , d. , meade , m. , & wesselius , p. r. 1985 , , 59 , 397 , w. 2002 , balta , 11 , 1 , p. r. , van duinen , r. j. , de jonge , a. r. w. , aalders , j. w. g. , luinge , w. , & wildeman , k. j. 1982 , a&as , 49 , 427 , c .- c . , gallagher , j. s. , peck , m. , faber , s. m. , & tinsley , b. m. 1980 , , 237 , 290 values as a function of galactic coordinates . the circular red points represent the unweighted mean values of @xmath0 in different coordinate bins with the rms error bars represented by the vertical lines . the galactic longitudes and latitudes are binned every @xmath79 and @xmath80 , respectively.,title="fig:",width=302 ] values as a function of galactic coordinates . the circular red points represent the unweighted mean values of @xmath0 in different coordinate bins with the rms error bars represented by the vertical lines . the galactic longitudes and latitudes are binned every @xmath79 and @xmath80 , respectively.,title="fig:",width=302 ] values obtained from equation ( [ rvweighted ] ) using the wegner s ( 2002 ) uv data versus the @xmath0 values computed with the same formula , but using our primary data . the line shows a 1-to-1 relationship . the right panel shows the comparison between two different calibrations of @xmath46 . the line marks the average difference level.,title="fig:",width=302 ] values obtained from equation ( [ rvweighted ] ) using the wegner s ( 2002 ) uv data versus the @xmath0 values computed with the same formula , but using our primary data . the line shows a 1-to-1 relationship . the right panel shows the comparison between two different calibrations of @xmath46 . the line marks the average difference level.,title="fig:",width=302 ] clcp6.5 cm @xmath74 & @xcite & [email protected] & + @xmath76 & @xcite & [email protected] & we adopt the same conservative errors estimate for all stars . + @xmath81 & @xcite & [email protected] & + @xmath73 & @xcite & [email protected] - [email protected] & + @xmath82 & @xcite & @xmath83 & + @xmath84 & @xcite & [email protected] - [email protected] & [ table1 ] lccccccccc bd-84617 & 37.0 & 8.4 & 1.22 & 3.23 & 0.24 & 0.12 & 3.94 & 0.47 & 0.34 + bd-84634 & 38.0 & 7.4 & 1.22 & 2.92 & 0.25 & 0.12 & 3.56 & 0.48 & 0.34 + bd-11471 & 213.4 & 1.4 & 0.74 & 2.87 & 0.44 & 0.21 & 2.13 & 0.50 & 0.36 + bd+233762 & 60.3 & @xmath850.3 & 1.05 & 3.30 & 0.30 & 0.15 & 3.47 & 0.50 & 0.36 + bd+23771 & 37.2 & @xmath851.4 & 0.93 & 2.65 & 0.35 & 0.17 & 2.46 & 0.50 & 0.36 + bd+243893 & 61.3 & @xmath850.5 & 0.65 & 3.32 & 0.45 & 0.22 & 2.16 & 0.47 & 0.34 + bd+341054 & 173.4 & @xmath850.2 & 0.49 & 3.78 & 0.59 & 0.28 & 1.85 & 0.47 & 0.34 + bd+341059 & 173.0 & 0.2 & 0.49 & 3.75 & 0.59 & 0.29 & 1.84 & 0.47 & 0.34 + bd+341150 & 175.1 & 2.4 & 0.44 & 2.58 & 0.69 & 0.33 & 1.13 & 0.47 & 0.34 + bd+341162 & 175.5 & 2.6 & 0.36 & 2.81 & 0.84 & 0.40 & 1.01 & 0.47 & 0.34 + bd+343631 & 69.2 & 6.9 & 0.13 & 4.62 & 2.20 & 1.05 & 0.60 & 0.48 & 0.34 + bd+354258 & 77.2 & @xmath854.7 & 0.29 & 2.39 & 1.07 & 0.51 & 0.69 & 0.48 & 0.34 + bd+361261 & 174.1 & 4.3 & 0.52 & 2.75 & 0.62 & 0.30 & 1.43 & 0.50 & 0.36 + bd+363882 & 73.5 & 2.2 & 0.64 & 3.43 & 0.50 & 0.24 & 2.19 & 0.50 & 0.36 + bd+364145 & 77.5 & @xmath852.0 & 0.96 & 2.79 & 0.31 & 0.15 & 2.68 & 0.47 & 0.34 + bd+373945 & 77.3 & @xmath850.2 & 1.07 & 3.24 & 0.30 & 0.14 & 3.46 & 0.50 & 0.36 + bd+374092 & 80.2 & @xmath854.2 & 0.55 & 2.90 & 0.59 & 0.28 & 1.60 & 0.50 & 0.36 + bd+391328 & 169.1 & 3.6 & 0.88 & 2.62 & 0.37 & 0.18 & 2.30 & 0.50 & 0.36 + bd+404179 & 79.0 & 1.2 & 0.88 & 3.27 & 0.34 & 0.16 & 2.87 & 0.47 & 0.34 + bd+421286 & 166.1 & 4.3 & 0.56 & 3.12 & 0.53 & 0.26 & 1.75 & 0.47 & 0.34 [ table2 ] l c c c c c c c c c c hd1544 & 119.3 & @xmath850.6 & 0.44 & 3.19 & 0.73 & 0.35 & 0.37 & 2.79 & 0.67 & 0.31 + hd2083 & 120.9 & 9.0 & 0.29 & 3.69 & 1.01 & 0.48 & 0.26 & 3.99 & 0.70 & 0.33 + hd2905 & 120.8 & 0.1 & 0.33 & 3.24 & 1.20 & 0.57 & 0.30 & 0.82 & 1.05 & 0.49 + hd7252 & 125.7 & @xmath851.9 & 0.35 & 3.01 & 0.86 & 0.41 & 0.32 & 3.17 & 0.68 & 0.32 + hd12867 & 133.0 & @xmath853.7 & 0.41 & 2.72 & 0.74 & 0.35 & 0.38 & 2.84 & 0.63 & 0.30 + hd13969 & 134.5 & @xmath853.8 & 0.56 & 2.73 & 0.54 & 0.26 & 0.54 & 2.88 & 0.46 & 0.21 + hd14092 & 134.7 & @xmath854.1 & 0.49 & 2.52 & 0.62 & 0.30 & 0.46 & 2.61 & 0.55 & 0.26 + hd14250 & 134.8 & @xmath853.7 & 0.58 & 2.60 & 0.52 & 0.25 & 0.55 & 2.59 & 0.49 & 0.23 + hd14357 & 135.0 & @xmath853.9 & 0.56 & 3.90 & 0.56 & 0.27 & 0.49 & 3.28 & 0.54 & 0.25 + hd14818 & 135.6 & @xmath853.9 & 0.48 & 2.63 & 0.84 & 0.40 & 0.46 & 2.17 & 0.68 & 0.31 + hd14947 & 135.0 & @xmath851.8 & 0.77 & 2.96 & 0.39 & 0.19 & 0.76 & 2.50 & 0.37 & 0.17 + hd14956 & 135.4 & @xmath852.9 & 0.89 & 2.54 & 0.45 & 0.22 & 0.88 & 2.44 & 0.36 & 0.17 + hd16429 & 135.7 & 1.1 & 0.92 & 3.12 & 0.35 & 0.17 & 0.86 & 2.72 & 0.33 & 0.16 + hd17114 & 137.3 & @xmath850.3 & 0.76 & 2.85 & 0.40 & 0.19 & 0.73 & 2.92 & 0.35 & 0.16 + hd17603 & 138.8 & @xmath852.1 & 0.92 & 2.71 & 0.44 & 0.21 & 0.92 & 2.66 & 0.31 & 0.15 + hd18352 & 137.7 & 2.1 & 0.48 & 2.91 & 0.63 & 0.30 & 0.45 & 3.07 & 0.52 & 0.24 + hd24431 & 148.8 & @xmath850.7 & 0.69 & 2.79 & 0.44 & 0.21 & 0.65 & 2.64 & 0.41 & 0.19 + hd24912 & 160.4 & @xmath8513.1 & 0.29 & 3.45 & 1.36 & 0.65 & 0.26 & 3.78 & 0.75 & 0.36 + hd30614 & 144.1 & 14.0 & 0.30 & 3.01 & 1.33 & 0.63 & 0.34 & 3.07 & 0.74 & 0.35 + hd34078 & 172.1 & @xmath852.3 & 0.52 & 3.44 & 0.57 & 0.27 & 0.49 & 3.20 & 0.49 & 0.23 [ table3 ]

we present determinations of the total to selective extinction ratio @xmath0 and visual extinction @xmath1 values for milky way stars using ultraviolet color excesses . we extend the analysis of @xcite by using non - equal weights derived from observational errors . we present a detailed discussion of various statistical errors . in addition , we estimate the level of systematic errors by considering different normalization of the extinction curve adopted by @xcite . our catalog of 782 @xmath0 and @xmath1 values and their errors is available in the electronic form on the world wide web . * key words : * catalogs dust , extinction galaxy : general ism : structure techniques : photometric

arxiv

an @xmath0-hypergeometric system @xmath1 is a @xmath4-module determined by an integral matrix @xmath0 and a complex parameter vector @xmath5 . these systems are also known as _ gkz - systems _ , as they were introduced in the late 1980 s by gelfand , graev , kapranov , and zelevinsky @xcite . their solutions occur naturally in mathematics and physics , including the study of roots of polynomials , picard fuchs equations for the variation of hodge structure of calabi - yau toric hypersurfaces , and generating functions for intersection numbers on moduli spaces of curves , see @xcite . the ( holonomic ) rank of @xmath1 coincides with the dimension of its solution space at a nonsingular point . in this article , we provide a combinatorial formula for the rank of @xmath1 in terms of certain lattice translates determined by @xmath0 and @xmath6 . for a fixed matrix @xmath0 , this computation yields a geometric stratification of the parameter space @xmath7 that refines its stratification by the rank of @xmath1 . let @xmath8 $ ] be an integer @xmath9-matrix with integral column span @xmath10 . assume further that @xmath0 is _ pointed _ , meaning that the origin is the only linear subspace of the cone @xmath11 . a subset @xmath12 of the column set of @xmath0 is called a _ face _ of @xmath0 , denoted @xmath13 , if @xmath14 is a face of the cone @xmath15 . let @xmath16 be variables and @xmath17 their associated partial differentiation operators . in the polynomial ring @xmath18 $ ] , let @xmath19 denote the toric ideal associated to @xmath0 , and let @xmath20 be its quotient ring . note that @xmath21 is isomorphic to the semigroup ring of @xmath0 , which is @xmath22 : = \bigoplus_{a\in { { \ensuremath{\mathbb{n}}}}a } { { \ensuremath{\mathbb{c}}}}\cdot t^a \end{aligned}\ ] ] with multiplication given by semigroup addition of exponents . the weyl algebra @xmath23=\delta_{ij } , [ x_i , x_j]=0=[\del_i,\del_j]\>\ ] ] is the ring of @xmath24-linear differential operators on @xmath25 $ ] . [ def - gkz ] the _ @xmath0-hypergeometric system _ with parameter @xmath26 is the left @xmath4-module @xmath27 where @xmath28 are _ euler operators _ associated to @xmath0 . the _ rank _ of a left @xmath4-module @xmath29 is @xmath30}m.\ ] ] the rank of a holonomic @xmath4-module is finite and equal to the dimension of its solution space of germs of holomorphic functions at a generic nonsingular point @xcite . in @xcite , gelfand , kapranov , and zelevinsky showed that when @xmath21 is cohen macaulay and standard @xmath31-graded , the @xmath0-hypergeometric system @xmath1 is holonomic of rank @xmath32 for all parameters @xmath6 , where @xmath32 is @xmath33 times the euclidean volume of the convex hull of @xmath0 and the origin . adolphson established further that @xmath1 is holonomic for all choices of @xmath0 and @xmath6 and that the holonomic rank of @xmath1 is generically given by @xmath32 @xcite . however , an example found by sturmfels and takayama showed that equality need not hold in general @xcite ( see also @xcite ) . at the same time , cattani , dandrea , and dickenstein produced an infinite family of such examples through a complete investigation of the rank of @xmath1 when @xmath2 defines a projective monomial curve @xcite . the relationship between @xmath32 and the rank of @xmath1 was made precise by matusevich , miller , and walther , who used euler koszul homology to study the holonomic _ generalized @xmath0-hypergeometric system _ @xmath34 associated to a toric module @xmath29 ( see definition [ def : toric ] ) . the euler koszul homology @xmath35 of @xmath29 with respect to @xmath6 is the homology of a twisted koszul complex on @xmath36 given by the sequence @xmath37 . this includes the @xmath0-hypergeometric system @xmath38 as the special case that @xmath39 . as in this special case , and for the purposes of this article , suppose that the generic rank of @xmath40 is @xmath32 . the matrix @xmath0 induces a natural @xmath41-grading on @xmath42 ; the _ quasidegree set _ of a finitely generated @xmath41-graded @xmath42-module @xmath43 is defined to be the zariski closure in @xmath7 of the set of vectors @xmath44 for which the graded piece @xmath45 is nonzero . in @xcite , an explicit description of the _ exceptional arrangement _ @xmath46 associated to @xmath29 is given in terms of the quasidegrees of certain @xmath47 modules involving @xmath29 ( see ) . this description shows that @xmath48 is a subspace arrangement in @xmath7 given by translates of linear subspaces that are generated by the faces of the cone @xmath15 , and that @xmath48 is empty exactly when @xmath49 is a maximal cohen macaulay @xmath21-module . it is also shown in @xcite that the rank of @xmath35 is upper semi - continuous as a function of @xmath6 . thus the exceptional arrangement @xmath48 is the nested union over @xmath50 of the zariski closed sets @xmath51 in particular , the rank of @xmath34 induces a stratification of @xmath48 , which we call its _ rank stratification_. the present article is a study of the rank stratification of @xmath48 when @xmath52 $ ] is @xmath41-graded such that the degree set @xmath53 of @xmath29 is a nontrivial @xmath54monoid . in particular , @xmath29 is weakly toric ( see definition [ def - wtoric ] ) . if @xmath55 , then @xmath29 is the semigroup ring @xmath21 from and @xmath56 is the @xmath0-hypergeometric system at @xmath6 . the module @xmath29 could also be a localization of @xmath21 along a subset of faces of @xmath0 . as @xmath29 will be fixed throughout this article , we will often not indicate dependence on @xmath29 in the notation . examination of the long exact sequence in euler koszul homology induced by the short exact sequence of weakly toric modules @xmath57 \rightarrow q \rightarrow 0\end{aligned}\ ] ] reveals that the _ rank jump _ of @xmath29 at @xmath6 , @xmath58 we define the _ ranking arrangement _ @xmath59 of @xmath29 to be the quasidegrees of @xmath60 . vanishing properties of euler koszul homology imply that the exceptional arrangement @xmath48 is contained in the ranking arrangement @xmath59 . we show in theorem [ thm - str comparison ] that @xmath59 is the union of @xmath48 and an explicit collection of hyperplanes . for a fixed @xmath61 , we then proceed to compute @xmath62 . in section [ sec : r - t mods ] , we combinatorially construct a _ finitely generated _ @xmath41-graded _ ranking toric module _ @xmath63 with @xmath64 . since @xmath62 is determined by the euler koszul homology of @xmath60 by , we see that @xmath63 contains the information essential to computing the rank jump @xmath62 . to outline the construction of the module @xmath63 , let @xmath65 be the polyhedral complex of faces of @xmath0 determined by the components of the ranking arrangement @xmath59 that contain @xmath6 . we call the collection of integral points @xmath66 the _ ranking lattices _ @xmath67 of @xmath29 at @xmath6 . this set is a union of translates of lattices generated by faces of @xmath0 , where the vectors in these lattice translates of @xmath68 in @xmath67 are precisely the degrees of @xmath60 which cause @xmath69 to lie in the ranking arrangement . since it contains full lattice translates , @xmath67 can not be the degree set of a finitely generated @xmath21-module . thus , to complete the construction of the degree set @xmath70 of @xmath63 , we intersect @xmath67 with an appropriate half space ( see definition [ def - cp ] ) . to give a flavor of our approach for @xmath71 , this is equivalent to intersecting @xmath72 with @xmath73 $ ] . by setting up the proper module structure , @xmath74 gives the @xmath41-graded degree set of the desired toric module @xmath63 with @xmath75 . after translating the computation of the rank jump @xmath62 to @xmath63 , we obtain a generalization of the formula given by okuyama in the case @xmath76 @xcite . [ thm - compute jump ] the rank jump @xmath62 of @xmath29 at @xmath6 can be computed from the combinatorics of the ranking lattices @xmath67 of @xmath29 at @xmath6 . in particular , the rank of the hypergeometric system is the same at parameters which share the same ranking lattices . the proof of theorem [ thm - compute jump ] can be found in section [ subsec : rank jumps ] as a special case of our main result , theorem [ thm - main ] . let @xmath77 and @xmath78 be subspace arrangements in @xmath7 . we say that a stratification @xmath79 of @xmath77 _ respects _ @xmath78 if for each irreducible component @xmath80 of @xmath78 and each stratum @xmath81 , either @xmath82 or @xmath83 . a _ ranking slab _ of @xmath29 is a stratum in the coarsest stratification of @xmath48 that respects the arrangements @xmath59 and the negatives of the quasidegrees of each of the @xmath47 modules that determine @xmath48 ( see definition [ def - ranking slab ] ) . proposition [ prop - zc e ] states that the parameters @xmath84 belong to the same ranking slab of @xmath29 exactly when their ranking lattices coincide , that is , @xmath85 . combining this with theorem [ thm - compute jump ] , we see that the rank of @xmath40 is constant on each ranking slab . [ cor - strat vs equiv ] the function @xmath86 is constant on each ranking slab . in particular , the stratification of the exceptional arrangement @xmath48 by ranking slabs refines its rank stratification . hence , like @xmath48 , each set @xmath87 is a union of translated linear subspaces of @xmath7 which are generated by faces of @xmath15 . in order for the ranking slab stratification of @xmath48 to refine its rank stratification , it must respect each of the arrangements appearing in its definition ; this can be seen from examples [ example - hidden ] , [ example - nonconstant slab ] , and [ plane - line-4diml ] . in particular , as @xmath88 , the exceptional arrangement @xmath48 does not generally contain enough information to determine its rank stratification . when @xmath39 , the ranking lattices @xmath67 are directly related to the combinatorial sets @xmath89 defined by saito , which determine the isomorphism classes of @xmath1 . in @xcite , various @xmath90-functions arising from an analysis of the symmetry algebra of @xmath0-hypergeometric systems are used to link these isomorphism classes to the sets @xmath89 . we conclude this paper with a shorter proof , replacing the use of @xmath90-functions with euler koszul homology . the following is a brief outline of this article . in section [ sec : defns ] , we summarize definitions and results on weakly toric modules and euler koszul homology , following @xcite . section [ sec : ekstr ] is a study of the structure of the euler koszul complex of maximal cohen macaulay toric face modules . the relationship between the exceptional and ranking arrangements of @xmath29 is made precise in section [ sec : exceptional arrangement ] . in section [ sec : r - t mods ] , we define the class of ranking toric modules , which play a pivotal role in calculating the rank jump @xmath62 . section [ sec : comb of rank ] contains our main theorem , theorem [ thm - main ] , which results in the computation of @xmath62 for a fixed parameter @xmath6 . we close with a discussion on the isomorphism classes of @xmath0-hypergeometric systems in section [ sec : isom classes ] . i am grateful to my advisor uli walther for many inspiring conversations and thoughtful suggestions throughout the duration of this work . i would also like to thank laura felicia matusevich for asking the question is rank constant on a slab ? " as well as helpful remarks on this text and conversations that led directly to the proof of theorem [ thm - isom classes ] . finally , my thanks to ezra miller for many valuable comments for improving this paper , including the addition of definition [ def - cellular definition ] , and to the referee for detailed notes that clarified this article . in this section , we summarize definitions found in the literature and set notation . most important are the definitions of a weakly toric module @xcite and euler koszul homology @xcite . let @xmath91 denote the columns of @xmath0 . for a face @xmath13 , let @xmath92 denote the complement of a face @xmath12 in the column set of @xmath0 . if @xmath12 is any subset of the columns of @xmath0 , the _ codimension _ of @xmath12 is @xmath93 , the codimension of the @xmath94-vector space generated by @xmath12 . the _ dimension _ of @xmath12 is @xmath95 . a face @xmath12 of @xmath0 is a _ facet _ of @xmath0 if @xmath96 . recall that the _ primitive integral support function _ of a facet @xmath13 is the unique linear functional @xmath97 such that @xmath98 @xmath99 for all @xmath100 , and @xmath101 exactly when @xmath102 . the _ volume _ of a face @xmath12 , denoted @xmath103 , is the integer @xmath104 times the euclidean volume in @xmath105 of the convex hull of @xmath12 and the origin . [ def : semigroup ring d ] let @xmath106 be the semigroup generated by the face @xmath12 and , as in , @xmath107\ ] ] is the corresponding semigroup ring , called a _ face ring _ of @xmath0 . let @xmath108 and @xmath109 . define @xmath110\ ] ] to be the polynomial ring in @xmath111 and @xmath112 = \delta_{ij},[x_i , x_j ] = 0 = [ \del_i,\del_j]\>\ ] ] to be the _ weyl algebra _ associated to @xmath12 . note that @xmath113 [ def : monoid ] let @xmath114 be variables . for a face @xmath13 , we say that a subset @xmath115 is an _ @xmath116module _ if @xmath117 . further , we call an @xmath116module @xmath118 an _ @xmath116monoid _ if it is closed under addition , that is , for all @xmath119 , @xmath120 . given an @xmath116module @xmath118 , define the @xmath121-module @xmath122 as a @xmath24-vector space with @xmath121-action given by @xmath123 . further , @xmath124 is equipped with a multiplicative structure given by @xmath125 for @xmath119 and extended @xmath24-linearly . by definition , @xmath116 is an @xmath116monoid and @xmath126 as rings . define a @xmath41-grading on @xmath127 by setting @xmath128 then @xmath124 is naturally a @xmath41-graded @xmath121-module by setting @xmath129 . the _ saturation _ of @xmath12 in @xmath68 is the semigroup @xmath130 . the _ saturation _ , or _ normalization _ , of @xmath121 is the semigroup ring of the saturation of @xmath12 in @xmath68 , which is given by @xmath131 as a @xmath41-graded @xmath121-module . note that @xmath132 is a cohen macaulay @xmath121-module @xcite . if @xmath43 is a @xmath41-graded @xmath42-module and @xmath133 , the _ degree set _ of @xmath43 , denoted @xmath134 , is the set of all @xmath133 such that @xmath135 . let @xmath136 denote the @xmath41-graded module with @xmath137-graded piece @xmath138 . we now recall the definitions of toric and weakly toric modules and their quasidegree sets , which can be found in ( * ? ? ? * definition 4.5 ) and ( * ? ? ? * section 5 ) , respectively . [ def : toric ] a @xmath41-graded @xmath42-module is _ toric _ if it has a filtration @xmath139 such that for each @xmath140 , @xmath141 is a @xmath41-graded translate of @xmath142 for some face @xmath143 . notice that toric modules are necessarily finitely generated @xmath42-modules . [ def - qdeg ] if @xmath43 is a finitely generated @xmath41-graded @xmath42-module , a vector @xmath144 is a _ quasidegree _ of @xmath43 , written @xmath145 , if @xmath146 lies in the zariski closure of @xmath134 under the natural embedding @xmath147 . notice that if @xmath43 is toric , then @xmath148 is a finite subspace arrangement in @xmath7 , consisting of translated subspaces generated by faces of @xmath0 , see ( * ? ? ? * lemma 2.5 ) . a partially ordered set @xmath149 is _ filtered _ if for each @xmath150 there exists @xmath151 with @xmath152 and @xmath153 . [ def - wtoric ] we say that a @xmath41-graded @xmath42-module @xmath29 is _ weakly toric _ if there is a filtered partially ordered set @xmath149 and a @xmath41-graded direct limit @xmath154 where @xmath155 is a toric @xmath42-module for each @xmath151 . we then define the _ quasidegree set _ of @xmath29 to be @xmath156 where each @xmath157 is defined by definition [ def - qdeg ] . [ ex - m wt ] if @xmath158 is an @xmath54module , then @xmath159 is weakly toric because it is a direct limit over @xmath160 of @xmath161 under the natural @xmath0-homogeneous inclusion @xmath162\cong { { \ensuremath{\mathbb{c}}}}[{{\ensuremath{\mathbb{z}}}}^d]$ ] . consider the matrix @xmath163 $ ] with face @xmath164 $ ] . the module @xmath165 $ ] is weakly toric with quasidegree set @xmath166\right ) = { { \ensuremath{\mathbb{c}}}}f\ ] ] because it is a filtered direct limit over @xmath167 of @xmath168 . similarly , the module @xmath169 $ ] is weakly toric with @xmath170\right ) = { { \ensuremath{\mathbb{c}}}}^2 $ ] . the quotient @xmath169/s_a$ ] is also weakly toric . its quasidegree set consists of the point @xmath171 $ ] and the union of lines in @xmath172 , where @xmath173 are the coordinates of @xmath174 . we now recall the definition of the euler koszul complex of a weakly toric module @xmath29 with respect to a parameter @xmath26 . for @xmath175 , each _ euler operator _ @xmath176 determines a @xmath41-graded @xmath4-linear endomorphism of @xmath36 , defined on a homogeneous @xmath177 by @xmath178 and extended @xmath24-linearly . this sequence @xmath179 of commuting endomorphisms determines a koszul complex @xmath180 on the left @xmath4-module @xmath36 , called the _ euler koszul complex _ of @xmath29 with parameter @xmath6 . @xmath181 euler koszul homology module _ of @xmath29 is @xmath182 . our object of study will be the _ generalized @xmath0-hypergeometric system _ @xmath34 associated to @xmath29 . the euler koszul complex defines an exact functor from the category of weakly toric modules with degree - preserving morphisms to the category of bounded complexes of @xmath41-graded left @xmath4-modules with degree - preserving morphisms , so short exact sequences of weakly toric modules yield long exact sequences of euler koszul homology . notice also that euler koszul homology behaves well under @xmath41-graded translations : for @xmath183 , @xmath184 we close this section by recording two important vanishing results for euler koszul homology . [ prop - hh qis 0 ] for a weakly toric module @xmath29 , the following are equivalent : 1 . @xmath185 for all @xmath186 , 2 . @xmath187 , 3 . @xmath188 see ( * ? ? ? * theorem 5.4 ) . [ thm - cm hh rel ] let @xmath29 be a weakly toric module . then @xmath185 for all @xmath189 and for all @xmath26 if and only if @xmath29 is a maximal cohen macaulay @xmath21-module . see ( * ? ? ? * theorem 6.6 ) for the toric case . the extension to the weakly toric case can be found in @xcite . theorem [ thm - cm hh rel ] provides a criterion for higher vanishing of euler koszul homology via maximal cohen macaulay @xmath21-modules . in this section , we provide a description of the euler koszul homology modules of maximal cohen macaulay @xmath121-modules for a face @xmath13 and use it to understand the images of maps between such modules . throughout this section , @xmath43 is a toric @xmath121-module for a face @xmath13 . recall the definitions of toric , @xmath121 , @xmath190 , and @xmath191 from definition [ def : semigroup ring d ] , and let @xmath192 [ not - hhf ] let @xmath193 be the lexicographically first subset of @xmath194 of cardinality @xmath195 such that @xmath196 is a set of linearly independent euler operators on @xmath197 . the existence of @xmath193 follows from the fact that the matrix @xmath0 has full rank . we use @xmath198 to denote the euler koszul complex on @xmath199 given by the operators @xmath200 , and set @xmath201 using the standard basis of @xmath10 , let @xmath202 and let @xmath203 denote a complex with trivial differentials . we show now that when @xmath204 , @xmath205 is quasi - isomorphic to a complex involving @xmath198 and @xmath203 . [ prop - str cm ek complex ] let @xmath13 and @xmath43 be a toric @xmath121-module . if @xmath206 , then there is a quasi - isomorphism of complexes @xmath207 \otimes_{{\ensuremath{\mathbb{c}}}}\kk_\bullet^f(n,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^\bullet{{\ensuremath{\mathbb{z}}}}f^\bot\right).\end{aligned}\ ] ] in particular , if @xmath43 is maximal cohen macaulay as an @xmath121-module , there is a decomposition @xmath208 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_0^f(n,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^\bullet{{\ensuremath{\mathbb{z}}}}f^\bot\right).\end{aligned}\ ] ] under the hypotheses of proposition [ prop - str cm ek complex ] , @xmath209 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_0^f(n,\beta^f)^{\binom{\codim(f)}{i}},\ ] ] for @xmath189 , as shown in @xcite . in particular , @xmath210 we show in proposition [ prop - str / rk image ] that surjections of maximal cohen macaulay toric modules for nested faces yield induced maps on euler koszul homology that respect the decompositions of . the additional information stored in @xmath203 of shows how images of collections of such surjections overlap , which will be vital to our calculation of @xmath62 in section [ sec : comb of rank ] . [ pf - gf ] fix a matrix @xmath211 such that the entries of each row of @xmath212 not corresponding to @xmath193 are zero and the rows of @xmath0 that do correspond to @xmath193 are identical in @xmath0 and @xmath213 . setting @xmath214 , @xmath21 and @xmath215 are isomorphic rings , and the matrix @xmath216 gives a bijection of their degree sets , sending @xmath54 to @xmath217 . this identification makes @xmath43 a @xmath218-graded @xmath219-module , where @xmath220 , and there is a quasi - isomorphism of complexes @xmath221 let @xmath222 and @xmath223 , and recall that @xmath224 . by the definition of @xmath216 , @xmath225 for @xmath226 because @xmath227 . let @xmath228 and @xmath229 denote the weyl algebra and the polynomial ring @xmath230 $ ] with an @xmath231-grading . since @xmath43 is an @xmath219-module , @xmath232 , and so there is an isomorphism @xmath233 \otimes_{{\ensuremath{\mathbb{c}}}}d_{f ' } \otimes_{r_{f ' } } n$ ] . hence the action of each element in @xmath234 on @xmath235 is 0 . if @xmath236 denotes the standard basis of @xmath237 , then the set @xmath238 generates @xmath239 by choice of @xmath216 . applying the isomorphism @xmath240 in the reverse direction , we obtain finally , if @xmath43 is maximal cohen macaulay as an @xmath121-module , @xmath241 for all @xmath189 by theorem [ thm - cm hh rel ] . [ remark - diffl in cm ek complex ] let @xmath242 and @xmath243 respectively denote the differentials of the euler koszul complexes @xmath244 and @xmath198 . under the hypotheses of proposition [ prop - str cm ek complex ] , if @xmath245 and @xmath246 \otimes_{{\ensuremath{\mathbb{c}}}}\kk^f_i(n,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^j{{\ensuremath{\mathbb{z}}}}f^\bot\right)$ ] , then @xmath247 is an element of @xmath248 \otimes_{{\ensuremath{\mathbb{c}}}}\kk^f_{i-1}(n,\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left(\textstyle\bigwedge^j{{\ensuremath{\mathbb{z}}}}f^\bot\right ) \subseteq \kk^{a'}_{q-1}(n,\beta).$ ] [ example - ek cm comp ] consider the matrix @xmath249.\ ] ] set @xmath250 \in{{\ensuremath{\mathbb{c}}}}^3 $ ] , and let @xmath251 denote the standard basis vectors in @xmath252 . notice that every face ring of @xmath0 is cohen macaulay because the semigroup generated by each face of @xmath0 is saturated . for the face @xmath253 , we choose @xmath254 to be the identity matrix . the proof of proposition [ prop - str cm ek complex ] shows that there is an isomorphism of complexes @xmath255\cdot e_i \ar[r]^{0\cdot } & { { \ensuremath{\mathbb{c}}}}[x]}\right),$ ] so the euler koszul homology of @xmath256 at @xmath6 is @xmath257 \cdot e_i\right).\ ] ] for the face @xmath258 $ ] of @xmath0 , @xmath259 is again already in the desired form , so take @xmath216 to be the identity matrix and write @xmath25\<\del_1,\del_2\>$ ] in place of @xmath260 \otimes_{{\ensuremath{\mathbb{c}}}}d_f \otimes_{r_f } s_f$ ] . then proposition [ prop - str cm ek complex ] implies that @xmath261\<\del_1,\del_2\ > & \text{if $ q=0$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\<\del_1,\del_2\>\cdot e_3 & \text{if $ q=1$,}\\ 0 & \text{otherwise . } \end{cases } \intertext{for $ g_1=[a_1]$ and $ g_{g_1}$ as the identity matrix , } \hh_q(s_{g_1},\beta ) & = \begin{cases } { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\ > & \text{if $ q=0$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\>\cdot e_2 \oplus { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\>\cdot e_3 & \text{if $ q=1$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\>\cdot e_2\wedge e_3 & \text{if $ q=2$,}\\ 0 & \text{otherwise . } \end{cases } \intertext{for the face $ g_2 = [ a_2]$ , setting $ g_{g_2 } = \left[\begin{smallmatrix } 1&\phantom{-}0&0\\ 1&-1&0\\ 0&\phantom{-}0&1 \end{smallmatrix}\right]$ yields the decomposition } \hh_q(s_{g_2},\beta ) & = \begin{cases } { { \ensuremath{\mathbb{c}}}}[x]\<\del_2\ > & \text{if $ q=0$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\<\del_2\>\cdot ( e_1-e_2 ) \oplus { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\>\cdot e_3 & \text{if $ q=1$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\<\del_2\>\cdot ( e_1-e_2)\wedge e_3 & \text{if $ q=2$,}\\ 0 & \text{otherwise . } \end{cases}\end{aligned}\ ] ] [ lemma - ek morph ] let @xmath262 be faces of @xmath0 , @xmath43 be a toric @xmath121-module , and @xmath263 be a toric @xmath264-module . regard @xmath43 and @xmath263 as toric @xmath21-modules via the natural maps @xmath265 . let @xmath266 be a morphism of @xmath21-modules . then there is a commutative diagram @xmath267^{\kk_\bullet(\pi,\beta ) } \ar[d ] & \kk_\bullet(l,\beta ) \ar[d ] \\ { { \ensuremath{\mathbb{c}}}}[x_{f^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk^{f}_\bullet(n,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}f^\bot \right ) \ar[r ] & { { \ensuremath{\mathbb{c}}}}[x_{g^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk^{g}_\bullet(l,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}g^\bot \right ) } \end{aligned}\ ] ] with vertical maps as in . by choice of @xmath193 and @xmath268 in notation [ not - hhf ] and the corresponding @xmath216 and @xmath269 , the diagram @xmath270^{\kk_\bullet(\pi,\beta ) } \ar[d ] & & \kk_\bullet(l,\beta ) \ar[d ] \\ \kk^{g_f a}_\bullet(n,\beta ) \ar[r ] & \kk^{g_f a}_\bullet(l,\beta ) \ar[r ] & \kk^{g_g a}_\bullet(l,\beta ) } \ ] ] commutes . hence the result follows from the proof of proposition [ prop - str cm ek complex ] . [ prop - str / rk image ] let @xmath262 be faces of @xmath0 , @xmath43 be a maximal cohen macaulay toric @xmath121-module , and @xmath263 be a maximal cohen macaulay toric @xmath264-module . regard @xmath43 and @xmath263 as toric @xmath21-modules via the natural maps @xmath265 . let @xmath271 be a surjection of @xmath21-modules . if @xmath272 , then @xmath273 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_0^g(l,\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left(\textstyle\bigwedge^\bullet{{\ensuremath{\mathbb{z}}}}f^\bot\right)\ ] ] as a submodule of @xmath274 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_0^g(l,\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left(\textstyle\bigwedge^\bullet{{\ensuremath{\mathbb{z}}}}g^\bot\right)$ ] . ( continuation of example [ example - ek cm comp ] ) [ example - ek cm comp i m ] the surjection of face rings given by @xmath275 induces the following image in euler koszul homology : @xmath276\<\del_1\ > & \text{if $ q=0$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\<\del_1\>\cdot e_3 & \text{if $ q=1$,}\\ 0 & \text{otherwise . } \end{cases}\ ] ] with @xmath224 , the image of @xmath277 is isomorphic to the image of @xmath278 . by proposition [ prop - str cm ek complex ] , there are decompositions @xmath279 \otimes_{{\ensuremath{\mathbb{c}}}}\hh^f_0(n,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}f^\bot \right ) \intertext{and } \hh^{a'}_\bullet(l,\beta ) & = { { \ensuremath{\mathbb{c}}}}[x_{f^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\hh^f_\bullet(l,\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}f^\bot \right),\end{aligned}\ ] ] so it is enough to find the image of @xmath280 as a submodule of @xmath281 . the result now follows because the sequence @xmath282^{\hh_0^f(\pi,\beta ) } & & \hh_0^f(l,\beta ) \ar[r ] & 0}\ ] ] is exact , @xmath283 \otimes_{{\ensuremath{\mathbb{c}}}}\hh^g_0(l,\beta)$ ] , and @xmath284 for @xmath189 . ( continuation of example [ example - ek cm comp i m ] ) [ example - ek cm comp i m intersect ] let @xmath285 for @xmath286 . then @xmath287 & \text{if $ q=0$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\cdot e_2 \oplus { { \ensuremath{\mathbb{c}}}}[x]\cdot e_3 & \text{if $ q=1$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\cdot e_2\wedge e_3 & \text{if $ q=2$,}\\ 0 & \text{otherwise , } \end{cases } \intertext{and } \image \hh_q(\pi_2,\beta ) & = \begin{cases } { { \ensuremath{\mathbb{c}}}}[x ] & \text{if $ q=0$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\cdot ( e_2-e_1 ) \oplus { { \ensuremath{\mathbb{c}}}}[x]\cdot e_3 & \text{if $ q=1$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\cdot ( e_2-e_1 ) \wedge e_3 & \text{if $ q=2$,}\\ 0 & \text{otherwise . } \end{cases}\end{aligned}\ ] ] the intersection of the images of euler koszul homology at @xmath6 applied to @xmath288 and @xmath289 is @xmath290\cap[\image \hh_q(\pi_2,\beta ) ] = \begin{cases } { { \ensuremath{\mathbb{c}}}}[x ] & \text{if $ q=0$,}\\ { { \ensuremath{\mathbb{c}}}}[x]\cdot e_3 & \text{if $ q=1$,}\\ 0 & \text{otherwise } \end{cases}\ ] ] because @xmath291 . we close this section with an observation that is vital to our rank jumps computations . for faces @xmath292 , set @xmath293 . let @xmath294 be maximal cohen macaulay toric @xmath142-modules , @xmath263 be a maximal cohen macaulay toric @xmath264-module , and @xmath295 be @xmath21-module surjections . suppose that @xmath272 . using the equality @xmath296^\bot$ ] , proposition [ prop - str / rk image ] implies that @xmath297 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_0^g(l,\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left(\textstyle\bigwedge^i { { \ensuremath{\mathbb{z}}}}[f\cup g]^\bot\right ) , \end{array}\ ] ] which has rank @xmath298 let @xmath299 be a nonempty @xmath54monoid ( see definition [ def : monoid ] ) , so that the nontrivial module @xmath300 $ ] is weakly toric ( see example [ ex - m wt ] ) and @xmath301 since @xmath302 is an @xmath54-monoid , the generic rank of @xmath40 is @xmath32 . rank jump _ of @xmath29 at @xmath6 is the nonnegative integer @xmath303 of parameters with nonzero rank jump . by @xcite and @xcite , the exceptional arrangement can be described in terms of certain @xmath47 modules involving @xmath29 , namely @xmath304 where @xmath305 . it follows that @xmath48 is a union of translates of linear subspaces spanned by the faces of @xmath0 , see ( * ? ? ? * corollary 9.3 ) . we begin our study of @xmath62 with the short exact sequence @xmath306 \rightarrow q \rightarrow 0.\end{aligned}\ ] ] while @xmath60 is not a noetherian @xmath21-module , it is a filtered limit of noetherian @xmath41-graded @xmath21-modules and is therefore weakly toric ( see definition [ def - wtoric ] ) . thus the _ ranking arrangement _ of @xmath29 @xmath307 is an infinite union of translates of linear subspaces of @xmath7 spanned by proper faces of @xmath0 . since @xmath308 $ ] is a maximal cohen macaulay @xmath21-module , theorem [ thm - cm hh rel ] implies that @xmath309,\beta)=0 $ ] for all @xmath189 . moreover , by ( * ? ? ? * theorem 4.2 ) , @xmath310,\beta)=\vol(a)$ ] for all @xmath6 . examination of the long exact sequence in euler koszul homology from reveals that @xmath311 this implies that for @xmath61 , @xmath312 is nonzero . therefore there is an inclusion @xmath313 . we make this relationship precise in theorem [ thm - str comparison ] . [ lemma - finite inside ] let @xmath133 . the number of irreducible components of @xmath314 which intersect @xmath315 is finite . view @xmath21 and its shifted saturation @xmath316 as graded submodules of @xmath308 $ ] . to see that the intersection @xmath317 involves only a finite number of irreducible components of @xmath314 , it is enough to show that the arrangement given by the quasidegrees of the module @xmath318 has finitely many irreducible components . this follows since @xmath316 is toric . recall from that @xmath53 . for @xmath183 and @xmath13 , let @xmath319 . [ lemma - mm05 ] let @xmath52 $ ] be a weakly toric module , @xmath183 , @xmath13 be a face of codimension at least two , and @xmath320 be an interior vector of @xmath116 . if @xmath12 is maximal among faces of @xmath0 not in @xmath321 , then for all sufficiently large positive integers @xmath322 , the vector @xmath323 is an exceptional degree of @xmath29 . this is ( * ? ? ? * lemma 14 ) when @xmath324 and @xmath0 is homogeneous . ( the matrix @xmath0 is called _ homogeneous _ when the vector @xmath325 is in the @xmath326-row span of @xmath0 . ) the same argument yields this generalization by @xmath41-graded local duality , see ( * ? ? ? * section 3.5 ) . [ thm - str comparison ] let @xmath52 $ ] be a weakly toric module . the ranking arrangement @xmath59 contains the exceptional arrangement @xmath48 and @xmath327 where @xmath328 is pure of codimension 1 . we must show that @xmath48 contains all irreducible components of @xmath59 of codimension at least two . to this end , let @xmath329 be such that @xmath330 is an irreducible component with @xmath331 . then there are submodules @xmath332 and @xmath333 such that @xmath334 and @xmath335 . in fact , there is a @xmath336 with @xmath337 and @xmath338 . we may further choose @xmath90 so that @xmath12 is maximal among faces of @xmath0 that are not in the set @xmath321 . to see this , first note that @xmath339 for all @xmath340 . indeed , for if @xmath341 , then there are @xmath342 and @xmath343 with @xmath344 , a contradiction . since @xmath338 is an irreducible component of @xmath345 , it suffices to show that @xmath90 can be chosen so that each facet @xmath346 of @xmath0 is in @xmath321 . first , if @xmath347 , then by lemma [ lemma - finite inside ] , there are at most a finite number of translates of @xmath348 that define components of @xmath345 and intersect @xmath349 ; write these as @xmath350 . if necessary , replace @xmath90 by a vector @xmath351 such that @xmath352 to assume that @xmath346 is in @xmath321 . note that after such a replacement , it is still true that @xmath353 by the previous paragraph , so @xmath354 . next , suppose that @xmath355 . if @xmath356 , then @xmath357 , an impossibility because @xmath358 defines an irreducible component of @xmath345 . thus it must be that @xmath359 . in this case , @xmath360 , so @xmath346 is in @xmath321 . hence every facet @xmath346 of @xmath0 is in @xmath321 , and the claim on the choice of @xmath90 is established . let @xmath361 be an interior vector of @xmath116 . lemma [ lemma - mm05 ] implies that for all sufficiently large integers @xmath322 , the vector @xmath362 . therefore @xmath363 . [ not - beta cmpt ] for @xmath26 , the _ @xmath6-components _ @xmath364 of the ranking arrangement of @xmath29 are the union of the irreducible components of @xmath59 which contain @xmath6 . since @xmath0 has a finite number of faces , @xmath364 has finitely many irreducible components . by ( * porism 9.5 ) , the exceptional arrangement @xmath365 of the @xmath0-hypergeometric system @xmath366 has codimension at least two . in the following example we show that there may be components of @xmath365 which are embedded in codimension 1 components of the ranking arrangement @xmath314 . in particular , the zariski closure of @xmath367 may not agree with @xmath365 . [ example - hidden ] let @xmath368\ ] ] with @xmath369 , and label the faces @xmath370 $ ] and @xmath371 $ ] . with @xmath372 $ ] , the exceptional arrangement of @xmath324 is properly contained in a hyperplane component of the ranking arrangement : @xmath373 we will discuss the rank jumps of @xmath1 in examples [ example - hidden2 ] and [ example - hidden3 ] . one goal of section [ sec : comb of rank ] is to understand the structure of the sets @xmath374 . we will achieve this by stratifying @xmath48 by ranking slabs ( see definition [ def - ranking slab ] ) . another description of ranking slabs ( via translates of certain lattices contained in the @xmath6-components @xmath364 ) will be given in proposition [ prop - zc e ] . let @xmath77 and @xmath78 be subspace arrangements in @xmath7 . we say that a stratification @xmath79 of @xmath77 _ respects _ @xmath78 if for each irreducible component @xmath80 of @xmath78 and each stratum @xmath81 , either @xmath82 or @xmath83 . [ def - ranking slab ] a _ ranking slab _ of @xmath29 is a stratum in the coarsest stratification of @xmath48 that respects each of the following arrangements : @xmath59 and @xmath375 for @xmath376 . since each of the arrangements used in definition [ def - ranking slab ] is determined by the quasidegrees of a weakly toric module , the closure of each ranking slab of @xmath29 is the translate of a linear subspace of @xmath7 that is generated by a face of @xmath0 . corollary [ cor - strat vs equiv ] states that @xmath86 is constant on each ranking slab , so each @xmath87 with @xmath186 is a union of translates of linear subspaces of @xmath7 that are spanned by faces of @xmath0 . it then follows that the stratification of @xmath48 by ranking slabs refines its rank stratification . while this is generally a strict refinement , examples [ example - hidden3 ] and [ example-4diml hidden ] show that the two stratifications may coincide for parameters close enough to the cone @xmath15 . we wish to emphasize that the rank jump @xmath62 is not simply determined by holes within the semigroup @xmath54 , as can be seen in example [ ex - first example ] . [ def : slab ] a _ slab _ is a set of parameters in @xmath7 that lie on a unique irreducible component of the exceptional arrangement @xmath48 @xcite . we will show by example that rank need not be constant on a slab . in example [ example-4diml hidden ] , this failure results from embedded " components of @xmath48 , while in example [ example - nonconstant slab ] , it is due to the hyperplanes of @xmath59 that strictly refine the arrangement stratification of @xmath48 . together with example [ plane - line-4diml ] , these examples show that each of the arrangements listed in definition [ def - ranking slab ] is necessary to determine such a geometric refinement of the rank stratification of @xmath48 . as in section [ sec : exceptional arrangement ] , let @xmath299 be a nonempty @xmath54monoid ( see definition [ def : monoid ] ) , so that @xmath300 $ ] is a nontrivial weakly toric module ( see example [ ex - m wt ] ) . for a fixed @xmath61 , we know from that @xmath60 can be used to compute the rank jump @xmath62 . however , this module contains a large amount of excess information that does not play a role in @xmath377 . to isolate the graded pieces of @xmath60 that impact @xmath62 , we will define weakly toric modules @xmath378 so that @xmath379{12 cm } \begin{enumerate } \item \label{prop1 } $ m \subseteq s^{\beta } \subseteq t^{\beta } \subseteq s_a[\del_a^{-1}],$ \item \label{prop2 } $ \cr_a(m,\beta ) = \qdeg\left(\dfrac{t^{\beta}}{s^{\beta}}\right),$ \item \label{prop3 } $ \beta \notin \qdeg\left(\dfrac{s_a[\del_a^{-1}]}{t^{\beta}}\right)$ , \item \label{prop4 } $ \beta \notin \qdeg\left(\dfrac{s^{\beta}}{m}\right)$ , and \item \label{prop5 } $ \pp^{\beta } = \deg\left(\dfrac{t^{\beta}}{s^{\beta}}\right)$ is a union of translates of $ \wt{{{\ensuremath{\mathbb{n}}}}f}$ for various $ f \preceq a$. \end{enumerate } \end{minipage}\end{aligned}\ ] ] in proposition [ prop - qis ] , we show that properties - of allow us to replace @xmath60 with @xmath380 when calculating @xmath62 . to use this module to actually compute @xmath62 , we will encounter other toric modules with structure similar to @xmath63 , which are called _ ranking toric modules_. property allows @xmath63 ( and similarly , any ranking toric module ) to be decomposed into _ simple ranking toric modules_. these modules are constructed so that their euler koszul homology modules have easily computable ranks . at the end of section [ subsec : simple rtm ] , we outline more specifically how simple ranking toric modules will play a role in our computation of @xmath62 . we now construct the class of _ ranking toric modules _ , which includes the module @xmath63 coming from . these modules will be constructed via their degree sets , which are unions of @xmath381modules for various @xmath13 . we begin by isolating the translated lattices contained in @xmath382 that lie in the @xmath6-components @xmath383 of the ranking arrangement of @xmath29 ( see notation [ not - beta cmpt ] ) . the union of these translated lattices will be denoted by @xmath67 . [ def - ep ] 1 . let @xmath65 be the set of faces of @xmath0 corresponding to the @xmath6-components @xmath364 of the ranking arrangement of @xmath29 . this set @xmath384 is a polyhedral cell complex , and @xmath364 is the union @xmath385 2 . for each @xmath386 , let @xmath387 lemma [ lemma - mm05 ] and theorem [ thm - str comparison ] together imply that @xmath388 is nonempty exactly when there is a containment @xmath389 . 3 . since @xmath302 is an @xmath54monoid , @xmath390 is closed under addition , so @xmath388 is @xmath68-stable . thus there is a finite set of @xmath68-orbit representatives @xmath391 such that @xmath392 is partitioned into @xmath68-orbits as a disjoint union over @xmath391 . notice that @xmath393 $ ] . the @xmath68-orbits @xmath394 in are the translated lattices that we will use to construct ranking toric modules . each is determined by the pair @xmath395 . we denote the collection of such pairs by @xmath396 5 . for a subset @xmath397 , let @xmath398 [ not - omitj ] many of the objects we define in this section are dependent upon a subset @xmath399 , and this dependence is indicated by the subscript @xmath400 . whenever we omit this subscript , it is understood that @xmath401 . by the upcoming proposition [ prop - zc e ] , two parameters @xmath84 belong to the same ranking slab exactly when @xmath85 . this is what will be used to show that the rank jump @xmath62 ) is constant on ranking slabs . [ lemma - zc e addition ] the zariski closure of the ranking lattices @xmath67 of @xmath29 at @xmath6 coincides with the @xmath6-components @xmath364 of the ranking arrangement . it is clear from the definitions that @xmath402 . for the reverse containment , if @xmath403 , then there exists a vector @xmath404 such that @xmath405 . this implies that @xmath406 is empty , so the claim now follows from the definition of quasidegree sets in definitions [ def - qdeg ] and [ def - wtoric ] . [ prop - zc e ] the parameters @xmath84 belong to the same ranking slab if and only if the ranking lattices of @xmath29 at @xmath6 and @xmath407 coincide , that is , if @xmath408 this is a consequence of lemma [ lemma - zc e addition ] , lemma [ lemma - mm05 ] , and theorem [ thm - str comparison ] . in light of proposition [ prop - zc e ] , use equality of ranking lattices to extend the ranking slab stratification of @xmath48 to the parameter space @xmath7 . one might try making the sets @xmath409 in definition [ def - ep ] the degree sets of ranking toric modules . however , while the natural map @xmath410 given by faces @xmath262 induces a vector space map @xmath411 , this induced map is not a morphism of @xmath121-modules because it sends units to zero . to overcome this , we introduce the lattice points in a certain polyhedron , denoted by @xmath412 , and intersect it with @xmath413 to produce the degree set of a ranking toric module . [ def - cp ] 1 . recall the primitive integral support functions @xmath414 from the beginning of section [ sec : defns ] . in order to construct a ranking toric module from @xmath409 , ( and achieve the various quasidegree sets proposed in ) , set @xmath415 for @xmath71 , notice that @xmath416 is simply the integral points in the cone @xmath417 after translation by @xmath6 . 2 . for a pair @xmath418 , let @xmath419 . \nonumber \intertext{the degree sets of ranking toric modules are of the form } \label{17 } \pp_j^{\beta } & \phantom{:}= \bigcup_{(f , b)\in j } \pp_{f , b}^{\beta } \ = \ \cc_a(\beta ) \cap \bbe_j^{\beta } \intertext{for $ j\subseteq\cj(\beta)$. the largest of these is } \label{eqn - p } \pp^{\beta } \ & : = \ \pp_{\cj(\beta)}^{\beta } \ = \ \cc_a(\beta ) \cap \bbe^{\beta } , \end{aligned}\ ] ] the degree set appearing in . ( continuation of example [ example - hidden ] ) [ example - hidden2 ] with @xmath420 \in\ee_a(s_a)$ ] and @xmath421 $ ] , the sets of definitions [ def - ep ] and [ def - ts and r - t ] are @xmath422 \sqcup [ b + { { \ensuremath{\mathbb{z}}}}g ] , \\ \text{and } \phantom{xx } \pp^{\beta } & = & [ \beta + { { \ensuremath{\mathbb{n}}}}f ] \sqcup [ b + { { \ensuremath{\mathbb{n}}}}g ] . \end{aligned}\ ] ] having defined the degree sets of ranking toric modules in , we now construct the modules themselves . along the way , we meet the modules that satisfy the requirements of . [ def - ts and r - t ] 1 . each ranking toric module will be a quotient of the module @xmath423 ( for some @xmath26 ) , where @xmath424 \phantom{xx } \text { and } \phantom{xx } t^{\beta}\ = \ { { \ensuremath{\mathbb{c}}}}\{\tt^{\beta}\}.\ ] ] notice that if @xmath425 , then @xmath426 . the simplest case occurs when @xmath55 and @xmath427 , so that @xmath428 . 2 . for @xmath429 , let @xmath430 we show in proposition [ prop - basics r - toric ] that @xmath423 and @xmath431 are indeed weakly toric modules ( see definition [ def - wtoric ] ) . when @xmath401 , these modules satisfy the properties . by notation [ not - omitj ] , @xmath432 . 3 . for a subset @xmath429 , the quotient @xmath433 has degree set @xmath434 , as recorded in proposition [ prop - basics r - toric ] . in proposition [ prop - qis ] , we show that @xmath60 in can be replaced by @xmath435 when computing @xmath62 . if a toric module @xmath43 is isomorphic to @xmath436 for a pair @xmath437 and a subset @xmath429 , we say that @xmath43 is a _ ranking toric module _ determined by @xmath400 . [ prop - basics r - toric ] let @xmath397 . there are containments of weakly toric modules : @xmath438 . \end{aligned}\ ] ] in particular , @xmath439 is a ranking toric module with degree set @xmath440 . by construction , we have the containment @xmath441 . since the intersection of @xmath67 and @xmath302 is empty , @xmath442 is empty as well . hence @xmath443 . the other containments in are obvious . it is clear from the definitions that the degree sets of all modules in question are closed under addition with elements of @xmath54 , so they are all weakly toric modules . for the second statement , since @xmath444 , @xmath436 is a finitely generated @xmath21-module and therefore @xmath436 is toric . by lemma [ lemma - zc e addition ] and the definition of @xmath445 , the arrangement @xmath446 coincides with the @xmath6-components @xmath364 of @xmath29 at @xmath6 . further , the construction of @xmath63 in definition [ def - ts and r - t ] is such that @xmath63 can replace @xmath60 in when calculating @xmath62 . [ prop - qis ] the euler koszul complexes @xmath447 and @xmath448 are quasi - isomorphic . in particular , @xmath449 consider the short exact sequences @xmath450}{m } \rightarrow \dfrac{s_a[\del_a^{-1}]}{t^{\beta } } \rightarrow 0 \quad \text{and } \quad 0 \rightarrow \dfrac{s^{\beta}}{m } \rightarrow \dfrac{t^{\beta}}{m } \rightarrow \dfrac{t^{\beta}}{s^{\beta } } \rightarrow 0.\end{aligned}\ ] ] the definition of @xmath412 ensures that @xmath6 is not a quasidegree of either @xmath451 or @xmath308/t^{\beta}$ ] . thus we obtain the result from long exact sequences in euler koszul homology and proposition [ prop - hh qis 0 ] . [ def - e k char ] the _ @xmath452 partial euler koszul characteristic _ of a weakly toric module @xmath43 is the ( nonnegative ) integer @xmath453 the main result of this article , theorem [ thm - main ] , states that the partial euler koszul characteristics of ranking toric modules are determined by the combinatorics of the ranking lattices @xmath67 of @xmath29 at @xmath6 . lemma [ lemma - chi p ] describes @xmath62 as the @xmath454 partial euler koszul characteristic of @xmath63 , so our results regarding the combinatorics of rank jumps are a consequence of theorem [ thm - main ] . [ lemma - chi p ] the @xmath455 partial euler koszul characteristic of every nontrivial ranking toric module for a pair @xmath437 is 0 . in particular , @xmath456 . this follows from ( * ? ? ? * theorem 4.2 ) and . the polyhedral structure of the degree sets of ranking toric modules plays an important role in our computation of their partial euler koszul characteristics . we will use that each face @xmath457 determines an @xmath121-module that is the quotient of a ranking toric module . modules of this type are called _ simple ranking toric modules_. [ def - rtoric simple ] 1 . for a subset @xmath429 , the ranking toric module @xmath436 is _ simple _ if there is a unique @xmath458 such that all pairs in @xmath400 are of the form @xmath395 . 2 . for each @xmath459 and @xmath429 , denote by @xmath460 the simple ranking toric module determined by the set @xmath461 . the degree set of this module is denoted by @xmath462 . 3 . call the parameter @xmath6 _ simple _ for @xmath29 if @xmath63 is a simple ranking toric module , or equivalently , if there is an @xmath459 such that @xmath463 ( see notation [ not - omitj ] ) . we show in theorem [ thm - rank simple r - toric ] that for @xmath13 , each simple ranking toric module @xmath460 is a maximal cohen macaulay toric @xmath121-module . thus the results of section [ sec : ekstr ] can be applied to compute the rank of their euler koszul homology modules . notice that by setting @xmath464 it follows from definition [ def - rtoric simple ] that @xmath465 . in particular , when @xmath401 , @xmath466 . [ prop - simple r - t ] for @xmath459 and @xmath429 , the simple ranking toric module @xmath460 of definition [ def - rtoric simple ] admits an @xmath121-module structure that is compatible with its @xmath21-module structure . by construction , @xmath462 is closed under addition with elements of @xmath116 . [ def - triangle ] we define a partial order @xmath467 on @xmath397 by @xmath468 if and only if @xmath469 for pairs @xmath470 . we let @xmath471 denote the subset of @xmath400 consisting of maximal elements with respect to @xmath467 . for @xmath429 , @xmath471 is the smallest subset of @xmath400 that determines a direct sum of simple ranking toric modules into which @xmath436 embeds . the calculation of the partial euler koszul characteristics of a ranking toric module @xmath436 will be achieved by homologically replacing it by an acyclic complex @xmath472 composed of simple ranking toric modules . we then examine the spectral sequences determined by the double complex @xmath473 to obtain a formula for the partial euler koszul characteristics of @xmath436 . we now define an equivalence relation on the union of the various @xmath68-orbit representatives of . we show in proposition [ prop - p decomp ] that for @xmath429 , the ranking toric module @xmath436 splits as direct sum over the equivalence classes of this relation . thus by additivity of rank , can be expressed as a sum involving simpler ranking toric modules . [ def - tilde ] 1 . let @xmath474 be the collection of all @xmath68-orbit representatives from . let @xmath475 be the equivalence relation on the elements of @xmath476 that is generated by the relations @xmath477 if there exist @xmath478 such that @xmath479 3 . let @xmath480 denote the set of equivalence classes of @xmath475 . 4 . for @xmath481 and @xmath429 , let @xmath482 hence for @xmath429 , there is a partition of @xmath440 over @xmath480 : @xmath483 . [ prop - p decomp ] for @xmath429 , there is a decomposition @xmath484 . for distinct @xmath485 , the sets @xmath486 and @xmath487 are disjoint by definition of @xmath475 . thus there is a decomposition @xmath488 . as a special case of proposition [ prop - p decomp ] , the simple ranking toric module @xmath489 can be expressed as the direct sum @xmath490 ( see ) . for @xmath401 , let the _ rank jump from @xmath491 of @xmath29 at @xmath6 _ be @xmath492 [ cor - j decomp ] the rank jump @xmath62 can be expressed as the sum @xmath493 . this follows from , proposition [ prop - p decomp ] , and the additivity of @xmath62 . as stated in corollary [ cor - j decomp ] , computing @xmath62 is reduced to finding @xmath494 for each @xmath481 . when working with examples , it is typically useful to consider each @xmath494 individually . in contrast , as we continue with the theory , it is more efficient for our notation to study @xmath62 directly . in sections [ subsec : simple case ] and [ subsec : non - simple case ] , replacing @xmath384 , @xmath63 , and @xmath62 by their corresponding @xmath491 counterparts calculates @xmath494 . we retain the notation of section [ sec : r - t mods ] . this section contains our main result , theorem [ thm - main ] , which states that for any subset @xmath429 , the partial euler koszul characteristics of the ranking toric module @xmath436 are determined by the combinatorics of @xmath409 ( refer to definitions [ def - e k char ] , [ def - ts and r - t ] , and [ def - ep ] ) . as a special case of this result , we will have computed @xmath456 in terms of the ranking lattices @xmath67 , resulting in a proof of theorem [ thm - compute jump ] . we begin by examining the partial euler koszul characteristics of simple ranking toric modules @xmath495 from definition [ def - rtoric simple ] . we will compute the partial euler koszul characteristics of a ranking toric module @xmath436 by homologically approximating it by a cellular resolution ( see definition [ def - cellular definition ] ) built from simple ranking toric modules . the next theorem shows that simple ranking toric modules are maximal cohen macaulay toric face modules , which will be useful in the general case . this allows us to compute the rank jump @xmath62 of @xmath29 at @xmath6 when @xmath6 is simple for @xmath29 , as in ( * ? ? ? * theorem 2.5 ) . [ thm - rank simple r - toric ] fix @xmath26 , @xmath459 , and @xmath429 . then the simple ranking toric module @xmath460 is a maximal cohen macaulay toric @xmath121-module . further , there is a decomposition @xmath496 \otimes_{{\ensuremath{\mathbb{c}}}}\hh^f_0(p_{f , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left(\textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}f^\bot\right ) , \end{aligned}\ ] ] and for all @xmath497 , @xmath498 fix a @xmath68-orbit representative @xmath499 , chosen so that @xmath500 . this implies that @xmath501 is a short exact sequence of toric modules . since @xmath502 the definition of @xmath412 ensures that @xmath503 . proposition [ prop - hh qis 0 ] and imply that induces the isomorphism @xmath504 . as @xmath505 , proposition [ prop - str cm ek complex ] gives the decomposition , in light of proposition [ prop - p decomp ] . by ( * ; * ? ? ? * lemma 3.3 ) , @xmath506 . now the additivity of rank and combine to complete the claim . [ cor - simple ] if @xmath61 is simple for @xmath29 , then the rank jump of @xmath29 at @xmath6 is @xmath507 \cdot \vol(f).\ ] ] since @xmath6 is simple for @xmath29 , @xmath463 for some @xmath13 . hence apply theorem [ thm - rank simple r - toric ] to corollary [ cor - j decomp ] , noting that @xmath508 . ( continuation of examples [ example - hidden ] and [ example - hidden2 ] ) [ example - hidden3 ] with @xmath509 $ ] , we have the set @xmath510 . both @xmath511 and @xmath512 are simple ranking toric modules . by corollary [ cor - simple ] , @xmath513\cdot 1 \ = \ 1 \quad \text{and } \quad j_{\widehat{b}}(\beta ) = 1\cdot [ 1 - 1]\cdot 1 \ = \ 0.\end{aligned}\ ] ] it now follows from proposition [ prop - p decomp ] that @xmath514 . a similar calculation shows that @xmath515 for any @xmath516 . [ example-4diml hidden ] let @xmath517\ ] ] with @xmath518 , and consider the faces @xmath370 $ ] and @xmath371 $ ] . note that the semigroup @xmath54 of example [ example - hidden ] embeds into the @xmath54 here . with @xmath519 $ ] , the exceptional arrangement of @xmath21 is @xmath520 , where @xmath521 thus the ranking slab stratification of @xmath365 is strictly finer than its arrangement stratification . further , this finer stratification coincides with the rank stratification of @xmath365 inside the cone @xmath15 . for @xmath522 , @xmath523 , while @xmath524 for @xmath525 $ ] . calculations similar to those of example [ example - hidden3 ] show that @xmath526$. } \end{cases}\ ] ] in particular , the rank of the @xmath0-hypergeometric system @xmath366 is not constant on the slab @xmath527\subseteq \ee_a(s_a)$ ] ( see definition [ def : slab ] ) . [ example-2 lines c4 ] let @xmath528 for @xmath529\ ] ] and consider the saturated faces @xmath530 $ ] and @xmath531 $ ] . here @xmath532 , @xmath533 , and @xmath534 . computations in macaulay 2 @xcite with reveal that @xmath535\cup[\beta'+{{\ensuremath{\mathbb{c}}}}f_2],\ ] ] where @xmath536 $ ] . with @xmath537 $ ] and @xmath538 , @xmath539\setminus\beta'$ , } \\ \beta+{{\ensuremath{\mathbb{c}}}}f_2 & \text{if $ \beta\in[\beta'+{{\ensuremath{\mathbb{c}}}}f_2]\setminus\beta'$ , } \end{cases}\ ] ] @xmath540\cup [ \beta+{{\ensuremath{\mathbb{n}}}}f_1]\cup [ \beta+{{\ensuremath{\mathbb{n}}}}f_2 ] & \text{if $ \beta=\beta'$}\\ [ \beta+b+{{\ensuremath{\mathbb{n}}}}f_1]\cup \beta+{{\ensuremath{\mathbb{n}}}}f_1 & \text{if $ \beta\in\wt{{{\ensuremath{\mathbb{n}}}}a}\cap[\beta'+{{\ensuremath{\mathbb{c}}}}f_1]\setminus\beta'$ , } \\ \beta+{{\ensuremath{\mathbb{n}}}}f_2 & \text{if $ \beta\in\wt{{{\ensuremath{\mathbb{n}}}}a}\cap[\beta'+{{\ensuremath{\mathbb{c}}}}f_2]\setminus\beta'$ , } \end{cases}\ ] ] and @xmath541\setminus\beta'$ , } \\ \ { \widehat{\beta } \ } & \text{if $ \beta\in\wt{{{\ensuremath{\mathbb{n}}}}a}\cap[\beta'+{{\ensuremath{\mathbb{c}}}}f_2]\setminus\beta'$. } \end{cases}\ ] ] by corollary [ cor - simple ] , for @xmath542\setminus\beta'$ ] , @xmath543\cdot\vol(f_2)\ = \ [ 3 - 1]\cdot 1\ = \ 2,\ ] ] while for @xmath544\setminus\beta'$ ] , @xmath545 , and @xmath546\cdot\vol(f_1)\ = \ 2 \cdot [ 3 - 1]\cdot 1\ = \ 4.\ ] ] to compute the rank jump of @xmath21 at @xmath407 , we must move to the general case . we will see in example [ example-2 lines c4 again ] that @xmath547 , which arises as the sum of the generic rank jumps along irreducible components of @xmath548 that is then corrected by error terms that arise from a spectral sequence calculation . we are now prepared to compute the partial euler koszul characteristics of ranking toric modules . the proof of our main theorem , theorem [ thm - main ] , will be given at the end of this section , after a sequence of lemmas . the definitions of @xmath549 , @xmath409 , and @xmath436 can be found in definitions [ def - ep ] and [ def - ts and r - t ] , respectively . [ thm - main ] for @xmath429 , the partial euler koszul characteristics of the ranking toric module @xmath436 are determined by the combinatorics of @xmath409 . we compute the partial euler koszul characteristics of the ranking toric module @xmath436 will be achieved by homologically approximating @xmath436 by simple ranking toric modules ; note that the ranks of the euler koszul homology modules of simple ranking toric modules have been computed in theorem [ thm - rank simple r - toric ] . [ def - cellular definition ] let @xmath550 be an oriented cell complex ( e.g. cw , simplicial , polyhedral ) . then @xmath550 has the cochain complex @xmath551 where @xmath552 is multiplication by some integer coeff@xmath553 . let @xmath554 be the category with the nonempty faces of @xmath550 as objects and morphisms @xmath555 fix an abelian category @xmath556 and suppose there is a covariant functor @xmath557 . let @xmath558 for each @xmath559 . a sequence of morphisms in @xmath556 @xmath560 is _ cellular _ and _ supported on _ @xmath550 if the @xmath561 component of @xmath562 is coeff@xmath553 @xmath563 . since @xmath556 is abelian , a cellular sequence is necessarily a complex . in a manner analogous to definition [ def - cellular definition ] , a cellular complex supported on @xmath550 can also be obtained from the chain complex @xmath564 of @xmath550 and a contravariant functor @xmath565 . further , we could replace @xmath564 in this construction with the _ reduced _ chain or cochain complexes of @xmath550 . we say that a complex in @xmath556 is _ cellular _ if it can be constructed from the underlying topological data of a cell complex and a functor @xmath557 . when @xmath550 is a simplicial or polyhedral cell complex , coeff@xmath553 of definition [ def - cellular definition ] is simply @xmath566 if the orientation of @xmath567 induces the orientation of @xmath568 and @xmath569 if it does not . the generality in which we define cellular complexes is alluded to in the introduction of @xcite and appears as ( * * definition 3.2 ) . an introduction to these complexes , in the polyhedral case , can be found in ( * ? ? ? * chapter 4 ) . recall from definition [ def - triangle ] that @xmath471 was defined so that it yields the smallest set of faces of @xmath0 that determines a direct sum of simple ranking toric modules into which @xmath436 embeds . [ not - cpx i ] we wish to take intersections of faces in the set @xmath471 . in order to keep track of which faces were involved in each intersection , set @xmath570 with @xmath571 , let @xmath572 be the standard @xmath573-simplex with vertices corresponding to the elements of @xmath574 . to the @xmath575-face of @xmath550 spanned by the vertices corresponding to the elements in @xmath576 , assign the ranking toric module @xmath577 . choosing the natural maps @xmath578 for @xmath579 induces a cellular complex supported on @xmath550 , @xmath580 with @xmath581 [ lemma - i acyclic ] the cohomology of the cellular complex @xmath472 of is concentrated in cohomological degree 0 and is isomorphic to @xmath436 . given @xmath582 , let @xmath583 be the faces @xmath584 such that @xmath585 . the degree @xmath44 part of @xmath472 computes the cohomology of the @xmath586-subsimplex of @xmath550 given by the vertices with labels corresponding to @xmath583 ; in particular , it is acyclic with @xmath587-cohomology @xmath588 . by construction , @xmath436 is a @xmath41-graded monomial module over the saturated semigroup ring @xmath589 , and it can be translated by some @xmath361 so that @xmath590 . after translation by @xmath44 , is similar to an _ irreducible resolution _ , as defined in ( * ? ? ? * ; * ? ? ? * definition 2.1 ) . we continue to view @xmath436 as an @xmath21-module , so we use maximal cohen macaulay toric face modules instead of irreducible quotients of @xmath589 . consider the @xmath41-graded double complex @xmath591 with @xmath592 let @xmath593 and @xmath594 denote the horizontal and vertical differentials of @xmath591 , respectively . by the exactness of , taking homology of @xmath591 with respect to @xmath593 yields @xmath595 we now apply the decomposition of these homologies given in theorem [ thm - rank simple r - toric ] to obtain a new description of the euler koszul homology of the ranking toric module @xmath436 . [ lemma - right ss ] the vertical spectral sequence obtained from the double complex @xmath596 \otimes_{{\ensuremath{\mathbb{c}}}}\kk^{f_s}_i(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^j { { \ensuremath{\mathbb{z}}}}f_s^\bot \right)\end{aligned}\ ] ] ( with differentials as in lemma [ lemma - ek morph ] ) has abutment @xmath597 by theorem [ thm - rank simple r - toric ] and lemma [ lemma - ek morph ] , the vertical differentials @xmath594 of @xmath598 are compatible with the quasi - isomorphism @xmath599 \otimes_{{\ensuremath{\mathbb{c}}}}\kk^{f_s}_\bullet(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^\bullet { { \ensuremath{\mathbb{z}}}}f_s^\bot \right).\end{aligned}\ ] ] since @xmath600 and @xmath601 converge to the same abutment , the result follows from . note that the first page of the spectral sequence in lemma [ lemma - right ss ] is @xmath602 \otimes_{{\ensuremath{\mathbb{c}}}}\hh^{f_s}_0(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right).\end{aligned}\ ] ] for @xmath603 , let @xmath604 denote the differential of @xmath605 , and let @xmath606 respectively denote the vertical and horizontal differentials of @xmath607 . if @xmath245 with @xmath608 , then by remark [ remark - diffl in cm ek complex ] , the element @xmath609 \otimes_{{\ensuremath{\mathbb{c}}}}\kk_i^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) \subseteq \ { ' e}_0^{p ,- q}\ ] ] has vertical differential @xmath610 we will use that is an element of @xmath611\otimes \kk_{i-1}^{f_s } ( p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{c}}}}\left ( \textstyle\bigwedge^j { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) \subseteq { ' e}_0^{p ,- q+1}\ ] ] to show that @xmath612 degenerates quickly . this is the main technical result of this article . [ lemma - ss terminates ] the spectral sequence @xmath613 of lemma [ lemma - right ss ] degenerates at the second page . for @xmath614 , let @xmath615 denote the image of @xmath616 in @xmath617 , if it exists . let @xmath618 denote the differential of @xmath619 , so @xmath620 . to see that @xmath621 , consider an element @xmath622 with @xmath623 . then there is an element @xmath624 such that @xmath625 , which is used to define @xmath626 ( recall that is independent of the choice of @xmath627 . ) we write @xmath628 as an element of . note that @xmath629 \otimes_{{\ensuremath{\mathbb{c}}}}\kk_{i-1}^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^j { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) , \nonumber \intertext{so $ \alpha^s_{ij}$ is in the kernel of $ { } _ v\delta$ for all $ s , i , j$. by \eqref{eq : nice 1st page } , $ \alpha^{s}_{ij}$ is in the image of $ { } _ v\delta$ whenever $ i>0$. hence without changing $ \oalpha$ , we may assume that for all $ s\in\cg^p_j$ , $ \alpha^{s}_{ij}=0 $ when $ i>0 $ , so that } \alpha & \in \bigoplus_{s\in\cg^p_j } { { \ensuremath{\mathbb{c}}}}[x_{f_s^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk_0^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) . \nonumber \intertext{as the differential $ { } _ h\delta$ is induced by \eqref{9 } , } { } _ h\delta(\alpha ) & \in \bigoplus_{s\in\cg^{p+1}_j } { { \ensuremath{\mathbb{c}}}}[x_{f_s^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk_0^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) . \nonumber \intertext{by hypothesis on $ \alpha$ and \eqref{31 } , there is an element } \label{34 } \eta & \in \bigoplus_{s\in\cg^{p+1}_j } { { \ensuremath{\mathbb{c}}}}[x_{f_s^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk_1^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) \intertext{such that $ { } _ v\delta(\eta)={}_h\delta(\alpha)$. set $ \zeta={}_h\delta(\eta)$ and note that $ \delta_2(\oalpha)=\ol{\zeta}$. using again that the differential $ { } _ h\delta$ is induced by \eqref{9 } , applied now to \eqref{34 } , we see that } \zeta & \in \bigoplus_{s\in\cg^{p+2}_j } { { \ensuremath{\mathbb{c}}}}[x_{f_s^c } ] \otimes_{{\ensuremath{\mathbb{c}}}}\kk_1^{f_s}(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right ) . \nonumber\end{aligned}\ ] ] since @xmath630 , @xmath631 is in the kernel of @xmath632 . hence implies that @xmath633 vanishes . [ lemma - formula ] for @xmath429 , the @xmath452 partial euler koszul characteristic of the ranking toric module @xmath436 is given by @xmath634 for @xmath635 , let @xmath636 by lemma [ lemma - ss terminates ] , @xmath637 . from the abutment , we see that @xmath638 for @xmath639 . since @xmath640 by lemma [ lemma - chi p ] , also @xmath641 . thus the @xmath452 partial euler koszul characteristic of @xmath436 can be expressed as @xmath642 now follows from the definition of @xmath643 and the quasi - isomorphism , as the isomorphic first pages of the spectral sequences there are and . we will compute the ranks of @xmath644 and the image of @xmath645 from in subsequent lemmas . the first is an immediate consequence of theorem [ thm - rank simple r - toric ] . [ lemma - rank - hqip ] if @xmath646 , then @xmath647 by definition of @xmath648 and additivity of rank , @xmath649 . now apply theorem [ thm - rank simple r - toric ] . the rank of the image of @xmath645 is determined combinatorially because the spectral sequence rows @xmath650 are cellular complexes . [ lemma-e cell rows ] the complexes @xmath650 are cellular with support @xmath651 of notation [ not - cpx i ] . in notation [ not - cpx i ] , we constructed the cellular complex @xmath472 from a labeling of the simplex @xmath572 . if we assign in this construction @xmath652 \otimes_{{\ensuremath{\mathbb{c}}}}\hh_q^f(p_f^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f^\bot \right)\ ] ] in place of @xmath460 and use the induced maps , we obtain the cellular complex @xmath650 with differential @xmath653 , see . the existence and compatibility of the differentials follows from lemma [ lemma - ek morph ] . [ lemma - rank image ] the rank of the image of @xmath645 is determined by the combinatorics of @xmath409 . by lemma [ lemma-e cell rows ] , the image of @xmath645 is determined by the @xmath575-coboundaries of @xmath550 and the corresponding labels of @xmath550 , which come from @xmath409 . if @xmath400 is simple for @xmath29 , the result follows from theorem [ thm - rank simple r - toric ] , so suppose that @xmath436 is not simple for @xmath29 . by lemma [ lemma - i acyclic ] , @xmath436 is the @xmath587-cohomology of the acyclic cellular complex @xmath472 . thus the abutment of the spectral sequences arising from the double complex @xmath473 is @xmath654 . by lemma [ lemma - right ss ] , the vertical spectral sequence obtained from the double complex @xmath655 of has the same abutment . since this spectral sequence degenerates on the second page by lemma [ lemma - ss terminates ] , lemma [ lemma - formula ] yields the formula , and by lemmas [ lemma - rank - hqip ] and [ lemma - rank image ] , the summands of are dependent only on the combinatorics of @xmath409 . recall formula : @xmath656 lemma [ lemma - rank - hqip ] computes the first summand , but lemma [ lemma - rank image ] does not explicitly state the rank of the image of @xmath645 . a method to do this is provided by proposition [ prop : alg ] . [ def : circuit ] for @xmath657 , a subset @xmath658 corresponds to a subcomplex @xmath659 of the simplex @xmath660 , as described in notation [ not - cpx i ] . if @xmath661 and there is a minimal generator of @xmath662 of the form @xmath663 $ ] , where all coefficients @xmath664 are nonzero , then we say that @xmath665 is a _ circuit _ for @xmath666 . for @xmath657 , let @xmath667 denote the set of circuits for @xmath666 , and set @xmath668 for @xmath669 , @xmath670 , and @xmath671 , set @xmath672 [ prop : alg ] let @xmath673 be the ranking toric module in . the rank of the image of @xmath645 from is equal to @xmath674 further , can be computed by combining theorem [ thm - rank simple r - toric ] and induction on the dimension of @xmath675 . before providing the proof of proposition [ prop : alg ] , we state two lemmas . [ lemma : easier comp ] for @xmath676 , @xmath677 where @xmath673 is the ranking toric module given by . the rank of the image of @xmath678 is @xmath679 by proposition [ prop - str / rk image ] . view the image of @xmath680 as a quotient of @xmath681 . if @xmath44 is one of its nonzero multigraded components , then it also appears in the degree set of another summand of @xmath682 . the collection of such degrees is exactly @xmath683 . [ lemma : harder comp ] for @xmath657 , @xmath684.\end{aligned}\ ] ] to see this , notice first that for @xmath685 , @xmath686 is generated by the images coming from circuits for @xmath666 . by proposition [ prop - str / rk image ] , given a fixed circuit @xmath665 for @xmath666 , the rank of @xmath687 can be computed as the rank of @xmath688 times the @xmath31-rank of @xmath689 = \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f(\lambda)^\bot . \end{aligned}\ ] ] by the same reasoning used to obtain , the rank of @xmath690 equals the rank of . the @xmath31-rank of is a binomial coefficient in the codimension in @xmath7 of the span of the vectors in @xmath691 , so the rank of is @xmath692 further , for a collection of circuits @xmath671 , @xmath693 gives the rank of the intersection over @xmath694 of the images of type , so the inclusion - exclusion principle yields . recall from that the domain of @xmath645 is the direct sum @xmath695 \otimes_{{\ensuremath{\mathbb{c}}}}\hh^{f_s}_0(p_{f_s , j}^{\beta},\beta ) \otimes_{{\ensuremath{\mathbb{z}}}}\left ( \textstyle\bigwedge^q { { \ensuremath{\mathbb{z}}}}f_s^\bot \right).\end{aligned}\ ] ] for @xmath696 , let @xmath697 denote the restriction of @xmath645 to the summands in @xmath698 . order the elements of @xmath699 , so that @xmath700 . \nonumber\end{aligned}\ ] ] lemmas [ lemma : easier comp ] and [ lemma : harder comp ] respectively computed the summands of , resulting in . thus it remains to show the second statement . if a ranking toric module has dimension 0 , then it is necessarily a simple ranking toric module , so theorem [ thm - rank simple r - toric ] computes the rank of its euler koszul homology modules . thus by induction on dimension , we can compute the summand in corresponding to @xmath669 if the dimension of @xmath701 is strictly less than the dimension of @xmath681 . if it is the case that the dimension of @xmath701 equals the dimension of @xmath681 , notice first that each pair @xmath702 has @xmath703 . this implies that @xmath701 is a direct sum ( as in proposition [ prop - p decomp ] ) of the simple ranking toric module @xmath704 and a lower - dimensional ranking toric module . therefore induction together with theorem [ thm - rank simple r - toric ] still completes the computation . finally , the same argument applies to computing the rank of @xmath705 for @xmath671 , since @xmath706 . by lemma [ lemma - chi p ] , our results on the partial euler koszul characteristics of ranking toric modules reveal the combinatorial nature of rank jumps of the generalized @xmath0-hypergeometric system @xmath34 . by and lemma [ lemma - chi p ] , @xmath456 , so the result is an immediate consequence of theorem [ thm - main ] and proposition [ prop : alg ] . [ example-2 cmpts ] if @xmath26 is such that @xmath707 , the proof of theorem [ thm - main ] and section [ subsec : compute ] show that the rank jump of @xmath29 at @xmath6 is @xmath708 \cdot \vol(f_i ) \right ) + |b_g^{\beta}| \cdot c^{\beta } \cdot\vol(g),\end{aligned}\ ] ] where @xmath709 and the constant @xmath710 is given by @xmath711 ( continuation of example [ example-2 lines c4 ] ) [ example-2 lines c4 again ] with @xmath712 $ ] , the set @xmath713 , and @xmath714 by proposition [ prop - p decomp ] . by , @xmath715 \cdot \vol(f_i ) \right ) \ + \ |\widehat{\beta ' } \cap b_g^{\beta}| \cdot c^{\beta } \cdot\vol(g)\\ & = & 2 + 2 + 1\cdot(-2)\cdot 1 \ = \ 2 , \end{aligned}\ ] ] and @xmath716 by corollary [ cor - simple ] . thus proposition [ prop - p decomp ] implies that the rank jump of the @xmath0-hypergeometric system @xmath717 is @xmath718 . when @xmath76 and @xmath719 for some @xmath481 , ( * ? ? * theorem 2.6 ) implies that the rank jump @xmath62 of @xmath29 at @xmath6 corresponds to the reduced homology of the lattice @xmath384 . the formula given by okuyama involves this homology and the volumes of the 1-dimensional faces of @xmath0 in @xmath384 . for higher - dimensional cases , the cellular structure of the complex @xmath720 of notation [ not - cpx i ] shows that , in general , more information than the reduced homology of @xmath384 is needed to compute @xmath62 , or even a single @xmath494 . recall from definition [ def - ranking slab ] that a ranking slab of @xmath29 is a stratum in the coarsest stratification of @xmath48 that respects a specified collection of subspace arrangements . we are now prepared to prove corollary [ cor - strat vs equiv ] , which states that the ranking slab stratification of @xmath48 refines its rank stratification . from this it follows that each @xmath374 is a union of ranking slabs , making each a union of translated linear subspaces of @xmath7 . if @xmath84 belong to the same ranking slab , then the ranking lattices @xmath721 coincide by proposition [ prop - zc e ] . by theorem [ thm - compute jump ] , the rank jumps @xmath62 and @xmath722 coincide as well . [ cor - eea union linear translates ] for all integers @xmath186 , @xmath87 is a union of translates of linear subspaces that are generated by faces of @xmath0 . this is an immediate consequence of theorem [ thm - str comparison ] and corollary [ cor - strat vs equiv ] . the following is the second example promised at the end of section [ sec : exceptional arrangement ] , showing that the rank of @xmath34 need not be constant on a slab ( see definition [ def : slab ] ) . further , this example shows that neither the arrangement stratification of @xmath48 nor its refinement given by the @xmath47 modules in determine its rank stratification . [ example - nonconstant slab ] consider the matrix @xmath723\ ] ] with @xmath724 , and label the faces @xmath725,$ ] @xmath726 $ ] , and @xmath727 $ ] . with @xmath728 $ ] and @xmath729 , \left[\begin{smallmatrix } 3 \\ 2 \\ 3 \end{smallmatrix}\right ] , \left[\begin{smallmatrix } 3 \\ 5 \\ 3 \end{smallmatrix}\right ] , \left[\begin{smallmatrix } 5 \\ 3 \\ 3 \end{smallmatrix}\right ] , \left[\begin{smallmatrix } 3 \\ 3 \\ 5 \end{smallmatrix}\right ] , \left[\begin{smallmatrix } 5 \\ 5 \\ 6 \end{smallmatrix}\right ] \right\}$ ] , the exceptional arrangement of @xmath21 is @xmath730 for @xmath731 , @xmath732\setminus \beta'$,}\\ \bigcup_{i=1}^3 [ \beta ' + \cc f_i ] & \text{if $ \beta = \beta'$,}\\ \end{cases}\ ] ] so by the proof of theorem [ thm - main ] , the rank jump of @xmath29 at @xmath61 is @xmath733\setminus\beta'$,}\\ 2 & \text{otherwise . } \end{cases}\ ] ] here the arrangement stratification of @xmath365 agrees with the one given by the @xmath47 modules that determine it , but @xmath86 is not constant on the slab @xmath734\subseteq\ee_a(s_a)$ ] . to show that all of the arrangements in the definition of ranking slabs ( definition [ def - ranking slab ] ) are necessary to obtain a refinement of the rank stratification of @xmath48 , we include the following example . here , @xmath62 changes where components of @xmath365 that correspond to different @xmath47 modules intersect . [ plane - line-4diml ] let @xmath735,\ ] ] @xmath736 $ ] , @xmath737 $ ] , and @xmath738 $ ] . here @xmath739 , and the exceptional arrangement of @xmath21 is @xmath740 \cup [ \beta ' + { { \ensuremath{\mathbb{c}}}}g],\ ] ] where @xmath741 by corollary [ cor - simple ] and example [ example-2 cmpts ] , @xmath742\setminus\beta'$ , } \\ 4 & \text{if $ \beta\in[\beta'+{{\ensuremath{\mathbb{c}}}}g]\setminus\beta'$. } \end{cases}\ ] ] we include a final example to show that @xmath62 is not determined simply by @xmath743 , the holes in the semigroup @xmath54 . [ ex - first example ] the matrix @xmath744\ ] ] has volume 16 . the exceptional arrangement of @xmath324 is the union of four lines and a point : @xmath745 + { { \ensuremath{\mathbb{c}}}}f \right ) \cup \left ( \left[\begin{smallmatrix } 1\\ 1\\ 0 \end{smallmatrix}\right ] + { { \ensuremath{\mathbb{c}}}}g \right ) \cup \left ( \left[\begin{smallmatrix } 1\\ 0\\ 0 \end{smallmatrix}\right ] + { { \ensuremath{\mathbb{c}}}}[a_5 ] \right ) \cup \left ( \left[\begin{smallmatrix } 0\\ 1\\ 0 \end{smallmatrix}\right ] + { { \ensuremath{\mathbb{c}}}}[a_6 ] \right ) \cup \left\ { \left[\begin{smallmatrix } 2\\ 2\\ 1 \end{smallmatrix}\right ] \right\},\ ] ] where @xmath746 $ ] and @xmath747 $ ] . the generic rank jumps along each component are as follows : @xmath748 + { { \ensuremath{\mathbb{c}}}}f \mapsto 3 , \quad \left[\begin{smallmatrix } 1\\ 0\\ 0 \end{smallmatrix}\right ] + { { \ensuremath{\mathbb{c}}}}[a_5 ] \mapsto 1 , \quad \left[\begin{smallmatrix } 1\\ 1\\ 0 \end{smallmatrix}\right ] + { { \ensuremath{\mathbb{c}}}}g \mapsto 3 , \quad \left[\begin{smallmatrix } 0\\ 1\\ 0 \end{smallmatrix}\right ] + { { \ensuremath{\mathbb{c}}}}[a_6 ] \mapsto 1 , \quad \text{and } \left[\begin{smallmatrix } 2\\ 2\\ 1 \end{smallmatrix}\right ] \mapsto 2 . \end{aligned}\ ] ] these generic rank jumps are achieved everywhere except at the points @xmath749 , \left[\begin{smallmatrix } 1\\ 0\\ 0 \end{smallmatrix}\right ] , \left[\begin{smallmatrix } 0\\ 1\\ 0 \end{smallmatrix}\right ] , \text { and } \left[\begin{smallmatrix } \phantom{-}0\\ \phantom{-}0\\ -1 \end{smallmatrix}\right].\ ] ] the point that may be unexpected in this collection is @xmath750 $ ] . both of the components of @xmath365 that contain @xmath90 have generic rank jumps of 1 ; however , @xmath751 . this is because the ranking arrangement of @xmath21 has three components that contain @xmath90 : @xmath752 \right ) \cup \left ( b + { { \ensuremath{\mathbb{c}}}}[a_6 ] \right ) \cup \left ( b + { { \ensuremath{\mathbb{c}}}}[a_1\ a_2\ a_3\ a_4 ] \right ) . \end{aligned}\ ] ] if the plane @xmath753)$ ] were not in @xmath754 , then the rank jump of @xmath21 at @xmath90 would only be 1 by . thus the hyperplane in , although unrelated to the holes in @xmath54 , accounts for the higher value of @xmath755 . the rank jumps at the other parameters are @xmath756 \right ) = j\left ( s_a , \left[\begin{smallmatrix } 0\\ 1\\ 0 \end{smallmatrix}\right ] \right ) = 3 \quad \text { and } \quad j\left ( s_a , \left[\begin{smallmatrix } 1\\ 1\\ 0 \end{smallmatrix}\right ] \right ) = 5.\ ] ] the algebraic upper semi - continuity of the rank of @xmath34 implies that most of the codimension 1 components of the ranking arrangement @xmath59 do not increase the rank of @xmath34 . it would interesting to know if the set of such hyperplanes can be identified . when @xmath324 , the results of section [ sec : comb of rank ] apply to the @xmath0-hypergeometric system @xmath366 . for a face @xmath567 of @xmath0 , @xmath757 is a finite set . it is shown in @xcite and @xcite that @xmath1 and @xmath758 are isomorphic as @xmath4-modules precisely when @xmath759 for all faces @xmath567 of @xmath0 . we will now use euler koszul homology to give a simple proof of one direction of this equivalence ; first we exhibit a complementary relationship between @xmath89 and @xmath760 ( see ) . [ obs - etau ] it is shown in theorem [ thm - compute jump ] that as @xmath567 runs through the faces of @xmath0 , the sets @xmath761 determine the rank jump of @xmath1 at @xmath6 . notice that @xmath762 is the complement of @xmath89 in the group @xmath763 . if @xmath770 is not an isomorphism , then @xmath771 by lemma [ lemma - nisom ] . by the definition of quasidegrees , there exist vectors @xmath772 and @xmath773 for some face @xmath567 such that @xmath774 , @xmath775 , and @xmath776 for all @xmath140 . hence @xmath777 , so @xmath778 . further , the condition @xmath779 for all @xmath140 implies that @xmath780 . for the if " direction , suppose that @xmath783 for all faces @xmath567 of @xmath0 . as stated in ( * ; * ? ? ? * proposition 2.2 ) , @xmath784 implies by definition that @xmath785 , so @xmath764 for some @xmath786 . there are unique vectors @xmath787 with disjoint support such that @xmath788 . in light of lemma [ lemma - netau ] , we may assume that both @xmath789 and @xmath790 are nonzero . we claim that at least one of @xmath791 or @xmath792 defines an isomorphism from @xmath1 to @xmath758 . if @xmath793 or @xmath794 defines an isomorphism , then the only - if " direction and lemma [ lemma - netau ] imply that @xmath791 or @xmath792 , respectively , give the desired isomorphism . we are left to consider the case when @xmath795 does not define an isomorphism from either domain . by lemma [ lemma - nisom ] , this is equivalent to @xmath796 from the right side of , we see that the nonempty face @xmath797 is such that @xmath798 , so @xmath799 . however , the shift @xmath800 in the left side of implies that @xmath801 . thus @xmath802 , which is a contradiction . it is not yet understood how the holomorphic solution space of @xmath1 varies as a function of @xmath6 ; different functions of @xmath6 suggest alternative behaviors . walther showed in @xcite that the reducibility of the monodromy of @xmath1 varies with @xmath6 in a lattice - like fashion . when the convex hull of @xmath0 and the origin is a simplex , saito used the sets @xmath89 to construct a basis of holomorphic solutions of @xmath1 with a common domain of convergence @xcite . thus theorem [ thm - compute jump ] and the complementary relationship in observation [ obs - etau ] between the @xmath89 and the ranking lattices of @xmath21 at @xmath6 suggest that the ranking slabs give the coarsest stratification over which there could be a constructible sheaf of solutions for the hypergeometric system . victor v. batyrev and duco van straten , _ generalized hypergeometric functions and rational curves on calabi yau complete intersections in toric varieties _ , ( english summary ) comm . math . phys . * 168 * ( 1995 ) , no . 3 , 493533 . i. m. gelfand , a. v. zelevinski , and m. m. kapranov , _ hypergeometric functions and toric varieties _ , funktsional i prilozhen . * 23 * ( 1989 ) , no . 2 , 1226 . correction in ibid , * 27 * ( 1993 ) , no . 4 , 91 . mutsumi saito and william n. traves , _ differential algebras on semigroup algebras _ , _ symbolic computation : solving equations in algebra , geometry , and engineering _ ( south hadley , ma , 2000 ) , 207226 , contemp . math , * 286 * , amer . soc . , providence , ri , 2001 . bernd sturmfels , _ solving algebraic equations in terms of @xmath0-hypergeometric series _ ( english summary ) , formal power series and algebraic combinatorics ( minneapolis , mn , 1996 ) , discrete math . * 210 * ( 2000 ) , no . 1 - 3 , 171181 . bernd sturmfels and nobuki takayama , _ grbner bases and hypergeometric functions _ , grbner bases and applications ( linz , 1998 ) , 246258 , london math . lecture note ser . , * 251 * , cambridge univ . press , cambridge , 1998 .

the holonomic rank of the @xmath0-hypergeometric system @xmath1 is the degree of the toric ideal @xmath2 for generic parameters ; in general , this is only a lower bound . to the semigroup ring of @xmath0 we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of nongeneric parameters to construct a combinatorial formula for the rank jump of @xmath1 . as consequences , we obtain a refinement of the stratification of the exceptional arrangement by the rank of @xmath1 and show that the zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space . these results hold for generalized @xmath0-hypergeometric systems as well , where the semigroup ring of @xmath0 is replaced by a nontrivial weakly toric module @xmath3 $ ] . we also provide a direct proof of the main result in @xcite regarding the isomorphism classes of @xmath1 .

arxiv

a quantum computer will allow to perform some algorithms much faster than in classical computers e.g. shor algorithm for the factorization the numbers @xcite . the basic elements in the quantum computation are qubits and quantum logical gates , which allow to construct any circuit to quantum algorithms . the good candidates to realization of qubits are semiconductor quantum dots with controlled electron numbers . the qubit state can be encoded using an electron charge or , which is also promising , an electron spin @xcite . the spin qubits are characterized by longer decoherence times necessary in the quantum computation @xcite . however to prepare that qubit one needs to apply a magnetic field and removed the degeneracy between spin up and down . the manipulation of the qubit can be done by electron spin resonance and the read - out via currents in spin - polarized leads @xcite . another concept to encode the qubit is based on the singlet - triplet states in a double quantum dot ( dqd ) . in this case the magnetic field is not necessary and the qubit preparation is performed by electrical control of the exchange interactions @xcite . the qubit states can be controlled by e.g. an external magnetic field @xcite , spin - orbit @xcite or hyperfine interaction @xcite . for the read - out of the qubit state one can use current measurement and the effect of pauli spin blockade @xcite . in the pauli blockade regime the current flows only for the singlet , which gives information about the qubit states . divincenzo _ et al _ @xcite suggested to build the qubit in more complex system , namely in three coherently couplet quantum dots ( tqd ) . the qubit states are encoded in the doublet subspace and can be controlled by exchange interactions . this subspace was pointed as a decoherence - free subspace ( dfs ) @xcite , which is immune to decoherence processes . another advantage of this proposal is the purely electrical control of the exchange interactions by gate potentials which act locally and provide much faster operations . in the tqd system , in the contrast to the dqd qubit , one can modify more than one exchange interaction between the spins and perform full unitary rotation of the qubit states @xcite . the three spin qubit has also more complicated energy spectrum which provides operations on more states in contrast to the two spin system . recently experimental efforts were undertaken @xcite to get coherent spin manipulations in a linear tqd system according to the scheme proposed by divincenzo _ et al _ @xcite . the initialization , coherent exchange and decoherence of the qubit states were shown in the doublet @xcite and doublet - quadruple subspace @xcite . the read - out of the qubit state was performed , like in dqd , by means of the pauli blockade @xcite . _ @xcite observed a quadruplet blockade effect which is based on reducing leakage current from quadruplet to triplet states in the presence of magnetic field . @xcite showed that divincenzo s proposal can be realized on double quantum dots with many levels and three spin system controlled by gate potentials . in this paper we demonstrate that tqd in a triangular geometry can work as a qubit . this kind of tqd was already fabricated experimentally by local anodic oxidation with the atomic force microscope @xcite and the electron - beam lithography @xcite . in the triangular tqd qubit exchange interactions between all spins are always on and very important is symmetry of the system . trif et al . @xcite and tsukerblat @xcite studied an influence of the electric field on the symmetry of triangular molecular magnets and spin configurations in the presence of a spin - orbit interaction . divincenzo s scheme to encode the qubit in triangular tqd was considered by hawrylak and korkusinski @xcite where one of the exchange coupling was modified by gate potential . recently georgeot and mila @xcite suggested to build the qubit on two opposite chiral states generated by a magnetic flux penetrating the triangular tqd . one can use also a special configuration of magnetic fields ( one in - plane and perpendicular to the tqd system ) to encode a qubit in chirality states @xcite . recent progres in theory and experiment with tqd system was reported in @xcite . our origin idea is to use the fully electrical control of the symmetry of tqd to encode and manipulate the qubit in the doublet subspace . the doublets are vulnerable to change the symmetry of tqd , which will be use to prepare and manipulate the qubit ( sec . [ preparation ] ) . the crucial aspect in quantum computations is to read - out the qubit states . here we propose a new detection method , namely , a doublet blockade effect which manifests itself in currents for a special configuration of the local potential gates . we show ( sec . [ detection ] ) that the doublet blockade is related with an asymmetry of a tunnel rates from source and drain electrodes to tqd and the inter - channel coulomb blockade . the method is fully compatible with purely electrical manipulations of the qubit . next we present studies of dynamics of the qubit and demonstrate the coherent and rabi oscillations ( sec . [ dynamics ] ) . the studies take into account relaxation and decoherence processes due to coupling with the electrodes as well as leakage from the doublet subspace in the measurement of current flowing through the system . we derive characteristic times which describe all relaxation processes . our model is general and can be used for a qubits encoded also in the linear tqd , which is a one of the cases of broken symmetry in the triangular tqd . our system is a triangular artificial molecule built of three coherently coupled quantum dots with a single electron spin on each dot ( see . fig.[fig1 ] ) . interactions between the spins are described by an effective heisenberg hamiltonian @xmath0 where the zeeman term is included to show splitting by an external magnetic field @xmath1 ( @xmath2 is the bohr magneton , _ g _ is the electron g - factor ) and @xmath3 is an exchange interaction between electrons on sites @xmath4 and @xmath5 . the exchange parameter can be calculated by heitler - london and hund - mulliken method . for a defined confinement potential one can find the parameter as a function of the interdot distance , the potential barrier and the magnetic field @xcite . for the system with three spins there are two subspaces , one of them is a quadruplet with the total spin @xmath6 and @xmath7 . the quadruplet states are given by : @xmath8 . the second subspace is formed by doublet states with @xmath9 and @xmath10 . the doublet state for @xmath11 can be expressed as : @xmath12 @xmath13 and @xmath14 denote a singlet and triplet state on the @xmath15 bond , respectively . here we assume large coulomb intradot interactions and ignore double electron occupancy . in the doublet subspace ( [ d1])-([d2 ] ) one can express the hamiltonian ( [ heisenberg ] ) as @xmath16 using the pauli matrix representation . the parameters are given by @xmath17 the eigenvalues of ( [ matrix2 ] ) are : @xmath18 where @xmath19 is the doublet splitting and @xmath20 describes the energy differences between the doublet and the quadruplet subspace . two other parameters @xmath21 and @xmath22 can be interpreted as an effective magnetic field in the _ z _ and _ x _ direction , respectively @xcite . in gaas / algaas quantum dots the exchange interaction @xmath20 is estimated in the range @xmath23 mev @xcite and in a molecular magnet as @xmath24 mev @xcite . the doublet splitting for a linear tqd is the order @xmath25 @xmath26ev @xcite . this parameter can be even larger @xmath27 @xmath26ev in si / sige quantum dots @xcite . in this paper we assume that the exchange couplings @xmath3 can be manipulated by local potential gates @xmath28 , which change potential barriers and modify electron hopping as well as local covalency between the quantum dots . the exchange coupling can be expressed in the linear approximation as @xmath29 , where @xmath30 describes sensitivity of the exchange coupling to the gate voltage . for our analysis of the symmetry breaking in tqd , it is more suitable to parameterize the gate potentials as @xmath31 $ ] with @xmath32 , some amplitude @xmath33 and angle @xmath34 . this parametrization corresponds to influence of an effective electric field @xmath35 on the bond polarization and covalency . for a small value of * e * the exchange couplings can be expressed as @xmath36,\end{aligned}\ ] ] where @xmath37 , @xmath38 is a parameter describing sensitivity of the exchange coupling to the electric field , @xmath39 - the elementary electron charge , @xmath40 - the vector showing the position of the @xmath4-th quantum dot , @xmath34 is the angle between @xmath35 and the axis @xmath41 ( see fig.[fig1 ] ) . a similar relation was obtained for the triangular molecule in the electric field which changed chirality of the spin system @xcite . let us stress that because the electric field is taken as the small parameter , single electron occupancy of each dot is conserved and the ground state is always the doublet . in the tqd system one can also consider superexchange processes through excited double occupied states . applying local potential gates to the quantum dots one can shift their energy levels and modify the superexchange couplings @xcite . because a parameter of inter - dot electron hopping is relatively small with respect to a intra - dot coulomb interaction , the modifications of the superexchange couplings are very small and will not be discussed in the paper . let us consider how to encode the qubit in the doublet subspace with the spin @xmath11 , expressed by @xmath42 . this state is isolated from the doublet @xmath43 and the quadruplets states for a moderate magnetic field , @xmath44 . the encoded qubit states @xmath45 and @xmath46 correspond to the doublets @xmath47 and @xmath48 , eq.([d1 ] ) and ( [ d2 ] ) ( the spin index @xmath49 is omitted to simplify the notation ) . in the further considerations the hyperfine and spin - orbit interactions are ignored . the qubit is prepared by a proper orientation @xmath34 of the effective electric field @xmath50 which changes the symmetry of the system . [ eigenvectors ] presents density matrix elements @xmath51 and @xmath52 as a function of @xmath34 . one can see that the qubit is prepared in the state @xmath47 for @xmath53 ( the electric field is oriented from the quantum dot 1 ) . for this symmetry the exchange parameters are @xmath54 and from eq . ( [ delta ] ) and ( [ gamma ] ) one gets @xmath55 and the mixing between the doublets @xmath56 . for @xmath57 the electric field points to the quantum dot 1 and @xmath58 , @xmath59 and @xmath56 . in this case the qubit is prepared in the state @xmath48 . we would like to emphasize that in the triangular tqd the both qubit states are equivalent and can be easily achieved only by change the symmetry of the system . this is the main advantage in comparison with the linear tqd where the qubit can be prepare usually only in one of the doublet state . ( solid black curve ) and @xmath52 ( dashed red curve ) vs @xmath34 for the ground state at a moderate value of the electric field @xmath60 . the left and right inserts shows direction of the electric field when the qubit is prepared in the states @xmath48 and @xmath47 , respectively . these two opposite directions of the electric field corresponds to a minimal ( thin line ) and maximal ( thick line ) bond polarization between the dots 2 and 3.,scaledwidth=30.0% ] now we show how one can perform one qubit operations by means of the electric field . after preparation of the qubit in one of the states @xmath47 or @xmath48 we change rapidly the angle @xmath34 to perform a dynamic rotation of the qubit state . the qubit dynamics is described by the time - dependent schrdinger equation with the hamiltonian ( [ matrix2 ] ) . we can show two basic quantum gates . for @xmath61 the pseudo - spin rotates around the x - axis on the bloch sphere , which is the pauli - x quantum gate and the solution of the schrdinger is given by @xmath62 $ ] is an unitary operator of rotation around the x - axis . second quantum gate we get for @xmath56 with the solution given by ( [ h2 ] ) but now instead @xmath63 we have @xmath64 $ ] which is an unitary operator of rotation around the z - axis . these two rotations can be use to get to any point on the bloch sphere . it is clearly seen that by modification of the parameters @xmath22 and @xmath21 one can get full control of the qubit operations ( see also @xcite ) . in the linear tqd the read - out of the qubit state is possible due to charge - spin conversion in the regime of the pauli spin blockade . a detuning voltage is applied between two outermost dots , which drives the system from single occupied configuration ( 1,1,1 ) to double occupied e.g. ( 2,0,1 ) . this transfer is possible for an electron with the opposite spin orientation and can be detected by a quantum point contact ( qpc ) @xcite . in this section we would like to show a new method to read - out the qubit state which is based on a measurement of currents flowing through the system . the detection is compatible with electrical control of qubit state and the charge - spin conversion is not necessary . we assume that tqd is coupled by tunnel junctions to the electrodes , where the first and the second dot are connected to the left and right electrode , respectively . application of this method in an experimental setup is a similar technical complexity as qpc . the electron transport through the tunnel junctions is studied within the sequential tunneling regime . transfer rates from the left ( l ) and the right ( r ) electrode to tqd are given by : @xmath65 here we assume that both tunnel barriers are characterized by the same parameter @xmath66 , the reduced planck constant is taken @xmath67 , @xmath68 and @xmath69 denote the states with two and three electrons with the corresponding energies @xmath70 and @xmath71 , @xmath72 is an electron creation operator on the dot 1 ( 2 ) with spin @xmath73 . _ f _ denotes the fermi distribution function , the electrochemical potentials in the left and the right electrode are @xmath74 and @xmath75 , where @xmath76 is the fermi energy and @xmath77 is an applied bias voltage . by analogy one can define transfer rates @xmath78 from tqd to the electrodes . we confine our considerations to a voltage window with transitions between the states with three and two electrons , but a similar situation one can expect for transitions between three and four electron states . two electron states @xmath68 can be either the singlet @xmath79 or triplet @xmath80 . for a high intra - dot coulomb interaction one can neglect double occupied states and confine considerations to the states with single electron occupancy only . the singlet can be then expressed as a linear superposition : @xmath81 , where @xmath82 denotes the singlet on the @xmath83 pair of dots . calculating the elements of the transfer matrices one can find net transfer rates between the doublet @xmath84 , @xmath85 and the singlet @xmath79 : latexmath:[\ ] ] @xmath195 has a lorentzian dependence and reaches its maximum for a resonance condition @xmath196 . in this case the doublet blockade is removed and higher current flows through the system . at the resonance condition ( @xmath196 ) . the notation and the parameters are the same as in fig . [ singdynamic ] . notice that the results are presented in the rotating frame , whereas fig . [ singdynamic ] presents the evolution in the laboratory frame . the relaxation rates are @xmath149 , and @xmath151 . ] [ singdynamicrez ] shows the density matrix elements , the currents and the oscillations on the bloch sphere for the driven case in the resonance ( @xmath196 ) . we take the same parameters as for fig . [ singdynamic ] but now the results are presented in the rotating frame . one can see the large rabi oscillations of @xmath51 and @xmath52 . the population of the states is changed alternately between @xmath47 and @xmath48 with the frequency @xmath197 . it is also seen in fig . [ singdynamicrez]c which presents the rotation of the pseudo - spin on the bloch sphere . the rotation is in the @xmath198-@xmath199 plane on the spiral with periodic transfers between the states @xmath51 and @xmath52 . in the stationary limit one gets @xmath200 and @xmath201 , which is a higher value than in the non - driven case with @xmath202 ( see fig . [ singdynamic ] ) . the relaxation times @xmath166 and @xmath167 are almost the same as for the time independent case due to their weak dependance on @xmath22 and @xmath21 . the current plots in fig . [ singdynamicrez]b ) present strong oscillations which corresponds to coherent switching between the doublet states ( rabi oscillations ) . the leakage current flowing through the right junction @xmath203 also shows some oscillations . the stationary current ( dashed line ) is larger than for the non - driven case as one may expect when the doublet blockade is removed . switched on at @xmath152 . panel a ) presents the time evolution of the density matrix elements , while b ) presents the current flowing through the left ( blue curve ) and right junction ( black curve ) . the black - dashed line corresponds to the current in the stationary limit . panel c ) shows the evolution of the pseudo - spin on the bloch sphere . the initial state is @xmath48 ( the black arrow ) , and the final state is represented by the green arrow . the parameters taken in the calculations : @xmath204 , @xmath205 the other parameters are the same as in fig.[blockadet ] . the relaxation rates for these parameters are @xmath206 , @xmath207 , @xmath208 , and @xmath209 . notice that in the present case the currents are smaller than those ones in fig.[singdynamic]b because now the transfer rates are different . ] one can expect similar dynamics when the two - electron ground state is triplet . here we still confine ourselves to the doublets with @xmath11 and ignore spin - flip processes . now in the voltage window ( see the inset in fig.[trypdynamic ] ) we have the states @xmath210 , @xmath211 and @xmath212 whereas @xmath213 is below the chemical potential in the left electrode . the master equation ( [ dme ] ) is rewritten in the form @xmath214 here the population of the states is built - up by the transfers @xmath215 , @xmath216 and @xmath217 from the right electrode . electrons escape from the system to the left electrode which is described by @xmath218 and @xmath219 . the currents flowing from the right and the left electrode are expressed as @xmath220 notice that now the role of the doublet states @xmath47 and @xmath48 is reversed , and current flows through @xmath48 while @xmath47 is the dark state and blocks electron transport - see eq.([dtot1 ] ) . in the bloch space we have : @xmath221 one can easily find the main contributions to the relaxation rates : @xmath222 , which is qualitatively similar to the previously considered case with the singlet state but now the transfer rates are different [ compare eq.([dtos])-([dtos2 ] ) with eq.([dtot1])-([dtot4 ] ) ] . the leakage to the triplet state is given by @xmath223 which has an additional term @xmath224 describing transfer from the triplet to quadruplet state . the last row in eq.([master - dt ] ) and ( [ bloch - trip ] ) describes another leakage process with the relaxation rate @xmath225 . this process changes the population of the triplet and the quadruplet state and indirectly influences dynamics in the doublet subspace . solving these equations we determine the occupation probabilities and consequently the currents : @xmath183 and @xmath180 . the results are presented in fig . [ trypdynamic ] for the time independent mixing term , @xmath226 for @xmath154 and @xmath48 as the initial state . on the top panel one can see coherent oscillations for @xmath51 and @xmath52 similar as for the case with the singlet state . however , due to the quadruplet state the dynamic of the doublets and their final occupation is different than for the singlet case . in the short time range the population @xmath227 and @xmath228 increases due to leakage from the doublet subspace with characteristics rates @xmath167 and @xmath229 . in the longer time scale the population @xmath227 goes to @xmath228 ( in the stationary limit @xmath230 ) . the influence of these states on the doublet dynamics is clearly seen in the currents @xmath231 and @xmath203 presented in fig . [ trypdynamic]b . @xmath183 shows different behavior than in the singlet case , because the doublet blockade is modified by the quadruplet state which gives an additional contribution to the current . the increase of the current @xmath180 at the short time scale is related with the leakage to the triplet state . for longer times @xmath180 is diminished but the quadruplet contribution makes the drop less pronounced than in the singlet case . the bottom panel , fig . [ trypdynamic]c , presents the doublet dynamics on the bloch sphere . the behavior of the bloch vector is similar as in the singlet case with some quantitative differences . the phase and the amplitude damping is smaller which is the result of the longer relaxation times in the considered case . in the considerations above we have taken into account only charge fluctuations on the evolution of the qubit . let us now extend the studies and include spin - flip processes . a spin of an electron captured on a quantum dot can interact with nuclear spins of many atoms confined in the area of the quantum dot , which can lead to decoherence of the qubit states . the decoherence processes due to hyperfine interaction in triangular spin clusters has been already investigated by troiani et al . @xcite . here we consider another decoherence process caused by the spin relaxation in the electrodes . the electrodes connected to tqd are paramagnetic and electron can be injected with spin up to the state @xmath232 or with spin down to @xmath233 . this stochastic process leads to mixing between two doublet subspaces . the evolution of the qubit is studied in the absence of the magnetic field . we assume that the qubit is prepared in the state @xmath210 with the spin @xmath141 and the singlet is the ground state for two electrons . similarly as in in the previous cases the qubit dynamics is govern by the master equation ( [ dme ] ) , but now we take into account states with different spin orientation @xmath234 , @xmath235 and @xmath79 @xmath236 where the transfer rates are the same for both spin orientations : @xmath237 and @xmath238 . one can see that the master equation ( [ master - spin - flip ] ) represents dynamics of two doublet subspaces with @xmath11 and @xmath239 . the subspaces are mixed with each other by transfers to the singlet state [ see fourth equation in ( [ master - spin - flip ] ) ] . these two subspaces correspond to two pseudo - spin vectors on two bloch spheres . each of the subspace is described by the master equation eq . ( [ bloch - sing ] ) but the relaxation rates are now : @xmath240 and the leakage process @xmath241 , which is twice larger than in the case without spin - flip due states degeneracy . we make the laplace transformation of the master equation ( [ master - spin - flip ] ) and find the relaxation rates from the poles of the polynomial @xmath242 . here the polynomial @xmath243 is the same as for the previous case [ described by eq . ( [ bloch - sing ] ) ] with the transfer rates including degeneracy of the doublet states . the second polynomial @xmath244 $ ] is related with the spin flip - processes which mix two doublet subspaces . from @xmath245 one finds the spin - flip relaxation times @xmath246 in the limit of weak coupling with the electrodes . the rate @xmath247 is the second largest rate , after leakage and describes a rapid relaxation process , which can be seen in the short time scale . @xmath248 corresponds the longest relaxation process which leads to total mixing of two doublet subspaces in the stationary limit . the dynamics in the doublet subspace including spin - flip processes is presented in fig . [ spinflip]a . the time evolution of @xmath249 is similar to the case without spin - flip ( compare with fig . [ singdynamic ] ) , but now its reduction in the short time scale is faster due to @xmath250 . at the same time the @xmath251 is built - up with the rate @xmath250 , and goes to the stationary limit @xmath252 with the rate @xmath253 . the qubit dynamics can be also seen in the spin current flowing through the left @xmath254 and right junction @xmath255 see fig.[spinflip ] b. the shape of the total current @xmath256 is very similar as in the case without spin - flip . to get more information one needs to measure the spin dependent currents @xmath257 . the dashed and dotted blue curves in fig . [ spinflip ] b show @xmath258 and @xmath259 with the characteristic times @xmath260 in the short and long time scale . the fast increase of the total current in the right electrode @xmath180 for the very short time scale is related with @xmath178 which now is two times shorter . ( blue dotted curve ) , @xmath261 ( blue dashed curve ) and right junction with spin @xmath262 ( black dashed curve ) . notice that @xmath263 . the total currents @xmath264 are plotted as a solid lines . the parameters taken in the calculations are the same as in fig . [ singdynamic ] . the relaxation rates are @xmath265 , @xmath150 , @xmath151 , @xmath266 , and @xmath267 . ] let us estimate the characteristic times calculated in the paper . in the first order of approximation the relaxation times are proportional to @xmath268 defined in equation ( [ transferrates ] ) . the tunnel rate @xmath66 in ( [ transferrates ] ) is the order of nev for sequential transport @xcite . however it can be much larger in the coherent regime , @xmath269 @xmath26ev @xcite . if we assume @xmath270 @xmath26ev for our system then the relaxation and decoherence time is @xmath271 ns and @xmath272 ns for the case with singlet . for triplet we have @xmath273 ns and @xmath274 ns . the leakage to the singlet and triplet states are @xmath275 ns and @xmath276 ns respectively . these relaxation times are the same order as the decoherence time @xmath277 ns due to hyperfine interaction in gaas - based quantum dots @xcite . for the spin - flip processes the relaxation times are : @xmath278 ns and @xmath279 ns . one can see that due to the long relaxation time @xmath280 the qubit conserves its spin coherence for a time needed for a read - out process . summarizing we have proposed the qubit controlled by a symmetry breaking effect in a triangular tqd system . the main result of the paper is the new method for read - out of the qubit state by the current measurement in the doublet blockade regime , and the analysis of the qubit dynamics in the presence of decoherence processes caused by interaction with the electrodes . we assumed that each dot contains one spin and the qubit was encoded in the doublet subspace . the qubit states has been controlled by the applied gate potentials which break the triangular symmetry . the calculations have been performed in the the heisenberg model where the exchange couplings are modified by the orientation @xmath34 of the electric field with respect to the triangular axes . for a specific @xmath34 one of the doublets is occupied and can be taken as an initial qubit state for further manipulations . by quick impulses of the electric field one can perform the pauli x - gate and z - gate operations . a composition of these two operations gives full unitary control of the single qubit . moreover we have demonstrated the new method to read - out of the qubit states using the electric transport through tqd and the doublet blockade effect . the method is compatible with pure electrical manipulations and the spin - to - charge conversion is not necessary . the doublet blockade effect is related with an asymmetry of tunnel rates between the doublet states and the electrodes . for some specific symmetry of tqd one of the doublet states is a dark one and the electron transport is blocked . we have considered two cases with the singlet and the triplet as a ground state for two electrons . for the singlet case the current is blocked due to the doublet @xmath48 , whereas for transport from the triplet the dark state is the doublet @xmath47 . the doublet blockade can be also used to detect the qubit states in the linear tqd . however to satisfy the blockade condition @xmath56 one of the electrodes must be connected to the central dot . moreover the blockade can be applied to dynamical initialization of the qubit state as well as to perform landau - zener passages @xcite . we have also considered the time dependent electron transport in the doublet blockade regime . our research gives information about dynamics of the qubit , the coherent oscillations and the relaxation processes due to presence of the electrodes . a role of the leakage processes from the doublet to two electron states has been studied as well . for the triplet case the leakage is larger than for singlet due to activation of the quadruplet state . we have also presented the driven case where the mixing parameter between the doublet state is time dependent @xmath281 . in the resonance condition @xmath196 the doublet blockade is partially removed and one can observe strong rabi oscillations . moreover we have investigated mixing of the doublet subspaces with @xmath11 and @xmath239 caused by the spin - flip processes in the electrodes . the total mixing time @xmath280 is very long what is promising for manipulation and read - out of the qubit . z. shi , c. b. simmons , d. r. ward , j. r. prance , r. t. mohr , t. s. koh , j. k. gamble , x. wu , d. e. savage , m. g. lagally , m. friesen , s. n. coppersmith , and m. a. eriksson , phys . rev . b * 88 * , 075416 ( 2013 ) .

we present a model of a qubit built of a three coherently coupled quantum dots with three spins in a triangular geometry . the qubit states are encoded in the doublet subspace and they are controlled by a gate voltage , which breaks the triangular symmetry of the system . we show how to prepare the qubit and to perform one qubit operations . a new type of the current blockade effect will be discussed . the blockade is related with an asymmetry of transfer rates from the electrodes to different doublet states and is used to read - out of the dynamics of the qubit state . our research also presents analysis of the rabi oscillations , decoherence and leakage processes in the doublets subspace .

arxiv

symbiotic systems ( ss ) are generally composed by a white dwarf ( wd ) , a red giant ( rg ) star , and by circumstellar and circumbinary nebulae . gas and dust radiation from the nebulae appear throughout a large frequency range , from radio to x - rays . z andromedae ( z and ) binary system consists of a cool giant of spectral type m4.5 and a hot compact component with a temperature of @xmath0 1.5 10@xmath1 k. many periods of activity have been reported during more than 100 years , in 1915 , 1939 , 1960 , 1984 , and 2000 as nonuniform eruptions of classical outbursts . sokolowski et al . ( 2006 ) claim that the _ outbursts sometimes come in pairs ( as in 1984 and 1986 ) _ , or in a series of eruptions with decreasing maximum brightness and different shapes , separated in time by periods slightly shorter than the orbital one ( kenyon & webbink 1984 ) . the main characteristics of the 1984 - 1986 event are the two maxima which appear in the profile of the visual magnitude ( fernndez - castro et al . 1995 , hereafter fc95 , fig . 1 ) leading to the ambiguous interpretation of the outburst as a double or a disturbed single one . fc95 suggested , on the basis of the line ratio analysis , that a drop in the wd radiation flux could provoke the observed trend of the light curve in 1986 . they claim that a shell of material was ejected during each of the _ two outbursts _ at 1984 and 1986 . the brightness of the 1984 - 1986 event was exceptionally low ( leibowitz & formiggini 2008 , fig . nevertheless , it was monitored at very close dates by the spectral observations of fc95 . the spectra are rich enough in number of lines to allow a detailed modeling of z and physical conditions . in this paper we revisit z and system focusing on the physical conditions in the nebulae and their fluctuations in the 1984 - 1986 years . our aim is to reveal unpredicted episodes by the detailed modeling of the spectra . in z and , the stellar wind of the cool giant has a velocity of @xmath0 25 ( sequist et al . bisikalo et al ( 2006 ) claim that varying the wind velocity from 25 to 30 changes the accretion regime from disk to wind accretion . this is accompanied by a jump in the accretion rate , increasing the hydrogen burning rate . an optical thick wind forms from the wd . this wind is revealed e.g. by the observations of uv and optical spectra during the 2000 - 2002 outburst ( e.g. sokoloski et al 2006 , tomov et al 2008 ) . the shocks which derive from wd and rg wind collision yield an increase in the luminosity of the system . this hypothesis is strengthened by fc95 who did not find evidence for an accretion disk at the 1984 - 1986 epoch but suggested collision episodes . collision of the winds ( girard & willson 1987 , bisikalo et al 2006 , angeloni et al 2010 , contini & angeloni 2011 ) leads to two shock fronts between the stars : the strong one , dominating the spectrum , propagates in reverse towards the wd , while the weakest one propagates towards the rg . moreover , a shock front expands out of the system throughout the circumbinary medium . we calculate the line spectra in sect . 2 in the frame of the wind collision model , adopting the code sumamarcel / suma / index.htm ] , which accounts consistently for shocks and photoionization . the input parameters are : the shock velocity , the preshock density 0 , the preshock magnetic field 0 , the colour temperature of the hot star , the ionization parameter @xmath2 relative to the black body ( bb ) flux reaching the nebula . the geometrical thickness of the emitting nebula @xmath3 , the dust - to - gas ratio @xmath4 , and the abundances of he , c , n , o , ne , mg , si , s , a , fe relative to h are also accounted for . 0=10@xmath5 gauss is adopted . the observations at different epochs of the continuum spectral energy distribution ( sed ) are reproduced in sect . 3 by models constrained by the fit of the line ratios . the adjustment factors determine the distance of the nebulae downstream of shock fronts from the system center , providing a detailed picture of the z and components . discussion and concluding remarks follow in sect . the trend of the spectral observations between 1978 and 1993 ( fc95 , figs . 4,5 ) gives a first hint about the 1984 - 1986 outburst in the frame of a longer activity period . we notice that : \1 ) the 1984 - 1986 outburst lies upon an event which developed about at 1979 and ended at 1988 . if this long event depends on the wd activity , then the temperature of the wd will not show dramatic changes in the 1984 - 1986 period . \2 ) the oi 1305 resonance line is most probably blended with an upper ionization level line because its behaviour is similar to that of high level lines the oi line is most probably emitted from a different nebula with conditions close to those of the ism . \3 ) the systematic decrease and increase of the cii 1336 line , whose trend is opposite to that of the high ionization lines , indicates that the cii minimum is strongly correlated with the trend of the physical parameters , e.g. the ionization parameter , and the shock velocity . the role of the wd temperature is less prominent . fc95 in their table 3 report an optical spectrum in july 1986 , observed at the same epoch as the uv spectrum presented in their table 2b , row 34 . we start by modeling the uv - optical combined spectrum observed at 11 july 1986 which is constrained by a relatively large number of lines from various ionization levels . this reveals the physical conditions in the emitting nebulae at that time . such conditions will be adopted as a first guess in the modeling of the uv spectra in the next epochs . [ cols="<,<,<,<,<,<,<,<,<",options="header " , ] @xmath6 2440000 + 3 ) : red ; ( in units of 10@xmath1 k ) : blue ; @xmath2 : green . dotted black lines define the outburst.,scaledwidth=98.0% ] 3 ) : red ; ( in units of 10@xmath1 k ) : blue ; @xmath2 : green . dotted black lines define the outburst.,scaledwidth=88.0% ] the visual light curve ( fig . 1 ) covers a frequency band accounting for lines and continuum in the optical range . the strongest lines are generally heii 4686 , , [ oiii]5007 , 4959 , etc . , while the optical spectrum ( fc95 ) shows relatively low forbidden lines . so we consider that the main contributors to the visual band are heii 4686 and , which depend strongly on the wd temperature and on the ionization parameter . and @xmath2 affect the spectra in the same way . namely , when they decrease , the high level lines decrease relatively to the low level ones . the increase of at the epochs of the peaks and the decrease at minima would indicate that the active phase is characterised by two different bursts . on the other hand , if does not change significantly and only @xmath2 changes , a disturbing dynamical and/or morphological event between the radiation source and the nebula is most likely the source of the visual curve minimum . the uv band accounts for lines from different levels which are variously modified by and @xmath2 fluctuation . moreover , the shock velocity and the compression downstream affect the stratification of the ions and , consequently , the line ratios . fc95 show that lines from different ionization levels have different trends throughout the 1984 - 1986 time period . we predict that the uv light curve throughout the outburst , would have a profile different from that reported in fig . 1 . in fig . 2 we present the physical parameters obtained by modeling the spectra throughout the 1984 - 1986 outburst . the profiles of the physical conditions are rather unexpected if compared with the visual light curve ( fig . the sudden increase of the density on april 1984 , accompanied by the sudden increase of the velocity , indicates that collision of the wd wind with a shell ejected from the red giant is occurring . the nebula downstream of the reverse shock is heated and ionized by both the shock and the photoionization flux from the hot star . in the next days the density and the shock velocity have opposite trends in agreement with conservation of mass at the shock front ( n1v1=n0v0 , from the rankine - hugoniot equations ) . in the meanwhile the temperature of the hot star is slightly increasing from @xmath0 120,000 k to a maximum of 170,000 k , corresponding to the outburst maximum , in agreement with fc95 . the drop of @xmath2 on 1984 toward july 1986 shows that the spectra during that epoch are dominated by a relatively strong shock , i.e. the reverse shock , which vanishes at a certain time , depending on the outburst characteristics ( chevalier 1982 ) . the second density peak at december 1985-january 1986 reveals that collision of the wd wind with another rg shell has occurred . the ionization parameter drop by more than two orders of magnitude after june 1985 , accompanied by the simultaneous variation of the dynamical parameters and 0 , suggests that the shock front has suddenly shifted farther from the wd . the matter downstream of the reverse shock front is fragmented by richtmyer - meshkov and kelvin - helmholtz instabilities . high density fragments between the stars eventually prevent the black body flux from reaching the nebula . however , the decline of the bb flux by fragment obstruction could indeed reduce the ionization parameter , but less sharply than predicted by the calculations . so , the whole picture is most likely explained by an unexpected disturbing episode such as the ejection of a new shell from the rg atmosphere and its collision with the wd wind at a relatively large distance from the wd ( sect . the gas composition at the epochs of collision with the shells reveals that the calculated siiii]/niv ] line ratios would better reproduce the data adopting si / h relative abundances lower than solar by a factor of 4 for models m@xmath7 , m@xmath8 and of @xmath0 10 for models m@xmath9 , m@xmath10 , m@xmath11 , m@xmath12 . this indicates that silicate grains are present after collision of the wd wind with the first rg shell . on the other hand si is in the gaseous phase with a solar si / h relative abundance at the epochs corresponding to models m@xmath13 , m@xmath14 , m@xmath15 , m@xmath16 . the grains had not enough time to form at collision with the next shell . grain formation time scales ( gail et al 1984 , scalo & slavsky 1980 ) are as long as @xmath0 1 year . however , in circumstellar shells the chemical equilibrium and consequently grain formation is disturbed by the uv radiation from the wd and by the velocity field ( gail et al . 3 we compare the observed continuum sed with model calculations . we refer to the sed of the continuum flux presented by fc95 in their fig . we have used the data in the uv and in the radio range presented by fc95 tables 2a and 4 , respectively . the dates corresponding to observations in the radio and those in the other frequency ranges are not exactly coincident , e.g. we adapted the radio fluxes observed in october 1986 to data observed in july 1986 . each diagram of fig . 3 presents the free - free + free - bound fluxes emitted from the nebula at a certain epoch . they are calculated by the same models ( table 2 ) which lead to the best fit of the line spectrum ( table 3 ) . moreover , the bb flux from the rg corresponding to a temperature of 3200 k ( fc95 ) , and the bb flux corresponding to the wd which was derived by modeling the line spectrum are also shown . reprocessed radiation of dust within the nebula calculated consistently with gas emission is added in fig . 3 diagrams . this flux is mostly hidden by the bb flux from the rg if dust - to - gas ratios of 10@xmath17 by number ( 4 10@xmath18 by mass for silicates ) are adopted . actually , the dust - to - gas ratios are constrained by the data in the mid - ir . 3 shows that the emission from the nebula appears in a small frequency range between the optical and the near - uv ( @xmath0 1 - 3 10@xmath19 hz ) , depending on the epoch . there are no data in the soft x - ray range and beyond it . we can predict that bremsstrahlung from the nebula downstream of shock fronts with @xmath20 160 would be the x - ray source . for lower velocities the sed in the soft x - ray range is the summed flux of bremsstrahlung from the nebulae and bb flux from the wd . the contribution to soft - x - rays up to the x - ray domain of bremsstrahlung from shocked nebulae was suggested for the symbiotic system ag dra by contini & angeloni ( 2011 ) . the fit of calculated to observed data is good enough to confirm a strong self - absorption of free - free radiation from the nebulae for frequencies @xmath2110@xmath22 hz . finally , fig . 3 diagrams show that the bb flux from the rg and from the nebula dominate the sed in the visual , while the flux from the wd appear in the uv and soft x - ray range . this indicates that strong variations in the wd temperature that generally accompany the outbursts can not be directly revealed by the visual light curve presented by fc95 , but only indirectly by the spectra emitted from the nebula . in previous works on ss , e.g. ch cyg ( contini et al 2009b ) , ag dra ( contini & angeloni 2011 ) , etc . we calculated the distance r of the nebulae from the center of the ss adjusting the continuum fluxes calculated by the models at the nebula to those observed at earth . we defined the adjusting factor @xmath23 by r@xmath24=10@xmath25 d@xmath24 , where is the filling factor and d the distance to earth . adopting a distance of z and to earth of 1.12 kpc ( viotti 1982 ) , the best fit to the data of the bremsstrahlung calculated from the nebulae downstream of the shock front , gives r= 1.9 10@xmath26 cm at april 1981 , r=1.4 10@xmath26 cm at april 1984 , r=9.4 10@xmath19 cm at december 1985 , and r= 2.7 10@xmath26 cm at july 1986 . the distances were calculated by a filling factor = 1 . considering that : @xmath27 ( r@xmath28/r)@xmath24 = @xmath2 n c ( where @xmath27 is the flux from the wd , r@xmath28 is the radius of the wd , n is the density of the gas and c the speed of light ) , the results suggest that the drop of the ionization parameter @xmath2 between november and december 1985 most probably derives from the sudden increase of the distance from the hot source . this is explained by the ejection of the next shell in the rg atmosphere . the collision between the wind from the wd and the two shells from the rg occurred within less that 2 years , from march 1984 to november 1985 . the shock front has therefore expanded from @xmath0 2 10@xmath26 to @xmath09 10@xmath19 cm , by an average velocity of @xmath0 130 , in rough agreement with the observations . the 1984 - 1986 outburst of z and is revisited by a detailed modeling of the spectra . comparing the visual light curve of z and during the 1984 - 1986 outburst with those in other epochs from 1895 up to 2007 september , the 1984 - 1986 event is remarkable for its double peaked structure and for its low brightness . we explain the burst double structure in the light of red giant pulsation , by the collision of the wd wind with two ejected shells . the minimum of the light curve between the two peaks is accompanied by the dip of the ionization parameter which is constrained by the detailed modeling the uv line spectra at november - december 1985 . the sharp drop of @xmath2 is due to the sudden increase of the distance between the wd and the downstream nebula reached by the wd black body flux , namely , the collision of the wd wind with a new shell in the rg atmosphere . since the shell was recently formed at the time of observations , the grains did not have enough time to develop . this is revealed by the solar relative abundance of si / h . we suggest that the outburst did not attain its maximum luminosity because the collision of the wind network was distorted by the oncoming of the next shell . the small peaks in the light curve before and after the 1984 - 1986 burst are most probably due to rg shell ejections . the periodicity of the maxima is in fact of @xmath0 300 - 400 days . during the 2000 - 2002 outburst of z and , broad wings ( up to 2000 ) developed in the line profile ( tomov et al 2008 ) . moreover , collimated bipolar jets appeared and disappeared throughout the 2006 outburst ( skopal et al 2009 ) . broad lines were observed in other ss , e.g. ch cyg , bi cru , ag dra , etc . they originate from the high velocity matter accompanying the outbursts . exceptionally broad and line profiles were explained by wd explosion ( contini et al . 2009 a , c ) , while collimated jets at some epochs reveal the presence of the accretion disk . both broad lines and jets were not observed during the 1984 - 1986 outburst of z and . we are grateful to sharon sadeh for helpful advise . angeloni , r. , contini , m. , ciroi , s. , rafanelli , p. 2007 , a&a , 471 , 825 angeloni , r. , contini , m. , ciroi , s. , rafanelli , p. 2010 , mnras , 402 , 207 bisikalo , d.v . , boyarchuk , a.a . , kilpio , e. yu , tomov , n.a . , tomova , m.t . 2006 , astron . , 50 , 722 chevalier , r.a . , 1982 , apj , 259 , l85 contini , m. 2003 , mnras , 339 , 125 contini , m. , angeloni , r. , 2011 new astr . , 16 , 199 contini , m. , angeloni , r. , rafanelli , p. 2009a , a&a , 496 , 759 contini , m. , angeloni , r. , rafanelli , p. 2009b , an , 330 , 816 contini , m. , angeloni , r. , rafanelli , p. 2009c , mnras , 396 , 807 fernndez - castro , t . , gonzlez - riestra , r. , cassatella , a. , taylor , a.r . , seaquist , e.r . apj , 1995 , 442 , 366 gail , h .- p . ; keller , r. ; sedlmayr , e. 1984 , a&a , 133 , 320 gail , h .- p . , gauger , a. ; goeres , a. ; henkel , r. ; sedlmayr , e. 1987 , mitag , 70 , 365 girard , t. , willson , l.a . , 1987 , a&a , 183,247 kenyon , s.j . , webbink , r.f . 1984 , apj , 279 , 252 leibowitz , e. m. ; formiggini , l. 2008 mnras . 385 , 445 nussbaumer , h. , schild , h. , schmid , h.m . , vogel , m. 1988 , a&a , 198 , 179 osterbrock , d.e . 1988 in astrophysics of gaseous nebulae and active galactic nuclei , university science books scalo , j.m . , slavsky , d.b . 1980 , apj , 239 , l73 saquist , e.r . , taylor , a.r . , button , s. 1984 , apj , 284 , 202 skopal , a. et al . 2009 , apj , 690 , 1222 sokoloski , j.l . 2006 apj , 636 , 1002 tomov , n.a . , tomova , m.t . , bisikalo , d.v . 2008 mnras , 389 , 829 viotti , r. , giangrande , a. , ricciardi , o. , cassatella , a. 1982 in the nature of symbiotic stars , ed . m. friedjung and r. viotti ( dordrecht : reidel ) , 125

the visual magnitude profile of the symbiotic system z and during the 1984 - 1986 activity period appears double peaked and the flux intensity is low compared to outbursts in other epochs . the detailed modeling of the observed spectra , accounting for the shells ejected by the red giant star , shows that the outburst is intrinsically single but distorted by the collision at different phases of the white dwarf wind with two close shells . binaries : symbiotic - stars : individual : z and

arxiv

networks with randomly distributed single walled carbon nanotubes ( swcnts ) are emerging as novel materials for various applications , particularly as electronic materials . @xcite this creates a need for the accurate determination of their fundamental electrical properties . swcnt networks can be viewed as a two - dimensional network of conducting one - dimensional rods.@xcite these rods are either metallic or semiconducting with large dimension ratio(length / diameter @xmath2 ) , which leads to the following interesting behavior.@xcite since the nanotube - nanotube junction resistance is much larger than the nanotube resistance itself,@xcite such a network can be seen as a system with randomly distributed barriers for electrical transport . for a sub - monolayer network , the junction resistance will dominate the overall resistance and the network resistance shows percolation behavior with nonlinear thickness dependence.@xcite for swcnt networks with thickness in the tens of nanometers , e.g. with tens of layers of swcnts , the conductance through the metallic tubes will dominate the conduction , and leads to metallic behavior of the overall resistance.@xcite the frequency - dependent and electric - field - dependent conductivity have been investigated for individual swcnt@xcite and some types of swcnt networks.@xcite burke and co - workers measured the microwave conductivity of individual swcnts and investigated their operation as a transistor at 2.6 ghz.@xcite for swcnt networks with 30 nm thickness , our previous work finds the ac conductivity is frequency - independent up to an onset frequency @xmath3 of about 10 ghz , beyond which it increases with an approximate power law behavior.@xcite for thicker films with thickness in the range of tens of micrometers , p. peit _ _ found that conductivities at dc and 10 ghz are almost the same.@xcite m. fuhrer _ _ observed the nonlinearity of the electric field - dependent conductivity of relatively thick nanotube networks , and they claimed that the charge carriers can be localized by disorder in the swcnts with an approximate length scale @xmath4 of 1.65@xmath5m.@xcite v. skakalova _ _ studied the current - voltage(_i - v _ ) characteristics of individual nanotubes and nanotube networks of varying thickness , and discussed the modulation of _ i - v _ characteristics by a gate - source voltage.@xcite however there is a lack of frequency and electric field dependent conductivity investigation for systematic variation of network densities , and also there is no correlation study between the onset frequency @xmath0 and the length scale determined from nonlinear transport . the frequency- and electric - field - dependent conductivity investigation of different network densities is not only helpful to build a comprehensive understanding of the electrical transport properties of swcnt networks , but is also important for the sake of applications of these networks.@xcite for example , when using swcnt networks to construct high speed transistors , or as transparent shielding materials , knowledge of their frequency dependent electric transport properties are required . also use of swcnt networks as inter - connects in circuits requires understanding of their electric field - dependent properties . in this paper , we investigated the frequency- and electric - field - dependent conductivity for swcnt networks with systematically varying densities . we find that the onset frequency @xmath0 extracted from the frequency dependence measurement increases with the film thickness , while the length scale extracted from the electric field dependent measurement decreases at the same time . using the measured @xmath0 and the extracted length scale @xmath1 for different films , we developed an empirical relation between the two transport measurement methods . swcnt networks on polyethylene terephthalate ( pet ) substrate are prepared by the spraying method.@xcite arc - discharged nanotube powder from carbon solution inc . was dispersed in water with sodium dodecyl sulphate surfactants . the tubes were purified with nitric acid and left with a variety of functional groups.@xcite after the nanotubes were sprayed on the substrate , the samples were rinsed in water thoroughly to wash away the surfactant . the network density , defined as the number of nanotubes per unit area , was controlled by the solution concentration(0.1 - 1 mg / ml ) and the spraying time(10 - 60 seconds ) . the nanotubes formed bundles with size of 4 - 7 nm and length of 2 - 4 @xmath5m.@xcite the thicknesses of the films range from submonolayer to 200 nm corresponding to room temperature sheet conductance ranging from @xmath6 to @xmath7 s / square . fig.[fig : figuresample ] shows sem images of films with two different coverage densities . it is convenient to characterize the films by their sheet conductance rather than conductivity , because of the inhomogeneous nature of the conducting pathways and thus the difficulty in assigning a film thickness , particularly for submonolayer films . = 2.5 in = 3.8 in the corbino reflectometry setup , which is ideal for broadband conductivity measurements of resistive materials,@xcite was used to investigate the conductivity of the swcnt networks . the measurement procedures are similar to our previous work.@xcite in order to measure the conductivity in as broad a frequency range as possible , two test instruments are used , the agilent e8364b network analyzer ( covering 10 mhz to 50 ghz ) and the agilent 4396b network analyzer ( covering 100 khz to 1.8 ghz ) , giving five and half decades of frequency coverage . the electric field dependent conductivity was measured with a standard 4-probe method in a well - shielded high precision dc transport probe.@xcite the data of conductance vs. frequency are shown in fig.[fig : figurecond](a ) . the films decrease in thickness from sample 1 ( 200 nm ) to sample 5 ( sub - monolayer ) . for all the films , the real parts of the conductance keep their dc value up to a characteristic frequency and start to increase at higher frequency . this kind of behavior has been widely observed for disordered systems.@xcite similar behavior has been found for carbon nanotube polymer composites close to the percolation threshold , in which the extended pair approximation model was applied to describe the observed phenomena.@xcite the carbon nanotube networks in fig . [ fig : figurecond](a ) have densities well above the percolation threshold . however , since the junction resistances between different tubes are much larger than the resistance of the tubes themselves,@xcite swcnt networks above the percolation threshold can still be seen as systems with randomly distributed barriers for electrical transport . in this case the extended pair approximation model can be used to describe the real part of the ac conductance : @xmath8 where @xmath9 is the dc conductance , @xmath10 , @xmath11 is a constant and @xmath0 is the onset frequency.@xcite = 3.2 in = 4.8 in the obtained frequency dependent conductance in fig.[fig : figurecond](a ) fit well to the extended pair approximation model . fig . [ fig : figurecond](b ) shows the relation between the fit film onset frequency and the dc sheet conductance . the onset frequency changes from @xmath12 to @xmath13 rad / sec as the sheet conductance increases from @xmath6 to @xmath7 s / square . the onset frequency of the ac conductance increases as the dc conductance increases . the solid line fig.[fig : figurecond].(b ) has slope one , implying a linear relation between the onset frequency and their dc sheet conductance , or @xmath14 , over about 4 decades . we also measured the frequency - dependent conductance of ultra - thin sub - monolayer nanotube networks , which are fabricated via the filtration method.@xcite we choose chloroform as the solvent instead of water to avoid the washing steps which can easily destroy the sub - monolayer film structure . the network density is controlled by the concentration and volume of the solvent used.@xcite the conductance of these films are measured by a 4284a precision lcr meter , covering the frequency range from 20 hz to 1 mhz . for these films just above the percolation threshold , their frequency dependent conductance also fit well to the extended pair approximation model . in fig . [ fig : swcntfig4 ] , we plot the onset frequency versus their dc sheet conductance . for comparison , we also present results of kilbride _ et al._,@xcite for polymer - nanotube composites as well as the data from fig.[fig : figurecond](b ) for our films prepared by the spray method . the three solid lines in fig . [ fig : swcntfig4 ] all have slope one ( @xmath15 ) . here an interesting result is that although the intercepts are very different , the slopes of the three different samples are essentially the same . the difference in intercept is due to the different types of nanotubes . different carbon nanotube sources have different bundle sizes and lengths , which causes the conductance difference . the polymer - cnt composite has much smaller dc conductance than the other two , due to the separation of swcnts by polymer that leads to charge transfer barriers . despite these differences in detail , there is a universal relationship between onset frequency and dc conductance . = 3.4 in = 2.8 in the linear proportionality @xmath16 is one of the characteristic features of ac conductance of disordered solids.@xcite for disordered systems , the universality of ac conductance has been studied by many researchers since its discovery in the 1950 s , and is usually referred to as taylor - isard scaling . @xcite this inspires us to investigate the scaling behavior of the conductance and frequency for swcnt networks . similar to kilbride _ s work@xcite , taking the dc conductance @xmath9 and the onset frequency @xmath0 as scaling parameters , we plotted the ac conductance data for all samples in fig . [ fig : swcntscaling ] . the data show a scaling behavior with all data sets falling on the same master curve described by @xmath17 and @xmath18 . = 3.4 in = 2.8 in in order to explain the observed onset frequency change with the swcnt volume fraction for polymer - nanotube composites close to percolation threshold , kilbride _ defined a characteristic length scale , the `` correlation length '' @xmath19.@xcite for our samples , which are well above the percolation threshold , we can similarly define the correlation length @xmath19 as the distance between connections in the sample , i.e. , the distance between junctions of the multiply - connected swcnt network . when one measures the conductance at a given frequency , there is a typical probed length scale . at low frequencies ( corresponding to long time periods ) , the carriers travel long distances in one - half ac cycle and the experiment investigates longer length scales . at high frequencies , the carriers travel short distances in one half ac - cycle and the probed length scale is shorter . in the absence of an applied dc electric field , and assuming a random walk , the probed length scale @xmath20.@xcite for low frequency , @xmath21 , the probed length scale spans multiple junctions of the swcnt network . the junction resistances between different tubes are much larger than the resistances of the tubes themselves . @xcite hence the conductance is small and equal to the dc conductance . as the frequency increases , @xmath22 becomes smaller than @xmath19 , @xmath23 , and the junction resistances have less contribution to the total resistance . the intrinsic properties of the swcnt then dominate the measurement . therefore the observed conductance increases as the frequency increases . the transition frequency is expected to be that where carriers scan an average distance of order of the correlation length . @xcite the swcnt film can be treated as a random walk network , in which the probed length scale at a given frequency goes as @xmath24 . then with the onset frequency @xmath0 of samples we are able to estimate their correlation length @xmath25 . as the density of the swcnt networks increases , there are more junctions and the average distance between connections , the correlation length @xmath19 , becomes smaller and thus the onset frequency @xmath0 increases . this is consistent with the behavior shown by the data in fig . [ fig : figurecond ] . from the measured frequency dependent conductivity , we observed that the correlation length @xmath19 varies as the thickness of the film increases . due to the large resistance of junctions between different tubes , the carriers in the swcnt networks are easier to move on small length scales than on larger length scales . this phenomenon inspires us to investigate whether the transport properties are nonlinear as a result . according to the works of m. s. fuhrer _ _ , the localization behavior of a swcnt network can be studied by measuring the electric field dependent nonlinearity of the conductivity . @xcite generally , increasing both the temperature and the electric field reduces the effects of localization . hence a system typically displays a characteristic electric field at which nonlinear conductivity begins to appear . through measurement of the characteristic field @xmath26 one can determine a length scale of the system : @xmath27 where @xmath1 corresponds to the size of the regions with good conductivity , and @xmath28 is the energy gained by a carrier in one conducting region on average.@xcite . for our swcnt films @xmath29 depends on the average distance between connections and thus should vary with sample thickness . for relatively thick films(@xmath30 ) with low sheet resistance , fuhrer _ et al . _ found a temperature - independent length scale @xmath31 m in a swcnt film.@xcite here we investigated the electric field - dependent conductance for a different class of films . the studied samples were cut into rectangular pieces with length @xmath32 mm and width @xmath33 mm . gold contact leads , which completely cover the width of the samples , were deposited on the surfaces of the samples to form a standard four probe pattern . the measured swcnt networks have dimensions of @xmath34mm@xmath353 mm , between the two voltage leads . because swcnt films have a large in - plane thermal conductivity , @xcite and the probe used to perform the measurement was specially designed to reduce heating effects@xcite we believe that the heating effect can be ignored in our measurement range . this was confirmed by observing the voltage response of pulsed current input of several samples in the time domain with an oscilloscope . = 3.2 in = 4.8 in the dc resistance vs. electric field of a submonolayer film(sample 5 ) at different temperatures is shown in fig.[fig : swcntfig5](a ) . we can see clearly that there is nonlinear behavior even at room temperature . as sketched in fig.[fig : swcntfig5](a ) , we extracted @xmath26 for each temperature , which is the characteristic electric field at which nonlinear conductance begins to appear.@xcite fig.[fig : swcntfig5](b ) shows the extracted @xmath26 vs temperature for this sample . as temperature increases , @xmath26 also increases . roughly , @xmath26 and temperature have a linear relationship and satisfy the equation @xmath36 , with @xmath37 . note that this is much larger than the @xmath4 obtained in ref.@xcite . this suggests that a different mechanism for hindering transport is active here , namely poor contacts between bundles of swcnts . to investigate the dependence of @xmath29 on the film thickness , we measured the electric field - dependent conductance for samples with various thicknesses . we found that the nonlinear behavior strongly depends on the thickness of the film . at low temperature , all the samples show nonlinear behavior of resistance on electric field , and the critical field @xmath26 is larger for thicker samples and smaller for thin samples . at room temperature , all the other samples are purely ohmic(linear ) in electric fields up to 100 v / cm , except for the thinnest sample , sample 5 , which showed electric field - dependent conductance at all temperatures . through the critical field @xmath26 , we extracted the @xmath29 for different density films , shown in fig.[fig : swcntfig6 ] as red - dots . clearly @xmath29 depends on the density of the swcnt networks . for the thicker film with larger swcnt density , which has larger sheet conductance , the @xmath29 is smaller . the lower density films with larger average distance between the junctions has larger @xmath29 . = 2.6 in although the localization behavior may be affected by the properties of individual swcnts @xcite , the above result shows that the observed behavior is more likely a consequence of poor electric conduction through inter - bundle junctions . as the thickness and density increase , there are more and more connections , which decreases the length scale @xmath29 , as shown in fig.[fig : swcntfig6 ] . when the film thickness increases to a large value , the density of the swcnts and connections of the film will finally saturate and stop increasing . for those very thick swcnt films ( also called mats ) , their correlation length @xmath19 , and also the length scale @xmath29 , as well as the onset frequency , will be independent of the thickness , but depend on the properties of individual bundles , which include both the disorder of the individual nanotube and also the geometry of the bundles . hence the difference of swcnt sources and purities can change the onset frequency . we believe this is the reason that our previous work found that the ac conductivity follows the extended pair approximation model with an onset frequency @xmath0 about 10 ghz , while peit _ _ found that swcnt networks have the same conductivities at dc and 10 ghz for swcnt networks with thickness in the range of tens of micrometers.@xcite the frequency dependent conductivity shows that the onset frequency @xmath0 increases with the sample thickness and gives an estimation of the correlation length @xmath19 of the sample , @xmath38 . @xmath19 and @xmath39 decrease as the sample thickness increases . the dc nonlinear conductivity measurement shows that the length scale @xmath29 of the samples also decreases with increasing sample thickness . qualitatively @xmath39 and @xmath29 have the same dependence on the sample thickness . based on the above consideration , it is reasonable to expect some correlation between the onset frequency @xmath0 and the length scale @xmath29 , and we proposed the following relation , @xmath40 where @xmath41 is a fitting parameter , which can be obtain from our experiment data , @xmath42 . fig.[fig : swcntfig6 ] shows the comparison of the nonlinear transport length scale @xmath29 measured directly through electric - field dependent dc conductance measurements(red - dot ) and the right - hand side of eq . [ eq : emperequation ] from the frequency dependent measurement(black - star ) . they are consistent with each other . in this way we obtain an empirical relation between the frequency - dependent and the electric field - dependent conductivity . the parameter @xmath41 should depend on the diffusion properties of carriers in the swcnt films.@xcite writing @xmath43 , where d is a diffusion constant and @xmath44 is the corresponding time scale , we take @xmath45 . assuming the einstein relation @xmath46 , we can estimate the mobility of carriers on the swcnt network , which is on the order of @xmath47 . this is much smaller than that of the individual single walled carbon nanotube , which can be on the order of @xmath48 at room temperature.@xcite in the swcnt networks , due to the randomly distributed barriers for electrical transport , the mobility is expected to be much smaller than the swcnt itself . the carbon nanotube film - based transistors have reported mobilities on the order of @xmath49,@xcite which is consistent with our estimation . to get a deeper physical understanding of these networks , more investigation and theoretical work are needed . for example , measuring the temperature dependence of the frequency - dependent conductivity for swcnt networks with various densities would be one approach.@xcite to conclude , we systematically studied the frequency and electric field dependent conductivity of swcnt films of various thicknesses . we found the poor interbundle junctions affect the transport behavior of the carriers in the swnct films , which causes strong thickness dependence of characteristic length scales . a thinner film has larger correlation length @xmath19 and length scale @xmath29 , so it has a smaller onset frequency @xmath0 , whereas a thicker film with smaller correlation length @xmath19 and length scale @xmath29 , has a larger onset frequency @xmath0 . an approximate empirical formula relating the onset frequency @xmath0 and nonlinear transport length scale establishes contact between the frequency and the electric field dependent conductivity , which helps us to understand the electric transport properties of the swcnt films . + the authors would like to thank m. s. fuhrer , c. j. lobb , weiqiang yu and bing liang for their help and discussions on this work . this work has been supported by center for nanophysics and advanced materials of the university of maryland and the national science foundation grant nos . ( dmr-0404029 ) , ( dmr-0322844 ) , and ( dmr-0302596 ) .

we present measurements of the frequency and electric field dependent conductivity of single walled carbon nanotube(swcnt ) networks of various densities . the ac conductivity as a function of frequency is consistent with the extended pair approximation model and increases with frequency above an onset frequency @xmath0 which varies over seven decades with a range of film thickness from sub - monolayer to 200 nm . the nonlinear electric field - dependent dc conductivity shows strong dependence on film thickness as well . measurement of the electric field dependence of the resistance r(e ) allows for the determination of a length scale @xmath1 possibly characterizing the distance between tube contacts , which is found to systematically decrease with increasing film thickness . the onset frequency @xmath0 of ac conductivity and the length scale @xmath1 of swcnt networks are found to be correlated , and a physically reasonable empirical formula relating them has been proposed . such studies will help the understanding of transport properties and benefit the applications of this material system .

arxiv

the quantum algorithms work much more efficiently than their classicalcounter parts due to quantum superposition and quantum interference . for example , consider the search of an item in an unsorted database containing @xmath0 elements . classical computation requires @xmath1 steps to carry out the search . however , grover showed that search can be carried out with only @xmath2 steps @xcite . thus , grover s algorithm represents a quadratic advantage over its classical counterpart . grover s algorithm has been realized using many physical systems like nmr @xcite , superconducting qubits @xcite and atom cavity qed systems @xcite . superconducting qubit cavity qed is an attractive approach for quantum information processing due to their strong coupling limit in microwave cavity as compared to atoms in cavity qed @xcite . squids have attracted much attention among the superconducting qubits , due to their design flexibility , large - scale integration , and compatibility to conventional electronics * hanx , ich , mooj*. recently , dicarlo et al . demonstrated the implementation of two - qubit grover and deutsch - jozsa algorithms @xcite and preparation and measurement of three - qubit entanglement @xcite using superconducting qubits . the goal of this work is to implement three - qubit grover s algorithm using four - level squids in cavity - qed . we consider a three - qubit phase gate , that reduces the number of quantum gates typically required for the realization of grover s algorithm . three - qubit grover s algorithm is probabilistic wlyang , as compared to two - qubit grover s algorithm . therefore , to achieve high success probability , we have to implement basic searching iteration several times . implementation of three - qubit grover search is much more complex as compared to two - qubit case . in our scheme , two lowest energy levels @xmath3 and @xmath4 of each squid represent logical states . the scheme is based on resonant , off - resonant interaction of cavity field with @xmath5 transition of squid and application of resonant microwave pulses . our scheme does not require adjustment of squid level spacing during the implementation of grover s search iteration , thus , decoherence caused by the adjustment of level spacing is suppressed . we do not require identical coupling constants of each squid with the resonator and direct coupling between the levels @xmath4 and @xmath6 @xcite . grover s iteration time becomes faster due to resonant and off - resonant interactions as compared to second order detuning or adiabatic passage . grover s iterations based on three - qubit quantum phase gate employed here , considerably simplify the implementation as compared to conventional gate decomposition method @xcite . more importantly , it reduces the possibility of error in comparison with a series of two - qubit gates . we also consider the effect of spontaneous decay rate from intermediate level @xmath7 and decay of cavity field during the implementation of grover s iterations . the basic idea of grover s algorithm is as follows ; we prepare input basis states in superposition state @xmath8 by applying walsh - hadamard transformation . first we , invert phase of the desired basis state through unitary operator ( called oracle ) and then invert all the basis states about the average . we consider the implementation of grover s algorithm in terms of quantum logic networks as shown in fig . 1 . any quantum logical network can be constructed using quantum phase gates and single - qubit quantum gates . the single - qubit quantum gate for @xmath9 qubit can be written in dirac notation as @xmath10for @xmath11 and @xmath12 , we have @xmath13 . here @xmath14 is the pauli rotation matrix whose function is to flip the state of qubit such that @xmath15 and @xmath16 for @xmath17 and @xmath18 , we have @xmath19 which transforms each qubit into superposition state i.e. , @xmath20 and @xmath21 @xmath22 the transformation for three - qubit quantum controlled phase gate can be expressed by @xmath23where @xmath24 , @xmath25 , and @xmath26 stand for basis @xmath6 or @xmath4 of the qubit and @xmath27 , @xmath28 , and @xmath29 are the kroneker delta functions . thus , three - qubit quantum phase gate induces a phase @xmath30 only when all three input qubit are in state @xmath31 . three - qubit quantum phase gate operator for @xmath32 can be written in dirac notation as @xmath33 the three - qubit controlled phase gate can be used instead of involving series of two - qubit gates . this method not only simplifies the implementation but also reduces the probability of error . figure . [ fig1 ] shows the circuit diagram of three - qubit grover s algorithm based on three - qubit phase gate and two - qubit gates @xcite . consider that the initial state of three qubits is @xmath34 . grover s algorithm can be carried out using the following three steps : _ _ part 1 _ _ * * ( w ) * * : apply walsh - hadamard transformation @xmath35 on each qubit . the resultant state is therefore given by @xmath36 _ part 2 _ ( * c * ) : in this step , consider the unitary operator @xmath37 ( called oracle ) which changes the sign of target state @xmath38 . the operator @xmath39 performs the unitary transformation which can be implemented using three - qubit phase gate @xmath40 and single - qubit gate @xmath41 @xmath42 as shown in fig . the sign change operators for eight possible target states are given by @xmath43now oracle applies one of @xmath44 operators on state given in eq . ( [ eq4 ] ) and changes the sign of target state . for example , our target state is @xmath45 , then by applying @xmath46 on state ( [ eq4 ] ) , we obtain the change of phase on target state @xmath47 i.e. , @xmath48 _ part 3 _ ( * n * ) : in this step , our goal is to find out the marked state @xmath45 . this can be accomplished through inversion about mean using the operator @xmath49 . it is clear from fig . fig1 that the operator @xmath50 can be written in terms of single - qubit gate and three qubit quantum phase gate @xmath51the combined operator @xmath52 is called grover s operator . when @xmath53 is applied to initial state @xmath54 @xmath55 @xmath56 times , then the probability of searching target state becomes maximum @xcite . here , we consider rf - squids as qubits that consist of a single josephson junction enclosed by superconducting loop . the corresponding hamiltonian is given by @xcite @xmath57where @xmath58 and @xmath59 are junction capacitance and loop inductance , respectively . the conjugate variables of the system are magnetic flux @xmath60 threading the ring and total charge @xmath61 on capacitor . the static external flux applied to the ring is @xmath62 and @xmath63 is the josephson coupling energy . here , @xmath64 is critical current of josephson junction and @xmath65 is the flux quantum . the squids are biased properly to achieve desired four - level structure as shown in fig . [ fig2 ] by varying the external flux @xcite . the single - mode of the cavity field is resonant with @xmath66 transition of squids 1 and 2 . the evolution of initial state @xmath67 and @xmath68 under the effect of hamiltonian ( [ eq8 ] ) can be written as @xcite @xmath69where @xmath70 ( @xmath71 ) is vacuum ( single photon ) state of the cavity field and @xmath72 ( @xmath73 ) is the coupling constant between the cavity field and @xmath74 transition of the squid @xmath75 and @xmath76 . the cavity field interacts off - resonantly with @xmath66 transition of squid 3 ( i.e. , @xmath77 as shown in fig . [ fig2 ] . here , @xmath78 is the detuning between @xmath66 transition frequency @xmath79 of squid 3 , @xmath80 is the frequency of resonator and @xmath81 is coupling constant between resonator mode and @xmath66 transition . in the presence of single photon inside the cavity , the evolution of initial state @xmath82 and @xmath83 are given by @xcite @xmath84it is clear that phase shifts @xmath85 and @xmath86 are induced to states @xmath87 and @xmath88 of squid 3 , respectively . however , states @xmath89 and @xmath90 remain unchanged . a classical microwave pulse resonant with @xmath91 ( @xmath92 ) is applied to each squid . the evolution of states can be written as @xcite @xmath93where @xmath94 is the rabi frequency between two levels @xmath95 and @xmath96 and @xmath97 is the phase associated with classical field . it may be mentioned that resonant interaction between pulse and squid can be carried out in a very short time by increasing the rabi frequency of pulse . three squids shown in fig . [ fig2 ] are initially prepared in state @xmath98 . for notation convenience , we denote ground level as @xmath4 and first excited state as @xmath6 for squid @xmath99 as shown in fig . [ fig2 ] . _ _ part 1 _ _ * * ( w ) * * : _ _ _ _ to accomplish _ _ part 1__of grover s algorithm , we apply single - qubit gate @xmath100 to each squid as shown in fig . the single - qubit gate @xmath101 is carried out through two - step process that involves an auxiliary level @xmath102 via method described in ref @xcite . we need two microwave pulses of different frequencies resonant to @xmath103 and @xmath104 transitions . the desired arbitrary single - qubit gate ( see eq . ( 1 ) ) can be achieved by choosing a proper interaction time ( i.e. , @xmath105 ) and phase @xmath97 of classical microwave pulse resonance to @xmath104 transition . we achieve @xmath101 by choosing @xmath106 and @xmath107 as a result we obtain state given by eq . ( [ eq4 ] ) . _ part 2 _ ( * c * ) * : * in order to implement @xmath108 apply single - qubit gate @xmath41 @xmath109 by choosing @xmath110 and @xmath12 as shown in fig . [ fig1 ] . then apply three - qubit quantum controlled phase gate . the procedure for realizing the three - qubit controlled phase gate is described as follows : initially , cavity is in a vacuum state @xmath70 and levels @xmath111 and @xmath88 of each squid are not occupied . _ step 1 . _ apply microwave pulse ( with @xmath112 and phase @xmath113 , where @xmath114 is pulse duration ) resonant to @xmath115 transition of the squid @xmath75 to occupy level @xmath116 cavity field interacts resonantly to the @xmath117 transition of squid @xmath75 for time interval @xmath118 such that the transformation @xmath119 is obtained . the overall step can be written as @xmath120 . however , the state @xmath121 remains unchanged . _ apply microwave pulse ( with @xmath122 and phase @xmath123 ) to the squid 1 while a microwave pulse ( with @xmath124 and phase @xmath18 ) to the squid 2 . as a result transformations @xmath125 for squid 1 while @xmath126 for squid 2 are obtained . _ after the above operation , when cavity is in a single photon state @xmath127 , only the level @xmath87 of squid 2 is populated . the cavity field interacts resonantly to @xmath128 transition of squid 2 for time @xmath129 we then obtain transformation @xmath130 while states @xmath131 , @xmath132 and , @xmath133 remain unchanged . then apply microwave pulse ( with @xmath134 and phase @xmath113 ) to squid 2 to transform state @xmath135 to @xmath136 . the overall step can be written as @xmath137 . however , states @xmath138 , @xmath139 and @xmath140 remain unchanged . the evolution of the system after above three steps is given by @xmath141 it must be noted here that we have shown the evolution of only four initial possible states out of eight states ( see eq . ( 4 ) ) since the evolution of other four states i.e. , @xmath142 and @xmath143 is trivial . _ step 4 . _ apply microwave pulse with @xmath144 and phase @xmath18 to squid 3 to obtain transformation @xmath145 @xmath146 . after the above operation only level @xmath111 of squid ( 3 ) is populated , when cavity is in a single photon state . now the cavity field interacts off - resonantly to @xmath147 transition of squid 3 . it is clear from eq . ( [ eq10 ] ) that for @xmath148 , state @xmath149 changes to @xmath150 . however , states @xmath151 , @xmath152 and @xmath153 remain unchanged . _ step 5 . _ apply microwave pulse ( with @xmath144 and phase @xmath123 ) to squid 3 , as a result state transformation @xmath154 is obtained . _ perform reverse of the operations mentioned in step 3 . apply microwave pulse ( with @xmath134 and phase @xmath155 ) to squid 2 . wait for time @xmath156 given in step 3 , during which cavity field interacts resonantly to @xmath157 transition of the squid 2 . over all transformation can easily be written as @xmath158 . however , states @xmath159 , @xmath139 and @xmath140 remain unchanged . the evolution of the system after applying steps 4 - 6 is given by @xmath160 _ step 7 . _ apply microwave pulse ( with @xmath122 and phase @xmath123 ) to squid 2 while a microwave pulse ( with @xmath161 and phase @xmath18 ) to squid 1 . the transformations @xmath162 for squid 2 while @xmath163 for squid 1 are obtained . _ now perform reverse operation of step 1 . first wait for time interval @xmath118 during which resonator interacts resonantly to the @xmath164 transition of squid 1 . then apply microwave pulse ( with @xmath165 and phase @xmath113 ) to squid 1 . the overall step can easily be written as @xmath166 . however , state @xmath167 remains unchanged . after applying steps 7 - 8 , the system evolves as @xmath168 after the application of three - qubit phase gate , the state given by eq . ( eq4 ) evolves to @xmath169 next apply single - qubit gate i.e. , @xmath41 @xmath170 which completes the part 2 ( c ) of the implementation scheme . _ part 3 _ ( * n * ) : in order to implement operator @xmath50 , apply single - qubit gate @xmath171 , then apply three - qubit quantum controlled phase gate by repeating the above mentioned 8 steps . finally , apply single - qubit hadamard gate @xmath172 . it is clear that grover s operator @xmath173 for eight objects ( @xmath174 ) can be implemented using four - level squids coupled to superconducting resonator . it may be pointed out that in order to implement a three - qubit phase gate using conventional decomposition method , it requires twenty five basic gates , i.e. , six two - qubit phase gates , twelve single - qubit hadamard gates , and seven single - qubit phase shift gates @xcite . if we assume that the realization of each basic gate requires a one - step operation only , then twenty five steps are required for a three - qubit phase gate . whereas in our scheme total number of eight steps are needed to implement three - qubit phase gate . as a result our proposed implementation scheme for grover s search algorithm which is based on three - qubit phase gate is faster than one based on two - qubit phase gate . after performing the required gate operations for grover s algorithm , we need to readout the computational results . this can be done by jointly detecting the states of the three qubits @xcite . the readout for flux qubit can be done by measuring the josephson inductance of a squid that is inductively coupled to the qubit @xcite . there are some interesting schemes to perform joint dispersive readouts for superconducting qubits @xcite . the implementation of our scheme also requires such joint dispersive readout for three squids . here , we discuss different types of imperfections which can arise during the implementation of grover s algorithm . the relevant parameters during the implementation are coupling constant @xmath175 decay rate @xmath176 of level @xmath88 and cavity decay rate @xmath177 we consider spontaneous decay from the intermediate level @xmath102 , during resonant interaction of cavity field with @xmath178 transition of squids 1 and 2 . here , we assume that under the condition that no photon from spontaneous emission is detected , the conditional hamiltonian for the evolution of system is given by @xcite @xmath179 the hamiltonian given by eq . ( [ eq13 ] ) is valid as long as the decay rate of level @xmath111 is much smaller than @xmath180 the decay rate @xmath176 of squid 1 and 2 can affect the performance of three - qubit phase gate . suppose each squid is initially prepare in generic state @xmath181 now the state of three qubits becomes @xmath182 where the coefficients @xmath183 @xmath184 @xmath185 and @xmath186 , satisfy the normalization condition . if we consider @xmath187 , then the state of the system after phase gate operation becomes @xmath188 however , if the decay of level @xmath102 is included during phase gate then the expression for @xmath189 becomes rather complex , therefore it is not reproduced here . the fidelity of three - qubit phase gate is given by @xmath190 here , @xmath191 with @xmath192 , @xmath193 , and @xmath194 in order to realize the effects of decay on the performance of three - qubit phase gate , average fidelity over all possible three - qubit initial states is calculated using the following : @xmath195 after carrying out integration , we obtain @xmath196 next , we show the plots of average fidelity as a function of @xmath197 in fig . it can easily be verified that for @xmath198 one has @xmath199 which leads to @xmath200 it is clear from fig . [ fig3 ] that average fidelity decreases due to the increase in the cavity decay rate . we also calculate the success probability of three - qubit phase gate which is given by @xmath201 if we consider @xmath202 then corresponding success probability of three - qubit phase gate reduces to @xmath203 of three - qubit controlled phase gate as a function of coupling constant @xmath204 , width=3 ] we take into account success rate of three - qubit phase gate ( eq . [ eq14 ] ) and grover search itself . we consider @xmath205 without loss of generality and perform simulation for finding the target state for @xmath206 ( ideal case ) , @xmath207 , and @xmath208 here , we consider the success probability for target state @xmath209 as shown in fig . [ fig4](a ) . it is clear from fig . fig4(a ) that success rate becomes closer to the ideal case when decay rate is sufficiently smaller than coupling constant i.e. , @xmath210 typical decoherence rate for squid is of the order @xmath211 while coupling constant can be achieved upto @xmath212 @xcite . this shows that for these parameters , three - qubit controlled phase gate and grover s iterations can be performed with high fidelity . probability of success is highest at @xmath213 iteration for ideal case . however , we should prefer @xmath214 iteration in the presence of dissipation because it has the highest value of fidelity . the effect of level decay on the fidelity of search state during iteration is shown in fig . [ fig4](b ) . it is clear from fig . [ fig4](b ) that the fidelity of state to be searched decreases much rapidly for the case of larger decay rate as compared to smaller decay rate as the number of iterations increase . from intermediate level @xmath215 ( a ) probability of finding the target state @xmath216 in case of @xmath217 , @xmath207 , and @xmath208 ( b ) fidelity of the state searched for @xmath218 and @xmath208,width=3 ] next we consider the effects of cavity decay during the implementation of grover s algorithm . during the implementation of three - qubit phase gate ( sec . iv ) , transition of squid 1 from level @xmath88 to @xmath219 would result in the emission of one photon ( see step @xmath75 in sec . then transition of squid 2 from level @xmath87 to @xmath88 would absorb this photon with unit probability in step @xmath99 and vice versa for step @xmath220 and @xmath221 in the absence of cavity decay the occupation probability of level @xmath87 and @xmath88 of squid 1 and 2 should be exactly one . however , if cavity relaxation is taken into account , then the occupation probabilities are expected to decay exponentially . under the assumption that no photon actually leaks out during implementation time , we can write the conditional hamiltonian as @xcite @xmath222 where , @xmath223 is the cavity decay rate and @xmath72 is the coupling constant . for @xmath224 we only need to focus on time evolution of the system governed by conditional hamiltonian ( eq . ( [ ch ] ) ) under the assumption of strong coupling limit . the fidelity and corresponding success probabilities for three - qubit phase gate can easily be obtained , which are given by @xmath225 and @xmath226 for @xmath227 , we have @xmath228 the success probability of target state @xmath229 for ideal case and in the presence of cavity decay is shown in fig . [ fig5](a ) . it can be seen that the probability of finding target state decreases in the presence of cavity decay rate . the effects of cavity decay rate on fidelity of grover s search iteration is also shown in fig . [ fig5](b ) . it is clear from fig . fig5(b ) that the fidelity of grover s search decreases as a function of iterations much rapidly for higher cavity decay rate . ( a ) probability of finding the target state @xmath216 in case of @xmath230 , @xmath231 and @xmath232 ( b ) fidelity of the state searched for @xmath233 and @xmath232,width=3 ] we have separately discussed the effects on fidelity from spontaneous decay and cavity decay , however , in a general case the system involves both these decays . thus we also consider the effect of these decays simultaneously . the results of our numerical simulation for corresponding average fidelity are shown in fig . it is clear from figs . [ fig4]-[fig6 ] that our proposed grover s iterations can be performed with high success probability and fidelity as long as cavity decay rate and decay rate of the intermediate level @xmath88 is small enough . the typical values of cavity decay rate is @xmath234 ( @xmath235 ) @xcite . it may be mentioned that more rigorous analysis is required for the case of very low @xmath236 resonators . of three - qubit controlled phase gate as a function of cavity and level decay , width=3 ] we have discussed the implementation of 3-qubit grover s algorithm using 4-level squids subjected to quantized and classical microwave fields . grover s algorithm involves three - qubit phase gates and single - qubit gates . here , we briefly estimate the total operational time for three - qubit controlled phase gate . the total implementation time is the sum of all interaction times involved in three - qubit controlled phase gate operation i.e. , @xmath237 on substituting the values of interaction times given in sec . @xmath238 we obtain @xmath239here , we consider without loss of generality @xmath240 which is the same as given in ref han . choosing @xmath241 , @xmath242 , the operational time for three - qubit phase gate comes out to be @xmath243 . the operation time for single - qubit gate is about @xmath244 @xcite which can be applied to each qubit , simultaneously . therefore , the estimated time for the implementation of three - qubit grover algorithm performing two iterations is approximately @xmath245 . we have considered the effect of decay for level @xmath246 under the assumption that the decay rate of level @xmath247 is much smaller then level @xmath246 . the typical values of the decay time for levels @xmath246 and @xmath248 are @xmath249 and @xmath250 0.16 ms as discussed in ref . @xcite . during the steps 1,2 , and 3 in phase gate operations level @xmath247 of squids 1 and 2 is occupied through the application of squid - pulse resonant interaction and squid - resonator resonant interaction as discussed in sec . operation time @xmath251 for squid 1 and @xmath252 for squid 2 , in these steps is equal to @xmath253 and @xmath254 , respectively , which can be shortened by increasing the rabi frequency @xmath255 and coupling constants @xmath256 and @xmath257 . for the typical choice of parameters as given in ref @xcite , we have @xmath258 , which is much shorter than @xmath259 . the squids can also be designed to have long relaxation time for level @xmath247 . thus decoherence due to relaxation of level @xmath111 can be negligibly small . as far as the decay of level @xmath260 is concerned , it may be pointed out that in our scheme direct coupling between levels @xmath261 and @xmath262 is not needed . the potential barrier between levels @xmath260 and @xmath262 can also be adjusted such that decay of level @xmath260 is negligibly small @xcite . therefore , storage time of each qubit can be made much longer . when levels @xmath111 and @xmath88 are manipulated by microwave pulses , resonant interaction between cavity field mode and @xmath263 transition of each control squid is unwanted . this effect can be minimized by setting the condition @xmath264 for squids 1 and 2 . during the application of microwave pulse , off - resonant interaction of cavity field with @xmath263 transition , in the presence of single photon inside the cavity is unwanted . it induces an unwanted phase which can effect the performance of the desired gate . this effect can be negligible under the condition @xmath265 for squid @xmath99 . however , if this effect is included during the steps @xmath266 and @xmath267 for three - qubit phase gate , then the corresponding fidelity can easily be obtained . here , we do not present the expression for @xmath268 owing to its complexity , however the fidelity is given by if off - resonant interaction during step @xmath266 and @xmath267 is not considered , then we have @xmath277 @xmath278 and@xmath279 which leads to @xmath280 the plot of the average fidelity as a function of rabi frequency @xmath281 is shown in fig . [ fig7 ] . we choose @xmath282 for this plot . it can be seen that the average fidelity increases as a function of rabi frequency @xmath283 applied to target squid . @xmath284it is clear from fig . [ fig7 ] that for @xmath285 , we have @xmath286 when slowly changing rabi frequencies are applied to satisfy the adiabatic passage , gate times becomes slow i.e. , of the order of 1 ms to a few microseconds @xcite . however , in our scheme we do not require slowly changing rabi frequency during the implementation of three - qubit phase gate , thus gate is significantly faster i.e. , of the order of 13ns . the fast pulses may introduce new imperfections , for example , accurately designing of the duration of pulses might not be easy . it may be mentioned here that adiabatic process is not always slow as an interesting proposal based on controllable stark - chirped rapid adiabatic passage has been proposed in a recent study @xcite . here , we would like to mention that the physical implementation of three - qubit grover s algorithm in cavity quantum electrodynamics ( qed ) has been proposed in a recent study by yang _ et al _ @xcite . the scheme is based upon the resonant interaction of three rydberg atoms initially prepared in a coherent superposition state traversing through a single - mode microwave cavity . as compared to the flying qubits , here we have considered stationary qubits defined through two lowest level of four level squids . our scheme is based on the generation and absorption of single photon in high q cavity using squids . the generation of single microwave photon in superconducting qubit have been reported , experimentally in some recent studies _ et al_. @xcite . in conclusion , we have proposed a scheme for the realization of three - qubit grover s algorithm based on three - qubit phase gate using four - level squids coupled to a single - mode superconducting resonator . in this proposal , adjustment of level spacing during the operation , slowly changing rabi frequencies ( to satisfy adiabatic passage ) , and the use of second - order detuning ( to achieve off - resonance raman coupling between two relevant levels ) are not required . thus , implementation time is significantly faster and has operation time of the order of nanoseconds . the coupling constants of each squid with the resonator are different . thus , an unavoidable non - uniformity in device parameters can be accommodated . we consider the effect of imperfections in the system which include the decay of the cavity field and the relevant level . we also incorporate the influence of unwanted off - resonant interaction during the gate implementation . our results show that the marked state can be searched with high fidelity even in the presence of imperfections in the system which is quite interesting . yang , c. p. , chu , shih - i . , han , s. : quantum information transfer and entanglement with squid qubits in cavity qed : a dark - state scheme with tolerance for nonuniform device parameter . phys . lett . * 92*(11 ) , 117902 ( 2004 ) ; leek , p. j. , filipp , s. , maurer , p. , baur , m. , bianchetti , r. , fink , j. m. , gppl , m. , steffen , l. , wallraff , a. : using sideband transitions for two - qubit operations in superconducting circuits . b * 79*(18 ) , 180511(r ) ( 2009 ) . wallraff , a. , schuster , d. i. , blais , a. , frunzio , l. , huang , r .- s . , majer , j. , kumar , s. , girvin , s. m. , schoelkopf , r. j.:strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics . nature ( london ) * 431 * , 162 - 167 ( 2004 ) . dicarlo , l. , chow , j. m. , gambetta , j. m. , bishop , l. s. , johnson , b. r. , schuster , d. i. , majer , j. , blais , a. , frunzio , l. , girvin , s. m. , schoelkopf , r. j. : demonstration of two - qubit algorithms with a superconducting quantum processor . nature , * 460 * , 240 - 244 ( 2009 ) . dicarlo , l. , reed , m. d. , sun , l. , johnson , b. r. , chow , j. m. , gambetta , j. m. , frunzio , l. , girvin , s. m. , devoret , m. h. , and schoelkopf , r. j. : preparation and measurement of three - qubit entanglement in a superconducting circuit . nature * 467 * , 574 - 578 ( 2010 ) . he , xiao - ling . , yang , c.p . , li , s. , luo , j. y. , han , s. : quantum logical gates with four - level superconducting quantum interference devices coupled to a superconducting resonator . rev . a * 82*(2),024301 ( 2010 ) . yang , c. p. , chu , shih - i . , han , s. : possible realization of entanglement , logical gates , and quantum - information transfer with superconducting - quantum - interference - device qubits in cavity qed . phys . a * 67*(4 ) , 042311 ( 2003 ) . blais , a. , huang , r. -s . , wallraff , a. , girvin , s. m. , schoelkopf , r. j. : cavity quantum electrodynamics for superconducting electrical circuits : an architecture for quantum computation . a * 69*(6 ) , 062320 ( 2004 ) . filipp , s. , maurer , p. , leek , p. j. , baur , m. , bianchetti , r. , fink , j. m. , gppl , m. , steffen , l. , gambetta , j. m. , blais , a. , wallraff , a. : two - qubit state tomography using a joint dispersive readout . lett . * 102*(20 ) , 200402 ( 2009 ) . zhang , p. , wang , z. d. , sun , j. d. , sun , c. p. : holonomic quantum computation using rf superconducting quantum interference devices coupled through a microwave cavity . a * 71*(4 ) , 042301 ( 2005 ) . wang , h. , hofheinz , m. , ansmann , m. , bialczak , r. c. , lucero , e. , neeley , m. , oconnell , a. d. , sank , d. , wenner , j. , cleland , a. n. , martinis , j. m. : measurement of the decay of fock states in a superconducting quantum circuit . lett . * 101*(24 ) , 240401 ( 2008 ) .

we present a scheme for the implementation of three - qubit grover s algorithm using four - level superconducting quantum interference devices ( squids ) coupled to a superconducting resonator . the scheme is based on resonant , off - resonant interaction of the cavity field with squids and application of classical microwave pulses . we show that adjustment of squid level spacings during the gate operations , adiabatic passage , and second - order detuning are not required that leads to faster implementation . we also show that the marked state can be searched with high fidelity even in the presence of unwanted off - resonant interactions , level decay , and cavity dissipation . keywords : quantum phase gate ; grover s algorithm ; superconducting quantum interference devices ( squids )

arxiv

in the 1970s many - body theory flourished . a topic that attracted much attention was the many - body effects on xps ( x - ray photoelectron spectroscopy ) spectra , from deep core levels in metals@xcite . related effects were singularities that appeared near edges in absorption and emission spectra@xcite . the cb ( conduction - band ) electrons can have many different effects on the xps peak from the core level . if the excitation could be considered adiabatic the screening of the core hole by the cb electrons would lead to a shift of the peak to higher kinetic energy ( lower binding energy of the core level ) . now the excitation is not adiabatic , the excitation is swift . the excitation frequency is large compared to the frequency components taking part in the screening of the core hole . this has the effect that the cb electrons are shaken up and are left in a nonequilibrium state . this is analogous to the following case : assume that a cork is floating in a glass of water . if we remove the cork very slowly the water will be left very quiet and without any ripples . if we on the other hand remove it very briskly the water will be left in an upset state with many surface waves excited . in a metal the water waves correspond to plasmons , collective excitations of the cb electrons . the shake - up effects will show up as plasmon replicas in the spectrum ; the main peak will be at the adiabatic position ; the first replica , a smaller copy of the main peak , will be at a shifted position to lower kinetic energy where the shift is equal to the plasma frequency ; the second replica is even smaller and shifted with two plasma frequencies ; this goes on until further peaks are absorbed by the background . the main peak corresponds to the system left in quasi equilibrium(fully equilibrium would demand that the core hole were filled by one of the electrons ) , the second to a state where one plasmon is excited , the third to a state where two plasmons are excited and so on . since all plasmons do not have identical energies , the plasmon curve shows dispersion , the replicas are not completely identical smaller versions of the main peak . in a metal also single - particle excitations can take place . these electron - hole pair excitations form a continuum , starting from zero frequency and upwards . these excitations lead to a deformation , including a low - energy tail , of both the main peak and the plasmon replicas . furthermore the finite life - time of the core hole causes a lorentzian broadening of all peaks and experimental uncertainties give a gaussian broadening . the low - energy tailing is a characteristic of a metallic system , i.e. a system where the chemical potential is inside an energy band and not in a band - gap , and can e.g. be used by experimentalists to find out where in a complex sample the core hole is situated . several parameters were introduced by doniach and sunji@xcite to characterize the xps line shapes and are still broadly used by experimentalists in the fitting procedure for xps spectra . in the present work we address pristine and doped graphene . in the pristine case the chemical potential is neither inside an energy band nor in a band - gap . the fermi surface is just two points in the brillouin zone . this makes this system special . as we will see there is still a low - energy tailing . in the doped case the chemical potential is inside an energy band and we would expect to find , and find , a tailing . however , the 2d ( two - dimensional ) character of the system means that the collective excitations are 2d plasmons with a completely different dispersion than in the ordinary 3d metallic systems . the 2d plasmons give contributions that start already from zero frequency and upwards . this means that they contribute to the tail and no distinct plasmon replicas are distinguishable . making any quantitative interpretation of graphene core - hole spectra using doniach and sunji fitting of the spectra is not feasible . it is more reasonable to calculate the spectra . one purpose of this work is to provide the reader with the tools needed for such a calculation . the material is organized in the following way : in sec . [ the ] we show how the core - hole spectra are derived for a 2d system . the results and comparison with experimental spectra are presented in sec . finally , we end with a brief summary and conclusion section , sec . we use a model that is based on the one used by langreth@xcite for the core - hole problem in the 1970s and here modified and extended to fit our problem . we have earlier@xcite with success used another modified version for the problem of exciton annihilation in quantum wells . in the excitation process the photoelectron leaves the system and a core hole is left behind . the shape of the xps spectrum depends on how fast the process is . if it is very slow one may use the adiabatic approximation in which one assumes that the electrons in the system have time to relax around the core hole during the process . when we derive the xps line shape we assume that the excitation process is very fast ; we use the sudden approximation in which the core - hole potential is turned on instantaneously . the electrons have not time , during the process , to settle down and reach equilibrium in the potential from the core hole . this results in shake - up effects in the form of single particle ( electron - hole pair ) excitations and collective ( plasmon ) excitations . the electrons contributing to the shake - up effects are the electrons in the valence and conduction bands . from here on we refer to them as the electrons . we use the assumption that the core hole does not recoil in the shake - up process and that there are no excitations within the core . we approximate the core - hole potential with a pure coulomb potential . the following model hamiltonian for the system is used : @xmath0 where @xmath1 , @xmath2 , @xmath3 , @xmath4 , and @xmath5 are the core - hole energy , the creation operator for the core hole , the annihilation operator for the core hole , the interaction potential for the interaction between the core hole and the electrons , and the hamiltonian for the electrons , respectively . the operators @xmath4 and @xmath5 contain creation and annihilation operators for the electrons and no core - hole operators . the core - hole number operator is @xmath6 . it commutes with the hamiltonian , i.e. , @xmath7 = 0 $ ] . in the ground state of the system there is no core hole and the electrons are in their ground state . let us introduce two hamiltonians @xmath8 where the first is the hamiltonian before the core hole has been created and the second the hamiltonian after the creation . let @xmath9 denote the ground state of @xmath10 . it is also the ground state of @xmath11 . then we have @xmath12 where @xmath13 is the ground state energy of the electron system . from now on we drop the subscript @xmath14 on the core - hole operators and introduce the functions @xmath15 these functions are connected to the time ordered and retarded green s functions , according to @xmath16 and @xmath17\left| 0 \right\rangle \\ \quad \quad = - i\theta \left ( t \right)\left [ { { g^ > } ( t ) - { g^ < } ( t ) } \right ] . \end{array } \label{equ6}\ ] ] the fourier transformed versions , @xmath18 have a direct physical meaning . their sum is the spectral function , @xmath19 is the density of states for an occupied core - hole state , and @xmath20 for an unoccupied . note that the creation and annihilation operators are for holes . we want the density of states for a core electron . thus we need to calculate @xmath20 . we first determine the time dependent form . we have @xmath21t/\hbar } } { c^\dag } \left| 0 \right\rangle \\ = { e^{ie_0^et/\hbar } } \left\langle 0 \right|c{e^ { - i\varepsilon nt/\hbar } } { e^ { - i\left [ { nv + { h_e } } \right]t/\hbar } } { c^\dag } \left| 0 \right\rangle \\ = { e^{i\left ( { e_0^e - \varepsilon } \right)t/\hbar } } \left\langle 0 \right|c{e^ { - i\left [ { v + { h_e } } \right]t/\hbar } } { c^\dag } \left| 0 \right\rangle \\ = { e^{i\left ( { e_0^e - \varepsilon } \right)t/\hbar } } { } _ e\left\langle 0 \right|{e^ { - i\left [ { v + { h_e } } \right]t/\hbar } } { \left| 0 \right\rangle _ e}{}_{ch}\left\langle 0 \right|1 - n{\left| 0 \right\rangle _ { ch}}\\ = { e^{i\left ( { e_0^e - \varepsilon } \right)t/\hbar } } \left\langle 0 \right|{e^ { - i\left [ { v + { h_e } } \right]t/\hbar } } \left| 0 \right\rangle , \end{array } \label{equ8}\ ] ] where we have let the operators operate to the left and right . we have made use of the general relation @xmath22}}$ ] , for operators @xmath23 and @xmath24 , and that the core - hole number operator commutes with all terms of the hamiltonian . it is now time to define our @xmath4 and the rest of the hamiltonian . the excitation spectrum of the electrons is given by the dynamical structure factor , which is related to the dielectric function according to @xmath25 where @xmath26 is the 2d fourier transform of the coulomb potential . the structure factor is nonzero in two regions ; one region is where electron - hole pairs are excited ; the other is on the plasmon dispersion curve . all these excitations are bosons . we assume that the bosons are independent . let us for simplicity start by assuming that the bosons have a distinct dispersion curve , @xmath27 . this is the case for plasmons and phonons . later we will generalize this to include contributions from the electron - hole pair continuum . we now get the following hamiltonian : @xmath28 where @xmath23 is the area of the 2d system and @xmath29 the coupling constant between the core hole and the boson excitation . the operators @xmath30 and @xmath31 are boson creation and annihilation operators , respectively , obeying the commutation relation @xmath32 = { \delta _ { { \bf{q}},{\bf{q}}'}}$ ] and @xmath33 are the density operators , where @xmath34 denotes the position of the core hole . we treat the core hole as a classical particle , i.e. , the core - hole operators are c - numbers . this is why the density operators do not contain any creation and/or annihilation operators . let us here take the opportunity to derive the results using the adiabatic approximation . since there is no kinetic energy terms for the core hole the hamiltonian can be diagonalized . this is achieved by using the following unitary transformation : @xmath35 , \label{equ12}\ ] ] where @xmath36 letting @xmath37 we find that @xmath38 = { f^\dag } ( - { \bf{q } } ) = f({\bf{q}})$ ] and @xmath39 = { f^\dag } ( { \bf{q}})$ ] . this gives @xmath40 the transformation of the hamiltonian gives @xmath41 } \\ - \frac{1}{{{a^{{1 \mathord{\left/ { \vphantom { 1 2 } } \right . \kern-\nulldelimiterspace } 2}}}}}\sum\limits_{\bf{q } } { g({\bf{q}})\rho _ { ch}^\dag ( { \bf{q } } ) } \left ( { { c_{\bf{q } } } + f({\bf{q } } ) + c _ { - { \bf{q}}}^\dag + { f^\dag } ( - { \bf{q } } ) } \right)\\ = \sum\limits_{\bf{q } } { \hbar { \omega _ { \bf{q}}}\left ( { c_{\bf{q}}^\dag { c_{\bf{q } } } + { 1 \mathord{\left/ { \vphantom { 1 2 } } \right . \kern-\nulldelimiterspace } 2 } } \right ) } - \frac{1}{{{a^{{1 \mathord{\left/ { \vphantom { 1 2 } } \right . \kern-\nulldelimiterspace } 2}}}}}\sum\limits_{\bf{q } } { g({\bf{q}})\rho _ { ch}^\dag ( { \bf{q } } ) } \left ( { { c_{\bf{q } } } + c _ { - { \bf{q}}}^\dag } \right)\\ + \sum\limits_{\bf{q } } { \hbar { \omega _ { \bf{q}}}\left ( { c_{\bf{q}}^\dag f({\bf{q } } ) + { c_{\bf{q}}}{f^\dag } ( { \bf{q } } ) + { f^\dag } ( { \bf{q}})f({\bf{q } } ) } \right ) } \\ - \frac{2}{{{a^{{1 \mathord{\left/ { \vphantom { 1 2 } } \right . \kern-\nulldelimiterspace } 2}}}}}\sum\limits_{\bf{q } } { g({\bf{q}})\rho _ { ch}^\dag ( { \bf{q } } ) } f({\bf{q } } ) . \end{array } \label{equ16}\ ] ] substituting for the expression of @xmath42 we find @xmath43 thus @xmath44 where @xmath45 we see that the transformation of the hamiltonian in eq.([equ10 ] ) has the effect that the interaction term is changed into a constant energy term . the interactions with the core hole produce an energy shift of the ground state . this is the relaxation energy ; the gain in energy when the electrons relax around the core hole . this is the shift of the xps peak ( towards higher kinetic energy ) one would get in the adiabatic approximation . we now return to the sudden approximation and our green s function @xmath46{t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } } \left| 0 \right\rangle $ ] . let us insert the identity on both sides of the exponential inside the ground state matrix element . @xmath47{t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } } u\left| 0 \right\rangle . \end{array } \label{equ20}\ ] ] now , since there is no boson excited in the ground state we may write @xmath48{t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } } \left| 0 \right\rangle = { e^{i\left ( { \sum\limits_{\bf{q } } { { \textstyle{1 \over 2}}\hbar { \omega _ { \bf{q } } } } } \right){t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } } \left| 0 \right\rangle \\ = { e^{ie_0^h{t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } } \left| 0 \right\rangle . \end{array } \label{equ21}\ ] ] thus we have @xmath49{t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } } u{e^{i\left [ { \sum\limits_{\bf{q } } { \hbar { \omega _ { \bf{q}}}\left ( { c_{\bf{q}}^\dag { c_{\bf{q } } } + { 1 \mathord{\left/ { \vphantom { 1 2 } } \right . \kern-\nulldelimiterspace } 2 } } \right ) } } \right]{t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } } \left| 0 \right\rangle \\ = { e^ { - i\left ( { \varepsilon + \delta \varepsilon } \right){t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } } \left\langle 0 \right|{u^\dag } \left ( 0 \right)u\left ( t \right)\left| 0 \right\rangle , \end{array } \label{equ22}\ ] ] where @xmath50 and we see that @xmath51 is also a unitary transformation . now , from eqs.([equ12 ] ) and ( [ equ23 ] ) we find @xmath52\;\\ = \exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } ( { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } - c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } ) } \right]\\ = \exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } ( - c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } + { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } ) } \right ] . \end{array } \label{equ24}\ ] ] if we now once again make use of the relation @xmath22}}$ ] we find @xmath53\exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } } \right]\\ \quad \quad \times \exp \left [ { { \textstyle{1 \over 2}}\sum\limits_{{\bf{k}},{\bf{l } } } { { f^\dag } ( { \bf{l } } ) } { f^\dag } ( { \bf{k}}){e^ { - i{\omega _ { \bf{l}}}t}}{e^{i{\omega _ { \bf{k}}}t}}\underbrace { \left [ { c _ { - { \bf{l}}}^\dag , { c_{\bf{k } } } } \right ] } _ { - { \delta _ { { \bf{k } } , - { \bf{l } } } } } } \right]\\ = \exp \left [ { - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right]\exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } } \right]\\ \quad \quad \times \exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { f^\dag } ( - { \bf{q } } ) } \right]\\ = \exp \left [ { - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right]\exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } } \right]\\ \quad \quad \times \exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } f({\bf{q } } ) } \right]\\ = \exp \left [ { - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right]\exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } } \right]\\ \quad \quad \times \exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right ] . \end{array } \label{equ25}\ ] ] we also need @xmath54\exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\\ \quad \quad \times \exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right ] . \end{array } \label{equ26}\ ] ] substituting the results of eqs.([equ25 ] ) and ( [ equ26 ] ) in eq.([equ22 ] ) gives @xmath55\exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\\ \quad \quad \times \exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\exp \left [ { - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right]\\ \quad \quad \times \exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } } \right]\exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\left| 0 \right\rangle . \end{array } \label{equ27}\ ] ] the factors that do not contain boson operators can be taken outside . then we can let the left most factor operate to the left . it produces a factor of unity . the rightmost factor may operate to the right and it also produces a factor of unity . we are left with @xmath56 here we make use of the relation @xmath57}}$ ] and find @xmath58\exp \left [ { - \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\\ \times \left\langle 0 \right|\exp \left [ { - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right]\\ \times \exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\\ \times \exp \left\ { { \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } , - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right ] } \right\}\left| 0 \right\rangle \\ = \exp \left [ { - i\left ( { \varepsilon + \delta \varepsilon } \right){t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } \right]\exp \left [ { - \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\\ \times \left\langle 0 \right|\exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\\ \times \exp \left\ { { \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } , - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right ] } \right\}\left| 0 \right\rangle \\ = \exp \left [ { - i\left ( { \varepsilon + \delta \varepsilon } \right){t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } \right]\exp \left [ { - \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\\ \times \left\langle 0 \right|\exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\\ \times \exp \left\ { { \sum\limits_{{\bf{k}},{\bf{l } } } { f({\bf{k}}){f^\dag } ( { \bf{l}}){e^ { - i{\omega _ { \bf{l}}}t } } } \underbrace { \left [ { { c _ { - { \bf{k}}}},c _ { - { \bf{l}}}^\dag } \right]}_{{\delta _ { { \bf{k}},{\bf{l } } } } } } \right\}\left| 0 \right\rangle \\ = \exp \left [ { - i\left ( { \varepsilon + \delta \varepsilon } \right){t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } \right]\exp \left [ { - \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\\ \times \exp \left [ { \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2}{e^ { - i{\omega _ { \bf{q}}}t } } } } \right]\\ \times \left\langle 0 \right|\exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\left| 0 \right\rangle \\ = { e^ { - i\left ( { \varepsilon + \delta \varepsilon } \right){t \mathord{\left/ { \vphantom { t \hbar } } \right . \kern-\nulldelimiterspace } \hbar } } } { e^ { - \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } } { e^{\sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2}{e^ { - i{\omega _ { \bf{q}}}t } } } } } . \end{array } \label{equ29}\ ] ] thus we have @xmath59 we want the fourier transformed version . it is @xmath60 where the so - called satellite generator is @xmath61 $ ] for doped graphene . we see the typical 2d plasmon dispersion . the plot is for the doping density @xmath62 but the result is very similar for any finite doping density . , width=226 ] in the unperturbed case , i.e. , when there is no interaction with the core hole we have @xmath63 so we can write @xmath64 where @xmath65 now we generalize this result to include all excitation processes in our system . we first rewrite eq . ( [ equ35 ] ) @xmath66 where @xmath67 is the boson propagator . now the boson propagator is replaced by @xmath68 times the dynamical structure factor of eq.([equ9 ] ) and the coupling constant by @xmath69 . thus we get @xmath70 $ ] for doped graphene . the plot is for the doping density @xmath62 but the result is very similar for any finite doping density.,width=207 ] the xps spectrum reflects the density of states of the core electron and becomes @xmath71}}{e^ { - a(t ) } } } \\ = \frac{1}{\pi } \int\limits_0^\infty { dt\cos \left [ { \left ( { \omega - d } \right)t + b\left ( t \right ) } \right]{e^ { - a(t ) } } } , \end{array } \label{equ38}\ ] ] where @xmath72,\\ b\left ( t \right ) = - \frac{{2{e^2}}}{{\pi \hbar } } \int\limits_0^\infty { \int\limits_0^\infty { dqd\omega } } \frac{1}{{{\omega ^2}}}{\mathop{\rm im}\nolimits } \frac { { - 1}}{{\varepsilon \left ( { q,\omega } \right)}}\sin \left ( { \omega t } \right),\\ d = \frac{{2{e^2}}}{{\pi \hbar } } \int\limits_0^\infty { \int\limits_0^\infty { dqd\omega } } \frac{1}{\omega } { \mathop{\rm im}\nolimits } \frac { { - 1}}{{\varepsilon \left ( { q,\omega } \right)}}. \end{array } \label{equ39}\ ] ] we give the energy relative to the core - level position when the interaction with the electrons are neglected . taking the finite life time of the core hole into account we arrive at the central theoretical result of this work , the xps spectrum . the xps spectrum can be written as @xmath73dt } , \label{equ40}\ ] ] where @xmath74\left [ { 1 - \cos \left ( { wt } \right ) } \right]dq } dw } , \end{array } \label{equ41}\ ] ] @xmath75\sin \left ( { wt } \right)dq } dw } , \label{equ42}\ ] ] and @xmath76dq } dw}. \label{equ43}\ ] ] $ ] for pristine graphene . the scaling of @xmath77 and @xmath78 is the same as in fig.[figu2].,width=207 ] all variables have been scaled according to @xmath79 and are now dimensionless . the two last relations will be used below . the quantity @xmath80 is the energy shift of the adiabatic peak . we have taken the finite life - time of the core hole into account by introducing the first factor of the integrand of eq.([equ40 ] ) , where @xmath81 is the fwhm ( full width at half maximum ) of the lorentz broadened peak . we have here assumed that the core - hole potential can be represented by a pure coulomb potential . the results are valid for a general 2d system . the particular system enters the problem through @xmath82 $ ] . for pristine graphene the dielectric function is@xcite @xmath83 where we have used the values@xcite @xmath84 m / s for the fermi velocity @xmath85 and @xmath86 for @xmath87 , respectively . the background dielectric constant @xmath87 is the result of high frequency electronic excitations to higher lying empty energy bands and from lower lying filled energy bands . when the graphene sheet is doped the dielectric function becomes much more complicated . however it has been derived by several groups@xcite . the dielectric function in a general point in the complex frequency plane , @xmath77 , away from the real axis is@xcite @xmath88 } \right\};}\\ { f\left ( { q , w } \right ) = { \rm{asin}}\left ( { \frac{{1 - w}}{q } } \right ) + { \rm{asin}}\left ( { \frac{{1 + w}}{q } } \right)}\\ { \qquad \qquad - \frac{{w - 1}}{q}\sqrt { 1 - { { \left ( { \frac{{w - 1}}{q } } \right)}^2 } } + \frac{{w + 1}}{q}\sqrt { 1 - { { \left ( { \frac{{w + 1}}{q } } \right)}^2 } } , } \end{array } \label{equ46}\ ] ] where @xmath89 is the density of states at the fermi level and @xmath90 is the doping concentration . in both eqs . ( [ equ45 ] ) and ( [ equ46 ] ) we let @xmath77 be real valued but add a small imaginary part to get the retarded forms of the dielectric functions . in figs.[figu1 ] and [ figu2 ] we show the contour plot and surface plot , respectively , of @xmath91 $ ] for doped graphene . the integrands in eqs.([equ41 ] ) , ( [ equ42 ] ) , and ( [ equ43 ] ) all have an extra factor @xmath92 for small @xmath77 which means that small energy transfers are important in the shake up structure of the spectra . in fig.[figu3 ] we give the corresponding surface plot for pristine graphene . here the excitations are of electron - hole pair type . we have taken the lorentzian broadening into account but how should we include the gaussian instrumental broadening ? here we use a trick . if all shake up processes were involving only one discrete frequency , @xmath93 , the @xmath94- , @xmath95- , and @xmath96-functions would become @xmath97,\\ { b_g}\left ( t \right ) = - \eta \sin \left ( { { w_0}t } \right),\\ { d_g } = \eta { w_0},\\ \eta = \int { \frac{{{d^2}q}}{{{{\left ( { 2\pi } \right)}^2}}}\frac{{{{\left| { g\left ( { \bf{q } } \right ) } \right|}^2}}}{{{{\left ( { \hbar { \omega _ 0 } } \right)}^2 } } } } , \end{array } \label{equ47}\ ] ] -xps ) c 1s core - level spectrum of a suspended graphene sheet obtained in ref.@xcite using the photon energy 480 ev , red circles . the solid curve is the theoretical result for an undoped free - standing graphene sheet . a linear background was subtracted from the experimental data . we used lorentzian and gaussian broadening of .4 ev and .666 ev , respectively , width=302 ] where @xmath98 is a strength constant and the spectrum would consist of a series of delta functions , @xmath99 but now for a gold supported graphene sheet . the theoretical curve was obtained assuming a doping density of @xmath100 . we used the same lorentzian and gaussian broadening as in fig.[figu5].,width=302 ] the amplitudes of the delta functions form a poisson distribution . why is it a poisson distribution ? the average number of bosons surrounding the core hole is @xmath98 . the bosons do not interact and the core hole does not recoil when a boson is excited . thus the probability that a boson is excited in a given instant does not depend on how many bosons are already excited . then , according to probability theory the probability for having exactly @xmath90 bosons excited at a certain time is given by the poisson distribution . when the strength parameter @xmath98 is large this turns into a gaussian distribution . it is close to a gaussian already at @xmath101 . the gaussian fwhm is given by @xmath102 . the trick is now to choose a small enough value for @xmath103 so that @xmath98 becomes large enough . then we add the resulting @xmath94- , @xmath95- , and @xmath96-functions to the original functions in eq.([equ40 ] ) . thus we get the final spectrum with both lorentzian and gaussian broadened structures from @xmath104 } } } \right . } \\ \quad \quad \quad \quad \times \cos \left\ { { \left [ { w - \left ( { d + { d_g } } \right ) } \right]t - \left [ { b\left ( t \right ) + { b_g}\left ( t \right ) } \right ] } \right\ } } \right ] . \end{array } \label{equ49}\ ] ] this is the relation we have used in finding the xps spectra presented in figs.[figu4]-[figu7 ] . in fig.[figu4 ] we compare our results to an experimental xps spectrum@xcite represented by red circles . the experiment was performed on a single quasi - free - standing epitaxial graphene layer on sic obtained by hydrogen intercalation@xcite . this leads to a virtually undoped graphene sheet . the photon energy used was 750 ev . the leftmost peak is from graphene c1s and the rightmost from the c1s core level in the sic substrate . the dashed curve is the result one would get if there were no interaction between the electrons and the core hole . another way to express this is to say that the electron wave functions are frozen . the energies are given relative to the position of this peak . the peak is symmetric and has the full broadening but no structure from shake - up effects . the thick solid curve is the adiabatic peak . this is what the spectrum would look like if the excitation were adiabatic , i.e. , the excitation were so slow that the electrons were left in a quasiequilibrium state ; the core - hole potential was fully screened . this peak is just the dashed peak shifted by the energy @xmath80 . the thin solid curve is the full result from eq.([equ49 ] ) . when we used the dielectric function from pristine graphene the curve emerged a little above the experimental tail . when we added a very small doping level , @xmath105 , the agreement with experiment was perfect . the reason for that a small amount of doping pushes down the tail a little is that there are some extra contributions at the top of the peak and the normalization then leads to a reduction away from the center of the peak . in figs.[figu5 ] and [ figu6 ] we compare our results for pristine and doped free - standing graphene to the experimental results in ref.@xcite . the experiments were performed using a photon energy of 480 ev on a suspended single graphene sheet and on a gold supported single graphene sheet . in the first sample the graphene is more or less undoped ; in the second gold provides p - doping . we subtracted a linear background from both experimental sets of data and used the same broadening parameters for both spectra . the main peak in the doped sample is broader than in the undoped but this extra broadening comes from different shake - up effects in the two samples . the theoretical spectrum was obtained assuming a doping density of @xmath100 . this number @xmath106 should not be taken too seriously . we varied the density in equidistant steps on a logarithmic scale and tried @xmath107 , @xmath108 and @xmath109 . the one in the middle gave the best fit with experiments . , from all spectra . the symmetric peak is the result from the fully adiabatic approximation and the result when there is no interaction between the core hole and the electron - hole system . the red dashed curve is the result from the undoped graphene sheet , the blue short - dashed from a sheet with doping concentration @xmath110 , and the green dotted curve from a sheet with doping concentration @xmath111 , respectively . the curve for doping concentration @xmath112 has been omitted since it is very close to the curve for pristine graphene . , width=302 ] finally , in fig.[figu7 ] we show the theoretical results for different doping levels . in order to more clearly see the effect doping has on the peak shape we have here removed all different shifts , @xmath80 , for the curves . the solid black curve is the noninteracting curve or the adiabatic curve ( it is the same for all doping levels now when the shift has been removed ) . the red dashed curve is the pristine result . the blue short - dashed curve is for the doping density @xmath110 , and the green dotted curve from a sheet with doping concentration @xmath111 . important doping effects show up first at doping concentrations exceeding @xmath112 . the reason is that some electron - hole - pair excitation - channels are blocked with doping due to the pauli exclusion principle ; this is compensated by new electron - hole - pair excitation - channels and the new plasmon - excitation channel . in a recent work@xcite one came to the same conclusion as we regarding the need to make a full calculation of the core - hole spectra in graphene instead of using the doniach and sunji fitting of the peaks . they treated the band - structure of graphene in a different way compared to our and relied on density functional theory . they seem to have assumed a constant 2d plasmon frequency and thereby obtained sharp plasmon structures in the tail region of the peaks . such structures are not observed in the experimental spectra . to summarize , we have presented a detailed derivation of the core - level spectra of 2d systems and presented numerical results for pristine and doped free - standing graphene . although , pristine graphene is not a metal its core - level spectrum shows a peak tailing , characteristic of metallic systems . the tailing increases with doping for doping concentrations exceeding @xmath112 . the peak shape changes with further increase of the doping concentration . this opens up for a complementary way to estimate the degree of doping of a sample . we have compared our results to three different experimental spectra from two experimental groups . the agreement is quite good which is very encouraging . we have furthermore introduced a convenient way to introduce the effect of gaussian instrumental broadening in the formalism . we thank professor leif i. johansson and professor chung - lin wu for providing us with their experimental data . p. h. citrin , g. k. wertheim , and y. baer , phys . lett . * 35 * , 885 ( 1975 ) . p. h. citrin , g. k. wertheim , and y. baer , phys . lett . * 41 * , 1425 ( 1978 ) . s. doniach and m. sunji , j. phys . c : solid state phys . * 3 * , 285 ( 1970 ) . g. d. mahan , phys . b * 11 * , 4814 ( 1975 ) . david c. langreth , phys . b * 1 * , 471 ( 1970 ) . bo e. sernelius , and erik sderstrm , j. phys.:condens . matter * 8 * 9071 ( 1996 ) . j. gonz@xmath113les , f. guinea , and m. a. h. vozmediano , _ nucl . phys . _ * 424 * , 595 ( 1994 ) . b. wunsch , t . stauber , f. sols , and f. guinea , _ new journal of physics _ * 8 * , 318 ( 2006 ) . bo e. sernelius , _ epl _ * 95 * , 57003 ( 2011 ) . e. h. hwang , and s. das sarma , phys . b * 75 * , 205418 ( 2007 ) . bo e. sernelius , phys . b * 85 * , 195427 ( 2012 ) . the spectrum was not published before but how the sample was prepared is described in ref.@xcite . j. hassan , c. virojanadara , a. meyer , i. g. ivanov , j. i. flege , s. watcharinyanon , j. falta , l. i. johansson , and e. janz ' en , mater . forum * 717 - 720 * , 605 ( 2012 ) . jhih - wei chen , chiang - lun wang , hung wei shiu , chi - yuan lin , chen - shiung chang , forest shih - sen chien , chia - hao chen , yi - chun chen , and chung - lin wu , applied physics express * 5 * , 085101 ( 2012 ) . v. despoja , and m. sunji , phys . b * 88 * , 245416 ( 2013 ) .

we calculate core - level spectra for pristine and doped free - standing graphene sheets . instructions for how to perform the calculations are given in detail . although pristine graphene is not metallic the core - level spectrum presents low - energy tailing which is characteristic of metallic systems . the peak shapes vary with doping level in a characteristic way . the spectra are compared to experiments and show good agreement . we compare to two different pristine samples and to one doped sample . the pristine samples are one with quasi - free - standing epitaxial graphene on sic obtained by hydrogen intercalation and one with a suspended graphene sheet . the doped sample is a gold supported graphene sheet . the gold substrate acts as an acceptor so the graphene sheet gets p - doped .

arxiv

selection , random genetic drift and mutations are the processes underlying darwinian evolution . for a long time population geneticists have analyzed the dynamics in the simplest setting consisting of two genotypes evolving under these processes @xcite . in those studies , a genotype represents an individual s genetic makeup , completely determining all relevant properties of the individual . a key concept is the so - called fitness of a genotype which represents the selection pressure for the individuals . the fitness defines the expected number of offspring an individual will produce . thus , selection acts on fitness differences preferring individuals with higher fitness over individuals with lower fitness . usually it is assumed that individuals have fixed fitnesses defined by their genotype alone @xcite . yet , experimental studies have revealed that many natural systems exhibit frequency - dependent selection @xcite , which means that an individual s fitness not only depends on its genotype , but also on its interactions with other individuals and hence on the frequency of the different genotypes in the population . although such frequency - dependent selection had already been studied early by crow and kimura @xcite , only recently has it received more attention @xcite . in these theoretical and computational studies , individuals interactions are represented by interaction matrices from game theory . this leads to a frequency dependence where the fitness depends directly on the interaction parameters in a linear way . however , fitness may depend on many diverse factors such as cooperation ( i.e. individuals acting together to increase their fitness @xcite ) and resource competition , so that certain systems may exhibit frequency - dependent fitness that is nonlinear . for example , in experiments certain hermaphrodites exhibit such nonlinear fitness - dependence @xcite . to the best of our knowledge the impact of such nonlinear dependencies on coevolutionary dynamics has not been investigated theoretically . in this article we show that nonlinear frequency dependence @xcite may induce new stable configurations of the evolutionary dynamics . furthermore , we study the impact of asymmetric mutation probabilities on the dynamics @xcite , which was also neglected in most models until now @xcite . as in previous works on coevolutionary dynamics we base our work on the moran process in a non - spatial environment which is a well established model to study evolutionary dynamics and was already used in many applications @xcite . the moran process is a stochastic birth - death process which keeps the population size constant @xcite . therefore , in a two - genotype model the system becomes effectively one - dimensional , so that the dynamics may be described by a one - dimensional markov chain with transition rates defined by the moran process . we derive the stationary probability distribution of the system dynamics via its fokker - planck equation @xcite . sharp maxima of the distribution reveal metastable points of the dynamics and a multitude of such maxima lead to stochastic switching dynamics between multiple stable points . the article is structured as follows . in section ii we introduce the model details and in section iii we derive the fokker - planck equation describing the probabilistic dynamics of the population . using this equation we derive the stationary probability distribution that describes the long - time behavior of the system . in section iv we analyze this probability distribution , which yields information about the impact of nonlinear frequency - dependent selection and of different mutation probabilities on the coevolutionary dynamics . in section v we give a summary and discuss our results . consider a population of @xmath0 individuals evolving in a non - spatial environment , meaning that the population is well - mixed so that each individual interacts with all other individuals at all times . in this population the individuals may assume one out of two genotypes @xmath1 and @xmath2 . the population sizes @xmath3 and @xmath4 ( @xmath5 ) evolve according to the time - continuous moran process described in the following , cf . the number of individuals @xmath3 of genotype @xmath1 completely determines the state of the system as @xmath6 . at all times the interactions of the individuals determine the actual ( frequency - dependent ) fitness , so that an individual s fitness of genotype @xmath1 or @xmath2 is defined by a fitness function @xmath7 or @xmath8 respectively . the fitness functions @xmath7 and @xmath8 may be any functions with the only condition that @xmath9 for all @xmath10 $ ] as negative fitness is not defined . at rate @xmath11 an individual of type @xmath1 produces an identical offspring which may mutate to genotype @xmath2 with probability @xmath12 . this applies analogously to genotype @xmath2 . then one individual of the population is chosen randomly to die , so that the population size @xmath0 stays constant and the variables @xmath13 change maximally by @xmath14 . this is the so - called moran process which was originally introduced for a population of two genotypes with fixed fitnesses and no mutations occurring , cf . however , this process is easily generalizable to more genotypes and frequency - dependent selection in the way described above , cf . @xcite . note that in our model the rate of reproduction of e.g. type @xmath1 is directly determined by the term @xmath15 as e.g. in @xcite . in other models the fitness is first normalized so that the rate of reproduction is given by @xmath16 @xcite , where @xmath17 \label{eq : meanfitness}\ ] ] is the population s mean fitness . while in both of these models the events occur with the same probability , the times between the events differ by a common factor determined by the mean fitness ( [ eq : meanfitness ] ) . thus , these models exhibit a quantitatively different time course , but the same event sequences . until now , usually linear functions have been considered for @xmath18 . however , in many applications nonlinear functions seem more appropriate @xcite . for example , cooperation effects in game theory induce functions linearly increasing in @xmath3 , so that @xmath19 @xcite . this would mean that fitness increases infinitely with the population size of genotype @xmath1 . yet , in any habitat there is only a limited amount of resources available for living . therefore , fitness should decline when the population of one genotype becomes too large as all of the individuals will compete for the same resources . we conclude that a fitness function including these two effects has to contain a nonlinear factor , e.g. @xmath20 where @xmath21 and @xmath22 . here we choose a quadratic nonlinear factor as a simple example . naturally , all other nonlinear functions could be applicable as well . note that linearly increasing fitness functions such as @xmath19 and @xmath23 lead to quadratic relations for the _ mean _ fitness @xmath24 ( see equation ( [ eq : meanfitness ] ) ) of the population which becomes @xmath25 the quadratic factor originates from the fact that the linear fitness function in ( [ eq : meanfitness ] ) is multiplied by the number of individuals leading to a quadratic dependence on @xmath26 , e.g. @xmath27 . this factor enters the replicator equation in evolutionary game theory , thus leading to a quadratic fitness dependence in standard game theoretic problems @xcite . however , this does not imply nonlinear interactions and thus fitness effects such as presented in equation ( [ eq : fitnessfunction ] ) . therefore , the fitness functions that can be generated by evolutionary game theory are a special case of the functions that occur in the theory presented here . fitness functions as for example presented in equation ( [ eq : fitnessfunction ] ) will as we show in this article lead to dynamics with more complex stability properties . usually , when analyzing the dynamics of such a model system as described above , it was assumed either that no mutations occur at all ( @xmath28 ) @xcite or that the mutation probabilities are equal ( @xmath29 ) @xcite . however , usually mutation probabilities can be highly diverse @xcite . this has been considered in some examples @xcite , but until now the effect of different mutation probabilities in the introduced model system has not been studied systematically . here , we explicitly study the effects caused by asymmetric mutation rates ( see e.g. figure [ fig : asymmetricexample ] ) . in the following we assume the fitness of the individuals to be given by @xmath30 and @xmath31 with the interaction functions @xmath32 and @xmath33 . the model system is effectively one - dimensional with the variable @xmath34 completely describing the system state @xcite . at each event @xmath35 may change by at most @xmath36 depending only on the actual state of the system , so that the dynamics are markovian . let the transition @xmath37 occur with the rate @xmath38 . then this rate is determined by the fitnesses and mutation probabilities in the following way : * each individual in the population receives offspring with rate @xmath39 with @xmath40 . this means that genotype @xmath1 receives one offspring with rate @xmath41 . * such an offspring increases the population @xmath1 with probability @xmath42 , and population @xmath2 in case it mutates with probability @xmath12 . * one individuum is chosen uniformly to die belonging to genotype @xmath2 with probability @xmath43 . * equivalently the number of individuals of genotype @xmath1 can increase if genotype @xmath2 receives one offspring , which mutates to genotype @xmath1 with probability @xmath44 , and one individuum of @xmath2 is chosen to die . taken these processes together the transition rate for @xmath37 is@xmath45 decreasing @xmath35 to @xmath46 is equivalent to increasing the number of individuals of genotype @xmath2 , so that the transition rate@xmath47 is obtained analogously . the master equation of the markov chain is then given by@xmath48 where we define @xmath49 . as also @xmath50 , the system has reflective boundaries at @xmath51 and @xmath52 . if @xmath53 ( @xmath54 ) then @xmath55 ( @xmath56 ) is an absorbing state of the dynamics it is possible to derive the exact solution to such an equation similarly to the derivations in @xcite which we do in appendix b. however , the obtained solutions are not very explicit and their implications are hard to grasp . it is thus more useful to advance to the fokker - planck equation @xcite describing the system in the limit of large @xmath0 . as previous works have shown , this approximation already leads to very good results for moderate population sizes of @xmath57 , cf . we use the transformation @xmath58,\qquad s=\frac{t}{n},\qquad{\tilde{\mu}}_{ij}=\mu_{ij}\cdot n,\\[0.1 cm ] \rho(x , s)=p_{xn}(sn)n,\qquad\tilde{g}_{j}(x)=g_{j}(xn)\cdot n \label{eq : rescaledinteraction}\end{aligned}\ ] ] yielding the fokker - planck equation . the scaling factors are derived in appendix a where the derivation of the fokker - planck equation in the scaling limit @xmath59 is shown in detail ( equation ( [ eq : appfpe ] ) ) . this leads to@xmath60x(1-x)-\delta{\tilde{\mu}}\right)\rho(x , s)\big]+\frac{\partial^{2}}{\partial x^{2}}\big[x(1-x)\rho(x , s)\big]\label{eq : fpe}\ ] ] with the normalization condition @xmath61 for all @xmath62 . here @xmath63 and @xmath64 are the rescaled interaction functions defined in ( [ eq : rescaledinteraction ] ) , @xmath65 is the rescaled mean mutation rate and @xmath66 is the rescaled mutation rate difference . these functions have to be rescaled to obtain a non - degenerate limit , the so - called weak selection regime . notice , that the drift term @xmath67x(1-x)-\delta{\tilde{\mu}}\ ] ] contains three different effects . the fitness difference at each point @xmath68 causes a selective drift , the mean mutation rate causes a drift directed to @xmath69 and the mutation rate difference causes a one - directional drift towards one of the boundaries @xmath70 or @xmath71 . the diffusion term@xmath72 reflects the undirected genetic drift . if fitness differences are on a scale of @xmath73 , then mutation , selection and genetic drift all act on the same scale @xcite . then the dynamics are characterized by an interplay of these different effects . on the other hand , in the strong selection regime where fitness differences are on a scale @xmath74 genetic drift becomes negligible and the dynamics in the bulk of the system become deterministic for large @xmath0 @xcite . however , as the factor @xmath75 vanishes for @xmath76 and @xmath77 the effects of mutations play an important role . clearly , without mutations @xmath70 and @xmath71 are absorbing states , but even in the strong selection regime they become non - absorbing for any positive mutation rate . this is reflected in the form of the drift term @xmath78 , as only the factors reflecting mutations @xmath79 remain non - zero in the limits @xmath76 and @xmath77 . for a one - dimensional fokker - planck equation ( [ eq : fpe ] ) the stationary distribution is @xcite@xmath80 with the potential@xmath81 which is defined up to a constant irrelevant to our calculations as it is canceled by the normalization@xmath82 thus , we obtain@xmath83dx\\ & = & \ln\left(x(1-x)\right)-{\tilde{\mu}}\ln(x(1-x))+\delta{\tilde{\mu}}\ln\left[\frac{x}{1-x}\right]-\int\left[{\tilde{g}_a}(x)-{\tilde{g}_b}(x)\right]dx\end{aligned}\ ] ] where we used@xmath84.\ ] ] the stationary solution becomes@xmath85dx}\label{eq : generalsolution}\ ] ] where the constant @xmath86 defined in ( [ eq : normalization ] ) has to be calculated numerically for given parameters . this stationary distribution contains contributions from the above mentioned four different effects : 1 . genetic drift is reflected by @xmath87 which diverges for @xmath88 at the boundaries . the mean mutation rate @xmath89 causes the balancing term @xmath90 . 3 . the asymmetry in the mutation probabilities causes the term @xmath91 which diverges at one boundary and vanishes at the other . all frequency - dependent selection effects are contained in the exponential factor @xmath92dx}$ ] which can take various shapes . what are the possible shapes for the stationary distribution @xmath93 in the form given by equation ( [ eq : generalsolution ] ) ? until now , the term @xmath91 representing the asymmetric mutation probabilities has not been considered to our knowledge and the interaction functions @xmath94 and @xmath95 in the selection term were considered to be at most linear in @xmath68 @xcite . we should therefore be interested in the effects of nonlinear interaction functions and asymmetric mutation rates . let us first analyze the dynamics for nonlinear interaction functions @xcite describing the effects of cooperation and limited resources as described by equation ( [ eq : fitnessfunction ] ) , so that@xmath96 already with @xmath97 such interaction functions induce dynamics stochastically switching between three metastable points . an example is shown in figure [ fig : firstexample ] , where the theoretically calculated stationary distribution from equation ( [ eq : generalsolution ] ) is shown together with data from simulations with a population of @xmath98 individuals which is enough to obtain almost perfect fitting ( cf . also figure [ fig : nplot ] ) . the fitness functions ( figure [ fig : firstexample]b ) both show at first an increase on increasing the number of individuals of genotype @xmath1 or @xmath2 from @xmath99 due to cooperation effects and then a strong decrease due to resource competion . as the resulting fitness functions are asymmetric , also the stationary distribution is asymmetric . there is a maximum at @xmath70 due to genetic drift . selection drives the dynamics towards a metastable state at @xmath100 , because for @xmath101 genotype @xmath1 is fitter than @xmath2 thus increasing in frequency and for @xmath102 genotype @xmath1 is less fit than @xmath2 thus decreasing in frequency ( cf . figure [ fig : firstexample]b ) . the maximum of the stationary distribution is thus exactly at the point where @xmath103 . for @xmath104 again genotype @xmath1 is fitter than @xmath2 and thus the dynamics are driven towards @xmath71 by selection as well as genetic drift . the mutational force induced by @xmath105 increases the height of the maximum at @xmath100 driving the system away from the maxima at @xmath70 and @xmath71 . so , in this example genetic drift , mutation and selection all significantly influence the dynamics . ) ( red , solid ) in perfect agreement with data from simulations with @xmath98 ( blue , @xmath106 ) . ( b ) shows the fitness functions of genotype @xmath1 ( blue , solid ) and @xmath2 ( red , dashed ) . interaction parameters are @xmath107 , @xmath108 , @xmath109 and @xmath110 ( see equation ( [ eq : fofk ] ) and ( [ eq : interactions ] ) ) while the mutation rate is @xmath105 ( @xmath97).,width=566 ] let us now study the influence of the mutation rates in detail . interestingly , for asymmetric mutation rates ( @xmath111 ) the factor @xmath91 always diverges in the interval @xmath112 $ ] . for @xmath113 it diverges for @xmath76 , otherwise for @xmath77 . this can cause the emergence of a maximum of the stationary distribution at @xmath70 ( or resp . @xmath71 ) . figure [ fig : asymmetricexample ] shows an example where due to asymmetric mutation rates a maximum occurs at @xmath71 , when for @xmath97 there is an absolute minimum at @xmath71 as this stable state minimizes the population s mean fitness . this means , that the system dynamics are mutation dominated near @xmath71 . furthermore , the maximum located at @xmath70 for @xmath97 is shifted for @xmath113 to values @xmath114 causing a minimum of the stationary distribution at @xmath70 . thus , in this example both selection and the asymmetry in mutation probabilities are the driving forces of the system s dynamics . ( red , solid ) and @xmath115 ( gray , dashed ) for a system with selective advantage for genotype @xmath2 as shown in ( b ) ( blue , solid : genotype @xmath1 ; red , dashed : genotype @xmath2 ) . solid and dashed lines in ( a ) show the theoretical curves from equation ( [ eq : generalsolution ] ) , crosses the data from simulations with @xmath98 ( @xmath106 : @xmath97 , @xmath116 : @xmath115 ) . interaction parameters are @xmath117 , @xmath118 , @xmath119 and @xmath120 ( see equation ( [ eq : interactions ] ) ) while mean mutation rate is @xmath121.,width=566 ] for low mutation rates the population has a high tendency to become dominated by one of the two genotypes for long times which is illustrated in figure [ fig : mutplot]a . thus , the dynamics stay close to the edges @xmath70 or @xmath71 waiting for one of the few mutations to occur . for low overall mutation rates differences in the mutation rates have only weak effects . high mutation rates cause the population to be drawn towards a mixture of both genotypes ( @xmath122 ) . even more , for high mutation rates the differences in the mutation rates @xmath123 can have an important influence on the dynamics . for instance figure [ fig : mutplot]b illustrates a shift of an existing stable point , which completely vanishes for high @xmath124 . we conclude that asymmetric mutation rates can cause the emergence of new stable states ( as in figure [ fig : asymmetricexample ] ) , the disappearance of existing ones ( as in figure [ fig : mutplot ] ) , and shift existing stable states to new positions . ) for three different symmetric ( @xmath97 ) mutation rates : @xmath121 ( blue , solid ) , @xmath125 ( red , long dashed ) and @xmath126 ( green , short dashed ) . this illustrates that for low mutation rates the dynamics stay close to the system s edges for long times . here , differences in the mutation rates only slightly affect the shape of the stationary distribution . on the other hand , ( b ) shows that for high mutation rates ( @xmath127 ) the dynamics are drawn stronger towards the middle . here , differences in the mutation rates have a stronger impact , as the three curves with @xmath97 ( blue , solid ) , @xmath128 ( red , long dashed ) and @xmath129 ( green , short dashed ) demonstrate . the increasing @xmath124 shifts the existing stable state towards the edge of the system until it vanishes . system parameters were @xmath98 and a symmetric interaction function according to equation ( [ eq : interactions ] ) with one stable state at @xmath130 with @xmath131 , @xmath132.,width=566 ] as we saw above , the fit of the theoretically calculated stationary distribution to simulation data is almost perfect for weak selection , i.e. the maximal fitness difference satisfies @xmath133 , cf . e.g. @xcite . to quantify the quality of the fit we measured the stationary distribution with the same parameters as above for different population sizes @xmath0 this is shown in figure [ fig : nplot ] demonstrating that for increasing @xmath0 the data gets ever closer to the theoretically obtained distribution . for this we define the empirical distribution @xmath134 where @xmath135 is the discrete process defined in equation ( [ eq : meq ] ) . @xmath136 is a time large enough for the process to reach stationarity and depends on the system size , as does the measurement time @xmath137 which is specified in figure [ fig : nplot ] . using this definition we quantify the quality of the fit using the distance measure @xmath138 which takes the mean distance of the measured empirical distribution @xmath139 from the theoretical distribution for every point except @xmath51 and @xmath52 where the density @xmath140 diverges . here , the the theoretical distribution is calculated by integrating the theoretical density over bins of the size @xmath141 around the points @xmath142 . figure [ fig : nplot]b shows that the mean distance @xmath143 decays with increasing @xmath0 . the decay of this quantity is dominated by the slow convergence at the domain boundaries @xmath51 and @xmath52 . therefore , we additionally defined the maximum distance measure @xmath144}\left\{\left|\pi_k-\int_{k / n-1/2n}^{k / n+1/2n}\rho^\ast(x)dx\right|\right\}\label{eq : maxdistance}\ ] ] which gives the maximum distance between the measured probability distribution from the theoretical distribution for all points except @xmath51 and @xmath52 due to the same argument as above . figure [ fig : nplot]b shows that the maximum distance @xmath145 decreases with increasing @xmath0 , however slower than the mean distance @xmath143 due to the divergence of @xmath146 at the boundaries . altogether we conclude , that the theoretical curve fits the data very well already for @xmath147 . however , near the domain boundaries special care has to be taken , as divergences of the theoretical curve lead to larger deviations . . ( a ) shows the theoretical curve from equation ( [ eq : generalsolution ] ) ( black , solid ) for the same system as in figure [ fig : firstexample ] . data points show empirical distributions as defined in equation ( [ eq : empiricaldensity ] ) obtained in simulations for @xmath148 ( blue , @xmath106 ) , @xmath149 ( red , @xmath150 ) , @xmath151 ( green , @xmath152 ) and @xmath98 ( gray , @xmath116 ) . ( b ) shows the distances @xmath143 and @xmath145 of simulation data and theoretical curve for different @xmath0 for the mean distance measure @xmath143 defined in equation ( [ eq : meandistance ] ) ( blue , @xmath106 ) and the maximum distance measure @xmath145 defined in equation ( [ eq : maxdistance ] ) ( red , @xmath150 ) , demonstrating that the distance decays for increasing @xmath0 . the solid line @xmath153 and the dashed line @xmath154 are added as a guide to the eye for the relation between measured distances and system size @xmath0 . the measured empirical distributions for both ( a ) and ( b ) were obtained by simulating the system dynamics from an initial state drawn from @xmath140 for a mixing time @xmath155 and then recording the density for a time @xmath156.,width=566 ] furthermore , as the fokker - planck approximation only holds for weak selection we study the quality of the solution obtained throught the fokker - planck equation in dependence of the selection strength . for this we introduce a scaling factor @xmath157 to the interaction functions , so that they become @xmath158 we find that the stationary solution of the fokker - planck equation well approximates the solution of the master equation @xmath159 for selection strengths up to @xmath157 of the order of 1 ( cf . figure [ fig : selectionplot ] ) . we derive the solution ( [ eq : mastersolutionintext ] ) of the master equation in appendix b. note that for the derivation of this solution no approximation is necessary and it hence fits data from simulations perfectly up to an error due to finite sampling in simulations . as figure [ fig : selectionplot ] illustrates , for strong selection @xmath160 the solution of the fokker - planck equation does not fit the exact solution of the master equation perfectly , but still catches the overall trend of the dynamics . the error , quantified by the mean distance measure analogous to ( [ eq : meandistance ] ) , increases with @xmath157 as a power law both in comparison with the solution from the master equation and with data from simulations , cf . figure [ fig : selectionplot]d . all in all , for weak selection the fokker - planck approximation works well while for strong selection it does not perfectly predict the stationary distribution , but still reflects the overall trend of the dynamics . if this approximation is not satisfying , the direct solution ( [ eq : mastersolutionintext ] ) of the master equation yields the exact distribution fitting the data for all selection strengths . ) obtained from the fokker - planck equation well fits the exact solution ( [ eq : mastersolutionintext ] ) from the master equation for weak selection , but not for strong selection . ( a ) shows the stationary solution from the fokker - planck equation ( red , solid ) for the system with interaction functions defined by equation ( [ eq : scaledinteractions ] ) for very weak selection @xmath161 together with the solution from the master equation ( blue , @xmath106 ) . the distributions of the same system for weak selection @xmath162 and strong selection @xmath163 are shown in ( b ) and ( c ) , respectively . all curves are plotted logarithmically to better allow a comparison of the deviations over all scales . ( c ) shows that the curve from the fokker - planck equation does not fit the exact master solution very well . this is quantified in ( d ) showing the mean distance @xmath143 between fokker - planck and master solution ( blue , @xmath106 ) and between fokker - planck solution and an empirical density obtained from simulations ( red , @xmath150 ) . the system parameters were @xmath98 , @xmath105 , @xmath131 and @xmath132 . the measured empirical distributions were obtained by simulating the system dynamics from an initial state drawn from @xmath140 for a mixing time @xmath164 and then recording the density for a time @xmath165.,width=566 ] we have shown above that the dynamics of a two - genotype system can exhibit multiple stable points induced by nonlinear selection . even more , there is no theoretical limit to the number of stable points in the system . assume for example @xmath121 and @xmath97 , so that mutation exactly balances genetic drift . then , the stationary distribution ( [ eq : generalsolution ] ) has a maximum at each point , where @xmath166 $ ] has a maximum . theoretically @xmath94 and @xmath95 can be any function with an arbitrary amount of extreme points in @xmath112 $ ] , so that there is no limit to the stationary distribution s number of maxima , if selection dominates the dynamics . however , for finite @xmath0 the number of possible maxima is naturally limited by @xmath167 . figure [ fig : multiplestability ] shows an example , where we used periodic interaction functions@xmath168 although this is not a realistic interaction function in most applications , it demonstrates what is theoretically possible in the introduced system . ) ( red , solid ) together with simulation data with @xmath98 ( blue , @xmath106 ) for the interaction functions given in ( [ eq : perfitfunc ] ) . the computed stationary distribution diverges for @xmath76 and @xmath77 while the simulation data remains finite due to the finite number of individuals . ( b ) shows a sample path which exhibits switching between the different maxima of the distribution . parameters are @xmath169 , @xmath170 , @xmath171 and @xmath97.,width=566 ] in this article we analyzed a two - genotype system in a very general setting with ( possibly ) asymmetric mutation probabilities and nonlinear fitness functions @xcite in finite populations . the underlying moran process is a well established model @xcite to gain an understanding of the interplay of selection , mutation and genetic drift in evolutionary dynamics . however , the moran process is studied mostly with symmetric mutation probabilities and at most linear interaction functions . we reasoned that neither need mutation probabilities be symmetric as experiments have shown , that mutation probabilities are often asymmetric @xcite nor can all interaction effects be described by linear interaction functions . for example cooperation in game theory leads to an interaction function increasing linearly in the frequency of the cooperating genotype . yet , in many applications also cooperators in the end compete for the same type of resource which is limited . therefore , due to limited resources a population being too large can not be sustained leading to a decrease in the fitness . there is no linear function that can reflect both of these effects at the same time . we derived the fokker - planck equation describing the dynamics of the number of individuals @xmath35 of genotype @xmath1 in the limit of large population sizes @xmath0 . we quantified the quality of the fokker - planck approach for an example ( cf . figure 4 ) where the difference of simulation data and theoretical solution became almost not detectable for population sizes larger than @xmath147 . actually , if the system exhibits absorbing states then the fokker - planck method does not work to study the corresponding quasi - stationary distributions . instead , wkb methods are more appropriate to describe the system dynamics as for example in @xcite , where fixation resulting from large fluctuations was studied . in our model system no such absorbing states exist , as long as the mutation rates are positive ( @xmath172 ) and therefore the fokker - planck equation is appropriate to describe the system dynamics . we identified the individual effects of selection , mean mutation rate and mutation difference as well as genetic drift and derived the stationary probability distribution as determined by the fokker - planck equation . analyzing the distribution , we found that asymmetries in the mutation probabilities may not only induce the shifting of existing stable points of the dynamics to new positions , but also lead to the emergence of new stable points . thus , a genotype that has a selective disadvantage can anyway have a stable dynamical state where its individuals dominate the population due to a higher mutational stability ( see figure [ fig : asymmetricexample ] ) . further we found , that dynamic fitness leads to multiple stable points of the dynamics induced by selection and also genetic drift . we showed an example ( figure [ fig : firstexample ] ) where three stable points exist , two caused by selection and one by genetic drift . we conclude that frequency - dependent fitness together with asymmetric mutation rates induces complex evolutionary dynamics , in particular if the interactions imply nonlinear fitness functions . theoretically , there is no limit for the number of stable points that the dynamics can exhibit ( see figure [ fig : multiplestability ] ) . all in all , we interprete our results such that in real biological systems multiple metastable equilibria may exist , whenever species interact in a way complex enough to imply a fitness that nonlinearly depends on frequency . as a consequence , one species may exhibit a certain frequency for a long time before a sudden shift occurs and then a new frequency prevails . such a change may thus occur even in the absence of changes of the environment ; it may be induced as well by a stochastic switching from one metastable state to another due to complex inter - species interactions . the moran process is a standard tool to gain theoretical insights into experimental data @xcite . of course , we do not propose here that any experimental setup may be exactly described by the moran process . however , we think that it should be feasible to develop an experiment where two different mutants evolve with asymmetric mutation rates . to find an experimental setup where the two genotypes also exhibit nonlinear fitness could however prove more difficult . rather , our study is a theoretical study indicating that nonlinear fitness may be the cause of multible stable states when observed in experimental data . for further studies on systems with more genotypes it should be useful to combine our considerations presented here with the work of traulsen et al . @xcite , where an analysis of systems with more than two genotypes was carried out . extending those results it may be possible to gain a better understanding of the effects of nonlinear interactions for many different genotypes . also , it may be interesting to study the effects of changing interactions , where the interactions change according to the system dynamics @xcite . thus , our study might serve as a promising starting point to investigate how nonlinear frequency dependencies impact evolutionary dynamics in complex environments . * acknowledgements * we thank steven strogatz for fruitful discussions during project initiation and stefan eule for helpful technical discussions . stefan grosskinsky acknowledges support by epsrc , grant no . ep / e501311/1 . the master equation ( [ eq : meq ] ) may be transformed to a fokker - planck equation in the limit of large @xmath0 @xcite . we introduce the transformation@xmath173 together with the rescaled functions@xmath174 we fix the scaling functions @xmath175 , @xmath176 and @xmath177 such that in the limit @xmath59 all terms in equation ( [ eq : meq ] ) remain finite so that mutation , selection and genetic drift all act on the same scale . we further define@xmath178 and substituting all this into the master equation ( [ eq : meq ] ) , we obtain@xmath179\rho({x_{-}},s)\right.\nonumber \\ & & \left[(1-\mu_{ba})(1+g_{b}({x_{+}}))(1-{x_{+}}){x_{+}}+\mu_{ab}(1+g_{a}({x_{+}})){x_{+}}^{2}\right]\rho({x_{+}},s)\nonumber \\ & & -\left[(1-\mu_{ab})(1+g_{a}(x))x(1-x)+\mu_{ba}(1+g_{b}(x))(1-x)^{2}\right.\nonumber \\ & & + \left.\left.(1-\mu_{ba})(1+g_{b}(x))x(1-x)+\mu_{ab}(1+g_{a}(x))x^{2}\right]\rho(x , s)\right\ } \label{eq : app1}\end{aligned}\ ] ] we choose @xmath180 and @xmath181 so that in the limit @xmath59 the terms stay finite . further we introduce the mean mutation rate @xmath65 and the mutation rate difference @xmath66 . to not overload the notation we drop the time argument @xmath182 of @xmath183 in the following calculation . this leads to @xmath184\right\ } \\ & & + n\left\ { { \tilde{g}_a}({x_{-}}){x_{-}}(1-{x_{-}})\rho({x_{-}})-{\tilde{g}_a}(x)x(1-x)\rho(x)+{\tilde{g}_b}({x_{+}}){x_{+}}(1-{x_{+}})\rho({x_{+}})-{\tilde{g}_b}(x)x(1-x)\rho(x)\right.\\ & & + \frac{{\tilde{\mu}}}{2}\left[(1 - 2{x_{-}})\rho({x_{-}})-(1 - 2{x_{+}})\rho({x_{+}})\right]\\ & & + \delta{\tilde{\mu}}\left[{x_{+}}\rho({x_{+}})-x\rho(x)-(1-{x_{-}})\rho({x_{-}})+(1-x)\rho(x)\right]\\ & & + \frac{{\tilde{\mu}}}{n}\left[\left({\tilde{g}_a}({x_{+}}){x_{+}}^{2}+{\tilde{g}_b}({x_{+}})({x_{+}}^{2}-{x_{+}})\right)\rho({x_{+}})-\left({\tilde{g}_a}(x)x^{2}+{\tilde{g}_b}(x)(x^{2}-x)\right)\rho(x)\right.\\ & & + \left.\left({\tilde{g}_a}({x_{-}})({x_{-}}^{2}-{x_{-}})+{\tilde{g}_b}({x_{-}})(1-{x_{-}})^{2}\right)\rho({x_{-}})-\left({\tilde{g}_a}(x)(x^{2}-x)+{\tilde{g}_b}(x)(1-x)^{2}\right)\rho(x)\right]\\ & & + \frac{\delta{\tilde{\mu}}}{n}\left[\left({\tilde{g}_a}({x_{+}}){x_{+}}^{2}-{\tilde{g}_b}({x_{+}})({x_{+}}^{2}-{x_{+}})\right)\rho({x_{+}})-\left({\tilde{g}_a}(x)x^{2}-{\tilde{g}_b}(x)(x^{2}-x)\right)\rho(x)\right.\\ & & + \left.\left.\left({\tilde{g}_a}({x_{-}})({x_{-}}^{2}-{x_{-}})-{\tilde{g}_b}({x_{-}})(1-{x_{-}})^{2}\right)\rho({x_{-}})-\left({\tilde{g}_a}(x)(x^{2}-x)-{\tilde{g}_b}(x)(1-x)^{2}\right)\rho(x)\right]\right\ } \end{aligned}\ ] ] in the limit @xmath59 the different terms with @xmath185 in front become second order derivatives with respect to @xmath68 , while the other terms become first order derivatives . the terms which have a @xmath141 factor vanish in the limit @xmath59 and thus the above equation becomes @xmath186x(1-x)\rho(x)\right)-\delta{\tilde{\mu}}\rho(x)\right]+\frac{\partial^{2}}{\partial x^{2}}\left[x(1-x)\rho(x)\right]\label{eq : appfpe}\ ] ] which is the focker - planck - equation of the system . we directly derive the stationary solution @xmath187 of the master equation ( [ eq : meq ] ) using the detailed balance equation @xmath188 which applies to any chain with only nearest neighbour transitions @xcite , cf . also @xcite . thus , using the rewritten balance equation @xmath189 iteratively , we obtain @xmath190 finally , we may use the normalization condition @xmath191 of the stationary distribution to eliminate the factor @xmath192 . we then obtain the exact stationary solution of the master equation @xmath193 which can be evaluated numerically for any transition rates @xmath194 and @xmath195 . for more details on the exact solution of the master equation in the moran process see for example the work by claussen and traulsen @xcite .

evolution is simultaneously driven by a number of processes such as mutation , competition and random sampling . understanding which of these processes is dominating the collective evolutionary dynamics in dependence on system properties is a fundamental aim of theoretical research . recent works quantitatively studied coevolutionary dynamics of competing species with a focus on linearly frequency - dependent interactions , derived from a game - theoretic viewpoint . however , several aspects of evolutionary dynamics , e.g. limited resources , may induce effectively nonlinear frequency dependencies . here we study the impact of nonlinear frequency dependence on evolutionary dynamics in a model class that covers linear frequency dependence as a special case . we focus on the simplest non - trivial setting of two genotypes and analyze the co - action of nonlinear frequency dependence with asymmetric mutation rates . we find that their co - action may induce novel metastable states as well as stochastic switching dynamics between them . our results reveal how the different mechanisms of mutation , selection and genetic drift contribute to the dynamics and the emergence of metastable states , suggesting that multistability is a generic feature in systems with frequency - dependent fitness . * keywords : * population dynamics ; dynamic fitness ; stochastic switching ; multistability

arxiv

the phase - field method has become the method of choice for simulating microstructure formation during solidification . it owes its popularity mainly to its algorithmic simplicity : the cumbersome problem of tracking moving solid - liquid interfaces or grain boundaries is avoided by describing the geometry in terms of one or several phase fields . the phase fields obey simple partial differential equations that can be easily coded by standard numerical methods . the foundations of the phase - field method and its application to solidification have been the subject of several recent review articles @xcite , and it seems of little use to repeat similar information here . instead , in this paper several topics are discussed where robust phase - field modelling tools are not yet available because some fundamental questions remain open . in sec . [ sec2 ] , the thin - interface limit of two - sided phase - field models is examined , and it is shown that the currently available approaches can not in general eliminate all effects linked to the finite interface thickness . in sec . [ sec3 ] , orientation - field models for polycrystalline solidification are discussed , and it is shown that the standard equation of motion usually written down for the orientation field is not appropriate for the evolution of coherent crystalline matter . finally , in sec . [ sec4 ] , the inclusion of microscopic fluctuations in the phase - field equations is reviewed , and it is shown that the standard approach can not be used in a straightforward way to investigate the process of nucleation . the common point of these topics is that they pose challenges or limitations for straightforward computations . indeed , a characteristic feature of the phase - field method is that its equations can often be written down following simple rules or intuition , but that their detailed properties ( which have to be known if quantitative simulations are desired ) become only apparent through a mathematical analysis that can be quite involved . therefore , it is not always easy to perceive the limits of applicability of the method . it is hoped that the present contribution will be helpful to point out some pitfalls and to stimulate further discussions that will facilitate the solution of these issues . the precision and performance of phase - field models have been greatly enhanced in the last decade by a detailed control of their properties . phase - field models are rooted in the mean - field description of spatially diffuse interfaces by order parameters . however , to be useful for simulating microstructure formation in solidification , phase - field models need to bridge the scale gap between the thickness of the physical solid - liquid interfaces and the typical scale of the microstructures . this is achieved by increasing the interface width in the model , sometimes by several orders of magnitude . obviously , this procedure magnifies any physical effect that is due to the diffuseness of the interface . therefore , to guarantee precise simulations , all these effects have to be controlled and , if possible , eliminated . the privileged tool to achieve this is the so - called _ thin - interface limit _ : the equations of the phase - field model are analysed under the assumption that the interface thickness is much smaller than any other physical length scale present in the problem , but otherwise arbitrary . the procedure of matched asymptotic expansions then yields the effective boundary conditions valid at the macroscale , which contain all effects of the finite interface thickness up to the order to which the expansions are carried out . this procedure was pioneered by karma and rappel , who analysed the symmetric model of solidification ( equal diffusion constants in the solid and the liquid ) and obtained a thin - interface correction to the expression of the kinetic coefficient @xcite . the use of this result has made it possible to carry out quantitative simulations of free dendritic growth of a pure substance , both at high and low undercoolings @xcite . it turned out , however , that the generalisation of this method to a model with arbitrary diffusivities is far from trivial @xcite , since several new thin - interface effects appear , which can not all be eliminated simultaneously . a solution to this problem was found later for the case of the one - sided model ( zero diffusivity in the solid ) with the introduction of the so - called antitrapping current @xcite , and it was shown that quantitative simulations of alloy solidification are possible with this model @xcite , including multi - phase @xcite and multi - component alloys @xcite . recently , several extensions of the antitrapping current were put forward to generalise the approach to the case of finite diffusivity in the solid @xcite , and simulations were presented which show that the approach works well for the instability of a steady - state planar interface @xcite and for free dendritic growth @xcite . however , as will be shown below , this is only a partial solution to the problem of developing a general quantitative model , since there is a second , independent thin - interface effect that can not be removed by an antitrapping current , namely , the kapitza resistance . for the sake of concreteness , consider the standard phase - field model for the solidification of a pure substance as discussed in refs . the evolution equation for the phase field reads @xmath0 where @xmath1 is the phase field , with @xmath2 and @xmath3 corresponding to solid and liquid , respectively , @xmath4 is the relaxation time of the phase field , @xmath5 is the interface thickness , and @xmath6 is a dimensionless coupling constant . the field @xmath7 is a dimensionless temperature defined by @xmath8 , where @xmath9 , @xmath10 and @xmath11 are the melting temperature , latent heat , and specific heat , respectively . it is assumed for simplicity that @xmath11 is the same in both phases . the temperature is governed by a diffusion equation with a source term , @xmath12 + \frac 12 \partial_t h(\phi ) . \label{diffusion}\ ] ] here , @xmath13 , which satisfies @xmath14 , is a function that describes the release or consumption of latent heat during the phase transition , and @xmath15 interpolates between the thermal diffusivities of the liquid and the solid , @xmath16 and @xmath17 , @xmath18 where the interpolation function @xmath19 satisfies @xmath20 and @xmath21 . for simplicity , crystalline anisotropy has not been included in the above model because it is not necessary for the present discussion . furthermore , the equations have been stated in the language of a two - sided thermal model , but with some modifications ( as detailed in refs . @xcite ) , they also apply to the isothermal solidification of a binary alloy . in this case , @xmath7 is a dimensionless chemical potential ( conjugate to the concentration of one of the alloy components ) , and @xmath15 is the chemical diffusivity . in the following , two simple one - dimensional solutions of these equations will be analysed . the first is a steady - state planar front that propagates with constant velocity @xmath22 in the positive @xmath23 direction into a liquid of undercooling @xmath24 ( @xmath25 for @xmath26 ) , and leaves behind a constant temperature . this solution only exists if the liquid is undercooled beyond the hypercooling limit , that is , @xmath27 . the sharp - interface solution to this problem is readily obtained and reads @xmath28 for an interface located at @xmath29 ( in the frame moving with the interface ) . here , @xmath30 and @xmath31 are the limit values of the temperature when the interface is approached from the solid and the liquid side , respectively . in the standard formulation of the free boundary problem of solidification , it is assumed that the temperature is the same on the two sides of the interface , @xmath32 . then , the use of the two boundary conditions @xmath33 , where @xmath34 is the linear kinetic coefficient , and @xmath35 ( the stefan boundary condition ) determines the solution , @xmath36 ( a simple consequence of heat conservation ) , and @xmath37 . the phase - field equations can be analysed and related to this sharp - interface solution by the method of matched asymptotic expansions in the limit where the interface thickness @xmath5 is much smaller than the diffusion length @xmath38 . this calculation has been presented in detail in refs . @xcite and will not be repeated here . the essential outcome is that , in general , the two asymptotes of the bulk phases do _ not _ correspond to the same temperature . the difference is given , to the lowest order , by @xmath39 , \label{tjump1}\ ] ] where @xmath40 is the equilibrium profile of the phase field . the physical interpretation of this temperature jump is _ trapping _ : when the diffusivity decreases upon solidification , the heat generated at the rear of the interface gets trapped . in the alloy version of the model , this is nothing but the well - known solute trapping effect . indeed , in sharp - interface models of alloy solidification the chemical potential exhibits a jump at the interface when solute trapping occurs . in the phase - field model , the temperature profile through the interface is determined by the interplay between the rejection of latent heat and the diffusion away from the interface ; therefore , it is natural that the heat source function @xmath13 and the diffusivity function @xmath15 appear in eq . ( [ tjump1 ] ) . whereas , thus , this discontinuity is physically correct , it generates problems for simulations . to see this , is is sufficient to rewrite eq . ( [ tjump1 ] ) in order to make the relevant scales apparent . since the only length scale in eq . ( [ pf ] ) is the interface thickness @xmath5 , the equilibrium solution @xmath41 is a function only of the reduced variable @xmath42 . using this together with the interpolation of @xmath15 given by eq . ( [ ddef ] ) , eq . ( [ tjump1 ] ) becomes @xmath43 @xmath44\ ; d\eta\qquad { \rm and } \label{fdef}\ ] ] @xmath45 the temperature jump is thus proportional to the velocity , the interface thickness , and the difference of the two integrals ; the latter depends only on the choice of the interpolation functions . if @xmath5 is the physical interface thickness ( a few angstroms ) , this effect is negligibly small , but if the interface thickness is increased by a large factor to make simulations feasible , this leads to potentially large errors in the simulations . as discussed in detail in refs . @xcite , it is not possible to eliminate this macroscopic discontinuity simply by the choice of appropriate interpolation functions , due to other constraints not discussed here . the solution put forward in ref . @xcite and further developed in ref . @xcite is the introduction of an antitrapping current : eq . ( [ diffusion ] ) is replaced by @xmath46 where the antitrapping current @xmath47 is given by @xmath48 where @xmath49 is a shorthand for the time derivative @xmath50 , @xmath51 is the unit normal vector to the interface , and @xmath52 is a new interpolation function . this term induces a current which is directed from the solid to the liquid , and proportional to the interface velocity ( through the factor @xmath49 ) . it thus `` pushes '' heat from the solid to the liquid side of the interface when the interface moves , and can be used to adjust the temperature jump at the interface . for the one - sided model ( @xmath53 ) with the standard choices @xmath54 and @xmath55 , it was shown that a constant @xmath56 leads to a vanishing jump in @xmath7 , because it modifies the function @xmath57 in eq . ( [ fdef ] ) such that @xmath58 . thus , continuity of the temperature between the two sides of the interface ( local equilibrium ) is restored for arbitrary @xmath5 and @xmath22 , as long as the asymptotic analysis remains valid . recently , several authors have put forward generalisations of this approach @xcite for arbitrary ratio of the diffusivities . for the case analysed above ( that is , the current far inside the solid vanishes ) , they reduce to the simple prescription that the same expression for the antitrapping current can be used , but with an additional prefactor that can be written as @xmath59 , @xmath60 indeed , the asymptotic analysis shows @xcite that in this way the temperature jump can be eliminated . however , this is not the only thin - interface effect that can arise in the two - sided case . to see this , consider now a different situation , namely an immobile interface in a temperature gradient . such an interface can be easily obtained in experiments by maintaining a pure substance between two walls which are held below and above the melting temperature , respectively . when the interface is stationary , @xmath61 by definition , and eq . ( [ diffusion ] ) implies that the system is crossed by a constant heat current flowing from the liquid into the solid , @xmath62 with @xmath63 a positive constant . as before , the centre of the interface is located at @xmath29 , and the solid is located in the domain @xmath64 . this situation can be analysed without performing a perturbation expansion , since it is sufficient to integrate eq . ( [ current ] ) to obtain a solution for @xmath7 , @xmath65 where @xmath66 is the temperature at @xmath29 . the sharp - interface solution for this case is simply given by @xmath67 matching the asymptotes of the phase - field and sharp - interface expressions , it is straightforward to show that there is again a temperature jump given by @xmath68,\ ] ] this time proportional to the _ current_. if the phase - field profile is replaced by its equilibrium shape , this can be rewritten as @xmath69 with @xmath70 this temperature jump corresponds to a surface thermal resistance , also called kapitza resistance , first found for an interface between liquid helium and metal @xcite . indeed , in a sharp - interface picture it is generally necessary to assign a surface resistance to an interface for a complete description of heat transfer , because transport through an interface can be decomposed into three elementary steps : transport in one bulk phase , crossing of the interface , and transport in the other phase . the surface resistance describes the kinetics associated with the crossing of the interface ( its inverse is sometimes referred to as the interfacial transfer coefficient ) . it is characterised either by the value of the resistance , @xmath71 , or by a length that is obtained by dividing this resistance by the conductivity of the liquid phase . here , this characteristic length is simply @xmath72 , which is of the order of the interface thickness . since this quantity is actually an interface excess of the inverse diffusivity ( in complete analogy to the interface excesses for equilibrium quantities obtained by the well - known gibbs construction ) , it can also be negative this does not violate the laws of thermodynamics because the _ local _ transport coefficients are strictly positive . if the surface resistance is finite , the temperature in the sharp - interface model is _ not _ continuous at the interface , but exhibits a jump that is proportional to the current crossing the interface . in the alloy version of the model , this corresponds to a jump in chemical potential that is proportional to the solute flux @xcite . such discontinuities have been thoroughly investigated @xcite , and can be measured in experiments @xcite and detected in molecular dynamics simulations @xcite for solid - liquid interfaces . thus , like the trapping effect , the surface resistance is a natural effect that is proportional to the interface thickness . if the interface thickness is to be upscaled , it should therefore also be eliminated . however , is is immediately clear that this effect can not be eliminated by any antitrapping current proportional to @xmath49 as given by eq . ( [ antinew ] ) : since the interface does not move , @xmath73 and the antitrapping current vanishes , independently of the current @xmath63 that crosses the interface . the authors of both refs . @xcite have recognised the importance of the current @xmath63 . they have developed generalised expressions for the antitrapping current with coefficients that depend on the value of @xmath63 . as long as the interface velocity remains non - zero , the formal asymptotic analysis shows that it is still possible to eliminate the temperature jump . however , for a fixed current @xmath63 , the expressions of the coefficients diverge when @xmath22 tends to zero , such that the asymptotic analysis is not valid in this limit . thus , it seems unlikely that this approach can be used as a robust method for simulations . in summary , there exist two independent thin - interface effects , one proportional to @xmath22 , and one proportional to @xmath63 . on a very fundamental level , this is just the consequence of the fact that the interface motion is driven by a diffusion equation , which has two independent boundary conditions . the corresponding physical quantities are the currents on the two sides of the interface , or one current and the velocity . a general solution to eliminate both thin - interface effects ( which are linearly independent ) does not seem to exist at this moment , but the above considerations can at least be used to obtain simple criteria when the prescription of eq . ( [ antinew ] ) can be used . indeed , eqs . ( [ tjump1a ] ) and ( [ tjump2 ] ) show that if @xmath74 ( note that , since @xmath7 is dimensionless , @xmath63 has the dimension of a velocity ) , the kapitza effect is much smaller than the trapping effect , and can thus be neglected . this is generally the case for equiaxed dendritic growth , in which the gradients outside the growing dendrite , which determine the growth speed , are much larger than the gradients inside the solid . indeed , it was shown in ref . @xcite that eq . ( [ antinew ] ) works well in this case . however , problems might arise in the case of alloy solidification in a temperature gradient or for multicomponent alloys with widely different solute diffusivities , since in this case large currents of heat or certain solutes may cross an interface whose velocity is controlled by a different diffusion field . such cases have to be critically examined before simulation results can be trusted . the size and shape of the crystalline grains formed upon solidification is one of the most important factors that determine materials properties . therefore , phase - field models that are to be helpful for materials design must be capable of dealing with the evolution of polycrystals , both during solidification of individual columnar or equiaxed grains from the melt and during the subsequent evolution of the grain structure after impingement . this can be achieved using the multi - phase - field approach @xcite , in which each grain is represented by a different phase field , even if they are of the same thermodynamic phase . the properties of each individual grain boundary or interface can then be specified separately @xcite , and it has been demonstrated that good quantitative control of the grain boundary properties can be achieved @xcite . the problem of handling several hundreds or even thousands of phase fields simultaneously can be solved by recognising that only a small number of fields are important at any given point of space ( see for example @xcite ) . an alternative approach is the orientation - field method . its starting point is the remark that it would be desirable , both for efficiency and simplicity , to formulate a model that works only with a small number of field variables . indeed , the orientation of a crystal can be described by one scalar quantity ( an angle ) in two dimensions , and three scalars in three dimension ( for instance , the euler angles ) . orientation - field models for pure substances in two dimensions that work with a single phase field , an orientation field ( the local angle of the crystalline structure with respect to a fixed coordinate system ) , and the temperature field were put forward in refs . @xcite , and generalised for alloy solidification @xcite and to three dimensions @xcite . while these models are elegant and simple in their formulation and therefore hugely appealing , it is pointed out here that the evolution equation of the angle field , which takes the form of a simple relaxation equation , does not correctly describe the microscopic evolution of the orientation field since it does not take into account the connectivity of matter and the resulting geometrical conservation laws . for simplicity , anisotropy and crystallographic effects will again be neglected , and it is sufficient to consider a two - dimensional system . the dimensionless free energy of the orientation - field model is @xcite @xmath75 d\vec r , \label{modeli}\ ] ] where now @xmath76 and @xmath2 in the liquid and the solid , respectively , @xmath77 and @xmath78 are positive constants , @xmath79 and @xmath80 are monotonous functions that satisfy @xmath81 and @xmath82 , and @xmath83 is the local free energy density , with @xmath7 the same dimensionless temperature field as previously ; the standard choice is @xmath84 . recently , an alternative model was developed @xcite , @xmath85 d\vec r , \label{modelii}\ ] ] where @xmath86 is a constant . in the following , these models will be called model i and model ii . they both have some features that distinguish them from standard phase - field models . model i contains a term proportional to @xmath87 , which has a singular derivative at @xmath88 . model ii has only a regular square gradient term in @xmath89 , but it is multiplied by a singular function of the phase field @xmath1 , which diverges in the limit @xmath90 ( the solid ) . these singular features are needed to create stable grain boundary solutions , that is , localised spatial regions where the phase field departs from its solid value and the angle field exhibits rapid variations . both models have a variational structure for the dynamics of the phase field and the angle field , that is @xmath91 @xmath92 which means that both @xmath1 and @xmath93 evolve such as to follow the gradient of the free energy , with @xmath94 and @xmath95 being the corresponding mobilities ( which may be functions of the fields ) . in the following , it will be shown that eq . ( [ thetaeq ] ) is incorrect for coherent crystalline matter . to illustrate the problems with this equation of motion , it is again useful to analyse a simple one - dimensional situation , which is a tricrystal . a slab of crystalline orientation @xmath96 is sandwiched between two crystals of identical orientation @xmath97 , as shown in the left side of fig . the two crystals on the sides of the system are assumed to be clamped to a substrate , that is , @xmath97 for all times . in both models , this initial condition evolves with time : the orientation of the central slab remains homogeneous , but changes with time to approach the orientation of the outer crystals . the final state is a uniform solid of orientation @xmath97 : the central slab has disappeared . of course , this process can take place since it corresponds to a minimisation of the free energy : the two grain boundaries with their positive grain boundary energy are eliminated . however , the pathway of this dynamics is not appropriate for the evolution of a coherent crystal . in fact , eq . ( [ thetaeq ] ) corresponds to the dynamics of matter which has orientational , but no positional order , such as a liquid crystal . indeed , if in model i the term proportional to @xmath87 is omitted or in model ii the singular coupling function is replaced by a regular one , the resulting model can be mapped to the standard landau - de gennes model for nematic liquid crystals in two dimensions @xcite . the free energies in eqs . ( [ modeli ] ) and ( [ modelii ] ) have been designed to stabilise grain boundaries , which do not exist in a nematic liquid crystal . the energetics of the models are thus quite different from liquid crystals . in contrast , the type of the dynamics has stayed the same . to understand where is the difference in dynamics between liquid crystals and crystals , consider the elongated molecules of a nematic liquid crystal characterised by a director field of a certain orientation @xmath96 . since the molecules have no bonds , it is possible to change the local orientation while keeping the centres of mass fixed , by just making each molecule rotate around its centre of mass ( of course , in a dense liquid crystal , this exact procedure is not possible because of steric exclusion , but the director can still be changed with only short - range displacements of the centres of the molecules ) . the system is thus free to _ locally _ change orientation in order to lower its free energy , and thus follows eq . ( [ thetaeq ] ) . this is obviously not the case in crystalline matter : it is not possible to rotate a unit cell without displacing the surrounding neighbours , because bonds ( or , more generally , the positional ordering of elements ) define a connectivity . it is easy to grasp that the evolution depicted in fig.[fig1 ] is impossible if the connectivity of the central slab is preserved . thus , a consistent evolution equation for @xmath93 has to take into account this connectivity , or , in other words , the evolution of the positions . this is , in general , a complicated undertaking . two elementary situations where it easy to obtain an equation are ( i ) rigid body rotation , in which case the ( advected ) time derivative of the local angle is given by the curl of the local velocity field , or ( ii ) purely elastic deformations of the solid , in which case the orientation is not an independent quantity but can be deduced from the elastic displacement field . here , a third possibility will be briefly discussed , namely , plastic deformation . this corresponds precisely to a change in the connectivity of matter . if the matter in question can be considered reasonably crystalline ( as opposed to , for example , an amorphous material ) , its geometry can formally always be described by a density of dislocations , which are singularities of the displacement field if a perfect crystal is taken as the reference state . if , furthermore , grain boundaries remain coherent ( that is , no grain boundary sliding takes place ) , the evolution of the local orientation can be linked to the motion of dislocations . a complete description is far outside of the scope of this article ; the interested reader is referred to ref . @xcite for a detailed introduction to the continuum theory of defects . here , only two simple examples will be qualitatively treated for illustration . consider again the tricrystal configuration . in the sketch shown in fig . [ fig2 ] , only one set of crystal planes is shown for clarity , and the central slab has a small misorientation with respect to the outer crystals . in this situation , the two low - angle grain boundaries consist of individual edge dislocations . the inner crystal can now rotate by an elementary process : take one of the edge dislocations of the left grain boundary ( marked by a circle ) and make it glide towards the other grain boundary . this process involves only local reconnection events . when the dislocation arrives at the right grain boundary , it can annihilate with a dislocation of the opposite sign . as a result , one dislocation has disappeared from each grain boundary . of course , this process can repeat itself until no dislocation is left , and the grain boundaries have disappeared . it should be stressed that this pathway for rearrangement exhibits large energy barriers , since the elastic energy of a single dislocation is much higher in the centre of the slab than at its original position within the grain boundary . therefore , if only thermal fluctuations are driving this process ( no external strains ) , it will be extremely slow . on a more quantitative level , the misorientation through a grain boundary is linked to the density of dislocations by simple geometrical arguments . therefore , it is natural that the misorientation is lowered when the dislocation density in the grain boundaries decreases . furthermore , it is obvious that the rotation rate of the central slab is proportional to the current of dislocations crossing the crystal . thus , a consistent equation of motion for the orientation should be based on the evolution of the dislocation density . however , the development of such an equation is a difficult task , because the motion of dislocations is determined by their complicated elastic interactions , as well as by external strain and interactions with other defects . despite intense activity on the phase - field modelling of defects , elasticity , and plasticity ( see @xcite for a recent overview ) , such an equation seems at present out of reach . let us now come back to the outcome of the simulations for the tricrystal configuration . the functional derivative of the gradient term in eq . ( [ modeli ] ) of model i generates a non - local diffusion equation for the angle field , which has to be regularised as described in ref . @xcite . for a constant mobility , the nonlocal interaction between the grain boundaries leads to a rotation rate that is almost independent of the distance between the grain boundaries . in model ii , the rotation rate of the central crystal decreases exponentially with the distance between the grain boundaries @xcite . in both cases , the central slab eventually disappears . while , quantitatively , neither of these evolutions is likely to be accurate , qualitatively the result is the same as the one achieved by dislocation motion . to see that there can be qualitative differences between the two dynamics , consider now a circular grain of orientation @xmath98 inserted in an infinite monocrystal of orientation @xmath97 . suppose that the misorientation ( which is equal to @xmath98 ) is small , such that the grain boundary is made of individual dislocations separated by a typical distance @xmath99 which is much larger than the lattice spacing . furthermore , suppose that the grain radius @xmath100 is large , @xmath101 , such that on the scale of the grain the boundary can still be described as a continuous line . for simplicity , disregard any anisotropy in the grain boundary energy or mobility . then , the grain will shrink by standard motion by curvature , and the dislocations will simply move towards the centre of the grain . note that the motion of the dislocations might not be strictly radial due to their coupling to the crystal structure ; however , this does not change the present discussion , as long as no annihilation of dislocations takes place . indeed , in this case , the total number of dislocations is conserved , and the dislocation density is simply proportional to @xmath102 , which _ increases _ with time as the grain shrinks . this means that the misorientation also increases with time , and if the outer crystal is fixed , the circular inner grain has to perform a rigid body rotation away from the orientation of the outer crystal . this seems surprising at first , since for low - angle grain boundaries the grain boundary energy is an increasing function of the misorientation . however , this process is perfectly possible if it leads to a decrease of the total energy of the grain boundary , which is given by @xmath103 , with @xmath104 the misorientation - dependent grain boundary energy . its time derivative is @xmath105,\ ] ] where @xmath106 is the derivative of @xmath107 with respect to the misorientation . the evolution can thus take place if the first term , which is always negative since @xmath108 , is large enough to outweigh the second one , which is positive . in that case , the geometrical constraints thus predict an increase of @xmath98 with time . the orientation - field models make exactly the opposite prediction : since the angle field evolves _ locally _ such as to lower the energy , the misorientation of the inner grain should _ decrease _ with time . recently , this situation was investigated by numerical simulations @xcite using the phase - field crystal model @xcite , which gives a faithful microscopic picture of dislocations . an increase of the misorientation with time was observed , consistent with the geometrical constraints . a previous study that had compared phase - field and molecular dynamics simulations @xcite and had reached different conclusions was limited to high misorientations , such that the above hypotheses were not satisfied . in conclusion , the simple relaxation equation for the angle field , eq . ( [ thetaeq ] ) is not consistent with the coherent crystalline structure of matter , and can sometimes lead to predictions that are even qualitatively wrong . for practical purposes , the quantitative importance of the committed errors might be small when the evolution of a large - scale grain structure is considered , but this has to be confirmed for each case at hand . it is worth mentioning that orientation - field models have been used to investigate the interplay between the positional and orientational degrees of freedom during the solidification of spherulites @xcite or in the presence of foreign - phase particles @xcite . these studies were performed with a vanishing orientational mobility @xmath95 in the solid , and are thus not affected by the problem pointed out here . indeed , in the interfacial region where the structure of the solid in not yet fully established , the concept of a rotational mobility is valid . many phase - field simulations include fluctuations , which are often introduced in a purely qualitative way to trigger instabilities or to create some disorder in the geometry of the microstructures . the role of fluctuations has been investigated more quantitatively in connection with the formation of sidebranches in free dendritic growth @xcite . the standard approach is to include fluctuations as langevin terms in the field equations , with coefficients deduced from the fluctuation - dissipation theorem . before proceeding further , this procedure will be summarised . after inclusion of noise , eqs . ( [ pf ] ) and ( [ diffusion ] ) for the solidification of a pure substance become ( see ref . @xcite for details ) @xmath109 @xmath110 where @xmath111 is assumed ( symmetric model ) , and lengths and times have been scaled by the interface thickness @xmath5 and the phase - field relaxation time @xmath4 , respectively . here , @xmath112 and @xmath113 are random fluctuations of the phase field and random microscopic heat currents , respectively . they are assumed to be @xmath114-correlated in space and time , @xmath115 @xmath116 with dimensionless amplitudes @xmath117 and @xmath118 given by @xmath119 @xmath120 where @xmath99 is the spatial dimension , and the quantity @xmath121 is determined by materials properties only , @xmath122 where @xmath123 , @xmath9 , @xmath11 , @xmath10 , and @xmath124 are boltzmann s constant , the melting temperature , the specific heat , the latent heat , and the capillary length , respectively . the latter is given by @xmath125 , where @xmath107 is the surface free energy . with the help of this expression for the capillary length , @xmath121 can be rewritten as @xmath126 , which makes its physical meaning more transparent : it is the ratio of the thermal energy and a capillary energy scale , and can thus be seen as a non - dimensional temperature . in a finite - difference discretization of timestep @xmath127 and grid spacing @xmath128 , the noise terms are implemented by drawing , at each grid point @xmath129 and for each time step @xmath130 , independent gaussian random variables of correlation @xmath131 where @xmath132 and @xmath133 are now kronecker symbols , and similarly for @xmath134 . this procedure was shown to yield the correct interface fluctuations at equilibrium in numerical simulations @xcite . an obvious question then arises , namely , can this method also be used to explore nucleation ? phase - field methods have been used recently to investigate homogeneous and heterogeneous nucleation , both in single - phase and multi - phase systems ( see , for example , @xcite ) . in particular , it was found that for high undercoolings , diffuse - interface models yield better agreement with experiments than classical nucleation theory , since the size of the nuclei is not much larger than the thickness of the diffuse interfaces ; therefore , the free energy barriers calculated in phase - field models can differ significantly from classical nucleation theory . is it sufficient , then , to add thermal noise as described above to obtain quantitative simulations of nucleation processes ? the answer to this question is negative . the reason is that , for strong noise , field equations like the phase - field model are renormalized by the fluctuations . this is a well - known fact in statistical field theory , but its implications do not yet seem to have been fully appreciated in the phase - field community . therefore , it is useful to briefly sketch a few calculations that can be found in textbooks ( see , for example , @xcite ) . they are , therefore , neither new nor complete ; however , they will prepare the ground for understanding the conclusions on the phase - field method at the end of this section . instead of the full phase - field model , consider a single equation for a scalar field @xmath1 that reads @xmath135 where @xmath136 is a non - conserved noise that is @xmath114-correlated , @xmath137 with @xmath138 a suitably non - dimensionalized temperature ( such as @xmath121 , see the discussion after eq . ( [ fexpt ] ) ) , and the deterministic part of the equation derives from the functional @xmath139d\vec r , \label{hamil}\ ] ] where @xmath140 is a local potential of the field @xmath1 ( lengths , times , and energies are dimensionless ) . it is important to stress that @xmath141 is _ not _ a free energy functional , but the hamiltonian of the field theory . ( [ langevin ] ) generates an evolution in which each microscopic field configuration appears with probability @xmath142 in the limit of infinite evolution time . here , @xmath143 is the partition function , @xmath144 and @xmath145 denotes a functional integration over the field @xmath1 . the free energy is then obtained by the standard formula @xmath146 . the free energy can be calculated exactly for the case of a quadratic potential , @xmath147 , where @xmath148 is a constant . to carry out the calculations , it is useful to consider a discrete version of the model . for simplicity , consider as the domain of integration @xmath22 a @xmath99-dimensional torus of size @xmath149 with periodic boundary conditions . when this system is discretized with the usual finite difference formulae using @xmath150 grid points in each direction and hence a grid spacing @xmath151 , the integral in eq . ( [ hamil ] ) becomes a sum over a finite number of variables . in one dimension , @xmath152 , \label{hamilex}\ ] ] with the convention that @xmath153 . for the discretized system , the functional integration in eq . ( [ funcint ] ) is replaced by a simple integration over the field variables at each grid point , @xmath154 since the hamiltonian of eq . ( [ hamilex ] ) is a quadratic form in the @xmath155 s , this is a @xmath150-dimensional gaussian integral which can be evaluated using standard formulae . the most convenient way is to use a discrete fourier transform to find the eigenvalues of the quadratic form . the final result for the free energy is ( up to a constant that can be dropped ) @xmath156 for dimensions @xmath157 , the same calculation can be repeated without difficulties , and the result is @xmath158 where the sum is now over an independent index @xmath159 for each dimension ( @xmath160 ) , and is normally taken over the first brillouin zone , @xmath161 . for an arbitrary potential @xmath140 , an exact calculation is generally impossible . statistical field theory has developed sophisticated approximation methods , in particular perturbation expansions . formally , every potential can be written as a perturbation of a quadratic potential . the perturbation expansion ( where the expansion parameter is the temperature , which sets the fluctuation strength ) is cumbersome and usually visualised in terms of diagrams @xcite . fortunately , the first order result can be understood in a relatively simple manner if we are interested in homogeneous systems . more precisely , consider the spatial average of the field , @xmath162 which is a fluctuating quantity . the probability distribution of @xmath163 can be written as @xmath164 where @xmath165 is the free energy density . to first order in the perturbation expansion , @xmath166 where the correction to the original ( `` bare '' ) potential @xmath167 is identical to the exact result for the quadratic potential , with the constant @xmath168 replaced by the second derivative of the bare potential , taken at @xmath163 . this results from a quadratic approximation ( second - order taylor expansion ) of the bare potential around @xmath163 . the result @xmath165 is a renormalized potential for @xmath163 . ) and from numerical simulations , for @xmath169 , @xmath170 , @xmath171 . only the part close to one of the potential wells is shown . the zero of @xmath172 was chosen at the minimum of the renormalized potential . the bin size for the histograms was @xmath173 . ] these calculations can be readily verified numerically . as an example , the standard double - well potential was used , @xmath174 ( usually called @xmath175-potential in the field - theory literature ) , and simulated in a two - dimensional system of size @xmath176 with a grid spacing of @xmath170 and @xmath169 , using the standard discretization method described above with a timestep @xmath177 , and an initial condition @xmath178 . in time intervals of @xmath179 , @xmath163 was calculated , and in total @xmath180 points were sampled . then , the free energy can be obtained by making a histogram of the values of @xmath163 , and taking the logarithm of the counts ( the normalisation contributes only a constant to @xmath172 and can be disregarded ) . the comparison between the simulation and the prediction of eq . ( [ theory ] ) in fig . [ fig3 ] shows excellent agreement . it can be seen that the minimum of the free energy density is shifted with respect to its `` bare '' value @xmath181 . this can be understood intuitively by the following reasoning . the system starts in the well of the `` bare '' potential , at @xmath181 . the random fluctuations push the system in both directions with equal probability , but since the potential is asymmetric , the restoring force is larger for fluctuations towards @xmath182 than towards @xmath183 ; therefore , smaller values are more likely to occur . in the example chosen here , the shift is small ( the minimum is close to @xmath184 ) , but for increasing temperature , the correction becomes larger and larger ( for an example of such simulations , see @xcite ) , and eventually a phase transition occurs ( the double well disappears ) ; in this regime , of course the first - order perturbation result is inaccurate . the correction also depends on the discretization . this is physically sound : a finer discretization introduces more degrees of freedom per unit volume in the discretized system , and hence allows for more fluctuation modes that contribute to the free energy . with a slight change of perspective , this can also be seen as the natural result of a coarse - graining procedure . indeed , if the free energy is calculated from a given microscopic model by coarse - graining ( averaging ) over cells with a certain size @xmath128 larger than the size of the microscopic elements , both the free energy density and the amplitude of the fluctuations that remain after the averaging ( which thus have a wavelength larger than @xmath128 ) depend on the choice of @xmath128 , as was recently demonstrated explicitly for a simple lattice gas model @xcite . however , a problem arises in the continuum picture : it is easy to verify that , when the grid spacing @xmath128 tends to zero , the sum in eq . ( [ theory ] ) diverges for @xmath185 . this is a classical example of an ultraviolet divergence . thus , eq . ( [ langevin ] ) has no continuum limit , and if it is written down in continuum language , it is implicitly understood that an ultraviolet cutoff must be specified . a reasonable physical value for a cutoff in condensed - matter systems is the size of an atom . let us now discuss the implications of these facts for phase - field modelling . even though the above calculation have not been carried out for the full model ( @xmath1 and @xmath7 ) , it is clear that renormalization occurs . if a phase - field model is seen as a simulation tool for a problem that is defined in terms of macroscopic parameters , the relevant quantities that need to be adjusted in the model are the renormalized ones . for instance , thermophysical properties are usually interpolated assuming that the phase field takes fixed values in the bulk phases ( @xmath186 ) . if , on average , this is no longer the case , such as in the example of fig . [ fig3 ] , these interpolations become incorrect . an obvious idea to cure this problem is to choose the `` bare '' potential such that the renormalized potential has the desired properties . for the @xmath175-potential , which is renormalizable , one may choose @xmath187 and determine the constants @xmath188 and @xmath189 by the two conditions @xmath190 and @xmath191 using eq . ( [ theory ] ) . for the example shown above , the values @xmath192 and @xmath193 indeed restore the correct bulk properties . however , in a quantitative phase - field model , the macroscopic properties not only of the bulk phases , but also of the interfaces need to be controlled . it is far from obvious that the above procedure , designed for homogeneous systems , will work . this is even more so for the critical nucleus needed to evaluate the nucleation barrier . it is instructive to examine some orders of magnitude . in nickel , the value of @xmath121 is @xmath194 @xcite , of order unity ; it can be expected that this value is of similar order of magnitude for other substances with microscopically rough interfaces . an inspection of eqs . ( [ fnoisedef][fudef ] ) reveals that _ if _ phase - field simulations are carried out with the `` natural '' interface thickness , which is of the order of the capillary length @xmath124 , the fluctuations are of order unity ( recall that @xmath117 and @xmath118 are equivalent to @xmath138 in the numerical example ) , and renormalization can not be neglected . this is a natural consequence of the fact that real solid - liquid interfaces do indeed exhibit very strong fluctuations , as evidenced from molecular dynamics simulations @xcite ; therefore , a mean - field approximation ( such as the phase - field model without noise ) is not accurate . in contrast , if ( as in refs . @xcite ) a much larger interface thickness is used , the fluctuation strength is greatly reduced , and the difference between `` bare '' and renormalized free energy is small . note , however , that even in this limit a sufficient refinement of the grid would create noticeable fluctuation corrections . we are thus faced with the conclusion ( opposite to the usual point of view in phase - field modelling ) that the use of the simple prescription of ref . @xcite is more precise for larger interface thickness and coarser grids . it is noted in passing that the concept of the sharp - interface limit , central for the asymptotic analysis in the deterministic case , has to be reexamined because a new length scale ( the microscopic cutoff for the fluctuations ) has been introduced . in conclusion , it is clear that the use of the phase - field method with fluctuations is subject to caution , at least on small length scales . to gain a better understanding , the fluctuation effects on the couplings of the phase - field variables need to be investigated furthermore , a good control of the discretization effects needs to be achieved ; the introduction of a simple cutoff will most likely be insufficient , since the renormalized free energy of eq . ( [ theory ] ) also depends on the grid structure . while a large body of results on these topics can certainly be found in the field - theory literature , the development of quantitative models for specific materials remains a challenging task . in this paper , some open questions concerning various aspects of phase - field modelling of solidification have been discussed , and potential future directions of research have been outlined . the selection of topics is necessarily incomplete , both concerning the problems and the potential solutions . for instance , the rapid development of the phase - field crystal approach @xcite and related methods currently opens up interesting new perspectives for the modelling of polycrystals , which are not discussed further here . the common point of the topics treated here is that they illustrate the dual nature of the phase - field method . on the one hand , it is a genuine representation of condensed - matter systems and their evolution in terms of order parameters on a mesoscopic scale . on the other hand , with the help of mathematical analysis , it can be turned into an efficient simulation tool for the solution of free boundary problems . as in the past , the development of more efficient and robust models for materials modelling will most likely benefit from the pursuit and confrontation of _ both _ of these two complementary viewpoints . therefore , the further development of the phase - field method remains an exciting research topic at the frontiers of physics , mathematics , and materials science . i thank jean - marc debierre , tristan ducousso , alphonse finel , lszl grnsy , herv henry , alain karma , yann le bouar , jesper mellenthin , tams pusztai , and james warren for stimulating discussions on these and many other topics . t. ducousso , _ tude de la solidification dirige par la mthode du champ de phase : comparaison thorie exprience pour un alliage binaire dilu _ , ph.d . thesis , universit paul czanne , marseille , france , 2009 .

three different topics in phase - field modelling of solidification are discussed , with particular emphasis on the limitations of the currently available modelling approaches . first , thin - interface limits of two - sided phase - field models are examined , and it is shown that the antitrapping current is in general not sufficient to remove all thin - interface effects . second , orientation - field models for polycrystalline solidification are analysed , and it is shown that the standard relaxational equation of motion for the orientation field is incorrect in coherent polycrystalline matter . third , it is pointed out that the standard procedure to incorporate fluctuations into the phase - field approach can not be used in a straightforward way for a quantitative description of nucleation .

arxiv

the fundamental atmospheric ( effective temperature , surface gravity , and metallicity ) and physical ( mass and age ) parameters of stars provide the major observational foundation for chemo - dynamical studies of the milky way and other galaxies in the local group . with the dawn of large spectroscopic surveys to study individual stars , such as segue @xcite , rave @xcite , gaia - eso @xcite , and hermes @xcite , these parameters are used to infer the characteristics of different populations of stars that comprise the milky way . stellar parameters determined by spectroscopic methods are of a key importance . the only way to accurately measure metallicity is through spectroscopy , which thus underlies photometric calibrations ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , while high - resolution spectroscopy is also used to correct the low - resolution results ( e.g. , * ? ? ? the atmospheric parameters can all be estimated from a spectrum in a consistent and efficient way . this also avoids the problem of reddening inherent in photometry since spectroscopic parameters are not sensitive to reddening . the spectroscopic parameters can then be used alone or in combination with photometric information to fit individual stars to theoretical isochrones or evolutionary tracks to determine the stellar mass , age , and distance of a star . a common method for deriving the spectroscopic atmospheric parameters is to use the information from fe and fe absorption lines under the assumption of hydrostatic equilibrium ( he ) and local thermodynamic equilibrium ( lte ) . many previous studies have used some variation of this technique ( e.g. , ionisation or excitation equilibrium ) to determine the stellar atmospheric parameters and abundances , and henceforth distances and kinematics , of fgk stars in the milky way . for example , some have used this procedure to estimate the effective temperature , surface gravity , and metallicity of a star ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , while others use photometric estimates of effective temperature in combination with the ionisation equilibrium of the abundance of iron in lte to estimate surface gravity and metallicity ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . however , both observational ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and theoretical evidence ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) suggest that systematic biases are present within such analyses due to the breakdown of the assumption of lte . more recently , @xcite and @xcite quantified the effects of non - local thermodynamic equilibrium ( nlte ) on the determination of surface gravity and metallicity , revealing very substantial systematic biases in the estimates at low metallicity and/or surface gravity . it is therefore extremely important to develop sophisticated methods , which reconcile these effects in order to derive accurate spectroscopic parameters . this is the first in a series of papers , in which we develop new , robust methods to determine the fundamental parameters of fgk stars and then apply these techniques to large stellar samples to study the chemical and dynamical properties of the different stellar populations of the milky way . in this work , we utilise the sample of stars selected from the rave survey originally published in ( * ? ? ? * hereafter r11 ) to formulate the methodology to derive very accurate atmospheric parameters . we consider several temperature scales and show that the balmer line method is the most reliable among the different methods presently available . further , we have developed the necessary tools to apply on - the - fly nlte corrections to fe lines , utilising the grid described in @xcite . we verify our method using a sample of standard stars with interferometric estimates of effective temperature and/or _ hipparcos _ parallaxes . we then perform a comprehensive comparison to standard 1d , lte techniques for the spectral analysis of stars , finding significant systematic biases . nlte effects in iron are most prominent in low - metallicity stars @xcite . we therefore chose the metal - poor sample from r11 for our study . these stars were originally selected for high - resolution observations based on data obtained by the rave survey in order to study the metal - poor thick disk of the milky way . spectral data for these stars were obtained using high - resolution echelle spectrographs at several facilities around the world . full details of the observations and data reduction of the spectra can be found in r11 . briefly , all spectrographs delivered a resolving power greater than 30,000 and covered the full optical wavelength range . further , nearly all spectra had signal - to - noise ratios greater than @xmath4 per pixel . the equivalent widths ( ews ) of both fe and fe lines , taken from the line lists of @xcite and @xcite , were measured using the ares code @xcite . however , during measurement quality checks , we found that the continuum was poorly estimated for some lines . we therefore determined ews for these affected lines using hand measurements . we computed the stellar parameters for each star using two different methods . in the first method , which is commonly used in the literature , we derived an effective temperature , @xmath5 , surface gravity , @xmath6 , metallicity , @xmath7 , and microturbulence , @xmath8 , from the ionisation and excitation equilibrium of fe in lte . this is hereafter denoted as the lte - fe method . we used an iterative procedure that utilised the ` moog ` analysis program @xcite and 1d , plane - parallel ` atlas - odf ` model atmospheres from kurucz computed under the assumption of lte and he . in our procedure , the stellar effective temperature was set by minimising the magnitude of the slope of the relationship between the abundance of iron from fe lines and the excitation potential of each line . similarly , the microturbulent velocity was found by minimising the slope between the abundance of iron from fe lines and the reduced ew of each line . the surface gravity was then estimated by minimising the difference between the abundance of iron measured from fe and fe lines . iterations continued until all of the criteria above were satisfied . finally , @xmath7 was chosen to equal the abundance of iron from the analysis . our results for this method are described in section [ sec - lte ] . the second method , denoted as the nlte - opt method , consists of two parts . first , we determined the optimal effective temperature estimate , @xmath9 , for each star ( see section [ sec - temp ] for more details ) . then , we utilised ` moog ` to compute a new surface gravity , @xmath10 , metallicity , @xmath11 , and microturbulence , @xmath12 . this was done using the same iterative techniques as the lte - fe method , that is the ionisation balance of the abundance of iron from fe and fe lines . there are , however , three important differences . first , the stellar effective temperature was held fixed to the optimal value , @xmath9 . second , we restricted the analysis to fe lines with excitation potentials above 2 ev , since these lines are less sensitive to 3d effects as compared to the low - excitation lines ( see the discussion in * ? ? ? third , the abundance of iron from each fe line was adjusted according to the nlte correction for that line at the stellar parameters of the current iteration in the procedure . the nlte corrections were determined using the nlte grid computed in @xcite and applied on - the - fly via a wrapper program to ` moog ` . note that the nlte calculations presented in @xcite were analogously calibrated using the ionisation equilibria of a handful of well - known stars . our extended sample , including more stars with direct measurements of surface gravity and effective temperature , provide support for the realism of this calibration . the grid extends down to @xmath13 . we imposed a routine which linearly extrapolated the nlte corrections to below this value . the results of extrapolations were checked against nlte grids presented in @xcite and no significant differences were found . further , @xcite found very small nlte corrections for fe lines . we therefore do not apply any correction to the fe lines . iterations continued until the difference between the average abundance of iron from the fe lines and the nlte - adjusted fe lines were in agreement ( within @xmath14 dex ) and the slope of the relationship between the reduced ew of the fe lines and their nlte - adjusted iron abundance was minimised . sections [ sec - temp ] and [ sec - nlte ] describe our final stellar parameter estimates for this method . the initial lte - fe stellar parameters for our sample stars are listed in table [ tab - par ] . residuals in the minimizations of this technique gave typical internal errors of 0.1 dex in both @xmath6 and @xmath7 and @xmath15 k in @xmath5 . as we show in the following sections , these small internal errors can be quite misleading as they are not representative of the actual accuracy of stellar parameter estimates . often , especially in metal - poor stars , estimates of @xmath16 , @xmath17 , and @xmath18 , that result from this method are far too low when compared to other , more accurate data ( cf . , r11 ) . @rrrrrrrrrrrrrrrrr@ & & & & & & & & + + star & @xmath16 & @xmath19 & @xmath17 & @xmath18 & @xmath20 & @xmath21 & @xmath22 & @xmath23 & @xmath24 & @xmath25 & @xmath26 & @xmath16 & @xmath19 & @xmath17 & @xmath18 & @xmath20 + & ( k ) & ( k ) & ( @xmath27 ) & ( @xmath27 ) & & & ( k ) & ( k ) & ( @xmath28 k ) & ( k ) & ( k ) & ( k ) & ( k ) & ( @xmath27 ) & ( @xmath27 ) & + c0023306 - 163143 & 5128 & 58 & 2.40 & -2.63 & 1.3 & & & & 5528 & 5400 & 100 & 5443 & 101 & 3.20 & -2.29 & 0.9 + c0315358 - 094743 & 4628 & 40 & 1.51 & -1.40 & 1.5 & & & & 4722 & 4800 & 100 & 4774 & 89 & 2.06 & -1.31 & 1.6 + c0408404 - 462531 & 4466 & 40 & 0.25 & -2.25 & 2.2 & & & & 4600 & & & 4600 & 90 & 1.03 & -2.10 & 2.1 + c0549576 - 334007 & 5151 & 50 & 2.53 & -1.94 & 1.3 & & & & 5379 & 5400 & 100 & 5393 & 82 & 3.16 & -1.70 & 1.1 + c1141088 - 453528 & 4439 & 40 & 0.39 & -2.42 & 2.1 & & & & 4592 & 4500 & 200 & 4562 & 123 & 1.10 & -2.28 & 1.9 + this table is published in its entirety in the electronic edition of the mnras . a portion is shown here for guidance regarding its form and content . it was found in @xcite and @xcite that taking into account nlte in the solution of excitation equilibrium does not lead to a significant improvement of the stellar effective temperature . this was also supported by our test calculations for a sub - sample of stars . fe lines formed in lte or nlte are still affected by convective surface inhomogeneities and overall different mean temperature / density stratifications , which are most prominent in strong low - excitation fe lines @xcite . using 1d hydrostatic models with either lte or nlte radiative transfer thus leads to effective temperature estimates that are too low when the excitation balance of fe lines is used ( see below ) . it is therefore important that the stellar effective temperature be estimated by other means . we used three different methods to compute the effective temperature . the first estimate , @xmath25 , was derived from the wings of the balmer lines , which is among the most reliable methods available for the effective temperature determination of fgk stars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the only restriction of this method is that for stars cooler than @xmath29 k , the wings of h lines become too weak to allow reliable determination of @xmath16 . profile fits of h@xmath30 and h@xmath31 lines were performed by careful visual inspection of different portions of the observed spectrum in the near and far wings of the balmer lines which were free of contaminant stellar lines . figures [ fig - bal3 ] and [ fig - bal5 ] show two example fits to h@xmath32 . note that the balmer lines were self - contained within a single order in each spectrum . therefore , we did not use neighbouring orders for the continuum normalisation . theoretical profiles were computed using the siu code with ` mafags - odf ` model atmospheres @xcite . same as ` atlas - odf ` ( section [ sec - par ] ) , the mafags models were computed with kurucz opacity distribution functions , thus the differences between the model atmosphere stratifications are expected to be minimal in our range of stellar parameters . for self - broadening of h lines , we used the @xcite theory . as shown by @xcite this method successfully reproduces the balmer line spectrum of the sun within @xmath33 k , and provides accurate stellar parameters that agree very well with _ hipparcos _ astrometry @xcite . the errors are obtained directly from profile fitting , and they are largely internal , @xmath34 to 100 k. in the spectrum of the metal - poor ( @xmath35 ) dwarf , j142911.4 - 053131 . the solid , red curve shows the best fit to the data ( at 5700 k ) , while the blue , dashed curves represent @xmath36 k around the fit.,width=317 ] in the spectrum of the metal - poor ( @xmath37 ) giant , j230222.8 - 683323 . the solid , red curve shows the best fit to the data ( at 5200 k ) , while the blue , dashed curves represent @xmath36 k around the fit.,width=317 ] a key advantage of the balmer lines is that they are insensitive to interstellar reddening , which affects photometric techniques ( see below ) . however , the balmer line effective temperature scale could be affected by systematic biases , caused by the physical limitation of the models . the influence of deviations from lte in the h line formation in application to cool metal - poor stars was studied by @xcite . comparing our results to the nlte estimates by @xcite for the stars in common , we obtain : @xmath38(our - m08 ) @xmath39 k ( hd 122563 metal - poor giant ) , @xmath38(our - m08 ) @xmath40 k ( hd 84937 , metal - poor turn - off ) . the difference is clearly within the @xmath16 uncertainties . on the other side , it should be kept in mind that the atomic data for nlte calculations for hydrogen are of insufficient quality and , at present , do not allow accurate quantitative assessment of nlte effects in h , as elaborately discussed by @xcite . likewise , the influence of granulation is difficult to assess . @xcite presented 3d effective temperature corrections for balmer lines for a few points on the hrd , for which 3d radiative - hydrodynamics simulations of stellar convection are available . for the sun consistent with the ` mafags - odf ` model atmospheres adopted here . ] , they find @xmath41 k , and for a typical metal - poor subdwarf with [ fe / h ] @xmath42 , @xmath43 of the order @xmath44 to @xmath45 k ( average over h@xmath30 , h@xmath31 , and h@xmath46 ) . however , in the absence of consistent 3d nlte calculations , it is not possible to tell whether 3d and nlte effects will amplify or cancel for fgk stars . thus , we do not apply any theoretical corrections to our balmer effective temperatures . currently , the only way to understand whether our balmer @xmath16 scale is affected by systematics is by comparing with independent methods , in particular interferometry . we , therefore , computed the balmer @xmath16 for several nearby stars with direct and indirect interferometric angular diameter measurements . the results are listed in table [ tab - inter ] , while we plot the difference between our balmer estimate and that from interferometry in figure [ fig - int ] . both @xmath16 scales show an agreement of @xmath47 k for stars with @xmath48 , while the balmer estimate is @xmath49 k warmer than @xmath50 at the lowest metallicities . these differences are well within the combined errors in the interferometric and balmer measurements . this suggests that deviations from 1d he _ and _ lte are either minimal , or affect both interferometric and balmer @xmath16 in exactly same way . also note , for the stars in common with @xcite , our estimates are fully consistent . .effective temperatures determined from direct interferometric measurements of angular diameters . [ cols= " > , > , > , > , > , > , > , < " , ] in figure [ fig - lte ] , we compare our final nlte - opt stellar parameters to those derived using the lte - fe method . the differences in the estimates of effective temperature , surface gravity , metallicity , and microturbulence all display clear trends with decreasing metallicity . the microturbulent velocity is underestimated by @xmath51 until @xmath52 , where @xmath8 becomes larger than @xmath12 . the differences between @xmath16 range from @xmath53 to @xmath0 k for metal - poor giants , and @xmath54 to @xmath55 k for dwarfs . the differences for @xmath56 and [ fe / h ] reach a factor of @xmath57 in surface gravity ( @xmath58 dex ) and a factor of @xmath59 in metallicity ( @xmath60 [ fe / h ] @xmath61 dex ) at [ fe / h ] @xmath62 . . the difference in effective temperature , surface gravity , and metallicity show a large systematic increase with decreasing metallicity . the dual trends seen in @xmath63 , @xmath64 , and @xmath65 are a result of the r11 effective temperature calibration , in which the authors found that stars with effective temperatures less than 4500 k only required a small correction to @xmath5 . therefore , these stars stand out in the plots.,width=317 ] figure [ fig - hrd ] illustrates how the different lte - fe and nlte - opt results can change the position of each star in the @xmath17 vs. @xmath66 plane . in addition , we have included several evolutionary tracks , computed using the garstec code @xcite , for comparison . generally , the nlte - opt estimates of surface gravity and effective temperature trace the morphology of the theoretical tracks much more accurately . several features are most notable . the nlte - opt parameters lead to far less stars that lie on or above the tip of the red giant branch , and more stars occupy the middle or lower portion of the rgb . also , stars at the turn - off and subgiant branch are now more consistent with stellar evolution calculations . figures [ fig - lte ] and [ fig - hrd ] further prompted us to determine the relative importance of the effective temperature scale versus the nlte corrections for gravities and metallicities in the nlte - opt method . we singled out the effect of the nlte corrections by deriving additional , lte - opt surface gravity and metallicity estimates using lte iron abundances combined with our @xmath9 estimate . note , as with the nlte - opt method , fe lines which have an excitation potential below 2 ev were excluded . the comparison between these lte - opt estimates and the final nlte - opt estimates are shown in figure [ fig - copt ] . as evident from this figure , solving for ionisation equilibrium in nlte also leads to _ systematic _ changes in the @xmath56 and [ fe / h ] , such that lte gravities are under - estimated by @xmath67 to @xmath68 dex , whereas the error in metallicity is about @xmath69 to @xmath70 dex . these effects are consistent with that seen in @xcite . we thus conclude that reliable effective temperatures are necessary to avoid substantial biases in a spectroscopic determination of @xmath56 and [ fe / h ] , such as displayed in figure [ fig - lte ] . we have shown here that , at present , excitation balance of fe lines with 1d hydrostatic model atmospheres in lte does not provide the correct effective temperature scale , supporting the results by bergemann et al . ( 2012 ) . on the contrary , balmer lines provide such a scale . furthermore , nlte effects on ionisation balance are necessary to eliminate the discrepancy between fe and fe lines , an effect that is present , regardless of the adopted @xmath16 . only in this way is it possible to determine accurate surface gravity and metallicity from fe lines . in this work , we explore several available methods to determine effective temperature , surface gravity , and metallicity for late - type stars . the methods include excitation and ionization balance of fe lines in lte and nlte , semi - empirically calibrated photometry ( r11 ) , and the infra - red flux method ( irfm ) . applying these methods to the large set of high - resolution spectra of metal - poor fgk stars selected from the rave survey , we then devise a new efficient strategy which provides robust estimates of their atmospheric parameters . the principal components of our method are ( i ) balmer lines to determine effective temperatures , ( ii ) nlte ionization balance of fe to determine @xmath17 and @xmath18 , and ( iii ) restriction of the fe lines to that with the lower level excitation potential greater than 2 ev to minimize the influence of 3d effects @xcite . a comparison of the new nlte - opt stellar parameters to that obtained from the widely - used method of 1d lte excitation - ionization of fe , lte - fe , reveals significant _ systematic biases _ in the latter . the difference between the nlte - opt and lte - fe parameters systematically increase with decreasing metallicity , and can be quite large for the metal - poor stars : from 200 to 400 k in @xmath16 , 0.5 to 1.5 dex in @xmath17 , and 0.1 to 0.5 dex in @xmath18 . these systematic trends are largely influenced by the difference in the estimate of the stellar effective temperature , and thus , a reliable effective temperature scale , such as the balmer scale , is of critical importance in any spectral parameter analysis . however , a disparity between the abundance of iron from fe and fe lines still remains . it is therefore necessary to include the nlte effects in fe lines to eliminate this discrepancy . the implications of the very large differences between the nlte - opt and lte - fe estimates of atmospheric parameters extend beyond that of just the characterisation of stars by their surface parameters and abundance analyses . spectroscopically derived parameters are often used to derive other fundamental stellar parameters such as mass , age and distance through comparison to stellar evolution models . the placement of a star along a given model will be largely influenced by the method used to determine the stellar parameters . for example , distance scales will change , which could affect the abundance gradients measured in the milky way ( e.g. , r11 ) , as well as the controversial identification of different components in the mw halo @xcite . we explore this in greater detail in the next paper of this series @xcite . we acknowledge valuable discussions with martin asplund , and are indebted to ulrike heiter for kindly providing interferometric temperatures for several _ gaia _ calibration stars . we also acknowledge the staff members of siding spring observatory , la silla observatory , apache point observatory , and las campanas observatory for their assistance in making the observations for this project possible . greg ruchti acknowledges support through grants from esf eurogenesis and max planck society for the firststars collaboration . aldo serenelli is partially supported by the european union international reintegration grant pirg - 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we present a comprehensive analysis of different techniques available for the spectroscopic analysis of fgk stars , and provide a recommended methodology which efficiently estimates accurate stellar atmospheric parameters for large samples of stars . our analysis includes a simultaneous equivalent width analysis of fe and fe spectral lines , and for the first time , utilises on - the - fly nlte corrections of individual fe lines . we further investigate several temperature scales , finding that estimates from balmer line measurements provide the most accurate effective temperatures at all metallicites . we apply our analysis to a large sample of both dwarf and giant stars selected from the rave survey . we then show that the difference between parameters determined by our method and that by standard 1d lte excitation - ionisation balance of fe reveals substantial systematic biases : up to @xmath0 k in effective temperature , @xmath1 dex in surface gravity , and @xmath2 dex in metallicity for stars with @xmath3 . this has large implications for the study of the stellar populations in the milky way . [ firstpage ] stars : abundances stars : late - type stars : population ii

arxiv

the effects of weak gravitational lensing by the large - scale structure have been detected in several samples of high redshift qsos , intermediate redshift galaxies , and batse grbs . in the case of point sources , qsos and grbs , weak lensing manifests itself as angular ( anti-)correlations between these sources and foreground inhomogeneously distributed mass @xcite , while in the case of galaxies weak lensing is detected through its coherent shear effect ( see @xcite for a recent review ) . in principle , there is another , more direct way of detecting weak lensing , which uses fluxes of standard candles . if the observed magnitudes of standard candles are corrected for cosmological distances then the effect of lensing can be seen : brighter sources will lie behind regions of mass density excess , while fainter ones will have mass deficits in their foregrounds . the best example of cosmological standard candle , supernovae type ia ( snia ) have been extensively observed with the purpose of determining the global geometry of the universe @xcite . nuisance effects like evolution , variations in individual sn , and gray dust extinction have been studied theoretically and observationally , and have either been corrected for or shown to be small . weak lensing , another nuisance effect has been addressed theoretically by several authors @xcite and found to be unimportant given the current uncertainties . for example , @xcite used ray tracing through cosmological simulations and found that the lensing induced dispersions on truly standard candles are @xmath3 and @xmath4 mag at redshift @xmath5 and @xmath6 , respectively , in a cobe - normalized cold dark matter universe with @xmath7 , @xmath8 , @xmath9km / s / mpc and @xmath10 . these are small variations compared to the current errors which are @xmath11 mag . even though weak lensing effects are estimated to be small for @xmath12 , they are predicted to be non - negligible for higher redshift sources , so it is not surprising that the highest redshift snia , sn1997ff at @xmath13 has been examined by several authors @xcite for the effects of weak lensing due to galaxies along the line of sight . present day high-@xmath0 snia samples are dominated by lower redshift sne , and so have not been examined for the effects of lensing . the main goal of this work is to determine if the observed fluxes of the cosmologically distant snia have suffered significantly from lensing induced ( de- ) amplifications . the largest homogeneous compilation of snia has been recently published by @xcite : table 15 of that paper contains 74 sne at @xmath14 . the authors use four different light curve fitting methods ( mlcs , @xmath15 , modified dm15 , and bayesian adapted template match ) to estimate distances to sne . the final quoted distance is the median of the estimates of the four individual methods , and the uncertainty is the median of the error of the contributing methods . the analysis presented in @xcite yields values of the global cosmological parameters ; if a flat model is assumed , then @xmath16 and @xmath17 . we use these values in all the analysis of the present paper . as tracers of foreground mass density we use apm galaxies @xcite . apm provides near full coverage of the sky in the northern and southern hemispheres , at @xmath18 . in our analysis we use only the central @xmath19 of apm plates . since the plate centres are separated by @xmath20 , there exist small portions of the sky that are not covered by any plate . as a result of these cuts , only 55 of the 74 sne lie on the usable parts of apm plates . the median redshift of the 55 sne is @xmath21 . is not in our sample : it fell in the cracks between the apm plates . ] since most of the sne have rather low redshifts , care must be taken to ensure that galaxies are foreground to the sne . furthermore , because sne span a large range of nearby redshifts , from @xmath22 to @xmath23 , the optimal lens redshift @xmath24 will depend on @xmath25 much more compared to a typical case where sources ( say , qsos ) are at @xmath26 and so the redshift of optimal lenses is roughly independent of @xmath25 . in our analysis we adjust @xmath24 for each sn source by selecting the appropriate limiting apparent magnitude , mag@xmath27 for apm galaxies on red plates . @xcite gives an empirical expression for the median redshift @xmath28 of a galaxy sample with a given faint magnitude flux cutoff . this median redshift can be equated with the optimal lens redshift @xmath24 , and hence the magnitude limit of the foreground galaxies can be determined for every sn separately . however , there is a small catch . for @xmath29 optimal @xmath30 . the galaxy redshift distribution whose median redshift @xmath31 has a considerable tail extending beyond @xmath32 . to avoid the problem of source / lens redshift overlap we use @xmath33 , where factor of 2 was chosen arbitrarily . we explore the dependence of the results on this factor in section [ robust ] . around every sn we draw a circle of radius @xmath34 , and count the number of galaxies , @xmath35 , in the appropriate magnitude range . this number is compared to the average number density in control circles , @xmath36 . fractional galaxy excess is @xmath37 . control circles are confined to the same apm plate as the sn , and to the same distance from the plate centre as the sn ( to reduce the effects of vignetting ) ; however , scattering the control circles randomly on the plate does not change the results significantly . for each sn we also calculate @xmath38 , where @xmath39 is the number of control circles , out of total @xmath40 , that have less galaxies in them than the circle around the sn . in other words , @xmath38 is the rank of the sn circle among its control ` peers ' . if sne are randomly distributed with respect to the foreground galaxies , then average @xmath41 . if sne have an excess ( deficit ) of galaxies in front of them then their @xmath42 will be greater ( less ) than 0.5 . analogous to the medians being more stable than averages , @xmath38 rank statistic is more stable than @xmath43 . , and @xmath17 , and extinction , k - corrected distances from @xcite . @xmath44 is a measure of the number density of foreground galaxies in circles of radius @xmath34 around sne ( see section [ anal ] for details ) . there are a total of 55 sources ( with magnitude errors ) , but only 50 ( filled points ) are used in the analysis . the 50 points are grouped into three bins , whose horizontal size is shown as thick horizontal `` error - bars '' . the corresponding vertical error - bars show the deviation of the mean of the points in each bin ( the rms is 4 times larger ) , and the intersection of the thick lines are the averages of the mag@xmath45 of sne in each bin . the thin slanting line is the best - fit to 50 sne , and has a slope @xmath46 . the significance of the correlation is @xmath2.,width=317 ] figure [ figone ] shows absolute magnitudes , @xmath47 , and @xmath38 ranks of 55 sne found on apm plates . the effect of flux dimming due to cosmological distances has been taken out , i.e. all the sne have been ` brought ' to the same redshift ; the constant magnitude offset on the vertical axis is irrelevant for this work . there are two sne whose magnitudes make them @xmath48 outliers , sn1997o , and sn1997bd represented by empty circles with dotted line error - bars . we exclude these from our analysis . as quoted in @xcite . sn1997o and sn1994h were excluded from the primary fit ( fit c ) by the analysis of @xcite . we do not exclude sn1994h from our analysis , but if we did it would improve the trend , as its coordinates on fig . [ figone ] are ( 0.427 ; -0.606 ) . ] . distance errors , @xmath49 were derived by @xcite , and listed in column 9 of their table 15 . the solid histogram is for 55 sn ; the dashed line represents 31 sn from the high-@xmath0 supernova search team . the 3 outlier sne with @xmath50 are not used in the present analysis.,width=317 ] the distance ( or , magnitude ) errors , as estimated by @xcite for the 55 sne are shown in fig . [ figthree ] , as the solid line histogram . the dashed line represents the subsample of 31 sne from high-@xmath0 supernova search team @xcite , whose errors appear to be generally smaller than those of the supernova cosmology project @xcite . we use sne from both the groups , but exclude three outliers in fig . [ figthree ] , whose @xmath51 . in fig . [ figone ] these are represented by empty circles with dashed line error - bars . thus , we exclude a total of 5 sne from our analysis , leaving us with 50 , shown as the solid points in fig . [ figone ] . these 50 sne exhibit a relation between their @xmath47 and @xmath38 , in the sense that brighter sne have an excess of galaxies in their foregrounds . for illustration purpose only , we bin the 50 sne into three bins ; fig . [ figone ] shows the extent of the bins and the deviation of the mean of the sne magnitudes in each bin as thick lines . the best - fit line to the 50 sne in fig . [ figone ] has a slope @xmath46 , and is shown as a thin slanting line . this fit does not include magnitude errors . to include the errors we do the following . we calculate the best - fit slope for 10,000 realizations of the data , where each data point s sn magnitude is replaced by a randomly picked magnitude from a gaussian distribution centred on the actual magnitude value , and having width equal to the quoted error . this procedure correctly incorporates the information contained in the errors , and produces a distribution of best - fit slopes , which is shown as a solid line histogram in fig . [ figextra ] . this distribution shows that the median best - fit slope is @xmath52 , while @xmath53 ( i.e. a case of no correlation ) is ruled out at 99.80% confidence level . had we used 53 sne ( i.e. had we not exluded the 3 sne with large distance errors ) , the median best - fit slope would gave been @xmath54 , while @xmath53 would have been ruled out at 99.37% confidence level . the corresponding distribution of @xmath55 slopes is shown as the dashed line in fig . [ figextra ] . ? the histograms show the distribution of derived @xmath55 values in 10,000 realizations of the data . the solid ( dashed ) line histogram represents results using 50 ( 53 ) sne . the short vertical lines at the bottom of the plot indicate the medians of the distributions.,width=317 ] next , we determine the likelihood of this relation arising by chance . to that end , we estimate the significance of the relation in two ways . first , we assign random positions to sne ( keeping these control sne on the same apm plate as the original source ) , and redo all the analysis . we repeat this 1000 times , and in 8 cases we find @xmath56 , implying statistical significance of @xmath57 . second , we take the list of observed @xmath47 values and randomly reassign them to observed sn sky positions . 10,000 randomized sne samples are created in this fashion , and only @xmath58 of these have @xmath59 . based on these two tests we conclude that the significance of the @xmath47@xmath38 relation is better than @xmath60 . these results are consistent with weak gravitational lensing , which would amplify sne found behind more nearby mass concentrations , as traced by apm galaxies . alternatively , the results could be due to the action of galactic dust , which will obscure certain directions of the sky making galaxies less numerous and sne fainter . we consider galactic dust further in section [ disc ] ; in sections [ model ] and [ montecarlo ] we proceed on the assumption that weak lensing is responsible to the @xmath47@xmath38 relation . the distribution of sne points in fig . [ figone ] depends on specific choices that we made for certain parameters , in particular we chose circles of radius @xmath34 and galaxy magnitude limit such that @xmath61 , where @xmath62 ( see section [ data ] ) . how would the results change if different choices were made ? in other words , how robust is our result , would it disappear had we picked a different set of parameters ? of circles around sne . the vertical axis shows @xmath55 , the slope of the best - fit line to the relation of the type shown in fig [ figone ] . the filled circles represent cases where galaxies were counted around sn ; stars symbols represent cases where galactic stars were used instead ( i.e. control experiment ) . the dotted lines denote , _ approximately _ , the levels of statistical confidence . the dashed vertical line indicates radius @xmath63 used in the analysis of section [ anal].,width=317 ] figure [ figthetatest ] shows the effect of changing @xmath63 . the vertical axis is @xmath55 , the best - fit to the @xmath47@xmath38 relation in each case . filled points represent cases where _ galaxies _ were counted and used to determine the @xmath38 rank , while star symbols represent cases where _ galactic stars _ were used instead . as expected , @xmath64 do not correlate with sne magnitudes , and the values of the best - fit slopes are near zero . however , because the apm star - galaxy classifier is not perfect , some ` stars ' are actually galaxies , which accounts for some signal being seen when using @xmath64 . horizontal dotted lines mark the approximate location of @xmath65 , and @xmath60 confidence levels . these are only approximate because every point in the plot will have its own significance level , but because the number of sne contributing to each point is the same in each case , and the total dispersion in sne magnitudes is the same , same @xmath55 values have about the same significance , regardless of @xmath63 . we note that for very small @xmath63 the galaxy numbers become very small , and poisson noise drowns out any @xmath47@xmath38 correlation that might exist , so the upturn in the values of the best - fit slope at small @xmath63 is probably not real . the dashed vertical line marks the @xmath63 value used in section [ anal ] . we conclude that significant @xmath47@xmath38 anti - correlations occur only with galaxies and not with galactic stars ( which serve as a control sample ) , and only for @xmath66 . . similar to fig . [ figthetatest ] ; see section [ robust ] for details.,width=317 ] figure [ figmagtest ] shows the effect of changing the median redshift , @xmath61 of the apm galaxies , or equivalently , mag@xmath27 . the hidden variable which is varied along the horizontal axis is @xmath67 . since each sn has a different value of @xmath24 and hence mag@xmath27 , depending on its @xmath25 , there is no unique way of labeling the horizontal axis by using galaxy magnitudes , or redshifts . we label that axis by assuming @xmath68 , the median of the sne redshift distribution . at the top of the plot we show how galaxy magnitude limit on the horizontal axis translates into the median galaxy redshift ( for a specific case of @xmath68 ) . we see that significant anti - correlation between @xmath47 and @xmath38 of foreground galaxies occurs for galaxies with @xmath69 . using galactic stars instead of galaxies produces no significant results . the value we used in section [ anal ] is shown with a vertical dashed line , and corresponds to @xmath70 . as in fig . [ figthetatest ] the upturn in the best - fit slope values at bright mag@xmath27 is probably due to poisson noise . the range of redshifts of apm galaxies that act as the best lenses for sne are @xmath71 ; more distant galaxies show no signal , as expected , if lensing is the correct interpretation of the data , because more distant galaxies are either too close to sne in redshift or are actually at the same @xmath0 . a number of other tests have been carried out as well . for example , instead of using a @xmath25-dependent mag@xmath27 , we tried fixed values of @xmath72 , which gave , @xmath73 , respectively , comparable to , but somewhat smaller than those seen in fig . [ figmagtest ] . this is not surprising : @xmath25-dependent galaxy magnitude limits pick optimal lens redshifts for each source , thus maximizing the observed lensing signature . we also reran the analysis with subsamples of the entire 50-source sample . we split the sne according to the teams : 20 supernova cosmology project snia gave a @xmath47@xmath38 anti - correlation significant at 95.8% , while the corresponding significance level for the 30 sne from the high-@xmath0 supernova search team is 96% . because the quality of ukst apm ( southern hemisphere ) plates is higher than poss apm ( northern hemisphere ) plates , and because there is some overlap between plates along the equator , we used ukst plates whenever possible . redoing the analysis using only the 36 ukst sne we get the best fit slope @xmath74 at a significance level of 99.6% , while the 14 poss sne had @xmath75 and the correlation was not significant , which is not surprising given the size of this subsample . splitting the whole sample into low and high redshift groups we get the following results : 25 sne with @xmath76 have a slope @xmath77 at 96.8% , while 25 sne with with @xmath78 have a slope @xmath79 at 88.9% . these tests suggest that the @xmath47@xmath38 relation has a physical origin ( weak lensing or galactic dust ) , and is not an artifact arising form one subset of the data . in this section we adopt the weak lensing interpretation of the @xmath47@xmath38 relation . our simple lensing model does not use the individual values of @xmath25 , and the optimal redshift distribution of the corresponding lenses . in lieu of these parameters we use the lensing optical depth of the matter traced by the galaxies , @xmath80 . for a fixed global geometry , @xmath80 depends on @xmath25 and the redshift distribution of the mass traced by the galaxies . we assume that the apm galaxies faithfully trace the mass up to some redshift , @xmath81 , then @xmath82^{1/2}\ ; \sigma_{crit}(z , z_s ) } } \ ; dz,\end{aligned}\ ] ] in that case , projected fractional mass excess is @xmath83 , where @xmath84 is the convergence , with respect to a smooth universe ( filled beam ) for the corresponding line of sight . in the weak lensing regime source amplification @xmath85 ; @xmath86 for an unlensed source . the smallest value that @xmath84 can attain is minus the total optical depth , @xmath87 , which corresponds to emptying out all the mass along the line of sight from the observer to @xmath81 . in addition to the value for @xmath80 , our model has three ingredients , ( 1 ) mass distribution , i.e. a probability distribution ( pdf ) for @xmath88 , which is related to @xmath84 pdf , ( 2 ) a biasing scheme , which relates @xmath88 to the projected fractional galaxy number density excess , @xmath43 , and ( 3 ) the dispersion in the intrinsic magnitudes of snia s . given this information we generate synthetic snia samples , with 50 sources each , then compute @xmath38 each sn , and the observed sn magnitude , @xmath47 . the slope of the best - fit line to the @xmath47@xmath38 relation , @xmath55 is then used to test how well a given model reproduces the observations . because our model assumes the same @xmath24 distribution for all sources , the value of @xmath55 to compare our models predictions to should be the one obtained by using a constant mag@xmath89 , i.e. @xmath90 ( see fourth paragraph in section [ robust ] ) . the specifics of the three model ingredients , and the associated model parameters are described later in this section . the goal of the modeling is to determine what set of parameters can reproduce the observations , i.e. have @xmath91 . we are particularly interested in what effect biasing has on the results . it is often suggested that the large amplitude of qso - galaxy correlations mentioned in the introduction is , at least in part , due to the fact that biasing is not a simple linear , one - to - one mapping from @xmath88 to @xmath43 . qso - galaxy correlation function ( @xmath92 ) , galaxy autocorrelation ( @xmath93 ) and the matter fluctuation power spectrum ( @xmath94 ) are related by @xmath95 where @xmath96 is the biasing parameter . this means that @xmath92 probes a _ combination _ of @xmath96 and @xmath94 . thus , for a given @xmath94 , @xmath92 can be enhanced with positive biasing , especially if it is non - linear on relevant spatial scales . the important difference between weak lensing induced qso - galaxy correlations and the effect on standard candles we have studied here is that in the latter case biasing plays a minor role . the slope of the @xmath47@xmath38 relation is most sensitive to the mass distribution . our modeling demonstrates this later ; first , we described the specifics of the three model ingredients . based on the results from cosmological n - body simulations @xcite the shape of the probability distribution function ( pdf ) of @xmath84 is very roughly gaussian , but asymmetric , with the most probable @xmath84 , which we call @xmath97 being less than 0 , i.e. most sources are deamplified . the tail of the @xmath84 distribution extends to high positive values of @xmath84 . we use these published results of @xcite to construct an approximate shape for our @xmath84 pdf : @xmath98 ^ 2/2\sigma_{\kappa_1 } ) , \quad { \rm for}\quad \kappa > \kappa_m\\ exp\;(-[\kappa-\kappa_m]^2/2\sigma_{\kappa_2 } ) , \quad { \rm for}\quad \kappa\le\kappa_m \end{array } \right . \label{stdpdf}\ ] ] the pdf is a combination of two half - gaussians , with two widths , @xmath99 and @xmath100 , which describe the high @xmath84 and the low @xmath84 sides of the pdf respectively . the location of the peak of the pdf , @xmath97 is adjusted such that the average @xmath84 is 0 , which implies @xmath101 @xmath97 is always negative , since the skewness of the pdf dictates that @xmath102 . from the numerically computed pdfs of @xcite we estimate that for a flat @xmath103 model , where the sources are at @xmath104 , and the smoothing scale is @xmath105 , the following approximations apply : @xmath106 , and @xmath107 . thus we now have a realistic @xmath84 pdf , appropriate for a standard cosmology with standard mass distribution . figure [ grid](d ) shows such a pdf as a short - dash line . in addition to the standard @xmath84 pdf described above , we also try an extreme form for @xmath84 pdf obtained using the `` bifocal lens '' mass distribution proposed by @xcite : @xmath108 where @xmath109 is the kronicker s delta function , @xmath110 , and @xmath111 . for a fixed @xmath80 , this pdf can produce more pronounced lensing effects than the standard pdf , if @xmath112 is assigned the smallest allowable value , and @xmath113 is made very large . this is because the standard pdf has a large probability near @xmath86 , while the bifocal pdf avoids such values altogether . the distribution of mass corresponding to eq . [ bifocals ] is unrealistic . one can make it somewhat realistic by allowing a range of @xmath113 and @xmath112 values . we do the following : for any one line of sight we randomly pick @xmath113 and @xmath112 from specified ranges , using flat priors , and eq . [ bifocals ] sets the probability distribution of the two amplifications . we use two sets of ranges : ( 1 ) @xmath114 and @xmath115 ( we will call this model bifocal i model ) , ( 2 ) @xmath116 and @xmath117 ( we will call this bifocal ii ) . the corresponding @xmath84 pdfs , obtained by considering many lines of sight are shown in fig . [ grid](d ) as a solid line for bifocal ii and as a long - dash line for bifocal i. in bifocal ii the minimum value of @xmath112 is the minimum allowed value ; the other limits on @xmath118 s were picked arbitrarily . for comparison , the standard @xmath84 pdf of section [ stdmass ] is also shown , as the short - dash line . optical depth of @xmath119 was assumed for all three . compared to a standard pdf , bifocal pdfs imply that most of the lines of sight are rather empty , and there are a few lines of sight with very high values of @xmath84 . projected fractional mass excess , @xmath88 is related to the observationally accessible quantity , @xmath43 through biasing . we chose `` power law '' biasing , motivated by numerical simulations of @xcite and @xcite , @xmath120 where the power law part allows the biasing to be non - linear , with different indexes depending on the sign of @xmath88 , while @xmath121 is a stochastic biasing component which is chosen randomly for each sn ; @xmath121 distribution has a gaussian shape and width @xmath122 . factor @xmath123 multiplying @xmath121 ensures that the dispersion in @xmath43 is reduced in underdense regions . qualitatively , the @xmath43 vs. @xmath88 relations produced by eq . [ biasing ] look similar to those in fig.1 of @xcite . we assume that the dispersion in magnitude about the perfect standard candle case has a gaussian shape , with width @xmath124 . each model has five independent parameters : + @xmath125 . we assume flat priors for the three biasing parameters , @xmath126 , @xmath127 , and @xmath122 , as well as for @xmath124 . each specific set of parameters together with one of the three mass distribution models generates a synthetic realization of the 50-source snia sample . from this entire ensemble of realizations , we only consider those that satisfy these observational constraints : ( 1 ) the total rms dispersion in the synthetic snia magnitudes ( which includes intrinsic and lensing induced contributions ) must be within 0.05 mag of the actual observed value of 0.3mag ; ( 2 ) the moments of the synthetic @xmath43 distribution ( average , standard deviation , skewness and kurtosis ) must be reasonably close to those of the observed distribution , which is characterized by ( 0.063 , 0.52 , 1.7 , 4.9 ) . we chose `` reasonably close '' to mean that synthetic values must not be more than a factor of 2 away from the actual values . because of these constraints , the range from which the values of @xmath126 , @xmath127 , @xmath122 , and @xmath124 parameters are picked is not relevant , as long as it not too restrictive . in other words , the observational constraints eliminate cases with very large values of these parameters . the data suggests that demagnified sne are not lost due to the flux limit . if they were , we would have less sne at small values of @xmath38 , and more at high values of @xmath38 , whereas in the data ( fig . [ figone ] ) the sources are roughly equally spread over the entire @xmath38 range . so our synthetic lensing models assume that we do not lose sne because of deamplification . of the @xmath47 vs. @xmath44 relation ) given different mass distribution models , and a range of optical depths . solid dots assume standard @xmath84 pdf of section [ stdmass ] , empty squares and star symbols represent bifocal i and ii models , respectively , discussed in section [ bifocalmass ] . each point was obtained using 10,000 synthetic realizations of snia 50-source sample . see section [ montecarlo ] for details.,width=317 ] for each synthetic realization of the 50-source snia sample we derive the corresponding @xmath47@xmath38 relation . the slope of the best - fit line , @xmath55 is recorded . for a given value of @xmath80 we generate 10,000 realizations . solid dots in fig . [ figoptdepth ] show the percentage of realizations that have @xmath55 smaller than the observed value of -0.3 , as a function of total optical depth , for the standard @xmath84 pdf . empty squares and star symbols represent the results for the bifocal i and ii . figure [ figoptdepth ] considers a range of optical depths . what is the appropriate value for @xmath80 ? if the source is at the median source redshift for our sample , @xmath68 , then apm galaxies sample the mass distribution up to a redshift of about 0.3 . in this case the lensing optical depth probed by apm galaxies is 0.011 ( left - most arrow in fig . [ figoptdepth ] ) . a somewhat more optimistic estimate for @xmath80 is obtained if we assume that galaxies trace the mass fluctuations up to the median source redshift , 0.47 , which gives @xmath128 ( middle arrow ) . the limiting case is obtained by considering the most distant source , at @xmath129 , and assuming that galaxies probe mass up to a redshift of 0.47 , which gives @xmath130 ( right - most arrow ) . the important conclusion from this figure is that if the standard mass distribution ( section [ stdmass ] ) is assumed , then the value of @xmath80 does not really matter , and the probability of reproducing the observations is @xmath131 , if @xmath80 is within a reasonable range . if , on the other hand , the bifocal pdf i or ii are assumed , the probability of reproducing observations depends sensitively on the assumed optical depth . if @xmath130 then these models yield 6 - 20% probability . the bifocal pdfs produce more discernible lensing effects , for the same @xmath80 because they have a wider range in @xmath118 , as seen in fig . [ grid](d ) . @xmath44 relation on the three biasing parameters ( see section [ biasscheme ] ) . short - dash lines are for the standard @xmath84 pdf model of section [ stdmass ] , while the solid lines are for the bifocal ii @xmath84 pdf described in section [ bifocalmass ] . both models assume optical depth @xmath132 . the contours are based on 50,000 realizations , and are spaced a factor of 3 apart . the shapes of the contours are partly determined by the observational constraints , described in section [ montecarlo ] ( second paragraph ) . it is evident that biasing parameters have little impact on the @xmath55 . _ panel ( d ) : _ short - dash and solid lines represent @xmath84 pdfs corresponding to the contours in panels ( a)-(c ) of the same line type . the minimum @xmath84 in the bifocal ii pdf is @xmath87 ( indicated by an arrow ) . the long - dash line represents bifocal i @xmath84 pdf . , width=317 ] the results presented in fig . [ figoptdepth ] marginalize over values of all the model parameters ( except @xmath80 ) , so the effects of the biasing parameters on @xmath55 are hidden . however , it turns out that these parameters do not correlate with @xmath55 . figure [ grid ] shows the dependence of @xmath55 on @xmath126 , @xmath127 and @xmath121 for the bifocal ii @xmath84 pdf model ( solid contour lines ) , and the standard @xmath84 pdf model ( dashed contour lines ) , both for @xmath132 . fifty thousand realizations were created for each model , and the corresponding contour lines plotted , with adjacent contours separated by a factor of 3 . it is apparent that for both models the dependence on the biasing parameters is weak . we conclude that _ the slope of the @xmath47@xmath44 relation depends on the amount and distribution of matter , i.e. on @xmath80 and the shape of the @xmath84 pdf model , but does not depend on the specifics of the biasing scheme . _ figure [ figoptdepth ] quantifies the dependence on amount and distribution of mass : as expected , higher optical depth results in more pronounced lensing effects . standard @xmath84 pdf produces less lensing for the same @xmath80 compared to a much broader bifocal pdf . figure [ grid ] demonstrates that biasing has little effect on the slope of @xmath47@xmath44 relation . in other words , the type of lensing signature considered here ( @xmath47@xmath44 relation for standard candles ) probes mass distribution , and is independent of biasing . we detect the signature of weak lensing in the current sample of 50 high redshift snia taken from two teams : high-@xmath0 supernova search team and supernova cosmology project . after correcting sne magnitudes for cosmological distances ( assuming @xcite values ) , we find that brighter sne are preferentially found behind regions overdense in foreground galaxies . this @xmath47@xmath44 relation has a slope @xmath133 to @xmath134 , when the angular radii of foreground regions are 5 - 15 arcmin . the statistical significance is @xmath2 ( see fig . [ figthetatest ] and [ figmagtest ] ) . the angular radii of 5 - 15 arcmin imply that the lensing structures are @xmath135 mpc across if they are located at a redshift of 0.1 , and so correspond to non - linearly evolved intermediate - scale structure . aside from the possibility that the observed @xmath47@xmath44 relation is a fluke , there is one other possible , non - lensing interpretation : galactic dust obscuration , which would make snia sources brighter and apm galaxies more numerous in the directions devoid of galactic dust . if this interpretation is correct , then ( 1 ) galactic extinction corrections applied to the sne @xcite are much too small , and ( 2 ) galactic dust is able to change projected apm galaxy number density by @xmath136 , since the observed rms in @xmath43 is @xmath137 . we can not comment on ( 1 ) , but can estimate the effect of ( 2 ) . how much excess / deficit in galaxy density can galactic dust create ? using @xcite data we calculate the average @xmath138 galactic extinction for the 50 sne sample to be 0.18 mag . the slope of the r - band apm galaxy number counts is @xmath139 , therefore typical `` dust - induced '' @xmath140 . this is an upper bound because extinction in the @xmath141 band will be smaller than in @xmath138 , and because extinction averaged over 5 - 15 arcmin radius patches will be smaller than estimates for individual sne , given that @xcite extinction map resolution is [email protected] fwhm . if galactic dust is indeed the cause of the @xmath47@xmath44 correlation , then @xmath47 and @xmath44 should separately correlate with sne extinction . in reality , @xcite extinction estimates do not show a correlation with either @xmath47 or @xmath44 . overall , galactic dust interpretation of the observed @xmath47@xmath44 relation seems unlikely , but can not be ruled out . note that dust _ intrinsic _ to the @xmath143 structures probed by the apm galaxies can not be invoked to explain the observations , because such dust would produce an effect opposite to the one detected here , i.e. @xmath55 would be positive . if dust is present in groups and clusters traced by the apm , it will diminish the amplitude of the effect we detect . presence of dust in groups was suggested by @xcite who found that faint qso candidates are anti - correlated with foreground groups . however , such an anti - correlation can also be explained by weak lensing @xcite . current observations indicate that groups and clusters do not contain significant amounts of dust @xcite . if the lensing interpretation in correct , then the standard models of mass distribution have some difficulty in reproducing @xmath55 ; the observed value would be detected only in @xmath131 of the cases . we investigate how @xmath55 is affected by the amount and distribution of mass along the light of sight to the sources , and galaxy biasing schemes . we find that larger optical depths and broader @xmath84 pdf result in steeper @xmath55 slopes ( fig . [ figoptdepth ] ) . optical depth is a function of global geometry , and no realistic cosmological model can give @xmath144 , which is what would be required to comfortably explain the results . broader @xmath84 pdfs mean that mass fluctuations are more extreme than the standard cosmological models allow , a scenario which is in apparant conflict with other means of determining mass fluctuations , like cosmic velocity flows and weak shear lensing . we also find that biasing has little effect on @xmath55 ( fig . [ grid ] ) . this insensitivity to biasing is in contrast to weak lensing induced qso - galaxy and grb - galaxy ( anti-)correlations , where biasing could , at least partly explain the higher than expected amplitude of the effect @xcite . we conclude that weak lensing of standard candles provides a cleaner probe of the mass distribution at @xmath145 , on @xmath146few mpc scales , then lensing induced angular correlations . in fact , snia can provide the perfect means of measuring mass inhomogeneities using gravitational lensing . a set of standard candles at known redshifts can be analyzed using the complementary techniques of weak magnification and weak shear lensing . the advantage of magnification lensing over the more commonly used shear lensing is that with the former one can chose the redshift range of the lenses , whereas the latter yields the cumulative effect of lensing along the entire line of sight to the source . therefore a large uniform set of intermediate redshift snia , such as the ones that would result from the snap mission and the lsst missionhome.html ] would be invaluable for the studies of mass clustering in the nearby universe . in addition to detecting intermediate and high - redshift sne for the purposes of estimating the global cosmological parameters and the equation of state of the dark energy , snap will also measure weak shear lensing signature due to large scale structure . combining magnification data ( of the type considered in this paper ) and shear information @xcite for a large set of sne will allow the study of mass distribution at @xmath147 with unprecedented accuracy and 3-dimensional spatial resolution . the final issue we address is the impact of weak lensing on the determination of global cosmological parameters , @xmath148 and @xmath149 using snia standard candles . in principle , weak lensing can affect the derived values of @xmath148 and @xmath149 , if ( 1 ) demagnified sne are preferentially lost from the sample due to faint flux cutoff , and/or ( 2 ) the @xmath84 pdf is asymmetric and the sn sample size is small ( see also @xcite ) . if demagnified sne were lost from the sample then the distribution of sne in @xmath44 would be skewed in the direction of larger @xmath44 values . if @xmath84 pdf is asymmetric , ( and it is , on scales considered here ) , then most sne in a small sample will be slightly demagnified compared to average , and will have their @xmath44 skewed in the direction of smaller values . so both ( 1 ) and ( 2 ) would make the distribution of sne in @xmath44 uneven , and would to some extent cancel each other . in the present sample of 50 , the distribution of sne in @xmath44 is indistinguishable from uniform , so the average @xmath47 corresponds to the average @xmath44 , and hence lensing effects by @xmath146mpc - size structures probably did not bias the derived values of @xmath148 and @xmath149 . the authors would like to thank peter garnavich and jason rhodes for their careful reading of the manuscript and valuable comments . 99 amanullah r. , mortsell e. , goobar a. 2003 , a&a , 397 , 819 benitez n. , martinez - gonzalez e. 1997 , apj , 477 , 27 benitez n. , riess a. , nugent p. , dickinson m. , chornock r. , filippenko a. 2002 , apj , 577 , l1 benitez n. , sanz j.l . , martinez - gonzalez e. 2001 , mnras , 320 , 241 boyle , b. j. , fong , r. & shanks , t. 1988 , mnras , 231 , 897 cen r. , ostriker j.p . 1993 , apj , 417 , 415 croom , s. m. , & shanks , t. 1999 , mnras , 307 , l17 dekel a. , lahav o. 1999 , apj , 520,24 ferguson , h.c . 1993 , mnras , 263 , 343 garnavich p.m. et al . 1998 , apj , 493 , l53 holz d.e . 1998 , apj , 506 , l1 holz d.e . , wald r.m . 1998 , phys . d , 58 , 063501 irwin m. , maddox s.j . , mcmahon r.g . 1994 , spectrum , 2 , 14 . see also http://www.ast.cam.ac.uk/~mike/apmcat/ jain b. , scranton r. , sheth r.k . 2003 , + preprint , http://xxx.lanl.gov/abs/astro-ph/0304203 jain b. , seljak u. , white s.d.m . 2000 , apj , 530 , 547 kovner i. 1991 , apj , 376 , 70 lewis g.f . , ibata r.a . 2001 , mnras , 324 , l25 maddox s. j. , efstathiou g. , sutherland w.j . 1996 , mnras , 283 , 1227 metcalf r.b . 1999 , mnras , 305 , 746 moertsell e. , gunnarsson c. , goobar a. 2001 , apj , 561 , 106 nollenberg , j.g . , williams , l.l.r . & maddox , s.j . 2003 , aj , 125 , 2927 refregier a. 2003 , preprint , + http://lanl.arxiv.org/abs/astro-ph/0307212 rhodes j. , refregier a. , massey r. & the snap collaboration , priprint , + http://lanl.arxiv.org/abs/astro-ph/0304417 rodrigues - 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the current sample of high - redshift supernova type ia , which combines results from two teams , high-@xmath0 supernova search team and supernova cosmology project , is analyzed for the effects of weak lensing . after correcting sne magnitudes for cosmological distances , assuming recently published , homogeneous distance and error estimates , we find that brighter sne are preferentially found behind regions ( 5 - 15 arcmin radius ) which are overdense in foreground , @xmath1 galaxies . this is consistent with the interpretation that sne fluxes are magnified by foreground galaxy excess , and demagnified by foreground galaxy deficit , compared to a smooth universe case . the difference between most magnified and most demagnified sne is about 0.3 - 0.4 mag . the effect is significant at @xmath2 level . simple modeling reveals that the slope of the relation between sn magnitude and foreground galaxy density depends on the amount and distribution of matter along the line of sight to the sources , but does not depend on the specifics of the galaxy biasing scheme . [ firstpage ] gravitational lensing supernovae : general

arxiv

lanthanide based heavy fermion ( hf ) metals constitute a major , long studied class of correlated electron materials @xcite . their behaviour is quite distinct from conventional clean metals , the basic physics being driven by strong spin - flip scattering from essentially localised @xmath3-levels , generating the large effective mass and attendant low - energy scale indicative of strong interactions . the low - temperature ( @xmath4 ) state is a lattice - coherent fermi liquid with well defined quasiparticles and coherently screened @xmath3-spins , crossing over with increasing @xmath4 to essentially incoherent screening via independent kondo scattering , before attaining characteristic clean metallic behaviour . physical properties of hf are in consequence typically ` anomalous ' : e.g. the resistivity @xmath5 shows a strong non - monotonic @xmath4-dependence , while optics often exhibit rich structure from the microwave to the near infrared , and pronounced evolution on low temperature scales @xcite . theoretical treatments of hf centre on the periodic anderson model ( pam ) , in which a non - interacting conduction band hybridizes locally with a correlated @xmath3-level in each unit cell of the lattice ; or on its strong coupling limit , the kondo lattice model . the absence of exact results ( save for some in one dimension , see e.g. @xcite ) has long spurred the search for suitable approximation schemes . one such framework , which has had a major impact in recent years , is provided by dynamical mean field theory ( dmft , for reviews see @xcite ) . formally exact in the large - dimensional limit , the self - energy within dmft becomes momentum independent and hence spatially local , but still retains full temporal dynamics ; such that all lattice models map onto an effective single - impurity model with a self - consistently determined host @xcite . that raises an immediate question , easier asked than answered : to what extent are the properties of real hf materials captured within a dmft approach to the pam ? to answer this clearly requires direct quantitative comparsion of theory to experiment . and a prerequisite to that in turn is a method to solve the pam which dmft does not _ per se _ provide . the latter has of course been studied extensively using a wide variety of techniques . full scale numerical methods include the numerical renormalization group ( nrg ) @xcite , quantum monte carlo @xcite and exact diagonalization @xcite , while theoretical approaches encompass finite - order perturbation theory in the interaction @xmath6 @xcite , iterated perturbation theory @xcite , the lattice non - crossing approximation @xcite and the average @xmath7-matrix approximation @xcite , large-@xmath8 mean - field theory / slave bosons @xcite , the gutzwiller variational approach @xcite and the recently developed local moment approach @xcite . all of these methods naturally have their own virtues . but most possess significant , well known limitations @xcite , be it the general inability of perturbative approaches ( and in practice quantum monto carlo ) to handle strong interactions ; failure to recover fermi liquid behaviour at low - energies as arises in nca - based approaches , restriction to the lowest - energy fermi liquid behaviour as in large-@xmath8/slave boson mean - field theories , finite - size effects limiting exact diagonalization , and so on . to enable viable comparison to experiment requires an approach that can adequately handle all experimentally relevant energy and/or temperature scales in the strongly correlated hf regime of primary interest ; and indeed ideally also across the full spectrum of interaction strengths , such that intermediate valence and related behaviour can likewise be treated . one such is employed here , the local moment approach ( lma ) @xcite . via study of the generic asymmetric pam , our essential aims are ( i ) to provide a many - body description of dynamical and transport properties of paramagnetic hf , notably single - particle dynamics , d.c . transport and optical conductivities ; as considered here . ( ii ) to make direct quantitative comparison with experiment . that is taken up in the following paper where comparison to transport / optical properties of @xmath9 , @xmath10 , @xmath11 and @xmath12 is made . some remarks on the lma are apposite at this point since the paper will focus mainly on results obtained using the approach , with minimal technical details . intrinsically non - perturbative and as such capable of handling strong interactions , the lma @xcite introduces the physically intuitive notion of local moments @xcite from the outset . this leads directly to a ` two - self - energy ' description in which , post mean - field level , the key correlated spin - flip dynamics is readily captured ; corresponding in physical terms to dynamical tunneling between initially degenerate local moment configurations , which lifts the erstwhile spin degeneracy and restores the local singlet symmetry characteristic of a fermi liquid state . as with all techniques for lattice models within dmft , the lma originated in study of the single - impurity anderson model ( aim ) @xcite , where results for dynamics are known to give good agreement with nrg calculations @xcite , and for static magnetic properties with known exact results @xcite the approach has recently been developed to encompass the anderson lattice ( pam ) ; initially for the particle - hole symmetric limit @xcite appropriate to the kondo insulating sub - class of heavy electron materials , where for all interaction strengths the system is an ` insulating fermi liquid ' that evolves continuously from its simple non - interacting limit of a hybridization - gap insulator @xcite . from this a rich description of transport and optical properties of kondo insulators arises @xcite , particularly in strong coupling where the system is characterized by an exponentially small indirect gap scale @xmath13 , such that dynamics / transport exhibit scaling as functions of @xmath14 . exploiting that scaling enables direct comparison to experiment with minimal use of ` bare ' material / model parameters ; and in particular for three classic kondo insulators @xmath15 , @xmath16 and @xmath17 , leads to what we regard as excellent agreement between theory and experiment on essentially all relevant energy and temperature scales @xcite . the particle - hole symmetric pam is of course special , confined as it is to the case of kondo insulators . most recently the lma has been non - trivially extended to handle the generic asymmetric pam @xcite and hence hf metals ( with the insulating symmetric limit recovered simply as a particular case ) . single - particle dynamics at @xmath18 were considered in @xcite , with a natural emphasis on the strongly correlated kondo lattice regime of localised @xmath3-electrons but general conduction ( ` @xmath19 ' ) band filling @xmath20 . the problem was found to be characterized by a _ single _ low - energy coherence scale @xmath1 the precise counterpart of the insulating indirect gap scale @xmath21 , and likewise exponentially small in strong coupling in terms of which dynamics exhibit one - parameter universal scaling as a function of @xmath22 , independently of either the interaction strength or local @xmath23 hybridization . with increasing @xmath24 dynamics cross over from the low - energy quasiparticle behaviour required by and symptomatic of the coherent fermi liquid state , to essentially incoherent single - impurity kondo scaling physics at high-@xmath24 but still in the @xmath24-scaling regime and as such incompatible @xcite with a two - scale ` exhaustion ' scenario @xcite . in this paper we extend the work of @xcite to finite temperature , thereby enabling access to d.c . transport and optics . our primary focus is again the strongly correlated hf regime and attendant issues of scaling / universality ( that play a key role in comparing to experiment ) , the paper being organised as follows . the model and a bare bones description of background theory is introduced in section 2 , together with preliminary consideration of transport / optics . results for the thermal evolution of single - particle dynamics and scattering rates , and the connection between the two , are given in section 3 . the d.c . resistivity is considered in section 4 , with particular emphasis in this context on the crossover from the low-@xmath25 coherent fermi liquid to the high-@xmath26 incoherent regime , and explicit connection to single - impurity scaling behaviour . optical conductivities on all relevant @xmath27- and @xmath4-scales are investigated in section 5 ; and the paper concludes with a brief summary . the hamiltonian for the pam is given by @xmath28 : = _ c_i , c^_i c^_i & -&t_(i , j),c^_i c^_i + _ i,(_f+ f^_i - f^_i- ) f^_if^_i + & + & v_i , ( f^_i c^_i + ) [ 2.1 ] the first two terms represent the uncorrelated conduction ( @xmath19 ) band , + @xmath29 ( @xmath30 ) ; with @xmath19-orbital site energies @xmath31 and nearest - neighbour hopping matrix element @xmath32 , rescaled as @xmath33 in the large dimensional limit where the coordination number @xmath34 @xcite . the third term describes the correlated @xmath3-levels , @xmath35 , with site energies @xmath36 and on site coulomb repulsion @xmath6 ; while the final term @xmath37 hybridizes the @xmath19- and @xmath3-levels locally via the matrix element @xmath38 , rendering the otherwise localised @xmath3-electrons itinerant . the model is thus characterized by four independent dimensionless parameters , @xmath39 and @xmath40 ( @xmath41 sets the scale for the width of the free conduction band and is taken as the basic unit of energy , @xmath42 ) . an equivalent and somewhat more convenient set of ` bare'/material parameters ( with @xmath43 ) is @xmath44 and @xmath45 , where @xmath46 . this parameter space is large , and as such encompasses a wide range of physical behaviour for the paramagnetic phases we consider . the system is of course generically metallic , with non - integral @xmath3-level and @xmath19-band occupancies ( @xmath47 and @xmath48 respectively ) . that in turn extends from the trivial case of weakly correlated , perturbative behaviour , through intermediate valence to the strongly correlated heavy fermion ( hf ) regime . it is naturally the latter , characterized by a low - energy coherence scale @xmath49 , that is of primary interest . the hf ( or kondo lattice ) regime corresponds to essentially localised @xmath3-electrons , @xmath50 , but with arbitrary conduction band filling @xmath51 , the latter being controlled by @xmath52 ( which determines the centre of gravity of the free ( @xmath53 ) conduction band relative to the fermi level ) . it arises when @xmath54 , for @xmath55 and @xmath56 ( whence @xmath57 ) ; where @xmath58 , with @xmath59 the free conduction electron density of states as specified below and @xmath60 the fermi level . the heavy fermion regime forms our main focus here ; intermediate valence behaviour will be discussed in an experimental context in the following paper . the exception to the above behaviour arises when @xmath61 . here the system is generically a kondo insulator ( see _ eg @xcite ) , with an indirect gap in both its @xmath18 single - particle spectrum _ and _ optical conductivity @xcite ; the canonical example being the particle - hole symmetric pam with @xmath62 and @xmath63 , where @xmath64 for all @xmath6 . just like its metallic counterpart arising for @xmath65 , the kondo insulator is however a fermi liquid , evolving continuously with increasing interaction strength from its non - interacting limit ( in this case a ` hybridization gap insulator ' @xcite ) . as such , the kondo insulating state is obtained simply as a particular limit of the underlying theory . _ a knowledge of local single - particle dynamics and their thermal evolution is well known to be sufficient within dmft @xcite to determine transport properties ( see section 2.2 below ) . our initial focus is thus on the local retarded green functions @xmath66 and likewise @xmath67 for the @xmath19-levels , with corresponding spectra @xmath68 ( and @xmath69 or @xmath3 ) . some brief comments on the free conduction band are first required ( @xmath53 in equation ( [ 2.1 ] ) , where the @xmath19- and @xmath3-subsystems decouple ) ; specified by the local propagator @xmath70 with corresponding density of states ( dos ) @xmath59 . this is given by g_0^c ( ) & = & ( ^+-_c ) [ 2.2a ] + & = & [ 2.2b ] with @xmath71 , where for arbitrary complex @xmath72 h(z ) = ^+_- d [ 2.3 ] denotes the hilbert transform with respect to @xmath73 ; such that from equation ( [ 2.2a ] ) , @xmath74 corresponds simply to a rigid shift of @xmath75 by @xmath52 . equation ( [ 2.2b ] ) defines the feenberg self - energy @xmath76 @xcite as used below , with @xmath77 $ ] alone ( since @xmath78 from equations ( 2.2 ) ) . the free conduction band is thus determined by the non - interacting dos @xmath79 which , modulo the rigid @xmath52-shift , reflects the underlying host bandstructure , @xmath80 . while the formalism below holds for an arbitrary @xmath73 , explicit results will later be given for the hypercubic lattice ( hcl ) , for which within dmft @xcite @xmath81 is an unbounded gaussian ; and the bethe lattice ( bl ) , with compact spectrum @xmath82 for @xmath83 @xcite . the hcl will in fact be the primary case , because the bloch states characteristic of it ultimately underlie the lattice coherence inherent to low - temperature metallic hf behaviour . the major simplifying feature of dmft is that the self - energy becomes momentum - independent and hence site - diagonal @xcite ; and since we are interested in the homogeneous paramagnetic phase , the local green functions @xmath84 ( @xmath85 ) are also site - independent . straightforward application of feenberg renormalized perturbation theory @xcite , then gives the @xmath86 as g^c()&=&[2.4a ] + g^f()&= & [ 2.4b ] + & = & [ 2.4c ] where @xmath87 is the retarded @xmath3-electron self - energy ( @xmath88 such that @xmath89 ) . in equations ( 2.4 ) , @xmath90 is the feenberg self - energy for the fully interacting case , with @xmath91 $ ] the _ same _ functional of @xmath92 as it is of @xmath93 in the @xmath53 limit . in consequence , @xmath92 is given using equations ( [ 2.4a ] ) , ( 2.2 ) , ( [ 2.3 ] ) as g^c()=h ( ) [ 2.5a ] where @xmath94 ( @xmath95 ) is given by ( ; t)=^+-_c- . [ 2.5b ] let us first point up the physical interpretation of equations ( 2.4 ) , ( 2.5 ) . @xmath92 is a _ local _ propagator , and as such familiarly expressed as @xmath96 ; with the @xmath97-resolved conduction electron propagator @xmath98^{-1}$ ] and the usual conduction electron self - energy @xmath99 thus defined . since @xmath80 , it follows directly that g^c ( ) = ^+_-d _ 0()g^c ( ; ) g^c ( ; ) _ [ 2.6 ] with ( @xmath100 ) @xmath101^{-1}$ ] . but equation ( [ 2.6 ] ) is precisely the form equation ( [ 2.5a ] ) ( with equation ( [ 2.3 ] ) for @xmath102 ) , showing that g^c ( ; ) = [ ( ; t)-]^-1 [ 2.7 ] with @xmath94 related to the conduction electron self - energy by ( ; t)=^+-_c - _ c(;t ) ; [ 2.8 ] and hence ( via equation ( [ 2.5b ] ) ) that @xmath103^{-1}$ ] in terms of the @xmath3-electron self - energy alone ( because the @xmath3-levels alone are correlated ) . for an arbitrary conduction band ( specified by @xmath104 ) equations ( 2.4 ) , ( 2.5 ) are central ; for given the self - energy @xmath87 , and hence @xmath94 from equation ( [ 2.5b ] ) , @xmath92 follows directly from the hilbert transform equation ( [ 2.5a ] ) , and @xmath105 in turn from equation ( [ 2.4c ] ) . that statement hides however the truly difficult part of the problem : obtaining the self - energy @xmath87 . this is not merely a calculational issue , e.g. the need to solve the problem iteratively and self - consistently ( any credible approximation to @xmath87 will in general be a functional of self - consistent propagators ) . it reflects by contrast the longstanding problem of obtaining an approximate @xmath87 that , ideally : ( i ) handles non - perturbatively the full range of interaction strengths , from weak coupling ( itself accessible by perturbation theory or simple variants thereof @xcite ) all the way to the strongly correlated kondo lattice regime that is dominated by spin - fluctuation physics and typified by an exponentially small coherence scale @xmath1 . ( ii ) respects the asymptotic dictates of fermi liquid behaviour on the lowest energy ( @xmath27 ) and/or @xmath4 scales on the order of @xmath106 itself yet can also handle the _ full _ @xmath27 and/or @xmath4 range ; including the non - trivial dynamics that arise on energy scales up to many multiples of @xmath1 yet which remain universal ( and the existence of which we find to dominate transport and optics ) , as well as the non - universal energy scales prescribed by the bare material parameters of the problem . the success of any theory naturally hinges on the inherent approximation to @xmath87 . in this paper we employ the local moment approach ( lma ) @xcite , for it is known to satisfy the above desiderata and to our knowledge is currently the only theory that does . it is based on an underlying two - self - energy description a natural consequence of the mean - field approach from which it starts , and from which the conventional single self - energy @xmath107 follows together with the concept of _ symmetry restoration _ that is central to the lma generally @xcite . full details of the lma for the pam , including discussion of its physical basis and content , are given in @xcite . in particular the generic asymmetric pam ( as considered here ) is detailed in @xcite for @xmath18 ; and extension of it to finite-@xmath4 , required to consider transport and optics , follows the approach of @xcite where the particle - hole symmetric pam appropriate to the case of kondo insulators was considered . for that reason further discussion of the approach is omitted here . the reader is instead directed to @xcite on the pam , from which appropriate results will be used when required ; and to @xcite for anderson impurity models _ per se _ where details of the lma , including its stengths and limitations in relation to other approaches , are fully discussed . as mentioned above , a knowledge of single - particle dynamics is sufficient within dmft to determine @xmath108 transport properties @xcite . this arises because the strict absence of vertex corrections in the skeleton expansion for the current - current correlation function means only the lowest - order conductivity bubble survives @xcite , and a formal result for it is thus readily obtained . denoting the trace of the conductivity tensor by @xmath109 ( @xmath110 of which , denoted by @xmath111 , provides an approximation to the isotropic conductivity of a 3-dimensional system ) , this may be cast in the form ( ; t)=_0 f(;t ) [ 2.9 ] with @xmath112 merely an overall scale factor ( @xmath113 is the lattice constant and @xmath114 is typically of order @xmath115 ) . the dimensionless dynamical conductivity @xmath116 naturally depends on the lattice type , and for a bloch decomposable lattice such as the hcl is given ( with @xmath43 ) by @xcite f_hcl(;t)=^_- d_1d^c(;_1 ) d^c(;_1 + ) _ [ 2.10 ] where @xmath117^{-1}$ ] is the fermi function . here ( as in equation ( [ 2.6 ] ) ) , the notation @xmath118 denotes an average with respect to the non - interacting conduction band dos @xmath73 ; and the spectral density @xmath119 with @xmath120^{-1}$ ] from equation ( [ 2.7 ] ) . physically , @xmath121 ( @xmath122 ) represents the @xmath27-dependent conduction electron scattering rate ( inverse scattering time ) arising from electron interactions , @xmath123 ( @xmath124 from equation ( [ 2.8 ] ) ) . it is given using equation ( [ 2.5b ] ) by = _ i(;t ) = [ 2.11 ] in terms of the @xmath3-electron self - energy ; a knowledge of which thus determines the scattering rates ( considered explicitly in section 3.1 ) , and in consequence the dynamical conductivity equation ( [ 2.10 ] ) ( noting that @xmath125 ^ 2 + [ \gamma_i(\om;t)]^2)$ ] ) . results for @xmath126 obtained using the lma will be considered in sections 4,5 . here we simply point out an exact result , not apparently well known , for the weight of the drude peak in the @xmath18 conductivity . at @xmath18 , scattering at the fermi level is absent since the system is a fermi liquid , i.e. @xmath127 and hence @xmath128 . the leading low - frequency behaviour of @xmath129 is given by ^r_f(;0 ) ~^r_f(0;0 ) - ( -1)[2.12 ] where @xmath130^{-1}$ ] is the usual quasiparticle weight / inverse mass renormalization ; hence ( from equation ( [ 2.5b ] ) ) @xmath131 , where ^*_f = _ f + ^r_f(0;0 ) [ 2.13 ] is the renormalized @xmath3-level energy . a straightforward evaluation of equation ( [ 2.10 ] ) for @xmath18 and @xmath132 then shows that @xmath133 contains a @xmath134 drude ` peak ' ( as it must , reflecting the total absence of fermi level scattering and a vanishing @xmath18 resistivity ) . denoted by @xmath135 , it is given explicitly by f_drude(;0 ) = ( ) _ 0(-_c+ ) [ 2.14a ] or equivalently f_drude(;0 ) = ( ) _ 0(-_c+ ) [ 2.14b ] where @xmath136 and _ l = zv^2 . [ 2.15 ] equations ( 2.14 ) are exact , and bear comment . in the trivial limit @xmath53 where ( equation(2.1 ) ) the @xmath3-levels decouple from the conduction band , the total drude weight is naturally @xmath137 , the free conduction band dos at the fermi level ( recall @xmath138 ) . for any @xmath139 , the luttinger integral theorem requires ( n_c+n_f ) = ^-_c+1/_f^*_- _ 0 ( ) d + ( -_f^ * ) [ 2.16 ] ( with @xmath140 merely the unit step function ) . this again is an exact result , proven in @xcite . it holds for _ any _ interaction @xmath6 , reflecting the adiabatic continuity to the non - interacting limit that is intrinsic to a fermi liquid ; and shows in general that ( any ) fixed total filling @xmath141 determines @xmath142 entering equations ( 2.14 ) . of particular interest is of course the strongly correlated hf regime , where @xmath143 . here @xmath1 in equation ( [ 2.15 ] ) ( @xmath144 with @xmath145 ) is the coherence scale : exponentially small in strong coupling ( because @xmath146 is ) , it is the _ single _ low - energy scale in terms of which all properties of the system exhibit universal scaling ( as shown in @xcite and pursued below ) . in the hf regime , @xmath20 itself is moreover given ( see @xcite ) by n_c = ^-_c_-_0 ( ) d[2.17a ] showing that @xmath147 and @xmath20 are in essence synonymous , @xmath148 being determined by @xmath147 alone . conjoining this with equation ( [ 2.16 ] ) gives n_f = ^-_c+1/_f^*_-_c _ 0 ( ) d + ( -_f^ * ) [ 2.17b ] so as @xmath143 ( the hf regime ) , @xmath149 is _ also _ determined by @xmath147 alone , and is typically of order unity . it is this that determines @xmath150 entering equation ( [ 2.14b ] ) for @xmath135 ; showing in turn that the net drude weight is itself @xmath151 , and hence exponentially diminished compared to the free conduction band limit . we add that kondo insulators , arising generically for @xmath152 as mentioned earlier , are also encompassed by the above . using @xmath153 , the luttinger theorem equation ( [ 2.16 ] ) shows that @xmath152 arises either for @xmath154 ( for an unbounded @xmath104 ) or for @xmath142 outside the band edges of a compact @xmath104 ; such that in either case the drude weight in equations ( 2.14 ) vanishes , symptomatic of the vanishing @xmath18 d.c . conductivity characteristic of the kondo insulating state . our focus above has naturally been on the canonical case of a bloch decomposable lattice . for a bethe lattice by contrast , @xmath116 is given @xcite by ( _ cf _ equation ( [ 2.10 ] ) ) f_bl(;t)=^_- d_1d^c(_1 ) d^c(_1 + ) [ 2.18 ] where @xmath155 ( @xmath156 ) is the local conduction band spectrum . in particular the d.c . conductivity at @xmath18 follows as @xmath157 ^ 2 $ ] ; which , using @xmath158 together with equations ( [ 2.2a ] ) , ( 2.5 ) , ( [ 2.12 ] ) , is given by f_bl(0;0 ) = [ _ 0(-_c+)]^2 . [ 2.19 ] in contrast to equations ( 2.14 ) there is thus no drude @xmath134-peak and the @xmath18 d.c . resistivity is in general finite , reflecting of course that the underlying one - particle states of the bl are not coherent bloch states . hence , aside from the case of kondo insulators where the bl ( like the hcl ) does capture the vanishing @xmath18 d.c . conductivity and indirect - gapped optics characteristic of the insulator @xcite , the ` joint density of states ' type formula equation ( [ 2.19 ] ) should not be taken seriously when considering transport / optics of real materials on sufficiently low @xmath4 and/or @xmath27 scales ( as discussed further in section 4 ) . we turn now to lma results for single - particle dynamics at finite-@xmath4 . our natural focus will be the strong coupling kondo lattice regime ( where @xmath143 ) , characterized by the low - energy lattice scale @xmath159 . this scale is of course a complicated function of the bare / material parameters , @xmath160 ( detailed lma results for it are given in @xcite , and nrg results in @xcite ) . that dependence is however a subsidiary issue in comparison to the fact that , because @xmath1 becomes exponentially small in strong coupling , physical properties exhibit scaling in terms of it ; i.e. depend universally on @xmath161 , independently of the interaction strength . universality in strong coupling single - particle dynamics at @xmath18 has been considered in @xcite for the generic pam ; the essential findings of which are first reprised for use below . ( i ) both the @xmath19-electron spectrum @xmath155 ( @xmath162 with @xmath43 ) and the @xmath3-electron spectrum @xmath163 ( with @xmath164 introduced in section 2 ) , exhibit universal scaling as a function of @xmath22 in a manner that is _ independent _ of both the interaction strength @xmath6 _ and _ hybridization matrix element @xmath38 . ( ii ) that scaling depends in general only on @xmath147 ( or equivalently the conduction band filling @xmath20 , see equation ( [ 2.17a ] ) ) which embodies the conduction band asymmetry ; and on @xmath165 reflecting the @xmath3-level asymmetry . more specifically , ( iii ) in the coherent fermi liquid regime arising for @xmath166 , the @xmath3- scaling spectra depend only on @xmath147 and are in fact independent of @xmath45 as well as @xmath6 and @xmath38 . in this low-@xmath24 regime the scaling spectra amount in essence to the quasiparticle behaviour ( equations ( 3.11 ) of @xcite ) required by the asymptotic dictates of low - energy fermi liquid theory . ( iv ) for @xmath167 by contrast the @xmath3- scaling spectra depend on the @xmath3-level asymmetry @xmath45 ( albeit rather weakly ) , but are now independent of @xmath147 and indeed also of the lattice type ; and the spectrum contains a long , logarithmically slowly decaying spectral tail . ( v ) the latter behaviour , which sets in progressively for @xmath168 , reflects in turn the crossover to incoherent effective single - impurity physics that one expects to arise for sufficiently high @xmath27 ( and/or @xmath4 ) : for @xmath167 the _ scaling form _ of the @xmath3-spectrum is found to be precisely that of an anderson impurity model ( aim ) . with increasing @xmath24 , dynamics thus cross over from the low - energy quasiparticle behaviour symptomatic of the lattice coherent fermi liquid state to single - impurity kondo scaling physics at high @xmath24 ( and that this crossover occurs in a single @xmath161 scaling regime is thus incompatible with the occurrence of ` two - scale exhaustion ' @xcite as explained in @xcite ) . figure [ fig1 ] summarises representative results for @xmath18 scaling dynamics ( irrelevant non - universal energy scales such as @xmath6 , @xmath41 ( @xmath169 ) or @xmath170 are of course projected out in scaling spectra @xcite ) . the main figure shows @xmath23 scaling spectra for the hcl as functions of the scaled frequency @xmath171 , for @xmath172 with @xmath173 ( dashed , and @xmath174 ) and @xmath175 ( solid , with @xmath176 ) . the @xmath177 example corresponds to the particle - hole ( p - h ) symmetric kondo insulator , whose spectra are thus gapped at the fermi level @xmath178 ( with @xmath159 here corresponding to the insulating gap scale @xcite ) . for the asymmetric conduction band @xmath179 by contrast , the gap ( which is well developed in strong coupling @xcite ) moves above the fermi level ; and a sharp lattice - kondo resonance symptomatic of the hf metal , straddling the fermi level and of width @xmath180 , takes its place in the @xmath3-spectra . the inset shows the @xmath3-spectra on a much larger @xmath24 scale ; displaying the @xmath31-independence of the slow logarithmic tails @xcite and reflecting the crossover to effective single - impurity behaviour ( which we emphasise arises whether the system is a hf metal or a kondo insulator ) . at finite temperatures , what one expects for the strong coupling scaling spectra is clear : they should now depend universally on @xmath181 _ and _ @xmath182 . that this arises correctly within the present lma is shown in figure [ fig2 ] . for a fixed @xmath183 , the @xmath3- and @xmath19-spectra are shown for progressively increasing interaction strengths @xmath184 and @xmath185 with @xmath186 ; for @xmath187 and @xmath172 ( corresponding results for the p - h symmetric limit have been obtained in @xcite ) . the inset shows the @xmath3-spectra on an absolute scale ( _ vs _ @xmath188 ) , where the exponential reduction of the @xmath1-scale with increasing @xmath6 is clearly seen from the change in the width of the resonance . the main figures by contrast show the spectra as functions of @xmath24 , from which the @xmath6-independent scaling collapse is evident ; repeating the calculations with different @xmath189 likewise shows the scaling to be independent of @xmath38 . this behaviour is not of course confined to the chosen @xmath26 , and figure [ fig3 ] shows the resultant lma scaling spectra for a range of @xmath26 ( again for the representative @xmath190 ) . figures [ fig3 ] and [ fig2 ] show clearly the thermal broadening and ultimate collapse of the @xmath3-resonance with increasing @xmath26 ; which is naturally accompanied by a redistribution of spectral weight leading to infilling of the ( @xmath191 ) spectral gap seen in figure [ fig1 ] for @xmath18 . in fact by @xmath192 this gap is already obliterated , and the lattice kondo resonance also significantly eroded . this behaviour is typical of the metallic hf state . by contrast , corresponding results for the p - h symmetric kondo insulator ( @xmath193 ) are shown in figure 4 of @xcite . in that case the insulating gap at the fermi level fills up with increasing temperature , and the fermi level @xmath194 in particular increases monotonically with increasing temperature ; in contrast to the the asymmetric hf spectra shown above where @xmath195 diminishes with @xmath26 . two further points regarding figure [ fig3 ] should be noted . first , the thermal evolution of the @xmath3- and @xmath19-spectra differ somewhat in terms of the persistence of a pseudogap the @xmath3-spectrum shows no sign of the gap by @xmath196 , while a weak pseudogap structure persists in the @xmath19-spectrum up to @xmath197 ; this reflects the rapid spread of spectral weight caused by the meltdown of the sharp resonance in the @xmath3-spectra , of which there is no counterpart in the @xmath19-spectra . second , the inset to figure [ fig3 ] shows the @xmath3-spectra on an enlarged frequency scale out to @xmath198 , from which it is seen that the high frequency behaviour of the finite-@xmath26 scaling spectra coincide with that for @xmath18 . this is physically natural , since one expects the dominant influence of temperature to be confined to frequencies @xmath199 . the corollary of course is that non - universal frequencies are affected only on non - universal , and thus in general physically irrelevant , temperature scales ( as shown in figure 5 of @xcite for the p - h symmetric case ) . we consider now the scattering rates @xmath200 that underlie the evolution of the conductivity , and are given explicitly in terms of the @xmath3-electron self - energy by equation ( [ 2.11 ] ) . since the system is a fermi liquid with @xmath201 , at @xmath18 there is of course no scattering at the fermi level , @xmath202 . the low frequency behaviour of the @xmath18 scattering rate can be understood qualitatively by using the low-@xmath27 expansion of @xmath203 ( equation ( [ 2.12 ] ) ) and simply neglecting the imaginary part @xmath204 ; leading to ^-1(;t=0 ) ( - _ f^ * ) [ 3.1 ] with @xmath205 the renormalized level . restoring the small but strictly non - vanishing @xmath206 naturally implies a narrow resonance centred on @xmath207 instead of a pure @xmath208-function . at finite temperature , we likewise expect the scattering rate to increase from zero in the neighbourhood of the fermi level , reflecting the finite-@xmath4 contribution to @xmath209 ; and that this will simultaneously lead to further , thermal broadening of the resonance at @xmath150 . the above picture is corroborated by lma results as shown in figure [ fig4 ] , displaying the @xmath22 dependence of @xmath200 ( in units of @xmath42 ) arising in strong coupling for @xmath187 and @xmath172 , for a range of temperatures @xmath210 between @xmath211 and @xmath212 . in this case the renormalized level is found to be @xmath213 , precisely where @xmath214 has a narrow resonance . with increasing temperature the resonance is indeed seen to broaden and decrease in intensity ; and we reiterate that this occurs for temperatures @xmath4 set by the scale @xmath1 the sole low - energy scale characteristic of the problem in strong coupling . excepting the lowest @xmath26 we also note that scattering rates in the vicinity of the fermi level are on the order of @xmath215 of the bandwidth @xmath41 , values some two or so orders of magnitude higher than for conventional clean metals ( and indicative of the higher d.c . resistivities that are typical of heavy fermion materials @xcite ) . neither is this behaviour confined to a narrow @xmath26 regime since even for @xmath216 the scattering rates decay very slowly with @xmath26 ; the fermi level scattering rate for example is readily shown to decay as @xmath217 . the scattering rates are also related to the @xmath3-electron scaling spectra considered above . for the kondo insulating p - h symmetric pam , it was shown in @xcite that the dimensionless scattering rate defined as = _ i(;t ) [ 3.2 ] coincides asymptotically with the @xmath3-spectral function , specifically ~_0 d^f ( ) [ 3.3 ] in the regime @xmath218 for any @xmath26 ( the spectral ` tails ' ) , and for all @xmath219 for sufficiently large @xmath216 . equation ( [ 3.3 ] ) is in fact readily shown to be quite general , and not dependent on p - h symmetry . that it holds for hf metals embodied in the asymmetric pam is illustrated in figure [ fig5 ] , where for @xmath187 and @xmath172 the strong coupling @xmath220 and @xmath221 _ vs _ @xmath24 are compared , for @xmath222 in the left panel and @xmath183 and @xmath223 in the right panel . the high - frequency behaviour of the scaling spectrum @xmath221 is itself known , being given ( here for @xmath224 explicitly ) by @xcite _ 0 d^f ( ) ( + ) [ 3.4 ] with @xmath113 a pure constant @xmath225 . these slowly decaying logarithmic tails are evident in figure [ fig5 ] , and as mentioned in section 3 embody the connection to effective incoherent single - impurity physics on high energy scales . they are independent of the interaction @xmath6 , local hybridization @xmath38 , underlying conduction band asymmetry @xmath31 , and even of the lattice type ; depending , albeit weakly , only on the @xmath3-level asymmetry @xcite . the above discussion of scattering rates leads naturally to consideration of transport ; beginning with the d.c . limit where ( section 2.2 ) the static conductivity @xmath226 , with @xmath116 given for the hypercubic lattice by equation ( [ 2.10 ] ) . in the strong coupling regime we expect static transport to exhibit universal scaling in terms of @xmath25 , and our aim here is to understand its thermal evolution across the full @xmath26 range . transport on non - universal temperature scales @xmath227 ( @xmath228 ) or @xmath229 , will be discussed briefly at the end of the section . for the p - h symmetric kondo insulator , lma results for the @xmath26-dependence of the scaling resistivity have been considered in @xcite ( in this case @xmath230 is equivalently the insulating gap scale ` @xmath13 ' ) . the @xmath18 resistivity is naturally infinite reflecting the gapped ground state , the scaling resistivity @xmath231 has an activated form @xmath232 for @xmath233 ( with @xmath234 a pure constant @xmath235 and hence a ` transport gap ' of @xmath236 ) ; and @xmath5 decreases monotonically with increasing @xmath26 , tending asymptotically to incoherent single - impurity scaling behaviour ( @xcite and figures [ fig7],[fig8 ] below ) . for the general case of heavy fermion metals the situation is of course quite different , and what one expects in qualitative terms well known @xcite . the @xmath18 resistivity vanishes , reflecting the absence of fermi level scattering and the underlying coherence generic to any bloch decomposable lattice . with increasing temperature @xmath5 increases ( initially as @xmath237 for @xmath233 @xcite ) , passes through a maximum at @xmath238 a classic signature of hf compounds @xcite and decreases thereafter in the strong coupling , kondo lattice regime of interest . figure [ fig6 ] shows lma results for @xmath5 _ vs _ @xmath26 for fixed @xmath239 , and with increasing interaction @xmath240 and @xmath185 for @xmath186 . the scaling collapse is clearly evident : while the low - energy scale @xmath241 itself diminishes exponentially on increasing @xmath6 , universal scaling of @xmath5 as a function of @xmath25 indeed arises in strong coupling , independent of interaction strength ( and likewise readily shown to be @xmath38-independent on repeating the calculations varying @xmath189 ) . this leads us first to comment briefly on the issue of ` the coherence scale ' , characterising the crossover from low - temperature lattice coherent behaviour to high - temperature effective single - impurity behaviour . experimentally , many such identifications of the low - energy scale are commonly employed . some groups use @xmath238 at which @xmath5 peaks , others identify the scale via the inflection points ( @xmath242 , marked by arrows in figure [ fig6 ] ) , via the leading @xmath237 behaviour of @xmath5 at low-@xmath4 , or via the onset of the ` log - linear ' regime @xcite ( shown in inset ( a ) to figure [ fig6 ] and seen in many experimental systems @xcite ) ; the inverse of the @xmath243 paramagnetic susceptibility , or the width of the lattice kondo resonance , are other possibilities . this leads to what at first sight might seem a plethora of low - energy scales . the key point however is that , because physical properties in strong coupling scale universally in terms of _ one _ low - energy scale , all the above definitions of ` the coherence scale ' are fundamentally equivalent : all are proportional to @xmath1 , and hence to each other in figure [ fig6 ] for example , the inflection points in @xmath5 lie at @xmath244 and @xmath245 , and the peak maximum at @xmath246 . as for single - particle dynamics and scattering rates considered in section 3 , the @xmath26-dependent scaling resistivity is independent of @xmath6 or @xmath38 ( as above ) but depends in general on @xmath147 ( reflecting the conduction band asymmetry and determining @xmath20 via equation ( [ 2.17a ] ) ) and @xmath45 ( reflecting the @xmath3-level asymmetry ) . to consider this figure [ fig7 ] shows the resultant scaling resistivities @xmath5 _ vs _ @xmath26 for @xmath172 , and a range of different @xmath147 @xmath247 and @xmath248 , corresponding respectively to conduction band fillings @xmath249 and @xmath250 . the @xmath177 example is the kondo insulator @xcite , with its characteristic diverging @xmath5 as @xmath251 . the others are all hf metals , and exhibit the same qualitative behaviour for all @xmath147 a positive slope for @xmath252 , going through the maximum and then decreasing monotonically for @xmath253 ; the coherence peak itself increasing monotonically with @xmath147 , albeit slowly such that @xmath254 for the @xmath147-range shown . qualitatively similar behaviour is found on varying the @xmath3-level asymmetry @xmath45 for fixed conduction band asymmetry embodied in @xmath147 , although quantitatively this effect is appreciably less . the significant @xmath147-dependence of @xmath255 seen in figure [ fig7 ] for @xmath256 is intuitively natural : the strong coupling kondo lattice regime corresponds to @xmath257 , but with variable conduction band filling ( @xmath20 ) controlled by @xmath147 ( equation ( [ 2.17a ] ) ) ; and on decreasing @xmath20 ( increasing @xmath147 ) one expects the static conductivity to diminish and hence an increased @xmath5 , as found . to understand the @xmath147-dependence , and in turn to enable connection to incoherent effective single - impurity behaviour at high-@xmath26 , we first consider an approximate evaluation of @xmath258 ( equation ( [ 2.10 ] ) ) ; in which the energy dependence of the free conduction band dos @xmath259 is neglected , @xmath260 being replaced by its fermi level value . employing this ` flat band ' approximation in equation ( [ 2.10 ] ) ( where it enters via the @xmath261 average ) leads to f_hcl(0;t ) [ _ 0(-_c)]^2 ^+_-d ( ; t ) [ _ 0(-_c)]^2 [ 4.1 ] expressed as a physically intuitive thermal average of the dimensionless scattering time @xmath262 ( equation ( [ 3.2 ] ) ) . for kondo insulators this approximation is qualitatively inadequate at low-@xmath26 @xcite , but as illustrated in figure [ fig7 ] ( inset ) it is entirely respectable for the hf metals and in particular recovers precisely the high-@xmath26 asymptotics of @xmath5 . as shown in section 3.1 , the large @xmath24 and/or @xmath26 dependence of the reduced scattering rate @xmath220 coincides with the @xmath3-spectral function @xmath221 ( equation ( [ 3.3 ] ) ) ; and in section 3 ( see also @xcite ) the latter were shown to have common spectral tails , independently of @xmath147 . this suggests that the primary effect of @xmath147 seen in figure [ fig7 ] for @xmath255 is contained in the @xmath263 ^ 2 $ ] of equation ( [ 4.1 ] ) . that this is so is seen in figure [ fig8 ] where the results of figure [ fig7 ] are now shown as @xmath264 _ vs _ @xmath26 , where ^(t)=. [ 4.2 ] for @xmath265 or so in practice , @xmath264 is seen in particular to be _ independent _ of the conduction band filling embodied in @xmath147 ; including we note the kondo insulator , whose ` high ' temperature resistivity is thus seen to be that of a regular heavy fermion metal . indeed as readily demonstrated , and evident in part from the above discussion , the behaviour seen in figure [ fig8 ] is barely dependent on the details ( @xmath266-dependence ) of the host bandstructure embodied in @xmath104 . [ h ] the obvious final question here concerns the high-@xmath26 form of @xmath264 for the pam . to that end we consider the anderson single - impurity model ( aim ) , with @xmath267 denoting as usual the change of resistivity due to addition of the impurity to the non - interacting host , and @xmath268 . this is given by @xcite ( _ _ cf__equations ( [ 4.1]),([4.2 ] ) ) = ^+_-d _ imp(;t ) [ 4.3 ] with the impurity scattering rate @xmath269 ; where @xmath270 is the impurity spectral function such that @xmath271 follows from the friedel sum rule @xcite in the singly occupied , strong coupling kondo regime of the aim . the lma scaling resistivity @xmath272 _ vs _ @xmath26 is also shown in figure [ fig8 ] , where @xmath273 and @xmath274 is the aim kondo scale ( with @xmath275 the impurity quasiparticle weight ) . from this it is seen that the high-@xmath26 scaling behaviour of @xmath264 for the pam is precisely that of the aim ; in particular the leading @xmath216 behaviour of the lma @xmath264 is readily shown analytically to be given by @xmath276 , which is exact in the kondo limit of the impurity model @xcite . this reflects again the crossover in the strong coupling pam from low - temperature lattice coherent behaviour to incoherent effective single - impurity scaling physics , here in the context of d.c . transport . as for its counterpart in the case of single - particle dynamics @xcite , we point out ( a ) that since this connection is established from scaling considerations it is entirely independent of how the scales @xmath1 and @xmath277 for the two distinct models ( pam and aim ) depend on the underlying bare / material parameters of the respective problems ; and ( b ) the fact that it arises in the @xmath278 scaling regime precludes a two - scale description of the crossover from lattice - coherent to incoherent effective single - impurity physics . our focus above has naturally been on the strong coupling , kondo lattice regime . we now look briefly at d.c . transport on non - universal scales . what one expects here is that when the temperature is a not insignificant fraction of the hybridization @xmath170 or bandwidth scale @xmath279 , kondo screening will be washed out , and hence @xmath5 should cross over from the logarithmically decreasing single - impurity form at @xmath280 ( figures [ fig7 ] and [ fig8 ] ) to conventional metallic behaviour @xmath281 at non - universal temperatures ; and thus as such must go through a minimum . that this indeed happens can be seen in the inset to figure [ fig8 ] where we show @xmath282 _ vs _ @xmath25 for @xmath187 , @xmath283 and @xmath284 ( solid line ) and @xmath285 ( point - dash ) . for the lower @xmath6 example , a minimum is seen at @xmath286 , which corresponds in ` absolute ' units ( @xmath41 ) to a temperature @xmath287 an appreciable fraction of the hybridization @xmath288 . the corresponding minimum does of course exist for the higher @xmath6 , but is pushed beyond @xmath289 ( and to concomitantly lower vales of @xmath5 ) ; and @xmath5 in this case lies on the universal scaling curve throughout the @xmath26-range shown in figure [ fig8 ] . a final point is worth noting here . for @xmath290 , the @xmath5 _ vs _ @xmath25 for the two @xmath6 s shown in figure [ fig8 ] ( inset ) are in essence coincident ; each lies on the universal scaling curve . what distinguishes different interaction strengths is of course the location of the minimum , occurring as it does on non - universal temperature scales . no real hf material is however in the universal scaling regime ` for ever ' with increasing @xmath4 the scaling regime will be exited sooner or later . and the temperature for which the experimental @xmath5 is a minimum ( once phonon contributions have been subtracted out ) can provide valuable information on the interaction strength , as we shall see in action in the following paper . we have considered almost exclusively the hypercubic lattice , for the obvious reason that its one - particle bloch states ultimately underlie the low - temperature lattice coherence of the interacting problem . for the bethe lattice , the strong coupling scaling resistivity @xmath291 ( with @xmath292 from equation ( [ 2.18 ] ) ) is shown _ vs _ @xmath26 in inset ( b ) to figure [ fig6 ] , for @xmath293 . in contrast to its counterpart for the hcl shown in the main figure , the @xmath222 resistivity is non - vanishing ( given by equation ( [ 2.19 ] ) ) , reflecting the absence of coherent bloch states for the bl . further , the _ high_-@xmath26 asymptote of the bl @xmath5 in the scaling regime is likewise non - zero ; being given by the @xmath18 value of the free ( @xmath53 ) conduction band resistivity , namely @xmath294 ^ 2 = \pi^2/[4(1-\epsilon_c^2)]$ ] as marked by a cross in figure [ fig6 ] inset ( and arising for the same physical reasons discussed for kondo insulators in @xcite ) . the qualitative contrast between @xmath5 for the canonically bloch decomposable hcl , and that for the bl , illustrates why the latter more specifically the associated ` joint density of states ' type formula equation ( [ 2.18 ] ) for @xmath295 that is not uncommonly employed in the literature gives a poor caricature of d.c . transport for hf metals in which the lattice coherence is of central importance . we turn now to the optical conductivity @xmath296 ( with @xmath126 given by equation ( [ 2.10 ] ) ) . in the strong coupling kondo lattice regime @xmath126 is of course independent of @xmath6 and @xmath189 , and a universal function of @xmath22 and @xmath25 for fixed @xmath147 and @xmath45 . lma results for @xmath126 are shown in figure [ fig9 ] , for @xmath179 and @xmath224 . the right panel shows the thermal evolution of the optical conductivity ( on a linear @xmath24-scale ) for temperatures @xmath297 and @xmath223 ; while the left panel ( on a log - log scale ) shows the behaviour for a lower range of temperatures up to @xmath298 . the latter in particular illustrates the thermal evolution of the optical drude peak , which at @xmath299 consists of an @xmath300 @xmath208-function given by equations ( 2.14 ) ( with net weight @xmath180 in strong coupling ) . on increasing @xmath26 from @xmath211 the drude peak naturally broadens , and is well fit by a lorentzian up to its half - width or so , after which it decays more slowly in @xmath24 . at the lowest @xmath26 shown the drude peak is well separated from the ` optical edge ' in @xmath126 seen at @xmath301 ( although we add that @xmath126 is strictly non - zero for all @xmath24 ) , and with increasing @xmath26 is seen to persist as an essentially separate entity up to @xmath302 or so ; after which it is progressively destroyed as expected , merging into an optical pseudogap in the neighbourhood of @xmath303 , which is reasonably well filled up by @xmath304 and all but gone by @xmath305 ( see figure [ fig9 ] , right panel ) . similar behaviour is naturally found on varying @xmath147 and/or @xmath45 . figure [ fig10 ] shows in particular the influence of @xmath147 ( varying conduction band filling ) on the optical pseudogap for a fixed temperature @xmath306 , from which it is seen that the pseudogap becomes shallower with increasing @xmath147 . the above behaviour should be compared to the p - h symmetric kondo insulator ( ki ) @xmath307 considered in @xcite . in that case the @xmath18 optical conductivity is characterized by an indirect gap @xmath308 , and there is of course no @xmath18 drude peak . instead a drude - like peak in the optical conductivity actually builds up on initially increasing @xmath26 from zero ( see figure 15 of @xcite ) , before being thermally broadened and subsumed into the optical pseudogap . for @xmath309 or so the low - frequency optics of the ki are thus very different from those of the hf metal , as expected . but for @xmath310 the optical behaviour of the two is qualitatively similar as shown by comparison of figure [ fig9 ] ( right panel ) and its counterpart for the ki , figure 14 of @xcite . this too is physically natural , since the infilling of the indirect optical gap on temperature scales @xmath192 means that the ki behaves to all intents and purposes as a hf metal ; as seen also in figures [ fig7 ] or [ fig8 ] for the static transport . a second point should be emphasised here , obvious though it is from the preceding discussion : whether for hf metals or kondo insulators , it is the low - energy scale @xmath159 that sets the intrinsic scale for both the @xmath27-dependence of the low - energy optical conductivity and its thermal evolution . and in strong coupling that scale is wholly distinct from the optical _ direct _ gap , @xmath311 . the latter arises at its simplest in the commonly employed renormalized band picture ( see e.g. @xcite ) , as the minimum direct gap for which optical transitions are allowed . in this effective single - particle description the imaginary part of the @xmath3-electron self - energy and hence all scattering is neglected entirely , and the corresponding real part @xmath312 is replaced by its leading low-@xmath27 behaviour equation ( [ 2.12 ] ) ( as also inherent to a slave boson mean - field approximation @xcite ) . the two branches of the renormalized bandstructure , denoted by @xmath313 with @xmath314 , then follow from the zeros of @xmath315^{-1 } = [ \gamma(\om)-\epsilon_{\mathbf k}]$ ] ( see equation ( [ 2.7 ] ) ) with the approximate @xmath316 @xmath317^{-1}$ ] from equation ( [ 2.5b ] ) ; and the resultant @xmath266-dependent direct gap @xmath318 $ ] is given by @xmath319^{1/2}$ ] with @xmath320 the usual renormalized level . the minimum direct gap , @xmath311 , occurs for @xmath321 ( @xmath322 in strong coupling ) and is thus _ [ 5.1 ] the corresponding result for the optical conductivity @xmath126 is readily determined from equation ( [ 2.10 ] ) . denoted by @xmath323 it is given for @xmath18 ( and all @xmath324 ) by f_o(;0 ) = [ 5.2 ] with @xmath325 the free conduction band dos and @xmath326 the unit step function ; and is thus non - zero only for frequencies @xmath327 _ above _ the direct gap ( which result is also readily shown to hold for _ all _ temperatures ) . two points should be noted here . first that the low - energy scale @xmath159 intrinsic to hfs or kis is qualitatively distinct from the direct gap @xmath311 . in fact since @xmath328 it follows that in strong coupling where the quasiparticle weight @xmath146 and hence @xmath1 becomes exponentially small , optics on the direct gap scale do not even lie in the @xmath329 scaling regime ; although neither do they occur on truly non - universal scales ( because @xmath330 ) and in that sense belong to the ` low - frequency ' optical spectrum . second , we emphasise the inherent naivet of interpreting optics in terms of renormalized single - particle interband transitions : it is scattering due to electron interactions that generates _ all _ the optical density below the direct gap scale . failure to include such , as in a renormalized band picture and regardless of how sophisticated the underlying band structure employed in practice inevitably leads to a qualitatively inadequate description of optics ( as illustrated explicitly in figure [ fig11 ] below ) . neither is this situation ameliorated in materials application by the introduction of _ ad hoc _ @xmath27-dependent broadening factors , for that simply avoids the basic underlying physics . lma results for optics on all frequency scales are given in figure [ fig11 ] , for @xmath190 ; where @xmath126 is shown _ vs _ @xmath22 on a log scale spanning five orders of magnitude , for the same range of temperatures @xmath25 employed in figure [ fig9 ] ( right panel ) . to encompass all @xmath27 including non - universal energies , the bare parameters @xmath6 and @xmath189 must of course be specified , @xmath331 and @xmath332 here being chosen for illustration ; although note that the optical conductivity as a function of @xmath22 remains ` universal ' ( independent of @xmath6 or @xmath189 ) up to large but finite values of @xmath24 determined by the particular @xmath6 and @xmath189 chosen , in this example @xmath333 ( a directly analogous situation for the resistivity @xmath5 is shown in the inset to figure [ fig8 ] ) . the inset to figure [ fig11 ] shows the renormalized bandstructure @xmath334 _ vs _ the free conduction band energy @xmath266 ; determined as above from solution of @xmath335 ( with @xmath336 the full @xmath337 ) . this enables the notional direct gap to be determined , @xmath338 here well separated from the low - energy coherence scale @xmath1 in strong coupling and indeed seen to occur for @xmath339 . the essential points from figure [ fig11 ] are clear . as expected and well known ( see e.g. @xcite ) , significant optical absorption occurs in the vicinity of the direct gap ; strongly broadened to low - energies due to electron interactions as above , and all but ` dead ' on non - universal energy scales ( e.g. the hybridization @xmath340 for the chosen bare parameters ) . regarding the thermal evolution of the optical conductivity note also that temperatures on the order of a few multiples of the coherence scale @xmath1 which control the thermal evolution of the low - energy optics have essentially no effect on frequencies of the order of the direct gap , reflecting the clean separation between @xmath1 and @xmath311 characteristic of strong coupling . as a corollary the direct gap should be thermally eroded only for @xmath341 ; as indeed seen in figure [ fig12 ] where ( for the same parameters as figure [ fig11 ] ) the thermal evolution of @xmath126 is shown for temperatures up to @xmath342 . significant thermal erosion sets in by about @xmath343 or so , and is well developed by the highest temperature shown . the clear scale separation between @xmath1 and @xmath311 will not however be captured properly if one is restricted to relatively low interactions and high temperatures as e.g. in quantum monte carlo @xcite , or from theories in which the quasiparticle weight @xmath146 is algebraically rather than exponentially small in the interaction strength , such as iterated perturbation theory @xcite . we have considered here the periodic anderson lattice , the canonical model for understanding heavy fermion metals , kondo insulators , intermediate valence and related materials . optical conductivities , d.c . transport and single - particle dynamics of the paramagnetic phase have been investigated , using the local moment approach within a dmft framework . for obvious physical reasons our main focus has been the strongly correlated kondo lattice regime , where we find the problem to be characterised by a single , exponentially small coherence scale @xmath1 ; in terms of which the frequency and temperature dependence of physical properties scale being universally dependent on @xmath344 and/or @xmath25 regardless of the interaction or hybridization strengths . all relevant energy / temperature scales are handled by the theory , from the low - energy coherent fermi liquid domain out to large ( and in the strict scaling limit arbitrarily large ) multiples of @xmath1 where incoherent many - body scattering dominates the physics ; followed by the crossover out of the scaling regime to non - universal , high energy / temperature scales dictated by ` bare ' model / material parameters . and while our emphasis has been on strong correlations we add that all interaction strengths from weak to strong coupling are encompassed by the lma @xcite , such that intermediate valence behaviour in particular can also be addressed . the first question posed in the introduction nonetheless remains : to what extent does the model , and our theory for it , capture experiment ? we turn to that in the following paper where direct comparison of theory and experiment is made for three heavy fermion materials and a classic intermediate valence compound . we are grateful to the epsrc for supporting this research .

the paramagnetic phase of heavy fermion systems is investigated , using a non - perturbative local moment approach to the asymmetric periodic anderson model within the framework of dynamical mean field theory . the natural focus is on the strong coupling kondo - lattice regime wherein single - particle spectra , scattering rates , d.c . transport and optics are found to exhibit @xmath0 scaling in terms of a single underlying low - energy coherence scale @xmath1 . dynamics / transport on all relevant ( @xmath2)-scales are encompassed , from the low - energy behaviour characteristic of the lattice coherent fermi liquid , through incoherent effective single - impurity physics likewise found to arise in the universal scaling regime , to non - universal high - energy scales ; and which description in turn enables viable quantitative comparison to experiment .

arxiv

the kinetics of polymer loop formation has been studied for several decades and recently has attracted renewed attention due to the particular importance in biology . the dna loop formation is a basic process that underlies genetic expression , replication , and recombination @xcite . for example , in _ e. coli _ the _ lac _ repressor ( laci)-mediated loop is crucial for the repressive regulation of _ lac _ genes . the hairpin loop formation is the elementary step in protein folding @xcite and structure formation in rna folding @xcite . a cell is crowded with a multitude of subcellular structures including globular proteins and rnas @xcite , with which dna is constantly interacting . a dna fragment about to loop is often subject to temporally fluctuating forces due to its dynamic environment including the other part of the chain . recently , the power spectrum of the fluctuating force exerted on cytoskeleton was measured to be an order of magnitude larger than that expected from thermal equilibrium condition @xcite . this indicates that the cell interior is an active and nonequilibrium medium . the advance of single molecule experiment techniques provides detailed information on the dna loop formation . finzi and gelles @xcite observed laci - mediated dna loop formation and dissociation by monitoring nano - scale brownian motion of the micron - sized particle attached to one end of the dna . et al . _ @xcite showed that in _ gal _ repressor and dna - bending protein hu mediated looping , mechanical constraints such as tension and torsion play a pivotal role . gemmen _ et al . _ @xcite studied effects of tension in the presence of two - site restriction enzymes which can cut the dna upon binding on two sites simultaneously . they found that the cleavage activity decreases approximately 10-fold as the tension increases from 30 fn to 700 fn . they also found that the optimum loop size decreases with the tension , which is qualitatively in agreement with theoretical predictions @xcite . more recently , chen _ et al . _ @xcite studied effects of tension in femtonewton range on the kinetics of laci - mediated dna looping . they found that small tension of 100 fn scale on the substrate dna can not only increases the looping time @xcite but also found that the looping time is greatly reduced in the presence of a fluctuating tension @xcite . these results suggest the ubiquitous roles of the static and temporally fluctuating tensions in regulation of the dna loop formation . yet , there appears to be no unifying conceptual or theoretical framework that explains a variety of experiments including these . theoretically , on the other hand , yan _ et al . _ @xcite developed a transfer matrix method to calculate semiflexible polymer end - to - end distance distribution function and loop formation probability ( or @xmath0-factor ) . they studied various effects of nonlinear elasticity arising from dna bending protein - induced kinks @xcite or thermal - fluctuation - induced bubbles on the @xmath0-factor . their study provides a valuable insight to understand dna bending on short length scale @xcite , which has attracted much attention recently @xcite . they also studied effects of tension on the @xmath0-factor @xcite , which is related to the free energy barrier for loop formation @xcite , thus to the loop formation rate . similar results are obtained by using an elastic theory of a semiflexible polymer @xcite . however , since the loop formation rate is not proportional to the @xmath0-factor alone but depends on the free energy given an arbitrary chain end - to - end distance , it is hard to quantitatively compare these theories to the experiment @xcite . independently , blumberg _ et al . _ @xcite studied effects of static tension on protein - mediated dna loop formation by modeling the dna conformation to be either one of two states , looped and unlooped states . in their appealing calculations of free energy change associated with the transition under the tension , they considered not only the stretching free energy of dna but also dna alignment constraint imposed by protein binding . they found that for the loop size larger than 100 base pair distance ( @xmath1 nm ) , a tension of 0.5 pn can increase the looping time by more than two order of magnitude . there is room for improvement in their approach , however , on the evaluation of the free energy that can be valid for short end - to - end distance of the chain as well as a description of detailed kinetic process using the mean first - passage time approach . in this paper , in an effort to understand the basic physical mechanism of the biopolymer looping in a coherent manner , we perform brownian dynamics simulation of semiflexible polymers treated as extensible wormlike chain , combined with one - dimensional theory of barrier crossing over the free energy of loop formation . for analytical understanding , we use , as an example , the mean - field wormlike chain model @xcite , which is shown to be a good approximation for the free energy for the chain lengths we consider here . with static tensions , we find that the looping time , defined as the mean first - passage time to cross the free energy barrier , steeply increases with the applied tension @xmath2 , in an agreement with our simulation results but distinct from the previous theoretical result @xcite . for the case of time - dependent tension , we consider dichotomically fluctuating tension , where the looping times are found to be reduced , consistent with the experiment @xcite . most importantly , we find so - called the resonant activation , where the looping time is the minimum at an optimal flipping time of the dichotomic force . in this exploratory study , we neglect the alignment constraint on the loop formation , which is minor effect for the chain lengths we consider here @xcite . in the following section , we describe our polymer model and simulation method , whose results are discussed in sec . [ sec : results ] . we summarize our results in sec . [ sec : conclusion ] . we consider the semiflexible polymer looping in the presence of static and fluctuating tension . the polymers are modeled as semiflexible chains of @xmath3 beads of diameter @xmath4 , with the interaction potential @xmath5 . here @xmath6 , where @xmath7 and @xmath8 are the stretching and bending energy @xmath9 @xmath10 where @xmath11 is the stretching stiffness , @xmath12 is the natural bond length , @xmath13 is the bending stiffness , @xmath14 is the position of the @xmath15th bead , and @xmath16 is the angle of the @xmath15th bond . .[fig : schematic],width=321 ] the dynamics of @xmath3 beads ( @xmath17 ) are described by the overdamped langevin equation @xmath18 where @xmath19 is the friction coefficient of the bead , and @xmath20 is the gaussian random noise with mean @xmath21 and variance @xmath22 . here , @xmath23 denotes ensemble average , @xmath24 and @xmath25 are the cartesian coordinate indices , @xmath26 is the boltzmann constant , and @xmath27 is the absolute temperature . the additional force @xmath28 is the tension which is applied only to the two end segments ( @xmath29 and @xmath3 ) along the chain end - to - end direction , @xmath30 ( see fig . [ fig : schematic ] ) . the bending stiffness of the short semiflexible chains we consider allows us to neglect the excluded volume effect . furthermore , we neglect the hydrodynamic interactions , which is found to be small in the short chains we consider here @xcite . we use the parameters @xmath12 , @xmath31 , and @xmath32 to fix the length , energy , and time scales , respectively . the dimensionless parameters in our simulation are @xmath33 , @xmath34 , @xmath35 , and @xmath36 . in our model , we consider @xmath37 as 10 nm , so with @xmath36 the persistence length @xmath38 of the chain is 50 nm . the diameter of the bead is set to be 5 nm to fit the hydrodynamic friction coefficient of a cylinder with 2 nm diameter and 10 nm length , which is about @xmath39 @xmath40 @xmath41 with the water viscosity @xmath42 @xmath40 @xmath43 @xmath41 at room temperature . the time unit is then 1.14 @xmath44s . the equations of motion are integrated by using a second - order stochastic runge - kutta algorithm @xcite with the time - step @xmath45 . in our simulation , we consider that the looping occurs whenever the chain end - to - end distance @xmath46 is shorter than a cutoff distance @xmath47 . the average were taken over at least 2,000 and 5,000 independent runs for static and fluctuating tension cases , respectively . we first study the effects of static tension on semiflexible polymer looping . here we consider the bead number @xmath48 , 18 , and 24 which respectively correspond to the chain lengths @xmath49 , 170 , and 230 nm , similar to the loop size considered in recent experiments @xcite . figure [ fig : n12_static1 ] shows the looping time @xmath50 as a function of tension @xmath2 for the chain of @xmath48 and @xmath51 . we apply the tension @xmath2 from -81 to 162 fn , comparable in magnitude to the typical entropic force on the double - stranded dna , @xmath52 @xmath53 80 fn . the @xmath2 is much smaller than the piconewton scale forces those are typically involved in active molecules @xcite e.g. , molecular motors in a cell . as @xmath2 changes over the scale as small as 100 fn , the looping time @xmath54 dramatically changes ; for example , @xmath54 increases about 5 times as the tension @xmath2 increases 120 fn . as a function of tension @xmath2 for the chain of @xmath48 . [ fig : n12_static1],width=321 ] to understand this behavior , we consider semiflexible polymer looping as a one - dimensional barrier crossing process @xcite , which is described by the langevin equation @xmath55 here , @xmath56 is the free energy of the chain given the chain end - to - end distance @xmath57 , where @xmath58 is the radial distribution function , and @xmath59 is a random force due to thermal fluctuation given by a gaussian and white noise that satisfies @xmath60 and @xmath61 . the shape of @xmath62 obtained by the @xmath3-beads simulation is shown in fig . [ fig : n12_mfwlc ] ( solid line with circles ) for the chain of @xmath48 . in this one - dimensional description , the looping time is the mean first - passage time ( mfpt ) for the variable @xmath57 to reach the cutoff distance @xmath47 starting from the initial chain end - to - end distance @xmath63 . it is given by @xcite @xmath64 where @xmath65 is the relative diffusion coefficient of two end beads , @xmath66 . then the looping time is @xmath67 , where @xmath68 represents the average over the initial equilibrium distribution . the mfpt in the presence of tension @xmath2 , also can be calculated by using the free energy , @xmath69 , where @xmath70 is the free energy without tension . for the @xmath58 , we use the mean - field wormlike chain model ( mf - wlc ) @xcite as an example . the radial distribution function @xmath58 of this model is @xmath71^{-\frac{9}{2 } } \exp[-\frac{3l}{4 l_{p } } \frac{1}{(1-(r / l)^2)}]$ ] . this formula yields a reasonable approximation to our simulation result for @xmath70 , except for @xmath72 , as shown in fig . [ fig : n12_mfwlc ] . the large deviation between two curves in the region near @xmath73 is because our model allows the chain extensibility while mf - wlc does not . however , according to the single molecule experiment using optical tweezers @xcite , the dna is extensible , manifesting overstretching transition subject to a strong tension , thus showing a significant deviation from the inextensible wlc . the optimal value @xmath11 to fit the force - extension curve is found to be @xmath74 2700 @xcite . however , the simulation using @xmath11 as large as @xmath75 demands computing time much longer than that using @xmath35 . unlike the free energy , the normalized looping time @xmath76 , the looping time in the presence of tension @xmath2 relative to the looping time in its absence , is expected to be quite insensitive to the value of @xmath11 @xcite . for this reason we adopt @xmath35 , which was also employed in a number of studies on the looping @xcite . of @xmath48 chain obtained from the simulation ( solid line with circles ) and from the mean - field wormlike chain model @xcite ( solid line).[fig : n12_mfwlc],width=321 ] to understand the effect of tension on the looping time , the mf - wlc provides a useful analytical model . figure [ fig : normalized_looping_static ] shows @xmath77 as a function of @xmath2 for the chain lengths @xmath3=12 , 18 , and 24 . here the symbols are simulation results and the dashed lines are from mfpt calculations ( eq . [ eq : mfpt ] ) . for the range of @xmath2 we study here , two results are in an excellent agreement . this figure also shows that the normalized looping time increases exponentially with @xmath2 . the reason is that , for the short chain lengths considered here , the free energy of the loop formation increases with @xmath78 , so @xmath54 increases approximately in exponential with @xmath2 . in contrast , the theory of ref . @xcite considered that , for low tension ( @xmath79 fn ) , @xmath54 increases exponentially with @xmath80 because of gaussian force - extension relation they used . as a function of tension @xmath2 for @xmath48 , @xmath81 and @xmath82 . the symbols are simulation data and the dashed lines are from mean first - passage time calculations using mean - field wormlike chain model @xcite.[fig : normalized_looping_static],width=359 ] the looping time in the presence of a tension @xmath2 can be written in dimensionless form @xmath83 where @xmath84 . @xmath85 is a function of a dimensionless scaling variable @xmath86 . it means for very large @xmath87 , a minute tension @xmath2 can dramatically change the looping time . indeed , fig . [ fig : normalized_looping_static ] shows that , for longer chains , the normalized looping time changes more sensitively with @xmath2 . if @xmath3 is 100 , corresponding to @xmath88 @xmath44 m , a change of the force as small as 4 fn can affect the looping time . this is a consequence of the cooperativity of the long polymer chains arising from the chain connectivity , which is previously addressed in a study of polymer translocation through membrane @xcite . the sensitivity of looping on tension is an emergent behavior that manifests beyond the complexity of real dna loop formation @xcite , e.g. , the details of chain conformations outside of the loop , orientation constraint , and associated proteins . finally , we compare our result with the recent experimental data @xcite for the dna loop size @xmath89 nm . as shown by the circles in fig . [ fig : cutoff_5 ] , the looping time rises steeply as a function of tension @xmath2 relative to the one with @xmath90 fn , which is the minimum tension used in the experiment . we also plot the corresponding relative mfpt by a solid curve . with the cutoff distance chosen to be the bead diameter , @xmath47=5 nm , our result is in good agreement with the experiment @xcite . because our model does not consider many details of dna loop formation , this agreement could be somehow fortuitous but encouraging . a theory including effectively the complexity of dna loop formation and further controlled experiments of various loop sizes are needed for quantitative comparison . ( @xmath90 fn ) as a function of tension @xmath2 with the cutoff distance @xmath91 nm for @xmath89 nm obtained by the mfpt calculation using mf - wlc model @xcite ( solid line ) . the circles are experimental data including error bar from ref . @xcite.[fig : cutoff_5],width=321 ] now suppose that tension temporally fluctuates due to the nonequilibrium noise inherent _ in vivo _ systems which can be generated by a variety of constituents of a cell , e.g. , protein like rna polymerase . also a dna fragment about to loop is influenced by the other part of the chain whose conformation is constantly fluctuating . as a simple example of nonequilibrium fluctuations , we consider a dichotomic noise , with which the tension @xmath92 flips between two level of the forces @xmath93 and @xmath94 with a mean flipping time @xmath95 . the @xmath92 is a noise with zero mean and its time correlation function is @xmath96 . we generate dichotomic noise @xmath92 using the algorithm described in @xcite . in the initial equilibration time , @xmath92 is given with either @xmath93 or @xmath94 with an equal probability 1/2 , and in a small time - step @xmath97 , @xmath92 can flip to the other value with the probability @xmath98 , which makes the mean flipping time be @xmath95 . in the presence of dichotomic tension as well as static tensions for the chain of @xmath48 . the dichotomic force amplitude is @xmath99 fn , and the static tensions amplitudes are @xmath100 ( circle ) and @xmath101 fn ( rectangular ) . the looping time shows a minimum at optimal flipping time @xmath102 and @xmath103 for @xmath100 and 161 fn , respectively . ( b ) the probability @xmath104 that the dichotomic force is negative ( i.e. , inward direction ) at the instant of looping as a function of @xmath95 . [ fig : ran12_static],width=321 ] figure [ fig : ran12_static ] @xmath105 shows the normalized looping times , @xmath106 , as a functions of @xmath95 for the chain length @xmath3=12 in the presence of fluctuating tension @xmath92 added to the static tension @xmath2 . we consider a value of dichotomic noise amplitude , @xmath99 fn and two values of static tensions amplitudes , @xmath100 ( circles ) and 161 fn ( squares ) , which are similar to those used in a recent experiment @xcite . for very short @xmath95 , the dichotomic forces are averaged out , so the looping times converge to the values without dichotomic force . they gradually decrease with @xmath95 until the minimum at @xmath107 and @xmath103 for @xmath100 and 161 fn , respectively . for very long @xmath95 , the dichotomic force rarely changes in a typical looping time , so the looping time goes to the average of the looping time with tension ( @xmath108 ) and the looping time with tension ( @xmath109 ) . the average is dominated by the looping time with ( @xmath108 ) , so that the looping time sharply rises with @xmath95 . related to this , we study the probability @xmath104 that the dichotomic tension @xmath92 is negative ( i.e. , is in inward direction ) at the instant of looping as a function of @xmath95 ( fig . [ fig : ran12_static ] @xmath110 ) . it has a maximum at @xmath111 and @xmath112 300 for @xmath100 and 161 fn , respectively , which means that , for @xmath95 near the @xmath113 , most of looping occurs when @xmath92 is in inward direction , and the looping time becomes the minimum . in the limits of very short or very long @xmath95 , @xmath92 is positive or negative with equal probabilities at the moment of looping . the maximum of @xmath104 is larger for larger @xmath2 . evidently the minimum looping time is closely associated with the maximum of the @xmath104 . the minimum of @xmath54 is found to occur when the flipping time @xmath95 is comparable to the diffusion time of a brownian _ particle _ in the free energy @xmath62 , @xmath114 , where @xmath115 is a numerical value of order unity that increases with the tension @xmath2 @xcite . this phenomenon is an extension of the resonant activation ( ra ) originally found in the single brownian particle crossing over a fluctuating barrier @xcite . indeed , we can regard the polymer looping in the presence of dichotomic tension as the process of a brownian _ particle _ crossing over a fluctuating _ free energy _ barrier . to study how the dichotomic force affects the ra phenomena , we consider different value of @xmath116 . figure [ fig : n12_t ] shows the normalized looping time with @xmath117 ( circles ) , 121 fn ( squares ) _ in the absence _ of static tension ( @xmath118 ) . the looping time has resonant minimum at the optimum flipping time @xmath119 which is smaller than the case with static tension @xmath2 . at the optimum flipping time , the looping time @xmath120 is smaller for larger @xmath116 , while , at very long @xmath95 , @xmath120 is larger for larger @xmath116 . therefore at certain @xmath95 in between , there is a crossing point . in the presence of dichotomic tension for the chain of @xmath48 . the dichotomic force amplitudes are @xmath99 ( circle ) and @xmath121 fn ( rectangular ) . the looping time shows a minimum at optimal flipping time @xmath119 . [ fig : n12_t],width=321 ] in contrast to the ra of single particle , the fluctuating barrier heights and thus the looping time depends sensitively on the chain length @xmath3 . we obtain the normalized looping time @xmath122 for different chain lengths @xmath87 in the presence of dichotomic force of amplitude @xmath117 fn and static tension @xmath100 fn ( fig . [ fig : compare ] ) . while the optimal flipping time @xmath123 increases with @xmath87 as implied by the relation @xmath124 , the minimum of @xmath125 decreases with @xmath87 . this is because , similar to the static tension case , the fluctuating barrier height will be a function of @xmath126 , so that the longer chain tends to be more susceptible to the tension @xmath116 , and have the lower relative looping time . in the presence of dichotomic tension as well as static tensions for the chains of @xmath48 ( circle ) , 18 ( rectangular ) , and 24 ( triangle ) . the dichotomic force amplitude is @xmath99 fn , and the static tension amplitude is @xmath100 fn . as @xmath3 increases , the optimal flipping times @xmath113 increase and the minimum values of normalized looping time @xmath127 decrease.[fig : compare],width=321 ] a recent experiment @xcite has shown that a fluctuating tension on dna greatly reduces the looping time for small @xmath95 , in consistency with our results . they attributed the phenomenon to an increase of the effective temperature . this may be reasonable for the small @xmath95 , where the dichotomic noise adds to the thermal noise , but could not lead to the nonmonotonic resonant behavior emerging over the entire range of @xmath95 . we have studied the effects of static or time - dependent tension on the semiflexible polymer looping using brownian dynamics simulation . for the case of static tension , we have found that a minute tension as small as 100 fn can dramatically change the looping time , especially for long chains . this sensitivity is a consequence of the cooperativity of the chain arising from chain connectivity . for the case of time - dependent tension , we considered dichotomically fluctuating tension , where the tension in average changes its sign in time @xmath95 . the looping time has a resonant minimum at an optimum flipping time @xmath123 , which is a nontrivial extension of the resonant activation ( ra ) of single brownian particle . the effect of time - dependent tension is also more significant for longer chains . our results are consistent with recent experiments for both static @xcite and a fluctuating tension @xcite cases . although we neglect the details of chain conformation outside of the loop and orientation constraints , etc . , our model could be a basic step to understand the loop formation process @xmath128 where biopolymers are constantly subject to forces . this study suggests a possibility that a biopolymer can self - organize optimally utilizing the ambient nonequilibrium fluctuations . further experiment using dichotomically fluctuating tension with various @xmath95 is called for to establish the ra phenomena in polymer loop formation . as the chain gets shorter , we observe small difference between the looping time with hydrodynamic interaction studied in @xcite and those without hydrodynamic interaction obtained by us using the path integral hyperdynamics method @xcite .

biopolymer looping is a dynamic process that occurs ubiquitously in cells for gene regulation , protein folding , etc . in cellular environments , biopolymers are often subject to tensions which are either static , or temporally fluctuating far away from equilibrium . we study the dynamics of semiflexible polymer looping in the presence of such tensions by using brownian dynamics simulation combined with an analytical theory . we show a minute tension dramatically changes the looping time , especially for long chains . considering a dichotomically flipping noise as a simple example of the nonequilibrium tension , we find the phenomenon of resonant activation , where the looping time can be the minimum at an optimal flipping time . we discuss our results in connection with recent experiments .

arxiv

tidal interactions play an important role in dynamical processes in the two - body problem in close binary systems : star star ( binary star ) or star planet . they can lead to such phenomena as synchronization and orbital circularization ( hut 1981 ; zahn 1977 ) as well as to the tidal capture ( press and teukolsky 1977 ) or disruption of an object ( star ) ( ivanov and novikov 2001 ) and the fall of the object onto the star ( rasio et al . 1996 ; penev et al . 2012 ; bolmont and mathis 2016 ) . in this paper we consider the tidal interaction of two bodies : a star and a point source . the point source can be both a star ( a neutron star , a white dwarf , etc . ) and a planet . below we will call the point source a planet by implying that this can also be a star . since the evolution time scales of the eccentricity or semimajor axis strongly depend on the orbital period of the binary system and for some stellar models can take values up to 108 yr or more for periods of about 5 days ( ivanov et al . 2013 ; chernov et al . 2013 ) , the evolution of the star itself should be taken into account on such time scales . as the star evolves , the orbital parameters change due to tidal interactions . in this paper we investigate the dynamical tides by taking into account the stellar evolution . for our study we chose three types of stars with masses of one , one and a half , and two solar masses . we consider all stars without allowance for their rotation and magnetic field and touch on the stellar physics itself superficially , as far as this problem requires . the star of one solar mass at lifetimes @xmath3 yr closely corresponds to our sun and has a radiative core and a convective envelope on the main sequence . the other two stars of one and a half and two solar masses are more massive and have a more complex structure . these stars have a convective core and a radiative envelope on the main sequence ( a more precise structure is presented below ) . the problem of determining the tidal evolution is reduced to the problem of determining the normal modes of stellar perturbations and to calculating the energy and angular momentum exchange in the star.planet system ( ivanov and papaloizou 2004 , 2010 ; papaloizou and ivanov 2010 ; lanza and mathis 2016 ) . the low - frequency g - modes of the stellar oscillations play an important role in the theory of dynamical tides . the tidal interactions are fairly intense at small periastron distances of the planet . for a periastron distance @xmath4 au , the dimensionless excitation frequency is @xmath5 , which corresponds to g - modes . a large number of exoplanets in stellar systems have been discovered in the last few years owing to the kepler , superwasp , and other observational programs . in particular , short - period massive planets with an orbital period of a few days , the so - called hot jupiters , have been detected ( winn 2015 ) . as a rule , the hot jupiters have low eccentricities , which points to the importance of tidal interactions ( ogilvie 2014 ) . the results of this paper can be directly applied to some of such systems . for example , the system ybp1194 is a solar twin ( brucalassi et al . a planet with a mass of 0.34@xmath6 and a period of only 6.9 days revolves around this star . for such a short - period planet the dynamical tides must be fairly intense and must affect the orbital evolution . predictions about the subsequent evolution of this planet can be made by analyzing this system . one of the results of this evolution is the fall of the planet onto the star . the possibility of such a fall has been considered in many papers ( see , e.g. , rasio et al . 1996 ; penev et al . 2012 ; weinberg et al . 2012 ; essick and weinberg 2016 ) . rasio et al . ( 1996 ) considered the possibility of the fall of the planet onto the star due to quasi - static tides and provided a plot for solar - like stars that shows the threshold , as a function of planetary mass and orbital period , below which the planet falls onto the star . penev et al . ( 2012 ) considered tides with a constant tidal quality factor @xmath7 specified phenomenologically . in reality , this factor will depend on the planet s orbital period ( ivanov et al . 2013 ) and stellar age . essick and weinberg ( 2016 ) took into account the energy dissipation due to nonlinear interaction of modes with one another . in contrast to our approach ( ivanov et al . 2013 ; chernov et al . ( 2013 ) , the simultaneous solution of a large number of ordinary differential equations for each stellar model is suggested , with only solar - type stars having been considered . in this paper we consider the evolution of stars with masses of one , one and a half , and two solar masses . data on the stars are presented in tables 1 3 . a novelty of this study is a consistent allowance for the stellar evolution . for each moment of the star s lifetime we calculated the spectra of normal modes , the overlap integrals ( press and teukolsky 1977 ) , which are a measure of the intensity of tidal interactions ( for a generalization to the case of a rotating star , see papaloizou and ivanov 2005 ; ivanov and papaloizou 2007 ) , and the time scales of the orbital parameters ( the tidal quality factor @xmath7 ) . the overlap integrals are directly related to the tidal resonance coefficients that were introduced by cowling ( 1941 ) ( see also rocca 1987 ; zahn 1970 ) . all of the quantities marked by a tilde are dimensionless ; the normalization is presented in appendix a. such a quantity as the overlap integral q is of great importance in the theory of dynamical tides . it is specified by the expression ( press and teukolsky 1977 ; zahn 1970 ) @xmath8 where m is the stellar mass , r is the stellar radius , and @xmath9 is the dimensionless overlap integral ; the remaining quantities and their dimensionless forms are defined in appendix a. the overlap integral serves as a measure of the intensity of the tidal forces and plays a crucial role in the physics of dynamical tides ( press and teukolsky 1977 ; zahn 1970 , 1975 , 1977 ) . here , we will consider only the quadrupole part of the tidal forces @xmath10 , because it is this part that makes the greatest contribution to the overlap integral ( press and teukolsky 1977 ) . the change in orbital semimajor axis and eccentricity with time is determined from the energy and angularmomentum conservation laws and is specified by the following formulas ( ivanov et al . 2013 ) : @xmath11 here , @xmath12 is the orbital eccentricity , @xmath13 is the semimajor axis , and the time scales of the change in semimajor axis @xmath14 and eccentricity @xmath15 in the case of low eccentricities ( @xmath16 ) are specified by the relations ( ivanov et al . 2013 ) @xmath17 where @xmath18 is the planetary mass . the rate of change of energy @xmath19 , @xmath20 in the case of a dense spectrum and moderately large viscosities is specified by eqs . ( 51 ) from ivanov et al . the spectrum is deemed to be dense enough if @xmath21 . for simplicity , we will consider here only the time scales of the changes in semimajor axis . using formulas from subsection 7.1 of the paper by ivanov et al . ( 2013 ) , we obtain @xmath22 where @xmath23 is the ratio of the planetary mass to the stellar mass , @xmath24 is the difference between adjacent frequencies , @xmath25 is the frequency number , and @xmath26 ( ivanov et al . . the orbital period of the planet around the star is specified by the relation @xmath27 below , using eqs . ( [ ta ] ) and ( [ porb ] ) and solving eq . ( [ tate ] ) , we will present the change in the planet s orbital period as a function of time by taking into account the stellar evolution . as is clear from eq . ( [ overlapsintegral ] ) , the eigenfunctions and , accordingly , eigenfrequencies of the stellar oscillations should be known to calculate the overlap integral . there exist three types of eigenfrequencies for a nonrotating star : the so - called p- , f- , and g - modes . we will be concerned only with the low - frequency g- modes . the properties of the g - modes are determined by the brunt - v@xmath28is@xmath29l@xmath29 frequency . it is specified as follows ( christensen - dalsgaard 1998 ) : @xmath30 where @xmath31 is the pressure , @xmath32 is the density , @xmath33 is the gravitational acceleration , @xmath34 is the adiabatic index . however , this definition of the brunt - v@xmath28is@xmath29l@xmath29 frequency is difficult to apply to many stars . this is related to the numerical errors , to the calculation of the derivative of the density , and to the fact that there exist regions in stars where both terms in parentheses can be of the same order of magnitude . therefore , a different definition is used ( brassard et al . 1991 ) : @xmath35,\end{aligned}\ ] ] where @xmath36 the temperature gradient @xmath37 @xmath38 is the mass fraction of atoms of type i , and @xmath39 it is convenient to divide the brunt - v@xmath28is@xmath29l@xmath29 frequency into two parts : the structural part ( structure terms ) brunt - v@xmath28is@xmath29l@xmath29 @xmath40\end{aligned}\ ] ] and the compositional part @xmath41 the compositional part is related to the change in stellar chemical composition due to nuclear reactions , and it is of great importance at the boundary of the convective and radiative regions . another important stellar characteristic is the acoustic frequency defined by the formula @xmath42 the low - frequency g - modes are excited at @xmath43 . the high - frequency p - modes are excited at @xmath44 . we will not consider the p - modes , because they play a secondary role in the theory of dynamical tides . thus , the problem is reduced to finding the eigenfunctions and eigenfrequencies of the stellar oscillations , calculating the overlap integrals , and calculating the characteristic orbital parameters as a function of time . the equations that describe the eigenfrequencies and eigenfunctions of a star in the adiabatic and cowling approximations are presented in appendix a. the stars were modeled with the mesa software package ( paxton et al . 2011 , 2013 , 2015 ) ; the data obtained were then interpolated by the steffen ( 1990 ) method to two million points . using these data , we solved the eigenfrequency and eigenfunction problem and calculated the overlap integrals . the fourth - order runge kutta method was used to solve the differential equations . the data for the stars and the overlap integrals were compared with those from ivanov et al . ( 2013 ) and chernov et al . in this section we present the results of our calculations of the change in the planet s orbital period for three types of stars with masses of one , one and a half , and two solar masses . the course of stellar evolution for these models is shown on the hertzsprung russell diagram depicted in fig . data for each star are presented in the tables in appendix b. the main phase of stellar evolution is the main - sequence phase . it is at this phase that any star spends most of its lifetime . this phase is indicated in fig . 1 by the nearly horizontal straight line . in this section we present the results of our calculations of the overlap integrals for various lifetimes of the star of one solar mass and the change in the planet s orbital period with time . the star was modeled from @xmath45 to @xmath46 yr . at times @xmath47 yr this star closely corresponds to our sun , and below we will call this star the sun . at the initial time the sun was completely convective , hydrogen burning has not yet begun , and its metallicity is z = 0.02 . figure 2 shows a dimensionless dependence of the solar density on radius at the initial and subsequent times . is@xmath29l@xmath29 frequency ( b ) for the sun versus radius for various times : the dashed , dotted , solid , and dash dotted lines are for the ages @xmath45 , @xmath48 , @xmath49 , and @xmath46 yr , respectively.,title="fig:",scaledwidth=45.0% ] is@xmath29l@xmath29 frequency ( b ) for the sun versus radius for various times : the dashed , dotted , solid , and dash dotted lines are for the ages @xmath45 , @xmath48 , @xmath49 , and @xmath46 yr , respectively.,title="fig:",scaledwidth=45.0% ] is@xmath29l@xmath29 frequency versus radius for the solar age @xmath49 : the solid , dashed , and dash - dotted lines indicate the total brunt - v@xmath28is@xmath29l@xmath29 frequency @xmath50 , the structural part @xmath51 , and the compositional part @xmath52 , respectively.,scaledwidth=45.0% ] the course of stellar evolution and the excitation of eigenmodes can be described as follows . at times @xmath53 yr the sun was completely convective ( @xmath54 ) and , therefore , no low - frequency g - modes were excited , but only the high - frequency f- and p - modes were excited . subsequently , as the sun evolves ( at times @xmath55 yr ) a radiative region ( radiative core ) where the brunt - v@xmath28is@xmath29l@xmath29 frequency is greater than zero ( @xmath56 ) appears and , consequently , the excitation of low - frequency g - modes begins . only the structural part makes a major contribution to the brunt - v@xmath28is@xmath29l@xmath29 frequency ; the compositional part is zero . this is because at such lifetimes of the sun hydrogen burning makes a minor contribution to the energetics and evolution dynamics of the sun . in the course of subsequent evolution the radiative region expands , the number of lowfrequency g - modes increases , and their spectrum becomes dense . the low - frequency g - modes begin to play an increasingly important role . at times @xmath57 yr hydrogen burning begins to make an increasingly large contribution to the total energy of the star and , consequently , the compositional part of the brunt - v@xmath28is@xmath29l@xmath29 frequency also begins to contribute to the total frequency . in the course of subsequent evolution , as hydrogen burns , this contribution becomes progressively larger . one of the problems being investigated here is to take into account the evolution of the brunt - v@xmath28is@xmath29l@xmath29 frequency and the influence of this evolution on the tidal interactions . the radiative region expands approximately to a radius @xmath58 . for the present sun the boundary between the radiative and convective regions is @xmath59 . in fig . 2 the brunt - v@xmath28is@xmath29l@xmath29 frequency is plotted against radius for various stellar evolution times . it can be seen from the plot that the brunt - v@xmath28is@xmath29l@xmath29 frequency is subject to significant changes as the solar age increases ; both the frequency spectrum and its density change accordingly , which can directly affect the tidal forces . in fig . 3 the brunt - v@xmath28is@xmath29l@xmath29 frequency is plotted against radius for the present sun . it follows from this plot that the brunt - v@xmath28is@xmath29l@xmath29 frequency exhibits a double - humped structure . the first and second humps are associated with the compositional and structural parts of the brunt - v@xmath28is@xmath29l@xmath29 frequency , respectively . at a solar age @xmath60 yr the two contributions become comparable . the overlap integrals ( 1 ) are plotted against frequency in the cowling approximation in fig . 4 . for a solar age @xmath53 yr , as has been said above , only the f- and p - modes exist ; therefore , there is no low - frequency part of the spectrum . it can be seen from the plot that the absolute value of the overlap integrals decreases in the low - frequency part of the spectrum as the sun evolves . this decrease is gradual , because the stellar structure does not change in the evolution time of the sun . on the main sequence the sun always has a radiative core and a convective envelope . in the high - frequency part of the spectrum , as the stellar age increases , the absolute value of the overlap integrals increases and a characteristic structure appears in the spectrumin the form of kinks . , @xmath48 , @xmath49 , and @xmath46 yr , respectively.,scaledwidth=45.0% ] figure 5 plots the change in the orbital period ( in days ) of a planet with a mass of one jupiter mass around the sun with time ( in years ) . the solar evolution models for which the overlap integrals were calculated are presented in table 1 . at lifetimes of the sun @xmath61 yr the spectrum at frequencies @xmath62 is insufficiently dense . at other lifetimes of the sun the spectrum may be deemed sufficiently dense with a good accuracy for all frequencies . therefore , eq . ( 4 ) may be used . equation ( 2 ) was integrated over time from @xmath63 to @xmath64 yr . the initial orbital periods of the planet around the sun were taken to be @xmath65 days ( solid curves ) . it follows from fig . 5 that in the main - sequence lifetime of the sun a planet with a mass of one jupiter mass with the initial orbital period @xmath66 days will reduce its orbital period to @xmath67 days due to dynamical tides . the time scale of the change in semimajor axis for this orbital period @xmath67 days and this solar age @xmath64 yr is @xmath68 yr . consequently , it can be said with confidence that this planet will fall onto the sun within several tens of millions of years . the same analysis showed that a planet with the initial orbital period @xmath69 days would reduce its orbital period to @xmath70 days and would fall onto the sun only within approximately half a billion years . for the initial orbital period @xmath69 days fig . 5 compares the changes in orbital period with allowance for the solar evolution ( solid curve ) , without allowance for the solar evolution ( dash dotted curve ) , and the one calculated using eq . ( 25 ) from essick and weinberg ( 2016 ) ( dotted curve ) . to calculate the change in orbital period without allowance for the solar evolution , we took the time scale at the solar age @xmath0 yr . all curves qualitatively coincide , and the influence of solar evolution may be disregarded in the estimation . this is because hydrogen burning in the sun is gradual , and no change in structure occurs . quantitatively , the dotted curve from essick and weinberg ( 2016 ) gives smaller changes in the planet s orbital period . this may be related to both the neglect of the change in the time scales of the orbital parameters as the sun evolves and to a different approach in this paper to the problem of the energy dissipation of tidally excited modes associated with their nonlinear interactions . in this subsection we present the results of our calculations of the overlap integrals and the change in orbital period for the star with a mass of one and a half solar masses . the star is considered from the time the protostar begins to collapse . this time is quite arbitrary and is @xmath72 yr ; the initial radius is @xmath73 . the initial metallicity of the star is @xmath74 . at lifetimes of the star up to @xmath75 yr it is completely convective ( @xmath54 ) . in such a star no lowfrequency g - modes are excited , and only the highfrequency f- and p - modes are present ; therefore , the effective energy and angular momentum transfer from the orbit to the star will be suppressed . as the stellar age increases further , hydrogen burning begins at the stellar center and a radiative core appears ( @xmath56 ) , which begins to expand , encompassing progressively newer layers . the star has a well - defined convective envelope and a radiative core . the radiative region expands up to @xmath76 for the age @xmath77 yr . a fairly dense spectrum of low - frequency g - modes appears very rapidly in such a star . at times @xmath78 yr the stellar structure changes completely . a convective core appears in the star , while its envelope becomes completely radiative . the star abruptly contracts from a radius @xmath79 to @xmath80 . the evolution time scales of the orbital parameters ( 2 ) increase sharply . the star passes from a solar - type star that has a radiative core and a convective envelope to a star with a convective core and a radiative envelope . is@xmath29l@xmath29 frequency ( b ) for the star of one and a half solar masses versus radius for various times : the dashed , dotted , solid , and dash dotted lines are for the ages @xmath81 , @xmath82 , @xmath83 , and @xmath84 yr , respectively.,title="fig:",scaledwidth=45.0% ] is@xmath29l@xmath29 frequency ( b ) for the star of one and a half solar masses versus radius for various times : the dashed , dotted , solid , and dash dotted lines are for the ages @xmath81 , @xmath82 , @xmath83 , and @xmath84 yr , respectively.,title="fig:",scaledwidth=45.0% ] in fig . 6 the stellar density and brunt - v@xmath28is@xmath29l@xmath29 frequency are plotted against radius for various times . from the figure we see how the central density increased as the star contracted . it can be seen from the figure for the brunt - v@xmath28is@xmath29l@xmath29 frequency that the star was initially completely convective ( dashed curve ) and subsequently a solar - type one ( the envelope was convective , and the core was radiative ; the dotted curve ) . at the subsequent times the star has a more complex structure , the core becomes convective , and the envelope is completely radiative ( solid curve ) . a thin convective envelope then again appears in its envelope ( dash dotted curve ) . is@xmath29l@xmath29 frequency versus radius for the star of one and a half solar masses for the age @xmath84 yr : the solid , dashed , and dash - dotted lines indicate the total brunt - v@xmath28is@xmath29l@xmath29 frequency @xmath50 , the structural part @xmath51 , and the compositional part @xmath52 , respectively.,scaledwidth=45.0% ] figure 7 shows the brunt - v@xmath28is@xmath29l@xmath29 frequency as well as its structural and compositional parts . it follows from the figure that the compositional part makes the greatest contribution at the boundary between the convective and radiative regions . , @xmath82 , @xmath83 , and @xmath84 yr , respectively.,scaledwidth=48.0% ] the overlap integrals are plotted against frequency in fig . . the dashed line does not extend to the lowfrequency part of the spectrum , because the star at this time is completely convective ( @xmath54 ) . it can also be seen from the figure that the absolute value of the overlap integrals rapidly drops by more than two orders of magnitude as the star s lifetime increases . such catastrophic changes are due to the change in stellar structure . in the high - frequency part the absolute value of the overlap integrals increases by an order of magnitude , and characteristic kinks appear in the spectrum . figure 9 plots the change in the orbital period ( in days ) of a planet with a mass of one jupiter mass around the star of one and a half solar masses with time ( in years ) . the stellar evolutionmodels for which the overlap integrals were calculated are presented in table 2 . equation ( 2 ) was integrated over time from @xmath85 to @xmath86 yr . for such a stellar lifetime interval the g - mode spectrum is sufficiently dense . the initial orbital periods of the planet around the sun are @xmath87 days ( solid curves ) . it follows from the figure that all planets with a mass equal to one jupiter mass and with an initial orbital period of 2 days or shorter will fall onto the star in the main - sequence lifetime of the star . a planet with the initial orbital period @xmath88 days will fall onto the star , while a planet with the initial orbital period @xmath89 days will reduce its orbital period to @xmath90 days in the main - sequence lifetime of the star . the dashed curve indicates the change in orbital period without allowance for the stellar evolution . the time scale of the orbital change was taken at the star s lifetime @xmath91 yr . it can be seen from fig . 9 that the planet does not fall onto the star in the mainsequence lifetime of the star without allowance for its evolution . the last star that we will consider is the star with a mass of two solar masses . the time the protostar begins to collapse is @xmath92 yr , the initial stellar radius is @xmath93 , and the stellar metallicity is @xmath74 . at initial times up to @xmath94 yr the star is completely convective ( @xmath54 ) . no low - frequency g - modes are excited ; only the high - frequency f- and p - modes are present . the dynamical tides play no significant role in such a star . however , as the star evolves further , at times @xmath94 yr a radiative region ( radiative core ) appears in the star , which begins to expand . the low - frequency g - modes are well excited in this region ( @xmath56 ) , and the dynamical tides begin to play an increasingly significant role as the star evolves further . such a star resembles solartype stars that have a radiative core and a convective envelope . the radiative region expands quite rapidly , and this region almost reaches the stellar surface @xmath95 already at times @xmath96 yr . the star becomes almost completely radiative ; the brunt - v@xmath28is@xmath29l@xmath29 frequency in the entire region becomes positive ( @xmath56 ) in the entire region , except for the stellar surface . a sufficiently dense g - mode spectrum is generated in such a star . in fig . 10 the stellar density and brunt - v@xmath28is@xmath29l@xmath29 frequency are plotted against radius for various times . is@xmath29l@xmath29 frequency ( b ) for the star of two solar masses versus radius for various times : the dashed , dotted , solid , and dash dotted lines are for the ages @xmath97 , @xmath98 , @xmath99 , and @xmath100 , respectively.,title="fig:",scaledwidth=45.0% ] is@xmath29l@xmath29 frequency ( b ) for the star of two solar masses versus radius for various times : the dashed , dotted , solid , and dash dotted lines are for the ages @xmath97 , @xmath98 , @xmath99 , and @xmath100 , respectively.,title="fig:",scaledwidth=45.0% ] as the lifetime increases further ( @xmath101 yr ) , a convective core ( @xmath54 ) appears in the star , which begins to expand as the star evolves . the compositional part of the brunt - v@xmath28is@xmath29l@xmath29 frequency @xmath52 begins to play an increasingly large role at the boundary between the convective and radiative regions ( fig . . the stellar structure does not undergo significant changes as the lifetime increases further . only the convective core evolves . initially , the convective region expands , and then its expansion is replaced by contraction ( fig . is@xmath29l@xmath29 frequency versus radius for the star of two solar masses for the age @xmath100 yr : the solid , dashed , and dash - dotted lines indicate the total brunt - v@xmath28is@xmath29l@xmath29 frequency @xmath50 , the structural part @xmath51 , and the compositional part @xmath52 , respectively.,scaledwidth=45.0% ] , @xmath98 , @xmath99 , and @xmath100 yr , respectively.,scaledwidth=45.0% ] the overlap integrals are plotted against frequency in fig . . the dashed line does not extend to the low - frequency part of the spectrum , because the star at this time is completely convective ( @xmath54 ) . as the star s lifetime increases , the absolute value of the overlap integral rapidly drops by more than an order of magnitude . such catastrophic changes are due to the change in stellar structure . when the stellar structure undergoes no significant changes , the overlap integrals change gradually and insignificantly ( see fig . 12 , the solid and dash - dotted lines ) . figure 13 plots the change in the orbital period ( in days ) of a planet with a mass of one jupiter mass around the star of two solar masses with time ( in years ) . the evolution models of the star of two solar masses for which the overlap integrals were calculated are presented in table 3 . equation ( 2 ) was integrated over time from @xmath102 to @xmath103 yr . for such a stellar lifetime interval the g - mode spectrum is sufficiently dense . the initial orbital periods of the planet around the sun were taken to be @xmath104 days ( solid curves ) . it follows from the figure that all planets with a mass equal to one jupiter mass and with an initial orbital period of 2 days or shorter will fall onto the star in the main - sequence lifetime of the star ( fig . a planet with the initial orbital period @xmath89 days will reduce its orbital period only to @xmath105 days in the main - sequence lifetime of the star . the dashed curve indicates the change in orbital period without allowance for the stellar evolution . the time scale of the orbital change was taken at the star s lifetime @xmath106 yr . as can be seen from fig . 13 , the planet does not fall onto the star in the main - sequence lifetime of the star without allowance for its evolution . q just as for the star of one and a half solar masses , the evolution time scales ( 3 ) of the planet decrease as the star begins to move off the main sequence . this is because the hydrogen reserves at the stellar center begin to be depleted , and the convective core becomes a helium one . this leads to an expansion of the star and , consequently , to an increase in the number of g - modes and the density of the spectrum , which leads to an increase in the pumping of orbital energy into the energy of stellar oscillations and to the fall of short - period planets onto the star . we considered the change in the orbital period of a planet around a star with a mass of one , one and a half , and two solar masses . the behavior of the overlap integrals for these models as a function of the main - sequence lifetime of the star was analyzed . we showed that for the same star with a different structure during its evolution the overlap integrals could differ by two or more orders of magnitude . we showed that all planets with a mass of one jupiter mass that revolve around the sun with an orbital period @xmath1 days or shorter should fall onto the stellar surface in the main - sequence lifetime of the star . all planets with a mass of one jupiter mass that revolve around the stars with a mass of one and a half and two solar masses with an orbital period @xmath2 or shorter will fall onto the stellar surface in the main - sequence lifetime of the star . such a fall may not occur without allowance for the evolution of the star itself . these results qualitatively agree with the previous results from bolmont and mathis ( 2016 ) and penev ( 2012 ) . calculations for the stellar models considered in this paper should be made for a more accurate quantitative estimate . i am grateful to p.b . ivanov who read the paper for a number of valuable remarks and to j. papaloizou for a fruitful discussion . this work was financially supported by the russian foundation for basic research ( project nos . 140200831 , 150208476 , 160201043 ) , grant no . nsh-6595.2016.2 from the president of the russian federation for state support of leading scientific schools , and program 7 of the presidium of the russian academy of sciences . the system of equations describing the adiabatic perturbations in stars is written in dimensionless form ( christensen - dalsgaard 1998 ) : @xmath107 where the following dimensionless variables are introduced : @xmath108 is the radius , @xmath109 is the radial perturbation , @xmath110 is the horizontal perturbation , @xmath111 and @xmath112 are the perturbations of the gravitational potential and its derivative , respectively . the subscript @xmath113 refers to an unperturbed quantity , and the prime @xmath114 refers to a perturbed quantity . a tilde over a quantity means that this quantity is dimensionless . the normalization is done as follows : @xmath115 where @xmath116 is the sound speed squared , @xmath31 is the pressure , @xmath32 is the density , and @xmath50 is the brunt - v@xmath28is@xmath29l@xmath29 frequency squared . four boundary conditions should be added to the four first - order differential equations . two boundary conditions are specified at the stellar center , and the other two are specified on the stellar surface ( christensen - dalsgaard 1998 ) . the boundary conditions at the stellar center @xmath117 are @xmath118 the boundary conditions on the stellar surface @xmath119 are @xmath120 there exists the cowling ( 1941 ) approximation , in which the perturbed gravitational potential is neglected , to investigate the low - frequency g - modes . this approximation holds good and simplifies considerably the system of equations for the adiabatic perturbations ( 14 ) . in the cowling approximation the system of equations ( 14 ) will be rewritten in a simpler form : @xmath121 in the cowling approximation the system of equations ( 18 ) ceases to depend explicitly on the stellar density . thus , in the low - frequency limit the radial and horizontal displacements do not depend on the stellar density . in contrast , the overlap integrals ( 1 ) directly depend on the density . in the cowling approximation not four but two boundary conditions should be added to the system of equations ( 18 ) . one boundary condition is specified at the stellar center , and the other one is specified on the stellar surface . the boundary condition at the stellar center @xmath117 is @xmath122 the boundary condition on the stellar surface @xmath123 is @xmath124 to determine the eigenfrequency spectrum of the system of equations ( 14 ) or ( 18 ) , this system should be redefined . this requires that the determinant composed of the eigenfunctions of the system of equations ( 14 ) or ( 18 ) be equal to zero . this will hold only for some frequencies ( eigenfrequencies ) of the system of equations ( 14 ) or ( 18 ) , which determines the perturbation spectrum . the tables present the models for which the overlap integrals ( 1 ) and the evolution time scales of the orbital parameters ( 2 ) were calculated . the notation is as follows : @xmath125 is the time in years , @xmath126 is the radius measured in solar radii , @xmath127 is the luminosity measured in solar luminosities , @xmath128 is effective surface temperature measured in kelvins , h1 is the hydrogen mass fraction at the stellar center , and @xmath129 is the mean stellar density measured in @xmath130 .

we investigate the change in the orbital period of a binary system due to dynamical tides by taking into account the evolution of a main - sequence star . three stars with masses of one , one and a half , and two solar masses are considered . a star of one solar mass at lifetimes @xmath0 yr closely corresponds to our sun . we show that a planet of one jupiter mass revolving around a star of one solar mass will fall onto the star in the main - sequence lifetime of the star due to dynamical tides if the initial orbital period of the planet is less than @xmath1 days . planets of one jupiter mass with an orbital period@xmath2 days or shorter will fall onto a star of one and a half and two solar masses in the mainsequence lifetime of the star .

arxiv

pushing the resolution limits of light microscopy , and understanding optical phenomena on scales below the diffraction limit , has been the driving force of what is known today as nano - optics @xcite . to overcome this limit , most of the early work was focused on near - field optical microscopy and related techniques @xcite . however , in recent years , new concepts in fluorescence microscopy have pushed the resolution of far - field imaging down to the nanometer range @xcite . most of these methods @xcite rely on the accurate localization of individual fluorescent markers , that are isolated from one another on the basis of one or more distinguishing optical characteristics , or by selective or random activation of a bright and a dark state @xcite . determining the location of an isolated fluorescent marker is only limited by photon noise , and not by the diffraction barrier . a key issue affecting these subwavelength imaging methods is the optical transparency of the media surrounding the light emitters . taking advantage of the transparency of cells , fluorescence microscopy uniquely provides noninvasive imaging of the interior of cells and allows the detection of specific cellular constituents through fluorescence tagging . however , certain biological tissues or soft - matter systems ( such as foams or colloidal suspensions ) look turbid due to intense scattering of photons traveling through them @xcite . the image formed at a given point in the observation plane consists in a superposition of multiple fields , each arising from a different scattering sequence in the medium . this gives rise to a chaotic intensity distribution with numerous bright and dark spots known as a speckle pattern , producing a blurred image carrying no apparent information about the source position @xcite . techniques to measure the distance between individual nano - objects without actually imaging their position exist @xcite , fluorescence resonance energy transfer ( fret ) being the most widespread example @xcite . it relies on the near - field energy transfer between two fluorophores ( donor and acceptor ) emitting at different wavelengths . the fret signal ( _ e.g. _ the ratio between the intensities emitted by the donor and the acceptor at different wavelengths ) depends on the donor - acceptor distance in the range @xmath0 nm . as such , it is not very sensitive to scattering problems . however , determining distances between two emitters in the range of 10 to 500 nm in a scattering medium still remains a challenging problem , not accessible either by fluorescence microscopy or fret techniques . our main goal here is to introduce a new approach to obtain information about the relative distance between two identical incoherent point sources in a disordered environment , based on the analysis of the fluctuations of the emitted light . this is an issue of much interest , for example , in the study of conformational changes in biomolecules in living tissues . sensing the distance between two incoherent sources in a complex medium could also provide an alternative to green s function retrieval techniques based on the correlations of the isotropic ambient noise measured at two receivers @xcite . in this paper , we propose a method to capture the interaction between two identical sources in a scattering environment , based only on the measurement of intensity fluctuations . the principle of the method is schematically illustrated in fig . 1 , and is based on the analysis of the intensity - intensity correlation function and the intensity fluctuations in the speckle pattern formed by two identical and mutually incoherent point sources . this approach permits , in principle , to monitor the relative distance between the sources in the range 10 - 500 nm , with a precision that is not limited by diffraction , but by the microstructure of the scattering medium . in application to green s function retrieval in complex media , the approach replaces the two - point field - field correlation of the background noise by a measurement at a single point of the intensity noise due to the two fluctuating sources . this might simplify the technique , in particular at visible or near - ir frequencies where time - domain field - field correlations are not easy to measure . the result in this paper also illustrate the fact that multiple scattering , that had long been considered as an unavoidable nuisance , can actually enhance the performance of sensing , imaging and communication techniques @xcite , as already demonstrated in the context of spatio - temporal focusing by time reversal @xcite , wavefront shaping of multiply scattered waves @xcite , or improvement of information capacity of telecommunication channels @xcite . we consider two point sources of light ( electric dipoles ) located at @xmath1 and @xmath2 in a disordered medium . the sources are characterized by their electric dipole moments @xmath3 and @xmath4 , that are fluctuating quantities of the form @xmath5 \exp(-i\omega t ) { { \bf u}}_k$ ] with @xmath6 a slowly varying random phase , @xmath7 a complex amplitude and @xmath8 a unit vector defining the orientation of the dipole moment . this corresponds to a classical model for a quasi - monochromatic temporally incoherent source , such as a fluorescent source emitting at frequency @xmath9 . we assume that the two sources are uncorrelated ( or mutually incoherent ) , so that @xmath10\exp[-i\phi_2(t)]}=0 $ ] , where the bar denotes averaging over the fluctuations of the sources . using the ( dyadic ) green function @xmath11 of the disordered medium , the electric fields at any point @xmath12 can be written : @xmath13 the intensity associated to this field is a time fluctuating and spatially varying quantity that forms a time - dependent speckle pattern . let us first consider the total power @xmath14 emitted by the two sources . it reads @xmath15 where @xmath16 is a sphere with radius @xmath17 that encloses the disordered medium and @xmath18 is the speed of light in vacuum . for a _ non absorbing _ medium , the following relation can be derived from the vector form of green s second identity @xcite ( the frequency dependence in the green function is dropped for simplicity ) : @xmath19 { { \bf p}}_2^ * \ . \label{eq : green_identity}\end{aligned}\ ] ] from eqs . ( [ eq : field_general]-[eq : green_identity ] ) , we obtain @xmath20 where the notation @xmath21 { { \bf u}}_{j^\prime}$ ] has been introduced for the sake of simplicity . we first assume that a temporal averaging over the fluctuations of the sources can be performed , in one configuration of the disordered medium ( frozen disorder ) . the fluctuation time scale of the emitted power can be associated to the coherence time , as usually defined for partially coherent sources @xcite . for fluorescent sources , this time is on the order of the lifetime @xmath22 of the excited state . for emission in the visible range , expected orders of magnitude are @xmath23 ns for dye molecules or quantum dots , and @xmath24s for rare - earth ions . the averaged power @xmath25 is simply the sum of the averaged power emitted by each source independently , since the terms with @xmath26 in eq . ( [ eq : p2 ] ) vanish upon time averaging . it reads as = ( |p_1|^2 _ 11 + |p_2|^2 _ 22 ) [ eq : pmoy ] where @xmath27 \mathrm{im } g_{jj}$ ] is the electric part of the local density of states ( ldos ) at point @xmath28 @xcite . however , a cross - term survives in the fluctuations of the total emitted power . indeed , calculating the variance of @xmath14 from eq . ( [ eq : p2 ] ) , one obtains @xcite @xmath29 \ . \label{eq : varp_time}\ ] ] the imaginary part of the two - point green function @xmath30 in eq . is known to enter the expression of field - field spatial correlations in random fields , such as blackbody radiation or volume speckle patterns @xcite , the description of time - reversed fields @xcite , and is at the core of green s function retrieval techniques based on ambient noise correlations @xcite . it is proportional to the cross density of states ( cdos ) , that was introduced in a different context for the description of spatial coherence in complex systems @xcite . physically , the cdos counts the number of photonic eigenmodes connecting two points ( in our case the source points ) at a given frequency @xcite . more precisely , the cdos connecting @xmath31 to @xmath28 is given by @xmath32 \mathrm{im } g_{jk}$ ] @xcite . using the cdos , and assuming that the two sources have the same amplitude ( @xmath33 ) , the variance of the total emitted power can be rewritten as @xmath34 this equation is the first result in this paper . it provides a direct relationship between the temporal fluctuations of the total power emitted by two incoherent sources and the cdos connecting the source points in an arbitrary environment . this suggests that a retrieval of the amplitude of the cdos ( or equivalently of the imaginary of the green function at two different points ) is possible in a structured medium from a measurement of temporal fluctuations of the emitted power emerging from two incoherent sources . such a measurement would resemble the green function retrieval from ambient noise correlations , based on the relationship between field - field correlations and the imaginary part of the green function given by the fluctuation - dissipation theorem , initially introduced in the context of electromagnetic thermal fluctuations @xcite . the generality of this relationship , also valid for field fluctuations in speckle patterns @xcite , has stimulated the development of green s function retrieval techniques in acoustics , seismology or with low - frequency electromagnetic waves @xcite . in this approach , the statistical isotropic ambient noise is detected by two receivers , while in the method suggested here the green function is encoded in the noise due to the fluctuations of the two sources . the possibility to measure power fluctuations instead of field - field correlations might be an advantage for electromanetic green s function retrieval in the visible or near - ir frequency range . equation ( [ eq : varp_time_2 ] ) also shows that the power fluctuations encode the interdistance between the sources . at this stage , since the cdos @xmath35 is specific to the sample under study and unknown , changes in power fluctuations could reflect changes in the interdistance , but the interdistance could not be determined without solving a difficult inverse problem . we will see how the problem can be simplified in the presence of multiple scattering in a disordered medium by peforming an ensemble averaging over the configurations of disorder . we now assume that in addition to a temporal averaging over the fluctuations of the sources , an average over the configurations of the disordered medium can be performed . a specific situation would be that of sources embedded in a dynamic medium , with configurational changes occurring on a time scale much larger than the characteristic time of the fluctuations of the sources . an equivalent situation is that of sources moving inside a frozen disordered medium , also on a sufficiently large time scale , as schematically shown in fig . if both the sources and the disordered medium are fixed ( i.e. the medium itself does not fluctuate ) , an artificial configurational averaging process could be induced by an external moving diffuser surrounding the medium , as shown in fig . 2(b ) . in all these situations both averaging processes can be performed independently and subsequently . the ( temporal ) variance of the total emitted power averaged over the configurations of the disordered medium is readily obtained from eq . : @xmath36 where we use the notation @xmath37 for configurational averaging . in practice , it is often convenient to work with normalized statistical quantities . for later convenience , we can introduce the contrast of the power fluctuations , that we define as _ [ eq : def_sigmap ] let us point out again the specific use of two non - commuting averaging processes in this definition , one over the temporal fluctuations of the sources and subsequently one over the configurations of the disordered medium . from eqs . and , the power contrast can be written in terms of the ldos and cdos , leading to _ [ eq : expr_sigmap ] in the next section we will derive a simple relationship between the power contrast @xmath38 and the speckle contrast @xmath39 deduced from the radiated intensity measured in a single direction , or equivalently at a single point in the far field speckle pattern . a measurement of the total emitted power @xmath14 requires a detection integrated over @xmath40 steradians . measuring fluctuations of the intensity radiated in a given direction , or intensity correlations between two different directions , could be a more convenient approach in practice . the time - averaged far - field intensity emerging in a given direction @xmath41 , per unit solid angle , and for a polarization state @xmath42 , is given by & = & + [ inocros ] where @xmath43 is the intensity radiated by the point source @xmath44 . the latter reads as i_j(*u * ) = _ r r^2 |*e*_*g*(*r*,*r*_j ) p_j * u*_j|^2 [ ijsigma ] with @xmath45 . the configurational average of the ( time averaged ) total emitted power and fluctuations can be rewritten as & = & _ 4 * u*[exact1 ] + & = & _ ^ _ 4 * u * * u*^ [ exact2 ] where @xmath46 means integration over the solid angle , @xmath47 runs over the two orthogonal polarization states in the far field , and @xmath48 . we now assume that the radiated field is a random statistically isotropic field , under the only constraints given by eqs . and . in this case one has @xmath49 and = ( 1 + _ ^ _ * u**u*^ ) [ avip1ip1 ] where @xmath50 is the usual kronecker delta and @xmath51 is a kronecker delta with respect to detection angles . the derivation of eq . is given in appendix a. at this stage , it is worth noticing that relation between the speckle intensity - intensity correlation function and the fluctuations of the total emitted power is the origin of the so - called @xmath52 correlation known for a speckle produced by a single source @xcite . if we consider the particular case of a single source at @xmath1 ( @xmath53 ) that does not fluctuate in time , the normalized intensity - intensity correlation function @xmath54 follows directly from eq . and is given by c(*u*,;*u*^,^ ) & = & -1 + & = & c_0 + ( c_0 + 1)_^ _ * u**u*^ [ c1]where c_0 & = & is the normalized variance of the ldos at the position of the source , that is at the origin of the infinite - range @xmath52 contribution to the speckle correlation function @xcite . therefore , for a single source , eq . provides another way of deriving the well - known results related to the @xmath52 correlation . moreover , from eq . , the speckle contrast ( defined as the normalized variance of the intensity in a specific direction and a given polarization channel @xmath47 ) is given by = 1 + 2 c_0 a result already obtained by shapiro @xcite , based on a microscopic diagrammatic approach for scalar waves . this result leads to non - universal corrections to the rayleigh statistical distribution of the intensity in speckle patterns produced by multiple scattering @xcite . for two fluctuating sources , assuming again that configurational changes in the disordered medium occur on time scales larger than the characteristic time of the source fluctuations , we also have [ the derivation is similar to that leading to eq . ( [ avip1ip1 ] ) , see appendix a ] : @xmath55 making use of eqs . , and for two sources with the same amplitude @xmath56 , we obtain @xmath57 \nonumber \\ & = \frac{\omega^4}{8 ^ 3 \ , \epsilon_0 ^ 2}|p|^4 \left\langle \rho_{12}^2\right\rangle \ . \label{eq : fluct_iu}\end{aligned}\ ] ] this equation shows that the intensities corresponding to two different speckle spots ( @xmath58 ) in the angular speckle pattern formed by two incoherent point sources are strongly correlated , with a correlation given by the fluctuations @xmath59 of the cdos . we end up with the surprising result that for two incoherent sources a cross - term survives averaging , and induces infinite range correlations in the speckle pattern . this correlation is formally very similar to that given by the @xmath52 contribution in the case of a single source , the fluctuations of the cdos replacing the fluctuations of the ldos . since the cdos is a two - point quantity , connecting in this specific situation the two source points , the speckle correlations encode the distance between the two sources . another implication of eq . is that in the presence of multiple scattering , cdos fluctuations can be accessed from measurements of the directional intensity , and not only from measurements of the total power @xmath14 as suggested initially by eq . . more precisely , we will now show that the speckle contrast measured in a single speckle spot ( hereafter denoted by @xmath60 ) contains the same information as the contrast @xmath61 in eq . , that assumed a measurement of the total emitted power . to proceed , we define the speckle contrast for a detection in a given polarization channel @xmath47 as @xmath62 in a similar way as the contrast of the total emitted power in eq . . from eq . , and the relation = one immediatly obtains _ s= = _ p [ sc2 ] or equivalently _ [ scfinal ] this equation is the second important result in this paper . it shows that for two incoherent sources in a disorder medium , the speckle contrast measured in a single speckle spot , and computed from both temporal averaging over the fluctuations of the sources _ and _ configurational averaging over disorder , is proportional to @xmath63 that characterizes the fluctuations of the cdos connecting the two sources.this means that the information on the interdistance between the sources is encoded in the speckle contrast @xmath60 measured at a single point . changes in the speckle contrast could therefore be used to detect changes in the interdistance between the two sources . compared to a measurement of the total emitted power @xmath14 , a measurement of the speckle contrast @xmath39 would require a simpler instrumentation , but at the cost of a substantial reduction of the signal level . note that a parallel detection of the fluctuations in several speckle spots could be performed using a ccd camera . in practice , a compromise between simplicity in instrumentation and signal - to - noise ratio should be found . the contrast in eq . ( [ scfinal ] ) would decrease to zero when increasing the interdistance . this change occurs on a range given by the width of the cdos ( green function ) considered as a function of the interdistance @xmath64 . this width depends on the microscopic structure of the disordered medium , providing in principle resolution capabilities beyond the free - space diffraction limit . the determination of the absolute value of the interdistance from the speckle contrast would require an expression of @xmath63 . the crucial issue in this case is to find the expression of @xmath65 in a disordered medium . this can be done at least in the weak scattering limit , as discussed in the next section . another consequence of eq . ( [ scfinal ] ) , or eq . , is that it provides a way to measure @xmath66 in a complex medium from the speckle noise recorded at a single point , which in the context of green s function retrieval might provide an alternative to the two - point field - field correlation technique , as discussed previously in section ii . when the sources are embedded in a weakly disordered , homogeneous and isotropic medium , it is possible to give an explicit approximate expression of the fluctuations of the cdos @xmath67 \langle ( \mathrm{im } g_{12})^2 \rangle$ ] . in the limit @xmath68 , where @xmath69 is the effective refractive index , @xmath70 ( with @xmath71 the wavelength in vacuum ) and @xmath72 the boltzmann scattering mean free path , the averaged green function is given by the well known result for homogeneous and isotropic media and we can write @xmath73 as @xmath74 where @xmath75 is the effective wavenumber . the description of the scattering medium by a complex effective wavenumber breaks down when the distance @xmath76 between the emitters approaches the size of the homogeneities ( more precisely the correlation length of disorder , see the discussion in ref . @xcite and references therein ) . although this might look like a severe limitation , this approximation is in practice very robust and has been shown to model accurately light diffusion through biological tissues . to lowest order , we can further approximate @xmath77 which , from eqs . , or , and provides an explicit expression of the intensity correlation function , or the speckle contrast , in terms of the relative distance between the sources . such an explicit expression should be of practical interest , e.g. , for the sensing of the interdistance between emitters embedded in biological tissues . let us point out that in terms of the detection of _ changes _ in the interdistance ( without measuring its absolute value ) , the method suggested in this paper does not requite any explicit expression of @xmath63 and relies only on eqs . ( [ eq : varp_time_2 ] ) , ( [ eq : expr_sigmap ] ) or ( [ scfinal ] ) . the results derived in this paper suggest the intriguing possibility that intensity measurements at only one point in a speckle pattern produced by two incoherent sources can provide information about the relative distance between the sources . moreover , this information is in principle not limited by diffraction . it can be extracted from the speckle contrast @xmath60 obtained after a proper time and configurational averaging process . the results also suggests an alternative approach to the green function retrieval technique . in the later the statistical isotropic ambient noise is detected by two receivers , while in the method suggested here the green function is encoded in the noise due to two fluctuating incoherent sources measured at a single point . finally , let us note that for fluorescent emitters , the quantum optical equivalent of eqs . , or might be established in terms of decay rate or photocount statistics of a coupled system , and fluctuations could be associated with super - radiant and sub - radiant states . this quantum treatment is left for future work . we acknowledge helpful discussions with m. fink and f. scheffold . this work was supported by the spanish mec through grant no . fis2012 - 36113 , by the comunidad de madrid through grant no . s2009/tic-1476 ( microseres project ) , by labex wifi ( laboratory of excellence within the french program `` investments for the future '' ) under references anr-10-labx-24 and anr-10-idex-0001 - 02 psl*. jjs acknowledges an ikerbasque visiting fellowship and rc and gc the hospitality of the dipc at donostia - san sebastian ( spain ) where part of this work was done . the far - field intensity radiated by two dipoles emerging in a given direction @xmath41 , per unit solid angle , and for a polarization state @xmath42 , can be written as i_(*u * ) = _ r r^2 | _ j=1 ^ 2 * e*_*g*(*r*,*r*_j ) * p*_j |^2 with @xmath45 . assuming that the @xmath40 solid angle is divided in a finite number @xmath78 , of ( solid angle ) pixels , the total power emitted by the two sources is given by p & = & _ 4 i(*u * ) * u*= _ _ u^n i_(*u * ) which can be written in compact matrix form , p & = & ^^ where @xmath79 and @xmath80 is a @xmath81 matrix ( @xmath82 comes from the two orthogonal polarizations of the field for each observation angle in the far - field ) . by using the singular value decomposition ( svd ) , the @xmath80 matrix can be factorized as & = & ^ + ^&= & ^2 ^ where @xmath83 is an unitary @xmath84 matrix , @xmath85 is a @xmath81 rectangular diagonal matrix with nonnegative real numbers on the diagonal and @xmath86 is a @xmath87 unitary matrix . equation in the main text can now be written as @xmath88= \frac{4\pi}{n } [ \hat{{{\bf p}}}^\dagger\mathbf{v } \mathbf{\sigma}^2 \mathbf{v}^\dagger \hat{{{\bf p } } } ] \nonumber \\ & = \frac{\mu_0 \ , \omega^3}{2 } \sum_{j , j ' = 1}^2 p_{j'}^*p_j \mathrm{im } g_{jj ' } \nonumber \\ & = \frac{\mu_0 \ , \omega^3}{2 } \begin{pmatrix } { { \bf p}}_1^\dagger & { { \bf p}}_2^\dagger \end{pmatrix } \begin{bmatrix } \mathrm{im}{{\bf g}}(11 ) & \mathrm{im}{{\bf g}}(12 ) \\ \mathrm{im}{{\bf g}}(21 ) & \mathrm{im}{{\bf g}}(22 ) \end{bmatrix } \begin{pmatrix } { { \bf p}}_1 \\ { { \bf p}}_2 \end{pmatrix } \label{eq : mat1}\end{aligned}\ ] ] where @xmath89 . in terms of these new matrices , the intensity at a given angle @xmath41 and polarization @xmath47 can be computed as @xmath90 where @xmath91 corresponds to the @xmath92 component of the dipole @xmath93 ( with @xmath94 $ ] and @xmath95 ) . if we assume that the radiated field is statistically isotropic , we can consider @xmath96 as a random unitary matrix statistically independent of the @xmath85 and @xmath97 matrices ( isotropy hypothesis ) . this is one of the key assumptions in the macroscopic scaling approach of transport theory @xcite . by using the averages over the unitary group ( evaluated by mello in ref . @xcite ) , u_b u_^ u_u_ b = _ b b _ * u**u* _ and the ensemble average of the intensity is given by @xmath98 which , from eq . , gives @xmath99 with @xmath100 and @xmath101 i.e. , as expected , under the isotropy hypothesis , the intensity at a given angle is simply proportional to the average of the total radiated power , @xmath102 . after averaging eq . ( [ eq : isum ] ) over the temporal fluctuations of the sources we have @xmath103 and @xmath104 is simply the sum of the individual intensities [ eq . ] i.e. the non - diagonal boxes of @xmath105 do not contribute ( no crosstalk term ) . the same applies for @xmath106 let us now consider eq . , & = & _ _*u*,*u* , [ exact3 ] with @xmath107 given by eq . . computation of eq . involves again @xmath103 together with averages of 4 elements of an unitary random matrix @xcite @xmath108 it is easy now to find : @xmath109 which , in the large @xmath110 limit , gives eq . in the main text . following a similar procedure , one also obtains @xmath111 which , in the large @xmath110 limit , gives eq . . e. betzig _ et al . _ , science , * 313 * 1642 ( 2006 ) ; s.t . hess , t.p.k . girirajan and m.d . mason , biophys . j. , * 91 * 4258 ( 2006 ) ; m.j . rust , m. bates and x. zhuang , nat . methods * 3 * , 793 ( 2006 ) ; e. toprak and p.r . selvin , annu . * 36 * , 349 ( 2007 ) ; j. antelman , c. wilking - chang , s. weiss and x. michalet , nano letters * 9 * 2199 ( 2009 ) ; a. pertsinidis , y. zhang and s. chu , nature * 466 * 647 ( 2010 ) . r. l. weaver and o. i. lobkis , phys . lett . * 87 * , 134301 ( 2001 ) ; b.a . van tiggelen , phys . * 91 * , 243904 ( 2003 ) ; a. derode , e. larose , m. tanter , j. de rosny , a. tourin , m. campillo , and m. fink , j. acoust . 113 * , 2973 ( 2003 ) ; m. campillo and a. paul , science * 299 * , 547 ( 2003 ) ; m. davy , m. fink and j. de rosny , phys . rev . lett . * 110 * , 203901 ( 2013 ) . vellekoop and a.p . mosk , opt . lett . * 32 * , 2309 ( 2007 ) ; z. yaqoob , d. psaltis , m.s . feld and c. yang , nature photon . * 2 * , 110 ( 2008 ) ; s. m. popoff , g. lerosey , r. carminati , m. fink , a.c . boccara and s. gigan phys . rev . lett . * 104 * , 100601 ( 2010 ) ; i.m . vellekoop , a. lagendijk and a.p . mosk , nature photon . * 4 * , 320 ( 2010 ) . g. j. foschini and m. gans , wireless personal commun . * 6 * , 311 ( 1998 ) ; i.e. telatar , eur . telecommun . , * 10 * , 585 ( 1999 ) ; s. h. simon , a. l. moustakas , m. stoytchev , and h. safar , phys . today * 54*(9 ) , 38 ( 2001 ) . the cdos defined here is a `` partial '' cdos defined from a particular matrix element @xmath114 of the tensor green function @xmath115 . in ref . @xcite , a `` global '' cdos was introduced from the trace of @xmath115 in order to wash out the polarization degrees of freedom . j. j. senz , l. s. froufe - prez , and a. garca - martn , in _ wave scattering in complex media : from theory to applications _ , ed . by s. skipetrov and b. van tiggelen , nato science series ii , vol . 107 ( kluwer , dordretch , 2003 ) , p. 175

we study the fluctuations of the light emitted by two identical incoherent point sources in a disordered environment . the intensity - intensity correlation function and the speckle contrast , obtained after proper temporal and configurational averaging , encode the relative distance between the two sources . this suggests the intriguing possibility that intensity measurements at only one point in a speckle pattern produced by two incoherent sources can provide information about the relative distance between the sources , with a precision that is not limited by diffraction . the theory also suggests an alternative approach to green s function retrieval technique , where the correlations of the isotropic ambient noise detected by two receivers are replaced by a measurement at a single point of the noise due to two fluctuating incoherent sources .

arxiv

in many studies of cell mechanics and dynamics , the cell is characterized as a viscoelastic body @xcite . it is an intriguing question to what extent such mechanical behaviour can be rationalized in terms of comparatively simple polymer physics models . in this respect , the comparison of cell rheological data and minimalistic _ in vitro _ reconstituted constructs of the cytoskeleton , such as pure actin solutions @xcite or crosslinked actin networks @xcite , has recently provided many new insights . important progress has also been achieved in the development of phenomenological mathematical descriptions . this includes approaches related to the tube model @xcite , tensegrity - based approaches @xcite , effective - medium models @xcite , and some others @xcite . in particular , the glassy wormlike chain ( ) model @xcite , a phenomenological extension of the standard model of semiflexible polymers @xcite has been successful in describing a plethora of rheological data for polymer solutions @xcite and living cells @xcite over many decades in time with a minimum of parameters . however , all these studies were primarily concerned with _ viscoelastic _ behaviour , while the latest investigations have underscored the glassy @xcite fragile @xcite , and inelastic @xcite character of the mechanical response of living cells . even for biopolymer networks _ in vitro _ , experiments operating in the nonlinear regime had so far to resort to special protocols that minimize plastic flow @xcite in order to make contact with dedicated theoretical models . the aim of the present contribution is to overcome this restriction by extending the to situations involving inelastic deformations . as a first step , we concentrate onto _ reversible inelastic _ behaviour , where the deformation does not alter the microscopic ground state . the protocol applied by trepat _ et al . _ @xcite provides a paradigmatic example . cells are subjected to a transient stretch such that , after some additional waiting time in the unstretched state , the ( linear ) material properties of the initial state are recovered . the simplification for the theoretical modelling results from the assumption that not only the macro - state but also the micro - state of the system may to a good approximation be treated as reversible under such conditions ; i.e. , we assume that the complete conformation of the polymer network , including the transiently broken bonds between adjacent polymers , is constrained to eventually return to its original equilibrium state . for the time - delayed hysteretic response of the network to such protocols one could thus still speak of a viscoelastic ( `` anelastic '' ) response in an operational sense , but we refrain from doing so in view of the fundamentally inelastic nature of the underlying stochastic process in contrast to the reversible softening effects observed in @xcite , for example . indeed , by simply allowing bonds to reform in new conformational states , the model developed below can readily be extended to arbitrary irreversible plastic deformations , as will be demonstrated elsewhere @xcite . before entering the discussion of our model , we would also like to point out that the proposed ( inelastic ) extension of the is strongly constrained by identifying the newly introduced parameters with those of the original ( viscoelastic ) model , where possible . despite its increased complexity , the extended model will therefore enable us to subject the underlying physical picture to a more stringent test than hitherto possible by comparing its predictions to dedicated experiments . moreover , unlike current state - of - the - art simulation studies @xcite it is not limited to rod networks but is firmly routed in a faithful mathematical description of the underlying brownian polymer dynamics . this paper is organized as follows . first , we review some basic facts about the in section [ sec : gwlc ] . next , in section [ sec : interaction ] , we introduce our extended reversible inelastic version , which we formulate using the notion of an effective interaction potential as in the original construction of the in @xcite . ( a preliminary account of the basic procedure and some of its cell - biological motivation including reversible bond - breaking kinetics has recently been given in a conference proceedings @xcite . ) sections [ sec : viscoelastic ] and [ sec : fluidization ] explain the physical mechanism underlying the mechanical response under pulsed and periodically pulsed loading , while section [ sec : remodelling ] illustrates its phenomenology . we demonstrate that the model exhibits the hallmarks of nonlinear cell mechanics : strain / stress stiffening , fluidization , and cyclic softening @xcite . section [ sec : intr_lengths ] investigates the relevance of the lately quantified structural heterogeneities in networks of semiflexible polymers @xcite for the mechanical properties , before we conclude and close with a brief outlook . the glassy wormlike chain ( ) is a phenomenological extension of the wormlike chain ( ) model , the well - established standard model of semiflexible polymers . a broad overview over and dynamics can be found elsewhere @xcite . the describes the mechanics of an isolated semiflexible polymer in an isothermal viscous solvent . in the weakly bending rod approximation , a solution of the stochastic differential equations of motion for the is possible _ via _ a mode decomposition ansatz for the transverse displacement of the polymer contour from the straight ground state . the individual modes labelled by an index @xmath0 are independent of each other and decay exponentially with rates @xmath1 . for convenience , we set the thermal energy @xmath2 , so that the bending rigidity can be identified with the persistence length @xmath3 , in the following . using this convention , the expression for the transverse susceptibility of a polymer of contour length @xmath4 ( with respect to a point force ) reads @xcite @xmath5 here , @xmath6 is an optional backbone tension , and @xmath7 the euler buckling force for a hinged rod of length @xmath4 . the different powers of @xmath0 in the denominator give notice of the competition of bending and stretching forces . in the , interactions of the polymer with the surrounding network ( e.g. excluded volume interactions or sticky contacts ) are reflected in an altered mode relaxation spectrum @xcite . the intuitive albeit not fully microscopic picture underlying the formulation of the is illustrated in figure [ fig : cartoon ] , which depicts a test polymer reversibly bound to the background network _ via _ potential wells at all topological contacts , the so - called entanglement points . consider now a generic point somewhere along the polymer backbone ( not an entanglement point ) . it can relax freely until the constrictions are being felt , which slow down the contributions from long wavelength bending modes . the translates this intuition into a simple prescription for the mode spectrum : the short - wavelength modes are directly taken over from the wormlike chain model , while modes of wavelength @xmath8 longer than the typical contour length @xmath9 between adjacent bonds are modified to account for the slowdown . motivated by the physical picture illustrated in figure [ fig : cartoon ] , the slowdown of the relaxation of a wavelength @xmath8 is expressed by an arrhenius factor @xmath10 $ ] for breaking @xmath11 potential energy barriers of height @xmath12 simultaneously . accordingly , the phenomenological recipe to turn a into a reads : @xmath13 & \lambda_n \geq \lambda \end{array}\right . . \ ] ] upon inserting this into ( [ eq : sus_wlc ] ) one obtains the susceptibility @xmath14 . the microscopic `` modulus '' for transverse point excitations of a generic backbone element on a test polymer is then defined as the inverse susceptibility , @xmath15 . an approximate expression for the macroscopic shear modulus is obtained along similar lines @xcite . in the original equilibrium theory , @xmath9 was assumed to be a constant on the order of the entanglement length of the network , @xmath16 . note , however , that @xmath17 is a geometric quantity ( which is determined by the polymer concentration and the persistence length ) while the contour length @xmath9 between closed bonds clearly depends on the state of the bond network . one therefore has to allow for an increase of @xmath9 with the number of broken bonds in non - equilibrium applications . this issue is explored in the following . all mechanical quantities calculated within the model crucially depend on the interaction length @xmath9 . in previous applications of the model @xcite it was assumed that @xmath9 remains constant equal to its equilibrium value and unaffected by the deformation of the sample . in other words , the equilibrium theory allowed for statistical bond fluctuations but not for a dynamical evolution of the parameters characterizing the thermodynamic state of the bond network . an obvious starting point for generalizations of the model to non - equilibrium situations is therefore to consider the number of closed bonds , and therefore also @xmath9 , as _ dynamic variables _ , dependent on the strain- and stress - history of the network . we now provide a mean - field description to account for such a dynamically evolving bond network . for clarity , we return to the intuitive picture underlying the where the ( possibly crosslinker- or molecular - motor - mediated ) complex interactions between the polymers are summarized into an effective interaction potential for a test segment against the background , as sketched in figure [ fig : potential ] . the same idea has also been used before in many related situations ( e.g.@xcite ) . between both states . the transition rates @xmath18 and @xmath19 between the bound and unbound state depend on the barrier heights @xmath12 and @xmath20 , respectively . an externally applied force can tilt the potential and favour binding or unbinding events.,width=9 ] the generic potential exhibits three essential features : a well - defined bound state , a well - defined unbound state , and a barrier in between ( figure [ fig : potential ] ) . the confining well corresponding to the unbound state represents the tube - like caging of the test polymer within the surrounding network @xcite . for ease of notation , we further introduce the dimensionless fraction of closed bonds or , _ bond fraction _ @xmath21 which is simply the ensemble - averaged fraction of sticky contacts that are actually in the bound state . the minimum bond distance @xmath22 is typically on the order of @xmath17 , but may be somewhat larger in situations where the bonds are mediated by crosslinker molecules or by partially sterically inaccessible sticky patches ( as e.g. for helical molecules @xcite ) . a quantitatively fully consistent way of calculating the dynamics of @xmath23 would involve solving the full fokker - planck equation for a -contact a formidable program to be pursued elsewhere @xcite . here , for the sake of our qualitative purposes , we chose to concentrate onto the physical essence of the discussion , and keep the mathematical structure as simple and transparent as possible . we therefore approximate the dynamics by a simple exponential relaxation as familiar from the standard example of reacting brownian particles , conventionally described by kramers theory with a bell - like force dependence @xcite . using this simplification and assuming a schematic interaction potential as depicted in figure [ fig : potential ] , the value of @xmath23 is determined by a competition of bond breaking and bond formation with reaction rates @xmath18 and @xmath19 , respectively . both rates are represented in the usual adiabatic approximation ( meaning that the equilibration _ inside _ the wells is much faster than the barrier crossing and external perturbations ) by @xcite @xmath24 where @xmath25 is an intrinsic characteristic brownian time scale for bond breaking and formation and @xmath26 are characterized by different brownian time scales depending on the width of the potential wells . at the present stage , we do not bother to distinguish these time scales nor to fine - tune their numerical values , and identify them , for the sake of simplicity , with the entanglement time scale @xmath27 in our numerical calculations . ] , and @xmath6 is the force pulling on the bond . noting that the fraction of open bonds is @xmath28 , we can then write down the following rate equation for the fraction of closed bonds @xmath29 the time dependence of @xmath30 leads via ( [ eq : nu_lambda ] ) to an implicit time dependence of the -parameter @xmath31 and thereby of all observables derived from the . the time - dependent force @xmath32 in ( [ eq : rate ] ) may be thought to result from an externally imposed stress protocol or from internal dynamical elements such as molecular motors setting the network under dynamic stress . hence , via an appropriately chosen set of slowly changing state parameters @xmath32 , @xmath33 , @xmath34 , the model can in principle accommodate for the active biological remodelling in living cells and tissues @xcite . ( for a discussion of the relation between the microscopic @xmath6 and the macroscopic stress , see @xcite . ) note that for constant force , the stationary value of @xmath23 , @xmath35\right)^{-1 } , \ ] ] obtained by setting @xmath36 , does not depend on @xmath12 , as it should be ( the steady state is independent of the transition state ) . on the bond fraction @xmath23 for various values of @xmath12 ( brown : @xmath37 , red : @xmath38 , blue : @xmath39 , green : @xmath40 , black : @xmath41 ) at a frequency @xmath42 and a force @xmath43 ( @xmath44 is the relaxation time of a mode of length @xmath22 and @xmath45 is the euler buckling force for a rod of length @xmath22 ) ; _ left : _ normalized real part @xmath46 ; _ right : _ loss angle.,title="fig:",width=7 ] on the bond fraction @xmath23 for various values of @xmath12 ( brown : @xmath37 , red : @xmath38 , blue : @xmath39 , green : @xmath40 , black : @xmath41 ) at a frequency @xmath42 and a force @xmath43 ( @xmath44 is the relaxation time of a mode of length @xmath22 and @xmath45 is the euler buckling force for a rod of length @xmath22 ) ; _ left : _ normalized real part @xmath46 ; _ right : _ loss angle.,title="fig:",width=7 ] whenever the bond kinetics can be disregarded ( @xmath47 ) , the viscoelastic properties are simply those of the bare @xcite , which can basically be characterized as short time dynamics followed by a slow , highly stretched logarithmic relaxation resembling power - law rheology with a non - universal exponent within the typical experimental time windows . strong perturbations ( stress or strain ) then give rise to a pronounced stiffening due to the steeply nonlinear response of semiflexible polymers under elongation @xcite . this well established behaviour can be understood as the canvas against which we aim to discern the characteristic mechanical signatures of the bond kinetics . an observable that will be of particular interest in the following is the complex microscopic `` modulus '' @xmath48 , introduced in the previous section . it is used to determine the linear as well as ( via a nonlinear update scheme , see [ sec : technical ] ) the nonlinear force response of the system . to understand how the time - dependence ( [ eq : rate ] ) of @xmath23 affects this important quantity , it is instructive to first examine the dependence of @xmath49 on @xmath23 . to this end , we formally consider @xmath23 temporarily as an independent variable instead of determining it from ( [ eq : rate ] ) . note that this approach nevertheless makes sense as for a fixed set of parameters , @xmath23 can still take any value between zero and one . this is due to the freedom of choosing an initial state , which can be imagined to have evolved from the prior deformation history . for fixed other parameters , both the real part @xmath46 ( figure [ fig : mod_lam]a ) and the imaginary part @xmath50 of @xmath49 increase monotonically with @xmath23 . we emphasize that @xmath50 is not simply proportional to @xmath46 and therefore the loss angle @xmath51 also depends on @xmath23 ( figure [ fig : mod_lam]b ) . for small @xmath23 , the loss angle is large , corresponding to fluid - like behaviour . with increasing @xmath23 , the system becomes more solid - like . increasing the barrier height @xmath12 makes the dependence of @xmath49 on @xmath23 more pronounced ( figure [ fig : mod_lam ] ) . conversely , as can be expected , the dependence on the barrier height @xmath12 vanishes with decreasing bond fraction @xmath23 . note that due to the boundary conditions the limit of a newtonian fluid ( @xmath52 ) is not recovered when formally taking the limit of vanishing bond fraction ( @xmath53 , @xmath54 ) . finally , the potential difference @xmath55 solely determines the dynamics of @xmath23 via ( [ eq : rate ] ) , and therefore influences the modulus @xmath49 solely through the equilibrium value for @xmath23 . after these general considerations , we now concentrate on the nonlinear and non - equilibrium dynamic response of the extended inelastic model , which results from the coupled relaxation of the viscoelastic polymer network _ and _ the transient bond network . a convenient way to characterize the mechanical properties of an inelastic material is a force - displacement curve . for a perfectly linear elastic ( hookean ) body , it would simply consist of a straight line , whereas a perfectly plastic body would feature as a rectangle delineated by a yield force @xmath56 and an arbitrary plastic strain value . for our qualitative purpose , we identify the average transverse displacement of the test polymer segment ( which is used to determine the force response , see [ sec : technical ] ) with the reaction coordinate @xmath57 of the schematic potential sketched in figure [ fig : potential ] ; and we identify the force @xmath6 entering the reaction rates with the backbone tension of the test polymer . as a characteristic length scale for the transverse displacements we use the width @xmath58 of the effective confinement potential , which is a measure of a typical mean - square displacement of the polymer in the unbound state . using this convention , we now consider the effect of a time - symmetric displacement pulse on the evolution of the force @xmath32 and bond fraction @xmath30 . for technical convenience , we use a gaussian shape for the displacement pulses , but the qualitative conclusions to be drawn are largely independent of the precise protocol . the duration of the displacement pulse , which sets the time scale for the dynamic response , is used as the unit of time in the following . here , we are not interested in short - time tension - propagation and single - polymer dynamics @xcite , hence a lower bound for pertinent pulse durations is provided by the interaction time scale @xmath59 . on the other hand , for pulse durations longer than @xmath60 the system deformation would be quasistatic so that no genuinely dynamic effects of the bond kinetics could be observed . for a small gaussian displacement pulse of relative amplitude @xmath61 ( c.f . figure [ fig : force_displacement]a and [ sec : technical ] ) , the shape of the curve predicted by our inelastic model shows all features of a _ viscoelastic _ medium ( figure [ fig : force_displacement]b , blue dashed curve ) . it starts with a nearly linear regime for relative displacements @xmath62 , where a weak stiffening sets in . due to dissipation in the medium , the path back to the initial state takes its course at lower force , which causes a weak hysteresis . this is essentially the viscoelastic response that already the bare model would have predicted . : gaussian deformation pulses ( @xmath63 ) with relative displacement amplitudes of @xmath61 ( dashed ) and @xmath64 ( solid ) ; corresponding force - displacement curves and bond fraction evolutions ( @xmath65 , @xmath66 ; dotted curve in @xmath65 : response of a without bond breaking ) ; order - of - magnitude estimate for the rate - dependence of the yield force , ( [ eq : rate_fc ] ) with const.@xmath67 , versus the numerically obtained maximum forces of the force - displacement curves for various mean force rates ( @xmath68 ) ; the minimum bond fraction ( @xmath68 , inset ) depends non - monotonically on the force rate ; @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 . , title="fig:",width=7 ] : gaussian deformation pulses ( @xmath63 ) with relative displacement amplitudes of @xmath61 ( dashed ) and @xmath64 ( solid ) ; corresponding force - displacement curves and bond fraction evolutions ( @xmath65 , @xmath66 ; dotted curve in @xmath65 : response of a without bond breaking ) ; order - of - magnitude estimate for the rate - dependence of the yield force , ( [ eq : rate_fc ] ) with const.@xmath67 , versus the numerically obtained maximum forces of the force - displacement curves for various mean force rates ( @xmath68 ) ; the minimum bond fraction ( @xmath68 , inset ) depends non - monotonically on the force rate ; @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 . , title="fig:",width=7 ] : gaussian deformation pulses ( @xmath63 ) with relative displacement amplitudes of @xmath61 ( dashed ) and @xmath64 ( solid ) ; corresponding force - displacement curves and bond fraction evolutions ( @xmath65 , @xmath66 ; dotted curve in @xmath65 : response of a without bond breaking ) ; order - of - magnitude estimate for the rate - dependence of the yield force , ( [ eq : rate_fc ] ) with const.@xmath67 , versus the numerically obtained maximum forces of the force - displacement curves for various mean force rates ( @xmath68 ) ; the minimum bond fraction ( @xmath68 , inset ) depends non - monotonically on the force rate ; @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 . , title="fig:",width=7 ] : gaussian deformation pulses ( @xmath63 ) with relative displacement amplitudes of @xmath61 ( dashed ) and @xmath64 ( solid ) ; corresponding force - displacement curves and bond fraction evolutions ( @xmath65 , @xmath66 ; dotted curve in @xmath65 : response of a without bond breaking ) ; order - of - magnitude estimate for the rate - dependence of the yield force , ( [ eq : rate_fc ] ) with const.@xmath67 , versus the numerically obtained maximum forces of the force - displacement curves for various mean force rates ( @xmath68 ) ; the minimum bond fraction ( @xmath68 , inset ) depends non - monotonically on the force rate ; @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 . , title="fig:",width=7 ] for a larger relative deformation amplitude of @xmath64 , however , the predictions of the bare and the inelastic diverge . the bare predicts a strong stiffening ( figure [ fig : force_displacement]b , red dotted curve ) , whereas the full model exhibits an initial stiffening regime followed by a pronounced softening ( figure [ fig : force_displacement]b , solid red curve ) . a very interesting observation is that `` softening '' in this case does not only mean a decrease of the slope of the force - extension curve , but that the slope actually becomes _ negative _ over an extended parameter region , once an operationally defined threshold force @xmath56 is exceeded . this `` flow state '' bears more resemblance to plastic flow than to viscoelastic relaxation . the reason for this qualitatively new phenomenon is the force - induced bond breaking . theoretically this is best exemplified by the time - dependence of the ( experimentally not easily accessible ) fraction @xmath30 of closed bonds ( figure [ fig : force_displacement]c ) . a glance at the bond fraction during the weak deformation scenario ( figure [ fig : force_displacement]c , blue dashed curve ) reveals how the limit of a bare is recovered , namely whenever the deformation is not sufficiently violent and persistent to significantly decrease the fraction of closed bonds . for the large deformation scenario , in contrast , the bond fraction stays only initially constant ( figure [ fig : force_displacement]c , solid red curve ) . as a consequence of the large strain the force rises steeply , as can be seen from the strong stiffening in figure [ fig : force_displacement]b ( solid red curve ) . at the yield force , the bond fraction suddenly decreases to nearly half of its initial value during a very short time . the decrease in the bond fraction is accompanied by a somewhat slower drop in the force , which is reflected in the sudden softening of the force - displacement curve . the bond fraction eventually recovers on a much slower time scale , which is roughly given by @xmath74 . note that even though the effects of the inelastic response dominate the stress - strain curve , the viscoelastic relaxations from the underlying model are still present . they could hardly be disentangled from the inelastic contributions , though , without an underlying faithful model of the viscoelastic response at hand . in summary , we observe a competition between force - amplitude dependent stress - stiffening and force - rate dependent yielding events . if the backbone force stays much smaller than the yield force , the adaptation of the bond network requires an adiabatically long time , on the order of the bond lifetime @xmath75 , so that plain viscoelastic behaviour is observed on the pulse time scale . in contrast , if the backbone tension @xmath6 reaches the yield force @xmath56 , the bond fraction declines sharply , thereby switching the response from the /-typical stiffening to a pronounced softening . the rate dependence of the yield force @xmath56 can be estimated by setting the time it takes to reach the yield force , @xmath76 , equal to the force - dependent time scale of bond opening , @xmath77 , from ( [ eq : kpm ] ) , @xmath78 here , lw@xmath79 is the positive real branch of the lambert w function . in figure [ fig : force_displacement]d , this estimate is compared with results from numerical evaluations for gaussian displacement pulses at different average rates . apart from the numerical errors , the slight deviations from the estimate may be attributed to the fact that the force rate is not constant for the gaussian protocol . they can be mostly eliminated by using force ramp protocols of various slopes instead of the gaussian displacement pulses . for not too low rates , the rate dependence can be approximated by the even simpler relation @xmath80 where the force rate has to be normalized by a suitable force scale . the minimum bond fraction reached during the application of the pulse , which we interpret as a measure of the degree of fluidization , depends _ non - monotonically _ on the rate ( figure [ fig : force_displacement]d , inset ) . qualitatively , this can be understood by noticing that two different factors influence the fluidization , namely the maximum force attained during deformation and the time over which the force is applied . the maximum force is simply the yield force @xmath56 , with the rate dependence in ( [ eq : rate_fc ] ) and figure [ fig : force_displacement]d , while the reciprocal rate itself sets the time scale . for high rates @xmath81 ( in our example ) the rate - dependence of the maximum force wins and the minimum @xmath23 reached decreases with increasing rate . for low rates , where the rate - dependence of the maximum force is much weaker ( figure [ fig : force_displacement]d ) the minimum @xmath23 decreases with increasing pulse duration , _ viz . _ decreasing rate . note that for slow pulses ( low rates ) , the bond fraction after the pulse may be quite different from the minimum @xmath23 , as significant recovery may already take place during the pulse . the substantial changes in the material properties accompanying bond breaking can be exemplified by monitoring the linear elastic modulus @xmath82 ( measured at a fixed frequency @xmath83 ) in response to a strain pulse ( figure [ fig : modulus]a ) . apart from the usual /-typical stress - stiffening below the threshold force @xmath84 , the modulus is seen to drop _ below _ the value it had before the deformation pulse , where it apparently saturates ( remember that the deformation vanishes for @xmath85 and that the recovery takes roughly a time @xmath86 ) . we thus observe what we call a `` passive '' , `` physical '' remodelling of the bond network as opposed to the `` active '' , `` biological '' remodelling of the cytoskeleton of living cells in response to external stimuli . these passive remodelling effects have recently been observed for human airway smooth muscle cells @xcite . ( a more quantitative discussion of the relation to the experimental observations will be given elsewhere . ) the fact that the deformation pulse leads to a decrease in @xmath46 and an increase in the loss angle @xmath87 ( figure [ fig : modulus]c ) suggests the notion of _ fluidization _ @xcite . and the bond fraction @xmath23 largely determine the storage modulus @xmath46 and the loss angle @xmath87 : normalized storage modulus @xmath88 ( @xmath63 ) and loss angle @xmath87 ( @xmath66 ) during displacement pulses of large and small amplitudes , 2.2 ( solid ) , 0.73 ( dashed ) ; the stronger deformation pulse causes a strong and persistent overshoot of the loss angle , indicating _ fluidization _ ; slow recovery after the pulse , ( @xmath65 , @xmath68 ) ; parameters as in figure [ fig : force_displacement ] , @xmath89.,title="fig:",width=7 ] and the bond fraction @xmath23 largely determine the storage modulus @xmath46 and the loss angle @xmath87 : normalized storage modulus @xmath88 ( @xmath63 ) and loss angle @xmath87 ( @xmath66 ) during displacement pulses of large and small amplitudes , 2.2 ( solid ) , 0.73 ( dashed ) ; the stronger deformation pulse causes a strong and persistent overshoot of the loss angle , indicating _ fluidization _ ; slow recovery after the pulse , ( @xmath65 , @xmath68 ) ; parameters as in figure [ fig : force_displacement ] , @xmath89.,title="fig:",width=7 ] and the bond fraction @xmath23 largely determine the storage modulus @xmath46 and the loss angle @xmath87 : normalized storage modulus @xmath88 ( @xmath63 ) and loss angle @xmath87 ( @xmath66 ) during displacement pulses of large and small amplitudes , 2.2 ( solid ) , 0.73 ( dashed ) ; the stronger deformation pulse causes a strong and persistent overshoot of the loss angle , indicating _ fluidization _ ; slow recovery after the pulse , ( @xmath65 , @xmath68 ) ; parameters as in figure [ fig : force_displacement ] , @xmath89.,title="fig:",width=7 ] and the bond fraction @xmath23 largely determine the storage modulus @xmath46 and the loss angle @xmath87 : normalized storage modulus @xmath88 ( @xmath63 ) and loss angle @xmath87 ( @xmath66 ) during displacement pulses of large and small amplitudes , 2.2 ( solid ) , 0.73 ( dashed ) ; the stronger deformation pulse causes a strong and persistent overshoot of the loss angle , indicating _ fluidization _ ; slow recovery after the pulse , ( @xmath65 , @xmath68 ) ; parameters as in figure [ fig : force_displacement ] , @xmath89.,title="fig:",width=7 ] the passive remodelling described here is a _ transient _ phenomenon , because , after cessation of the external perturbations , the bond fraction will eventually recover its equilibrium value . this indicates that also the change in the system properties is a transient effect , which is demonstrated by the recovery from fluidization in figure [ fig : modulus ] b & d. thus , while the fluidization resembles a plastic process on short time scales , on long times scales , the phenomenology is more similar to pseudo- and superelasticity as observed in shape - memory alloys @xcite . to demonstrate that the transient passive remodelling also affects the nonlinear material properties , we consider a series of three pulses , next ( figure [ fig : stress_strain ] ) . the force response to such a protocol is depicted in figure [ fig : stress_strain]b . the force - displacement curve for the second stretching ( second left branch of the solid curve ) is less steep than for the first stretching ( very left branch of the solid curve ) and the yield force is lower . this indicates a cyclic softening , or viscoelastic shake - down effect , closely related to the fluidization of the network by strain . the strength of the shake - down depends on the fraction of transiently broken bonds , and hence on the rate as well as on the amplitude of the applied deformation ( c.f . figure [ fig : stress_strain]c , solid red curve ) . upon repeated application of deformation pulses the force - displacement curves settle on a limit - curve that is essentially preconditioned by the initial deformation pulse and the inelastic work it performed on the sample . for more gentle protocols that only break a smaller fraction of bonds , the initial fluidization would be not as pronounced and one would obtain a gradual shake - down , which converges to a limit curve only after many cycles . ) and response ( @xmath65 , @xmath66 ) for three subsequent gaussian strain pulses ; force - displacement curves ( @xmath65 ) for a single minimum interaction distance ( @xmath90 , solid ) and for a distribution of @xmath22 according to @xcite with mean @xmath91 ( dashed ) ; the bond fraction @xmath23 ( @xmath66 ) for @xmath92 ( dash - dotted ) , @xmath93 ( solid ) , and @xmath94 ( dashed ) ; other parameters as in figure [ fig : force_displacement ] . , title="fig:",width=14 ] ) and response ( @xmath65 , @xmath66 ) for three subsequent gaussian strain pulses ; force - displacement curves ( @xmath65 ) for a single minimum interaction distance ( @xmath90 , solid ) and for a distribution of @xmath22 according to @xcite with mean @xmath91 ( dashed ) ; the bond fraction @xmath23 ( @xmath66 ) for @xmath92 ( dash - dotted ) , @xmath93 ( solid ) , and @xmath94 ( dashed ) ; other parameters as in figure [ fig : force_displacement ] . , title="fig:",width=7 ] ) and response ( @xmath65 , @xmath66 ) for three subsequent gaussian strain pulses ; force - displacement curves ( @xmath65 ) for a single minimum interaction distance ( @xmath90 , solid ) and for a distribution of @xmath22 according to @xcite with mean @xmath91 ( dashed ) ; the bond fraction @xmath23 ( @xmath66 ) for @xmath92 ( dash - dotted ) , @xmath93 ( solid ) , and @xmath94 ( dashed ) ; other parameters as in figure [ fig : force_displacement ] . , title="fig:",width=7 ] so far , we have assumed that there is one well - defined characteristic length scale @xmath22 for the polymer interactions , which is on the order of the entanglement length @xmath17 and interpreted as the minimum average contour distance between adjacent bonds of the test polymer with the background network . this is clearly a mean - field assumption . recent combined experimental and theoretical studies have established that the local tube diameter and entanglement length in pure semiflexible polymer solutions actually exhibit a skewed leptocurtic distribution with broad tails @xcite . to include this information in the above analysis is not straightforward , since the elastic interactions between regions of different entanglement length are not known , _ a priori_. for the sake of a first qualitative estimate , the simplest procedure seems to be to average the above results over a distribution of @xmath22 , corresponding to a parallel connection of independent entanglement elements . qualitatively , this renders the abrupt transition from stiffening to softening somewhat smoother ( c.f . figure [ fig : stress_strain]b , dashed curve ) . moreover , the initial stiffening is slightly more pronounced . this is due to the contributions of @xmath95 , which exhibit both stronger stiffening and fluidization . nevertheless , all qualitative features like stiffening and remodelling effects and a `` flow state '' indicating fluidization or inelastic shakedown can still be well discerned . this is consistent with the assumption that pure semiflexible polymer solutions are at least qualitatively well described by a single entanglement length @xcite . we have presented a theoretical framework for a polymer - based description of the inelastic nonlinear mechanical behaviour of sticky biopolymer networks . we represented the polymer network on a mean - field level by a test polymer described by the phenomenologically highly successful model . beyond the equilibrium statistical fluctuations captured by the original model , we additionally allowed for a dynamically evolving thermodynamic state variable @xmath31 characterizing the network of thermo - reversible sticky bonds , which is updated dynamically upon bond breaking according to an appropriate rate equation . within our approximate treatment we could obtain a number of robust and qualitatively interesting results , which are not sensitive to the precise parameter values chosen , nor to the choice of the transverse rather than the longitudinal susceptibility in ( [ eq : sus_wlc ] ) . for sufficiently strong deformations , changes in the mean fraction of closed bonds , reflected in @xmath31 , can influence material properties to a degree at which the material behaves qualitatively different from the usual viscoelastic paradigm . in particular , our theory predicts a pronounced fluidization response , which develops upon strong deformations on top of the intrinsic viscoelastic stiffening response provided by the individual polymers . after cessation of loading , the system slowly recovers its initial state . these observations are in qualitative agreement with recently published experimental results obtained for living cells @xcite , and a more quantitative comparison with dedicated measurement results for cells and _ in vitro _ biopolymer networks will be the subject of future work . we found the yield force for the onset of the fluidization to be sensitive to the deformation rate . moreover , the fluidization response was shown to be accompanied by a cyclic softening or shake - down effect . taking into account the spatial heterogeneity of biopolymer solutions by a distribution of entanglement lengths leads to a smoothing of the force - displacement curves without affecting their qualitative characteristics . one may still expect qualitatively new effects in situations with unusually broad distributions of entanglement lengths , such as for strongly heterogeneous ( e.g. phase separating ) networks . a parameter that was found to be very important for cells @xcite but has not been discussed much in this contribution is the _ prestressing force _ @xmath96 ( see appendix [ sec : technical ] ) , which is present in adhering cells even in the absence of external driving . experimentally , it was established that increasing prestress is correlated with a higher stiffness of adhering cells @xcite which in turn is correlated with a higher stiffness of the substrate @xcite . when naively treating the prestressing force just as an additive contribution to the overall force , our model predicts the opposite : the additional force breaks more bonds and the network becomes softer and more fluid - like . to reconcile these apparently contradicting trends , one can appeal to the notion of _ homeostasis _ , which essentially amounts to postulating that the cell actively adapts its structure such that a certain set of thermodynamic state variables remain in a physiologically sensible range @xcite . this basically implies that the cell will respond to stiff substrates or persistent external stresses by a biological remodelling that corresponds to an increase of @xmath12 and @xmath55 , and possibly to a decrease of @xmath22 . by adapting also the internal prestress @xmath96 , the cell may then avoid , apart from the structural collapse , an equally undesirable loss of flexibility . we note , however , that internal stresses are indeed observed to disrupt the cytoskeleton , as implied by the simple physical theory presented here , if they are not permanently balanced by a substrate @xcite . even though the nonlinear effects presented in this contribution depend crucially on the bond kinetics , it should not be overlooked that the is an equally indispensable ingredient of the complete theory . first , it provides the constitutive relation that gives the force in response to an infinitesimal displacement , which in turn governs the bond kinetics . secondly , the time - dependent linear susceptibility strongly filters the dynamic remodelling of the bond network in practical applications , in which one rarely has access to the microstate of the underlying bond network . in summary , the extended model thus integrates experimentally confirmed features of the response such as slow relaxation , power - law rheology , viscoelastic hysteresis , and shear stiffening with a simple bond kinetics scheme , which allows less intuitive complex nonlinear physical remodelling effects like fluidization and inelastic shake - down to be addressed . finally , we want to point out that it is straightforward to extend the above analysis in a natural way to account for irreversible plastic deformations @xcite . to this end , one has to allow for the possibility that the transiently broken bonds reform somewhere else than at their original sites ( as always assumed in the foregoing discussion ) after strong deformations with finite residual displacement . this can be included by accommodating an additional term acting as a `` slip rate '' in ( [ eq : rate ] ) @xcite . we acknowledge many fruitful discussions with j. glaser , s. sturm , and k. hallatschek , and financial support from the deutsche forschungsgemeinschaft ( dfg ) through for 877 and , within the german excellence initiative , the leipzig school of natural sciences building with molecules and nano - objects . the deformation pulse protocol used in this paper is @xmath97 to obtain the full nonlinear response of the ( extended ) to the displacement protocol ( [ eq : strain_gauss ] ) , we make use of the superposition principle : we know that the fourier transform of @xmath98 is the linear response to a small displacement step . after decomposing a finite displacement into infinitesimal displacement steps and a partial integration , we write @xmath99 note that the right hand side of ( [ eq : force ] ) depends on @xmath100 , rendering it a highly nonlinear implicit equation . using ( [ eq : strain_gauss ] ) and integrating by parts , we obtain @xmath101 the next step is to identify a connection between the force in ( [ eq : rate ] ) and @xmath102(t ) . a finite prestressing force @xmath96 is introduced mainly for technical reasons , namely to avoid the unphysical region of negative ( i.e. compressive ) backbone stress , which would buckle the polymers . while the physics of the prestressed network can essentially be mapped back to that of a network without prestress by a renormalization of the parameters @xmath12 and @xmath55 , prestress may also be seen as an important feature : indeed , the cytoskeleton of adhered cells is well known to be under permanent prestress , and suspended cells seem prone to spontaneous shape oscillations that could be indicative of a propensity of the cells to set themselves under prestress @xcite . in the context and on the longer time scales of the biological remodelling of the cytoskeleton it would therefore make sense to think of @xmath96 as a dynamic force generated by molecular motors and polymerization forces . for the following , however , we take the prestress to be constant so that the force entering ( [ eq : rate ] ) is @xmath103 we now face the problem that ( [ eq : force ] ) is an implicit equation for @xmath100 , which depends on @xmath30 , which in turn depends via @xmath32 on @xmath104 . we therefore use a two - step euler scheme : as initial values for @xmath6 and @xmath23 , we choose the prestressing force , @xmath105 , and the steady - state value under the prestressing force , @xmath106 , respectively . we then choose a sufficiently small time step @xmath107 and apply the following iterative rule : @xmath108 where @xmath109 . in the limit @xmath110 , this procedure converges to the exact solution . ke kasza , f nakamura , s hu , p kollmannsberger , n bonakdar , b fabry , tp stossel , n wang and da weitz 2009 filamin a is essential for active cell stiffening but not passive stiffening under external force * 96 43264335 . * eh zhou , x trepat , cy park , g lenormand , mn oliver , sm mijailovich , c hardin , da weitz , jp butler and jj fredberg 2009 universal behavior of the osmotically compressed cell and its analogy to the colloidal glass transition * 106 10632 . * n wang , i m tolic - norrelykke , jx chen , sm mijailovich , jp butler , jj fredberg and d stamenovic 2002 cell prestress . i. stiffness and prestress are closely associated in adherent contractile cells * 282 c606c616 . *

we propose a physical model for the nonlinear inelastic mechanics of sticky biopolymer networks with potential applications to inelastic cell mechanics . it consists in a minimal extension of the glassy wormlike chain ( ) model , which has recently been highly successful as a quantitative mathematical description of the viscoelastic properties of biopolymer networks and cells . to extend its scope to nonequilibrium situations , where the thermodynamic state variables may evolve dynamically , the is furnished with an explicit representation of the kinetics of breaking and reforming sticky bonds . in spite of its simplicity the model exhibits many experimentally established non - trivial features such as power - law rheology , stress stiffening , fluidization , and cyclic softening effects .

arxiv

recently , topological properties of time - reversal - invariant band insulators in two and three dimensions have been extensively studied@xcite . a class of insulators preserving the time reversal symmetry is called topological insulators characterized by non - trivial topological [email protected] topological insulators have been intensively studied because of the existence and potential applications of robust surface metallic states . both in two and three dimensions , the topological phases are typically realized in the systems with strong spin - orbit interaction@xcite . all the known topological insulators contain heavy or rare metal elements , such as bismuth or iridium , which poses constraints on the search for topological materials . irrespective of constitutents , ubiquitous mutual coulomb repulsions among electrons have been proposed to generate effective spin - orbit couplings @xcite . it has been proposed that an extended hubbard model on the honeycomb lattice can generate an effective spin - orbit interaction from a spontaneous symmetry breaking at the hartree - fock mean - field level leading to a topologically non - trivial phase@xcite . since the honeycomb - lattice system , which is dirac semimetals in the non - interacting limit , becomes a topologically nontrivial insulator driven by the coulomb interaction , this phase is often called a topological mott insulator ( tmi ) . this phenomenon is quite unusual not only because an emergent spin - orbit interaction appears from the electronic mutual coulomb interaction , but also it shows an unconventional quantum criticality that depends on the electron band dispersion near the fermi point@xcite . however , this proposed topological phase by utilizing the ubiquitous coulomb repulsions has not been achieved in real materials even though the tmi is proposed not only in various solids @xcite but also in cold atoms loaded in optical lattices @xcite . even in simple theoretical models such as extended hubbard models , it is not clear whether the tmis become stable against competitions with other orders and quantum fluctuations . reliable examination of stable topological mott orders in the extended hubbard model is hampered by competing symmetry breakings such as cdws . couplings driving the topological mott transitions are also relevant to formations of a cdw , which has not been satisfactorily discussed in the previous studies . since the emergence of the tmi in the honeycomb lattice requires the coulomb repulsion between the next nearest neighbor sites , the long - period cdw instability must be considered on equal footing , which is not captured in the small - unit - cell mean - field ansatz employed in the previous studies . examining charge fluctuations with finite momentum over entire brillouin zones is an alternative way to clarify the competitions among tmis and cdws , as studied by employing functional renormalization group methods @xcite . however , first order thermal or quantum phase transitions not characterized by diverging order - parameter fluctuations are hardly captured by such theoretical methods . the most plausible symmetry breking competing with tmis indeed occurs as a first order quantum phase transition as discussed later . the quantum many - body fluctuations beyond the mean - field approximation severely affects the stability of the tmi . the stability of the tmi and estimation of the critical value of interaction on the honeycomb lattice has mainly been considered by mean - field calculations which can not treat the correlation effect satisfactorily . however , there exists a reliable limit where the tmi becomes stable : for infinitesimally small relevant coulomb repulsions , the quadratic band crossing with vanishing fermi velocities cause the leading instability toward the tmi , as extensively examined by using perturbative renormalization group methods@xcite . however , examining the instabilities toward the tmi in dirac semimetals requires elaborate theoretical treatments . in this study , for clarification of the competitions among tmis and other symmetry breakings , we first examine the long - period cdw at the level of mean - field approximation that turns out to be much more stable compared to that of short period . indeed , this cdw severly competes the tmi on the honeycomb lattice . the tmi on the honeycomb lattice studied in the literatures is consequently taken over by the cdw . we , however , found a prescription to stabilize the tmis on the honeycomb lattice : by reducing the fermi velocity of the dirac cones , the tmi tends to be stabilized . we examine the realization of the tmis in the extended hubbard model on the honeycomb lattice by controlling the fermi velocity and employing a variational monte carlo method@xcite with many variational parameters@xcite , multi - variable variational monte carlo ( mvmc)@xcite , together with the mean - field approximation . finally , we found that , by suppressing the fermi velocity to a tenth of that of the original honeycomb lattice , the tmi emerges in an extended parameter region as a spontaneous symmetry breaking even when we take many - body and quantum fluctuations into account . this paper is organized as follows . in section [ sec : model and method ] , we introduce an extended hubbard model and explain the order parameter of tmi . we also introduce the mvmc method . in section [ sec : stability ] , we first show how the long - range cdw becomes stable over the tmi phase in standard honeycomb lattice models . then we pursue the stabilization of tmi by modulating fermi velocity at the dirac cone at the mean - field level . finally we study by the mvmc method the effect of on - site coulomb interaction which was expected to unchange the stability of the tmi phase at the level of mean - field approximation . section [ sec : dis ] is devoted to proposal for realization of tmis in real materials such as twisted bilayer graphene . in this section , we study ground states of an extended hubbard model on the honeycomb lattice at half filling defined by @xmath0 where the single particle parts of @xmath1 are defined as @xmath2 and @xmath3 is the spin - orbit interaction . here @xmath4 is a creation ( annihilation ) operator for a @xmath5- spin electron , @xmath6 is an electron density operator , @xmath7 represents the hopping of electrons between site @xmath8 and @xmath9 , and @xmath10 are on - site ( off - site ) coulomb repulsion . bracket @xmath11 denotes the next - neighbor pair , @xmath12 is the strength of the spin - orbit interaction and @xmath13 is the @xmath14=@xmath15-spin operator . in eq.([hso ] ) , the @xmath16-th site is in the middle of the next nearest neighboring pair @xmath8 and @xmath9 as shown in fig . [ fig : honeycomb ] , and @xmath17 is the vector from the site @xmath8 to @xmath9 . we start with the hopping matrix @xmath7 in eq.([eq : h0 ] ) for the bond connecting a pair of the nearest - neighbor sites @xmath18 , @xmath19 as the simplest extended hubbard model on the honeycomb lattice . later , we will examine the effect of third neighbor hoppings @xmath20 . we take @xmath21 as the unit of energy and set @xmath22 throughout the rest of this paper . for the non - interacting limit , @xmath23 , the system becomes a topological insulator when @xmath12 is nonzero , which is identical to the topological phase of the kane - mele model @xcite . for the off - site coulomb repulsion , we mainly consider the second neighbor interaction ( @xmath24 ) , which is necessary for the emergence of the correlation - induced topological insulator @xcite . the second neighbor coulomb repulsions @xmath25 effectively generate the spin - orbit interactions , which are identical to @xmath12 , and induce topological insulator phases even for @xmath26 . we note that the coulomb repulsion of on - site or the nearest neighbor site do not affect the stability of the tmi phase at the level of the mean - field approximation . indeed , our mvmc results show that this is essentially true beyond the level of mean - field approximation which will be discussed in the later section . therefore , for the moment , we focus only on the effect of @xmath27 for the consideration of interaction effects . . next nearest neighbor interaction @xmath25 and the loop current @xmath28 flowing between the next nearest neighbor are shown by red and blue arrows . , width=245 ] here the tmi is the broken symmetry phase characterized by the order parameter @xmath28 defined by @xmath29 where the self - consistent mean fields for the second neighbor bonds are given by @xmath30 here , @xmath31 denotes the expectation value for next nearest neighbor bonds and the order parameter @xmath28 is physically interpreted as spin dependent loop currents flowing within a hexagons constituting the honeycomb lattice . at the level of the mean - field approximation , this quantum phase transition is understood by decoupling two - body electron correlation term of the next nearest neighbor bond , @xmath32 , as @xmath33 we also note that this phase transition to tmi , which is proposed not only on the honeycomb lattice but also in several other lattice models , belongs to an unconventional universality class , which depends on the dimension of the system and the dispersion of the electron band@xcite . in this section , we pursue the topological mott phase transition by employing the mean - field analysis and the variational monte carlo method . for the latter method , we use a trial wave function of the gutzwiller - jastrow form , @xmath34 with a one body part , @xmath35 where @xmath36 is the variational parameters and @xmath37 is the number of the electrons in the system . though this form of the wave function restricts itself to the hilbert subspace with the zero total @xmath38-component of @xmath14=@xmath15 , @xmath39 , it can describe topological phases on the honeycomb lattice as long as we use complex variables for @xmath36 . here , @xmath40 and @xmath41 are the gutzwiller and jastrow factors defined as @xmath42 and @xmath43 respectively , with which the effects of electron correlations are taken into account beyond the level of mean - field approximation . the expectation value , @xmath44 , is minimized with respect to variational parameters , @xmath36 , @xmath45 , and @xmath46 , by using the monte - carlo sampling and using the stochastic reconfiguration method by calculating gradient of the energy and the overlap matrix in the parameter space@xcite . we optimize the parameters by typically 2000 stochastic reconfiguration steps . in the present implementation of the variational monte carlo method with complex variables , the feasible system size for the calculation is about up to 300 , from which we speculate properties in the thermodynamic limit . for this purpose , we perform the size extrapolation using the standard formula @xmath47 where @xmath48 is the linear dimension of the system size . we note that the order of taking the limit in the right hand side of eq.([zeta_limit ] ) is also important for the validity of the extrapolation , which is known as the textbook prescription for the defining spontaneous symmetry breakings . another practical way to determine the spontaneous symmetry breakings is in principle the finite size scaling of the correlation for the order parameter . however , the latter finite size scaling is not practically easy . the correlation of @xmath28 becomes too small because @xmath28 itself is about the order of @xmath49 and the correlation becomes the order of @xmath50 which becomes comparable to the statistical error of the monte carlo sampling . for the size extrapolation , we fit the data of the finite size calculations by a polynomial of @xmath51 , that is , we assume the size dependence by @xmath52 the above assumption for the finite size scaling is based on a practical observation and an analogy to the finite size scaling in the spin wave theory @xcite . as a practical observation , @xmath53 for the limit @xmath54 and @xmath55 is scaled by @xmath51 . we examine the ground states of the extended hubbard model on the honeycomb lattice by tuning the on - site coulomb repulsion @xmath56 and the second neighbor coulomb repulsion @xmath25 . even for the parameter sets favorable to the tmis that have been studied in the pioneering works on the tmi @xcite , we show that the tmi is not stabilized when we take into account other competing orders overlooked in the literature . however , here , we reveal that , by tuning the fermi velocities of the dirac cones , the tmi is indeed stabilized . in this section , we consider long - period cdw states and show that 6-sublattice order is stable when @xmath25 becomes dominat . this state is schematically shown in fig.[fig : cdw6sub](a ) , where the electron density per site is disproportionated into four inequivalent values . when we pick up very rich sites or very poor sites , they constitute triangular lattices . the moderately rich or poor sites constitute the honeycomb lattices . sites labelled by 1 ( red ) , 2 ( light red ) , 3 ( light blue ) and 4 ( blue ) correspond to the sites where the electron densities are very rich , rich , poor and very poor , respectively . ( b ) the growth of order parameter of the cdw state . ( c ) resulting phase diagram of honeycomb lattice for @xmath56 and @xmath25 at the level of mean - field approximation . we do not find the region of stable tmi phase . , width=302 ] figure [ fig : cdw6sub](b ) shows the growth of the order parameters of the 6-sublattice cdw state for several different parameters of @xmath56 . in this calculation , we have used the mean - field approximation , where we defined the mean field by @xmath57 where @xmath58 is defined through @xmath59 here , the condition of half filling is satisfied by the following identity , @xmath60 as can be seen from fig.[fig : cdw6sub](b ) , the order parameter grows above @xmath61 even if we set @xmath62 . this is a quite serious problem for the stabilization of the tmi since the critical value of @xmath25 for the tmi at the mean - field approximation is also about @xmath63 . in addition , the energy gain due to the formation of the topological mott order is much smaller than that for the cdw state . when we consider larger values of @xmath56 , then the system becomes an antiferromagnetic state . furthermore , the mean - field approximation often overestimates the ordered phase , because it does not take into account fluctuation effects seriously . actually , by using the mvmc method , we could not find the region where the topological insulator becomes the ground state in the parameter space of @xmath56 and @xmath27 . the resulting phase diagram by the mean - field approximation is shown in fig . [ fig : cdw6sub ] ( c ) . there the antiferromagnetic phase is denoted as nel , where the spin on different sites of the bipartite aligns in the anti - parallel direction . as examined above , we found that the cdw state dominates and could not find parameter regions where the tmi is stable . however , we find that the tmi becomes stable by modulating the fermi velocity at the dirac cones in the electronic band dispersion actually it is confirmed that , when the fermi velocity is 0 and then the dirac cones change to the quadratic band crossing points , the phase transition from a zero - gap semiconductors to a tmi occurs with infinitesimal coulomb repulsions @xcite . for the honeycomb lattice , it is possible to change dirac semimetals to quadratic band crossings by introducing the third neighbor hopping @xmath20 ( schematically shown in fig.[fig : t3 ] ( a ) ) . then the part of the hamiltonian @xmath64 is replaced with @xmath65 where the third neighbor hoppings are given as @xmath66 here the tnn stands for the third neighbor . we find that the fermi velocity linearly decreases by introducing @xmath20 and becomes @xmath67 at @xmath68 as @xmath69 , which is also shown in fig.[fig : t3 ] ( b ) . though tuning nominal value of @xmath70 in the graphene is difficult , tuning of the fermi velocity at the dirac cone has been proposed in bilayer graphene by changing the relative orientation angle between two layers@xcite , which is effectively equivalent to the tuning of @xmath70 . and fermi velocity @xmath71 at dirac point . here , @xmath71 is shown by the ratio to the fermi velocity at @xmath72 . , width=321 ] figure [ fig : mfphase ] shows the phase diagram calculated by the mean - field approximation for @xmath73 , @xmath74 , and @xmath75 . compared to the nel ordered phase and the tmi , the cdw phase is not largely affected by the fermi velocity . therefore , the region of tmi recovers . ( a ) 0.3 , ( b ) 0.4 and ( c ) 0.45 . sm denotes the semimetal . , width=321 ] next , we examine the stability by using the mvmc method . in our mvmc method , we impose the translational symmetry on the variational parameters to reduce the computational cost , as @xmath76 here , @xmath77 is taken as any bravais vector of the 6 sublattice unit cell illustrated in fig.[fig : cdw6sub](a ) , in order to examine possible spontaneous symmetry breakings including the cdw shown in fig.[fig : cdw6sub](a ) and the tmi on an equal footing . therefore the number of the sites for each calculation is taken as @xmath78 . we have calculated for @xmath79 , where about 2500 variational parameters are used for the calculation of the largest size . to perform the extrapolation to the small external filed limit , @xmath80 , we have also calculated for several different strengths of the spin - orbit interaction , @xmath81 , and @xmath82 . figure [ fig : vmcdata](a ) shows the numerical results for the order parameter of the tmi at @xmath83 and @xmath84 . the sudden drops in @xmath28 around @xmath85 signals the emergence of the 6-sublattice cdw state . indeed , when @xmath27 further increases , @xmath28 vanishes . this is physically quite natural because @xmath28 is interpreted as the loop current and hard to be stabilized inside the cdw phase where electrons are locked at specific sites . for @xmath86 at @xmath84 . ( b ) size extrapolation at @xmath87 for several different values of @xmath27 . here , four different sizes ( @xmath79 ) are calculated . lines are results of quadratic fittings . ( c)extrapolation of @xmath12 is shown for different values of @xmath27 . lines are results of quadratic fitting . , width=321 ] same calculations are carried out for @xmath88 , which is employed in the size extrapolation of @xmath28 . figure [ fig : vmcdata](b ) shows the size extrapolation at @xmath87 as a typical example . since the cdw becomes dominant at @xmath85 , the results for @xmath89 are shown in fig.[fig : vmcdata](b ) . when @xmath25 is small , @xmath28 empirically follows the size dependence @xmath90 with a constant @xmath91 as we see in fig.5(b ) . this extrapolation is performed for four different values of @xmath12 , and then we get the values for @xmath92 . these data are shown in fig . [ fig : vmcdata](c ) , where the second extrapolation to @xmath54 is performed . final results for @xmath93 are shown in fig . [ fig : vmcresult](a ) , where the results of @xmath94 are also shown . the errorbars are defined from the errors of the extrapolation , where the largest error of the first extrapolation is added to the errors of the second extrapolation . we note that the error bars arising from the statistical errors of the monte carlo sampling are much smaller . relatively large errorbars for small @xmath27 at @xmath94 is possiblly because of the existence of the critical point at a finite value of @xmath27 , which is about @xmath95 . when @xmath27 exceeds this critical value and @xmath28 in the thermodynamic limit remains nonzero , the error becomes smaller as can be seen from fig . [ fig : vmcresult](a ) . for @xmath83 , the order grows from small value of @xmath96 . however , at @xmath94 , non - zero @xmath28 can not be detected for small values of @xmath27 . though the estimate of the universality class from these data is difficult , theoretically it is expected to belong to that of the gross - neveu model@xcite , and our result does not contradict this criticality . for @xmath97 , we do not find the value of @xmath27 where @xmath28 in the thermodynamic limit remains nonzero , and therefore phase transitions is not expected . this is shown in fig . [ fig : vmcresult](b ) , where the size extrapolation at @xmath87 is shown . there , @xmath28 becomes @xmath67 at @xmath98 for all @xmath27 , which is completely different from the behavior at @xmath94 and @xmath83 . the resulting phase diagram is shown in fig . [ fig : ueffect](a ) . although we expect that the mvmc results show larger critical values @xmath99 for the transition at @xmath100 , the estimated results indicate @xmath101 slightly smaller than the mean - field results shown in fig.[fig : mfphase ] . the reason that @xmath101 becomes smaller at @xmath100 in the mvmc results is probably an artifact arising from a peculiar size dependence near the essential singularity at @xmath102 , as we see even in the mean - field calculation shown in fig . [ fig : ueffect](b ) where , the size dependence in the mean - field calculation are shown . the possible errors in in the estimate of @xmath28 is as large as @xmath103 and the resultant errors in the estimate of @xmath101 is around 0.1 . therefore , the stability of the tmi phase over the cdw and nel phases in the region @xmath104 for @xmath100 is robust . here we note that the boundary between the cdw and tmi phases around @xmath105 does not change when we take into account the quantum fluctuations ( by calculating with the vmc method ) , because it is a strong first - order transition . are shown . ( b)size extrapolation at @xmath106 and @xmath87 for several different values of @xmath27 . here , four different sizes ( @xmath79 ) are calculated . lines are results of quadratic fittings . , width=321 ] now we discuss the effect of the on - site coulomb repulsion . though the onsite coulomb repulsion does not affect the value of @xmath28 and therefore stability of the tmi in the mean - field approximation , our mvmc result shows that increasing @xmath56 decreases the value of @xmath28 if @xmath27 is fixed as shown in fig . [ fig : ueffect](c ) . while the increasing @xmath56 quantitatively decreases the value of @xmath28 , its effect does not destroy the stability of the tmi phase at @xmath107 at least if @xmath108 . it also suppresses the emergence of the cdw . therefore it may help to enlarge the region of the tmi phase . the same effect is expected when we consider the coulomb repulsion for the nearest neighbor sites @xmath109 . that is , it decreases the value of @xmath28 beyond the level of the mean - field approximation but does not essentially affect the phase transition . however , we also note @xmath109 may cause another type of cdw , and stabilization of tmi should be examined against this cdw phase when @xmath109 is large . and @xmath70 at @xmath110 obtained by mvmc calculations . ( b)size dependence of mean - field calculation for several different values of @xmath27 at @xmath83 . ( c ) @xmath27 dependence of order parameter of tmi for @xmath111 and @xmath112 at @xmath113 . decrease in @xmath28 is observed by introducing @xmath56 . , width=321 ] as a qualitative difference from the mean - field approximation , we found that effect of @xmath56 decreases the value of the order parameter of the tmi . in the case of the border between the semimetal and the nel ordered phases , it is expected that @xmath114 , namely the critical values for @xmath56 becomes larger by treating fluctuation effects carefully . furthermore , we expect that @xmath114 becomes a function of @xmath27 , which may enhance the fluctuation of the nel order and suppress the phase transition . in our calculation , we did not find nel ordered states or mixed states of tmi and nel ordered states . on the other hand , the cdw phase is expected to be much more stable against fluctuations and the mean - field solution gives a reasonably good description because of a large scale of the energy gain for the cdw phase in comparison to other phases as adequately shown even by the mean - field approximation . at the boundary of the tmi to nel ordered phases , the universality class may change as suggested in the kane - mele - hubbard model@xcite . here , we discuss the realization of the tmi in the real solids . a primary candidate of tmis is graphene . as a well - known fact , graphene is nothing but a two - dimensional honeycomb network of carbon atoms and hosts dirac electrons . however , it is also a well - known fact that , in free - standing graphene and graphene on substrates , significant single - particle excitation gaps have not been observed yet@xcite . below , we examine possible routes toward the realization of the tmi in graphene - related systems . first of all , as already studied above , the suppression of the fermi velocity of the dirac electrons is crucial for the stabilization of the tmis . as extensively studied in the literature @xcite , twisted bilayer graphene ( tblg ) offers dirac electrons with tunable fermi velocities . by choosing stacking procedures , the quadratic band crossing , in other words , the zero fermi velocity limit , is also achieved , which has been already observed in experiments@xcite . next , we need to clarify competitions with any other possible symmetry breakings in the tblg with the fermi velocity smaller than that of graphene . the suppressed fermi velocities may possibly cause instabilities towards not only the tmis but also other competing orders as discussed in this paper . for clarification of the competition we need an _ ab initio _ estimation of effective coulomb repulsions which directly correspond to the coulomb repulsions in the extended hubbard model@xcite . the _ ab initio _ study on the effective coulomb repulsions employs a many - body perturbation scheme called constrained random phase approximation ( crpa)@xcite . the crpa estimation gives the following values for the coulomb repulsions : the on - site and off - site coulomb repulsions are given as @xmath115 , @xmath116 , and @xmath117 with @xmath118 ev for free - standing graphene . if we neglect longer - ranged coulomb repulsions , we expect the nel or cdw orders by employing these crpa estimates of @xmath119 , and @xmath25 in graphene and tblg . therefore , the free - standing graphene and tblg do not offer a suitable platform for the tmis . however , by choosing dielectric substrates , the strength of the coulomb repulsions , @xmath120 , and @xmath121 , is suppressed due to enhancement of dielectric constant as @xmath122 , and @xmath123 , where @xmath124 is defined by dielectric constants of each materials as @xmath125 . ( here , we ignored the possible reduction of the effective dielectric constants at small distances . ) then , we may approach the parameter region , where tmis become stable , as shown in fig.[fig : ueffect](b ) . if we neglect @xmath126 , dielectric substrates with @xmath127 are enough to stabilize the tmis . even when the nearest - neighbor coulomb repulsion @xmath126 is taken into account , tmi is expected to be stable as long as @xmath126 is not strong . in the above discussion , we neglected further neighbor coulomb repulsions , namely , the third neighbor coulomb repulsions @xmath128 , the fourth neighbor ones @xmath129 , and so on . to justify the above discussion , we need to screen the further neighbor coulomb repulsions by utilizing dielectric responses of the substrates and/or adatoms . here we note that the screening from atomic orbitals on the same and neighboring sites effectively reduces the coulomb repulsions as extensively studied by using crpa@xcite . the relative strength of the on - site and second neighbor coulomb repulsions , @xmath130 , may also be controllable by utilizing the adatoms , which are expected to efficiently screen the on - site coulomb repulsions if the adatoms is located just on top of the carbon atoms . if we combine the control of the @xmath130 with the suppression of the further - neighbor coulomb repulsions , the above discussion may be relevant . by utilizing the adatoms , the nearest - neighbor coulomb repulsions @xmath126 are also expected to be well - screened by adatoms on nearest - neighbor carbon - carbon bonds . the suppression of @xmath126 is helpfull for suppressing the cdws competing with the tmi . finally , we estimate the single - particle excitations gap @xmath131 induced by the tmi . the excitation gap is crucial for actual applications of the tmi as a spintronics platform . if we set @xmath132 and expect the tblg with @xmath133 ev , we obtain the excitation gap up to 0.1 ev , where we use the mean - field estimation of the gap @xmath134@xcite with the value of the mvmc result for @xmath28 and assumed this formula is valid in the presence of electron correlation . the estimated gap scale is substantially larger than the room temperatures . our theoretical results support that topological insulators with such a large excitation gap @xmath135 ev are possibly obtained by using abundant carbon atoms . in this paper , we have studied the realization of the tmi phase for the electronic systems on a honeycomb lattice by using the mean - field calculation and mvmc method . we found that the cdw of the 6 sublattice unit cell is much more stable than the previously estimated cdw with smaller unit cells for the simplest case where the electronic transfer is limited to the nearest neighbor pair . for the stabilization of the tmi we need to suppress the fermi velocity at the dirac point than the standard dirac dispersion for the case with only the nearest neighbor transfer . in the case of the honeycomb lattice , this is realized by introducing the third neighbor hopping @xmath20 and we have given quantitative criteria for the emergence of the tmi . related real material is a bilayer graphene where the fermi velocity is tuned by changing the rotation angle between two parallel layers@xcite . actually , the quadratic band crossing is realized when the rotation angle is @xmath67 ( known as the ab stacking bilayer graphene@xcite ) , which is mimicked by @xmath136 . since smaller values of fermi velocity stabilizes the tmi at smaller values of @xmath27 , its effective control may offer a breakthrough in the realization of two dimensional tmis . we need further analyses for experimental methods of controlling the stability of the tmis and _ ab initio _ quantitative estimates of the stability for bilayer graphenes , which are intriguing future subjects of our study . the authors thank financial support by grant - in - aid for scientific research ( no . 22340090 ) , from mext , japan . the authors thank t. misawa and d. tahara for fruitful discussions . a part of this research was supported by the strategic programs for innovative research ( spire ) , mext ( grant number hp130007 and hp140215 ) , and the computational materials science initiative ( cmsi ) , japan .

realization and design of topological insulators emerging from electron correlations , called topological mott insulators ( tmis ) , is pursued by using mean - field approximations as well as multi - variable variational monte carlo ( mvmc ) methods for dirac electrons on honeycomb lattices . the topological insulator phases predicted in the previous studies by the mean - field approximation for an extended hubbard model on the honeycomb lattice turn out to disappear , when we consider the possibility of a long - period charge - density - wave ( cdw ) order taking over the tmi phase . nevertheless , we further show that the tmi phase is still stabilized when we are able to tune the fermi velocity of the dirac point of the electron band . beyond the limitation of the mean - field calculation , we apply the newly developed mvmc to make accurate predictions after including the many - body and quantum fluctuations . by taking the extrapolation to the thermodynamic and weak external field limit , we present realistic criteria for the emergence of the topological insulator caused by the electron correlations . by suppressing the fermi velocity to a tenth of that of the original honeycomb lattice , the topological insulator emerges in an extended region as a spontaneous symmetry breaking surviving competitions with other orders . we discuss experimental ways to realize it in a bilayer graphenesystem .

arxiv

graphene has garnered interest from broad spectrum of communities , ranging from those aiming at atomic scale circuit devices to those searching for new topological phases . both communities sought after ways to gap the massless dirac spectrum . the realization of a gate - induced band - gap in the bernal stacked bi - layer graphene @xcite following the prediction in ref . @xcite brought the holy grail of graphene based transistor one step closer to reality . however , the sub - gap conductance measured by @xcite with weak temperature dependence well below the optically measured gap as large as 250 mev@xcite introduced a new puzzle and obstacle : the gapped bilayer is not as insulating as it should be . dominant transport along physical edge of the samples proposed earlier by @xcite have been ruled out by corbino geometry measurements@xcite , which observed two - dimensional variable range hopping type temperature dependence , independent of geometry . in this paper we predict existence of topological gapless channel of transport along recently imaged ab - ba tilt boundary network @xcite which solves the puzzle . the predicted topological edge state holds the promise of the first realization of topological surface(edge ) state hosted by structural topological defect . though there has been much theoretical interest in topological gapless modes hosted by structural topological defects@xcite no such topological gapless mode has been observed so far . the lattice dislocations in three dimensional crystals previously discussed occur deep in the sample that is not directly accessible . however , the tilt boundary of interest have recently been observed@xcite . the tilt boundary is a structural topological line defect along which each neighboring layer is displaced by one inter - atomic spacing . such defect can occur due to the third dimension added by the stacking of the graphene layers ; it forms a boundary between two inequivalent stacking structures frequently referred to as ab and ba . here we show that the tilt boundaries host gapless modes of topological origin and form the first example of a naked structural defect hosting topological electronic states . ( @xmath2 ) sublattice sites . @xmath3 represent hopping matrices for a tight - binding model . , scaledwidth=50.0% ] topological aspects of gapped multi - layer graphene have been previously discussed@xcite and it was pointed out that they should exhibit quantum valley hall effect with corresponding edge states . however , to this date there has been no experimental detection of proposed edge state @xcite . moreover , little is known about how the topological aspects of gapped multi - layer graphene relates to topological insulators @xcite . the idea of classifying different topological insulator ( superconductor ) candidates based on symmetries @xcite have played a key role in the field of topological insulators . in particular the observation that additional symmetries such as the crystalline symmetries can enlarge the possibilities of topological phases@xcite led to the discovery of three - dimensional topological crystalline insulators@xcite . on one hand we propose feasible experiments to detect topological edge states at naturally occurring tilt boundaries . at the same time , we make first concrete application of the spt approach @xcite for two dimensional ( 2d ) system and study a large class of gapped graphene systems placing the quantum valley hall insulator in the larger context and predicting conditions for topological superconductors . the rest of the paper is organized as follows . in section [ sec : micro ] we show that a ab - ba tilt boundary in gated bilayer graphene supports gapless edge states through explicit microscopic calculations . specifically we consider an abrupt boundary in tight - binding model and then investigate the effect of strain using ab - initio calculation . in section [ sec : topo ] we show that these edge - states are protected by no valley mixing , electron number conservation , and time reversal ( @xmath1 ) symmetries within the framework of spt . hence we identify chirally stacked gated @xmath4-layer graphene layers as _ time - reversal symmetric _ @xmath0-type spt . in section [ sec : expt ] we discuss experimental implications . finally in section [ sec : summary ] we summarize the results and comment on practical implications . + c + fig . [ fig : bilayer ] and fig . [ fig : domainwall](a ) show tilt boundaries of interest in gapped bernal stacked bi - layer graphene . in the case sketched , strain is concentrated at the tilt boundary with the top layer stretched by one inter - atomic spacing with respect to the bottom layer . for a general orientation , tilt boundaries can involve both strain and shear . as the tilt boundaries in layered graphene form a type of topological line defects in structure , they can be characterized using the tangent vector @xmath5 and the burger s vector @xmath6 . the tangent vector @xmath5 points along the tilt boundary which can point along any direction with respect to the burger s vector @xmath6 . when the tilt boundary only involves strain as in the case depicted in fig . [ fig : bilayer ] and fig . [ fig : domainwall](a ) , the @xmath6 is perpendicular to @xmath5 . in the opposite extreme limit of @xmath7 , shear is concentrated at the boundary . independent of the angle between @xmath6 and @xmath5 , the burger s vector magnitude is the inter - atomic spacing i.e. @xmath8 for a bilayer system , as it is shown explicitly for the strain tilt boundary in fig . [ fig : domainwall](a ) . since @xmath9 is a fraction of the bravis lattice primitive vector magnitude @xmath10 , the bilayer domain boundary is a partial dislocation from quasi two - dimensional view . in a general mult - layer a vertical array of these partial dislocations form a tilt - boundary . in typical samples , the domain wall separating the ab and ba stacked domains has substantial width spanning 5 - 20 nm and the angle between @xmath6 and @xmath5 ranges between @xmath11 and @xmath12 @xcite . [ fig : bilayer ] allow us to makes two important microscopic observations about tilt - boundaries . ( 1 ) as the boundary requires a shift of one - layer with respect to the other by one inter - atomic spacing along the bond direction , there are three natural directions for the tilt - boundary to run for each fixed angle between @xmath6 and @xmath5 . ( 2 ) the boundary is arm - chair for @xmath13 ( pure shear ) whereas it is zigzag for @xmath14 ( pure strain case shown in fig . [ fig : bilayer ] ) . based on observation ( 1 ) we expect a given type of tilt boundary to possibly form a triangular network seen in experiments@xcite . the observation ( 2 ) combined with earlier microscopic studies of boundary condition effects on edge states in ref . @xcite implies that electronic spectrum at tilt - boundaries with @xmath13 will be gapped though the gap magnitude will be small when the tilt boundary is spread over finite width . in this section , we consider the electronic structure of tilt boundaries with @xmath14 and return to more general case in the section [ sec : expt ] . as it is shown schematically in fig . [ fig : bilayer ] , the bulk of each domain is gapped in the presence of inter - layer hopping and the external electric field . the latter is important for breaking the inversion symmetry between the layers and gapping otherwise touching bands @xcite . below we present two separate microscopic calculations of the ab - ba tilt boundary electronic structure for @xmath14 , which shows gapless edge states . we consider a tight - binding hamiltonian with nearest - neighbor intra- and inter - layer hopping . for the ab - stacking region ( see fig . [ fig : domainwall](a ) ) , @xmath15 where @xmath16 is a layer index , @xmath17 and @xmath18 are intra - layer and interlayer nearest neighbor hopping respectively , and @xmath19 is the chemical potential difference between two layers due to the gate voltage . ( m , n ) labels the position of the two site unit - cell with @xmath20 and @xmath21 annihilating electrons at the two sites of layer @xmath22 . for the ba stacked region , the only change is in the inter - layer term with @xmath23 replacing the inter - layer term in @xmath24 . as we address the effect of strain through ab - initio simulation , we focus here on a sharp tilt boundary as shown in fig . [ fig : domainwall ] . we plot the energy spectrum in fig . [ fig : domainwall](b ) with the model parameters set to be @xmath25ev , @xmath26ev , and @xmath27ev . the size of the system was 200 unit - cell in each direction under periodic boundary condition with each domain spanning 100 unit - cell width separated by two sharp tilt boundaries . from the spectra it is clear that @xmath28 and @xmath29 valleys each have two edge states per spin . further investigation of the wave function shows right and left moving edge states associated with the given valley are spatially separated between the two edges : the edges offer valley filtering ( see fig.[fig : domainwall](c ) ) . . the fermi level is indicated with the blue dashed line and the bands with marked linear dispersion relation intersecting near the fermi level are outlined by the purple ( light ) curves . ( b ) top and side views of the charge distribution for a state near the @xmath30 point of ( a ) . the yellow rectangles indicate the ab - ba domain wall structure . , scaledwidth=48.0% ] the electronic structure in realistic tilt boundary will be affected by both the span over which the lattice structure transition from ab to ba stacking , and the strain concentrated at the tilt boundary . to address these issues we further carried out first - principles calculation using density functional theory ( dft ) within the local density approximation ( lda ) @xcite . we constructed a periodic supercell with two tilt boundaries characterized by burgers vectors identical to the ones considered within tight binding calculation , but the domain wall is set to have finite width over which the one unit - cell mismatch is spread . we have tried several configurations with domains and tilt boundaries of different widths to find qualitatively similar results . in the rest of this paper we focus on a representative example with 3.1 nm wide domain wall between 1.5 nm wide domains . the choice of narrow width for the domains was due to the limitation in the simulation capability . to this 2-d system , we then applied a slightly exaggerative perpendicular electric field of 5 v / nm . by relaxing the domain boundaries until the forces reaches below 0.5 ev / nm , we took the effect of both strain and the width into account . we have tried several configurations with domains and tilt boundaries of different width but otherwise similar setting to find qualitatively similar results . the dft - lda simulation results are presented in fig . [ fig : ldaband ] , which confirms the existence of the gapless edge states predicted by the tight - binding model . the electronic structure along the extended direction of edge is shown in fig . [ fig : ldaband](a ) , focusing on the region near @xmath28 point . in this figure two gapless 1d dirac dispersion is clearly resolved from the gapped bulk specta , with two distinct dirac points @xmath30 and @xmath31 in the vicinity of the @xmath28 point . the strain concentrated at the tilt boundary causes energy splitting of @xmath30 and @xmath31 states ; the energy of 1d dirac points is increased by compressive strain and decreased by tensile strain . this energy splitting will become negligibly small in realistic tilt boundaries with much wider span , as the strain will become smaller . [ fig : ldaband](a ) shows that the existence of gapless edge states found in our tight - binding calculation are robust against long range perturbations such as tilt boundary width or subtle bond - length variation inside the domain wall , as well as the strain at the tilt boundary . we now turn to the spatial distribution of the gapless edge states . for illustration , [ fig : ldaband ] ( b ) shows the wavefunction amplitude of the state slightly above the @xmath30 point . as expected from the tight - binding results , the charge distribution is prominent inside the domain wall but rapidly decays away from the tilt boundary . the dirac points of edge states and the decaying feature of edge states always exist , making us believe it is a universal feature . meanwhile , the charge for the states near @xmath30 is highly localized on the layer subject to compressive strain , regardless of the direction of the applied electric field . the situation for the states in vicinity of @xmath31 is similar except that the charge prefer to highly localized on the layer under tensile strain . hence forming a layer selective contact to an isolated edge state could be a mechanism for valley filtering@xcite . finally , we comment on the so far ignored effect of interaction . if the edge boundaries have substantial width , forward scattering part of the coulomb interaction will be the dominant correlation effect to the edge states and lead to luttinger liquid behavior @xcite . in order to address the robustness of the edge states , we investigate topological aspects of the low energy effective theory in the continuum limit . we first show that the ab - ba tilt boundary can be mapped to a gate - polarity boundary of uniform bi - layer . based on this mapping and results of refs . @xcite on the gate - polarity boundary , we discuss the valley chern number of the tilt boundary edge states . we than apply the notion of spt @xcite and identify chirally - stacked multi - layer graphene as a realization of @xmath32-type spt , protected by time - reversal ( @xmath1 ) , absence of valley mixing , charge conservation symmetries . this identification enables us to address effects of symmetry changes : topological quantum phase transitions . there are recent studies of such perturbations for specific cases such as rashba spin - orbit coupling @xcite and magnetic ordering @xcite . through our first application of spt classification scheme by @xcite to a concrete physical system of multi - layer graphene , we obtain an exhaustive systematic study of topological quantum phase transition possibilities . the low energy effective hamiltonian near the @xmath28 valley for uniformly ab or ba stacked bi - layer is @xmath33 where @xmath34 or @xmath35 sign should be used for ab or ba stacking respectively . in eq . , @xmath36 , @xmath37 s are pauli matrices acting on the sub - lattice indices , and @xmath38 s are pauli matrices acting on the layer indices . the effective hamiltonian near @xmath29 is @xmath39 . now it is straight forward to show that ba stacking is equivalent to ab stacking subject to the opposite gate polarity . at zero field the hamiltonian of the ab stacked bilayer can be transformed to that of the ba stacking by interchanging the two layers via the following unitary transformation : @xmath40 , with @xmath41 . for a gated bi - layer however , the gate polarity has to flip since @xmath42 hence at the level of low energy effective theory , the tilt boundary between ab and ba stacking under uniform external field is equivalent to the gate polarity domain wall of structurally uniform bi - layer proposed by @xcite . the above equivalence combined with earlier results on valley chern number of gated chirally stacked multi - layer offers the topological origin of the helical edge states observed in the microscopic calculation of section [ sec : micro ] . first for bi - layer and then for general @xmath4-layers , it was shown that low energy effective theory of chiral stacked @xmath4-layer under uniform vertical electric field @xmath19 has finite chern number per spin for each valley of equal magnitude and opposite sign @xcite : @xmath43 the chern numbers in eq . can be obtained by integrating the berry curvature over momenta @xmath44 continuing the linearized dispersion to infinity . this combined with the equivalence relation of eq . means the valley chern numbers change sign at the tilt boundary . such sign change leads to @xmath45 across the tilt boundary and @xmath4 branches of valley helical edge states @xcite , as long as the two valleys @xmath28 and @xmath29 remain distinct . hence the two valley helical edge states per spin observed in section [ sec : micro ] originate from the valley chern number change across the tilt boundary , as in the gate polarity boundary edge states @xcite . hence , our prediction is the tilt boundaries will be the first experimentally observed crystalline topological defects to host topological gapless mode due to change in the chern number . we now apply the procedure for identifying the class of spt based on symmetries of free fermion hamiltonian developed by @xcite , which predicts possible number of protected edge ( surface ) states . this procedure allows us to consider additional symmetries in the multi - layer graphene in addition to the @xmath46 , @xmath1 , and @xmath47 taken into account in the pioneering work by @xcite , and by @xcite . the procedure consists of three steps : ( 1 ) find a gapless dirac hamiltonian ( by keeping the kinetic term only ) with the same symmetries . then we find all the symmetry preserving mass terms that can gap out the gapless part and are amenable to classification using clifford algebra . this is based on the assumption that the spt order is robust as long as the energy gap stays finite and the symmetries remain the same and hence any gapped hamiltonian can be adiabatically transformed into a gapped dirac hamiltonian . ( 2 ) express the hamiltonian and the conserved quantities associated with symmetries in the majorana basis . this leads to the clifford algebra ( i.e. real representation of the dirac algebra ) associated with the gapless part of the hamiltonian . ( 3 ) find the space of mass matrices that anti - commutes with all the generators of this clifford algebra . the resulting space may have disconnected pieces , the number of which gives the classification of the spt . two mass matrices are topologically distinct if and only if they belong to two different pieces . applying this procedure to chiral multi - layer graphene will enable us to study phase transitions into different spts upon symmetry changes . for chiral multi - layer graphene system , we assume no valley mixing , electron number conservation @xmath48 , and time reversal ( @xmath1 ) symmetries . the relevant gapless dirac hamiltonian is : @xmath49 where @xmath50 , and @xmath51 , and @xmath52 , in which @xmath53 is given by the number of layers . this hamiltonian can be written in the majorana fermion basis using the following decomposition @xmath54 where @xmath55 denotes the majorana fermion satisfying @xmath56 where @xmath57 ( @xmath58 ) denotes the a , or b sublattice ( @xmath28 or @xmath29 valley ) indices , and @xmath59 denotes the flavor of the majorana fermions ( + or - ) . in the majorana fermion basis the hamiltonian is represented as follows : @xmath60 where @xmath61 is a real anti - symmetric matrix ( differential operator ) , and @xmath62 is an eight component vector whose components are @xmath63 . now we express the conserved quantities associated with the symmetries of the hamiltonian in the majorana fermion basis . first , no valley mixing combined with total electron number conservation symmetry leads to separate conservation of the electron number at each valley @xmath64 and @xmath65 . hence the total electron number @xmath66 , and the valley polarization @xmath67 are conserved . in the majorana fermion basis , @xmath68 where @xmath69 ( @xmath70 ) pauli matrices act on the majorana flavors ( valley indices ) . so defined @xmath71 and @xmath72 satisfy @xmath73 . under time reversal symmetry @xmath1 , @xmath28 and @xmath29 valley indices are exchanged i.e. , @xmath74 . hence @xmath1 acts like @xmath75 in the valley basis with the matrix part of the time reversal operator satisfying @xmath76 . on the other hand , no valley mixing implies the hamiltonian is invariant under @xmath77 transformation which acts like @xmath78 in the valley basis . hence , in the presence of no valley mixing symmetry , we can define a new time reversal operator @xmath79 , which acts like @xmath80 . in terms of majorana fermions @xmath81 in order to find the relevant clifford algebra , we need to form anti - commuting generators in terms of @xmath61 in eq . combined with symmetries , @xmath82 , @xmath83 , and @xmath84 however , the symmetries require @xmath85=\left[\mathcal{a},\hat{q}_{v}\right]=0 $ ] and @xmath86=0 . moreover , symmetry operators satisfy @xmath87 , @xmath88 , and @xmath89 relations . using these relations , we find the full set of generators of the relevant clifford algebra as @xmath90 , @xmath91 , and @xmath92 as @xmath93 for @xmath94 . the resulting full set of anti - commutation relations is @xmath95 and it defines a clifford algebra cliff(3,2 ) . now , we will find the space of mass matrices , @xmath96 , that can gap out dirac hamiltonian associated with this clifford algebra in order to obtain spt classification . the mass term with matrix representation @xmath97 should satisfy the following algebra : @xmath98 where we have normalized @xmath99 . solving the above equation yields the allowed space for the mass matrix , @xmath100 . it has been shown @xcite that @xmath96 which solves eq . for the case of cliff@xmath101 is @xmath102 the spt classification is then given by the number of disconnected pieces in the space of mass matrix @xmath100 i.e. its zeroth homotopy group : @xmath103 . using eq . , it can be verified that @xmath104 @xcite . consequently , each class of the time reversal invariant multilayer graphene in the absence of intervalley scattering is indexed by a @xmath32-valued number : in this case the valley chern number . [ tab1 ] [ cols="^,^,^,^,^",options="header " , ] now we are in a good position to consider symmetry changes . important to note here that spontaneously ordered phases can be considered alongside systems under external field , as once a system is deep inside the ordered phase it can be treated within mean - field theory . we first consider the symmetry reduction possibilities while maintaining spin degeneracy ( see table [ table : spinless ] ) . if we only break the time reversal symmetry , the system is characterized by two independent topological indices @xmath105 , hence the classification is given by @xmath106 . we refer to these spt phases as intra - valley quantum ( anomalous ) hall ( qah ) states . such phases may be realized by placing trigonally - strained graphene @xcite under an external magnetic field , as the sign of the pseudo - magnetic field caused by strain is opposite for the two valleys . further reducing the symmetry by introducing inter - valley scattering leads to inter - valley qah state indexed by a single integer @xmath32 : the total chern number . breaking electron number conservation turns the above insulators into superconductors . following the procedure above , we obtain the same classification for the topological superconductors in 2d , resulting in topological valley superconductor ( tvsc ) and intra- or inter - valley topological superconductors . table [ table : spinless ] summarizes all symmetry reduction possibilities and their classifications starting form gated multi - layer graphene . now we consider extending our classification to take the electron spin into account as a dynamical degree of freedom . this will allow us to consider interaction effects at the level of spin ordering . with both spin and valley degrees of freedom , there are three symmetry operators related to time reversal : @xmath107 exchanges two valleys , @xmath108 which acts on the spin indices flips spin , and @xmath109 does both . if both @xmath110 and @xmath111 are broken , the classification reduces to that of spinless electrons ( see upper part of table [ table : spinful ] ) . however , taking any of these symmetry operators into account leads to new classes ( see lower part of table [ table : spinful ] ) . when all @xmath112 , @xmath110 ( and as a result @xmath111 ) and no valley - mixing symmetries are imposed the state corresponds to the so - called `` layer antiferromagnetic '' ( laf ) phase predicted in refs . @xcite and possibly occurring as a ground state in bi - layer graphene at neutrality point @xcite . in this phase the product of spin and valley of edge quasi - particles is locked to their momentum . therefore , one may index this state by its _ spin - valley chern number _ @xmath113 @xcite . in this phase , for each valley , the edge states associated with two spins are counter - propagating . as long as these two counter - propagating modes do not couple , there can be any number of them per each valley , leading to @xmath114 spin - valley hall conductivity @xcite . however there is a form of symmetry allowed coupling between a pair of sets of counter propagating edges ; this coupling can gap out the edge modes @xcite . therefore , the number of symmetry protected edge modes for each valley is @xmath113 mod @xmath115 . hence , we obtain a @xmath116 classification for this phase labeled by @xmath117 as supposed to @xmath32 classification which would be implied by refs . @xcite . another interesting possibility is breaking @xmath112 , and @xmath110 , while respecting their product @xmath111 and no - valley mixing symmetry . this leads to the quantum spin hall ( qsh ) phase , in which @xmath118 , and @xmath119 due to @xmath111 symmetry , while there is no constraint on the @xmath120 . hence , unlike usual 2d qsh states @xcite , we obtain @xmath32 . this is because multiple type of time reversal operators can be defined @xcite in the presence of no valley mixing symmetry . this and other spt possibilities are summarized in table [ table : spinful ] . with the classification at hand , we return to what it implies for the fate of the gapless edge states at the tilt boundary of chirally stacked and gated n - layer graphene . in general , a topological phase transition between two phases within the same class requires the bulk gap to close and reopen ( as in inter - plateaux transitions in integer quantum hall effect ) . gapless edge states are guranteed at a physical boundary between such two phases . the edge states at the ab - ba tilt boundary we established in the section ii are examples of such edge states . hence the edge states will remain gapless as long as the time ( valley ) reversal and charge conservation symmetry are maintained . on the other hand , symmetry change can either yield a trivial phase which does not support an edge state or a different type of spt with different type of edge states . according to tables i and ii , ruining the no valley mixing symmetry is the only way to render the system trivial and gap the edge states . however this requires large momentum transfer which generally requires fine tuning unless the unit - cell becomes enlarged either for the entire system @xcite or for the edge through arm - chair edge , or short - range disorder such as a vacancy breaks @xmath121-@xmath122 sub - lattice symmetry @xcite . as both rarely occur , we anticipate gapless edge states at most tilt boundaries . in particular , the natural zigzag boundary formation for the tilt boundaries make such edge states more robust than the edge states in gate polarity boundaries @xcite . among symmetry change possibilities leading to another spt , transition from qvh with spin degeneracy to laf phase where spin is a dynamic degree of freedom is of particular interest as laf is suspected to be the ground state of bi - layer graphene near neutrality point @xcite . upon this phase transition the nature of edge states change from spin degenerate valley helical states ( qvh ) to spin - valley hall edge states ( laf ) . in such transitions , the bulk gap has to close and reopen ; this is indeed seen in the experiment of @xcite . in this section , we discuss the experimental implications of our findings . specifically we propose the transport through the network of tilt boundaries as a solution to the long standing mystery of sub - gap transport@xcite . further we propose feasible experiments to test the proposal . in order to discuss the topological transport through the network of tilt boundaries observed in refs . @xcite we should first discuss the effect of the arbitrary angle between the burger s vector @xmath6 and the tangent vector @xmath5 . microscopic study of tilt boundaries at various angles between @xmath6 and @xmath5 will be presented in the future@xcite . however , it has been known that a single domain bernal stacked gapped bilayer ribbon should support gapless edge state for zig - zag edges , but not for arm - chair edges @xcite . this is because the arm - chair edge enlarges the unit - cell along the direction parallel to the boundary and makes the projection of k and k@xmath123 valley identical . however , @xcite showed that the polarity boundary edge states only develop barely visible gap which is orders of magnitude less then the bulk gap even for a sharp boundary , and the gap decreases quickly when the polarity boundary becomes smooth . these arguments apply to our tilt boundary and the edge state will develop a small gap when @xmath13 . however , given the large width of the observed tilt boundaries we expect that all straight tilt boundaries will have nearly gapless edge states except those with small angles between @xmath6 and @xmath5 . when the tilt boundary meanders and changes directions , likely there will be portions with small gap segmenting the gapless regions and the transport will occur through hopping between the gapless regions . the observed 2d network of such tilt boundaries would yield 2d variable range hopping temperature dependence @xmath124@xcite at low temperatures governed by the 2d connectivity and the small characteristic gaps of gapped regions . this explains the observed temperature @xmath1 dependence of resistance at low temperatures @xcite . we propose following experiments to test our proposal . ( 1 ) four terminal transport measurements with two of the contacts , say contacts 1 and 3 , at two ends of a tilt boundary . this would yield highly anisotropic transport proving dominant transport along the tilt boundary i.e. , @xmath125 . ( 2 ) scanning tunneling spectroscopy measurements of local density of states . this should measure a gapless spectrum at the tilt boundary but exhibit a gapped spectrum with the gap magnitude of the optical gap away from the tilt boundary . ( 3 ) thermoelectric imaging . the mid - gap density of state at tilt boundaries would appear in scanning thermopower images . unpublished thermopower imaging data by @xcite indeed show a network with local density of state near fermi energy , that is reminiscent of the tilt boundary network . ( 4 ) edge current imaging using scanning squid which can detect magnetic field generated by edge currents . we showed that spin - degenerate tilt boundaries of gated multi - layer graphene support topological gapless edge states protected by three symmetries : time ( valley ) reversal , no - valley - mixing , and electron number conservation . we demonstrated the existence of gapless edge states through a tight - binding model calculation and a first principal calculation , where the latter took strain effects into account . we then addressed the symmetry protection of the edge states and consequences of symmetry changes within the framework of spt @xcite . the framework of spt allowed us to place the 2d topological phase supporting the edge states , namely qvh , among various topological insulator / superconductor phases alongside previously postulated qah , laf and qsh . while previous literature postulated qvh , qah , laf and qsh to be all supporting number of edge states growing with the number of layers @xmath4 ( i.e. @xmath0-type in the language of classification ) , we found that the symmetry of laf only protects odd number of edge modes for each valley . hence laf is a @xmath116-topological insulator much like quantum spin hall insulator@xcite . transition between these different spt s require closing and re - opening of the bulks gap as already been observed in ref . @xcite . we predict the naturally occurring tilt boundary @xcite to be the first topological structural defect hosting topologically protected gapless mode of transport , most importantly , our findings on tilt boundaries combined with the recent observations @xcite solve the long standing mystery of sub - gap transport @xcite . our explanation can be tested through proposed transport , scanning tunneling spectroscopy and thermopower imaging experiments , and scanning squid experiments . experimental confirmation of the tilt - boundary transport origin of the sub - gap transport will open doors to control the sub - gap transport and enable device application of gated multi - layer graphene systems . : we thank paul mceuen for numerous discussions and sharing his unpublished data . we thank joe stroscio for sharing his unpublished data on thermopower imaging . we thank e.j . mele for useful discussions and sharing the preprint @xcite and j. sethna for useful discussions regarding structural topological defect aspect of the tilt boundary . e .- a.k . and d.n . were supported by nsf award eec-0646547 through cornell center for nanoscience . e .- a.k . and a.v . were supported in part by nsf career grant dmr-0955822 . was also supported in part by nsf grant dmr-1120296 through cornell center for materials research . y.l . and l.y . were supported by nsf grant no . the computational resources have been provided by lonestar of teragrid at the texas advanced computing center ( tacc ) . 45ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1038/nmat2082 [ * * , ( ) ] link:\doibase 10.1103/physrevb.74.161403 [ * * , ( ) ] http://dx.doi.org/10.1038/nature08105 [ * * , ( ) ] http://dx.doi.org/10.1038/nphys1822 [ * * , ( ) ] link:\doibase 10.1021/nl102459 t [ * * , ( ) ] , @noop `` , '' @noop ( ) , @noop * * , ( ) http://dx.doi.org/10.1038/nphys1220 [ * * , ( ) ] link:\doibase 10.1103/physrevx.2.031013 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.115120 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.99.236809 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.036804 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.95.226801 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.96.106802 [ * * , ( ) ] link:\doibase 10.1103/physrevb.78.195125 [ * * , ( ) ] link:\doibase 10.1103/physrevb.85.085103 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.106.106802 [ * * , ( ) ] http://dx.doi.org/10.1038/nmat3449 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.075418 [ * * , ( ) ] link:\doibase 10.1103/physrev.136.b864 [ * * , ( ) ] link:\doibase 10.1103/physrev.140.a1133 [ * * , ( ) ] http://dx.doi.org/10.1038/nphys547 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.104.216406 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.256801 [ * * , ( ) ] link:\doibase 10.1103/physrevb.83.155447 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.195438 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.106.156801 [ * * , ( ) ] in link:\doibase 10.1063/1.3149495 [ _ _ ] , , vol . , ( ) pp . , link:\doibase 10.1126/science.1191700 [ * * , ( ) ] http://dx.doi.org/10.1038/nphys1420 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.205106 [ * * , ( ) ] http://dx.doi.org/10.1038/nnano.2011.251 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.95.146802 [ * * , ( ) ] link:\doibase 10.1103/physrevb.86.205116 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.186809 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.266402 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.89.266603 [ * * , ( ) ] @noop `` , '' link:\doibase 10.1103/physrevlett.100.026802 [ * * , ( ) ] @noop * * , ( ) @noop * * @noop `` , '' @noop `` , ''

despite much interest in engineering new topological surface(edge ) states using structural defects , such topological surface states have not been observed yet . we show that recently imaged tilt boundaries in gated multi - layer graphene should support topologically protected gapless edge states . we approach the problem from two perspectives : the microscopic perspective of a tight - binding model and an ab - initio calculation on a bilayer , and the symmetry protected topological ( spt ) states perspective for a general multi - layer . hence we establish the tilt boundary edge states as the first concrete example of edge states of symmetry enriched @xmath0-type spt , protected by no valley mixing , electron number conservation , and time reversal @xmath1 symmetries . further we discuss possible phase transitions between distinct spt s upon symmetry changes . combined with recently imaged tilt boundary network , our findings offer a natural explanation for the long standing puzzle of sub - gap conductance in gated bilayer graphene , which can be tested through future transport experiments on tilt boundaries . in particular , the tilt boundaries offer an opportunity for in - situ imaging of topological edge transport

arxiv

macromolecules containing ionizable groups when dissolve in a polar solvent such as water , dissociate into charged macromolecules and counterions ( ions of opposite charge ) . depending on acidic or basic property of their monomers , ionizable polymers in solution can be classified into polyelectrolytes and polyampholytes . polyelectrolytes contain a single sign of charged monomers and polyampholytes bear charged monomers of both signs . these macromolecules are often water - soluble and have numerous industrial and medical applications . many biological macromolecules such as dna , rna , and proteins are charged polymers . in polymer science , charged polymers has been an important subject during last several decades @xcite . contrary to a polyelectrolyte chain in which the intra - chain electrostatic interactions are repulsive and tend to swell the chain , in a polyampholyte chain attractive interactions between charged monomers of opposite sign tend to decrease the chain size . oppositely charged monomers can be distributed randomly along a polyampholyte chain or charges of one sign can be arranged in long blocks . with the same ratio of positively and negatively charged monomers ( isoelectric condition ) , behaviors of a single polyampholyte and the solution of polyampholytes depend noticeably on the sequence of charged monomers on the chains . for example , it has been shown that the sequence of charged amino acids ( charge distribution ) along ionically complementary peptides affect the aggregation behavior and self - assembling process in the solution of such peptides @xcite . also , using monte carlo simulations it has been shown that charged monomers sequence of charge - symmetric polyampholytes affect their adsorption properties to a charged surface @xcite . the properties of the system of polymers anchored on a surface are of great interest both in industrial and biological applications and academic research . with a sufficiently strong repulsion between the polymers , the chains become stretched and the structure obtained is known as a polymer brush . planar and curved brushes formed by grafted homopolymers have extensively been investigated by various theoretical methods @xcite . the anchored polymers of a brush may be consisting of charged monomers . in this case , the brush is known as a polyelectrolyte or a polyampholyte brush depending on the charged monomers of the chains being of the same sign or being composed of both signs . in a brush of charged polymers , electrostatic interactions introduce additional length scales such as bjerrum length and debye screening length to the system . in a polyelectrolyte brush , the repulsion of electrostatic origin between the chains can be sufficiently strong even at low grafting densities , making it easy for the system to access the brush regime . polyelectrolyte brushes have been investigated extensively using both theoretical @xcite and computer simulation methods @xcite . at high enough grafting densities and charge fractions of polyelectrolyte chains , most of counterions are trapped inside the polyelectrolyte brush and competition between osmotic pressure of the counterions and elasticity of the chains determines the brush thickness . this regime of a polyelectrolyte brush is known as the osmotic regime in which some theoretical scaling methods predict no dependence of the brush thickness to the grafting density @xcite . however , other scaling method that takes into account the excluded volume effects and nonlinear elasticity of polyelectrolyte chains and is in agreement with experiment and simulation , predicts a linear dependence of the brush thickness on the grafting density @xcite . also , it has been shown that diffusion of a fraction of counterions outside the polyelectrolyte brush leads to a logarithmic dependence of the average brush thickness on the grafting density @xcite . electrostatic interactions in a polyelectrolyte brush cause most of the counterions to be trapped inside the brush and help the chains to be more stretched and the brush to be more aligned . however , in a brush of overally neutral polyampholyte chains , most of counterions are outside the brush and the electrostatic correlations tend to decrease the chains size and the brush thickness . at a given value of the grafting density , the average thickness and equilibrium properties of such a polyampholyte brush are mainly determined by the chains properties such as fraction and sequence of charged monomers and the bending energy . brushes formed by grafted diblock polyampholytes have been investigated by lattice mean field modeling @xcite and computer simulation @xcite . the effect of chain stiffness , charge density and grafting density on spherical brushes of diblock polyampholytes and interaction between colloids with grafted diblock polyampholytes have been studied using monte carlo simulations @xcite . also , using molecular dynamics ( md ) simulations , the effects of various parameters such as charged monomers sequence , grafting density and salt concentration on the average thickness and equilibrium conformations of planar semiflexible polyampholyte brushes have been investigated @xcite . in this paper , we study planar brushes of flexible , semiflexible and rodlike diblock polyampholytes using md simulations in a wide range of the grafting density . we find that in all cases the average brush thickness linearly depends on the grafting density regardless of the chains different flexibility . our results also show that the strength of this dependence is considerably weaker in the case of the brush of flexible polyampholytes than two other cases . despite mentioned same functionality obtained for the average thickness versus the grafting density for brushes of different polyampholytes we find that histograms of their equilibrium conformations are noticeably different . inter - chain correlations are too weak in the brush of flexible polyampholytes and the brush properties are dominantly determined by single chain behavior . in this case , dependence of the equilibrium conformation of the brush on the grafting density is very weak . in the cases of the brushes of semiflexible and rodlike polyampholytes however , because of the combination of electrostatic correlations and strong excluded volume effects , collective behavior of the chains is dominant and dependence of equilibrium conformations on the grafting density is strong . in these cases , we also observe separation of the anchored chains into two coexisting fractions . using a simple scaling method which is consistently applicable for the brush of flexible chains , we describe theoretically the linear dependence of the brush thickness on the grafting density . the rest of the paper is organized as follows . in section [ simulation ] we describe our model and simulation method in detail and present the results of md simulations . our scaling analysis to describe dependence of the average thickness of flexible diblock polyampholyte brushes on the grafting density is presented in section [ theory ] . in section [ discussion ] we conclude the paper and present a short discussion . in our simulations which are performed with the md simulation package espresso @xcite , each brush is modeled by @xmath0 diblock polyampholyte bead - spring chains of length @xmath1 ( 24 spherical monomers ) which are end - grafted onto an uncharged surface at @xmath2 . the positions of anchored monomers which are fixed during the simulation , form an square lattice on the grafting surface ( @xmath3 plane ) with lattice spacing @xmath4 in which @xmath5 is the grafting density of the chains . the fraction @xmath6 of the monomers of each chain are charged and the chains consist of an alternating sequence of charged and neutral monomers . each chain contains the same number of positively and negatively charged monomers with charges @xmath7 and @xmath8 respectively ( see fig . [ fig1 ] ) . excluded volume interaction between particles is modeled by a shifted lennard - jones potential , @xmath9 in which @xmath10 and @xmath11 are the usual lennard - jones parameters and the cutoff radius is @xmath12 . successive monomers of each chain are bonded to each other by a fene ( finite extensible nonlinear elastic ) potential @xcite , @xmath13 with bond strength @xmath14 and maximum bond length @xmath15 . bending elasticity of the chains is modeled by a bond angle potential , @xmath16 in which @xmath17 is the angle between two successive bond vectors and @xmath18 is the bending energy of the chains . the value of the persistence length , @xmath19 , of the chains depends on the value of @xmath18 as @xmath20 . to model brushes of flexible , semiflexible and rodlike chains , we use four different values of @xmath18 namely @xmath21 , @xmath22 , @xmath23 and @xmath24 respectively . the simulation box is of volume @xmath25 in which @xmath26 is the box width in @xmath27 and @xmath28 directions and @xmath29 is its height in @xmath30 direction and the grafting density is given by @xmath31 . we consider @xmath32 monovalent counterions to neutralize the chains charge . positive and negative monovalent counterions are modeled by equal number of spherical lennard - jones particles of diameter @xmath11 with charges @xmath7 and @xmath8 respectively . all the particles interact repulsively with the grafting surface at short distances with the shifted lennard - jones potential introduced in eq . in addition , a similar repulsive potential is applied at the top boundary of the simulation box and in our simulations @xmath33 . all the charged particles interact with each other with the coulomb interaction @xmath34 in which @xmath35 and @xmath36 are charges of particles @xmath37 and @xmath38 in units of elementary charge @xmath7 and @xmath39 is separation between them . the bjerrum length , @xmath40 , which determines the strength of the coulomb interaction relative to the thermal energy , @xmath41 , is given by @xmath42 , where @xmath43 is the dielectric constant of the solvent and we set @xmath44 in our simulations . periodic boundary conditions are applied only in two dimensions ( @xmath27 and @xmath28 ) . to calculate coulomb forces and energies , we use the so - called @xmath45 technique introduced by strebel and sperb @xcite and modified for laterally periodic systems ( @xmath46 ) by arnold and holm @xcite . the temperature in our simulations is kept fixed at @xmath47 using a langevin thermostat . for each value of the bending energy , @xmath18 , we do simulations of the brush at dimensionless grafting densities @xmath48 . in the beginning of each simulation , all of the chains are straight and perpendicular to the grafting surface and all the ions are randomly distributed inside the simulation box . we equilibrate the system for @xmath49 md time steps which is enough for all values of the grafting density mentioned above and then calculate thermal averages over @xmath50 independent configurations of the system selected from @xmath51 additional md steps after equilibration . md time step in our simulations is @xmath52 in which @xmath53 is the md time scale and @xmath54 is the mass of the particles . we calculate the average brush thickness which can be measured by taking the first moment of the monomer density profile @xmath55 in which @xmath56 is the number density of monomers as a function of the distance from the grafting surface . for a better monitoring of the statistics of the chains conformations , we calculate the histogram of the mean end - to - end distance of the chains , @xmath57 , in which @xmath58 and @xmath59 is the end - to - end vector of chain @xmath37 . we also calculate the histogram of the average distance of the end monomers of the chains from the grafting surface , @xmath60 , in which @xmath61 and @xmath62 is the @xmath30 component of the end monomer of chain @xmath37 . with the same method discussed in ref . @xcite , it has been checked that our results are not affected by finite - size effects ( see sec . [ discussion ] ) . the average thickness versus the grafting density for brushes of flexible , semiflexible and rodlike diblock polyampholytes are shown in fig . dependence of the average thickness on the grafting density can be described well by a linear function for all values of the bending energy , @xmath18 , that we use in our simulations . also , it can be seen that at all values of the grafting density the average brush thickness decreases with increasing the flexibility of the chains . linear fits to the average thickness of the brushes of semiflexible and rodlike chains for which the persistence length , @xmath19 , exceeds their contour length , @xmath63 , are of approximately the same slope ( see solid and dashed lines in fig . [ fig2 ] ) . the slopes of the linear dependence in cases of two brushes of flexible chains are also approximately the same and differ by a factor of @xmath64 from those of two other cases ( dotted and dash - dotted lines in fig . [ fig2 ] ) . to analyze such dependencies of the brushes thickness on the grafting density , we look at the equilibrium conformations statistics of the chains at different values of @xmath5 . in figs . [ fig3 ] and [ fig4 ] the histograms @xmath57 and @xmath60 at three different grafting densities are shown for brushes of flexible , semiflexible and rodlike chains . because of very similar behaviors of the histograms in @xmath65 and @xmath66 cases , the histograms of the brush with @xmath65 are not shown for clarity of the figures . as it can be seen in fig . [ fig3 ] , in the case of the brush of flexible diblock polyampholytes ( @xmath67 ) , dependence of the histogram profile on the grafting density is very weak . also , in this case , the contribution of large values of @xmath39 ( @xmath68 ) in the histogram is negligible which shows that the chains are mostly coiled at all grafting densities ( see a sample configuration of the brush at @xmath69 in fig . [ fig1]a ) . in the case of the brush of semiflexible chains ( @xmath70 ) , with increasing the grafting density , the values of the histogram corresponding to smaller values of @xmath39 become nonzero showing that polyampholyte chains take buckled conformations at high grafting densities @xcite . as it is expected , the histogram @xmath57 of the brush of rodlike chains ( @xmath71 ) exhibit no noticeable buckling of the chains . the histograms @xmath60 in fig . [ fig4 ] show that in the brush of flexible chains at all grafting densities the end monomers are mostly distributed near the grafting surface showing that the positive blocks of the chains are mostly turned back towards the anchored negative blocks . the profile of this histogram also does nt depend noticeably on the grafting density . the histograms @xmath60 for brushes of semiflexible and rodlike chains show that two maxima appear and their height increase with increasing the grafting density . by combining the information obtained from histograms @xmath57 and @xmath60 for the brush of rodlike chains it can be understood that at high grafting densities a fraction of the chains which are perpendicular to the grafting surface coexist with the remaining fraction which fluctuate in the vicinity of the grafting surface . in the case of the brush of semiflexible chains also , the fraction of perpendicular chains to the brush surface coexist with the chains which are buckled towards the grafting surface @xcite . histograms @xmath57 and @xmath60 show that conformations of the brushes of semiflexible and rodlike chains change noticeably with changing the grafting density despite the case of the brush of flexible chains . as mentioned in the previous section , our md simulations show that the brush thickness is a linear function of the grafting density regardless of different values of grafted diblock polyampholytes bending energy . also , as it is shown in fig . [ fig2 ] , in cases of the brushes of flexible chains the slope of the fitted line to the average thickness versus the grafting density is considerably smaller than that in two other cases . we use here a simple scaling theory similar to that of the solution of charge - symmetric diblock polyampholytes @xcite to describe the linear dependence of the brush thickness on the grafting density . this scaling analysis is applicable to the brushes of flexible diblock polyampholytes . consider @xmath72 flexible diblock polyampholyte chains end - grafted to a flat surface of area @xmath73 . the degree of polymerization of each chain and the fraction of charged monomers are @xmath74 and @xmath75 respectively . electrostatic attractive interactions between oppositely charged blocks of the chains lead them to form a dense layer of positive and negative monomers of average thickness @xmath76 near the grafting surface ( see fig . [ fig5 ] ) . inside the layer , to use the blob concept we define correlation length @xmath77 as a length scale that electrostatic interactions do nt perturb the chains statistics at smaller length scales and are dominant over thermal fluctuations at larger length scales . let suppose that the number of monomers inside the correlation blob is @xmath78 and we have @xmath79 where @xmath80 and @xmath81 are the kuhn length and the flory exponent respectively . accordingly , the layer is a melt of correlation blobs in which positive blobs with a high probability are surrounded by negative blobs . electrostatic interaction between any two neighboring blobs is of the order of the thermal energy , @xmath41 @xmath82 thus we obtain @xmath83 . as a result , the local monomer concentration inside the blob is @xmath84 considering that correlation blobs are space filling , the global monomer concentration , @xmath85 , approximately equals to the local monomer concentration , @xmath86 . thus , dependence of the thickness of a brush of diblock polyampholytes on the grafting density can be obtained as @xmath87 the main prediction of this equation in the range of its validity is as follows . the brush thickness , @xmath76 , is a linear function of the grafting density , @xmath5 , irrespective of the values of the system parameters . only the slope of this linear function , @xmath88 , depends on the values of the system parameters . one should note here that despite the bulk solution of flexible diblock polyampholytes , in a polyampholyte brush the fact that the chains are end grafted to the surface introduces an additional length scale to the system , namely @xmath89 . in the scaling analysis presented here , if the size of the correlation length , @xmath77 , exceeds the separation between the grafting points , @xmath90 , the scaling approach becomes inconsistent . although linear dependence of the brush thickness on the grafting density is observed at all values of the bending energy used in our simulations , the condition @xmath91 is valid only in the case of the brush of flexible chains with @xmath65 . in this case the values of @xmath92 corresponding to lowest and highest values of the grafting density we have used in our simulations are 0.19 and 0.45 respectively . for the set of parameters used in our simulations , in the case of the brush with @xmath65 the upper limit of the validity of the scaling method ( @xmath93 ) corresponds to the grafting density @xmath94 which is quite far from our range of the grafting density . for a brush of flexible chains with @xmath65 using the values @xmath95 and @xmath96 gives the slope @xmath97 for the brush thickness versus the grafting density for the used set of system parameters . as it is shown in fig . [ fig2 ] , the value of @xmath98 obtained from our simulations of flexible chains is @xmath99 which is in reasonable agreement with prediction of the scaling method . our scaling method is not consistently applicable for the brush of flexible chains with @xmath100 because the range of the grafting density used in our simulations is higher than the validity range of the condition @xmath91 in this case . this method is not also applicable for the brushes of semiflexible and rodlike chains because in these cases the kuhn length exceeds the contour length of the chains . the fact that the average thickness linearly depends on the grafting density at all values of @xmath18 used in our simulations shows that although the scaling method presented here can not be used to describe all the simulation results , this linear dependence persists over a wide range of the system parameters . brushes of flexible , semiflexible and rodlike diblock polyampholytes have been studied using md simulations and a scaling analysis has been presented to describe the results of the simulation of flexible chins brush . the average thickness as a function of the grafting density and histograms of equilibrium conformations of the brushes are obtained . strong dependence of the system conformations on the grafting density and separation of the chains into two coexisting fractions at high grafting densities have been observed in cases of the brushes of semiflexible and rodlike chains . in cases of the brushes of flexible chains , single - chain behavior is dominant and dependence of the brush conformations on the grafting density is very weak . in spite of above mentioned differences , it has been observed that dependence of the average brush thickness on the grafting density is linear for brushes of all different chains . this linear dependence resulted from our md simulations has been described well using a simple scaling method in the case of the brush of flexible chains . brushes of polyelectrolytes and polyampholytes are dense assembly of these macromolecules in which the interplay between electrostatic correlations , strong excluded volume effects and bending elasticity of the chains determine equilibrium properties of the system . the main differences between brushes of polyelectrolyte and polyampholyte chains originates from opposite trends of inter- and intra - chain electrostatic interactions and different rules of counterions osmotic pressure in these brushes . in a brush of polyelectrolyte chains most of counterions are contained inside the brush and their osmotic pressure tends to increase the brush thickness . in a brush of overally neutral polyampholytes however counterions are outside the brush and have no effect on the brush thickness . linear dependence of polyelectrolyte brush thickness on the grafting density has theoretically been described @xcite . the results of a recent simulation of semiflexible polyampholytes @xcite and our simulations and theoretical analysis here show that linear dependence of the average thickness on the grafting density is also the case in the brush of polyampholyte chains . amount of the flexibility of the chains which control the strength of excluded volume effects and inter - chain correlations , causes the equilibrium conformations of the brushes of three different polyampholytes to be different . in brushes of semiflexible and rodlike polyampholytes , strong dependence of the equilibrium conformations on the grafting density and separation of the chains into two coexisting fractions at high grafting densities are resulted from strong electrostatic and excluded volume correlations between the chains . similar phenomenon has been observed in the brush of rodlike polyelectrolytes @xcite . to pay attention to possible finite - size effects in our simulations , we have calculated the average size of the chains lateral fluctuations as well as the histograms @xmath60 and @xmath57 for a larger brush containing @xmath101 semiflexible chains at the grafting density @xmath102 @xcite . the average size of lateral fluctuations is defined as @xmath103 , where @xmath104 and @xmath105 denotes averaging over equilibrium configurations . we have found that the histograms @xmath57 and @xmath60 do not change noticeably with increasing the system size @xcite . also , we have found that the value of @xmath106 is smaller than the lateral size , @xmath26 , of the simulation box of a brush containing @xmath0 chains at the same grafting density . this result shows that in simulation of the brush of @xmath0 polyampholytes , the chains do not overlap with their own images . accordingly , to avoid time consuming simulations of larger brushes , we concentrated on the simulations of the brushes containing @xmath0 chains . j. rhe , m. ballauff , m. biesalski , p. dziezok , f. grhn , d. johannsmann , n. houbenov , n. hugenberg , r. konradi , s. minko , m. motornov , r. r. netz , m. schmidt , c. seidel , m. stamm , t. stephan , d. usov , h. zhang , adv . sci . * 165 * , 79 ( 2004 ) .

planar brushes of flexible , semiflexible and rodlike diblock polyampholytes are studied using molecular dynamics simulations in a wide range of the grafting density . simulations show linear dependence of the average thickness on the grafting density in all cases regardless of different flexibility of anchored chains and the brushes different equilibrium conformations . slopes of fitted lines to the average thickness of the brushes of semiflexible and rodlike polyampholytes versus the grafting density are approximately the same and differ considerably from that of the brushes of flexible chains . the average thickness of the brush of flexible diblock polyampholytes as a function of the grafting density is also obtained using a simple scaling analysis which is in good agreement with our simulations .

arxiv

the light flavor scalar mesons present a remarkable exception for the naive quark model , and the structures of those mesons have not been unambiguously determined yet . the numerous candidates with @xmath1 below @xmath2 can not be accommodated in one @xmath3 nonet , some are supposed to be glueballs , molecular states and tetraquark states ( or their special superpositions ) @xcite . the @xmath4 and @xmath5 are good candidates for the @xmath6 molecular states @xcite , however , their cousins @xmath7 and @xmath8 lie considerably higher than the corresponding thresholds , it is difficult to identify them as the @xmath9 and @xmath10 molecular states , respectively . there may be different dynamics which dominate the @xmath11 mesons below and above @xmath12 respectively , and result in two scalar nonets below @xmath13 @xcite . the strong attractions between the diquark states @xmath14 and @xmath15 in relative @xmath16-wave may result in a nonet tetraquark states manifest below @xmath12 , while the conventional @xmath17 @xmath3 nonet have masses about @xmath18 , and the well established @xmath19 and @xmath20 @xmath3 nonets with @xmath21 and @xmath22 respectively lie in the same region . furthermore , there are enough candidates for the @xmath17 @xmath3 nonet mesons , @xmath23 , @xmath24 , @xmath25 , @xmath26 and @xmath27 @xcite . in the tetraquark scenario , the structures of the nonet scalar mesons in the ideal mixing limit can be symbolically written as @xcite @xmath28 the four light isospin-@xmath29 @xmath30 resonances near @xmath31 , known as the @xmath8 mesons , have not been firmly established yet , there are still controversy about their existence due to the large width and nearby @xmath30 threshold @xcite . in general , we may expect constructing the tetraquark currents and studying the nonet scalar mesons below @xmath12 as the tetraquark states with the qcd sum rules @xcite . for the conventional mesons and baryons , the `` single - pole + continuum states '' model works well in representing the phenomenological spectral densities , the continuum states are usually approximated by the contributions from the asymptotic quarks and gluons , the borel windows are rather large and reliable qcd sum rules can be obtained . however , for the light flavor multiquark states , we can not obtain a borel window to satisfy the two criteria ( pole dominance and convergence of the operator product expansion ) of the qcd sum rules @xcite . in ref.@xcite , t. v. brito et al take the quarks as the basic quantum fields , and study the scalar mesons @xmath7 , @xmath8 , @xmath5 and @xmath4 as the diquak - antidiquark states with the qcd sum rules , and can not obtain borel windows to satisfy the two criteria , and resort to a compromise between the two criteria . for the heavy tetraquark states and molecular states , the two criteria can be satisfied , but the borel windows are rather small @xcite . we can take the colored diquarks as point particles and describe them as the basic scalar , pseudoscalar , vector , axial - vector and tensor fields respectively to overcome the embarrassment @xcite . in this article , we construct the color singlet tetraquark currents with the scalar diquark fields , parameterize the nonperturbative effects with new vacuum condensates besides the gluon condensate , and perform the standard procedure of the qcd sum rules to study the nonet scalar mesons below @xmath12 . the qcd sum rules are `` new '' because the interpolating currents are constructed from the basic diquark fields instead of the quark and gluon fields . whether or not the colored diquarks can be taken as basic constituents is of great importance , because it provides a new spectroscopy for the mesons and baryons @xcite . the article is arranged as follows : we derive the new qcd sum rules for the nonet scalar mesons in sect.2 ; in sect.3 , we present the numerical results and discussions ; and sect.4 is reserved for our conclusions . in the following , we write down the interpolating currents for the nonet scalar mesons below @xmath12 , @xmath32 where @xmath33 the @xmath34 are color indices , the @xmath35 is the charge conjugation matrix , the @xmath36 , @xmath37 and @xmath38 are basic scalar diquark fields , while the @xmath39 , @xmath40 and @xmath41 are the corresponding scalar two - quark currents . in this article , we take the isospin limit for the @xmath42 and @xmath43 quarks , and denote the fields @xmath36 and @xmath37 as @xmath44 . for the general color antitriplet bilinear quark - quark fields @xmath45 , @xmath40 and @xmath41 are recovered . ] and color singlet bilinear quark - antiquark fields @xmath46 , where the flavor , color and spin indexes are not shown explicitly for simplicity , we can project them into a local and a nonlocal part , after bosonization , the two parts are translated into a basic quantum field and a bound state amplitude , respectively , @xmath47 where the @xmath48 and @xmath49 denote the diquark and meson fields respectively , the @xmath50 and @xmath51 denote the corresponding bethe - salpeter amplitudes respectively @xcite . in ref.@xcite , we study the structures of the pseudoscalar mesons @xmath52 , @xmath53 and the scalar diquarks @xmath54 , @xmath55 , @xmath56 in the framework of the coupled rainbow schwinger - dyson equation and ladder bethe - salpeter equation using a confining effective potential , and observe that the dominant dirac spinor structure of the bethe - salpeter amplitudes of the scalar diquarks is @xmath57 . if we take the local limit for the nonlocal bethe - salpeter amplitudes , the dimension-1 scalar diquark fields @xmath54 , @xmath55 and @xmath56 are proportional to the dimension-3 scalar two - quark currents @xmath58 , @xmath59 and @xmath60 , respectively . a dimension-1 quantity @xmath61 can be introduced to represent the hadronization @xmath62 , @xmath63 and @xmath64 . the attractive interaction of one - gluon exchange favors formation of the diquarks in color antitriplet @xmath65 , flavor antitriplet @xmath66 and spin singlet @xmath67 @xcite . lattice qcd studies of the light flavors indicate that the strong attraction in the scalar diquark channels favors the formation of good diquarks , the weaker attraction ( the quark - quark correlation is rather weak ) in the axial - vector diquark channels maybe form bad diquarks , the energy gap between the axial - vector and scalar diquarks is about @xmath68 of the @xmath69-nucleon mass splitting , i.e. @xmath70 @xcite , which is also expected from the hypersplitting color - spin interaction @xmath71 @xcite . on the other hand , the studies based on the random instanton liquid model indicate that the instanton induced quark - quark interactions are weakly repulsive in the vector and axial - vector channels , strongly repulsive in the pseudoscalar channel , and strongly attractive in the scalar and tensor channels @xcite . so it is sensible to use the scalar diquark fields to construct the tetraquark currents . the two - point correlation functions @xmath72 can be written as @xmath73 denotes @xmath74 , @xmath75 , @xmath76 and @xmath77 . we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators @xmath78 into the correlation functions @xmath79 to obtain the hadronic representation @xcite . isolating the ground state contributions from the pole terms of the nonet scalar mesons , we obtain the results , @xmath80 where the @xmath81 are the ground state masses , the @xmath82 are corresponding pole residues defined by @xmath83 , the thresholds @xmath84 in the channels @xmath85 , @xmath8 , @xmath7 respectively , the @xmath86 are the thresholds for the higher resonances and continuum states @xmath87 , and the @xmath88 are the corresponding hadronic spectral densities . we introduce the following new lagrangian @xmath89 , @xmath90 where @xmath91 , and carry out the operator product expansion with the basic diquark fields @xmath92 , @xmath93 and @xmath16 instead of the quark fields @xmath42 , @xmath43 and @xmath94 , and we have neglected the terms concerning the heavy diquark fields in the lagrangian . in the qcd , the basic quantum fields are the quark and gluon fields , the attractive interactions in the color antitriplet @xmath65 , flavor antitriplet @xmath95 and spin singlet @xmath67 quark - quark channels favor formation of the scalar diquarks , we can absorb some effects of the quark - gluon interactions into the effective diquark masses , which are characterized by the correlation length @xmath96 . at the distance @xmath97 , the @xmath98 diquark state combines with one quark or one @xmath99 antidiquark to form a baryon state or a tetraquark state , while at the distance @xmath100 , the @xmath98 diquark states dissociate into asymptotic quarks and gluons gradually , the strength of the quark - quark correlations is very weak . just like the quarks , the diquarks have three colors , and can be gauged with the same @xmath0 color group to embody the residual quark - gluon interactions . in calculations , we take into account all diagrams like the typical ones shown in fig.1 , introduce new vacuum diquark condensates @xmath101 and @xmath102 besides the gluon condensate to parameterize the nonperturbative qcd vacuum , and consider the vacuum condensates up to dimension four . in the qcd sum rules , the high dimensional vacuum condensates are usually suppressed by large denominators or additional powers of the inverse borel parameters @xmath103 , the net contributions are very small . for example , in the present case the contributions of the dimension-6 vacuum condensates can be estimated as @xmath104 , which is a tiny quantity . if additional suppressions originate from the denominators are taken into account , the contributions are even smaller , and can be safely neglected . once the analytical results are obtained , then we can take the dualities below the thresholds @xmath105 and perform the borel transform with respect to the variable @xmath106 , finally we obtain the following sum rules , @xmath107 where the @xmath108 denote the qcd spectral densities @xmath109 , @xmath110 and @xmath111 , @xmath112\delta(s-\widehat{m}_{q / s}^2)\ , , \end{aligned}\ ] ] @xmath113 , @xmath114 , @xmath115 in the channels @xmath85 , @xmath8 , @xmath7 respectively , and the @xmath116 is the borel parameter . the threshold parameters @xmath117 are different from the corresponding @xmath118 in eq.(6 ) , because we absorb some qcd interactions into the effective diquark masses . differentiate eq.(8 ) with respect to @xmath103 , then eliminate the pole residues @xmath119 , we can obtain the sum rules for the masses of the nonet scalar mesons , @xmath120 we estimate the vacuum diquark condensates @xmath101 and @xmath102 with the assumption of the vacuum saturation , which works well in the large @xmath121 limit , @xmath122 where the @xmath61 is a quantity has dimension of mass and can be taken as the confinement energy scale @xmath123 . at the energy scale @xmath124 , @xmath125 , @xmath126 , @xmath127 @xcite , and @xmath128 . the quark condensates play a special role being responsible for the spontaneous breaking of the chiral symmetry , and relate with the masses and decay constants of the light pseudoscalar mesons through the gell - mann - oakes - renner relation @xcite . the values of other vacuum condensates , such as the mixed condensates , the four quark condensates and the gluon condensates , can not be obtained from the first principles , we usually calculate them with the lattice qcd , the instanton models , or determine them empirically by fitting the qcd sum rules to the experimental data . in this article , we introduce the diquark condensates @xmath101 and @xmath102 to parameterize the nonperturbative qcd vacuum , and assume that they relate with the four quark condensates ( therefore they have implicit relations with the spontaneous breaking of the chiral symmetry ) , and take the dimension - one parameter @xmath61 to be the confinement energy scale , as the scalar mesons @xmath5 , @xmath4 , @xmath8 and @xmath7 are bound states which consist of the confined quarks and gluons , and some parameters are needed to embody the confinement . on the other hand , we can understand the parameter @xmath123 as a fitted value , which happens to be the confinement energy scale , the crude estimation works well . we take the updated values of the diquark masses from the qcd sum rules for consistency , where the interpolating currents @xmath39 , @xmath40 and @xmath41 are used @xcite , @xmath129 and @xmath130 ; the scalar diquarks were originally studied with the qcd sum rules about twenty years ago @xcite . there have been several theoretical approaches to estimate the diquark masses , for example , the simple constituent diquark mass plus hyperfine spin - spin interaction model @xcite . the @xmath5 and @xmath4 are well established , and the existence of the @xmath7 meson is confirmed , although there are controversy about its mass and width , the values listed in the review of particle physics are @xmath131 and @xmath132 respectively @xcite . as far as the @xmath8 are concerned , there are still controversy about their existence , we take them as the @xmath16-wave isospin-@xmath29 @xmath30 resonance with the breit - wigner mass about @xmath133 . the e791 collaboration observed a low - mass scalar @xmath30 resonance with the breit - wigner mass @xmath134 and width @xmath135 respectively in the decay @xmath136 @xcite , and the bes collaboration observed a clear low mass enhancement in the invariant @xmath30 mass distribution in the decay @xmath137 with the breit - wigner mass @xmath138 and width @xmath139 , respectively @xcite . recently , the bes collaboration reported the charged @xmath8 in the decay @xmath140 with the breit - wigner mass @xmath141 and width @xmath142 , respectively @xcite . it is sensible to estimate @xmath143 . on the other hand , the qcd sum rules for the tetraquark states indicate that @xmath144 and @xmath145 @xcite . assuming the energy gap between the ground and first radial excited tetraquark states is about @xmath146 , we can tentatively determine the threshold parameters @xmath147 , @xmath148 , and @xmath149 . the convergence behavior of the operator product expansion is very good , the contributions from the different terms have the hierarchy : perturbative - term @xmath150 . in calculation , we take uniform minimum value for the borel parameters @xmath151 . the perturbative continuum @xmath152 is suppressed by the factor @xmath153 , the contributions from the pole terms are very large , see fig.2 . in this article , we take uniform maximum value for the borel parameters , @xmath154 , the contributions from the pole terms are about @xmath155 , @xmath156 and @xmath157 in the channels @xmath158 , @xmath8 and @xmath7 , respectively . the two criteria ( pole dominance and convergence of the operator product expansion ) of the qcd sum rules are well satisfied . if the conventional quark currents are chosen , the multiquark states i.e. tetraquark states , pentaquark states , hexaquark states , etc , have the spectral densities @xmath159 with the largest @xmath160 , the integral @xmath161 converges slowly @xcite . if one do nt want to release the criterion of pole dominance , we have to either postpone the threshold parameter @xmath105 to very large value or choose very small value for the borel parameter @xmath162 . with large value of the threshold parameter @xmath105 , for example , @xmath163 , here @xmath164 stands for the ground state , the contributions from the high resonance states and continuum states are included in , we can not use single - pole ( or ground state ) approximation for the spectral densities ; on the other hand , with very small value of the borel parameter @xmath162 , the borel window @xmath165 shrinks to zero or very small values . the numerical values of the masses and pole residues are presented in table 1 and figs.3 - 4 . from table 1 , we can see that the present predictions are compatible with ( or not in conflict with ) the experimental data @xcite and theoretical estimations @xcite . the scalar tetraquark currents @xmath166 , @xmath76 and @xmath77 maybe have non - vanishing couplings with the scattering states @xmath9 , @xmath167 , @xmath30 , @xmath168 , @xmath169 , @xmath170 , etc , for example , @xmath171 if the couplings denoted by the @xmath172 and @xmath173 are strong enough , the contaminations from the continuum states are expected to be large . in the following , we study the contributions of the intermediate pseudoscalar meson loops to the correlation function @xmath174 in details as an example , @xmath175 where @xmath176\left [ ( p - q)^2-m_{\pi}^2\right ] } \ , , \nonumber\\ \sigma_{kk}(p)&=&\int~{d^4q\over(2\pi)^4}\frac{1}{\left [ q^2-m_{k}^2\right]\left [ ( p - q)^2-m_{k}^2\right ] } \ , , \end{aligned}\ ] ] the @xmath177 , @xmath178 are the strong coupling constants between the @xmath5 and the pseudoscalar meson pairs @xmath9 , @xmath167 respectively , and the @xmath179 , @xmath180 , @xmath181 , @xmath182 are the scattering amplitudes among the pseudoscalar meson pairs @xmath9 and @xmath167 . the couplings @xmath183 and @xmath173 are complicated functions of the @xmath184 , @xmath185 , @xmath186 , @xmath179 , @xmath187 , @xmath181 , @xmath188 , @xmath177 and @xmath178 , the explicit expressions are difficult to obtain . we should bear in mind that the intermediate meson loops contribute a self - energy to the scalar meson @xmath5 , and therefore the scalar meson @xmath5 develops a breit - wigner width . in fact , the scalar mesons @xmath85 , @xmath8 and @xmath7 below @xmath12 can be generated dynamically from the unitaried scattering amplitudes of the pseudoscalar mesons @xcite . we can take into account those meson loops effectively by taking the following replacement for the hadronic spectral density , @xmath189 here we neglect the complicated renormalization procedure , take the physical values , and ignore the energy scale dependence of the mass @xmath184 and pole residue @xmath186 for simplicity ; the approximation works well . the qcd sum rules for other scalar mesons are treated with the same routine . in ref.@xcite , dai et al perform the renormalization procedure in details to take into account the contributions from the continuum states . the widths listed in the review of particle physics are @xmath190 , @xmath191 , @xmath192 and @xmath193 , respectively @xcite . taking into account the finite widths , we can obtain the modified masses from the corresponding qcd sum rules , which are shown in table 1 and fig.5 . in calculation , we observe that the narrow width @xmath194 modifies the mass @xmath195 slightly and the effect can be neglected safely , while the broad widths @xmath196 and @xmath197 reduce the masses @xmath198 and @xmath199 about @xmath200 and @xmath201 , respectively . comparing with the experimental data @xcite , the modified masses are better . the scalar tetraquark currents @xmath166 , @xmath76 and @xmath77 maybe also have non - vanishing couplings with the higher resonances @xmath23 , @xmath24 , @xmath25 , @xmath26 and @xmath27 , which are supposed to be the @xmath202 @xmath3 states or glueballs , the couplings should be very small . furthermore , the threshold parameters are @xmath203 , @xmath204 and @xmath205 in the channels @xmath85 , @xmath8 and @xmath7 , respectively , the contaminations should be very small . . the masses and the pole residues of the nonet @xmath11 tetraquark states . the star denotes the modified masses from the sum rules where the finite widths shown in the bracket are taken into account in the hadronic spectral densities . [ cols="^,^,^",options="header " , ] , @xmath206 and @xmath35 correspond to the channels @xmath158 , @xmath8 and @xmath7 respectively . , width=302 ] , @xmath206 and @xmath35 correspond to the channels @xmath158 , @xmath8 and @xmath7 respectively . , width=302 ] , @xmath206 and @xmath35 correspond to the channels @xmath158 , @xmath8 and @xmath7 respectively . , width=302 ] , @xmath206 and @xmath35 correspond to the channels @xmath158 , @xmath8 and @xmath7 respectively . the values in the bracket denote the finite widths in the hadronic spectral densities . , width=302 ] in this article , we take the scalar diquarks as the point particles and describe them as the basic quantum fields , then introduce the @xmath0 color gauge interactions , and construct the tetraquark currents which consist of the scalar fields to study the nonet scalar mesons as tetraquark states with the new qcd sum rules . the numerical values are compatible with ( or not in conflict with ) the experimental data and theoretical estimations . comparing with the conventional quark currents , the diquark currents have the outstanding advantage to satisfy the two criteria of the qcd sum rules more easily , the new sum rules can be extended to study other multiquark states . this work is supported by national natural science foundation of china , grant number 11075053 , and the fundamental research funds for the central universities . j. weinstein and n. isgur , phys . * 48 * ( 1982 ) 659 ; f. e. close , n. isgur and s. kumana , nucl . phys . * b389 * ( 1993 ) 513 ; n. n. achasov , v. v. gubin and v. i. shevchenko , phys . rev . * d56 * ( 1997 ) 203 . z. g. wang , eur . j. * c62 * ( 2009 ) 375 ; z. g. wang , eur . j. * c63 * ( 2009 ) 115 ; r. d. matheus , s. narison , m. nielsen and j. m. richard , phys . * d75 * ( 2007 ) 014005 ; z. g. wang , eur . phys . j. * c67 * ( 2010 ) 411 .

in this article , we take the scalar diquarks as point particles and describe them as basic quantum fields , then introduce the @xmath0 color gauge interaction and new vacuum condensates to study the nonet scalar mesons as tetraquark states with the qcd sum rules . comparing with the conventional quark currents , the diquark currents have the outstanding advantage to satisfy the two criteria of the qcd sum rules more easily . + zhi - gang wang + department of physics , north china electric power university , baoding 071003 , p. r. china pacs numbers : 12.38.lg ; 13.25.jx ; 14.40.cs * key words : * nonet scalar mesons , qcd sum rules

arxiv

ime - domain simulation is an important approach for power system dynamic analysis . however , the complete system model , or interchangeably the long - term stability model , typically includes different components where each component requires several differential and algebraic equations ( dae ) to represent , at the same time , these dynamics involve different time scales from millisecond to minute . as a result , the total number of dae of a real power system can be formidably large and complex such that time domain simulation over long time intervals is expensive@xcite . these constraints are even more stringent in the context of on - line stability assessment . intense efforts have been made to accelerate the simulation of long - term stability model . one approach is to use a larger time step size to filter out the fast dynamics or use automatic adjustment of step size according to system behavior in time - domain simulation @xcite@xcite@xcite from the aspect of numerical method . another approach is to implement the quasi steady - state ( qss ) model in long - term stability analysis @xcite@xcite from the aspect of model approximation . nevertheless , the qss model suffers from numerical difficulties when the model gets close to singularities which were addressed in @xcite-@xcite . moreover , the qss model can not provide correct approximations of the long - term stability model consistently as numerical examples shown in @xcite@xcite . in addition , sufficient conditions of the qss model were developed in @xcite which pointed to a direction to improve the qss model . as a result , the qss model requires improvements in both model development and numerical implementation . this paper contributes to the latter one . in this paper , we apply pseudo - transient continuation ( @xmath0 ) which is a theoretical - based numerical method in power system long - term stability analysis . pseudo - transient continuation method can be implemented directly in the long - term stability model to accelerate simulation speed compared with conventional implicit integration method . on the other hand , the method can also be applied in the qss model to overcome possible numerical difficulties due to good stability property . this paper is organized as follows . section [ sectiondyptc ] briefly reviews general pseudo - transient continuation method in dae system . section [ sectionptcinpowersystem ] includes a introduction about power system models followed by implementation of pseudo - transient continuation method in the long - term stability model and the qss model respectively . section [ sectionnumerical ] presents three numerical examples to show the feasibility of the method . and conclusions are stated in section [ sectionconclusion ] . pseudo - transient continuation is a physically - motivated method and can be used in temporal integration . the method follows the solution of dynamical system accurately in early stages until the steady state is approaching . the time step is thereafter increased by sacrificing temporal accuracy to gain rapid convergence to steady state @xcite . if only the steady state of a dynamical system instead of intermediate trajectories is of interest , pseudo - transient continuation method is a better choice than accurate step - by - step integration . on the other hand , compared with methods that solve nonlinear equations for steady state such as line - search and trust region methods , pseudo - transient continuation method can avoid converging to nonphysical solutions or stagnating when the jacobian matrix is singular . this is particularly the case when the system has complex features such as discontinuities which exist in power system models . therefore , @xmath0 method can be regarded as a middle ground between integrating accurately and calculating the steady state directly . @xmath0 method can help reach the steady state quickly while maintain good accuracy for the intermittent trajectories . for ode dynamics , sufficient conditions for convergence of @xmath0 were given in @xcite . the results were further extended the semi - explicit index - one dae system in @xcite . we recall the basic algorithm here . we consider the following semi - explicit index - one dae system : @xmath1 with initial value @xmath2 . here @xmath3 , @xmath4 , @xmath5^t \in \re^{n_1+n_2}$ ] , and @xmath6 where @xmath7 is a nonsingular scaling matrix . we assume the initial condition for ( [ dae ] ) is consistent , i.e. @xmath8 and seek to find the equilibrium point @xmath9 such that @xmath10 and satisfies @xmath11 . as stated before , the step - by - step integration is too time consuming if the intermediate states are not of interest . on the other hand , newton s method for @xmath12 alone usually fails as the initial condition is not sufficiently near the equilibrium point . the @xmath0 procedure is defined by the iteration : @xmath13 where @xmath14 is adjusted to efficiently find @xmath9 rather than to enforce temporal accuracy . the convergence results in @xcite@xcite assume that the time step is updated with `` switched evolution relaxation''(ser ) : @xmath15 the algorithm is shown as below : set @xmath16 and @xmath17 . evaluate @xmath18 . while @xmath19 is too large . solve @xmath20 . set @xmath21 . evaluate @xmath18 . update @xmath22 according to ( [ delta ] ) . step 2.a is a newton step which is typically solved by an iterative method which terminates on small linear residuals while it may also be solved by inexact newton iteration . note that the failure of @xmath0 usually can be well signaled by reaching the bound on the total number of iterations @xcite . the convergence of @xmath0 for smooth @xmath23 was proved in @xcite under the assumptions that the dae has index one in a certain uniform sense , that it has a global solution in time , and that the solution converges to a steady state . the result were further extended to nonsmooth @xmath23 in @xcite with @xmath24 in ( [ ptc ] ) replaced by a generalized derivative . next we explain why @xmath0 has a better stability property . firstly , conventional integration methods insist on a small norm of the linear residual at each step and will either converge , diverge to infinity , or stagnate at a point where iteration matrix is singular . however , @xmath0 method will accept an increase in the residual , responding to that increase by decreasing @xmath22@xcite . in addition , @xmath0 stems from backward euler method which is an attractive choice when stability is the desired property instead of accuracy . one may think of @xmath0 method as a predictor - corrector method where the simple predictor is from previous time step and the corrector is backward euler with newton iteration @xcite . to see this , consider the implicit euler step from @xmath25 with @xmath26 , @xmath27 thus , @xmath28 is the root of the following equation : @xmath29 the newton s method to find the root of the above equation is : @xmath30 if we take @xmath31 , then by binomial inverse theorem , the first newton iterate is : @xmath32 which is exactly @xmath0 step . as @xmath0 method has a better stability property , it can be applied in the qss model when conventional integration method fails to converge . in this section , we firstly introduce power system models in long - term stability analysis . then we apply @xmath0 method to the long - term stability model with modifications . finally , we present an algorithm of @xmath0 method in the qss model to overcome possible numerical difficulties . the long - term stability model for calculating system dynamic response relative to a disturbance can be described as : @xmath33 equation ( [ algebraic eqn ] ) describes the transmission system and the internal static behaviors of passive devices , and ( [ fast ode ] ) describes the internal dynamics of devices such as generators , their associated control systems , certain loads , and other dynamically modeled components . @xmath34 and @xmath35 are continuous functions , and vector @xmath36 and @xmath37 are the corresponding short - term state variables and algebraic variables . besides , equations ( [ slow ode ] ) and ( [ slow dde ] ) describe long - term dynamics including exponential recovery load , turbine governor , load tap changer ( ltc ) , over excitation limiter ( oxl ) , etc . @xmath38 and @xmath39 are the continuous and discrete long - term state variables respectively , and @xmath40 is the maximum time constant among devices . note that shunt compensation switching and ltc operation are typical discrete events captured by ( [ slow dde ] ) and @xmath41 is shunt susceptance and the transformer ratio correspondingly . transitions of @xmath41 depend on system variables , thus @xmath41 change values from @xmath42 to @xmath43 at distinct times @xmath44 where @xmath45 , otherwise , these variables remain constants . since short - term dynamics have much smaller time constants compared with those of long - term dynamics , @xmath46 and @xmath41 are also termed as slow state variables , and @xmath47 are termed as fast state variables . if we represent the long - term stability model and the qss model in @xmath48 time scale , where @xmath49 , and we denote @xmath50 as @xmath51 , then the long - term stability model of power system can be represented as : @xmath52 where the study region @xmath53 , and @xmath54 , @xmath55 , @xmath56 , @xmath57 . at the same time , the qss model can be represented as : @xmath58 moreover , the long - term stability model ( [ complete ] ) can be regarded as two decoupled systems ( [ couple1 ] ) and ( [ couple2 ] ) showed as below when @xmath41 jump from @xmath42 to @xmath43 : @xmath59 and @xmath60 discrete variables @xmath41 are updated first and then system ( [ couple2 ] ) works with fixed parameters @xmath41 . similarly , when @xmath41 jump from @xmath42 to @xmath43 , the qss model ( [ qss ] ) can be decoupled as : @xmath61 and @xmath62 assuming @xmath63 is nonsingular , then the long - term stability model ( [ couple2 ] ) with @xmath41 fixed as parameters is a semi - explicit index-1 dae system . @xmath0 method requires initial condition to satisfy the algebraic equations , however , the discrete equation ( [ couple1 ] ) will violate this condition whenever it works . as a result , we need to modify the original @xmath0 method for its implementation in long - term stability model . in power system long - term stability model , @xmath64^t$ ] , @xmath65^t\in\re^{p+m+n}$ ] , @xmath66 , where @xmath67 is the identiy matrix of size @xmath68 . in order to make the initial condition of @xmath0 consistent , we switch back to implicit integration method whenever discrete variables jump and set the step length to be @xmath69 . moreover , @xmath0 method is implemented for the post - fault system starting from @xmath70several seconds after the contingency . in examples of this paper , @xmath70 was set to be @xmath71 . the proposed algorithm is shown as below . run the long - term stability model up to @xmath70 by implicit integration method . set the value @xmath72 at @xmath70 as the initial condition @xmath73 of @xmath0 , and set @xmath17 . while @xmath74 is too large . if discrete variables jump at @xmath44 update @xmath41 according to ( [ couple1 ] ) . set @xmath75 , @xmath17 . solve the newton step @xmath76 . set @xmath77 . evaluate @xmath78 . otherwise set @xmath79 . solve the newton step @xmath80 . set @xmath77 . evaluate @xmath78 . update @xmath22 according to ( [ delta ] ) . note that @xmath81 and @xmath82 depend on the specific integration method used . for instance , if implicit trapezoidal method is used , then @xmath83 @xmath84 assuming @xmath85 is nonsingular , then the qss model ( [ coupleqss2 ] ) with @xmath41 fixed as parameters is a semi - explicit index-1 dae system . @xmath0 method requires initial condition to satisfy the algebraic equations , however , the discrete equation ( [ coupleqss1 ] ) will violate this condition whenever it works . as a result , we need to modify the original @xmath0 method for its implementation in the qss model . in the qss model , @xmath64^t$ ] , @xmath65^t\in\re^{p+m+n}$ ] , @xmath86 , where @xmath67 is the identity matrix of size @xmath87 . in order to make the initial condition of @xmath0 consistent , we switch back to implicit integration method whenever discrete variables jump and set the step length to be @xmath69 . besides , the qss model is implemented at @xmath88when short - term dynamics settle down after the contingency . usually , @xmath88 can be set as 30s . the proposed algorithm is shown as below . run the long - term stability model up to @xmath88 by implicit integration method . set the value @xmath72 at @xmath88 as the initial condition @xmath73 of the qss model , and set @xmath17 . start to run the qss model . if the qss model has a numerical difficulty by using implicit integration method , then go to step 3 , otherwise , continue with the qss model . while @xmath74 is too large . if discrete variables jump at @xmath44 update @xmath41 according to ( [ coupleqss1 ] ) . set @xmath75 , @xmath17 . solve the newton step @xmath76 . set @xmath77 . evaluate @xmath78 . otherwise set @xmath79 . solve the newton step @xmath80 . set @xmath77 . evaluate @xmath78 . update @xmath22 according to ( [ delta ] ) . similarly , @xmath81 and @xmath82 depend on the specific integration method used . if implicit trapezoidal method is used , then @xmath89 @xmath90 in this section , three examples are to be presented . the first two examples were the same 145-bus system in which the qss model met numerical difficulties during simulation while the long - term stability model was stable in long - term time scale . firstly , @xmath0 method was implemented in the long - term stability model and the speed was more than 7 times faster than the trapezoidal method . secondly , when the qss model by trapezoidal method met difficulty , @xmath0 method was implemented in the qss model and provided correct approximations and the speed was still more than 5 times faster than the long - term stability model by trapezoidal method . and in the last example which was a 14-bus system , the long - term stability model was unstable and @xmath0 method successfully captured the instability which was signaled by reaching the bound of maximum iteration in the newton step . all simulations were done in psat-2.1.6@xcite . the system was a 145-bus system @xcite . there were exciters and power system stabilizers for each of generator 1 - 20 . and there were turbine governors for each of generator 30 - 40 . besides , there were three load tap changers at lines between bus 73 - 74 , bus 73 - 81 and bus 90 - 92 respectively . the contingency was a line loss between bus 1 - 6 . in this case , the post - fault system was stable after the contingency in the long - term time scale and it took 122.39s for the time domain simulation of the long - term stability model by implicit trapezoidal method . however , @xmath0 method only took 16.12s for the simulation of the long - term stability model which was about @xmath91 of the time consumed by trapezoidal method . [ tdsptc ] shows that the trajectories by @xmath0 method followed closely to the trajectories by trapezoidal method , and finally both converged to the same long - term stable equilibrium point . thus @xmath0 method provided correct approximations for the long - term stability model in terms of trajectories and stability assessment . method . @xmath0 method provided correct approximations.,width=172 ] method . @xmath0 method provided correct approximations.,width=172 ] method . @xmath0 method provided correct approximations.,width=172 ] method . @xmath0 method provided correct approximations.,width=172 ] in this example , the system was the same as the last case . however , the qss model met numerical difficulties at 40s when implicit trapezoidal method was used , thus @xmath0 method was implemented in the qss model starting from 40s . [ tdsptcqss ] shows that the trajectories by @xmath0 method converged to the long - term stable equilibrium point which the long - term stability model converged to , and also provided good accuracy for the intermittent trajectories . it took 21.75s for @xmath0 method which was about @xmath92 of the time consumed by the long - term stability model using implicit trapezoidal method . method . @xmath0 method overcame numerical difficulties and provided correct approximations.,width=172 ] method . @xmath0 method overcame numerical difficulties and provided correct approximations.,width=172 ] method . @xmath0 method overcame numerical difficulties and provided correct approximations.,width=172 ] method . @xmath0 method overcame numerical difficulties and provided correct approximations.,width=172 ] in this case , the 14-bus system was long - term unstable due to wild oscillations of fast variables . the system was modified based on the 14-bus test system in psat-2.1.6@xcite . there were three exponential recovery loads at bus 9 , 10 and 14 respectively and two turbine governors at generator 1 and 3 . besides , there were over excitation limiters at all generators and three load tap changers at lines between bus 4 - 9 , bus 12 - 13 and bus 2 - 4 . the system suffered from long - term instabilities and simulation by implicit trapezoidal method could not continue after 101.22s . @xmath0 method also stopped at 103.34s when the bound on the total number of iterations for the newton step was reached . thus @xmath0 method was able to capture instabilities of the long - term stability model . method . @xmath0 method was able to capture instabilities of the long - term stability model.,width=172 ] method . @xmath0 method was able to capture instabilities of the long - term stability model.,width=172 ] method . @xmath0 method was able to capture instabilities of the long - term stability model.,width=172 ] method . @xmath0 method was able to capture instabilities of the long - term stability model.,width=172 ] in this paper , modified @xmath0 methods for the long - term stability model and the qss model are given for power system long - term stability analysis with illustrative numerical examples . we make use of the fast asymptotic convergence of @xmath0 method in the long - term stability model to achieve fast simulation speed . on the other hand , we take advantage of good stability property of @xmath0 method in the qss model to overcome numerical difficulties . numerical examples show that @xmath0 can successfully provide correct stability assessment for the long - term stability model and overcome numerical difficulties in the qss model , as well as offer good accuracy for the intermediate trajectories . @xmath0 can be regarded as a good in - between method with respect to integration and steady state calculation , thus serves as an alternative method in power system long - term stability analysis . this work was supported by the consortium for electric reliability technology solutions provided by u.s . department no . de - fc26 - 09nt43321 . kurita , a. , h. okubo , k. oki , s. agematsu , d. b. klapper , n. w. miller , w. w. price jr , j. j. sanchez - gasca , k. a. wirgau , and t. d. younkins , _ multiple time - scale power system dynamic simulation _ , ieee transactions on power systems , vol 216 - 223 , 1993 . t. v. cutsem , m. e. grenier , d. lefebvre , _ combined detailed and quasi steady - state time simulations for large - disturbance analysis_. international journal of electrical power and energy systems , vol . 28 , issue 9 , pp . 634 - 642 , november 2006 . x. z. wang , h. d. chiang , _ numerical investigations on quasi steady - state model for voltage stability : limitations and nonlinear analysis_. submitted to international transactions on electrical energy systems . x. z. wang , h. d. chiang , _ analytical studies of quasi steady - state model in power system long - term stability analysis_. to appear in ieee transactions on circuits and systems i : regular papers . doi 10.1109/tcsi.2013.2284171 . c. t. kelley , d. e. keyes _ convergence analysis of pseudo - transient continuation_. siam journal on numerical analysis , vol . 508 - 523 , april 1998 . v. vittal , d. martin , r. chu , j. fish , j. c. giri , c. k. tang , f. eugenio villaseca , and r. g. farmer . _ transient stability test systems for direct stability methods_. ieee transactions on power systems , vol . 7 , no . 37 - 43 , feb . 1992

in this paper , pseudo - transient continuation method has been modified and implemented in power system long - term stability analysis . this method is a middle ground between integration and steady state calculation , thus is a good compromise between accuracy and efficiency . pseudo - transient continuation method can be applied in the long - term stability model directly to accelerate simulation speed and can also be implemented in the qss model to overcome numerical difficulties . numerical examples show that pseudo - transient continuation method can provide correct approximations for the long - term stability model in terms of trajectories and stability assessment . pseudo - transient continuation , long - term stability model , quasi steady - state model , long - term stability .

arxiv

in recent years , the problem of anisotropic quantum scattering in two spatial dimensions ( 2d ) attracts increasing interest . it is stimulated by the spectacular proposals for prospects to create exotic and highly correlated quantum systems with dipolar gases @xcite . particularly , there were considered anisotropic superfluidity @xcite , 2d dipolar fermions @xcite , and few - body dipolar complexes @xcite . the recent experimental production of ultracold polar molecules in the confined geometry of optical traps @xcite has opened up ways to realize these phenomena . noteworthy also is a rather long history of research of 2d quantum effects in condensed matter physics . one can note superfluid films @xcite , high - temperature superconductivity @xcite , 2d materials , such as graphene @xcite , and even possibilities for topological quantum computation @xcite . unique opportunities for modeling these 2d effects in a highly controlled environment have recently appeared with the development of experimental techniques for creating quasi-2d bose and fermi ultracold gases @xcite . interest in the processes and effects in 2d - geometry has stimulated the theory of elementary quantum two - body systems and processes in the plane . special consideration should be given to the anisotropy and long - range character of the dipole - dipole interaction . actually , usual partial - wave analysis becomes inefficient for describing the dipole - dipole scattering due to the strong anisotropic coupling of different partial - waves in the asymptotic region @xcite . recently , considerable progress in the analysis of the 2d and quasi-2d ( q2d ) scattering of dipoles has been achieved @xcite . thus , the 2d dipolar scattering in the threshold and semiclassical regimes was studied in the case of the dipole polarization directed orthogonally to the scattering plane @xcite . an arbitrary angle of polarization was considered in @xcite . in this work , we develop a method for quantitative analysis of the 2d quantum scattering on a long - range strongly anisotropic scatterer . particularly , it permits the description of the 2d collisions of unpolarized dipoles . our approach is based on the method suggested in @xcite for the few - dimensional scattering which was successfully applied to the dipole - dipole scattering induced by an elliptically polarized laser field in the 3d free - space @xcite . the key elements of the method are described in section ii . in section iii , we apply the method to the 2d scattering on the cylindrical potential with the elliptical base and the 2d dipole - dipole scattering of unpolarized dipoles . we reproduce the threshold formula @xcite for the scattering amplitude on the cylinder potential with the circular base and the results of @xcite for the 2d scattering of polarized dipoles . high efficiency of the method has been found in all problems being considered . the last section contains the concluding remarks . some important details of the computational scheme and illustration of the convergence are given in appendices . the quantum scattering on the anisotropic potential @xmath0 in the plane is described by the 2d schrdinger equation in polar coordinates @xmath1 @xmath2 with the scattering boundary conditions @xmath3 in the asymptotic region @xmath4 and the hamiltonian of the system @xmath5 the unknown wave function @xmath6 and the scattering amplitude @xmath7 are searched for the fixed momentum @xmath8 defined by the colliding energy @xmath9 ( @xmath10 and the direction @xmath11 of the incident wave ( defined by the angle @xmath12 and for the scattering angle @xmath13 . here @xmath14 is the reduced mass of the system . in the polar coordinates , the angular part of the kinetic energy operator in @xmath15 has a simple form @xmath16 . the interaction potential @xmath17 can be anisotropic in the general case , i.e. to be strongly dependent on @xmath13 . it is clear that varying the direction of the incident wave @xmath11 can be replaced by the rotation @xmath18 of the interaction potential by the angle @xmath19 for the fixed direction of the incident wave , which we choose to be coincident with the x - axis . thus , in the case of anisotropic potential @xmath17 the task is to solve the problem ( [ eq1 ] ) with the interaction potential @xmath20 for all possible @xmath19 and fixed @xmath9 with the scattering boundary conditions @xmath21 if the scattering amplitude @xmath7 is found , one can calculate the differential scattering cross section @xmath22 where @xmath23 , as well as the total cross section @xmath24 by averaging over all possible orientations @xmath19 of the scatterer and integration over the scattering angle @xmath13 . to integrate the problem ( [ eq1]),([eq2 ] ) , we use the method suggested in @xcite to solving a few - dimensional scattering problem and applied in @xcite for the dipole - dipole scattering in the 3d free - space . following the ideas of these works we choose the eigenfunctions @xmath25 of the operator @xmath26 as a fourier basis for the angular - grid representation of the searched wave - function @xmath27 . we introduce the uniform grid @xmath28 ) over the @xmath13 and @xmath19-variables and search the wave function as expansion @xmath29 where @xmath30 is the inverse matrix to the @xmath31 square matrix @xmath32 defined on the angular grid , we use the completeness relation for the fourier basis @xmath33 , which in our grid representation reads @xmath34 . ] . in the representation ( [ eq7 ] ) the unknown coefficients @xmath35 are defined by the values of the searched wave function on the angular grid @xmath36 , any local interaction is diagonal @xmath37 and the angular part @xmath38 of the kinetic energy operator has a simple form @xmath39 note that the presence in the interaction potential of the `` nonlocal '' angular part ( i.e. the integration or differentiation over angular variable ) leads to destroying the diagonal structure in ( [ eq8 ] ) . thus , the 2d schrdinger equation ( [ eq1 ] ) is reduced in the angular - grid representation ( [ eq7 ] ) to the system of coupled ordinary differential equations of the second order : @xmath40 since the wave function @xmath41 must be finite at the origin @xmath42 , the `` left - side '' boundary condition for the functions @xmath35 reads as @xmath43 in the asymptotic region @xmath4 the scattering boundary condition ( [ eq3 ] ) accepts the form @xmath44 by using the fourier expansion for the plane wave @xmath45 and the scattering amplitude @xmath46 are the first kind bessel functions of integer order . their asymptotic behavior @xcite : @xmath47{}\sqrt { \frac{2}{\pi z } } \cos \left ( { z-\frac{m\pi } { 2}-\frac{\pi } { 4 } } \right)+e^{\left| { imz } \right|}{\rm o}\left ( { \left| z \right|^{-1 } } \right),\\ \left ( { \left| { \arg ( z ) } \right|<\pi } \right)\end{aligned}\ ] ] ] @xmath48 @xmath49 we eliminate the angular dependence from the asymptotic equation ( [ eq12 ] ) and represent the `` right - side '' boundary condition for the functions @xmath50 in the form @xmath51 to solve the boundary - value problem ( [ eq10]),([eq11 ] ) and ( [ eq15 ] ) , we introduce the grid over the @xmath52 and reduce the system of differential equations ( [ eq10 ] ) by using the finite - difference approximation of the sixth order to the system of @xmath53 algebraic equations @xmath54 with the band - structure of the matrix @xmath55 with the width @xmath56 of the band . by using the asymptotic equations ( [ eq15 ] ) in the last two points @xmath57 and @xmath58 one can eliminate the unknown vector @xmath59 from equation ( [ eq15 ] ) and rewrite the `` right - side '' boundary condition in the form @xmath60 analogously , one can eliminate unknown constant from expression ( [ eq11 ] ) by considering asymptotic equations ( [ eq11 ] ) at the first points @xmath61 and @xmath62 . the acquired `` left - side '' boundary condition reads @xmath63 thus , the scattering problem is reduced to the boundary value problem ( [ my_eq15]-[my_eq17 ] ) @xmath64 which can be efficiently solved with standard computational techniques such as the sweeping method @xcite or the lu - decomposition @xcite . the detailed structure of the matrix of the coefficients @xmath65 is discussed in appendix a. after the solving of eq.([my_eq19 ] ) and finding the wave function @xmath66 the scattering amplitude @xmath67 is constructed according to eqs.([eq15 ] ) and ( [ eq14 ] ) . first , we have analyzed the 2d scattering on the cylindrical potential barrier with the elliptical base @xmath68 the case of the circular base @xmath69 was considered in @xcite , where analytic formula for the scattering amplitude @xmath70+i\frac{\pi } { 2}}\ ] ] was obtained at the zero - energy limit @xmath71 . here and @xmath72 is the euler constant . we have analyzed the scattering on the potential barrier with circular base @xmath73 for arbitrary momentum @xmath74 . the results of calculation presented in figs . [ fig1 ] and [ fig3 ] confirm the convergence of the scattering amplitude @xmath67 to the analytical value ( [ eq17 ] ) at @xmath75 . in this subsection all the calculations were performed in the units @xmath76 . in the limiting case of the infinitely high potential barrier ( [ eq16 ] ) with the circular base @xmath77 the asymptotic formula ( [ eq17 ] ) becomes exact for arbitrary @xmath74 . this is confirmed by investigation presented in table [ tab1 ] which illustrates the convergence of the numerical values @xmath78 with increasing ( @xmath79 ) and narrowing ( @xmath80 ) of the potential barrier to the analytic result ( [ eq17 ] ) . in the limit case @xmath79 and @xmath81 we obtain @xmath82 for the scattering length @xmath83 extracted from the calculated amplitude @xmath84 by the formula ( [ eq17 ] ) , what is in agreement to the estimate given in @xcite . the range of applicability of eq . ( [ eq17 ] ) was investigated recently in @xcite . [ cols="^,^,^,^,^ " , ] the boundary - value problem ( [ eq10]),([eq11 ] ) and ( [ eq15 ] ) , obtained in section ii in the angular - grid representation ( [ eq7 ] ) , reads in a matrix form as : @xmath85\bm { \psi } ( \rho ) \,+\\ \hspace{5 cm } + \,\hat{h}^{(0 ) } \bm { \psi } ( \rho ) = 0 \\ \bm { \psi } \xrightarrow[{\rho \to 0}]{}const\cdot \sqrt \rho \\ \frac{2\pi}{(2 m + 1)\sqrt \rho } \sum\limits_{j=0}^{2 m } { e ^{-im\phi_j } \psi _ j ( \rho ) } = i^mj_m ( q\rho ) \sqrt { 2\pi } + \\ \hspace{5 cm } + \,\frac{f_m(\phi_q)}{\sqrt { -i\rho } } e^{iq\rho } \,,\\ \end{array } } \right.\ ] ] where and @xmath86 . in this representation the angular dependence is built into the matrix @xmath87 and the interaction is included into the diagonal matrix @xmath88 of values of the potential in the angular grid nodes . the constant matrix @xmath87 couples all equations in a system and does not depend on the radial variable . there is no need to compute any matrix elements of the potential , what essentially minimizes the computational costs . for solving boundary value problem ( [ eq23 ] ) the seven - point finite - difference approximation for second derivatives of sixth order @xmath89 is applied in the points @xmath90 of the radial grid , where @xmath91 . as a result , the system ( [ eq23 ] ) reduces to the system of linear algebraic equations ( [ my_eq16 ] ) with the matrix @xmath55 whose band structure reads @xmath92 where the @xmath93 coefficients are the square @xmath94 matrices , and the elements @xmath95 of the right - side of ( [ eq25 ] ) are the @xmath96 dimensional vectors . after employing the `` right - side '' boundary condition in the form ( [ my_eq16 ] ) in the last three grid points @xmath97 and , analogously , the `` left - side '' boundary condition in the form ( [ my_eq17 ] ) in the first two points @xmath98 , the detailed structure of the matrix @xmath55 is represented as @xmath99 the block structure of the system ( [ eq26 ] ) provides several significant advantages . the block matrix can be stored in a packaged form , which allows the use of optimal resource . the system ( [ eq26 ] ) can be efficiently solved by a fast implicit matrix algorithm based on the idea of the block sweep method @xcite . in the table [ tab4 ] we illustrate the convergence of the calculated scattering amplitude @xmath67 over the number of angular grid points @xmath96 for the scatterers with weak and essential anisotropy at @xmath100 in the potential barrier ( [ eq16 ] ) . for the case @xmath101 we reach the accuracy of four significant digits in the scattering amplitude on the angular grids with @xmath102 . for stronger anisotropy @xmath103 the accuracy of two significant digits was reached at @xmath104 . 33 m. a. baranov , phys . rep . * 464 * , 71 ( 2008 ) . i. bloch , j. dalibard and w. zwerger , rev . phys . * 80 * , 885 ( 2008 ) . c. ticknor , r. m. wilson and j. l. bohn , phys . lett . * 106 * , 065301 ( 2011 ) . g. m. brunn and e. taylor , phys . * 101 * , 245301 ( 2008 ) . j. c. cremon , g. m. brunn and s. m. reimann , phys . lett . * 105 * , 255301 ( 2010 ) . al . , science * 322 * , 231 ( 2008 ) ; s.ospelkaus et . al . , science * 327 * , 853 ( 2010 ) . l. d. carr et . new j. phys * 11 * , 055049 ( 2009 ) . m. h. g. de miranda et . al . , nat . phys . * 7 * , 502 ( 2011 ) . p. minnhagen , rev . phys . * 59 * , 1001 ( 1987 ) . a. lee , n. nagaosa and x .- g . wen , rev . mod . phys . * 78 * , 17 ( 2006 ) . k. s. novoselov , rev . * 83 * , 837 ( 2011 ) . c. nayak , s. h. simon , a. stern , m. freedman and s. d. sarma , rev . 80 * , 1083 ( 2008 ) . k. martiyanov , v. makhalov and a. turlapov , phys . * 105 * , 030404 ( 2010 ) ; a. turlapov , jetp letters * 95 * , 96 ( 2012 ) . m. marinescu and l. you , phys . * 81 * , 4596 ( 1998 ) ; b. deb and l. you . a * 64 * , 022717 ( 2001 ) v. s. melezhik and chi - yu hu , phys . lett . * 90 * , 083201 ( 2003 ) . c. ticknor , phys . a * 80 * , 052702 ( 2009 ) . c. ticknor , phys . a * 84 * , 032702 ( 2011 ) . c. ticknor , phys . a * 81 * , 042708 ( 2010 ) . j. p. dincao and c. h. greene , phys . a * 83 * , 030702 ( 2011 ) . z. li , s. v. alyabishev and r. v. krems , phys . * 100 * , 073202 ( 2008 ) . v. s. melezhik , j. comput . phys . * 92 * , 67 - 81 ( 1991 ) . l. d. landau and e. m. lifshitz , in _ quantum mechanics : non - relativistic theory _ , vol . 3 ( pergamon press , 1977 ) 3rd ed . , chap .

[ txt : abstract ] we study the quantum scattering in two spatial dimensions ( 2d ) without the usual partial - wave formalism . the analysis beyond the partial - wave approximation allows a quantitative treatment of the anisotropic scattering with a strong coupling of different angular momenta nonvanishing even at the zero - energy limit . high efficiency of our method is demonstrated for the 2d scattering on the cylindrical potential with the elliptical base and dipole - dipole collisions in the plane . we reproduce the result for the 2d scattering of polarized dipoles in binary collisions obtained recently by ticknor [ phys . rev . a * 84 * , 032702 ( 2011 ) ] and explore the 2d collisions of unpolarized dipoles .

arxiv

within a decade of being identified as a supernova remnant ( snr ) , [email protected] ( also called g65.3 + 5.7 ) was the subject of several observational papers published in quick succession . however , once its radio and optical images and its optical and x - ray spectra were recorded , the remnant was ignored . now , a quarter of a century later , [email protected] has once again become interesting , this time because it helps to explain an unusual class of supernova remnants called thermal composite snrs ( also known as mixed morphology snrs ) . both of these terms describe the combination of a shell - like radio continuum morphology with a centrally - bright x - ray morphology in which the x - ray emission is due to thermal , rather than synchrotron , processes . compared with shell - type and plerionic snrs , these remnants have more mysterious origins . various workers have suggested that thermal composite snrs lack x - ray shells because they have evolved into the radiative phase ( and thus have cool , x - ray dim shells ) , or that their centers are x - ray bright because thermal conduction , other forms of entropy mixing , or cloudlet evaporation has enhanced the central densities . ejecta enrichment or dust destruction could increase the metallicities in the centers , also enhancing the x - ray emission . alternatively , a thermal composite morphology may be due to collisions with molecular clouds . interested readers are directed to @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , and @xcite for additional discussions of these processes . currently , there are few observations of the `` smoking gun '' for the radiative phase evolution hypothesis : h shells . the h shell on w44 is the only possible example ( @xcite , with interpretations in @xcite and @xcite ) . here we demonstrate that another thermal composite supernova remnant , [email protected] , is clearly in the radiative phase . ironically , this demonstration is not performed by discovering that a previously known thermal composite snr is in the radiative phase , but is done by demonstrating that a previously known radiative phase snr is a thermal composite . in section 2 , we discuss published optical observations , indicating that [email protected] has entered the shell formation phase ; we also discuss published radio and low spatial resolution x - ray observations , showing that [email protected] has a shell - like radio continuum morphology and emits thermal x - rays . with these characteristics , [email protected] meets two of the three requirements for the thermal composite classification . the third requirement , that the remnant has a centrally filled x - ray morphology , is established in section 3 , through the analysis of archival _ rosat _ observations . furthermore , model fits to the _ rosat _ data imply that the temperature peaks in the center and decreases mildly with radius , as is found for other thermal composite supernova remnants . in section 4 , we summarize the results . the first observations of [email protected] , those made with the red prints of the palomar sky survey @xcite , only found two bright filaments , s91 and s94 , but failed to recognize the entire outline of the remnant . the true extent of the remnant remained unknown until @xcite s optical emission line survey of the galactic plane revealed filaments circumscribing a 4@xmath5@xmath6 3.3@xmath5 ellipse . @xcite s radio frequency data also suggested a similarly sized shell - type remnant . even before its true extent was revealed , @xcite used their h@xmath7 , [ n@xmath8 , [ s ] ( 6717 ) , and [ s ] ( 6731 ) measurements to show that the s91 and s94 filaments were dense ( @xmath9 @xmath10 ) shell fragments belonging to a large supernova remnant . @xcite made additional optical emission line measurements of the s91 filament and of one of the brightest [ o@xmath11 filaments . both regions have large [ o / h@xmath12 ratios , which , when compared with the shock models of @xcite , indicate that the photons were emitted by thick ( @xmath13 @xmath14 ) postshock cooling regions behind moderate velocity ( @xmath15 to 120 km s@xmath16 ) shocks . given its [ o / h@xmath12 ratio , s91 s relatively modest [ o@xmath11/h@xmath12 ratio indicates that the cooling and recombination behind the shock is `` complete '' @xcite . the cooling zone along @xcite s second pointing direction appears to be well developed , but incomplete . given that the second pointing direction was chosen for its anomalously large [ o@xmath11/h@xmath12 ratio , this portion may be less complete than most of the remnant s shell . similarly , @xcite found regions of incomplete cooling as well as regions of nearly complete cooling . the shell temperature found from the older [ o@xmath11 and [ s measurements is @xmath1738,000 k ( fesen , blair & kirshner 1985 ; sabbadin & dodorico 1976 ) . @xcite calculated shock velocities of 90 to 140 km sec@xmath16 , which are roughly consistent with @xcite s kinematically determined expansion velocity of @xmath18 km s@xmath16 . in summary , [email protected] is sufficiently evolved for the shock to be weak and the gas behind it to be relatively cool and recombined . multiple techniques have been employed to estimate the remnant s size and distance . @xcite applied the @xmath19-d relation to their 1420 mhz data obtaining a diameter of @xmath20 pc and a distance of @xmath21 pc . @xcite applied the galactic kinematic relationship to her fabry - perot interferometric h@xmath7 data in order to estimate the distance as 800 pc and the minor and major axes of this elliptically shaped remnant as 56 and 64 pc , respectively . with a distance of @xmath21 pc and galactic latitude of 5.7@xmath5 , the remnant lies @xmath22 pc above the galactic disk . furthermore , [email protected] s age has been estimated as @xmath23 yr ( @xcite , using radiative phase snr relations ) . + while others were observing the remnant with optical and radio - frequency detectors , @xcite and @xcite were observing it with x - ray detectors . @xcite s _ heao 1 _ ( bandpass = 0.2 to 2.5 kev , fwhm = 3@xmath5 ) spectrum exhibits line emission , the hallmark of thermal emission from hot plasma . the model which best fits the _ heao 1 _ spectrum is a raymond and smith thermal model with a temperature between @xmath24 and @xmath25 k @xcite . although they were not able to resolve the emission spatially with the _ heao 1 _ data , they were able to show that the emission region is not point - like . in the following section , we extend upon this work . we use archival _ rosat _ data to map the remnant , confirming that the emitting gas is extended and finding that it fills the remnant s outline , lacks a bright shell , and exhibits a slowly decreasing temperature gradient . between august 1995 and september 1997 , the remnant was mapped with a series of overlapping _ rosat _ pspc observations . we processed the _ rosat _ pspc data ( removed the contamination , scattered solar x - rays , long - term enhancements , after - pulses , particle background , and bright point sources ) and mosaiced the pointings according to the extended object analysis procedures described in @xcite and @xcite . the resulting images in the _ rosat _ 1l2 band ( sensitive to 0.11 to 0.284 kev photons ) , 45 band ( sensitive to 0.44 to 1.21 kev photons ) , and 67 band ( sensitive to 0.73 to 2.04 kev photons ) are displayed in figure [ rosatimages.fig ] , along with a 1l2 band image overlayed with contours from the 67 band map . with the exception of a ridge of band 1l2 emission along the northeast edge and roughly coincident with bright optical filaments , the periphery is x - ray dim . most of the projected interior shows x - ray emission . because of its generally `` centrally filled '' x - ray morphology , thermal x - ray emission , and `` shell - type '' radio continuum morphology , the remnant should now be classified as a thermal composite . comparison of the soft and hard band images shows that the x - ray emission originating in the snr interior is slightly harder than that originating nearer to the edges . we quantify this trend with a spectral model . the gas is assumed to be near collisional ionizational equilibrium , as is observed both in similar remnants such as w44 ( harrus et al . 1997 ; shelton , kuntz & petre 2004 ) , 3c391 ( chen & slane 2001 ) , and 0045 - 734 in the smc ( yokogawa et al 2002 ) , and in simulated remnants which have evolved long enough to have centrally - peaked x - ray morphologies @xcite . given s spectral resolution , the differences between equilibrium and nearly equilibrium spectra are inconsequential , as are differences between newer and older versions of the @xcite spectral code . in general , the model spectra are calculated and handled according to the techniques described in @xcite . thus , we used the @xcite spectral code to create a suite of collisional ionizational equilibrium spectral models . the electron temperature ( @xmath26 ) ranged from @xmath27 k to @xmath28 k , with increments in log(@xmath26 ) of 0.05 . in order to account for the absorption by the interstellar medium , we multiplied our suite of model spectra by @xmath29 , where @xmath30 is the product of the absorption column density ( @xmath31 ) and the absorption cross section @xcite . @xmath31 ranged from @xmath32 @xmath14 to @xmath33 @xmath14 , with increments in log(@xmath31 ) of 0.10 . the resulting spectra were convolved with the _ rosat _ pspc response matrix , yielding the countrates in each _ rosat _ pspc band . we prepared the data for comparison with the calculated models by excluding counts from regions of the x - ray image that were contaminated by bright point sources or dominated by the galactic background emission , dividing the the remaining snr area into ten , concentric , nearly equal area annuli . these annuli were centered on ra = 19@xmath2 33@xmath3 41@xmath4 , dec = 31@xmath5 15 57 in j2000 coordinates and had outer radii of 26 , 39 , 48 , 56 , 64 , 71 , 79 , 86 , 94 , and 122 arcmin , respectively . regions beyond the snr and regions contaminated by point sources ( including the slightly extended hard source centered on ra = 19@xmath2 36@xmath3 46@xmath4 , dec = 30@xmath5 40 07 ) were excluded from the annuli areas . as a result , in order to create approximately equal area masks , the outermost annuli were extended to relatively large radii . we then calculated the countrates in each of the _ rosat _ pspc bands for each annulus . the background countrate for each band was calculated from the cleaned data for the portion of the observed sky that lies outside the snr s footprint and subtracted from the countrate for each annulus , resulting in the `` observed countrates '' . the standard chi - squared test was used to compare the observed countrates in the _ rosat _ 1l , 4 , 5 , and 6 bands for each annulus with the countrates predicted for the spectral models . band 2 was omitted , owing to complications of an absorption edge in the detector and band 7 was omitted owing to its low countrate . the best fitting models were then selected for each annulus . figure [ spectrum.fig ] displays the observed and model spectra for the seventh annulus ( spanning the region between 79 and 86 arcmin from the remnant s center ) . the best - fit temperatures range from 2.5 to @xmath34 k and decrease slightly with radius ( see figure [ temp.fig ] ) . the shallow temperature gradient is a common feature of thermal composite snrs ( i.e. , 3c400.2 @xcite , kes 27 @xcite , w44 @xcite ) . in the best fit models for the various regions , the column densities ranged from @xmath35 to @xmath36 @xmath14 , but did not exhibit a radial trend . the snr s apparent flux of 0.11 to 2.04 kev photons is @xmath37 ergs @xmath14 s@xmath16 . if there had been no interstellar scattering , the flux would have been @xmath38 ergs @xmath14 s@xmath16 . assuming that the remnant is 900 pc distant , its luminosity is @xmath39 ergs s@xmath16 and the electron density of the x - ray emitting gas is @xmath40 @xmath10 . based on the theoretical predictions and hydrocode simulations of the archetypical thermal composite supernova remnant , w44 ( @xcite , @xcite ) , we expect the ambient density to be about 10 times the density within the remnant s center . thus , the ambient density would be roughly 0.1 @xmath10 , a reasonable value at [email protected] s location 90 pc above the galactic plane . the electron density is lower than that of w44 , causing the luminosity to be lower as well . in addition to displaying a centrally concentrated snr morphology , the band 45 and band 67 x - ray images also reveal a `` slightly extended point source '' centered on ra = 19@xmath2 36@xmath3 46@xmath4 , dec = 30@xmath5 40 07 and extending approximately 6.5 arcmin in radius in the harder band map and less far in the band 45 map . the 1l2 band map does not show excess emission at or near this location . the observed spectrum is too noisy for meaningful fitting to model spectra and there is no known pulsar at this location . although the princeton - arecibo survey for millisecond pulsar s did not find a pulsar coincident with the observed bright spot , it did find a pulsar , j1931 + 30 , within the remnant s outline @xcite . we did not find excess x - ray emission at its location , ra = 19@xmath2 31@xmath3 28@xmath4 , dec = 30@xmath5 35. these _ rosat _ observations , in combination with existing radio observations , establish [email protected] s membership in the thermal composite ( mixed morphology ) class of supernova remnants . the radio continuum shells and thermal x - ray bright centers of this puzzling class of remnants have inspired several theoretical explanations . in one scenario , these remnants have evolved beyond the adiabatic phase into the radiative phase . as a result , their peripheries are now cool and not x - ray emissive . this scenario can explain why these remnants lack x - ray bright shells , while other explanations are required in order to account for the extreme brightnesses of these remnants centers . here , we note that crucial support for the evolutionary hypothesis is provided by optical evidence of nearly complete cooling behind [email protected] s relatively slow shockfront . + if @xcite s age estimate is correct , then [email protected] is the oldest known thermal composite supernova remnant . the fact that several other thermal composite supernova remnants neighbor molecular clouds has led to the suggestion that collisions with molecular clouds cause the characteristic thermal composite morphology . in order to test this suggestion on [email protected] , we searched the @xcite co survey data and the @xcite h survey data near @xmath41 at the remnant s systemic velocity ( @xmath42 km s@xmath16 , @xcite ) . confusion with galactic material made the search inconclusive . the possibility of a snr - cloud interaction could potentially be addressed with a targeted search for co emission features outlining the remnant . + r.l.s . wishes to thank the national research council and nasa s long term space astrophysics grant ( nag5 - 10807 ) for financial support and the anonomous referee for his / her helpful comments . k.d.k . wishes to thank gsrp for financial support . this research has made use of data obtained from the high energy astrophysics science archive research center ( heasarc ) , provided by nasa s goddard space flight center .

this paper presents archival _ rosat _ pspc observations of the [email protected] supernova remnant ( also known as g65.3 + 5.7 ) . little material obscures this remnant and so it was well observed , even at the softest end of _ rosat _ s bandpass ( @xmath1 to 0.28 kev ) . these soft x - ray images reveal the remnant s centrally - filled morphology which , in combination with existing radio frequency observations , places [email protected] in the thermal composite ( mixed morphology ) class of supernova remnants . not only might [email protected] be the oldest known thermal composite supernova remnant , but owing to its optically revealed cool , dense shell , this remnant supports the proposal that thermal composite supernova remnants lack x - ray bright shells because they have evolved beyond the adiabatic phase . these observations also reveal a slightly extended point source centered on ra = 19@xmath2 36@xmath3 46@xmath4 , dec = 30@xmath5 40 07 and extending 6.5 arcmin in radius in the band 67 map . the source of this emission has yet to be discovered , as there is no known pulsar at this location .

arxiv

permutations of the integers @xmath5 = \{1,\ldots , n\}$ ] are a basic building block for the succinct encoding of integer functions @xcite , strings @xcite , and binary relations @xcite , among others . a permutation @xmath0 is trivially representable in @xmath6 bits , which is within @xmath7 bits of the information theory lower bound of @xmath8 bits . and @xmath9=\{1,\ldots , x\}$ ] . ] in many interesting applications , efficient computation of both the permutation @xmath2 and its inverse @xmath3 is required . the lower bound of @xmath8 bits yields a lower bound of @xmath10 comparisons to sort such a permutation in the comparison model . yet , a large body of research has been dedicated to finding better sorting algorithms which can take advantage of specificities of each permutation to sort . trivial examples are permutations sorted such as the identity , or containing sorted blocks @xcite ( e.g. @xmath11 or @xmath12 ) , or containing sorted subsequences @xcite ( e.g. @xmath13 ) : algorithms performing only @xmath7 comparisons on such permutations , yet still @xmath14 comparisons in the worst case , are achievable and obviously preferable . less trivial examples are classes of permutations whose structure makes them interesting for applications : see mannila s seminal paper @xcite and estivil - castro and wood s review @xcite for more details . each sorting algorithm in the comparison model yields an encoding scheme for permutations : it suffices to note the result of each comparison performed to uniquely identify the permutation sorted , and hence to encode it . since an adaptive sorting algorithm performs @xmath15 comparisons on many classes of permutations , each adaptive algorithm yields a _ compression scheme _ for permutations , at the cost of losing a constant factor on some other `` bad '' classes of permutations . we show in section [ sec : applications ] some examples of applications where only `` easy '' permutations arise . yet such compression schemes do not necessarily support in reasonable time the inverse of the permutation , or even the simple application of the permutation : this is the topic of our study . we describe several encodings of permutations so that on interesting classes of instances the encoding uses @xmath15 bits while supporting the operations @xmath2 and @xmath3 in time @xmath16 . later , we apply our compression schemes to various scenarios , such as the encoding of integer functions , text indexes , and others , yielding original compression schemes for these abstract data types . the _ entropy _ of a sequence of positive integers @xmath17 adding up to @xmath1 is @xmath18 . by convexity of the logarithm , @xmath19 . [ def : entrop ] [ [ sec : sequences ] ] * succinct encodings of sequences * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + let @xmath20 $ ] be a sequence over an alphabet @xmath21 $ ] . this includes bitmaps when @xmath22 ( where , for convenience , the alphabet will be @xmath23 ) . we will make use of succinct representations of @xmath24 that support operations @xmath25 and @xmath26 : @xmath27 gives the number of occurrences of @xmath28 in @xmath29 $ ] and @xmath30 gives the position in @xmath24 of the @xmath31th occurrence of @xmath28 . for the case @xmath22 , @xmath24 requires @xmath1 bits of space and @xmath25 and @xmath26 can be supported in constant time using @xmath32 bits on top of @xmath24 @xcite . the extra space is more precisely @xmath33 for some parameter @xmath34 , which is chosen to be , say , @xmath35 to achieve the given bounds . in this paper , we will sometimes apply the technique over sequences of length @xmath36 ( @xmath1 will be the length of the permutations ) . still , we will maintain the value of @xmath34 as a function of @xmath1 , not @xmath37 , which ensures that the extra space will be of the form @xmath38 , i.e. , it will tend to zero when divided by @xmath37 as @xmath1 grows , even if @xmath37 stays constant . all of our @xmath39 terms involving several variables in this paper can be interpreted in this strong sense : asymptotic in @xmath1 . thus we will write the above space simply as @xmath40 . raman _ et al . _ @xcite devised a bitmap representation that takes @xmath41 bits , while maintaining the constant time for the operations . here @xmath42 , where @xmath43 is the number of occurrences of symbol @xmath28 in @xmath24 , is the so - called _ zero - order entropy _ of @xmath24 . for the binary case this simplifies to @xmath44 , where @xmath45 is the number of bits set in @xmath24 . grossi _ et al . _ @xcite extended the result to larger alphabets using the so - called _ wavelet tree _ , which decomposes a sequence into several bitmaps . by representing those bitmaps in plain form , one can represent @xmath24 using @xmath46 bits of space , and answer @xmath47 $ ] , as well as @xmath25 and @xmath26 queries on @xmath24 , in time @xmath48 . by , instead , using raman _ s representation for the bitmaps , one achieves @xmath49 bits of space , and the same times . et al . _ @xcite used multiary wavelet trees to maintain the same compressed space , while improving the times for all the operations to @xmath50 . [ [ sec : meas - disord - perm ] ] * measures of disorder in permutations * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + various previous studies on the presortedness in sorting considered in particular the following measures of order on an input array to be sorted . among others , mehlhorn @xcite and guibas _ et al . _ @xcite considered the number of pairs in the wrong order , knuth @xcite considered the number of ascending substrings ( runs ) , cook and kim @xcite , and later mannila @xcite considered the number of elements which have to be removed to leave a sorted list , mannila @xcite considered the smallest number of exchanges of arbitrary elements needed to bring the input into ascending order , skiena @xcite considered the number of encroaching sequences , obtained by distributing the input elements into sorted sequences built by additions to both ends , and levcopoulos and petersson @xcite considered shuffled upsequences and shuffled monotone sequences . estivil - castro and wood @xcite list them all and some others . we first introduce a compression method that takes advantage of ( ascending ) runs in the permutation . then we consider a stricter variant of the runs , which allows for further compression in applications when those runs arise , and in particular allows the representation size to be sublinear in @xmath1 . next , we consider a more general type of runs , which need not be contiguous . one of the best known sorting algorithm is merge sort , based on a simple linear procedure to merge two already sorted arrays , resulting in a worst case complexity of @xmath14 . yet , checking in linear time for _ down - step _ positions in the array , where an element is followed by a smaller one , partitions the original arrays into ascending runs which are already sorted . this can speed up the algorithm when the array is partially sorted @xcite . we use this same observation to encode permutations . a _ down step _ of a permutation @xmath0 over @xmath5 $ ] is a position @xmath51 such that @xmath52 . a _ run _ in a permutation @xmath0 is a maximal range of consecutive positions @xmath53 which does not contain any down step . let @xmath54 be the list of consecutive down steps in @xmath0 . then the number of runs of @xmath0 is noted @xmath55 , and the sequence of the lengths of the runs is noted @xmath56 . for example , permutation @xmath11 contains @xmath57 runs , of lengths @xmath58 . whereas previous analyses @xcite of adaptive sorting algorithms considered only the number @xmath59 of runs , we refine them to consider the distribution @xmath60 of the sizes of the runs . there is an encoding scheme using at most @xmath61 bits to encode a permutation @xmath0 over @xmath5 $ ] covered by @xmath59 runs of lengths @xmath60 . it supports @xmath2 and @xmath3 in time @xmath62 for any value of @xmath63 $ ] . if @xmath51 is chosen uniformly at random in @xmath5 $ ] then the average time is @xmath64 . [ thm : main ] the hu - tucker algorithm @xcite ( see also knuth @xcite ) produces in @xmath65 time a prefix - free code from a sequence of frequencies @xmath66 adding up to @xmath1 , so that ( 1 ) the @xmath51-th lexicographically smallest code is that for frequency @xmath67 , and ( 2 ) if @xmath68 is the bit length of the code assigned to the @xmath51-th sequence element , then @xmath69 is minimal and moreover @xmath70 ( * ? ? ? * , eq . ( 27 ) ) . we first determine @xmath60 in @xmath7 time , and then apply the hu - tucker algorithm to @xmath60 . we arrange the set of codes produced in a binary trie ( equivalent to a huffman tree @xcite ) , where each leaf corresponds to a run and points to its two endpoints in @xmath0 . because of property ( 1 ) , reading the leaves left - to - right yields the runs also in left - to - right order . now we convert this trie into a wavelet - tree - like structure @xcite without altering its shape , as follows . starting from the root , first process recursively each child . for the leaves do nothing . once both children of an internal node have been processed , the invariant is that they point to the contiguous area in @xmath0 covering all their leaves , and that this area of @xmath0 has already been sorted . now we merge the areas of the two children in time proportional to the new area created ( which , again , is contiguous in @xmath0 because of property ( 1 ) ) . as we do the merging , each time we take an element from the left child we append a 0 bit to a bitmap we create for the node , and a 1 bit when we take an element from the right list . when we finish , we have the following facts : ( 1 ) @xmath0 has been sorted , ( 2 ) the time for sorting has been @xmath71 plus the total number of bits appended to all bitmaps , ( 3 ) each of the @xmath67 elements of leaf @xmath51 ( at depth @xmath68 ) has been merged @xmath68 times , contributing @xmath68 bits to the bitmaps of its ancestors , and thus the total number of bits is @xmath72 . therefore , the total number of bits in the hu - tucker - shaped wavelet tree is at most @xmath73 . to this we must add the @xmath74 bits of the tree pointers . we preprocess all the bitmaps for @xmath25 and @xmath26 queries so as to spend @xmath75 extra bits ( [ sec : sequences ] ) . to compute @xmath3 we start at offset @xmath51 at the root bitmap @xmath76 , with position @xmath77 , and bitmap size @xmath78 . if @xmath79 = 0 $ ] we go down to the left child with @xmath80 and @xmath81 . otherwise we go down to the right child with @xmath82 , @xmath83 , and @xmath84 . when we reach a leaf , the answer is @xmath85 . to compute @xmath2 we do the reverse process , but we must first determine the leaf @xmath86 and offset @xmath31 within @xmath86 corresponding to position @xmath51 : we start at the root bitmap @xmath76 , with bitmap size @xmath78 and position @xmath87 . if @xmath88 we go down to the left child with @xmath89 . otherwise we go down to the right child with @xmath90 and @xmath91 . we eventually reach leaf @xmath86 , and the offset within @xmath86 is @xmath31 . we now start an upward traversal using the nodes that are already in the recursion stack ( those will be limited to @xmath92 soon ) . if @xmath86 is a left child of its parent @xmath93 , then we set @xmath94 , else we set @xmath95 , where @xmath76 is the bitmap of @xmath93 . then we set @xmath96 until reaching the root , where @xmath97 . in both cases the time is @xmath98 , where @xmath37 is the depth of the leaf arrived at . if @xmath51 is chosen uniformly at random in @xmath5 $ ] , then the average cost is @xmath99 . however , the worst case can be @xmath100 in a fully skewed tree . we can ensure @xmath101 in the worst case while maintaining the average case by slightly rebalancing the hu - tucker tree : if there exist nodes at depth @xmath102 , we rebalance their subtrees , so as to guarantee maximum depth @xmath103 . this affects only marginally the size of the structure . a node at depth @xmath37 can not add up to a frequency higher than @xmath104 ( see next paragraph ) . added over all the possible @xmath59 nodes we have a total frequency of @xmath105 . therefore , by rebalancing those subtrees we add at most @xmath106 bits . this is @xmath107 if @xmath108 , and otherwise the cost was @xmath109 anyway . for the same reasons the average time stays @xmath64 as it increases at most by @xmath110 . the bound on the frequency at depth @xmath37 is proved as follows . consider the node @xmath86 at depth @xmath37 , and its grandparent @xmath93 . then the uncle of @xmath86 can not have smaller frequency than @xmath86 . otherwise we could improve the already optimal hu - tucker tree by executing either a single ( if @xmath86 is left - left or right - right grandchild of @xmath93 ) or double ( if @xmath86 is left - right or right - left grandchild of @xmath93 ) avl - like rotation that decreases the depth of @xmath86 by 1 and increases that of the uncle of @xmath86 by 1 . thus the overall frequency at least doubles whenever we go up two nodes from @xmath86 , and this holds recursively . thus the weight of @xmath86 is at most @xmath111 . the general result of the theorem can be simplified when the distribution @xmath60 is not particularly favorable . there is an encoding scheme using at most @xmath112 bits to encode a permutation @xmath0 over @xmath5 $ ] with a set of @xmath59 runs . it supports @xmath2 and @xmath3 in time @xmath62 for any value of @xmath63 $ ] . [ cor : mainbal ] as a corollary , we obtain a new proof of a well - known result on adaptive algorithms telling that one can sort in time @xmath113 @xcite , now refined to consider the entropy of the partition and not only its size . we can sort an array of length @xmath1 covered by @xmath59 runs of lengths @xmath60 in time @xmath114 , which is worst - case optimal in the comparison model among all permutations with @xmath59 runs of lengths @xmath60 so that @xmath115 . [ cor : mainsort ] some classes of permutations can be covered by a small number of runs of a stricter type . we present an encoding scheme which uses @xmath107 bits for encoding the permutations from those classes , and still @xmath116 bits for all others . a _ strict run _ in a permutation @xmath0 is a maximal range of positions satisfying @xmath117 . the _ head _ of such run is its first position . the number of strict runs of @xmath0 is noted @xmath118 , and the sequence of the lengths of the strict runs is noted @xmath119 . we will call @xmath120 the sequence of run lengths of the sequence formed by the strict run heads of @xmath0 . for example , permutation @xmath12 contains @xmath121 strict runs , of lengths @xmath122 . the run heads are @xmath123 , and contain 2 runs , of lengths @xmath124 . instead , @xmath11 contains @xmath125 strict runs , all of length 1 . there is an encoding scheme using at most @xmath126 bits to encode a permutation @xmath0 over @xmath5 $ ] covered by @xmath118 strict runs and by @xmath127 runs , and with @xmath120 being the @xmath59 run lengths in the permutation of strict run heads . it supports @xmath2 and @xmath3 in time @xmath62 for any value of @xmath63 $ ] . if @xmath51 is chosen uniformly at random in @xmath5 $ ] then the average time is @xmath128 . [ thm : strict ] we first set up a bitmap @xmath129 marking with a 1 bit the beginning of the strict runs . set up a second bitmap @xmath130 such that @xmath131 = r[\pi^{-1}(i)]$ ] . now we create a new permutation @xmath132 of @xmath133 $ ] which collapses the strict runs of @xmath0 , @xmath134 . all this takes @xmath7 time and the bitmaps take @xmath135 bits using raman _ s technique , where @xmath25 and @xmath26 are solved in constant time ( [ sec : sequences ] ) . now build the structure of thm . [ thm : main ] for @xmath132 . the number of down steps in @xmath0 is the same as for the sequence of strict run heads in @xmath0 , and in turn the same as the down steps in @xmath132 . so the number of runs in @xmath132 is also @xmath59 and their lengths are @xmath120 . thus we get at most @xmath136 bits to encode @xmath132 , and can compute @xmath132 and its inverse in @xmath137 worst case and @xmath128 average time . to compute @xmath2 , we find @xmath138 and then compute @xmath139 . the final answer is @xmath140 . to compute @xmath3 , we find @xmath141 and then compute @xmath142 . the final answer is @xmath143 . this adds only constant time on top of that to compute @xmath132 and its inverse . once again , we might simplify the results when the distribution @xmath120 is not particularly favorable , and we also obtain interesting algorithmic results on sorting . there is an encoding scheme using at most @xmath144 bits to encode a permutation @xmath0 over @xmath5 $ ] covered by @xmath118 strict runs and by @xmath127 runs . it supports @xmath2 and @xmath3 in time @xmath62 for any value of @xmath63 $ ] . [ cor : strictbal ] we can sort a permutation of @xmath5 $ ] , covered by @xmath118 strict runs and by @xmath59 runs , and @xmath120 being the run lengths of the strict run heads , in time @xmath145 , which is worst - case optimal , in the comparison model , among all permutations sharing these @xmath59 , @xmath118 , and @xmath120 values , such that @xmath146 . [ cor : strictsort ] levcopoulos and petersson @xcite introduced the more sophisticated concept of partitions formed by interleaved runs , such as _ shuffled upsequences _ we discuss here the advantage of considering permutations formed by shuffling a small number of runs . a decomposition of a permutation @xmath0 over @xmath5 $ ] into _ shuffled upsequences _ is a set of , not necessarily consecutive , subsequences of increasing numbers that have to be removed from @xmath0 in order to reduce it to the empty sequence . the minimum number of shuffled upsequences in such a decomposition of @xmath0 is noted @xmath147 , and the sequence of the lengths of the involved shuffled upsequences , in arbitrary order , is noted @xmath148 . for example , permutation @xmath13 contains @xmath149 shuffled upsequences of lengths @xmath150 , but @xmath151 runs , all of length 2 . whereas the decomposition of a permutation into runs or strict runs can be computed in linear time , the decomposition into shuffled upsequences requires a bit more time . fredman @xcite gave an algorithm to compute the size of an optimal partition , claiming a worst case complexity of @xmath14 . in fact his algorithm is adaptive and takes @xmath152 time . we give here a variant of his algorithm which computes the partition itself within the same complexity , and we achieve even better time on favorable sequences @xmath148 . [ lem : partitioninsus ] given a permutation @xmath0 over @xmath5 $ ] covered by @xmath147 shuffled upsequences of lengths @xmath148 , there is an algorithm finding such a partition in time @xmath153 . initialize a sequence @xmath154 , and a splay tree @xmath155 @xcite with the node @xmath156 , ordered by the rightmost value of the sequence contained by each node . for each further element @xmath2 , search for the sequence with the maximum ending point smaller than @xmath2 . if any , add @xmath2 to this sequence , otherwise create a new sequence and add it to @xmath155 . fredman @xcite already proved that this algorithm computes an optimal partition . the adaptive complexity results from the mere observation that the splay tree ( a simple sorted array in fredman s proof ) contains at most @xmath147 elements , and that the node corresponding to a subsequence is accessed once per element in it . hence the total access time is @xmath153 ( * ? ? ? * thm . 2 ) . the complete description of the permutation requires to encode the computation of both the partitioning algorithm and the sorting one , and this time the encoding cost of partitioning is as important as that of merging . [ thm : sus ] there is an encoding scheme using at most @xmath157 bits to encode a permutation @xmath0 over @xmath5 $ ] covered by @xmath147 shuffled upsequences of lengths @xmath148 . it supports the operations @xmath2 and @xmath3 in time @xmath158 for any value of @xmath63 $ ] . if @xmath51 is chosen uniformly at random in @xmath5 $ ] the average time is @xmath159 . partition the permutation @xmath0 into @xmath147 shuffled upsequences using lemma [ lem : partitioninsus ] , resulting in a string @xmath24 of length @xmath1 over alphabet @xmath160 $ ] which indicates for each element of the permutation @xmath0 the label of the upsequence it belongs to . encode @xmath24 with a wavelet tree using raman _ et al . _ s compression for the bitmaps , so as to achieve @xmath161 bits of space and support retrieval of any @xmath47 $ ] , as well as symbol @xmath25 and @xmath26 on @xmath24 , in time @xmath158 ( [ sec : sequences ] ) . store also an array @xmath162 $ ] so that @xmath163 $ ] is the accumulated length of all the upsequences with label less than @xmath37 . array @xmath164 requires @xmath165 bits . finally , consider the permutation @xmath132 formed by the upsequences taken in label order : @xmath132 has at most @xmath147 runs and hence can be encoded using @xmath166 bits using thm . [ thm : main ] , as @xmath148 in @xmath0 corresponds to @xmath60 in @xmath132 . this supports @xmath167 and @xmath168 in time @xmath158 . now @xmath169+rank_{s[i]}(s , i))$ ] can be computed in time @xmath158 . similarly , @xmath170)$ ] , where @xmath37 is such that @xmath163 < ( \pi')^{-1}(i ) \le a[\ell+1]$ ] , can also be computed in @xmath158 time . thus the whole structure uses @xmath171 bits and supports @xmath2 and @xmath3 in time @xmath158 . the obstacles to achieve the claimed average time are the operations on the wavelet tree of @xmath24 , and the binary search in @xmath164 . the former can be reduced to @xmath172 by using the improved wavelet tree representation by ferragina _ et al . _ ( [ sec : sequences ] ) . the latter is reduced to constant time by representing @xmath164 with a bitmap @xmath173 $ ] with the bits set at the values @xmath163 + 1 $ ] , so that @xmath163 = select_1(a',\ell)-1 $ ] , and the binary search is replaced by @xmath174 . with raman _ _ s structure ( [ sec : sequences ] ) , @xmath175 needs @xmath176 bits and operates in constant time . again , we might prefer a simplified result when @xmath148 has no interesting distribution , and we also achieve an improved result on sorting , better than the known @xmath152 . [ cor : susbal ] there is an encoding scheme using at most @xmath177 bits to encode a permutation @xmath0 over @xmath5 $ ] covered by @xmath147 shuffled upsequences . it supports the operations @xmath2 and @xmath3 in time @xmath158 for any value of @xmath63 $ ] . [ cor : sortsus ] we can sort an array of length @xmath1 , covered by @xmath147 shuffled upsequences of lenghts @xmath148 , in time @xmath153 , which is worst - case optimal , in the comparison model , among all permutations decomposable into @xmath147 shuffled upsequences of lenghts @xmath148 such that @xmath178 . consider a full - text inverted index which gives the word positions of any word in a text . this is a popular data structure for natural language text retrieval @xcite , as it permits for example solving phrase queries without accessing the text . for each different text word , an increasing list of its text positions is stored . let @xmath1 be the total number of words in a text collection @xmath179 $ ] and @xmath59 the vocabulary size ( i.e. , number of different words ) . an uncompressed inverted index requires @xmath180 bits . it has been shown @xcite that , by @xmath181-encoding the differences between consecutive entries in the inverted lists , the total space reduces to @xmath182 , where @xmath183 is the zero - order entropy of the text if seen as a sequence of words ( [ sec : sequences ] ) . we note that the empirical law by heaps @xcite , well accepted in information retrieval , establishes that @xmath59 is small : @xmath184 for some constant @xmath185 depending on the text type . several successful methods to compress natural language text take words as symbols and use zero - order encoding , and thus the size they can achieve is lower bounded by @xmath186 @xcite . if we add the differentially encoded inverted index in order to be able of searching the compressed text , the total space is at least @xmath187 . now , the concatenation of the @xmath59 inverted lists can be seen as a permutation of @xmath5 $ ] with @xmath59 runs , and therefore thm . [ thm : main ] lets us encode it in @xmath188 bits . within the same space we can add @xmath59 numbers telling where the runs begin , in an array @xmath189 $ ] . now , in order to retrieve the list of the @xmath51-th word , we simply obtain @xmath190 ) , \pi(v[i]+1 ) , \ldots , \pi(v[i+1]-1)$ ] , each in @xmath62 time . moreover we can extract any random position from a list , which enables binary - search - based strategies for list intersection @xcite . in addition , we can also obtain a text passage from the ( inverse ) permutation : to find out @xmath191 $ ] , @xmath192 gives its position in the inverted lists , and a binary search on @xmath193 finds the interval @xmath194 \le \pi^{-1}(j ) < v[i+1]$ ] , to output that @xmath191 = i$]th word , in @xmath137 time . this result is very interesting , as it constitutes a true word - based _ self - index _ @xcite ( i.e. , a compressed text index that contains the text ) . similar results have been recently obtained with rather different methods @xcite . the cleanest one is to build a wavelet tree over @xmath155 with compression @xcite , which achieves @xmath195 bits of space , and permits obtaining @xmath196 $ ] , as well as extracting the @xmath31th element of the inverted list of the @xmath51th word with @xmath197 , all in time @xmath198 . yet , one advantage of our approach is that the extraction of @xmath37 consecutive entries @xmath199)$ ] takes @xmath200 time if we do the process for all the entries as a block : start at range @xmath201 $ ] at the root bitmap @xmath76 , with position @xmath77 , and bitmap size @xmath78 . go down to both left and right children : to the left with @xmath201 \leftarrow [ rank_0(b , i),rank_0(b , i')]$ ] , same @xmath202 , and @xmath89 ; to the right with @xmath201 \leftarrow [ rank_1(b , i),rank_1(b , i')]$ ] , @xmath83 , and @xmath91 . stop when the range @xmath201 $ ] becomes empty or when we reach a leaf , in which case report all answers @xmath203 , @xmath204 . by representing the inverted list as @xmath205 , we can extract long inverted lists faster than the existing methods . [ cor : invfile ] there exists a representation for a text @xmath179 $ ] of integers in @xmath206 $ ] ( regarded as word identifiers ) , with zero - order entropy @xmath207 , that takes @xmath208 bits of space , and can retrieve the text position of the @xmath31th occurrence of the @xmath51th text word , as well as the value @xmath191 $ ] , in @xmath137 time . it can also retrieve any range of @xmath37 successive occurrences of the @xmath51th text word in time @xmath209 . we could , instead , represent the inverted list as @xmath0 , so as to extract long text passages efficiently , but the wavelet tree representation can achieve the same result . another interesting functionality that both representations share , and which is useful for other list intersection algorithms @xcite , is that to obtain the first entry of a list which is larger than @xmath210 . this is done with @xmath25 and @xmath26 on the wavelet tree representation . in our permutation representation , we can also achieve it in @xmath62 time by finding out the position of a number @xmath210 within a given run . the algorithm is similar to those in thm . [ thm : main ] that descend to a leaf while maintaining the offset within the node , except that the decision on whether to descend left or right depends on the leaf we want to arrive at and not on the bitmap content ( this is actually the algorithm to compute @xmath25 on binary wavelet trees @xcite ) . finally , we note that our inverted index data structure supports in small time all the operations required to solve conjunctive queries on binary relations . suffix arrays are used to index texts that can not be handled with inverted lists . given a text @xmath179 $ ] of @xmath1 symbols over an alphabet of size @xmath59 , the _ suffix _ array @xmath211 $ ] is a permutation of @xmath5 $ ] so that @xmath212,n]$ ] is lexicographically smaller than @xmath213,n]$ ] . as suffix arrays take much space , several compressed data structures have been developed for them @xcite . one of interest for us is the _ compressed suffix array ( csa ) _ of sadakane @xcite . it builds over a permutation @xmath214 of @xmath5 $ ] , which satisfies @xmath215 = ( a[i]~\textrm{mod}~n ) + 1 $ ] ( and thus lets us move virtually one position forward in the text ) @xcite . it turns out that , using just @xmath214 and @xmath74 extra bits , one can @xmath216 _ count _ the number of times a pattern @xmath217 $ ] occurs in @xmath155 using @xmath218 applications of @xmath214 ; @xmath219 _ locate _ any such occurrence using @xmath220 applications of @xmath214 , by spending @xmath221 extra bits of space ; and @xmath222 _ extract _ a text substring @xmath223 $ ] using at most @xmath224 applications of @xmath214 . hence this is another self - index , and its main burden of space is that to represent permutation @xmath214 . sadakane shows that @xmath214 has at most @xmath59 runs , and gives a representation that accesses @xmath225 $ ] in constant time by using @xmath226 bits of space . it was shown later @xcite that the space is actually @xmath227 bits , for any @xmath228 and constant @xmath229 . here @xmath230 is the @xmath231th order empirical entropy of @xmath155 @xcite . with thm . [ thm : main ] we can encode @xmath214 using @xmath232 bits of space , whose extra terms aside from entropy are better than sadakane s . those extra terms can be very significant in practice . the price is that the time to access @xmath214 is @xmath137 instead of constant . on the other hand , an interesting extra functionality is that to compute @xmath233 , which lets us move ( virtually ) one position backward in @xmath155 . this allows , for example , displaying the text context around an occurrence without having to spend any extra space . still , although interesting , the result is not competitive with recent developments @xcite . an interesting point is that @xmath214 contains @xmath234 strict runs , for any @xmath231 @xcite . therefore , cor . [ cor : strictbal ] lets us represent it using @xmath235 bits of space . for @xmath231 limited as above , this is at most @xmath236 bits , which is similar to the space achieved by another self - index @xcite , yet again it is slightly superseded by its time performance . munro _ et al . _ @xcite described how to represent a permutation @xmath0 as the concatenation of its cycles , completed by a bitvector of @xmath1 bits coding the lengths of the cycles . as the cycle representation is itself a permutation of @xmath5 $ ] , we can use any of the permutation encodings described in [ sec : compr - techn ] to encode it , adding the binary vector encoding the lengths of the cycles . it is important to note that , for a specific permutation @xmath0 , the difficulty to compress its cycle encoding @xmath132 is not the same as the difficulty to encode the original permutation @xmath0 . given a permutation @xmath0 with @xmath28 cycles of lengths @xmath237 , there are several ways to encode it as a permutation @xmath132 , depending on the starting point of each cycle ( @xmath238 } n_i$ ] choices ) and the order of the cycles in the encoding ( @xmath239 choices ) . as a consequence , each permutation @xmath0 with @xmath28 cycles of lengths @xmath237 can be encoded by any of the @xmath238 } i\times n_i$ ] corresponding permutations . any of the encodings from theorems [ thm : main ] , [ thm : strict ] and [ thm : sus ] can be combined with an additional cost of at most @xmath240 bits to encode a permutation @xmath0 over @xmath5 $ ] composed of @xmath28 cycles of lengths @xmath237 to support the operation @xmath4 for any value of @xmath241 , in time and space function of the order in the permutation encoding of the cycles of @xmath0 . the space `` wasted '' by such a permutation representation of the cycles of @xmath0 is @xmath242 bits . to recover some of this space , one can define a canonical cycle encoding by starting the encoding of each cycle with its smallest value , and by ordering the cycles in order of their starting point . this canonical encoding always starts with a @xmath243 and creates at least one shuffled upsequence of length @xmath28 : it can be compressed as a permutation over @xmath244 $ ] with at least one shuffled upsequence of length @xmath245 through thm [ thm : sus ] . munro and rao @xcite extended the results on permutations to arbitrary functions from @xmath5 $ ] to @xmath5 $ ] , and to their iterated application @xmath246 , the function iterated @xmath231 times starting at @xmath51 . their encoding is based on the decomposition of the function into a bijective part , represented as a permutation , and an injective part , represented as a forest of trees whose roots are elements of the permutation : the summary of the concept is that an integer function is just a `` hairy permutation '' . combining the representation of permutations from @xcite with any representation of trees supporting the level - ancestor operator and an iterator of the descendants at a given level yields a representation of an integer function @xmath247 using @xmath248 bits to support @xmath246 in @xmath249 time , for any fixed @xmath250 , integer @xmath241 and @xmath63 $ ] . janssen _ et al . _ @xcite defined the _ degree entropy _ of an ordered tree @xmath155 with @xmath1 nodes , having @xmath67 nodes with @xmath51 children , as @xmath251 , and proposed a succinct data structure for @xmath155 using @xmath252 bits to encode the tree and support , among others , the level - ancestor operator . obviously , the definition and encoding can be generalized to a forest of @xmath231 trees by simply adding one node whose @xmath231 children are the roots of the @xmath231 trees . encoding the injective parts of the function using janssen _ et al . _ @xcite succinct encoding , and the bijective parts of the function using one of our permutation encodings , yields a compressed representation of any integer function which supports its application and the application of its iterated variants in small time . [ cor : fun ] there is a representation of a function @xmath253\rightarrow[n]$ ] that uses @xmath254 bits to support @xmath246 in @xmath255 time , for any integer @xmath231 and for any @xmath63 $ ] , where @xmath155 is the forest representing the injective part of the function , and @xmath59 is the number of runs in the bijective part of the function . bentley and yao @xcite , when introducing a family of search algorithms adaptive to the position of the element searched ( aka the `` unbounded search '' problem ) , did so through the definition of a family of adaptive codes for unbounded integers , hence proving that the link between algorithms and encodings was not limited to the complexity lower bounds suggested by information theory . in this paper , we have considered the relation between the difficulty measures of adaptive sorting algorithms and some measures of `` entropy '' for compression techniques on permutations . in particular , we have shown that some concepts originally defined for adaptive sorting algorithms , such as runs and shuffled upsequences , are useful in terms of the compression of permutations ; and conversely , that concepts originally defined for data compression , such as the entropy of the sets of sizes of runs , are a useful addition to the set of difficulty measures that one can consider in the study of adaptive algorithms . it is easy to generalize our results on runs and strict runs to take advantage of permutations which are a mix of up and down runs or strict runs ( e.g. @xmath256 , with only a linear extra computational and/or space cost . the generalization of our results on shuffled upsequences to sms @xcite , permutations containing mixes of subsequences sorted in increasing and decreasing orders ( e.g. @xmath257 ) is sligthly more problematic , because it is np hard to optimally decompose a permutation into such subsequences @xcite , but any approximation scheme @xcite would yield a good encoding . j. barbay and g. navarro . compressed representations of permutations , and applications . technical report tr / dcc-2008 - 18 , department of computer science ( dcc ) , university of chile , december 2008 .

we explore various techniques to compress a permutation @xmath0 over @xmath1 integers , taking advantage of ordered subsequences in @xmath0 , while supporting its application @xmath2 and the application of its inverse @xmath3 in small time . our compression schemes yield several interesting byproducts , in many cases matching , improving or extending the best existing results on applications such as the encoding of a permutation in order to support iterated applications @xmath4 of it , of integer functions , and of inverted lists and suffix arrays . jrmy barbay gonzalo navarro

arxiv

about ten years ago , a peculiar dynamical phenomenon was discovered in populations of identical phase oscillators : under nonlocal symmetric coupling , the coexistence of coherent ( synchronized ) and incoherent oscillators was observed @xcite . this highly counterintuitive phenomenon was given the name chimera state after the greek mythological creature made up of different animals @xcite . since then the study of chimera states has been the focus of extensive research in a wide number of models , from kuramoto phase oscillators @xcite to periodic and chaotic maps @xcite , as well as stuart - landau oscillators @xcite . the first experimental evidence of chimera states was found in populations of coupled chemical oscillators as well as in optical coupled - map lattices realized by liquid - crystal light modulators @xcite . recently , moreover , martens and coauthors showed that chimeras emerge naturally from a competition between two antagonistic synchronization patterns in a mechanical experiment involving two subpopulations of identical metronomes coupled in a hierarchical network @xcite . in the context of neuroscience , a similar effort has been undertaken by several groups , since it is believed that chimera states might explain the phenomenon of unihemispheric sleep observed in many birds and dolphins which sleep with one eye open , meaning that one hemisphere of the brain is synchronouns whereas the other is asynchronous @xcite . the purpose of this paper is to make a contribution in this direction , by identifying for the first time a variety of single and multi - chimera states in networks of non - locally coupled neurons represented by hindmarsh rose oscillators . recently , multi - chimera states were discovered on a ring of nonlocally coupled fitzhugh - nagumo ( fhn ) oscillators @xcite . the fhn model is a 2dimensional ( 2d ) simplification of the physiologically realistic hodgkin - huxley model @xcite and is therefore computationally a lot simpler to handle . however , it fails to reproduce several important dynamical behaviors shown by real neurons , like rapid firing or regular and chaotic bursting . this can be overcome by replacing the fhn with another well - known more realistic model for single neuron dynamics , the hindmarsh - rose ( hr ) model @xcite , which we will be used throughout this work both in its 2d and 3d versions . in section 2 of this paper , we first treat the case of 2d - hr oscillators represented by two first order ordinary differential equations ( odes ) describing the interaction of a membrane potential and a single variable related to ionic currents across the membrane under periodic boundary conditions . we review the dynamics in the 2d plane in terms of its fixed points and limit cycles , coupling each of the @xmath0 oscillators to @xmath1 nearest neighbors symmetrically on both sides , in the manner adopted in @xcite , through which chimeras were discovered in fhn oscillators . we identify parameter values for which chimeras appear in this setting and note the variety of oscillating patterns that are possible due to the bistability features of the 2d model . in particular , we identify a new `` mixed oscillatory state '' ( mos ) , in which the desynchronized neurons are uniformly distributed among those attracted by a stable stationary state . furthermore , we also discover chimera like patterns in the more natural setting where only the membrane potential variables are coupled with @xmath2 of the same type . next , we turn in section 3 to the more realistic 3d - hr model where a third variable is added representing a slowly varying current , through which the system can also exhibit bursting modes . here , we choose a different type of coupling where the two first variables are coupled symmetrically to @xmath2 of their own kind and observe only states of complete synchronization as well as mos in which desynchronized oscillators are interspersed among neurons that oscillate in synchrony . however , when coupling is allowed only among the first ( membrane potential ) variables chimera states are discovered in cases where spiking occurs within sufficiently long bursting intervals . finally , the paper ends with our conclusions in section 4 . following the particular type of setting proposed in @xcite we consider @xmath0 nonlocally coupled hindmarsh - rose oscillators , where the interconnections between neurons exist with @xmath3 nearest neighbors only on either side as follows : @xmath4 \label{eq:01 } \\ \dot y_k & = & 1 - 5x_k^2-y_k+ \frac{\sigma_y}{2r}\sum_{j = k - r}^{j = k+r } [ b_{yx}(x_j - x_k)+b_{yy}(y_j - y_k ) ] . \label{eq:02 } \end{aligned}\ ] ] in the above equations @xmath5 is the membrane potential of the @xmath6-th neuron , @xmath7 represents various physical quantities related to electrical conductances of the relevant ion currents across the membrane , @xmath8 , @xmath9 and @xmath10 are constant parameters , and @xmath11 is the external stimulus current . each oscillator is coupled with its @xmath12 nearest neighbors on both sides with coupling strengths @xmath13 . this induces nonlocality in the form of a ring topology established by considering periodic boundary conditions for our systems of odes . as in @xcite , our system contains not only direct @xmath14 and @xmath15 coupling , but also cross - coupling between variables @xmath16 and @xmath17 . this feature is modeled by a rotational coupling matrix : @xmath18 depending on a coupling phase @xmath19 . in what follows , we study the collective behavior of the above system and investigate , in particular , the existence of chimera states in relation to the values of all network parameters : @xmath0 , @xmath3 , @xmath19 , @xmath20 and @xmath21 . more specifically , we consider two cases : ( @xmath22 ) direct and cross - coupling of equal strength in both variables ( @xmath23 ) and ( @xmath24 ) direct coupling in the @xmath16 variable only ( @xmath25 , @xmath26 ) . similarly to @xcite we shall use initial conditions randomly distributed on the unit circle ( @xmath27 ) . at @xmath28 of eqs . ( [ eq:01]),([eq:02 ] ) ( left ) and selected time series ( right ) for @xmath29 . ( a ) @xmath30 , ( b ) @xmath31 and ( c ) @xmath32 . @xmath33 and @xmath34.,scaledwidth=70.0% ] typical spatial patterns for case ( @xmath22 ) are shown on the left panel of fig . [ fig : fig1 ] , where the @xmath16 variable is plotted over the index number @xmath6 at a time snapshot chosen after a sufficiently long simulation of the system . in this figure the effect of different values of the phase @xmath19 is demonstrated while the number of oscillators @xmath0 and their nearest neighbors @xmath3 , as well as the coupling strengths @xmath23 are kept constant . for example , for @xmath30 ( fig . [ fig : fig1](a ) ) an interesting novel type of dynamics is observed that we shall call `` mixed oscillatory state '' ( mos ) , whereby nearly half of the @xmath5 are attracted to a fixed point ( at this snapshot they are all at a value near @xmath35 ) , while the other half are oscillating interspersed among the stationary ones . from the respective time series ( fig . [ fig : fig1](a ) , right ) it is clear that the former correspond to spiking neurons whereas the latter to quiescent ones . this interesting mos phenomenon is due to the fact that , in the standard parameter values we have chosen , the uncoupled hr oscillators are characterized by the property of _ bistability_. clearly , as shown in the phase portrait of fig . [ fig : fig2 ] , each oscillator possesses three fixed points : the leftmost fixed point is a stable node corresponding to the resting state of the neuron while the other two correspond to a saddle point and an unstable node and are therefore repelling . for @xmath11 ( which is the case here ) a stable limit cycle also exists which attracts many of the neurons into oscillatory motion , rendering the system bistable and producing the dynamics observed in fig . [ fig : fig1](a ) . now , when a positive current @xmath36 is applied , the @xmath16-nullcline is lowered such that the saddle point and the stable node collide and finally vanish . in this case the full system enters a stable limit cycle associated with typical spiking behaviour . similar complex patterns including mos and chimeras have been observed in this regime as well . . three fixed points coexist with a stable limit cycle.,scaledwidth=40.0% ] for @xmath31 , on the other hand , there is a `` shift '' in the dynamics of the individual neurons into the spiking regime , as seen in fig . [ fig : fig1](b ) ( right ) . the corresponding spatial pattern has a wave - like form of period 2 . then , for @xmath32 a classical chimera state with two incoherent domains is observed ( ( fig . [ fig : fig1](c ) , left ) . diagonal coupling ( @xmath37 , @xmath38 ) is , therefore , identified as the necessary condition to achieve chimera states . by contrast , it is interesting to note that in nonlocally coupled fitz - hugh nagumo oscillators @xcite it has been shown both analytically and numerically that chimera states occur for _ off - diagonal _ coupling . by decreasing the range of coupling @xmath3 and increasing the system size @xmath0 , chimera states occur with multiple domains of incoherence and coherence for @xmath32 ( fig . [ fig : fig3](c , d ) ) , and , accordingly , periodic spatial patterns with larger wave numbers arise for @xmath31 , as seen in fig . [ fig : fig3](a , b ) . this is in agreement with previous works reported in @xcite . next we consider the case ( @xmath24 ) where the coupling term is restricted to the @xmath16 variable . this case is important since incorporating the coupling in the voltage membrane ( @xmath16 ) alone is more realistic from a biophysiological point of view . in fig . [ fig : fig4 ] spatial plots ( left ) and the corresponding @xmath39-plane ( right ) for increasing coupling strength are shown . chimera states ( fig . [ fig : fig4 ] ( b , c ) ) are observed as an intermediate pattern between desynchronization ( fig . [ fig : fig4 ] ( a ) ) and complete synchronization ( fig . [ fig : fig4 ] ( d ) ) . of eqs . ( [ eq:01]),([eq:02 ] ) at @xmath28 for @xmath23 . @xmath40 ( top ) and @xmath41 ( bottom ) whereas @xmath31 ( left ) and @xmath32 ( right ) . in ( a ) and ( b ) @xmath42 , in ( c ) @xmath43 , and in ( d ) @xmath44.,scaledwidth=50.0% ] ( left ) and in the @xmath45-plane ( right ) of eqs . ( [ eq:01]),([eq:02 ] ) at @xmath28 for @xmath25 , @xmath33 and @xmath34 . ( a ) @xmath46 , ( b ) @xmath47 , ( c ) @xmath48 and ( d ) @xmath49.,scaledwidth=40.0% ] in order to complete our study of the hindmarsh - rose model we shall consider , in this section , its three - dimensional version . the corresponding equations read : @xmath50 the extra variable @xmath51 represents a slowly varying current , which changes the applied current @xmath52 to @xmath53 and guarantees firing frequency adaptation ( governed by the parameter @xmath54 ) as well as the ability to produce typical bursting modes , which the 2d model can not reproduce . parameter @xmath55 controls the transition between spiking and bursting , parameter @xmath56 determines the spiking frequency ( in the spiking regime ) and the number of spikes per bursting ( in the bursting regime ) , while @xmath57 sets the resting potential of the system . the parameters of the fast @xmath58 system are set to the same values used in the two - dimensional version ( @xmath8 , @xmath10 ) and typical values are used for the parameters of the @xmath51-equation ( @xmath59 , @xmath60 , @xmath61 ) . the 3d hindmarsh - rose model exhibits a rich variety of bifurcation scenarios in the @xmath62 parameter plane @xcite . thus , we prepare all nodes in the spiking regime ( with corresponding parameter values @xmath9 and @xmath63 ) and , as in section 2 , we use initial conditions randomly distributed on the unit sphere ( @xmath64 ) . first let us consider direct coupling in both variables @xmath16 and @xmath17 and vary the value of the equal coupling strengths @xmath23 , while @xmath0 and @xmath3 are kept constant . naturally , the interaction of @xmath0 spiking neurons will lead to a change in their dynamics , as we discuss in what follows . for low values of @xmath65 we observe the occurrence of a type of mos where nearly half of the neurons spike regularly in a synchronous fashion , while the rest are unsynchronized and spike in an irregular fashion . this is illustrated in the respective time series in the right panel of fig . [ fig : fig5](a ) at higher values of the coupling strength the the system is fully synchronized ( fig . [ fig : fig5](b ) ) . of eqs . ( [ eq:03])-([eq:05 ] ) at @xmath66 ( left ) and selected time series ( right ) for : ( a ) @xmath67 and ( b ) @xmath68 . @xmath33 and @xmath34.,scaledwidth=50.0% ] next we check the case of coupling in the @xmath16 variable alone ( @xmath69 , @xmath25 ) . figure [ fig : fig6 ] displays characteristic synchronization patterns obtained when we increase @xmath20 ( left panel ) and selected time series ( right panel ) . at low values of the coupling strength all neurons remain in the regular spiking regime and desynchronization alternates with complete synchronization as @xmath20 increases ( fig . [ fig : fig6](a , b , c ) ) . for intermediate values of the coupling strength , chimera states with one incoherent domain are to be observed . these are associated with a change in the dynamics of the individual neurons , which now produce irregular bursts ( fig . [ fig : fig6](d ) ) . the number of spikes in each burst increases at higher @xmath20 values and the system is again fully synchronized ( fig . [ fig : fig6](e ) ) . extensive simulations show that the chimera states disappear and reappear again by varying @xmath20 which is most likely due to the system s multistability and sensitive dependence on initial conditions . of eqs . ( [ eq:03])-([eq:05 ] ) at @xmath66 ( left ) and selected time series for @xmath25 , @xmath33 and @xmath34 . ( a ) @xmath70 , ( b ) @xmath71 , @xmath72 , ( d ) @xmath73 , and ( e ) @xmath74 . , in this paper , we have identified the occurrence of chimera states for various coupling schemes in networks of 2d and 3d hindmarsh - rose models . this , together with recent reports on multiple chimera states in nonlocally coupled fitzhugh - nagumo oscillators , provide strong evidence that this counterintuitive phenomenon is very relevant as far as neuroscience applications are concerned . chimera states are strongly related to the phenomenon of synchronization . during the last years , synchronization phenomena have been intensely studied in the framework of complex systems @xcite . moreover , it is a well - established fact the key ingredient for the occurrence of chimera states is nonlocal coupling . the human brain is an excellent example of a complex system where nonlocal connectivity is compatible with reality . therefore , the study of chimera states in systems modelling neuron dynamics is both significant and relevant as far as applications are concerned . moreover , the present work is also important from a theoretical point of view , since it verifies the existence of chimera states in coupled bistable elements , while , up to now , it was known to arise in oscillator models possessing a single attracting state of the limit cycle type . finally , we have identified a novel type of mixed oscillatory state ( mos ) , in which desynchronized neurons are interspersed among those that are either stationary or oscillate in synchrony . as a continuation of this work , it is very interesting to see whether chimeras and mos states appear also in networks of _ populations _ of hindmarsh - rose oscillators , which are currently under investigation . the authors acknowledge support by the european union ( european social fund esf ) and greek national funds through the operational program `` education and lifelong learning '' of the national strategic reference framework ( nsrf ) - research funding program : thales . investing in knowledge society through the european social fund . funding was also provided by ninds r01 - 40596 . hagerstrom , a. m. , thomas , e. , roy , r. , hvel , p. , omelchenko , i. & schll , e. [ 2012 ] `` chimeras in coupled - map lattices : experiments on a liquid crystal spatial light modulator system '' , _ nature physics _ * 8 * , 658 . omelchenko , i. , omelchenko , o. e. , hvel , p. & schll , e. [ 2013 ] `` when nonlocal coupling between oscillators becomes stronger : patched synchrony or multichimera states '' , _ phys . lett . _ * 110 * , 224101 . rattenborg , n. c. , amlaner , c. j. & lim , s. l. [ 2000 ] `` behavioral , neurophysiological and evolutionary perspectives on unihemispheric sleep , '' _ neuroscience and biobehavioral reviews _ * 24 * , pp . 817-842 .

we have identified the occurrence of chimera states for various coupling schemes in networks of two - dimensional and three - dimensional hindmarsh - rose oscillators , which represent realistic models of neuronal ensembles . this result , together with recent studies on multiple chimera states in nonlocally coupled fitzhugh - nagumo oscillators , provide strong evidence that the phenomenon of chimeras may indeed be relevant in neuroscience applications . moreover , our work verifies the existence of chimera states in coupled bistable elements , whereas to date chimeras were known to arise in models possessing a single stable limit cycle . finally , we have identified an interesting class of mixed oscillatory states , in which desynchronized neurons are uniformly interspersed among the remaining ones that are either stationary or oscillate in synchronized motion .

arxiv

kauffman networks are disordered dynamical systems proposed by kauffman in 1969 as a model for genetic regulatory systems @xcite . they attracted the interest of physicists in the 80 s @xcite , due to their analogy with the disordered systems studied in statistical mechanics , such as the mean field spin glass @xcite . a dynamical phase transition was found and studied in the framework of mean field theory . in this and in the next paper @xcite we deal with some structural properties of the networks that determine their attractors . in the present paper we introduce the relevant elements , a notion that was suggested by flyvbjerg @xcite and flyvbjerg and kjaer @xcite , and we study their probability distribution . in the next one we describe how the relevant elements are subdivided into asymptotically non communicating , independent modules . the modular organization of random boolean networks was already suggested by kauffman @xcite , and it was used by flyvbjerg and kjaer to study analytically the attractors in @xmath4 networks . we shall show that it is possible to describe the phase transition in random boolean networks in terms of the scaling of the number of relevant elements with system size , or in terms of a percolation transition in the set of the relevant elements . the interest of this approach is that some consequences about the statistical properties of the attractors can be directly drawn . in @xcite we computed the properties of the attractors in the framework of the annealed approximation , introduced by derrida and pomeau @xcite , but we observed that the results of this approximation are reliable only when the system is chaotic enough , becoming exact for a random map . the study of the relevant elements is complementary to this approach , and we sketch the lines of a new approximation scheme that works better in the frozen phase and on the critical line . this region in parameter space is the most interesting one , since , according to kauffman , it reproduces some features of real cells , and is also the less understood , since neither approximate computations nor simulations @xcite give a precise picture of the properties of the attractors for systems of large size . in next section we define the model , discussing some old results together with open problems . in section 3 we define the relevant elements and in section 4 we give an approximate argument predicting the scaling of their number with system size in the different phases of the model . in the following section we present our numerical results , starting from the magnetization and the stable elements ( section 5.1 ) and then discussing the distribution of the relevant elements and its connection with the properties of the attractors , respectively in the chaotic phase ( section 5.2 ) and on the critical line ( section 5.3 ) . the discussion of the results is postponed to our following paper @xcite , concerning the modular organization of the relevant elements on the critical line . kauffman model is defined as follows . we consider a set of @xmath1 elements @xmath5 and we associate to each of them a binary variable , @xmath6 . in the biological interpretation proposed by kauffman each element of the network represents one gene and the binary variable @xmath7 represents its state of activation . each element is under the control of @xmath8 elements , in the sense that its state at time @xmath9 is determined by the states at time @xmath10 of the @xmath8 control genes , @xmath11 and by a response function of @xmath8 binary variables , @xmath12 , that specifies how the element @xmath13 responds to the signals coming from its control variables . the control elements are chosen in @xmath14 with uniform probability . the response functions are also extracted at random , and it s believed that the properties of the model do not depend on the details of their distribution @xcite . the rule most generally used in teh literature is the following : for each of the possible inputs @xmath15 we extract independently the value of @xmath16 , and we call @xmath17 the probability that @xmath16 is equal to 0 . the dynamics of the system obey the equation _ i(t+1)=f_i(_j_1(i),_j_k(i ) ) . this evolution law is deterministic , but the system is disordered because the control rules ( elements and functions ) are chosen at random from the beginning and kept fixed : thus we deal with a statistical ensemble of deterministic dynamical systems , and we are interested in the statistical properties of systems of large size . for finite @xmath1 , every trajectory becomes periodic after a long enough transient time , and the configuration space is partitioned into the attraction basins of the different periodic orbits . we are interested in the probability distributions of the number , the length and the size of the attraction basin of the periodic orbits , as well as in that of transient times . in the biological metaphor , given a set of rules ( a genome ) an attractor represents a possible cellular type , its length represents the duration of the cellular cycle , and the number of attractors represents the number of cells that can be formed with a given genome . it was observed already in the first simulations that two dynamical regimes are present , and that the line separating them has properties reminiscent of those of real cells @xcite . in the so - called chaotic phase ( large connectivity , @xmath17 close to @xmath18 ) the average length of the cycles increases exponentially with system size . the limit case of the chaotic phase , @xmath19 , was already known as random map in the mathematical literature , and was studied in detail by derrida and flyvbjerg @xcite , who pointed out interesting analogies between this system and the mean field spin glass @xcite concerning the distribution of the weights of the attraction basins . in the frozen phase , on the other hand , the typical length of the cycles does not increase with @xmath1 . the limit case of this phase , @xmath4 , was analytically studied to some extent by flyvbjerg and kjaer @xcite , who introduced in that context the concept of relevant elements ( though without using this name ) . the first description of this dynamical phase transition in terms of an order parameter was given by derrida and pomeau @xcite . they studied the evolution of the hamming distance between configurations in the kauffman networks approximating it with a markovian stochastic process . such approximation ( the so - called annealed approximation ) was then shown to be exact in the infinite size limit , concerning the average value of the distance @xcite . below a critical line in parameter space the average distance goes to zero in the infinite size limit ( _ frozen phase _ ) and above it the distance goes to a finite value ( _ chaotic phase _ ) . the position of the phase transition depends only on the parameter @xmath20 , representing the probability that the responses to two different signals are different one has @xmath21 , so its value is comprised between zero and 1/2 , but for @xmath4 @xmath20 can be taken as an independent parameter in [ 0,1 ] ] , and is given by the equation @xmath22 . the properties of the attractors can be easily computed from the knowledge of the whole stationary distribution of the distance , and this can also be obtained within the annealed approximation @xcite , but the validity of this approximation in this more general case is not guaranteed . comparison with simulations shows that the agreement is satisfactory in the chaotic phase , while the approximation fails on the critical line . in the chaotic phase it is possible to compute the value of the exponent of the typical length of a cycle , @xmath23 , in good agreement with numerical results , but the distribution of cycle lengths is much broader than it is expected . the annealed approximation predicts also that the distribution of the weights of the attraction basins is universal in the whole chaotic phase , and equal to the one obtained by derrida and flyvbjerg in the case of the random map @xcite . the corrections to this prediction appear small , if any , even for @xmath24 . finally , the number of different cycles in a network is expected to be linear in @xmath1 , but it is very hard to test numerically this prediction . the annealed approximation makes also predictions about the critical line of the model @xcite . it predicts that the properties of the attractors are universal on the critical line @xmath25 ( with the exceptions of the points @xmath26 and @xmath27 , which are not transition points ) . in particular , the typical length of the cycles should increase as @xmath0 all along the critical line . numerical results are not clear under this respect @xcite : it seems that the rescaled cycle length @xmath28 has a limit distribution if @xmath29 is small ( roughly , smaller than 2 ) but for larger values the distribution becomes broader and broader as @xmath1 increases @xcite , so that it is possible to define an effective length scale increasing much faster with system size ( as a stretched exponential ) . the distribution of the number of cycles has exactly the same characteristics . these results cast doubts on the validity of the biological analogy proposed by kauffman , that relies very much on the fact that in critical networks the typical number of cycles scales as @xmath0 , reminiscent of the fact that the number of cell types of multicellular organisms very far apart in the filogenetic tree scales as the square root of the number of the genes , and that in critical networks the typical length of the cycles increases as a power law of system size , also consistently with the behavior of cell cycles time . thus it is interesting to understand how these distributions look like in the limit of very large systems . another reason of interest of the present approach is that it allows to understand the limits of the annealed approximation . in our interpretation the annealed approximation is valid as far as the system loses memory of the details of its evolution . this , of course , does not happen if in a realization of a random network some structural properties that are able to influence its asymptotic dynamics emerge . thus the approach presented here is complementary to the one used in @xcite . let us start recalling the definition of the stable elements @xcite . these are elements that evolve to a constant state , independent of the initial configuration . flyvbjerg defined them and computed their fraction @xmath30 using the annealed approximation , which becomes exact in the infinite size limit . we now recall briefly , for future convenience , the main steps of this calculation . let us suppose that an element is controlled by @xmath31 stable elements and @xmath13 stable ones . then it will be stable if the control function does not depend on the unstable arguments when the stable arguments assume their fixed values . otherwise it will be unstable . when all the @xmath13 unstable elements are different ( this can always be taken to be the case if @xmath8 is finite and @xmath1 grows ) , the probability @xmath32 to choose a constant control function of @xmath13 binary variables is given by @xmath33 , with @xmath34 . in the framework of the annealed approximation , extracting at random connections and response functions at each time step , we get the following equation for the fraction of variables that are stable at time @xmath10 : [ stable ] s(t+1)=(s(t))= _ i=0^k ki s(t)^k - i(1-s(t))^i p_i . this equation can be shown to be exact in the infinite size limit . the fixed point of this map ( which can be interpreted as a self - consistency equation for the fraction of stable variables ) has only the trivial solution @xmath35 in the frozen phase , in other words all the elements are stable except eventually a number increasing less than linearly with @xmath1 . in the chaotic phase this solution becomes unstable and another solution less than 1 appears . this happens when @xmath36 . since @xmath37 ( it is just the probability that the response to two different signals are different ) this condition is equivalent to the condition obtained from the study of the hamming distance . the existence of the stable variables is due to the finite connectivity of the network ( @xmath38 goes to zero very fast when @xmath8 increases ) . these variables do not take part in the asymptotic dynamics . among the remaining unstable variables , some are irrelevant for the dynamics , either because they do not send signals to any other variable , or because they send signals , but the response functions are independent of this signal when the stable variables have attained their fixed values . the remaining variables , that are unstable and control some unstable variable , are what we call the relevant variables . they are the only ones that can influence the long time behavior of the system . to be more clear we now describe the algorithm that we used to identify the relevant variables . as a first step , we have to identify the stable variables . these are the variables that assume the same constant state in every limit cycle , and identifying them is computationally very hard , but very simple in principle . we then eliminate from the system the stable variables , reducing the response functions to functions of the unstable variables alone . some of the connections left are still irrelevant , and we have to eliminate them ( a connection between the elements @xmath13 and @xmath39 is irrelevant if the reduced response function @xmath40 does not depend on the argument @xmath41 for all the configurations of the remaining @xmath42 control variables ) . at this point we iterate a procedure to eliminate the irrelevant variables . at each iteration we eliminate the variables that do not send any signal to anyone of the variables that are left , until we remain with a set that can not be further reduced . this is the set of the relevant variables . measuring the number of relevant variables is computationally a very hard task . in order to identify the stable variables , in fact , we should find all the cycles in the network , and , to be rigorous , we should simulate a number of trajectories of the same order of the number of configurations in the system . of course this is not feasible and we run only 200 ( in some case 300 ) randomly chosen trajectories in every network . thus we overestimate the number of stable elements . nevertheless , the number of stable elements changes very little when we simulate more initial conditions and we think that the error that we make is not very large . however , for every network we simulate some hundreds of trajectories and every trajectories has to be followed until the closing time . this grows exponentially with system size in the chaotic phase . on the critical line the typical closing time increase roughly as a power law of system size , but the distribution becomes broader and broader and the average closing time is more and more dominated by rare events . the average depends thus on the number of samples generated and on the cutoff of the closing time , _ i.e. _ the maximum time that we are disposed to wait to look for a cycle . to reduce the bias determined by the cutoff , we had to run simulations lasting a time which increases roughly as a stretched exponential of system size on the critical line . thus it is not possible to simulate systems of more than about one hundred elements in the chaotic phase and one thousands of elements on the critical line . the mean field analysis @xcite shows that the fraction of relevant variables vanishes in the frozen phase and on the critical line , but does not tell how the number of relevant variables scales with @xmath1 as @xmath1 grows . in order to clarify this point , we have to go beyond the mean field picture . in the special case of @xmath4 , belonging to the frozen phase for every @xmath43 , there are detailed analytical results about the distribution of the relevant variables @xcite . we propose here a rough argument that generalizes those results to the whole frozen phase and predicts that the typical number of relevant elements scales as @xmath0 on the critical line . though this argument is based on some approximations which we can not control , its results coincide for @xmath4 with the exact results by flyvbjerg and kjaer . let us suppose that we add a new element to a system with @xmath1 elements , @xmath44 of which are relevant , while @xmath45 are stable and @xmath46 are indifferent , _ i.e. _ neither stable nor relevant . the probability that the new element is relevant can be computed as a function of @xmath44 and @xmath45 , within some approximations that we are going to discuss in a while . this probability is equal to the fraction of relevant elements in the system with @xmath47 elements , given that the relevant elements are @xmath44 and the stable ones are @xmath45 in the system with @xmath1 elements . we can then average over @xmath44 and @xmath45 in order to get an equation connecting @xmath48 to the moments of the distribution of @xmath44 in the system with @xmath1 elements . since in the frozen phase and on the critical line @xmath49 vanishes , it will be enough to consider the first two moments of the distribution , and the resulting equation can be solved asymptotically in @xmath1 . the weakness of this approach lies on the assumptions that allow us to express the probability that the new element is relevant as a function of @xmath44 and @xmath45 , as it will become soon clear . we compute now this probability . to this aim , we need two steps : 1 . as a first step , we have to extract the @xmath8 control elements and the response function of the new element . as a consequence , the new element can be stable , unstable or , if it receives an input from itself and this input is relevant in the sense discussed above , relevant . the evaluation of the stability is perfectly equivalent to the mean field argument , but this stability is only temporary because it can be altered by the second step described below . thus we call a new element that is stable ( unstable ) after the first step a _ temporarily _ stable ( unstable ) element . then we have to send to the old system the signal of the new element . for each of the @xmath50 old control connections we have a probability @xmath51 that the connection is broken and the old control element is substituted by the new element . this step perturbs the elements that control the new element and modifies its temporary stability . we have no chance to take this into account , unless we use some drastic approximations . in the second step , three situations can occur . 1 . if the new element was relevant in the first step , the new step can not modify this condition . 2 . if the new element was unstable , it can not become stable through the feedback of its signal . so it will be relevant or indifferent , depending on whether it sends an input to at least one relevant element or not . 3 . if the new element was stable , its signal can destabilize some of the elements that control it and thus it can become relevant through a feedback mechanism , very hard to investigate analytically . to compute the probability of case 3 , we should know the organization of the network in the very detail and not only the number of relevant and stable elements . we propose to bypass this difficulty considering a different event : we will consider the new element relevant if it receives a signal from a previously relevant element or from itself . this is the simplest way to get a closed equation for the average number of relevant elements . in this way we make two errors of opposite sign : on one hand we overestimate the probability that a temporarily unstable element becomes relevant , on the other one we underestimate the probability that the new element is temporarily unstable and we neglect the probability that a temporarily stable element becomes relevant through a feedback loop . we think that this method captures at least the qualitative behavior of the number of relevant elements . we have then to compare the estimate given by this approximation to the simulations , because the approximation is not under control . we present this argument because its results agree with both the numerical results and with the analytical calculations for @xmath4 and because we believe that it is possible to improve this method and to keep the approximation under control . since we are interested in the frozen phase , where the fraction of unstable elements vanishes in the infinite size limit , we can neglect the eventuality that the new element is controlled by more than two elements that were relevant in the old system . the results are consistent with this assumption . with these approximations we obtain the following equation for the probability that the new element is relevant : r_n+1=_n=0^n \{r_n = n } [k(n+1n+1 ) (1-n+1n+1)^k-1.[rilev ] + + ._2k2(n+1n+1)^2 (1-n+1n+1)^k-2 ] , where @xmath52 represents the probability that a boolean function of two arguments is not constant and in terms of @xmath20 is given by _ 2=1-p^4-(1-p)^4=(2 - 2 ) . in the frozen phase it is sufficient to consider that the new element receives only one signal from the previously relevant elements . so , posing @xmath53 , the equation for the new fraction of relevant elements , @xmath54 , is r_n+1cr_n+cn . the first term represents a new element that receives a relevant signal from one of the previously relevant elements , the second term represents a new element that receives its own relevant signal . thus the average number of relevant elements is independent on @xmath1 and its asymptotic value is r_n = c1-c.[froz ] this number diverges on the critical line @xmath55 . in this case , we have to consider also the eventuality that the new element receives a signal from two of the previously relevant elements . expanding to the second order in @xmath56 , and using the fact that @xmath57 , we get the equation r_n+1r_n - (k-14k ) r^2_n+1n , whence , in the asymptotic regime where the variations of @xmath58 are of order @xmath59 , we finally get r^2_n(k-14k)1n . this means that the scale of the number of relevant elements grows , on the chaotic phase , as @xmath0 . we stress here that these computations are valid because of the finite connectivity of the system . if we perform the limit @xmath19 on the above result , we get that the scale of the number of relevant elements grow as @xmath60 . if , instead , we apply the limit @xmath19 prior to the limit @xmath61 we get the trivial critical point @xmath62 , where all the elements are stable after one time step , while for every other @xmath20 value all the elements are relevant . thus , the two limits do not commute . in fact , the equation ( [ stable ] ) for the fraction of stable variables and all the computations performed in this section are valid only if we can neglect that the same element is chosen more than once to control a given element , _ i.e. _ for @xmath63 . the result ( [ froz ] ) coincides for @xmath4 with the analytical computation by flyvbjerg and kjaer @xcite , thus suggesting that the distribution of relevant elements is independent on @xmath1 in the whole frozen phase , and depends on the two parameters @xmath8 and @xmath20 only through their product . this picture agrees with the results of the annealed approximation , which predicts that the distribution of the number of different elements in two asymptotic configurations is independent on @xmath1 and depends only on the product of the parameters @xmath8 and @xmath20 in the frozen phase @xcite . our simulations confirm that on the critical line the number of relevant elements scales as @xmath0 ( see figure [ fig_ril4 ] ) . also the annealed approximation is consistent with this result , since it predicts that the number of elements whose state is different in two asymptotic configurations has to be rescaled with @xmath0 on the critical line @xcite . on the other hand the number of unstable elements grows much faster with @xmath1 ( numerically it is found that it goes as @xmath64 , see below ) but this discrepancy is only apparent , since the asymptotic hamming distance is related more to the number of relevant elements than to this quantity . for later convenience ( see our next paper ) it is also interesting to compute the effective connectivity , defined as the average value of the relevant connections between relevant elements . let us compute it by imposing the condition that the network has @xmath44 relevant elements . the effective connectivity is equal to the average number of connections between the new element and the other relevant elements of the older system , with the condition that the new element is relevant . from equation ( [ rilev ] ) we have , at the leading order in @xmath65 : [ ceff ] k_eff(r)= & & c(1-)^k-1 + 2_2k2()^2(1-)^k-2c(1-)^k-1+_2k2()^2(1-)^k-2 + & & 1+a(k , ) . this equation shows that the effective connectivity minus 1 goes to zero as @xmath65 in the frozen phase ( where @xmath66 ) and on the critical line ( where @xmath67 ) . for a fixed system size , the effective connectivity increases linearly with the number of relevant elements . the distribution of this variable , shown in figure [ fig_magneti ] for @xmath24 and @xmath70 , has many peaks , corresponding to simple rational values . this perhaps reflects the fact that the relevant elements are divided into asymptotically independent modules , so that a cycle can be decomposed into several independent shorter cycles . this subject will be further discussed in our second paper . our results have to be compared to the analytical work by derrida and flyvbjerg @xcite . they defined the magnetization of element @xmath13 at time @xmath10 on a given network as the activity of the element at time @xmath10 averaged over many initial configurations and could compute analytically its stationary distribution , in the limit @xmath61 , using the annealed approximation , that can be shown to be exact for this purpose . the picture they got is different from ours , in particular we see peaks much higher than theirs . for instance the peak at @xmath71 , which gives information on the size of the stable core of the network , is about 10 times larger then expected , and the first moments of the magnetization , that can be computed analytically , are larger than the predicted values . thus we performed other simulations that strongly suggest that these discrepancies are finite size effects , and we present an argument that explains their origin . in order to investigate larger systems we had to change the definition of the magnetization . the definition ( [ mag ] ) is numerically cumbersome , since the measure takes place only after that a cycle has been found , and this means , for chaotic systems , that we have to wait a time exponentially increasing with @xmath1 . thus we neglected this condition and we measured the magnetization of the variable @xmath13 at time @xmath10 as the average activity of the variable with respect to different initial conditions ( this definition coincides with the one used by derrida and flyvbjerg ) . for very large @xmath10 , when all trajectories have reached a limit cycle , this quantity tends to the asymptotic value m_i=_w_m_i^,[magna]where @xmath72 is the weight of the basin of cycle @xmath69 and @xmath68 is defined in ( [ mag ] ) . we observed that @xmath73 reaches a stationary value ( within some precision ) much earlier than the typical time at which the trajectories reach their limit cycles . at first sight surprisingly , the time after which @xmath73 reaches its stationary distribution does decrease with system size instead of increasing ( see figure [ fig_score ] ) . we measured the second and fourth moment of the magnetization in a system with @xmath24 and @xmath74 , and we found a large positive correction to the infinite size values computed by derrida and flyvbjerg @xcite . the values found coincide within the statistical error with those obtained from equation ( [ mag ] ) for a system of small size for which we did an explicit comparison . these values can be fitted to the sum of the infinite size value , that we got from @xcite , plus an exponentially decreasing term . the exponent of the best fit turned out to be the same for both the moments that we measured : we found @xmath75 , and @xmath76 . the measure of the magnetization allows to identify the stable elements as the elements with @xmath77 equal either to 0 or to 1 . the two definitions of the magnetization gave roughly the same number of stable elements in the cases where we could compare the results , but with the second method we could consider much larger systems ( we recall that the difference between the two methods is that in the first case a cycle has been reached while in the second one the system is still in some transient configuration ) . the second method was used only to study finite size effects , since it does not allow to identify the relevant elements ( see below ) . both the methods overestimate the number of stable elements , since it could happen that an element appearing stable in our sample of trajectories ( some hundreds ) oscillates in a cycle that is not reached by any of them . we checked that the results do not change qualitatively if we consider a larger number of trajectories . the fraction of stable nodes measured in simulations with @xmath24 and @xmath1 ranging from 50 to 200 have been compared to the prediction of the mean field theory by flyvbjerg . the networks with @xmath78 have a stable core about 10 times larger than the mean field value ( in this case we measured the magnetization using both the above definitions , while for larger systems only equation ( [ magna ] ) was used ) . the corrections to the mean field value , that is exact in the infinite size limit , appear to decay exponentially with a rate identical , within statistical errors , to the decay rate of the corrections to the moments of the magnetization : we found @xmath79 . for every size of the systems which we simulated the stable core is then much larger than it would be in an infinite system . on this ground , we may expect very important finite size effects concerning the dynamical properties of the system . summarizing , the distribution of the magnetization for finite systems has the following characteristic : 1 ) the asymptotic value is reached after a time that _ decreases _ with system size ; 2 ) the corrections to the infinite size values are very large ; and 3 ) these corrections decrease exponentially with system size . these apparently strange finite size effects have a simple interpretation : they arise as a consequence of the periodic dynamic of the random networks . the mean field values of the magnetization and of the stable core are computed within the annealed approximation without taking into account the fact that the asymptotic dynamic is periodic . as we proposed in @xcite , the existence of limit cycles must be taken into account in the framework of the annealed approximation in this way : if at time @xmath10 all the configurations generated are different ( _ i.e. _ the trajectory is still open ) we treat the quantities of interest ( distance , magnetization or stable core ) as a markovian stochastic process ; if one configuration has been found twice ( the trajectory is closed ) we impose the condition that all quantities are periodic . thus the master equation for the distribution for the number of stable variables is , in the framework of the annealed approximation : & & \ { s(t+1)=s,o(t+1)s(t)=s , o(t)}= n s((s))^s(1-(s))^n - s(1-_n(s , t ) ) + & & \ { s(t+1)=s,s(t)=s , o(t)}= ns((s))^s(1-(s))^n - s_n(s , t ) + & & \ { s(t+1)=s,s(t)=s , } = _ ss , [ stable2 ] where @xmath80 is the number of stable elements , @xmath30 , @xmath81 stands for the condition that the trajectory is open at time @xmath10 ( no configuration has been visited twice ) , @xmath82 stands for the condition complementary to @xmath81 ( the trajectory has closed on a previously visited configuration ) and @xmath83 is the probability that a trajectory open at time @xmath10 and with @xmath45 stable elements at time @xmath9 closes at that time . finally , @xmath84 is given by equation ( [ stable ] ) . we do nt know how to compute @xmath83 , but it is clear that this is an increasing function of @xmath45 for fixed @xmath10 : the more elements are stable , the more it is likely that the trajectory closes . the infinite size value of the stable core is given by equation ( [ stable ] ) , that represents the evolution of the most probable number of stable variables . it is clear that the corrections to this value are positive , and that they go to zero as soon as the closing time becomes much larger than the time necessary for the stable core to reach its stationary value in an infinite system ( where all trajectories are still open ) . thus we expect that these corrections vanish as a power law of the typical length of the cycles : in the chaotic phase this means that the finite size corrections due to this effect vanish exponentially with system size , as we observed simulating systems with @xmath24 and @xmath74 . lastly , this argument implies that the time after which the distribution of the stable elements becomes stationary is shorter in an infinite system than in a small system , where the evolution of @xmath80 is coupled to the closure of the periodic orbits . thus the correction of the annealed approximation to take into account the existence of periodic attractors can account for all the features of the finite size effects that we observed . after having identified the stable elements we detect the relevant elements using the algorithm described in the second section and we study how this quantity influences the dynamical properties of the network . the main results are that the average cycle length grows almost exponentially with the number of relevant variables in some range of this variable and the average weight of the attraction basins has apparently a non monotonic behavior versus the number of relevant variables . this qualitative features are the same both in the chaotic phase and on the critical line , but the ranges of @xmath44 in which these things happen are quite different in the two cases . we start discussing the situation in the chaotic phase . the simulations were done generating at random @xmath85 sample networks and running 200 trajectories on each of them . the parameters considered in this section are @xmath24 , @xmath74 and system size @xmath1 ranging from 30 to 60 elements . figure [ fig_ril ] shows the density of the distribution of the fraction @xmath49 of relevant variables , @xmath86 . the density relative to the most probable value increases with system size , and it appears that @xmath49 tends to be delta - distributed in the infinite size limit , as it is expected on the ground of the annealed master equation ( [ stable2 ] ) . we observe an excess of networks with very few relevant elements ( _ i.e. _ very many stable elements ) , consistently with the finite size effects discussed in last section . this excess seems to disappear in the infinite size limit . then we show the average length of the cycles in networks with @xmath44 relevant elements ( figure [ perril ] ) . this quantity increases almost exponentially with @xmath44 when @xmath56 is large , while its behavior is different for small @xmath54 . the crossover takes place at about @xmath87 . thus the number of relevant elements turns out to have a very important influence on the typical length of the cycles we have also measured the conditional distribution of the length of the cycles in networks with @xmath44 relevant elements . when @xmath44 is close to @xmath1 the distribution decays as a stretched exponential with an exponent smaller than one , very close to the one found in the unconditioned distribution . thus the deviation of the unconditioned distribution from the prediction of the annealed approximation , that predicts a much narrower distribution , is not a consequence of the existence of the relevant elements . y_2=_w_^2 , where @xmath72 is the attraction basin of cycle @xmath89 . we used the method proposed by derrida and flyvbjerg @xcite , that is based on the fact that @xmath88 is equal to the probability that two trajectories chosen at random end up on the same attractor . from our data ( not shown ) it appears that @xmath88 has a non monotonic behavior as a function of @xmath54 : for very small @xmath54 it decreases from the value 1 , corresponding to @xmath90 , reaches a minimum and rapidly increases . at large @xmath54 , @xmath88 does not seem to be correlated with @xmath54 ( at least within the statistical error , that is rather large ) . we will see in the next paper that the decreasing behavior at small @xmath54 can be interpreted as an effect of the modular organization of kauffman networks . we simulated systems with @xmath91 and the critical value @xmath92 . systems size ranges from 120 to 1600 . concerning the statistical properties of the attractors , these networks have a behavior very similar to that of the more studied @xmath93 , @xmath74 networks @xcite . in these networks , the number of relevant elements appears to scale as @xmath0 , in agreement with the argument presented in section 4 . the number of unstable elements , on the other hand , appears to scale as @xmath64 . this implies that the probability to extract at random an element which is relevant , scaling as @xmath94 , is approximately proportional to the square of the probability to extract at random an element which is relevant ( @xmath95 ) . these scaling laws can be observed both looking at the average quantities and looking at the whole distribution . the average number of unstable variables is found to follow the power law @xmath96 , with @xmath97 . we then define the rescaled variable @xmath98 , and we compare its probability density for various system sizes . as it can be seen in figure [ fig_unst ] the different curves superimpose within the statistical errors . this suggests that @xmath99 has a well defined probability density in the infinite size limit , although our data are rather noisy to state this point without doubts . we can distinguish in the distribution three different ranges with different characteristics : at vanishingly small values of @xmath99 ( ranging from @xmath100 up to @xmath101 ) the density decreases very fast . at intermediate values , roughly up to @xmath102 , it looks to decrease approximately as a power law with a small exponent ( the best fit exponents that we found range from 0.25 to 0.40 , showing some tendency to increase with system size ) . asymptotically , for large @xmath99 , the best fit is a stretched exponential , @xmath103 , with an exponent compatible with @xmath104 for all the systems that with studied with @xmath1 larger than 240 . the number of relevant variables was studied in a similar way . its average value increases as a power law of @xmath1 , @xmath105 , with @xmath106 . the rescaled variable @xmath107 looks to have a well defined distribution in the infinite size limit , as it is shown in figure [ fig_ril4 ] , where the probability density of @xmath108 is plotted for system sizes ranging from 120 to 1600 . for large @xmath109 the density of the distribution is well fitted by a stretched exponential , @xmath110 , with the exponent @xmath111 compatible with the value @xmath112 for system size larger than 240 . the average length of the cycles increases exponentially as a function of the number of relevant elements , for @xmath54 large , and more slowly for @xmath54 small , just as it happens in the chaotic phase . figure [ fig_perril4 ] shows on a logarithmic scale the behavior of the average length of the cycles as a function of the rescaled number of relevant elements , @xmath107 , for different system sizes at the critical point @xmath91 , @xmath92 . the average weight of the attraction basins , @xmath88 , has a non monotonic behavior as a function of the number of relevant elements , as it happens in the chaotic phase . the value of @xmath113 is one for @xmath114 , then decreases to a minimum value and increases very slowly , as it is shown in figure [ fig_yril4 ] , where @xmath115 is plotted against @xmath107 , for @xmath91 , @xmath92 and different system sizes . nevertheless , there are two important differences with respect to the chaotic phase : first , the range where @xmath113 is a decreasing function is much wider on the critical line than in the chaotic phase ; then , on the critical line the curves corresponding to a smaller @xmath1 value are lower , while in the chaotic phase the contrary holds . as a consequence , if we average @xmath113 over @xmath44 on the critical line , we get a quantity vanishing in the infinite size limit @xcite , while the average weight of the attraction basins is finite and very close to the random map value in the chaotic phase @xcite . this difference and the non monotonic @xmath44 behavior of @xmath113 have a clear interpretation in the framework of the modular organization of kauffman networks @xcite . we thus postpone to that paper the discussion of our results . kauffman ( 1969 ) , metabolic stability and epigenesis in random constructed genetic nets , _ j. theor . _ * 22 * , 437 - 467 . kauffman ( 1993 ) , _ origins of order : self - organization and selection in evolution _ , oxford university press derrida b. and y. pomeau ( 1986 ) , random networks of automata : a simple annealed approximation , _ biophys . lett . _ * 1*(*2 * ) , 45 - 49 b. derrida and g. weisbuch ( 1986 ) , _ j. physique _ * 47 * 1297 . hilorst and m. nijmajer ( 1987 ) , _ j. phys . paris , * 48 * _ , 185 . b. derrida and h. flyvbjerg ( 1986 ) , multivalley structure in kauffman s model : analogy with spin glasses , _ j.phys.a : math.gen._ * 19 * , l1003-l1008 h. flyvbjerg ( 1988 ) , an order parameter for networks of automata , _ a : math.gen . _ * 21 * l955-l960 h. flyvbjerg , n.j . kjaer ( 1988 ) , exact solution of kauffman model with connectivity one , _ j.phys . a : math.gen . _ * 21*(*7 * ) , 1695 - 1718 m. mezard , g. parisi , m.a . virasoro ( 1987 ) , _ spin glass theory and beyond _ , singapore : world scientific u. bastolla and g. parisi ( 1997 ) , the modular structure of kauffman networks , _ physica d _ u. bastolla and g. parisi ( 1996 ) , closing probabilities in the kauffman model : an annealed computation , _ physica d _ 98 , 1 . u. bastolla and g. parisi ( 1996 ) , the critical line of kauffman networks , _ journal of theoretical biology _ , in press . b. derrida , h. flyvbjerg ( 1986 ) , the random map model : a disordered model with deterministic dynamics , _ journal de physique _ * 48 * , 971 - 978 a. bhattacharijya and s. liang ( 1996 ) , power - law distribution in the kauffman net , _ physica d _ b. derrida and h. flyvbjerg ( 1987 ) , distribution of local magnetizations in random networks of automata , _ j.phys.a : math.gen._ * 20 * , l1107-l1112 .

this is the first of two papers about the structure of kauffman networks . in this paper we define the relevant elements of random networks of automata , following previous work by flyvbjerg @xcite and flyvbjerg and kjaer @xcite , and we study numerically their probability distribution in the chaotic phase and on the critical line of the model . a simple approximate argument predicts that their number scales as @xmath0 on the critical line , while it is linear with @xmath1 in the chaotic phase and independent on system size in the frozen phase . this argument is confirmed by numerical results . the study of the relevant elements gives useful information about the properties of the attractors in critical networks , where the pictures coming from either approximate computation methods or from simulations are not very clear . 48 pt -1.5 truecm -1.2 truecm 16.5 cm 23 cm @xmath2dipartimento di fisica , universit `` la sapienza '' , p.le aldo moro 2 , i-00185 roma italy @xmath3hlrz , forschungszentrum jlich , d-52425 jlich germany keywords : disordered systems , genetic regulatory networks , random boolean networks , cellular automata

arxiv

turbulence is a strong nonequilibrium phenomenon , exhibiting unpredictable behavior of the velocity field , which can result from the multiple degrees of freedom and nonlinearity of fluid systems @xcite . this situation complicates our understanding of turbulence , making it one of the unresolved problems in modern physics . the studies on turbulence in quantum fluids have possibility to shed new light on our understanding of turbulence . in quantum turbulence ( qt ) for quantum fluids such as superfluid helium and one - component atomic bose - einstein condensate ( bec ) @xcite , many quantized vortices with quantized circulation are nucleated , forming tangles . this situation is drastically different from classical turbulence ( ct ) realized in classical fluids because the circulation of velocity field is not quantized in this system . therefore , the element of qt is more obvious than that in ct , and qt is believed to be useful for the understanding of ct . recently , a new trend begins to appear in turbulence study in atomic becs , which is turbulence in multi - component becs @xcite . atomic becs have various characteristic features , one of which is a realization of multicomponent becs @xcite . in this system , there exists not only a velocity field but also a ( quasi ) spin field . owing to the spin degrees of freedom , various topological excitations such as monopole , skyrmion , knot , domain wall , and vortex therefore , in this system , one expects novel turbulence , in which both the velocity and spin fields are much disturbed and various topological excitations are generated . thus , this kind of turbulence should give us new observations for turbulence not found in conventional systems . previously , we studied turbulence in a spin-1 ferromagnetic spinor bec , which is a typical multicomponent bec @xcite . in this turbulence , the spin field is disturbed , so that we call it spin turbulence ( st ) . in our previous study , we focused on the spectrum of the spin - dependent interaction energy corresponding to the spin correlation , finding the characteristic @xmath1 power law . however , this observation sees only one side of st . in st , the velocity and spin fields interact with each other , so that a coupled turbulence with two fields can be realized , lending the possibility of showing a property of the velocity field not seen in conventional ct and qt . studying the both sides of turbulence completes the story . in this paper , we treat this problem , focusing on the spectrum of the superflow kinetic energy in a spin-1 ferromagnetic spinor bec . the spectrum of the kinetic energy is theoretically and numerically found to show a kolmogorov spectrum through the interaction between the spin and velocity fields . the kolmogorov spectrum refers to the @xmath0 power law in the kinetic energy spectrum , which is known to appear in ct and qt @xcite . this spectrum is considered to be related to the vortex dynamics , so that it is significant to confirm it for understanding the turbulence addressed here . furthermore , when the @xmath0 power law is sustained , the spectrum of the spin - dependent interaction energy exhibits a @xmath1 power law simultaneously . therefore , we obtain the coupled turbulence with the disturbed spin and superfluid velocity fields sustaining the two power laws , anew calling it spin - superflow turbulence ( sst ) instead of st . we consider a bec of spin-1 bosonic atoms with mass @xmath2 at zero temperature without trapping and magnetic fields . this system is well described by the macroscopic wave functions @xmath3 ( @xmath4 ) with magnetic quantum number @xmath5 , which obey the spin-1 spinor gorss - pitaevskii ( gp ) equations @xcite given by @xmath6 in this paper , roman indices that appear twice are to be summed over @xmath7 , 0 , and 1 , and greek indices are to be summed over @xmath8 , @xmath9 , and @xmath10 . the parameters @xmath11 and @xmath12 are the coefficients of the spin - independent and spin - dependent interactions , which are expressed by @xmath13 and @xmath14 , respectively . here , @xmath15 and @xmath16 are the @xmath17-wave scattering lengths corresponding to the total spin-@xmath18 and spin-2 channels . the total density @xmath19 and the spin density vector @xmath20 ( @xmath21 ) are given by @xmath22 and @xmath23 , where @xmath24 are the spin-1 matrices . the sign of the coefficient @xmath12 drastically changes the spin dynamics . in this paper , we consider the ferromagnetic interaction @xmath25 . in this paper , we focus on the energy spectra for the superflow kinetic and the spin - dependent interaction energy . the kinetic energy of superfluid velocity @xmath26 per unit mass is given by @xmath27 where @xmath28 and @xmath29 is a total particle number . the superfluid velocity @xmath30 is given by @xmath31 by using the fourier series @xmath32 , we define the spectrum for the kinetic energy per unit mass : @xmath33 where @xmath34 and @xmath35 are given by @xmath36 and @xmath37 , respectively , with system size @xmath38 and spatial dimension @xmath39 . similarly , the spectrum of the spin - dependent interaction energy per unit mass is defined by @xmath40 where @xmath41 is defined by @xmath42 } $ ] with the fourier transformation @xmath43 } = \int \cdot \hspace{0.5mm}e^{-i\bm{k}\cdot \bm{r } } d\bm{r}/v$ ] and @xmath44 . we discuss the possibility for a kolmogorov spectrum in sst with hydrodynamic equations obtained from eq . ( [ gp ] ) . these hydrodynamic equations are derived in @xcite , being composed of the equations of the total density @xmath19 , the superfluid velocity @xmath26 , the spin vector @xmath45 , and the nematic tensor @xmath46 with @xmath47/2 $ ] . we apply the following three approximations to these hydrodynamic equations : ( i ) the macroscopic wave functions are expressed by the fully magnetized state , ( ii ) the total density is almost uniform ( @xmath48 ) , and ( iii ) the magnitude of velocity @xmath26 is much smaller than the density sound velocity @xmath49 . these similar approximations are discussed in our previous papers @xcite . the approximation ( i ) leads to the relation between the spin vector and nematic tensor @xcite : @xmath50 thus , by eliminating the nematic tensor in the hydrodynamic equations of @xcite , we obtain the following equations @xmath51 @xmath52 @xmath53 with the approximations ( ii ) and ( iii ) , eqs . ( [ 1 ] ) - ( [ 3 ] ) become @xmath54 , \label{spin_motion}\end{aligned}\ ] ] @xmath55 in eq . ( [ velocity_motion ] ) , the inertial term @xmath56 is smaller than the other terms @xcite , but we retain this term for the following explanation . we apply a kolmogorov - type dimensional scaling analysis @xcite to eqs . ( [ spin_motion ] ) and ( [ velocity_motion ] ) , obtaining the @xmath0 power law . we consider the scale transformation @xmath57 and @xmath58 . then , if @xmath59 and @xmath60 are transformed to @xmath61 and @xmath62 , eqs . ( [ spin_motion ] ) and ( [ velocity_motion ] ) are invariant . thus , the velocity field satisfies @xmath63 with a nondimensional coefficient @xmath64 . then the spatial and temporal dependence of @xmath65 is same as that of @xmath26 , which leads to @xmath66 with @xmath67 . thus , in sst , the spectrum of the superflow kinetic energy can be determined by the kinetic energy flux @xmath68 and the coefficient @xmath69 , which , by using a kolmogorov - type dimensional analysis , leads to @xmath70 the coefficient @xmath71 is nondimensional , which corresponds to the kolmogorov constant in ct . therefore , the spectrum of the kinetic energy of sst can obey the kolmogorov spectrum . applying a similar analysis to the spin field , we obtain the relation @xmath72 with a dimensional coefficient @xmath73 , which leads to the @xmath1 power law given by @xmath74 with the spin - dependent interaction energy flux @xmath75 and a dimensional coefficient @xmath76 . this was discussed in the previous study @xcite . we note that this @xmath0 power law in sst is much different from that in ct . in three ( two)-dimensional ct , there is the direct ( inverse ) energy cascade , where the @xmath0 power law is generated by the inertial term @xmath56 in the navier - stokes equation @xcite . on the contrary , in sst , the spatial gradient of the spin vector in eq . ( [ velocity_motion ] ) leads to the @xmath0 power law because , in eq . ( [ velocity_motion ] ) , the order estimation @xcite finds that the inertial term is smaller than the nonlinear spin term in the range @xmath77 with the density coherence length @xmath78 . this suggests that the mechanism responsible for the @xmath0 power law in sst should be different from that in ct . finally , the assumptions used in the derivation of the @xmath0 and @xmath1 power laws are discussed . we use five assumpotions , which are ( i ) the macroscopic wave functions are expressed by the fully magnetized state , ( ii ) the total density is almost uniform ( @xmath48 ) , ( iii ) the magnitude of velocity @xmath26 is much smaller than the density sound velocity @xmath79 , ( iv ) the spin and velocity fields are scale invariant , and ( v ) the energy flux is independent of the wave number . it is difficult to theoretically confim the validity of these assumptions . then , we consider that the validity may be indirectly confirmed if the @xmath0 and @xmath1 power laws based on these assumptions appear in the numerical result . in the following , we show our numerical method and results to confirm these theoretical considerations . [ b ] components and the rotations for the velocity and spin fields in sst at @xmath80 . ( a ) and ( b ) show the @xmath8 components for the velocity @xmath26 and spin @xmath81 fields , and ( c ) and ( d ) show the @xmath10 component of their rotation . the system size is @xmath82.,width=321 ] [ t ] in our numerical calculation , we introduce a phenomenological small - scale energy dissipation term into eq . ( [ gp ] ) for the following reasons . in three - dimensional ct , an energy cascade from low to high wave number is assumed , and the energy in the high - wave - number region is considered to dissipate @xcite . unless this kind of dissipation takes place , energy accumulates in the high - wave - number region , which can break the power law in the spectrum . based on this description of the energy cascade in three - dimensional ct , in the previous study of qt in the one - component becs @xcite , a phenomenological small - scale dissipation was added to the one - component gp equation given by @xmath83 @xmath84},\end{aligned}\ ] ] where the macroscopic wave function and the interaction coefficient for the one - component bec are denoted by @xmath85 and @xmath86 , respectively . @xmath87 is the fourier component @xmath88 } $ ] of this wave function . the function @xmath89 is defined by @xmath90 with the step function @xmath91 and the strength of dissipation @xmath92 , which dissipates the energy in the high - wave - number region larger than the wave number @xmath93 corresponding to the coherence length in the one - component gp equation . also , this dissipation reduces the particle number , so that , in the study of ref . @xcite , the chemical potential is adjusted for it s conservation . thus the chemical potential @xmath94 has a time dependence . we introduce a similar phenomenological dissipation term into the spin-1 spinor gp equations ( [ gp ] ) . in a spin-1 spinor bec , there are two characteristic lengths : the density coherence length @xmath95 and the spin coherence length @xmath96 , which are defined by @xmath97 and @xmath98 . in the usual experiments , @xmath99 is larger than unity , which leads to the condition @xmath100 . the size of the spin structure such as the spin domain wall and the spin vortex is on the order of @xmath101 . with reference to qt in the one - component bec , we expect that the energy dissipation occurs for wave numbers greater than the wave number @xmath102 corresponding to the spin coherence length @xmath103 . therefore , we add a phenomenological small - scale energy dissipation to eq . ( [ gp ] ) , which is given by @xmath104 @xmath105 } , \label{dgpk2}\end{aligned}\ ] ] where @xmath106 and @xmath107 is defined by @xmath108}$ ] and @xmath109 . in the previous study , the chemical potential has a time dependence in order to conserve the total particle number . however , in our calculation , we do not adjust the chemical potential @xmath110 , because the total particle number hardly decreases in sst . we describe the parameters and the initial state in our numerical calculation of eqs . ( [ dgpk1 ] ) and ( [ dgpk2 ] ) . all our numerical results are obtained in a two - dimensional system , whose size is @xmath82 . to generate sst , we prepare the dynamically unstable state as the initial state , which is the counterflow state @xcite . this state is given by @xmath111 with a relative velocity @xmath112 . in our numerical calculation , we set @xmath113 . the dependence of numerical result on @xmath112 is discussed in sec . vi a. the usual experimental ratio @xmath114 for @xmath115 is about @xmath116 @xcite , but , in our numerical calculation , we set @xmath117 with a positive @xmath11 and negative @xmath12 to grow the dynamical instability quickly . we use the strength of the dissipation @xmath118 and the chemical potential @xmath119 . in the initial state , we add a small white noise contribution to cause the counterflow instability . figures [ fig1](a ) and 1(b ) show the distribution of the @xmath8 components of the velocity and spin fields in sst at @xmath120 with @xmath121 . through the counterflow instability , these two fields are disturbed . the @xmath10 components of the rotation for these two fields are shown in figs . [ fig1](c ) and 1(d ) , where the rotations are also disturbed @xcite . in a spin-1 spinor bec , the superfluid velocity is related to the spin field through the mermin - ho relation , so that the vortical field can be continuous , which is much different from the one - component bec @xcite . actually , as seen in fig . [ fig1](c ) , the vortical field @xmath122_{z } $ ] has a smooth spatial dependence . the time development of the spectrum of the spin - dependent interaction and the superflow kinetic energy is shown in fig . [ fig2 ] . in the early stage of the instability , as shown in fig . [ fig2](a ) , these spectra have a peak corresponding to the most unstable wave number @xmath123 for the counterflow instability , which was theoretically obtained by using the bogoliubov - de gennes equations for this initial state @xcite . this wave number @xmath124 is the energy injection scale . as time progresses , the energy is transferred from low to high wave number , as seen in fig . [ fig2](b ) . after time @xmath125 , the spectra of the velocity and the spin - dependent interaction energy exhibit the @xmath0 and @xmath1 power laws in figs . [ fig2](c ) and ( d ) . in fig . [ fig2 ] ( d ) , the range @xmath126 of these scaling laws is not so wide , but the scaling behavior is consistent with our consideration in sec . therefore , we conclude that our numerical result confirms the -5/3 and @xmath1 power laws . we comment on the scaling range . this narrow scaling range may come from how to excite to the initial state . as shown in fig . [ fig2](a ) , the energy is injected at the wave number @xmath127 , which roughly determines the lower limit of the scaling range . therefore , if we use another method with lower @xmath124 for generating sst , the scaling range may be wider . in our calculation , the @xmath0 power law applies in the range @xmath128 . for @xmath129 , we confirm that the spectrum of the kinetic energy begins to deviate from the @xmath0 power law as shown in fig . 3 ; this is caused by the shortage of energy in the low - wave - number region and the energy accumulation near @xmath102 . on the other hand , the @xmath1 power law in the spectrum of spin - dependent interaction energy sustains even for @xmath129 . as time sufficiently passes , the spectrum in low wave number region grows , and this spectrum has a configuration similar to that in our previous study @xcite . this growth is discussed in sec . b. finally , we comment on what happens if the energy dissipation is absent . we perform numerical calculation without the dissipation , which shows two following behaviors : ( i ) the spectrum of superflow kinetic energy still shows the @xmath0 power law , but the time period sustaining this power law becomes shorter , and ( ii ) the fluctuation of the spectrum is larger . these similar behaviors were observed in the previous study for qt of one - component bec too @xcite . [ t ] ( a ) 800 and ( b ) 1500 . the dotted lines in the upper and lower graphs are proportional to @xmath130 and @xmath131 , respectively.,width=321 ] we now consider what structure of velocity field leads to the kolmogorov @xmath0 power law in sst . in qt , the vortical velocity field seems to be important for the kolmogorov spectrum . then , to investigate the vortical flow of @xmath132 in sst , we decompose the vector @xmath133 into incompressible @xmath134 and compressible @xmath135 parts @xcite . the helmholtz theorem leads to @xmath136 , where the relations @xmath137 and @xmath138 are satisfied . thus , the superflow kinetic energy per unit mass is expressed by @xmath139 with @xmath140 ( @xmath141 ) . using the fourier series @xmath142 ( @xmath141 ) , we can define the spectra for each kinetic energy per unit mass as @xmath143 we calculate the time dependence of @xmath144 in fig . [ fig4](a ) ; one can see that the incompressible superflow kinetic energy is much larger than the compressible one . this can be caused by ( i ) the condition @xmath145 under which the total density is hard to disturb , and ( ii ) the dissipation , which prevents the total density modulation from accumulating . figure [ fig4](b ) shows the spectrum @xmath146 of the compressible kinetic energy at @xmath147 , which deviates from the @xmath0 power law in comparison with @xmath148 in fig . [ fig2](d ) . therefore , the vortical structure of @xmath133 is significant for the kolmogorov spectrum in sst , which is similar to the situation of qt . [ t ] ( @xmath149 ) and ( b ) spectrum for compressible superflow kinetic energy at @xmath150 . the dotted line in ( b ) is proportional to @xmath131.,width=321 ] we numerically confirm that the @xmath0 power law in sst is different from that in ct . in sec . iii , we point out that the @xmath0 power law in sst originates from the nonlinear spin term of eq . ( [ velocity_motion ] ) . the magnitude of the inertial and nonlinear spin terms is dependent on the scale , so that we perform the fourier transform for these two terms , comparing the magnitude of the fourier components in order to confirm the argument of sec . specifically , we numerically calculate the following quantities : @xmath151 @xmath152 where @xmath153 and @xmath154 are defined by @xmath155 , \end{aligned}\ ] ] @xmath156.\end{aligned}\ ] ] figure 5 shows the wave number dependence of @xmath157 and @xmath158 . one sees that the nonlinear spin term @xmath159 is larger than the inertial term @xmath160 , and these ratios are about @xmath161 in the scaling range @xmath126 of fig . 2 where the @xmath0 power law appears . thus we can consider that the kolmogorov spectrum in sst is generated by the nonlinear spin term . [ t ] ( @xmath162 ) at @xmath163 ( a ) 500 and ( b ) 700 . , width=321 ] we now discuss the dependence of the spectra of the superflow kinetic and spin - dependent interaction energy on @xmath112 . when the relative velocity @xmath112 is large , the period sustaining the kolmogorov spectrum in the superflow kinetic energy is numerically confirmed to become short . this can be caused by the following two reasons . one is ( i ) modulation of total density , and the other is ( ii ) high energy injection wave number . as for ( i ) , when the relative velocity is large , the total density is easy to fluctuate , which leads to the growth of the compressible velocity field . actually , we numerically confirm the increase of the compressible kinetic energy . in our derivation of the power law , the uniformity of the total density is assumed , so that this increase shortens the time period sustaining the kolmogorov spectrum . as for ( ii ) , the most unstable wave number becomes high when the relative velocity is large @xcite , which leads to the high injection wave number @xmath124 . as discussed in sec . v. a , this generates the narrow scaling range , which disturbs the appearance of the power law . because of the above reasons , in our calculation , we set @xmath113 . on the other hand , in the spectrum of spin - dependent interaction energy , the @xmath1 power law can appear even under the condition of large relative velocity . this was confirmed in our previous study @xcite . at present , we do not understood why this spectrum can exhibit the @xmath1 power law independently of the relative velocity . in this paper , we perform the two - dimensional numerical calculation for sst , so that the possibility of inverse cascade in sst is discussed . as briefly described in sec . iii , in three - dimensional ct , the kinetic energy is transported from low wave numbers to high ones , which is called direct energy cascade . however , in two - dimensional ct , the energy inverse cascade occurs , where the kinetic energy is transported from high wave numbers from low ones . this inverse cascade is caused by the conservation of ensthropy as well as that of kinetic energy in two - dimensional fluid system @xcite . at present , we consider that the inverse energy cascade for superflow kinetic energy does not occur in the two - dimensional sst because the enstrophy in ferromagnetic spin-1 spinor bec is not conserved . actually , our numerical calculation does not exhibit the sign of the energy inverse cascade in the superflow kinetic energy . on the other hand , as for the spin - dependent interaction energy , the inverse energy cascade may occur because our previous study and fig . [ fig3 ] seem to exhibit the growth of spectrum of spin - dependent interaction energy in low wave number region as the time sufficiently passes . this growth may be caused by the ferromagnetic interaction since it tends to align the spin density vector and make the spin domain . however , as shown in fig . [ fig2 ] , the spectrum apparently seems to exhibit the sign of direct energy cascade . thus , at the moment , we does not sufficiently understand the direction of energy cascade for the spin - dependent interaction energy . we theoretically and numerically studied sst in a spin-1 ferromagnetic spinor bec at zero temperature by using the spin-1 spinor gp equations , finding that both the @xmath0 and @xmath1 power laws appear in the spectrum of the superflow kinetic and the spin - dependent interaction energy . first , we discussed the possibility for the kolmogorov spectrum in sst , pointing out that this spectrum can be generated by the nonlinear spin term . second , we showed the numerical results of the spin-1 spinor gp equation with the phenomenological small - scale energy dissipation , whereby sst in the two - dimensional system was obtained by the counterflow instability . our numerical results indicated that both the @xmath0 and @xmath1 power laws appeared . furthermore , we estimated the magnitude of the inertial term and the nonlinear spin term of eq . ( [ velocity_motion ] ) in the wave number space , numerically confirming that the kolmogorov spectrum in sst can be generated by the latter term . k. f. was supported by a grant - in - aid for jsps fellows grant number 262524 . m. t. was supported by jsps kakenhi grant number 26400366 and mext kakenhi `` fluctuation @xmath164 structure '' grant number 26103526 . 99 p. a. davidson , _ turbulence : an introduction for scientists and engineers _ ( oxford university press , oxford , 2004 ) . u. frisch , _ turbulence _ ( cambridge university press , cambridge , 1995 ) . , edited by w. p. halperin and m. tsubota ( elsevier , amsterdam , 2008 ) , vol . w. f. vinen , j. low temp . phys . * 161 * , 419 ( 2010 ) . l. skrbek and k. r. sreenivasan , phys . fluids * 24 * , 011301 ( 2012 ) . c. nore , m. abid , and m. e. brachet , phys . * 78 * , 3896 ( 1997 ) ; phys . fluids * 9 * , 2644 ( 1997 ) . n. g. parker and c. s. adams , phys . rev . lett . * 95 * , 145301 ( 2005 ) . m. kobayashi and m. tsubota , phys . lett . * 94 * , 065302 ( 2005 ) ; j. phys . . jpn . * 74 * , 3248 ( 2005 ) . m. kobayashi and m. tsubota , phys . a * 76 * , 045603 ( 2007 ) . horng , c .- h . hsueh , s .- w . su , y .- m . kao , and s .- c . gou , phys . a * 80 * , 023618 ( 2009 ) . e. a. l. henn , j. a. seman , g. roati , k. m. f. magalhaes , and v. s. bagnato , phys . * 103 * , 045301 ( 2009 ) . a. c. white , c. f. barenghi , n. p. proukakis , a. j. you d , and d. h. wacks , phys . lett . * 104 * , 075301 ( 2010 ) . b. nowak , j. schole , d. sexty , and t. gasenzer , phys . a * 85 * , 043627 ( 2012 ) . m. t. reeves , b. p. anderson , and a. s. bradley , phys . a * 86 * , 053621 ( 2012 ) . k. e. wilson , e. c. samson , z. l. newman , t. w. neely , and b. p. anderson , annu . * 1 * , 261 ( 2013 ) . h. takeuchi , s. ishino , and m. tsubota , phys . lett . * 105 * , 205301 ( 2010 ) . k. fujimoto and m. tsubota , phys . a * 85 * , 033642 ( 2012 ) . k. fujimoto and m. tsubota , phys . a * 88 * , 063628 ( 2013 ) . m. karl , b. nowak , and t. gasenzer , sci . rep . * 3 * , 2394 ( 2013 ) . m. karl , b. nowak , and t. gasenzer , phys . a * 88 * , 063615 ( 2013 ) . d. kobyakov , a. bezett , e. lundh , m. marklund , and v. bychkov , phys . rev . a * 89 * , 013631 ( 2014 ) . b. villase@xmath165or , r. zamora - zamora , d. bernal , and v. romero - roch@xmath166n , phys . rev . a * 89 * , 033611 ( 2014 ) . k. kasamatsu , m. tsubota , and m. ueda , int . j. mod b * 19 * , 1835 ( 2005 ) . y. kawaguchi and m. ueda , phys . rep . * 520 * , 253 ( 2013 ) . d. m. stamper - kurn and m. ueda , rev . * 85 * , 1191 ( 2013 ) . t. ohmi and k. machida , j. phys . . jpn . * 67 * , 1822 ( 1998 ) . ho , phys . rev . lett . * 81 * , 742 ( 1998 ) . e. yukawa and m. ueda , phys . a * 86 * , 063614 ( 2012 ) . as shown in fig . 1 , the velocity is smaller than @xmath79 , whose order is about @xmath167 . the magnitude @xmath168 of the spin vector is of the order of unity because of the ferromagnetic interaction . then , by using the nondimensional form of eq . ( [ velocity_motion ] ) , we can estimate the order of each term . m. ottaviani and j. a. krommes , phys . * 69 * , 2923 ( 1992 ) . t. watanabe , h. fujisaka , and t. iwayama , phys . e * 55 * , 5575 ( 1997 ) . in contrast to a one - component bec , the superfluid velocity and the vorticity in fig . ( [ fig1 ] ) do not diverge . this is because the vortex core is filled with other components . however , if the instability is much stronger , the velocity may diverge since the other components may escape from the vortex core . e. g. m. van kempen , s. j. j. m. f. kokkelmans , d. j. heinzen , b. j. verhaar , phys . lett . * 88 * , 093201 ( 2002 ) .

spin - superflow turbulence ( sst ) in spin-1 ferromagnetic spinor bose - einstein condensates is theoretically and numerically studied by using the spin-1 spinor gross - pitaevskii ( gp ) equations . sst is turbulence in which the disturbed spin and superfluid velocity fields are coupled . applying the kolmogorov - type dimensional scaling analysis to the hydrodynamic equations of spin and velocity fields , we theoretically find that the @xmath0 and @xmath1 power laws can appear in spectra of the superflow kinetic and the spin - dependent interaction energy , respectively . our numerical calculation of the gp equations with a phenomenological small - scale energy dissipation confirms sst with the coexistence of disturbed spin and superfluid velocity field with two power laws .

arxiv

low - luminosity active galactic nuclei ( llagns ) , operationally defined as a h@xmath0 luminosity of @xmath1 , reside in about 40 per cent of nearby bright galaxies , and occupy majority of the population of agns @xcite . its low luminosities are thought to be caused by very low accretion rate on to super - massive black holes . such accretion has often been explained by the model of optically - thin disc such as an advection - dominated accretion flow ( adaf ; @xcite ) , rather than optically - thick disc ( ` standard disc ' ; ) . it is predicted that the adaf has a ` submillimetre bump ' in its spectral - energy distribution ( sed ) , while the standard disc is responsible for a big - blue bump in the sed of luminous agns such as quasars . since a high brightness temperature of @xmath2 k in radio bands is expected from the adaf model , the first imaging investigations for accretion discs must be promising with future very - long - baseline - interferometry ( vlbi ) instruments that will provide micro - arcsec resolutions @xcite . therefore , the observational evidence of the adaf is crucial in the llagn radio sources . although the adaf model successfully explains the broadband spectra of llagns , there is a significant gap between observations and the adaf spectrum especially at low - frequency radio bands ( e.g. , @xcite ) . this indicates that additional unknown emission does exist , and putative emission from the accretion disc may be buried under it . the submillimetre bump means a highly inverted spectrum at centimetre - to - submillimetre wavelengths of spectral index @xmath3 , where @xmath4 , @xmath5 is flux density at frequency @xmath6 @xcite . observations for llagns have been carried out exclusively at centimetre bands where high sensitivities are available , because most of llagns are very faint radio sources . about half of low - luminosity seyfert galaxies and low - ionization nuclear emission - line regions ( liners ; @xcite ) hold a compact radio core @xcite , and at least 25 per cent of transition objects , which are hypothesized to be liner/-nucleus composite systems @xcite , also hold a compact core @xcite , at 15 or 8.4 ghz in 0.152.5 arcsec resolutions . most of the cores show nearly flat spectra ( @xmath7 ) in 0.5-arcsec resolution : that is evidence for jet domination @xcite . although slightly inverted spectra have been found in unresolved compact cores of several llagns in milli - arcsec resolutions of vlbis , their radio - to - x - ray luminosity ratios suggest that another significant radio component seems to contribute in addition to original fluxes from the adaf model @xcite . to explain this radio excess , an jet adaf model has been proposed @xcite . thus , at centimetre bands , the contamination that is believed to be from jets has prevented the adaf from being revealed . in the present paper , we report millimetre survey and submillimetre data analyses for many llagns . although technically difficult , a high - frequency observation is very promising to detect the inverted spectrum of the adaf because of following two advantages : ( 1 ) spectral measurements at high frequencies are less affected by non - thermal jets , which generally show power - law spectra ( @xmath8@xmath9 ) , and ( 2 ) the adaf predicts larger flux densities at higher frequencies . in fact , it has been obsevationally confirmed that flux densities at millimetre wavelengths are 510 times larger than those at centimetre wavelengths in sgr a * ( e.g. @xcite ) , which is the nearest low - accretion rate system from us . however , contamination from diffuse dust may be harmful at @xmath10 ghz when we use a poor spatial resolution . therefore , the use of a beam as small as possible and the estimations of every conceivable contamination are essential . the present paper is structured as follows . sample selection is described in section [ section : sample ] . observations and data reduction are described in section [ section : obs&reduction ] . in section [ section : results ] , we report these results and the estimations of diffuse contamination in our millimetre measurements . our results of submillimetre photometry are utilized only for estimation for dust contribution . the origins of the spectra are discussed in section [ section : discussion ] from the relation between the high - frequency radio spectrum and low - frequency radio core power . finally , we summarize in section [ section : summary ] . our sample , ` vlbi - detected llagn sample ' , covers 20 out of 25 all - known llagn radio sources that have been detected with vlbis ( column [ 1 ] of table [ table1 ] ) . the other five llagns are ngc 4235 , ngc 4450 @xcite , ngc 524 , ngc 5354 , and ngc 5846 @xcite , which were recently newly vlbi - detected , but whose reports had not yet been published when we plan the survey . it is difficult to estimate resultant selection bias including into our sample , because of multiplicity in selection criteria across the past vlbi surveys . however , at least we can say that 16 out of 20 sources of our sample are all known llagns that are at @xmath11 mpc with @xmath12 mjy at 15 ghz , and the other four sources ( ngc 266 , ngc 3147 , ngc 3226 , and ngc 4772 ) are more distant . it may be , therefore , as a moderately good distance - limited and radio - flux - limited sample for llagns with a compact radio core . we made millimetre continuum observations for 16 out of 20 targets using the nobeyama millimetre array ( nma ) , at the nobeyama radio observatory ( nro ) , with d configuration that is the most compact array configuration . we had precluded the other four sources from our observation list because they had already been observed at @xmath13 mm in the past ( ngc 266 , @xcite ; ngc 3031 , @xcite ; ngc 4258 , doi et al . in prep . ; ngc 4486 , @xcite ) . our campaign spent more than 20 observational days between 2002 november 28 and 2003 may 25 . most of weak targets were observed over several days for integration . visibility data were obtained with double - sided - band receiving system at centre frequencies of 89.725 and 101.725 ghz , which were doppler - tracked . we used the ultra wide - band correlator ( uwbc ; * ? ? ? * ) , which can process a bandwidth of 1 ghz per each sideband , i.e. , 2 ghz in total . the wide bandwidth gives the nma a very high sensitivity for continuum observations . even if the systemic velocities of our targets and galaxies rotations of several hundred km s@xmath14 are accounted for , the observing band can avoid possible contaminations from several significant line emissions : @xmath15co(@xmath16 ) , c@xmath17o(@xmath16 ) , hcn(@xmath16 ) , hco@xmath18(@xmath16 ) , and sio(@xmath19 ) . a system noise temperature , @xmath20 , was typically about 150 k in the double - sided bands . for observations of each target , we made scans of a reference calibrator close to the target every 20 or 25 minutes for gain calibration . bandpass calibrators of very bright quasars were scanned once a day . the data were reduced using the uvproc - ii package @xcite , developed at the nro , by standard manners , including flagging bad data , baseline correction , opacity correction , bandpass calibration and gain calibration . to achieve higher sensitivity , visibilities of both the sidebands were combined in the same weight , which resulted in a centre frequency of 95.725 ghz . each daily image was individually made in natural weighting and deconvolved using the aips software , developed at the national radio astronomy observatory ( nrao ) . statistically significant variability was detected from the daily images in several llagns , which will be reported in a separate paper . in the present paper , the visibilities of all days were combined by weighting with @xmath21 , and then imaged . the half - power beam widths ( hpbws ) of synthesized beams were typically @xmath22 arcsec , corresponding to @xmath23660 pc at a mean distance of 18.9 mpc in our sample . the flux scales of the gain calibrators were derived with the uncertainty to 10 per cent by relative comparisons to the flux - known calibrators , such as uranus , neptune , or mars . these were scanned quasi - simultaneously when they were at almost the same elevations . the rms of noise on the images were estimated from statistics on off - source blank sky with the imean task of the aips . source identifications and measurements of flux densities were done on image domain by elliptical gaussian profile fitting with the jmfit task of the aips . we derived total errors in the flux measurements from root sum square of the errors in the gaussian fitting ( including thermal noise ) and the flux scaling . we attempted to measure flux densities at submillimetre band from the nuclear regions using archival data . we searched the data that had been obtained at 347.38 ghz ( @xmath24 ) using submillimetre common user bolometer array ( scuba ; @xcite ) on the james clerk maxwell telescope ( jcmt ) . the data of 14 out of 20 objects of the vlbi - detected sample were available to the public on 2004 . the jcmt measurements for five objects have already been reported ( ngc 266 , @xcite ; ngc 4258 , doi et al . in prep . ; ngc 4374 , @xcite ; ngc 4472 , @xcite ; ngc 4486 , @xcite ) . four objects had been observed in jiggle - map mode . we reduced these data using the scuba user reduction facility ( surf ) package . standard reduction procedures were used , including flat fielding , flagging of transient spikes , correction for extinction , pointing correction , sky removal and flux - density scaling . flux calibrators ( uranus , mars , or crl 618 ) gave us flux scaling factors ( we assumed its uncertainty to @xmath2315 per cent ) and an effective beam size of 15.1 arcsec . the derivations of the rms of noise on images , source identifications , measurements of flux densities , and total error estimations were done in the same manners as the nma reductions described in section [ subsection:3 mm ] . the other five objects had been observed in photometry mode . we followed standard reduction procedures , including flat fielding , flagging of transient spikes , correction for extinction , sky removal , averaging , and flux - density scaling , using the surf package . the photometry can measure a flux density from a beam - size region at the nucleus , i.e. , an intensity . we derived total errors in flux measurements from root sum square of the errors of the image noise and the flux scale . [ cols="<,>,^,^,>,>,^,^,^ " , ] [ table4 ] note . column are : * ( 1 ) * galaxy name ; * ( 2 ) * core power detected in vlbi observations at 5 ghz , except for ngc 4258 at 22 ghz ; * ( 3 ) * presence of jet feature in the vlbi image . ` s ' represents sub - pc - scale jets . ` p ' represents pc - scale jets ; * ( 4 ) * reference for the vlbi observation , as listed below ; * ( 5 ) * presence of jet feature in vla image . ` k ' represents kpc - scale jets ; * ( 6 ) * reference for the vla observation , as listed below ; * ( 7 ) * spectral category , the same as table [ table3 ] . the sources showing steep spectra have exclusively high vlbi - core powers , @xmath25 w hz@xmath14 . note . reference : * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ; * * : @xcite ) . the dot - dashed line represents the radio power limit that the adaf on a black hole with @xmath26 m@xmath27 can generate . ] we have suggested in the previous section that jets and the adaf seem to coexist in llagns . however , only an unresolved structure has been found in milli - arcsec images for most of the vlbi - detected llagns @xcite . on the other hand , pc / sub - pc - scale jet structures have been found in several llagns . we find from the core power ranking in table [ table4 ] that the presence of vlbi - detectable pc - scale jets depends on the core radio powers . the pc - scale jets have been found exclusively in high - core - power sources . in low - core - power sources , no pc - scale jet has been detected in vlbi images even with moderately high dynamic ranges of @xmath28 @xcite . can not the low - core - power sources generate high - brightness jets ? the cases of ngc 3031 and ngc 4258 are very suggestive . sub - pc - scale jets have been revealed in their nuclei ( e.g. , * ? ? ? * ; * ? ? ? * ) , nevertheless , they are low - core - power llagns . note that the two llagns are the nearest two sources in the vlbi - detected sample ( table [ table3 ] ) . we speculate that all low - core - power sources also have sub - pc - scale or smaller jets , and that such structures would be revealed if they were at less than several mpc like ngc 3031 and ngc 4258 . the detectability of small - scale jets might merely depend on distance . if that is true , the 5-ghz vlbi - core power should contain jet contributions significantly . additionally , ngc 3031 and ngc 4258 are very suggestive cases also about coexistence with the adaf disc . although they show clear evidence of jets , their high - frequency spectra are highly inverted , @xmath29 and @xmath30 , respectively . these are the highest two spectral indices in the vlbi - detected llagns with negligible diffuse contaminations . if their millimetre emissions originate in the adaf discs , the two llagns would be examples for the coexistence of the jets and adaf . we mention apparent exceptions . ngc 266 and ngc 3147 show steep spectra , but have no detectable jet in spite of their high core powers ( table [ table4 ] ) . since the two llagns are the most two distant llagns in our sample , it could be somewhat hard to reveal their putative small - scale jet structures . their high radio powers might be also responsible for the dominance of such small - scale jets . ngc 5866 show a definitely steep spectrum , in spite of a low core power . its nuclear activity type , ` transition ' , might be related to the jet domination ; another transition object , ngc 4552 , also shows a definitely steep spectrum . however , our limited sample of transition objects can not allow us to discuss it in detail . there appears a clear difference in the spectral properties between type 1s ( s1.5 , s1.9 , and l1.9 ) and type 2s ( s2 , l2 , and t2 ) , as seen in table [ table3 ] . most of the type 1s show inverted spectra , while most of type 2s show steep spectra . we suggest this difference conflicts with the unified scheme @xcite . in this scheme , many observed properties are determined basically only by a viewing angle with respect to the obscuring dusty torus surrounding the nucleus . type-1 seyferts , with broad permitted optical emission lines , are those in which the torus is viewed closer to face - on orientation . type-2 seyferts lack broad permitted lines , except in polarized flux , and thought to be seen with the line of sight passing through the torus in an orientation closer to edge - on . this classification has been expediently used in liners and transitions ( e.g. , @xcite ) . since the radio emission is transparent through the dusty torus , luminosities and spectral indices of type 1s and 2s should look just the same to the observer unless affected by relativistic beaming . however , radio powers of type 2s tend to be significantly larger than those of type 1s in our sample . it seems that the difference is due to larger contribution of jets in type 2s , nevertheless , which would be less affected by relativistic beaming than type 1s . there are also the other reports about differences between type 1s and type 2s in llagns . @xcite found that low - luminosity type-1 seyferts rather show somewhat stronger centimetre radio emission than low - luminosity type-2 seyferts . @xcite have reported that type-2 llagns are underluminous in an x - ray band compared to type-1 ones with the same h@xmath0 luminosity . there are ` intrinsically true type-2 seyferts ( without broad lines even in polarized flux ) ' especially in the low - luminosity range ( @xcite , and reference therein ) . thus , the discrepancy in the unified scheme is commonly seen in llagns . we suggest a possible idea for the discrepancy with the unified scheme in terms of radio properties of llagns . if outflows carry accreting mass away and decrease a density at the innermost disc , then the spectral - peak frequency could decrease to several tens of ghz @xcite . although such a central engine could generate strong jets , neither high - frequency inverted spectra nor a lot of up - scattered photons that will ionize broad - line clouds can be produced . that is , llagns with strong jets would tend to have weak broad - line emission , which could not have been detected in a limited sensitivity of the optical survey , then they could be classified into type 2s , even if the obscuring torus is in a face - on orientation . we have found evidence for inverted high - frequency radio spectra in a number of llagns . such a high - frequency property is potentially the long - sought signatures of the adaf . it is an apparent observational confirmation for the submillimetre bump theoretically - predicted in the adaf model . in past observations , the radio spectra of llagns had been investigated exclusively at centimetre bands where high sensitivities are available . @xcite reported the radio spectra of 16 llagns that had been observed with the vla at 515 ghz ; @xcite reported the radio spectra of three llagns that had been observed with the vlba at 1.78.4 ghz ; @xcite reported the radio spectra measured with the vlba for three llagns at 2.315 ghz and three llagns at 1.743 ghz . the frequency range of submillimetre - bump hunting has greatly expanded by our high - frequency radio survey . our findings will encourage further llagn surveys at higher - frequency ranges with , e.g. , the coming atacama large millimetre and submillimetre array ( alma ) , to prove the adaf . the existence of the adaf will give radio astronomers a chance to make the direct imaging for accretion discs . their area of study had been limited to the nonthermal phenomena in relativistic jets of agns , although optical and x - ray astronomers have observed the direct fluxes from the accretion discs as unresolved sources . radio astronomers have vlbi techniques , which provide the greatest spatial resolutions all over the electromagnetic wavelengths . a high - frequency vlbi instrument will report the first image of morphology of accretion discs in micro - arcsec resolutions . the vsop-2 , a next generation space vlbi mission currently being planned , will provide a 38 micro - arcsec resolution at 43 ghz and a brightness - temperature sensitivity of 10@xmath31 k @xcite . because an electron temperature of 10@xmath32 k at the region within several tens schwarzschild radius is predicted by the adaf model , the vsop-2 can reveal disc structures of more than a dozen nearby llagns . if the accretion disc were an optically - thick standard disc , it will have never been detected with the vlbi because of low brightness temperature . from our results , both the jets and disc seem to coexist in the core of llagns , and their flux ratios are different from object to object . it suggests that relative radiative efficiencies between the jet and disc would determined by some kind of physical parameter(s ) in the central engine . from the theoretical point of view @xcite , it has been suggested that radiatively inefficient flows of low accretion rate tend to stimulate strong convective instabilities and powerful outflows . the central engines of llagns probably can produce jets . we will discuss the physical parameters that control jet productions in our llagn sample in a separate paper ( doi et al . in prep ) . our millimetre survey and analyses of submillimetre archival data for the 20 vlbi - detected llagns reveal high - frequency radio spectra of llagns with compact cores as follows : ( i ) : : at least half of the sources show inverted spectra between 15 and 96 ghz . we used the published data at 15 ghz with the vla in a 0.15-arcsec resolution and our measurements at 96 ghz with the nma in a 7-arcsec resolution . ( ii ) : : submillimetre fluxes measured with the scuba on the jcmt may be from the dust associated with host galaxies . these data have been useful to estimate dust contamination in the nma measurements . ( iii ) : : we confirm little diffuse contamination in the nma measurements from dust , free free , and extended nonthermal emissions for llagns showing inverted spectra . therefore , the inverted spectra are not caused by resolution effect , but intrinsic properties of the cores . ( iv ) : : in our sample of 20 llagns , radio cores with inverted high frequency radio spectra reside in at least ten llagns . ( v ) : : we find an inverse correlation between the high - frequency radio spectrum and the low - frequency radio core power . such a property can be explained by an idea that flux ratios between the jet showing steep spectrum and the adaf disc showing inverted spectrum are different from llagn to llagn . the disc components could be seen only if the jets are faint . consequently , we have found evidence for inverted high - frequency radio spectra in a number of llagns . such a high - frequency property is potentially the long - sought signatures of the adaf . we acknowledge s. okumura for helpful advices in observations with the nma at the nobeyama radio observatory . the nobeyama radio observatory ( nro ) is a branch of the national astronomical observatory of japan ( naoj ) , which belongs to the national institutes of natural sciences ( nins ) . we have made extensive use of the nvss and first data from the national radio astronomy observatory that is a facility of the national science foundation operated under cooperative agreement by associated universities , inc . we have also made extensive use of jcmt archival data from canadian astronomy data centre , which is operated by the dominion astrophysical observatory for the national research council of canada s herzberg institute of astrophysics .

we investigate the high - frequency radio spectra of twenty low - luminosity active galactic nuclei ( llagns ) with compact radio cores . our millimetre survey with the nobeyama millimetre array ( nma ) and analyses of submillimetre archival data that had been obtained with the submillimetre common user bolometer array ( scuba ) on the james clerk maxwell telescope ( jcmt ) reveal the following properties . at least half of the llagns show inverted spectra between 15 and 96 ghz ; we use published data at 15 ghz with the very large array ( vla ) in a 0.15-arcsec resolution and our measurements at 96 ghz with the nma in a 7-arcsec resolution . the inverted spectra are not artificially made due to their unmatched beam sizes , because of little diffuse contamination from dust , regions , or extended jets in these llagns . such high - frequency inverted spectra are apparently consistent with a ` submillimetre bump ' , which is predicted by an advection - dominated accretion flow ( adaf ) model . we find a strong correlation between the high - frequency spectral index and low - frequency core power measured with very - long - baseline - interferometry ( vlbi ) instruments . the inverted spectra were found exclusively in low - core - power sources , while steep spectra were in high - core - power ones with prominent pc - scale jets . this suggests that the adaf and nonthermal jets may coexists . the flux ratios between disc and jet seem to be different from llagn to llagn ; disc components can be seen in nuclear radio spectra only if the jets are faint . [ firstpage ] galaxies : active galaxies : seyfert radio continuum : galaxies submillimetre .

arxiv

scattering , absorption , and reradiation of photons by dust grains affect the propagation of starlight and play a key role in regulating the energy balance in the interstellar medium . studies of these processes have provided a great deal of insight into the physical and chemical properties of dust grains ( for reviews of dust properties , see mathis 1990a ; witt 2003 ) . it is also important to disentangle the effects of intervening dust from many kinds of astronomical observations , ranging from photometry and spectroscopy of galactic stars and external galaxies to mapping small - scale anisotropy in the cosmic microwave background radiation ( cmbr ) . the degree of absorption and scattering by dust depends , in general , on the wavelength of the incident radiation ( whitford 1958 ) . this dependence can be quantified in terms of the normalized dust extinction law @xmath20 ( or reddening law ) , where extinction is the sum of absorption plus scattering . the ratio of total to selective extinction , @xmath21 , is a commonly - used measure of the slope of the extinction / reddening law . using data along many different galactic sight - lines , cardelli , clayton , & mathis ( 1989 , hereafter ccm ) found a tight correlation between the overall shape of the reddening law and the slope @xmath0 . ccm devised an @xmath0-based , one - parameter family of empirical fitting functions to characterize the observed range of extinction law shapes ; this parameterization was later refined by fitzpatrick ( 1999 ) . galactic dust displays a wide range of behavior . while diffuse interstellar dust is observed to have @xmath22 _ on average _ ( savage & mathis 1979 ; ccm ) , this value is by no means universal . significant deviations from this canonical @xmath22 value ( and corresponding differences in the overall shape of the extinction / reddening law ) are known to exist for a variety of interstellar dust clouds , and these variations appear to be generally correlated with environment ( fitzpatrick 1999 ) . for example , @xmath0 has been found to range from 4 to 5 in dense molecular clouds ( mathis 1990b ; larson , whittet , & hough 1996 ; whittet et al . 2001 ) , whereas there are indications that more diffuse , high - latitude cirrus clouds may have @xmath0 values as small as @xmath23 ( fitzpatrick & massa 1990 ; larson et al . 1996 ; szomoru & guhathakurta 1999 ) . studies based on stellar photometry from the optical gravitational lens experiment ( ogle ) and massive compact halo objects ( macho ) projects point to @xmath0 being substantially smaller than 3 in the direction of the galactic bulge ( popowski 2000 ; popowski , cook , & becker 2003 ; udalski 2003 ; sumi 2004 ; popowski 2004 ) . barbaro et al . ( 2001 ) find departures from the ccm parameterization in the ultraviolet portion of the galactic extinction law . it is worth noting that the above studies sample dust at a wide range of distances from the sun : some probe the diffuse interstellar medium in its immediate vicinity whereas others , especially those at low galactic latitudes , can probe dust at much larger distances well outside the solar neighborhood ( e.g. , the rcra cloud studied by szomoru & guhathakurta 1999 ) . some studies have suggested that @xmath0 variations are tied to variations in the size distribution of dust grains from one line of sight to another . in diffuse cirrus clouds , a relative abundance of small grains might explain the steep rise of the extinction curve into the ultraviolet ( ccm ; larson et al.1996 ) . by contrast , a high abundance of large grains , which grow readily by coagulation in dense molecular clouds , may explain the larger than average values of @xmath0 observed in these regions ( whittet et al . whittet et al . suggest that , in dense molecular clouds , @xmath0 remains close to the standard value of 3.1 except for lines of sight with unusually high extinction ( @xmath24 ) . this may indicate a more complicated relationship between @xmath0 and @xmath25 . moreover , it has been argued that chemical composition can also play a role in determining the shape of the extinction law ( rhoads , malhotra , & kochanski 2004 ) . to improve our understanding of the dependence of @xmath0 on environment , measurements are necessary along many sight - lines through a broad range of cloud types . clusters with differential extinction / reddening have long been used to measure the @xmath0 of intervening dust ( mihalas & routly 1968 ) . the traditional method of spectral typing individual cluster stars ( e.g. , on the morgan - keenan system ) , in conjunction with photometric measurements , allows for a direct and precise measurement of extinction and reddening . however , this method of measuring @xmath0 requires high - quality , flux - calibrated spectra and a good understanding of the effects of metallicity on the energy output of stars . our study focuses instead on the use of broad - band photometry in light of the fact that an extensive suite of photometric data sets is currently available . in this paper , we propose three methods for measuring the @xmath0 of dust in the foreground of differentially - reddened globular clusters . the first two methods are closely related to each other and rely on accurate three - band optical photometry of cluster stars . the third relies on two - band optical photometry and a map of the dust thermal emission ( schlegel , finkbeiner , & davis 1998 , hereafter sfd ) and is particularly well suited to wide - field data . all three methods are based on the premise that the width of the blue horizontal branch ( hb ) is minimized when the appropriate @xmath0 value is used to correct the photometry for extinction and reddening . the methods are applied to ngc 4833 , a low - latitude galactic globular cluster with variable extinction across its face . realistic monte carlo simulations of the cluster data set are used to estimate the accuracy with which @xmath0 can be recovered . nearly two decades ago , low et al . ( 1984 ) noticed diffuse background emission in the infrared astronomical satellite ( iras ) 60 and 100@xmath26 m maps and termed it ` infrared cirrus ' due to its complex texture . the cirrus has been associated with thermal emission from dust grains in the diffuse interstellar medium ( beichman 1987 ) . combining the angular resolution of the iras observations with the photometric accuracy of the cosmic background explorer ( cobe)/diffuse infrared background experiment ( dirbe ) data , sfd derived all - sky maps of the dust column density and mean temperature . one of the primary uses of these maps has been to correct extragalactic sources for extinction and reddening . the sfd maps show structure in the cirrus down to the smallest angular scales resolved by iras ( @xmath27 ) . gautier et al . ( 1992 ) described the structure in the iras data in terms of its angular power spectrum , @xmath28 where @xmath29 is the fourier power , @xmath30 is the reciprocal of the angular scale , and @xmath31 is the spectral index . similarly , guhathakurta & cutri ( 1994 ) examined both iras 100@xmath26 m maps and optical ccd surface photometry of reprocessed starlight from dust grains : they found that power on small scales was dominated by stars and galaxies , and since it was impossible to completely separate out these components , they merely cleared out the obvious compact `` objects '' ( stars and galaxies ) and smoothed the residual image over arcminute scales ; the resulting power spectrum had an index @xmath32 . more recently , kiss et al . ( 2003 ) observed 13 fields with infrared space observatory ( iso)/isophot and found that the power spectrum index varies from field to field over the range @xmath33 for angular scales larger than @xmath34 . this work extends the finding of @xmath35 by herbstmeier et al . ( 1998 ) in their earlier isophot - based study . these last two studies indicate that @xmath36 is not universal . in this study , the scatter in the hb of the differentially - reddened globular cluster ngc 4833 is used to measure the angular power spectrum of dust on scales smaller than the angular resolution of the thermal emission maps ( e.g. , iras , dirbe , isophot ) . for uncrowded data such as ours , the smallest angular scale down to which this method can be applied is set by the surface density of hb tracers which in turn determines the typical nearest - neighbor separation ; in crowding - limited situations , the smallest angular scale is a few times the size of the stellar point spread function : few arcseconds for ground - based images and sub - arcsecond for _ hubble space telescope _ images . because interstellar dust causes extinction and reddening of starlight and reprocesses the energy , dust patchiness has a profound impact on the accuracy of many astronomical measurements even at high galactic latitudes . thermal emission from cold dust can confuse cmbr anisotropy measurements . measurements of large - scale structure in galaxy surveys may be influenced by spatial non - uniformities in the foreground dust . studies of stellar populations are at risk because measurements of @xmath37 can be impacted by reddening . extinction can affect the photometry of distance indicators and thereby produce systematic errors in the distance - scale ladder . it is therefore important to quantify extinction / reddening variations on the smallest scales possible . while the sfd study does an excellent job of characterizing moderate- to large - scale dust variations , their maps miss power on scales smaller than a few arcminutes . + + the photometry of ngc 4833 is presented in 2 . the construction of simulated cluster data sets in discussed in 3 . the @xmath0 measurement methods are outlined in 4 , along with their application to ngc 4833 and simulated data sets to estimate the accuracy of each method . in 5 , we describe the angular power spectrum of cirrus in the direction of ngc 4833 . possible future extensions of this work are discussed in 6 . the main conclusions of this paper are summarized in 7 . the globular cluster ngc 4833 is located near the equatorial plane of our galaxy ( @xmath38 ) in a dusty region of the constellation musca . the cluster has a mean reddening of @xmath39 mag and reddening variations of order @xmath40 mag across the field ( melbourne et al . figure 1 of melbourne et al . shows an image of the dusty features in the region around the cluster including the coalsack nebula . the johnson - cousins @xmath1 stellar photometry of ngc 4833 used in this study are drawn from melbourne et al . a detailed description of the data set is given in that paper so only a brief discussion appears here . the ccd images of the cluster were obtained in 1996 using the 0.9-m telescope at the cerro tololo inter - american observatory . the data reduction was carried out with the daophot ii ( stetson 1994 ) stellar photometry package . figure [ fig : fullcmd ] shows the ( @xmath4 , @xmath5 ) and ( @xmath6 , @xmath5 ) cmds ( left and right panels , respectively ) for the full @xmath41 field . boxes are drawn in the former cmd around the 260 objects that are considered to be candidate cluster hb stars for the purposes of this study ( possible non - hb interlopers in the sample are discussed in 4.1.1 ) . extinction / reddening variations across the field affect the width of the principal sequences of the cmds . figure [ fig : radcmd ] shows that the hb is relatively tight and narrow for the central @xmath42 region of the cluster , but increases in width for stars distributed over a larger area in the outer region of the cluster . this suggests that extinction / reddening variations across the field , rather than photometric errors , are the cause of the large scatter in the hb ; if anything , photometric errors tend to be slightly worse in the inner region of the cluster because of crowding . a color - color diagram of cluster hb stars is plotted in the top panel of figure [ fig : cc ] , while the bottom panel shows the same after correcting for reddening on a star - by - star basis . the @xmath43 value for each star is taken from the sfd dust map , and an assumed @xmath0 value of 3.0 is plugged into the ccm parameterization of the dust extinction law ( eqn . [ eqn : cardel ] ) to derive @xmath44 and the corresponding @xmath45 value . the reddening - corrected data ( and the uncorrected data , for that matter ) show a linear relationship between @xmath4 and @xmath6 . a least - squares fit ( solid line ) is made to the reddening - corrected data : @xmath46 this equation , in conjunction with the two - polynomial fiducial fit to the hb in the ( @xmath4 , @xmath5 ) cmd , is used to simulate @xmath1 photometry of hb stars ( 3.2 ) . figure [ fig : photerr ] shows the typical @xmath47 photometric errors in the @xmath1 bands for ngc 4833 stars as a function of their apparent visual magnitude ( melbourne et al . the photometric errors , estimated using the daophot ii software package ( stetson 1994 ) , range from @xmath48 mag at the bright end of the hb ( @xmath49 ) to @xmath50 mag at the faint end ( @xmath51 ) , and contribute noticeably to the width of the hb . the characteristic photometric error is defined to be that at @xmath52 , @xmath53 mag , and this is what is referred to throughout the rest of this paper . a key aspect of our study is the use of simulated cluster data sets to fine tune and test the @xmath0 measurement methods ( 4 ) . these simulated data sets are designed to mimic the general properties of the ngc 4833 data set as closely as possible . since we are also interested in applying these @xmath0 measurement methods to a broad range of data , the main set of simulations are constructed over a two - dimensional grid of parameter space : three _ input _ @xmath0 values and three levels of photometric precision . at each of these 9 ( @xmath54 ) grid points , 100 monte - carlo realizations are carried out , a total of 900 simulated clusters are constructed each containing 260 hb stars . the rest of this section describes the step - by - step procedure used to construct each simulated data set . three input @xmath0 values , 2.0 , 3.0 and 4.0 , are chosen for the simulated cluster data sets . these mimic a range of extinction law properties for the foreground dust and span the range of values observed in lines of sight with moderate to light reddening ( 1.1 ) . the ccm parameterization of the galactic extinction laws : @xmath55 is adopted here . for a given choice of @xmath0 , the visual extinction @xmath25 is translated to its corresponding @xmath43 and @xmath45 color excesses . each simulated star is assigned @xmath56- and @xmath5-band photometry via a random drawing from a fiducial hb curve in the cmd . the fiducial is based on stitching together two second - order polynomials that are fit to the extinction / reddening - corrected ( @xmath4 , @xmath5 ) hb of ngc 4833 , where the @xmath43 correction for each star is obtained from the sfd dust map and @xmath0 is assumed to be 3.0 . examples of such two - polynomial fiducial hbs are shown in figures [ fig : av_method ] and [ fig : schlegel_method ] . the `` sfd - corrected '' cluster hb used for the fiducial fit is similar to the one shown in the left - middle panel of figure [ fig : cmd6 ] , except for the fact that the one illustrated has had the mean extinction / reddening reapplied to it . when drawing 260 stars from the fiducial , the run of stellar density along the simulated hb is made to mimic that observed along the cluster hb . this is done by dividing the fiducial into sections and demanding that the number of stars in each section match that in the corresponding section of the cluster hb . simulated @xmath57-band photometry is obtained from the linear fit to the ( @xmath4 , @xmath6 ) color - color diagram of cluster hb stars ( fig . [ fig : cc ] , eqn . [ eqn : cc ] ) . the width of the hb can be used to quantify the effect of extinction / reddening variations . figure [ fig : hbscat ] shows the rms scatter of hb stars in ngc 4833 as a function of their @xmath5 magnitude ( solid line ) . the rms scatter is calculated based on the _ shortest distance _ of each hb star from the fiducial curve ( i.e. , distance to the nearest tangent point on the curve ) . the fiducial consists of two second - order polynomials fit to hb stars within @xmath58 of the cluster center . the dotted line in figure [ fig : hbscat ] is a model prediction for the hb scatter that would result from photometric errors alone , the dashed line shows the effect of large - scale extinction / reddening variations from sfd plus photometric error , and the dot - dashed line the effect of small- and large - scale extinction / reddening variations plus photometric error . it is clear from the scatter at the bright end of the hb ( @xmath59 ) that only the dot - dashed line matches the ngc 4833 data . the following subsections describe in turn the simulations behind the three model lines . a special set of 100 photometric - error - only ( `` @xmath60-only '' ) simulations containing _ no _ reddening variations is constructed for illustration purposes only ; this set is not used in any of the @xmath0 determination exercises described in 4 . each monte carlo realization in this special set involves three steps : ( * 1 * ) an artificial hb is constructed as described in 3.2 above ; ( * 2 * ) the mean extinction and reddening of the cluster , @xmath61 and @xmath39 ( based on @xmath62 ) , are applied to all simulated stars to bring them into the same range of apparent @xmath1 magnitudes as the ngc 4833 hb stars ; and ( * 3 * ) gaussian photometric errors are added assuming @xmath63 . the details of step ( 3 ) are given in 3.4 below . the dotted line in figure [ fig : hbscat ] is the scatter measured in @xmath64 simulated hb stars . the mismatch with the ngc 4833 data ( solid line ) demonstrates that photometric errors alone can not explain the width of the cluster s hb and there must be additional sources of scatter in the hb . ( the dotted line is much smoother than the solid line because it is based on 100@xmath65 as many stars . ) the obvious additional source of hb scatter is differential extinction / reddening across the cluster . large - scale variations based on the sfd map are explored in this section . their maps provide @xmath43 color excesses as a function of sky position , but have an angular resolution of @xmath66 and therefore contain power only on scales larger than this . the sfd map includes ngc 4833 and the region around it , and indicates that the mean reddening along this line of sight is @xmath67 mag with variations on the order of @xmath40 mag . following the procedure described above for the `` @xmath60-only '' simulations ( 3.3.1 ) , another special set of 100 simulations is constructed containing only large - scale extinction / reddening variations and photometric errors . again , these special `` sfd+@xmath60 '' simulations are created for illustrative purposes alone . the only difference is in step ( 2 ) of the procedure : instead of applying the same ( mean ) extinction and reddening values to all simulated hb stars , each star is randomly assigned the sky position of one of the 260 hb stars in ngc 4833 and the corresponding sfd @xmath43 value , and associated @xmath25 and @xmath45 values based on @xmath62 , are applied to the @xmath1 magnitudes of the simulated star . by assigning actual sky positions to the simulated hb stars , we ensure that they mimic the spatial distribution of ngc 4833 s hb stars and range of large - scale extinction / reddening variations across the cluster data set . the dashed line in figure [ fig : hbscat ] , based on @xmath64 simulated hb stars , shows the scatter resulting from a realistic amount of large - scale extinction / reddening variations and photometric error . even this line falls a little short of explaining the hb scatter observed in the cluster ( solid line ) . the most probable cause of this shortfall is the coarse resolution of the sfd map which averages over structure in the dust distribution on angular scales smaller than @xmath66 . small - scale variations are incorporated into the simulated data sets by adding random gaussian fluctuations with an amplitude proportional to the sfd reddening value at that location : @xmath68\ ] ] where @xmath69 is a random number ( for the @xmath9-th star ) drawn from a gaussian distribution with zero mean and width @xmath70 . this brings us back to the main set of simulations used for testing the @xmath0 determination methods in 4 . they are constructed using a procedure similar to that used for the special set of `` sfd+@xmath60 '' simulations ( 3.3.2 ) but with two changes . first , the sfd reddening values in step ( 2 ) of the procedure are replaced by @xmath71 from eqn . [ eqn : smallscale ] above . second , a range of @xmath0 values ( 3.1 ) and photometric error levels ( 3.4 ) are used . with the above choice of gaussian width for the small - scale dust variations , @xmath70 , and for @xmath63 and @xmath62 , the scatter in the simulated hb approaches that of ngc 4833 s hb ( dot - dashed vs. solid lines in fig . [ fig : hbscat ] ) . this is a reasonable estimate of the amount of smoothing of dust fine - scale structure in the sfd maps ( 5.1 ; fig . [ fig : powspec ] ) , and is well within the 10% uncertainty in reddening values estimated by the authors . the final step in the construction of the main set of simulated hb data sets is the inclusion of photometric error . since the photometric uncertainty is a function of stellar brightness , this step must be carried out after the inclusion of ( large- and small - scale ) dust effects so that each star is at the appropriate apparent brightness . gaussian random noise is added to the @xmath1 magnitudes of each star . three levels of photometric error are simulated : ( 1 ) @xmath63 , with the latter being the daophot - based errors shown in figure [ fig : photerr ] ; ( 2 ) @xmath72 ; and ( 3 ) @xmath73 . we employ three methods for measuring @xmath0 in the direction of differentially - reddened globular clusters . each method relies on the premise that the width of the blue hb will be minimized when the appropriate @xmath0 value is used to correct for extinction and reddening . a description of each method and its application to the ngc 4833 and simulated cluster data sets follows . the `` @xmath25 rms '' method relies on three - color photometry of differentially - reddened globular clusters . figure [ fig : av_method ] shows the ( @xmath4 , @xmath5 ) and ( @xmath6 , @xmath5 ) cmds of a simulated hb ( left and right panels , respectively ) . a fiducial , consisting of two second - order polynomials stitched together ( solid curve ; 3.2 ) , is fit to the hb stars within @xmath58 of the cluster center where differential extinction / reddening is relatively small ; this is done independently for the ( @xmath4 , @xmath5 ) and ( @xmath6 , @xmath5 ) cmds . the relative visual extinction of each star is calculated to be the vertical distance between the star and the point where the reddening vector drawn from the star intersects the fiducial ( fig . [ fig : av_method ] inset ) ; it is designated @xmath74 and @xmath75 for the ( @xmath4 , @xmath5 ) and ( @xmath6 , @xmath5 ) cmds , respectively . '' is used to denote the relative visual extinction of a star based on a single fiducial / cmd , whereas `` @xmath76 '' is used to denote the _ difference _ between two @xmath3 values , say between two cmds for a given star or between a pair of stars . ] the @xmath74 measurement is based on an assumed / test value of @xmath0 , the slope of the reddening vector in the ( @xmath4 , @xmath5 ) cmd . for the @xmath75 calculation , the ccm parameterization given in eqn . [ eqn : cardel ] is used to translate @xmath0 to @xmath44 , the slope in the ( @xmath6 , @xmath5 ) cmd . the difference between the two @xmath3 values is defined to be @xmath77 . for a given test value of @xmath0 , the rms scatter of @xmath78 is computed over all hb stars . this is done in two iterations , with 3-@xmath79 rejection of outliers after the first iteration to eliminate any non - hb stars in the sample ( e.g. , cluster blue stragglers or galactic field main sequence turnoff stars in the foreground encircled symbols in upper panels of fig . [ fig : radcmd ] ) . non - hb interlopers are expected to have large deviations from the hb fiducial with effective @xmath3 values that are different between the two cmds . the calculation of the rms of @xmath78 is repeated for a series of test @xmath0 values between 0.5 and 5.5 in steps of 0.25 . the @xmath3 parameters from the two cmds are expected to approach each other when the test value approaches the true @xmath0 of the dust in the foreground of the cluster . in other words , the rms of @xmath78 should go through a minimum at the correct @xmath0 value . a polynomial fit is made to the rms of @xmath78 vs.test @xmath0 data points to interpolate to the minimum . figure [ fig : av_results ] shows the rms scatter in @xmath78 vs. test @xmath0 value , the result of applying the @xmath25 rms method to ngc 4833 ( filled diamonds ) and three sample simulated cluster data sets ( open triangles ) . the simulated data sets are all based on an input @xmath0 value of 3.0 , but have different levels of photometric error : 10% , 50% , and 100% that of the ngc 4833 data set ( 3.4 ) , with the rms sequences arranged in order of @xmath80 increasing upwards . as expected , each sequence of rms of @xmath78 vs. test @xmath0 goes through a clear minimum ( large open square ) near @xmath62 . the depth / sharpness of the rms minimum decreases with increasing photometric error and the power of the method diminishes . the results for all @xmath81 simulated cluster data sets are summarized in table [ tbl : sim ] . for @xmath73 , the @xmath25 rms method reproduces the correct @xmath0 to within @xmath82 , as measured by the standard deviation of measured ( output ) @xmath0 values for 100 simulated data sets . the scatter in output @xmath0 increases to @xmath83 for @xmath63 . the ( absolute ) precision of the method appears to decrease with increasing input @xmath0 , again judging from the scatter of output @xmath0 values , although the fractional error @xmath84 remains roughly constant . it is important to note that the output @xmath0 from this method tends to be biased low for moderate to large photometric errors and input @xmath85 . for example , simulated data sets with input @xmath62 and @xmath63 ( a good match to the real cluster data ) have a mean output @xmath86 . this bias of @xmath87 in @xmath88 is significant compared to the error in the mean , @xmath89 , and must be taken into account in the determination of @xmath0 for an actual cluster data set . the cause of the bias towards low @xmath0 may be explained as follows . photometric error produces scatter in the hb which , unlike the scatter caused by variable extinction / reddening , is _ uncorrelated _ between the two cmds i.e . , the resulting values of @xmath74 and @xmath75 are different . these errors are particularly important at faint magnitudes where the hb is vertical and the effect is a spread in color . all else being equal , choosing a smaller @xmath0 ( and @xmath44 ) translates these color errors into smaller @xmath3 values and hence to smaller ( @xmath76)@xmath90 differences as well . this biases the rms minimum towards lower @xmath0 values . the rms of @xmath78 for the ngc 4833 data set displays a significant minimum at @xmath91 ( open square associated with filled diamonds in fig . [ fig : av_results ] ) , where the error bar is taken directly from the scatter in output @xmath0 for comparable simulated data sets . correcting for the bias in the measured @xmath88 for such simulations , our best guess for the @xmath0 towards ngc 4833 is @xmath92 . the `` @xmath25 slope '' method is closely related to the @xmath25 rms method in that it relies on the same @xmath74 and @xmath75 values determined for each star in 4.1.1 above . instead of computing the difference between the two @xmath3 values as in the @xmath25 rms method , one is plotted against the other and a linear fit made . this is illustrated in figure [ fig : av_slope_method ] for a simulated cluster data set created with ( input ) @xmath62 ; four test @xmath0 values are shown . the linear fit in each panel is effectively a bivariate one : one of the dashed lines is a fit treating the @xmath93- and @xmath94-axes as the independent and dependent variables , respectively , and vice versa for the other dashed line ; their slopes are averaged to get the solid line . as expected , the slope of the linear fit approaches unity when the test @xmath0 approaches the true ( input ) @xmath0 value . as in 4.1.1 , a polynomial fit is made to the slope vs. test @xmath0 data to interpolate to unit slope . figure [ fig : slope_results ] shows results from the application of the @xmath25 slope method : the slope of the relation between @xmath74 and @xmath75 is plotted versus test @xmath0 . the format of this figure is the same as for figure [ fig : av_results ] , except that a slope of unity ( rather than an rms minimum ) corresponds to the best - fit @xmath0 value . the @xmath25 slope method results for ngc 4833 and simulated data sets are summarized in table [ tbl : sim ] . applying this method to @xmath81 simulated data sets , we learn that , like the @xmath25 rms method , the power of the @xmath25 slope method decreases with increasing photometric errors . the scatter in output @xmath0 is @xmath82 for @xmath73 , but increases to @xmath95 for @xmath63 . some of the 9 simulation categories display a small but significant bias ( positive or negative ) in the measured ( output ) @xmath88 , but not the category directly relevant to ngc 4833 . the cause of this bias is not as readily apparent as for the `` @xmath25 rms '' method ( 4.1.2 ) . for the ngc 4833 data set , the @xmath25 slope method gives @xmath96 . the `` optical / ir '' method is different from the previous two methods in that it relies on two - color photometry ( i.e. , a single cmd which , in our case , is @xmath5 vs. @xmath4 ) and the sfd reddening map . using a test @xmath0 value , @xmath25 values are derived for all hb stars by scaling the corresponding sfd @xmath43 values . these color and magnitude corrections are applied on a star - by - star basis . figure [ fig : schlegel_method ] shows the ( @xmath4 , @xmath5 ) cmd of a simulated hb before ( left ) and after ( right ) making the extinction / reddening corrections . the latter hb is slightly tighter , but not dramatically so because the sfd reddening values smooth over small - scale dust variations . the rms scatter of the corrected hb stars is computed with respect to a best - fit fiducial curve ( two second - order polynomials solid line in fig . [ fig : schlegel_method ] ) based on `` shortest distance '' ( see 3.3 ) . this process is repeated for a range of test @xmath0 values as for the other two methods . the rms scatter is expected to go through a minimum when the test @xmath0 approaches the true @xmath0 value . as in 4.1.1 , a polynomial fit is made to the hb scatter vs. test @xmath0 data to interpolate to the minimum . figure [ fig : schlegel_results ] shows results from the application of the optical / ir method : rms scatter of hb stars versus test @xmath0 . the format of this figure is the same as for figure [ fig : av_results ] . the optical / ir results for ngc 4833 and simulated data sets are summarized in table [ tbl : sim ] . tests of the method on @xmath81 simulated cluster data sets indicate that the method has a low level of predictive power ( @xmath97 in @xmath0 ) over the full range of photometric errors simulated . this is because the main source of error here is uncertainty in the sfd reddening values , including small - scale dust variations that get smoothed out by the relatively coarse resolution of their map . the optical / ir method gives @xmath98 for ngc 4833 . our best estimate of the extinction law slope in the direction of ngc 4833 is @xmath7 , based on a weighted average of the values from the three determination methods ( see table [ tbl : sim ] ) . before averaging , the output @xmath0 value from the @xmath25 rms method is corrected for measurement bias : @xmath99 ( 4.1.2 ) . the weights are proportional to @xmath100 ^ 2 $ ] where @xmath101 is the uncertainty in output @xmath0 for a given determination method as derived from the simulations . figure [ fig : cmd6 ] illustrates the effectiveness of the star - by - star extinction / reddening corrections based on the @xmath0 determination methods described earlier in this section . the top two panels show the uncorrected ngc 4833 hb in the ( @xmath4 , @xmath5 ) and ( @xmath6 , @xmath5 ) cmds ( left and right panels , respectively ) . the photometry in the middle two panels has been corrected for extinction / reddening based on the sfd @xmath43 maps , using a best - fit @xmath62 and the corresponding @xmath44 from eqn . [ eqn : cardel ] to translate the @xmath4 color excess to @xmath25 and @xmath45 . note , the mean extinction and reddening of the cluster have been added back to the data for ease of comparison with the top panels . the hb is somewhat tighter after the sfd - based correction but the effect is fairly subtle , a result of missing small - scale dust structure in the sfd map and possibly other sources of uncertainty as well . the correction in the bottom two panels of figure [ fig : cmd6 ] is based on information derived from @xmath1 photometry of the cluster hb . the relative visual extinction measures derived from the two cmds , @xmath74 and @xmath75 ( 4.1.1 ) , are averaged to obtain a @xmath102 value for each hb star . this is translated to color excesses in the two cmds using the best - fit @xmath62 and its corresponding @xmath44 ( eqn . [ eqn : cardel ] ) . the dramatic tightening of the hb indicates that the scatter is strongly correlated between ( @xmath4 , @xmath5 ) and ( @xmath6 , @xmath5 ) cmds e.g . , if a star is displaced to the right of / below the hb fiducial in one cmd , it tends to be displaced in the same direction and by the same amount ( or by a proportional amount in color ) in the other cmd . the most natural explanation of this correlated hb scatter is of course differential extinction / reddening in the ngc 4833 data set . there is a handful of stars at each extremity of the hb in the bottom two panels of figure [ fig : cmd6 ] that do not follow the tight sequence delineated by the rest of the stars in fact , their scatter is unchanged relative to the uncorrected cmds in the top panels . this is because the polynomial hb fiducial used to determine @xmath3 does not extend all the way to the ends of the sequence ( to avoid extrapolation of the fit ) ; as a result no @xmath3 measurement or correction is available for these stars . a couple of stars near the middle of the hb continue to show large deviations in the @xmath3-corrected cmds but these are probably non - hb stars ( see 4.1.1 and fig . [ fig : radcmd ] ) . before closing out this discussion , two additional sources of hb scatter are considered . the first of these is photometric error . errors in @xmath56 and @xmath57 magnitudes produce uncorrelated shifts between the two cmds . errors in @xmath5 produce shifts with slope @xmath103 in the ( @xmath4 , @xmath5 ) cmd , more or less orthogonal to the reddening vector , and shifts of slope @xmath104 in the ( @xmath6 , @xmath5 ) cmd , roughly along the reddening vector though its slope @xmath44 is significantly steeper than unity . only at the very top of the hb where it becomes horizontal can @xmath5 magnitude errors mimic differential extinction / reddening effects . to summarize , photometric errors can not explain the correlated scatter seen in ngc 4833 s hb . moreover , as will be demonstrated in 5 , the difference in @xmath3 between pairs of hb stars scales with their angular separation in a manner consistent with the dust power spectrum measured at larger angular separations ; photometric errors have no natural way of explaining this trend either . another factor to consider is _ intrinsic _ scatter in the hb caused by astrophysical effects such as variations in chemical abundance , age , and mass loss between cluster stars . it is possible that these can produce correlated scatter between the two cmds and thereby mimic differential extinction / reddening . however , this scenario can not explain the scaling of the @xmath3 differences with angular separation ( 5 ) . if there was a radial gradient in intrinsic stellar properties within the cluster , one might expect to see a shift of the hb sequence in the cmd between inner and outer regions . instead , figure [ fig : radcmd ] shows no such shift ( the increased scatter in the hb in the outer cmd is best explained in terms of dust variations ; see 2.1 ) . in earlier sections , we demonstrated the effects of spatial extinction / reddening variations on the photometry of ngc 4833 . we now turn to a statistical description of the strength of these extinction / reddening variations as a function of angular scale . figure [ fig : dustmap ] is a map of the _ relative _ visual extinction across the face of ngc 4833 . each point on the map represents the sky position of a hb star in the cluster . the size of each point is scaled by the measured strength of the relative visual extinction @xmath74 at that location ; it varies from @xmath105 to @xmath106 mag ( smallest@xmath107largest ) . as described in 4.1.1 , @xmath74 is the vertical distance between the star s cmd location and the point where the reddening vector drawn from the star intersects the fiducial hb ; the fiducial is a fit to the central region of the cluster , and therefore corresponds to a mean visual extinction of @xmath108 . in keeping with the findings of melbourne et al . ( 2000 ) , there is a clear overall n - s gradient in @xmath74 across ngc 4833 , but the extinction can vary significantly from point to point even for points relatively close to each other . these variations can be characterized in terms of the angular power spectrum of the foreground dust complex . a direct measurement of the power spectrum is difficult however because of the sparse and irregular spatial sampling of the data points . the following approach is adopted instead . first , the `` variance spectrum '' as a function of angular scale is calculated from the cluster @xmath74 data ( 5.1 ) . next , a variance spectrum and conventional power spectrum are measured from the uniformly - sampled sfd maps , and an empirical transformation is calculated from the former to the latter ( 5.2 ) . finally , this empirical transformation is applied to the cluster variance spectrum to convert it to a power spectrum , and this allows us to study the composite power spectrum over a wide range of angular scales ( 5.3 ) . the best - fit value of @xmath62 for ngc 4833 is used to determine the relative visual extinction corresponding to the @xmath9-th hb star , @xmath109 . the difference in relative @xmath25 between stars @xmath9 and @xmath10 , @xmath110 , and their angular separation @xmath11 are then calculated for all possible pairs of hb stars . figure [ fig : powspec ] ( open diamonds ) shows the variance ( mean square deviation ) of @xmath111 as a function of angular separation @xmath112 in a log - log plot ; the radial bins are chosen so as to include roughly equal numbers of pairs in each bin . the variance decreases with decreasing @xmath112 before levelling off at log@xmath113\bigr]=-2.3 $ ] for @xmath114 . the behavior at low @xmath112 may be attributed to photometric errors and possibly the natural width of the hb . in order to quantify and compensate for the effect of photometric error on the variance spectrum , we resort to the special set of `` @xmath115-only '' simulations containing _ no _ reddening variations ( 3.3.1 ) . the variance of this `` @xmath60-only '' simulated data set , analyzed in the same way as the real cluster data set , is : log@xmath116\bigr]=-2.4 $ ] , and is of course independent of stellar angular separation ( dashed horizontal line in fig . [ fig : powspec ] ) . the variance due to photometric errors is subtracted from the variance spectrum of ngc 4833 , equivalent to statistical subtraction of errors in quadrature . the resulting variance spectrum is shown as filled diamonds in figure [ fig : powspec ] . a power - law fit to the corrected variance spectrum : @xmath117 yields an index of @xmath14 , over the radial range @xmath118@xmath119 . the upper end of this range corresponds to the nyquist frequency of the @xmath41 area covered by the ngc 4833 data set . the first radial bin of the corrected variance spectrum ( @xmath120 ) is excluded from the power - law fit because it is strongly affected by uncertainties in the photometric error correction . the cluster variance spectrum is used to make a rough estimate of how much small - scale power is missing from the sfd dust map . the sfd map , like the iras 100@xmath26 m data on which they are based , have an angular resolution of about @xmath121 , or @xmath122 . this point in the cluster variance spectrum corresponds to var@xmath123=4.4\times10^{-3}$ ] . the rms scatter at this point is @xmath124}=6.6\times10^{-2}$ ] . since the mean visual extinction @xmath2 in the direction of ngc 4833 , the rms is about 7% of the mean . this provides a good sanity check : it corroborates the result from the analysis of the cluster hb width ( 3.3.2 ) that the sfd map is missing small - scale variations on the order of 6% of the reported reddening value . it also gives an indication of how accurately one can correct for extinction / reddening using the sfd map , with general implications for precision photometry in dusty regions of the sky . in this section , the sfd map is used to investigate the relationship between the variance spectrum and conventional angular power spectrum . this is done by extracting a @xmath125 sfd reddening image centered on ngc 4833 . the @xmath126 reddening values are converted to @xmath25 using the best - fit @xmath0 of 3.0 . the difference in visual extinction between pixels @xmath9 and @xmath10 , @xmath127 , is calculated for all possible pairs of pixels . the variance of @xmath128 is computed as a function of angular separation @xmath112 . the ` @xmath129 ' symbols in figure [ fig : pwvr ] show @xmath130 for the sfd map , where @xmath19 . the two - dimensional power spectrum of dust fluctuations is computed via a fast fourier transform ( fft ) of the sfd reddening image . this is then azimuthally averaged to produce a one - dimensional angular power spectrum @xmath131 , shown by asterisks in figure [ fig : pwvr ] . over the radial range @xmath132 to @xmath133 [ @xmath134 to @xmath135 , @xmath136 and @xmath131 have comparable logarithmic slopes ( power - law index : @xmath137 ) such that : @xmath138 with a best - fit scale factor of @xmath139 . the above transformation ( eqn . [ eqn : var_to_ps ] ) is applied to the photometric - error - corrected variance spectrum derived from the ngc 4833 @xmath140 data ( 5.1 ) to convert it to an angular power spectrum . the result is shown as filled diamonds in figure [ fig : pwvr ] . this small - scale power spectrum derived from optical data matches smoothly onto the power spectrum derived from the sfd map at larger angular scales . a power law with an index @xmath141 provides an adequate fit to the data over the full range of projected separations shown in figure [ fig : pwvr ] , @xmath17 to @xmath142 . our best - fit index @xmath143 is significantly shallower than the index of @xmath144 found by gautier et al . ( 1992 ) and guhathakurta & cutri ( 1994 ) and at the shallow end of the range of indices found by kiss et al . ( 2003 ) in their isophot - based study of 13 fields : @xmath145 to @xmath146 ( @xmath82 ) . the shallower the power - spectrum index , the larger the _ fraction _ of power missed by the sfd map on small scales relative to that on larger scales . another possible implication relates to kiss et al.s finding that shallower @xmath143 values are associated either with low hi column density , @xmath147 @xmath148 , and/or warm dust . for example , in the region of the draco nebula , they find that the map of 90@xmath26 m emission ( dominated by warmer dust ) yields @xmath149 while the 170@xmath26 m map ( dominated by cooler dust ) yields @xmath150 . while ngc 4833 has an extended blue hb , the methods presented in this paper should be usable for clump hbs or the rgb or any other tight cmd feature for that matter . for instance , law et al . ( 2003 ) used the width of the rgb to measure differential reddening in the direction of three galactic globular clusters , von braun et al . ( 2002 ) used the main sequence for their study , and udalski ( 2003 ) used clump stars to probe dust along the line of sight to the galactic bulge . our methods should work for any three bands , not just @xmath1 . additional bands ( beyond three ) can provide : ( 1 ) a longer wavelength baseline and therefore more leverage in @xmath0 determination ; ( 2 ) a larger number of independent measures of the star - by - star extinction / reddening which can be averaged to beat down the errors ; and ( 3 ) most importantly , an empirical check of the ccm parameterization of the shapes of galactic extinction laws . there is at least one generalization of the methodology presented here that might be worth exploring in the future . in determining @xmath3 , we effectively use @xmath5 magnitude to mark where along the hb sequence the extinction / reddening - corrected star lies . a more general definition of a `` locator '' parameter , to mark the position of the unreddened star along any given cmd sequence , may be particularly useful . for example , multi - wavelength data sets may not have any single observable in common across all of the cmds in question ( unlike the @xmath5 mag for our two cmds ) , and a color parameter may be more sensitive than @xmath5 mag when using a not - so - vertical cmd feature ( such as the upper portion of ngc 4833 s hb ) . several high - quality , multi - band ( three or more ) , ground- and space - based photometric data sets have long been available for many star clusters , so the methods presented here can be readily implemented to probe the nature of dust along many lines of sight . these include milky way clusters , as well as others near enough to allow clean photometry in relatively uncrowded fields , such as _ hubble space telescope _ images of clusters in , and even somewhat beyond , the magellanic clouds . many of these clusters have detectable extinction / reddening variations across them e.g . , @xmath151 centauri , ngc 6388 , ngc 6441 ( law et al . 2003 ) , m10 , and m12 ( von braun et al . the sloan digital sky survey ( sdss , stoughton et al . 2002 ) five - band ( @xmath152 ) data set , with its high photometric accuracy , minimal systematic errors , excellent uniformity / homogeneity , and large areal coverage , is opening up lines of sight towards differentially - reddened star clusters across the entire high - latitude north galactic cap for this type of analysis . near - infrared @xmath153 photometry from the two micron all - sky survey ( 2mass ) data should also prove very fruitful , both for probing highly - reddened lines of sight and for extending the sdss wavelength coverage . making @xmath0 and power spectrum measurements along many sight - lines should lead to a better understanding of grain formation and chemistry and the effect of interstellar dust on astronomical observations . while our @xmath54 grid of simulations covers some amount of parameter space in photometric error and @xmath0 , it is fairly limited in scope : it can not simply be translated to any arbitrary cluster data set . another data set may be different from the ngc 4833 data set in terms of photometric bandpasses , level of photometric error , areal coverage , detailed morphology of cmd features , degree of extinction / reddening variations , and/or extinction law slope . new simulations must be carried out , matched to the exact parameters of each new cluster data set in question , in order to test the efficacy of the @xmath0 measurement methods presented here . * we have demonstrated the use of three methods , `` @xmath25 rms '' , `` @xmath25 slope '' , and `` optical / ir '' methods , for determining the dust extinction law slope @xmath0 in the direction of differentially - reddened galactic globular clusters , and have tested the methods on an extensive suite of simulated cluster data sets . * for cluster data sets with low photometric error , @xmath154 mag , the `` @xmath25 rms '' and `` @xmath25 slope '' methods can be used to determine @xmath0 to an accuracy of @xmath1550.3 . for cluster data sets with @xmath156 mag , the two methods yield @xmath0 to within @xmath83 , with some systematic biases . * the `` optical / ir '' method generally provides relatively imprecise estimates of @xmath0 , @xmath157 , over the full range of photometric errors explored . * combining the results from all three methods gives a mean extinction law slope of @xmath7 for the line of sight towards the low - latitute galactic globular cluster ngc 4833 . * the scatter in the cluster hb is used to estimate the amount of small - scale structure in the dust complex in the foreground of ngc 4833 . the schlegel et al . ( 1998 ) iras+dirbe - based map of the dust thermal emission averages over small - scale reddening variations that are @xmath158% of the mean reddening value . * star - to - star variations in relative visual extinction across the face of ngc 4833 provide a measure of the foreground dust angular power spectrum for projected separations in the range @xmath17@xmath18 . this small - scale power spectrum derived from cluster optical data matches smoothly onto the larger - scale power spectrum derived from the sfd reddening map of the region . the overall power spectrum is well fit by a power law : @xmath15 , where @xmath30 is the reciprocal of the angular scale @xmath112 , with spectral index @xmath137 . we would like to thank neil balmforth , david burstein , sandra faber , and peter stetson for comments that significantly improved this work . we would also like to thank ata sarajedini for providing the observations of ngc 4833 . cc|c|ccc + & & 2.0 & @xmath159 & @xmath160 & @xmath161 + 0.003 & 10 & 3.0 & @xmath162 & @xmath163 & @xmath164 + & & 4.0 & @xmath165 & @xmath166 & @xmath167 + & & 2.0 & @xmath168 & @xmath169 & @xmath170 + 0.015 & 50 & 3.0 & @xmath171 & @xmath172 & @xmath173 + & & 4.0 & @xmath174 & @xmath175 & @xmath176 + & & 2.0 & @xmath177 & @xmath178 & @xmath179 + 0.030 & 100 & 3.0 & @xmath180 & @xmath181 & @xmath182 + & & 4.0 & @xmath183 & @xmath184 & @xmath185 + + 0.030 & & & @xmath186 & @xmath187 & @xmath188

we present three methods for measuring the slope of the galactic dust extinction law , @xmath0 , and a method for measuring the fine - scale structure of dust clouds in the direction of differentially - reddened globular clusters . we apply these techniques to @xmath1 photometry of stars in the low - latitude galactic globular cluster ngc 4833 which displays spatially - variable extinction / reddening about a mean @xmath2 . an extensive suite of monte carlo simulations is used to characterize the efficacy of the methods . the essence of the first two methods is to determine , for an assumed value of @xmath0 , the _ relative _ visual extinction @xmath3 of each cluster horizontal branch ( hb ) star with respect to an empirical hb locus ; the locus is derived from the color - magnitude diagram ( cmd ) of a subset of stars in a small region near the cluster center for which differential extinction / reddening are relatively small . a star - by - star comparison of @xmath3 from the ( @xmath4 , @xmath5 ) cmd with that from the ( @xmath6 , @xmath5 ) cmd is used to find the optimal @xmath0 . in the third method , @xmath0 is determined by minimizing the scatter in the hb in the ( @xmath4 , @xmath5 ) cmd after correcting the photometry for extinction and reddening using the schlegel , finkbeiner , & davis ( 1998 ) dust maps . the weighted average of the results from the three methods gives @xmath7 for the dust along the line of sight to ngc 4833 . the fine - scale structure of the dust is quantified via the difference , @xmath8 , between pairs of cluster hb stars ( @xmath9,@xmath10 ) as a function of their angular separation @xmath11 . the variance ( mean square scatter ) of @xmath12 is found to have a power - law dependence on angular scale : @xmath13 , with @xmath14 . this translates into an angular power spectrum @xmath15 , with the index @xmath16 for @xmath17@xmath18 , where @xmath19 . the dust angular power spectrum on small scales ( from optical data ) matches smoothly onto the larger - scale power spectrum derived from schlegel et al.s far - infrared map of the dust thermal emission .

arxiv

heavy - fermion systems @xcite are characterized by a hierarchy of distinctive energy scales @xcite . the kondo scale , @xmath1 with bandwidth @xmath2 and superexchange @xmath3 , marks the screening of local magnetic moments . this screening is a many - body effect which entangles the spins of the conduction electrons and local moments @xcite . below the coherence temperature , which is believed to track the kondo scale @xcite , the paramagnetic ( pm ) heavy - fermion liquid @xcite emerges and corresponds to a coherent , bloch like , superposition of the screening clouds of the individual magnetic moments . even in the kondo limit , where charge fluctuations of the impurity spins are completely suppressed , this paramagnetic state is characterized by a large fermi surface with luttinger volume including both the magnetic moments and conduction electrons @xcite . the coherence temperature of this metallic state is small or , equivalently , the effective mass large . + kondo screening competes with the ruderman - kittel - kasuya - yosida ( rkky ) interaction , which indirectly couples the local moments via the magnetic polarization of the conduction electrons . the rkky energy scale is set by @xmath4 where @xmath5 corresponds to the spin susceptibility of the conduction electrons @xcite . + the competition between kondo screening - favoring paramagnetic ground states - and the rkky interaction - favoring magnetically ordered states - is at the heart of quantum phase transitions @xcite , the detailed understanding of which is still under debate ( for recent reviews see ref . ) . + here , two radically different scenarios have been put forward to describe this quantum phase transition . in the _ standard _ hertz - millis picture @xcite , the quasi - particles of the heavy - fermion liquid remain intact across the transition and undergo a spin - density wave transition . in particular , neutron scattering experiments of the heavy - fermion system @xmath6 show that fluctuations of the antiferromagnetic order parameter are responsible for the magnetic phase transition and that the transition is well understood in terms of the hertz - millis approach @xcite . + on the other hand , since many experimental observations such as the almost wave vector independent spin susceptibility in @xmath7 @xcite , or the jump in the low - temperature hall coefficient in @xmath8 @xcite are not accounted for by this theory alternative scenarios have been put forward @xcite . in those scenarios , the quantum critical point is linked to the very breakdown of the quasi - particle of the heavy - fermion state @xcite , and a topological reorganization of the fermi surface across the transition is expected @xcite . + recent experiments on @xmath9 @xcite or @xmath10 @xcite show that a change in fermi surface ( fs ) topology must not necessarily occur only at the magnetic order - disorder quantum critical point ( qcp ) . in fact , even in ybrh@xmath11si@xmath11 it has since been shown that the fermi surface reconstruction can be shifted to either side of the qcp via application of positive or negative chemical pressure @xcite . in this paper , we address the above questions through an explicit calculation of the fermi surface topology in the framework of the kondo lattice model ( klm ) . in its simplest form the klm describes an array of localized magnetic moments of spin @xmath12 , arising from atomic @xmath13-orbitals , that are coupled antiferromagnetically ( af ) via the exchange interaction @xmath3 to a metallic host of mobile conduction electrons . + we present detailed dynamical cluster approximation ( dca ) calculations aimed at the investigation of the klm ground state . for the simulations within the magnetically ordered phase , we have extended the dca to allow for symmetry breaking antiferromagnetic order . we map out the magnetic phase diagram as a function of @xmath14 and conduction electron density @xmath15 , with particular interest in the single - particle spectral function and the evolution of the fermi surface . the outline is as follows . the model and the dca implementation is discussed in sec . [ sec : section2 ] . results for the case of half - band filling and hole - doping are discussed in sec . [ sec : section3 ] and [ sec : section4 ] . section [ sec : section5 ] is devoted to a summary . this paper is an extension to our previous work , where part of the results have already been published @xcite . the kondo lattice model ( klm ) we consider reads @xmath16 the operator @xmath17 denotes creation of an electron in a bloch state with wave vector @xmath18 and a z - component of spin @xmath19 . the spin @xmath12 degrees of freedom , coupled via @xmath20 , are represented with the aid of the pauli spin matrices @xmath21 by @xmath22 and the equivalent definition for @xmath23 using the localized orbital creation operators @xmath24 . the chemical potential is denoted by @xmath25 . the definition of the klm excludes charge fluctuations on the @xmath13-orbitals and as such a strict constraint of one electron per localized @xmath13-orbital has to be included . for an extensive review of this model we refer the reader to ref . . + particle - hole symmetry at half - filling is given if hopping is restricted to nearest neighbors on the square lattice and the chemical potential is set to zero . we introduce a next - nearest neighbor hopping with matrix element @xmath26 to give a modified dispersion @xmath27 -2 t ' \left [ \cos(k_{x}+k_{y } ) + \cos(k_{x}-k_{y})\right]$ ] . as we will see , the suppression of particle - hole symmetry due to a finite value of @xmath26 leads to dramatic changes in the scaling of the quasi - particle gap at low values of @xmath14 and at half - band filling . we have considered the value @xmath28 . this choice guarantees that the spin susceptibility of the host metallic state at small dopings away from half - filling is peaked at wave vector @xmath29 . thus , antiferromagnetic order as opposed to an incommensurate spin state is favored . + to solve the model we have used the dca @xcite approach which retains _ temporal _ fluctuations and hence accounts for the kondo effect but neglects spatial fluctuations on a length scale larger than the cluster size . the approach relies on the coarse - graining of momentum space , and momentum conservation holds only between the @xmath18-space patches . by gradually defining smaller sized patches the dca allows to restore the @xmath18-dependency of the self - energy . + the standard formulation of dca is naturally translationally invariant . to measure orders beyond translation invariance the dca can be generalized to lattices with a supercell containing @xmath30-unit cells of the original lattice @xcite . a general unit cell is addressed by @xmath31 , @xmath32 denoting the supercell and @xmath33 the relative points with @xmath34 . in this work we opted for @xmath35 which is the minimum requirement for the formation of antiferromagnetic order . the reduced brillouin zone ( rbz ) is then spanned by @xmath36 and @xmath37 and we have set the lattice constant to unity . this supercell as well as the rbz ( also coined magnetic brillouin zone ( mbz ) ) are plotted in fig . [ fig : fig1 ] . + the self - energy and green function are to be understood as spin dependent matrix functions , with an index for the unit cell within the supercell as well as an orbital index for the @xmath38- and @xmath13-orbitals in each unit cell . the @xmath18-space discretization into @xmath39 patches with momentum conservation only between patches yields the coarse - grained lattice green function @xmath40 and ensures that the self - energy is only dependent on the coarse - grained momentum @xmath41 : @xmath42 . here , the reciprocal vector @xmath43 denotes the center of a patch and the original @xmath18 vectors are given by @xmath44 . the self - energy is extracted from a real - space cluster calculation with periodic boundaries yielding the quantized @xmath43 values . let @xmath45 be the bath ( non - interacting ) green function of the cluster problem and @xmath46 the full cluster green function . hence , @xmath47 and self - consistency requires that : @xmath48 + and @xmath49 connect the af unit cells . the two inequivalent @xmath38-orbitals are denoted by filled ( empty ) circles and the localized @xmath13-orbitals with arrows . the reciprocal - space lattice vectors @xmath50 and @xmath51 span the magnetic brillouin zone . the mbz is uniformly discretized in @xmath52 patches , corresponding to real - space clusters each containing @xmath39 magnetic unit cells . the color coding refers to constant momentum dependency of the self - energy and cluster green function . the dashed square denotes the extended brillouin zone . ] the non - interacting lattice green function is denoted by @xmath53 . + to summarize , our implementation of dca approximates the fourier space of a lattice with two - point basis by patching . regarding the klm this leads to real - space clusters of size @xmath39 , each encompassing @xmath54 @xmath38- and @xmath13-orbitals . in the present work cluster sizes @xmath55 and @xmath52 are considered ( fig . [ fig : fig1 ] ) . + the lattice green function @xmath56 with @xmath57 can equally be expressed as @xmath58 in an extended brillouin zone scheme , with @xmath59 and @xmath60 : @xmath61_{\mu \nu}\;. \label{eqn : eqn4}\end{aligned}\ ] ] in order to implement the dca one has to be able to solve , for a given bath , the kondo model on the effective cluster . the method we have chosen is the auxiliary field quantum monte carlo ( qmc ) version of the hirsch - fye algorithm , following precisely the same realization as in ref . . the implementation details of the self - consistency cycle can be found in ref . . the performance of the qmc cluster solver has been enhanced by almost an order of magnitude by implementing the method of delayed updates @xcite . during the qmc markov process the full updates of the green function are delayed until a sufficiently large number of local changes in the auxiliary field have been accumulated . this enhances the performance of the whole qmc algorithm since considerable cpu time is spent in the updating section . to efficiently extract spectral information from the imaginary time discrete qmc data a stochastic analytic continuation scheme is employed @xcite . + ground state properties of the model are accessed by extrapolation of the inverse temperatures @xmath62 to infinity . this is a demanding task since all relevant energy scales , the kondo scale , the coherence scale and the rkky scale , become dramatically smaller with decreasing @xmath14 . in the paramagnetic phase and below the coherence scale a clear hybridization gap is apparent in the single particle spectral function and we use this criterion to estimate the coherence temperature . since the computational time required by the qmc cluster solver @xcite increases proportional to @xmath63 , this limits us in the resolution of energy scales to values of @xmath64 . it is important to note that for small dopings away from half - band filling and for the considered cluster sizes , the negative sign problem is not severe and hence is not the limiting factor . we consider the klm at half - filling in two cases : either with or without particle - hole symmetry . that is @xmath65 and @xmath66 respectively . at @xmath67 lattice qmc simulations do not suffer from the negative sign problem , and it is well established that the kondo screened phase gives way to an af ordered phase at @xmath68 @xcite . this magnetic order - disorder transition occurs between insulating states and it is reasonable to assume that it belongs to the three - dimensional @xmath69 universality class @xcite . here , the dynamical exponent takes the value of unity such that the correlation lengths in imaginary time and in real space are locked in together and diverge at the critical point . due to the very small cluster sizes considered in the dca it is clear that we will not capture the physics of this transition . in fact as soon as the correlation length exceeds the size of the dca cluster , symmetry breaking signaled by a finite value of the staggered moment @xmath70 sets in and mean - field exponents are expected . the normalization is chosen such that the staggered magnetization of the fully polarized state takes the value of unity . the above quantity is plotted in fig . [ fig : fig2 ] at @xmath65 and @xmath55 as apparent , the critical value of @xmath14 overestimates ( @xmath71 . ) the _ exact _ lattice qmc result . switching off nesting by including a finite value of @xmath72 shifts the magnetic order - disorder transition to lower values of @xmath14 . + to improve on this result , we can systematically enhance the cluster size . however , our major interest lies in the single - particle spectral function . as we will see below , and well within the magnetically ordered phase , this quantity compares very well with the _ exact _ lattice qmc results . of the local moment spins in the ground state as a function of coupling @xmath14 with next - nearest neighbor hopping @xmath73 and @xmath74 , respectively . the low temperature limit is reached by performing simulations at various inverse temperatures @xmath62 . convergence to the ground state has been achieved for the following inverse temperatures : @xmath75 ( @xmath76 ) , @xmath77 ( @xmath78 ) , @xmath79 ( @xmath80 ) and @xmath81 ( @xmath82 ) . ] we have calculated the single - particle spectral function @xmath83 $ ] of the conduction electrons along a path of high symmetry in the extended brillouin zone . in all our spectral plots , on the energy axis , @xmath84 values are given relative to the chemical potential @xmath25 . + in fig . [ fig : fig3 ] we plot the spectrum for @xmath85 which was seen to be in the paramagnetic phase ( see fig . [ fig : fig2 ] ) . the lower band runs very flat around @xmath86 , with relatively low spectral weight ( note the logarithmic scale of the color chart ) in comparison to the other parts of the band which are mostly unchanged from the non - interacting case . this feature is associated with kondo screening of the impurity spins and the resultant large effective mass of the composite quasi - particles . since no band crosses the fermi energy ( @xmath87 ) we classify this region of parameter space as a kondo insulator . the observed dispersion relation is well described already in the framework of the large-@xmath88 mean - field theory of the kondo lattice model @xcite . at the particle - hole symmetric point this approximation , which in contrast to the dca+qmc results of fig . [ fig : fig3 ] neglects the constraint of no double occupancy of the @xmath13-orbitals , yields the dispersion relation of hybridized bands : @xmath89 . \label{eqn : eqn6}\ ] ] within the mean - field theory the quasi - particle gap ( see eq . [ eqn : eqn7 ] ) , @xmath90 , tracks the kondo scale . + at half - filling ( @xmath91 ) and with particle - hole symmetry ( @xmath74 ) close to the magnetic phase transition on the paramagnetic side , @xmath85 . ] at @xmath92 we have measured a non - vanishing staggered magnetization @xmath93 . the spectral function ( fig . [ fig : fig4 ] ) for this lightly af ordered simulation now includes additional low - energy band structures : in the upper band around @xmath86 and in the lower band around @xmath94 . these _ shadow bands _ arise due to the scattering of the heavy quasi - particle off the magnetic fluctuations centered at wave vector @xmath95 . + at half - filling ( @xmath91 ) and with particle - hole symmetry ( @xmath74 ) close to the magnetic phase transition on the af side , @xmath92 ( a ) and closeup ( b ) . the onset of magnetic order and concomitant emergence of the mbz is signaled by the appearance of shadow bands . the staggered magnetization for this point is measured to be @xmath93 . ] at @xmath96 we make a comparison with the spectrum obtained via qmc lattice simulations using the projective auxiliary field algorithm of ref . for the same parameters . we see in fig . [ fig : fig5 ] the qmc lattice simulation results @xcite on the left and our dca result on the right , both only for the photoemission spectrum @xmath97 . the agreement of the dca spectrum with the blankenbecler - scalapino - sugar ( bss ) result is excellent . as in the bss lattice calculations and for the particle - hole symmetric case , the conduction ( valence ) band maximum ( minimum ) is located at @xmath98 ( @xmath99 ) for all considered values of @xmath14 . + with the @xmath100 spectral function . data ( a ) taken from ref . which used a projective auxiliary field monte carlo ( bss ) method for the klm on a @xmath101 lattice and our dca results ( b ) at @xmath77 , both for the same parameter set - @xmath74 , @xmath102 , @xmath96.,title="fig : " ] + it is noteworthy that already with the smallest possible cluster capable of capturing af order , @xmath55 , we are able to produce a single - particle spectrum which is essentially the same as the lattice qmc result . this confirms that the dca is indeed a well suited approximation for use with the klm : the essence of the competition between rkky - mediated spacial magnetic order and the time - displaced correlations responsible for kondo screening is successfully distilled to a small cluster dynamically embedded in the mean - field of the remaining bath electrons . we now set @xmath73 , but remain at half - filling via careful adjustment of the chemical potential @xmath25 . with decreasing @xmath14 we can divide the phase diagram into four regions on the basis of the characteristic spectrum in each region . for @xmath103 ( fig . [ fig : fig6](a ) ) the spectrum is qualitatively identical to the pm phase in the @xmath74 case such that the model is a kondo insulator with an indirect gap . the minimum of the valence band lies at @xmath94 and the maximum of the conduction band is at @xmath86 . + for @xmath104 ( fig . [ fig : fig6](b ) ) the system is weakly af ordered , and the spectrum reflects this in the development of shadow bands , but is otherwise qualitatively the same as before . at @xmath105 ( fig . [ fig : fig6](c ) ) , well inside the af phase , the shadow bands are more pronounced and now the minimum of the valence band has shifted to @xmath106 . the final plot in the series ( fig . [ fig : fig6](d ) ) , with @xmath107 shows that the form of the lower band has also changed such that the maximum of this band now lies at @xmath108 . + in figs . [ fig : fig7](a - h ) we show the evolution of this lower band between @xmath94 and @xmath86 with an enlargement of the energy axis around the fermi energy . the local minimum energy dip at @xmath108 becomes less pronounced as the heavy - fermion band flattens until by the time we reach @xmath96 this dip has turned into a bump such that the lower band maximum has shifted from @xmath86 to @xmath108 . this important change in band structure appears to be continuous . for smaller coupling , @xmath109 , @xmath110 , and @xmath111 the lower band maximum remains at @xmath108 and becomes more pronounced . assuming a rigid band picture , this evolution of the band structure maps onto a topology change of the fermi surface at small dopings away from half filling . we will see by explicit calculations at finite dopings that this topology change indeed occurs . + the position of the minima and maxima in the valence and conduction bands of the spectra for @xmath73 are summarized schematically in fig . [ fig : fig8 ] . and for half - filling . the squares represent the first brillouin zone ( for the pm phase ) or the extended brillouin zone ( for the af phase ) , with the bottom left corner and top right corner of each square given by @xmath112 and @xmath86 , respectively.,title="fig : " ] + the quasi - particle gap corresponds to the energy difference between the top of the conduction band and the bottom of the valence band . to be more precise , @xmath113 here , @xmath114 corresponds to the ground state energy of the @xmath115 particle system with total momentum @xmath18 and @xmath116 . technically , the energy differences are at best extracted from the low temperature single - particle green functions : @xmath117 as a function of kondo coupling @xmath14 . at @xmath118 , the magnetic order - disorder transition takes place at @xmath119 . in the coupling range @xmath120 kondo screening coexists with magnetism and is at the origin of the quasiparticle gap . ] in the large @xmath14 limit , each impurity spin traps a conduction electron in a kondo singlet . the wave function corresponds to a direct product of such kondo singlets and the quasi - particle gap is set by the energy scale @xmath121 required to break a kondo singlet . hence , in the large @xmath14 limit the quasi - particle gap does not depend on the details of the band structure . the situation is more subtle in the weak - coupling limit , where the underlying nesting properties of the fermi surface play a crucial role . assuming static impurity spins locked into an antiferromagnetic order , @xmath122 one obtains the single particle dispersion relation @xmath123 with @xmath124 . in one dimension and at @xmath74 nesting leads to @xmath125 and magnetic order opens a quasi - particle gap set by @xmath126 . both the dca results presented in fig . [ fig : fig9 ] as well as the bss results of ref . support this point of view . in one - dimension , this scaling of the quasi - particle gap is also observed @xcite . at @xmath66 nesting is not present and magnetic ordering can only partially gap the fermi surface . since the dca results of fig . ( [ fig : fig9 ] ) support a finite quasi - particle gap in the magnetically ordered phase , the above frozen - spin ansatz fails and points to one of the main result of our work , namely that within the magnetically ordered phase kondo screening and the heavy quasi - particles are still present . even though we can not track the functional form of the gap at @xmath127 and at weak couplings , we interpret the dca results in terms of a quasi - particle gap which tracks the kondo scale from weak to strong coupling as in the large-@xmath88 mean - field calculations @xcite . in this section we concentrate on the hole - doped klm at @xmath66 . we will first map out the magnetic phase diagram in the doping versus coupling plane and then study the evolution of single - particle spectral function from the previously discussed half - filled case to the heavily doped paramagnetic heavy - fermion metallic state . of the @xmath13-electrons as a function of @xmath15 at different constant couplings @xmath14 . ] [ fig : fig10 ] shows the staggered magnetization as a function of conduction electron density for @xmath82 , @xmath110 , @xmath128 and @xmath129 . the magnetically ordered state found at half - filling initially survives when doping with holes . at all coupling values a continuous magnetic phase transition is observed with the af order decreasing gradually as the system is doped and vanishing smoothly at a quantum critical point . the results are summarized in the magnetic phase diagram shown in fig . [ fig : fig11 ] . with decreasing values of @xmath14 and increasing dopings the rkky interaction progressively dominates over the kondo scale and the magnetic metallic state is stabilized . checks were made by varying the temperature , @xmath130 , of the simulations to ensure that the results can be considered to be ground state . below @xmath131 we found the coherence scale to be too low to guarantee convergence . as pointed out previously , the limiting computational factor is the cubic scaling of the hirsch - fye @xcite algorithm , @xmath132 and not the negative sign problem . ( color - coded ) as a function of coupling @xmath14 and conduction electron occupancy @xmath15 . triangles : pm region , large fs . squares : af , large fs . circles : af , small fs . here @xmath66 and the calculations are carried out with the @xmath133 cluster . below , the fs topologies corresponding to the numbered regions are shown schematically.,title="fig : " ] + we have calculated and followed the evolution of the single - particle spectrum , plotted in an extended brillouin zone scheme . to begin with we set @xmath80 and show results in fig . [ fig : fig12 ] for the spectral function , as it is here that a change in the fermi surface topology becomes evident . starting from the previously discussed half - filled case ( fig . [ fig : fig12](a ) ) a rigid band approximation produces hole pockets around the @xmath134 points in the brillouin zone . this is confirmed by explicit calculations at small dopings away from half - filling as demonstrated in fig . [ fig : fig12](b ) . with further doping this low energy band flattens out progressively becoming almost flat by @xmath135 ( fig . [ fig : fig12](d ) ) . doping further gives rise to a fermi surface with holes centered around @xmath136 . since at @xmath137 ( fig . [ fig : fig12](e ) ) and @xmath138 ( fig . [ fig : fig12](f ) ) we still have non - zero magnetizations of @xmath139 and @xmath140 , respectively , shadow features centered around @xmath94 are expected . the weight of those shadow features progressively diminishes as the staggered magnetization vanishes . in particular , in fig . [ fig : fig12](g ) the magnetization is very small ( @xmath141 ) and indistinguishable from zero in the final figure of the series . + across the transition from @xmath142 to @xmath143 the quasiparticle weight @xmath144 as obtained from the behaviour of @xmath145 at large imaginary times decreases albeit it shows no sign of singularity . this observation stands in agreement with results obtained at half - band filling @xcite and excludes the occurrence of a kondo breakdown . + this evolution of the single - particle spectral function at @xmath146 points to three distinct fermi surface topologies , sketched in fig . [ fig : fig11 ] . in the paramagnetic phase ( fermi surface ( 1 ) in fig . [ fig : fig11 ] ) the fermi surface consists of hole pockets around the @xmath147 points in the brillouin zone . even though in our simulations charge fluctuations of the @xmath13-sites are completely prohibited , the fermi surface volume accounts for both the conduction electrons and impurity spins . this fermi surface topology maps onto that of the corresponding non - interacting periodic anderson model with total particle density given by @xmath148 and is coined large fermi surface . in the antiferromagnetic metallic phase close to the magnetic order - disorder transition , ( fermi surface ( 2 ) in fig . [ fig : fig11 ] ) the fermi surface merely corresponds to a backfolding of the paramagnetic fermi surface as expected in a generic spin - density wave transition . here , a heavy quasi - particle with crystal momentum @xmath18 can scatter off a magnon with momentum @xmath149 to produce a shadow feature at @xmath150 . it is only within the magnetically ordered phase that we observe the topology change of the fermi surface to hole pockets centered around @xmath151 . due to the antiferromagnetic order and the accompanied reduced brillouin zone , this fermi surface topology satisfies the luttinger sum rule . we coin it a small fermi surface in the sense that it is linked to that obtained in a frozen @xmath13-spin mean - field calculation as presented in eq . ( [ eqn : eqn9 ] ) . of the conduction electrons for @xmath105 and @xmath73 , at @xmath152 . the shadow features present in ( a ) vanish as the occupation number @xmath15 is reduced . ] we have equally plotted the single - particle spectral function at higher values of @xmath105 as a function of doping ( see fig . [ fig : fig13 ] ) . for this coupling strength , and at half - band filling , the conduction band maximum is located at @xmath86 with accompanying shadow bands at @xmath94 . the fermi surface obtained upon doping follows from a rigid band approximation , and yields the topology shown in fig . [ fig : fig11 ] ( 2 ) . the transition to the paramagnetic state shows up in the vanishing of the shadow features at @xmath153 . + we note that our results are confirmed by simulations carried out using the larger cluster size @xmath52 , see fig . [ fig : fig14 ] . in particular , we observe the small fermi surface topology in the strong af region with small coupling @xmath14 and evidence for a large fermi surface in the weakly ordered region ( @xmath154 ) . of the conduction electrons obtained using cluster size @xmath52 . confirming that the small fermi surface topology in the lightly doped af phase with coupling @xmath80 is not an artifact of the smaller cluster results . the staggered magnetisation of the @xmath13-electrons is @xmath155 . ] we resort to a mean - field modelling in order to propose a scenario for the detailed nature of the topological transition of the fs within the ordered phase . within this model the topology change will correspond to two lifshitz transitions . aspects of the single - particle spectral function can be well accounted for within the following mean - field hamiltonian @xcite : @xmath156 here @xmath157 , ( @xmath25 ) are lagrange multipliers fixing the @xmath13- ( @xmath38- ) particle number to unity ( @xmath15 ) and @xmath158 , @xmath159 corresponds to the staggered magnetization of the conduction and @xmath13-electrons . the order parameter for kondo screening is @xmath160 . the large-@xmath88 mean - field saddle point corresponds to the choice @xmath161 and @xmath162 . as already mentioned this saddle point gives a good account of the hybridized bands we observe numerically in the paramagnetic phase . in the magnetic phase , one could speculate that kondo screening breaks down such that the @xmath13-spins are frozen and do not participate in the luttinger volume . this corresponds to the parameter set @xmath163 , @xmath164 and @xmath165 and leads to the dispersion relation of eq . ( [ eqn : eqn8 ] ) . throughout the phase diagram of fig . [ fig : fig11 ] the single - particle spectral function never shows features following this scenario . in fact , many aspects of the spectral function in the magnetically ordered phase can be understood by choosing @xmath166 , @xmath167 and @xmath168 . this explicitly accounts for a heavy - fermion band in the ordered phase . it is in this sense that we claim the absence of kondo breakdown within our model calculations . [ fig : fig15 ] plots the four - band energy dispersion relation @xmath169 obtained from the mean - field calculation in the magnetically ordered phase at @xmath73 , @xmath80 and at band fillings @xmath170 ( fig . [ fig : fig15](a , b ) ) and @xmath171 ( fig . [ fig : fig15](c , d ) ) corresponding to the fermi surface topologies labeled ( 2 ) and ( 3 ) , respectively , in fig . [ fig : fig11 ] . in each case we have used the dca value for the magnetization and varied @xmath172 to obtain the best qualitative fit to the respective dca spectrum . the change in the topology of the fermi surface stems from a delicate interplay of the relative magnitudes of the magnetization @xmath173 and _ kondo screening _ parameter @xmath172 . + with @xmath80 ( @xmath73 ) . ( a ) @xmath174 , @xmath170 . the line intensity is proportional to the quasi - particle spectral weight , @xmath175 . the fermi surface topology ( see closeup ( b ) ) results from backfolding of the large-@xmath88 mean - field bands . this corresponds to the fermi surface ( 2 ) in fig.[fig : fig11 ] . + ( d ) @xmath176 , @xmath171 . the fermi surface topology ( see closeup ( c ) ) corresponds to that of hole pockets centered around @xmath177 as depicted by the fermi surface ( 3 ) in fig . [ fig : fig11 ] . ] in the dca calculations , we are unable to study the precise nature of the topology change of the fermi surface within the magnetically ordered phase since the energy scales are too small . however we can draw on the mean - field model to gain some insight . [ fig : fig16 ] plots the dispersion relation along the @xmath94 to @xmath86 line in the brillouin zone as a function of @xmath172 . we would like to emphasize the following points . i ) the transition from hole pockets around the wave vector @xmath178 to @xmath179 corresponds to two lifshitz transitions in which the hole pockets around @xmath178 disappears and those around @xmath180 emerge . the luttinger sum rule requires an intermediate phase with the presence of pockets both at @xmath180 and at @xmath178 . this situation is explicitly seen in fig . [ fig : fig16](b ) at @xmath181 . ii ) starting from the antiferromagnetic state with hole pockets at @xmath177 there is an overall flatting of the band prior to the change in the fermi surface topology . hence on both sides of the transition , an enhanced effective mass is expected . owing to point i ) the effective mass does not diverge . iii ) the two lifshitz transition scenario suggested by this mean - field modeling , leads to a continuous change of the hall coefficient . however , since a very _ small _ variation of the hybridization @xmath172 suffices to change the fermi surface topology one can expect , as a function of this control parameter , a rapid variation of the hall coefficient as it is observed in experiment @xcite . of the dca results ( @xmath80 , @xmath73 @xmath182 , @xmath183 , @xmath170 ) . as apparent on the scale of the closeup ( b ) , hole pockets around @xmath180 as well as around @xmath178 are present at @xmath184 . ] the evolution of the fermi surface across the magnetic order - disorder transition has been investigated by other groups and especially in the framework of a gutzwiller projected mean - field wave function corresponding to the ground state of the single particle hamiltonian of eq . [ eqn : eqn9 ] . both a variational quantum monte carlo calculation @xcite as well as a gutzwiller approximation @xcite show a rich phase diagram which bears some similarities but also important differences with the present dca calculation . at small doping away from half - filling , the magnetic transition as a function of the coupling @xmath14 is continuous and of sdw type as marked by the back - folding of the fermi surface . within the magnetically ordered phase a first order transition occurs between hole ( af@xmath185 ) and electron ( af@xmath186 ) like fermi surfaces . at larger hole dopings , the magnetic transition is of first order and the fermi surface abruptly changes from large to small ( af@xmath186 ) . the calculations in refs . are carried out at @xmath187 . at finite values of @xmath188 one expects the af@xmath186 fermi - surface topology to correspond to hole pockets centered around @xmath189 as shown in fig . [ fig : fig12 ] . on the other hand , the af@xmath185 fermi surface arises from a back - folding of the large fermi surface ( fig . [ fig : fig13 ] ) . with this identification , our dca results bear some similarity with the variational calculation in the sense that the same fermi surface topologies are realized . however , in our calculations we see no sign of first order transitions and no direct transitions from the large paramagnetic fermi - surface topology to the af@xmath186 phase . it is equally important to note that in both the variational and dca approaches no kondo breakdown is apparent . in particular the variational parameter @xmath190 which encodes hybridization between the @xmath13- and @xmath38-electrons never vanishes @xcite . on the other hand , @xmath100 cdmft calculations of the three - dimensional periodic anderson model on a two site cluster and finite sized baths @xcite have put forward the idea that the magnetic order - disorder transition is driven by an orbital selective mott transition or in other words a kondo breakdown @xcite . in particular even in the presence of spin symmetry broken baths allowing for antiferromagnetic ordering a big enhancement of the @xmath13-effective mass is apparent in the vicinity of the magnetic transition . in the paramagnetic phase , the low energy decoupling of the @xmath13- and @xmath38-electrons stem from the vanishing of the hybridization function at low energies . our results , which makes no approximation of the number of bath degrees of freedom and which allow to treat larger cluster sizes do not support this point of view . in particular as shown in ref . no singularity in the quasiparticle weight is apparent across the transition . equally , fig . [ fig : fig12 ] which plots the single particle spectral function at finite doping shows no singularity in @xmath191 . we have presented detailed dca+qmc results for the kondo lattice model on a square lattice . our dca approach allows for antiferromagnetic order such that spectral functions can be computed across magnetic transitions and in magnetically ordered phases . at the particle - hole symmetric point ( half - band filling and @xmath67 ) we can compare this approach to previous _ lattice qmc auxiliary field simulations @xcite . unsurprisingly , with the _ small _ cluster sizes considered in the dca approach the critical value of @xmath14 at which the magnetic order - disorder transition occurs is substantially overestimated . the important point however is that the dca+qmc approximation gives an extremely good account of the single - particle spectral function both in the paramagnetic and antiferromagnetic phases . hence , and as far as we can test against benchmark results , the combination of static magnetic ordering and dynamical kondo screening , as realized in the dca , provides a very good approximation of the underlying physics . + as opposed to _ exact _ lattice qmc auxiliary field simulations , which fail away from the particle - hole symmetric point due to the so - called negative sign problem , the dca+qmc approach allows us to take steps away from this symmetry either by doping or by introducing a finite value of @xmath26 . it is worthwhile noting that for cluster sizes up to @xmath0 orbitals the limiting factor is the cubic scaling of the hirsch - fye algorithm rather than the negative sign problem which turns out to be very mild in the considered parameter range . + our major findings are summarized by the following points . + i ) we observe no kondo breakdown throughout the phase diagram even deep in the antiferromagnetic ordered state where the staggered magnetization takes large values . this has the most dramatic effect at half - band filling away from the particle - hole symmetric point and at small values of @xmath14 where magnetic order is robust . here , magnetism alone will not account for the insulating state and the observed quasi - particle gap can only be interpreted in terms of kondo screening . the absence of kondo breakdown throughout the phase diagram is equally confirmed by the single - particle spectral function which always shows a feature reminiscent of the heavy - fermion band.@xcite + ii ) the transition from the paramagnetic to the antiferromagnetic state is continuous and is associated with the backfolding of the large heavy fermion fermi surface . + iii ) within the antiferromagnetic metallic phase there is a fermi surface topology change from hole pockets centered around @xmath86 at _ small _ values of the magnetization @xmath159 to one centered around @xmath192 at _ larger _ values of @xmath159 . the latter fermi surface is adiabatically connected to one where the @xmath13-spins are frozen and in which kondo screening is completely absent . @xcite this transition comes about by _ continuously deforming _ a heavy - fermion band such that the effective mass grows substantially on both sides of the transition . within a mean - field modelling of the quantum monte carlo results the transition in the fermi surface topology corresponds to two continuous lifshitz transitions . as a function of decreasing mean - field kondo screening parameter @xmath172 and at constant hole doping , the volume of the hole pocket at @xmath86 decreases at the expense of the increase of volume of the hole pocket at @xmath108 . owing to the luttinger sum rule , the total volume of the hole pockets remains constant . it is important to note that the energy scales required to resolve the nature of this transition are presently orders of magnitudes smaller than accessible within the dca with a quantum monte carlo solver . + + we would like to thank the dfg for financial support under the grant numbers as120/6 - 1 ( for1162 ) and as120/4 - 3 . the numerical calculations were carried out at the lrz - munich as well as at the jsc - jlich . we thank those institutions for generous allocation of cpu time .

we report the results of extensive dynamical cluster approximation calculations , based on a quantum monte carlo solver , for the two - dimensional kondo lattice model . our particular cluster implementation renders possible the simulation of spontaneous antiferromagnetic symmetry breaking . by explicitly computing the single - particle spectral function both in the paramagnetic and antiferromagnetic phases , we follow the evolution of the fermi surface across this magnetic transition . the results , computed for clusters up to @xmath0 orbitals , show clear evidence for the existence of three distinct fermi surface topologies . the transition from the paramagnetic metallic phase to the antiferromagnetic metal is continuous ; kondo screening does not break down and we observe a back - folding of the paramagnetic heavy fermion band . within the antiferromagnetic phase and when the ordered moment becomes _ large _ the fermi surface evolves to one which is adiabatically connected to a fermi surface where the local moments are frozen in an antiferromagnetic order .

arxiv

the recent discovery at the large hadron collider ( lhc ) of a resonance at 125 gev compatible with the expectations for the higgs particle @xcite represents a major step towards understanding the origin of the mass of fundamental particles . eventually , this should also affect the other subfield in which mass has a pivotal role , _ i.e. _ gravitation . this is particularly relevant in models in which the higgs field has nonminimal coupling to the general relativity sector , as invoked in various extensions of the standard model . nonminimal coupling between the higgs and spacetime curvature may be beneficial to have the higgs responsible for inflation @xcite , and as a suppression mechanism for the contribution to dark energy expected from quantum fields @xcite . upper bounds on the gravitational interaction of higgs bosons from the lhc experiments have been recently discussed @xcite . bounds on the crosstalk between the higgs particle and gravity may also be obtained by considering strong - gravity astrophysical objects , as proposed in @xcite in the case of active galactic nuclei ( agn ) and primordial black holes . the presence of a strong spacetime curvature deforms the vacuum expectaction value of the higgs field and therefore the mass of fundamental particles such as the electron . nucleons instead should be minimally affected by the strong curvature since most of their mass arises from the gluonic fields that , being massless , are not coupled to the higgs field at tree level . peculiar wavelength shifts are therefore predicted which should be present for electronic transitions and strongly suppressed for molecular transitions in which the main role is played by the nuclei themselves , such as in vibrational or rotational spectroscopy . due to the vanishing of the ricci scalar for spherically symmetric objects , attention was focused on the possibility of couplings to the only non - null curvature invariant , the kreschmann invariant , defined as @xmath0 , where @xmath1 is the riemann curvature tensor . this invariant plays an important role in quadratic theories of gravity @xcite , and more in general in modified @xmath2 theories @xcite and einstein - gauss - bonnet models of gravity @xcite . while agns would provide a strong - gravity setting near their black holes , their complex structure and the presence of turbulence and high - energy interactions near the accretion region induce uncontrollable systematic effects which hinder the possibility for extracting bounds on a higgs - kreschmann coupling as this relies upon the simultaneous observation of atomic and molecular transitions . to our knowledge , no neutron stars appear to show both molecular and atomic lines in their spectra , while white dwarfs have both . although their surface gravity is much weaker than around agns and neutron stars , many features can be controlled more precisely , thus providing a quieter environment to search for the putative higgs shift . white dwarfs have been known since the 19th century and in addition to their interest for astronomical and cosmological problems including understanding the late stages of stellar evolution , determining the galaxy s age , and the nature of ia supernovae , they have had a prominent role in fundamental physics since the early 20th century . @xcite made the first attempt to verify general relativity by measuring the gravitational redshift of sirius b. @xcite studied the consequences of fermi - dirac statistics for stars , introducing his celebrated limit . bounds on the distance dependence of the newtonian gravitational constant have been discussed comparing observations and models for the white dwarf sirius b @xcite and those in the hyades @xcite . more recently @xcite proposed using white dwarfs to study the dependence of the fine structure constant on gravity . here we show that white dwarfs can be used to obtain limits on the coupling of the higgs field to a specific curvature invariant , by means of spectroscopic observations of a carbon - rich white dwarf , bpm 27606 , using the southern african large telescope ( salt ) . the analysis is complemented by considering data taken from the hst archive on a second white dwarf , procyon b , in which caii and mgii lines , in addition to the c@xmath3 bands , are also present . the search for coupling between the higgs ( or any scalar field permeating the whole universe ) and spacetime curvature arises naturally within the framework of field theory in curved spacetime @xcite . the lagrangian density for an interacting scalar field in a generic spacetime characterized by the metric tensor @xmath4 is written as @xcite : @xmath5,\ ] ] where @xmath6 and @xmath7 are the mass parameter and the self - coupling quartic coefficient of the higgs field , respectively . 1 we have also introduced the determinant of the metric @xmath8 as @xmath9 , and @xmath10 , the coupling constant between the higgs field @xmath11 and the ricci scalar @xmath12 . the coupling constant @xmath10 is a free parameter in any model so far imagined to describe scenarios of scalar fields coupled to gravity , and it is therefore important to extract this coefficient , or upper bounds , from phenomenological analyses . the higgs field develops , under spontaneous symmetry breaking , a vacuum expectation value @xmath13 in flat spacetime , and the masses of the fundamental fermions are proportional to @xmath14 via the yukawa coefficients of the fermion - higgs lagrangian density term , @xmath15 . the effective mass parameter of the higgs field gets an extra - term due to the scalar curvature as @xmath16 , and the vacuum expectation value of the higgs field will become spacetime dependent through the curvature scalar as : @xmath17 where the approximation holds in a weak - curvature limit . this implies that the mass @xmath18 of fundamental fermions , such as the electron , will be simply changed proportionally to the higgs vacuum expectation value @xmath19 in other words , the presence of coupling of the higgs field to space - time curvature adds to its inertial mass a contribution , which acts as a mass renormalization due to curved space - time ( however , for an opposite interpretation of this mass shift see @xcite ) . this mass shift is not present for protons and neutrons , due to the fact that their mass is primarily due to the gluonic fields which , being massless , are unaffected by the higgs field . as discussed in more detail in @xcite , this implies that all molecular transitions only depending on the nuclei mass , such as vibrational and rotational spectra , should be unaffected by the higgs - curvature coupling at leading order . unfortunately , the ricci scalar outside spherically symmetric masses is zero , so we can not use this coupling to infer possible mass shifts for the electrons . the only non - zero curvature scalar outside spherically symmetric masses is the kreschmann invariant , and we will assume in the following considerations bounds to the kreschmann coupling * @xmath20 * to the higgs field in a lagrangian of the form @xmath21,\ ] ] where @xmath22 is the planck length , whose value is @xmath23 m in conventional quantum gravity models , or larger values such as the one corresponding to models with _ early _ unifications of gravity to the other fundamental interactions @xcite . in the latter case the planck length occurs at the tev scale via extra - dimensions , @xmath24 m , and in the following we will consider both these extreme situations . analogously to the case of the ricci scalar , the mass parameter in the higgs term then gets normalized as @xmath25 , where we have introduced the compton wavelength corresponding to the higgs mass , @xmath26 , equal to @xmath27 m if we assume @xmath28 gev . notice that , due to the subattometer scale values of @xmath29 and @xmath30 , an extremely large value of @xmath20 is necessary for having mass shifts of order unity or lower to compensate for kreschmann invariants due to macroscopic curvature of any spacetime of astrophysical interest . in fact , we get a relative mass shift , for instance in the case of the electron , equal to @xmath31 in the case of the schwarzschild metric the kretschmann invariant is @xmath32 , with @xmath33 the schwarzschild radius @xmath34 , and @xmath35 the distance from the center of the mass @xmath36 . as a benchmark , for a solar mass white dwarf , @xmath37 with a earth radius @xmath38 , we get @xmath39 m and @xmath40 . with @xmath23 m and the abovementioned value of @xmath29 we obtain @xmath41 in mksa units . for atomic transitions due to relocations of the electron in states with different principal quantum number , we expect that the mass shift affects the spectroscopy with a scaling of the transition wavelengths as @xmath42 , and therefore any evidence for a wavelength shift in the electronic transitions not accompanied by the same shift for transitions determined by the mass of the nuclei may be a distintive signature of higgs - shifts . therefore we need to detect emission or absorption wavelengths of both electronic and nuclear nature from a strong - gravity source , and make a comparison with either laboratory spectra or spectra gathered from weak - gravity astrophysical sources . the spectrum of the c@xmath3 molecule has been the subject of extensive experimental and theoretical studies in molecular spectroscopy , and has been found in various astrophysical contexts including white dwarfs . in the visible region , the most prominent features of the c@xmath3 spectrum are the swan bands , involving vibronic transitions between the electronic states d@xmath43-a@xmath44 @xcite . for these transitions , in the presence of a higgs - shift the electronic energy levels , proportional to the electron mass , should be shifted , while the vibrational levels , proportional to the nucleon mass , should stay constant . we therefore expect that the separation between different terms of the same swan band should stay constant , the only effect of the higgs shift being an overall shift of all the wavelengths . in the following we therefore focus on these specific spectra as gathered from two white dwarfs . in the bruce proper motion survey , @xcite found bpm 27606 , which has a dme common proper motion companion . @xcite indicated that it is a white dwarf from _ ubv _ photometry . @xcite discovered the strong c@xmath3 swan bands which establish its spectral class as dq in the current classification scheme , and discussed its kinematical properties @xcite . more detailed descriptions of the spectrum were given by @xcite and @xcite . atmospheric analyses @xcite showed that bpm 27606 has a helium dominated atmosphere ( c : he @xmath45 ) and effective temperature @xmath46 k and more recent studies of dq atmospheres ( _ e.g. _ @xcite ) leave this essentially unchanged . in recent years many new dq white dwarfs have been found ( _ e.g. _ @xcite ) and several new properties about them have been discovered . these include rapid variability of some of the hot ones @xcite , rotation @xcite , and strong magnetic fields @xcite . an additional problem that remains unsolved is the physics of the blueshifts in the swan bands of the dqs . these were already measured in the milder cases of bpm 27606 and l 879 - 14 @xcite , but for cooler dqs this becomes more extreme , such as for lhs1126 @xcite . @xcite and @xcite have reviewed possible mechanisms , but this subject is clearly hampered by the lack of information on the behaviour of the spectra of ci and c@xmath3 under white dwarf conditions . bpm 27606 is a particularly good star for studying these effects because it is in the common proper motion system with cd-51@xmath47 13128 . this allows its true velocity to be known within fairly restrictive limits and the pair has a known distance from the trigonometric parallax . in addition , the shifts in the lines are relatively small ( @xmath48 ) and its magnetic field is not large . @xcite have reported a magnetic field measurement of @xmath49 mg from circular polarization of the ch bands near @xmath74300 . however the magnetic field may not be this high . from our new spectra this seems inconsistent with the lack of splitting of the @xmath74771 ci line and h@xmath50 , which indicates a magnetic field @xmath51 g. although variability is possible , the line is single on all of our spectra as it was in the 1978 spectra of @xcite . the observations were made with the robert stobie spectrograph ( rss ) attached to the southern african large telescope ( salt ) , which is described by @xcite . the rss @xcite ) employs volume space holographic transmission gratings ( vphgs ) and three e2v44 - 82 2048 @xmath52 4096 ccds with 15 @xmath6 m pixels separated by gaps of 1.5 mm width , or about 10 - 15 at the dispersions used here . all of the observations of bpm 27606 were obtained using a 1.0 arcsec @xmath52 8 arcmin slit rotated by 71@xmath47 so that both the white dwarf and its bright companion could be observed simultaneously . a 2 @xmath52 2 pixel binning with a gain of 1.0 e@xmath53/adu gives a readout noise of 3.3 e@xmath53/pixel . the h@xmath54 region ( @xmath55 6085 - 6925 ) which also covers the @xmath56 bandhead was observed 2013 may 2 with the pg2300 grating and a pc04600 filter . three 900 seconds exposures were obtained along with ar comparison spectra . the fwhm measured from comparison lines is 1.3 . an identical set of exposures were secured 2014 may 12 to improve the signal - to - noise ratio . the @xmath55 4700 - 5380 wavelength region which covers the 5135 bandhead was observed 2013 may 19 ( two exposures ) and 2013 may 21 ( four exposures ) . all exposure times were 812 seconds with the pg3000 grating , no filter , and cuar comparisons were used , giving a fwhm of comparison lines of 1.0 . the 4737 and 4382 bandheads ( @xmath55 4326 - 5004 ) were observed 2013 may 5 and three 1042 second exposures using the pg3000 grating and no filter with a cuar wavelength comparison . the resulting comparison line has a fwhm=1.1 . in 2014 , two additional spectral regions were observed . three 900 seconds exposures were secured 2014 may 13 of @xmath553540 - 4323 with the pg3000 grating , no filter and a thar comparison producing a fwhm resolution of 1.3 from companion lines . three 663 seconds exposures covering @xmath555050 - 6010 were made 2014 may 22 using the pg2300 grating with an ar comparison giving a fwhm of 1.7 for the comparison lines . starting from the bias subtracted and flattened images provided by the salt pipeline @xcite , the spectra were wavelength calibrated using the longslit menu in iraf . fifth order polynomials were used for the wavelength calibrations in two dimensions , background subtraction , and the spectra were extracted using apsum . special care was taken to use portions of the frames adjacent to the spectra , to minimize the effects of the curved lines in the rss . the final individual spectra were summed using imcombine and the ccdsum option to remove cosmic rays . examples of the resulting spectra are shown in figures 1 - 5 . the heliocentric velocity of the m2ve companion , measured from our spectra using the h@xmath57 emission lines , was @xmath58 km / s , which we adopt here . this can be compared with @xmath59 km / s @xcite and @xmath60 km / s @xcite . the observed wavelengths of the major bandheads of c@xmath3 are shown in table 1 where they are compared with the wavelengths given by @xcite . these wavelengths refer to the minimum intensity at each bandhead from our spectra , and have been corrected to heliocentric values . we have analyzed the distance between two consecutive lines corresponding to the same @xmath61 , for the observed wavelengths @xmath62 and for the laboratory measured wavelengths @xmath63 , evaluated both the overall average separation as @xmath64 $ ] and its dispersion as @xmath65 ^ 2 $ ] , obtaining the values of @xmath66 and @xmath67 . the fact that @xmath68 shows that , within the instrumental error , the spacing of the vibrational transitions is the same for the laboratory and the observed wavelengths . this provides a reliable anchor to study possible shifts in the bandheads due to the electron mass shift induced by the higgs field . the stability of the vibrational transitions is further assured by their insensitivity to pressure - induced shifts since these are expected to be the same , in a linear approximation suitable for low pressures , for different vibronic bands @xcite . due to the lack of recent detailed spectral scans of bpm 27606 , its effective temperature is estimated from photometry . @xcite used intermediate band strmgren uby data from @xcite and @xmath69 models which yield an effective temperature @xmath70 k and @xmath71 . using models of @xcite these colours give @xmath72 and @xmath73 . @xcite found @xmath74 k , @xmath75 , and @xmath76 in line , within few standard deviations , with our findings . broadband ubv photometry , 2mass near infrared colours @xcite and an atmosphere model by @xcite bracket these values , so for the present estimates we adopt @xmath77 k and @xmath78 . to estimate the mass , we first determine the white dwarf radius using the relationship @xmath79 with @xmath80 from @xcite . the visual magnitude @xmath81 is the mean of @xcite and @xcite . for the parallax we adopt @xmath82 , the mean of three parallax measurements : from hipparcos @xcite , the yale parallax catalogue @xcite , and the photometric parallax using the colour magnitude diagrams of reid ( http://www.stsci.edu ~ inr / cmd.html ) for the dm companion this gives @xmath83 which implies a mass @xmath84 according to the hamada and salpeter carbon or chandrashekhar @xmath85 mass - radius relations . these values give a surface gravity of @xmath86 or @xmath87 . the corresponding gravitational redshift would be @xmath88 . the leading source of systematic effects in the c@xmath3 bands is pressure shifts in the dominant he background atmosphere , and in this section we estimate their order of magnitude . we interpolate in the dq models of @xcite for the atmospheric parameters at rosseland mean optical depth @xmath89 for @xmath90 k and @xmath91 for the order of magnitude of conditions in the line forming region . this is @xmath92 k and @xmath93 . as the models use @xmath94 , and for bpm 27606 @xmath95 , the number density is scaled to be @xmath96 . @xcite measured pressure shifts for the five @xmath97 bandheads observed here in he under conditions close to those in bpm 27606 ( @xmath98 k and @xmath99 ) . although these measurements can not explain the large blueshifts @xmath100 -200 in the cooler peculiar dq stars @xcite , the shifts in bpm 27606 are much smaller than this and the laboratory measurements are of similar size , so it seems reasonable that for this star they can be used to estimate the pressure shifts . if the line broadening in the swan bands resembles van der waals broadening , the pressure shifts measured by @xcite , @xmath101 , would scale as @xmath102 with @xmath103 expressed in k and @xmath104 in @xmath105 . for the estimated conditions at @xmath89 , @xmath106 , which is used to correct the measured wavelengths of the @xmath97 bandheads given in table 2 as @xmath107 , where @xmath108 is the gravitational redshift . the major source of systematic error in equation 6 is the choice of @xmath89 for the model atmosphere . this turns out to be relatively insensitive , as increasing the rosseland optical depth to @xmath109 would multiply @xmath110 in tables 2 and 3 by a factor 1.2 . we also detect a weak feature at @xmath111 . if this is the @xmath112 line , @xmath113 , no pressure shifts are expected , and this gives a gravitational redshift @xmath114 km / s . we do not detect the @xmath115 line in our spectra . the ch(0,0 ) band at 4314.2 appears to be present as a weak feature at @xmath116 and helps to confirm the presence of hydrogen . a weak ci line can be seen in figure 2 ( see also fig . 3 for details ) . the measured heliocentric wavelength neglecting pressure shifts is 4771.1 . the laboratory wavelength of the strongest component of the @xmath117 multiplet is 4771.75 @xcite . there are currently no data on pressure shifts of ci lines . the magnetic field on the surface of bpm 27606 is estimated looking at the resolution of the ci line . the classical zeeman effect yields the relationship @xmath118 ( in cgs units ) . the fact that only a single line with fwhm @xmath119 is visible ( although with a low signal - to - noise ratio ) , yields an upper bound on the magnetic field of @xmath120 g , which makes negligible any correction to the estimates above . using detailed calculations by @xcite of the magnetic splitting of this line also suggest such a low magnetic field . the orbital motion is estimated considering the presence of the common proper motion companion to bpm 27606 with @xmath121 , of spectral type m2ve ( @xmath122 ) at a projected separation of 28.45 arcsec , which at 15 pc yields a separation of @xmath123 au . with a white dwarf mass of @xmath124 and taking this to be the semimajor axis of a circular orbit , the orbital period would be @xmath125 years and the projected orbital velocity would be @xmath126 km / s , which is the order of magnitude of an additional source of uncertainty in the gravitational redshift determination . for bpm 27606 we conclude that an upper bound to the wavelength difference between the molecular c@xmath3 bands and the h@xmath127 and ci atomic lines is 1.3 . this is obtained by summing up the variance of the estimated pressure shifts ( the square of the error in the second column of table 2 , corrected by the 1.44 hammond scaling factor ) and the square of the final difference between the processed and the laboratory wavelength ( last column in table 2 ) , all other systematic errors being negligible with respect to these sources . if we disqualify the troubling ci line , it is 0.7 . although the corrections for pressure shifts in the c@xmath3 bands seems satisfactory , one must consider the uncertainties in their measurements which dominate the error budget in our estimate of the difference and is of the order of @xmath128 2 . procyon b is another white dwarf that has both atomic and molecular features in its spectrum and is in a well known binary . its orbit and mass have been long studied ( _ e.g. _ @xcite ; @xcite ; @xcite ) . @xcite obtained masses of @xmath129 and @xmath130 for the two components . although procyon b was long known to be a white dwarf , its spectrum could not be studied due to its proximity to its bright ( @xmath131 + 0.34 ) primary ( separation @xmath132 5 arcsec ) . @xcite secured spectra using the stis instrument on the hst in 1998 february ( proposal 7398 ; pi h.l . shipman ) where the observations and data reductions are detailed . here we adopt the atmospheric analysis in @xcite which gives @xmath133 k , @xmath134 and @xmath135 . these parameters show that procyon b lies close to the carbon white dwarf mass - radius relation and predict a surface gravity of @xmath136 and a gravitational redshift of @xmath137 . from the hst archives we used images 04g802010 , 04g802020 , 04g8020j0 , and 04g802090 which have suitable signal to noise ratio . we measured the wavelengths for @xmath97 bands , mgii and caii lines . figure 6 shows the @xmath97 bands near 4,737 and figure 7 shows the caii lines . the remainder of the spectrum is shown in @xcite . the orbital velocity from @xcite for the radial motion of procyon b in 1998 is @xmath138 . the models of @xcite using the @xmath139 and @xmath9 above indicate at @xmath89 , t@xmath140=6,750 k , and n@xmath141 . the @xmath97 pressure shifts of @xcite discussed in section 3.2 are thus multiplied by 1.3 . the pressure shifts of the observed caii and mg lines produced by he are available and in both cases are redshifts . for caii , theoretical values @xcite and laboratory measurements @xcite agree relatively well . @xcite calculated pressure shifts for mgii and these were scaled to the above atmospheric parameters assuming van der waals broadening . table 3 summarizes the laboratory and measured wavelengths of features in the spectrum of procyon b along with the corrections due to the orbital motion , @xmath142 , the estimated pressure shifts @xmath110 , and the gravitational redshift , @xmath108 . the resulting corrected wavelength of each line , @xmath143 , and the residual @xmath144 are given . these are consistent with the results for bpm 27606 . by repeating the analysis as for the latter , and using the hammond scaling factor of 1.3 , the difference between atomic and molecular lines is formally 0.5 , with the same uncertainties due to pressure shifts as before . based on the absence of relative shifts between the electronic and the vibrational transitions of the swan bands , we are able to assess upper bounds on the higgs - kreschmann coupling . the expected wavelength shift is @xmath145 where the kreschmann invariant is , for the estimated values of the mass and radius of bpm 27606 , @xmath146 . by assuming @xmath147 m , and the value of @xmath148 quoted in section 2 , we obtain @xmath149 , which may be inverted yielding an upper bound , for an average wavelength of @xmath150 @xmath151 where @xmath152 ( in ) is the estimated wavelength resolution . if instead we use @xmath153 m as in models with extra dimension and quantum gravity at the fermi scale @xcite , the bounds are a bit more constraining , as @xmath154 at @xmath155 . both examples of upper bounds are expressed in meter - kilogram - second - ampere ( mksa ) units of the systeme international ( si ) , and it is worth converting them into natural units for the benefit of a comparison to the bounds already estimated from the observation of the higgs particle at lhc @xcite . the action term for the higgs field including its coupling to spacetime , taking into account explicitly @xmath156 and @xmath157 , is expressed in si units in terms of , for instance , the mass term as @xmath158 . then the scalar field has dimensions @xmath159=m^{1/2 } t^{-1/2}$ ] , and the higgs - curvature term satisfying @xmath160 implies that @xmath10 is dimensionless . if the analysis is repeated for the action term expressed in natural units ( nu ) , with @xmath161 , the dimensions of the scalar field change accordingly , and from the mass term it is simple to infer that @xmath162 , implying the following relationship between the @xmath10 parameter evaluated in the two units systems , @xmath163 . analogous considerations may be repeated for the higgs - kreschmann coupling in which the curvature term has the same dimensions since @xmath12 is replaced by @xmath164 . this implies that in natural units the bounds on the higgs - kreschmann coupling become respectively , for the two extreme choices of @xmath30 , @xmath165 ( for @xmath147 m ) and @xmath166 ( for @xmath153 m ) . this second bound on the higgs - kreschmann coupling is quantitatively comparable to the ones assessed on the higgs - ricci coupling through measurements at the lhc as reported in @xcite and @xcite , although in their analyses the usual @xmath147 m is assumed . with respect to bounds from table - top experiments based on tests of the superposition principle for gravitational interactions as discussed in @xcite , the bounds derived in this paper represent an improvement by ten orders of magnitude , as shown in table 4 . we have shown that observations in a somewhat controlled environment like the one provided by carbon - rich white dwarfs may be used to give upper bounds on the coupling between the higgs field and a specific invariant of the curvature of the spacetime , such as the kreschmann invariant . the existence of nonminimal couplings between the higgs and spacetime curvature is crucial to various proposals in which the higgs also plays the role of the inflaton @xcite , and as a mechanism to suppress the dark energy contribution of quantum fields to the level compatible with the astrophysical observations based on snia @xcite . this methodology is complementary to the upper bounds recently discussed arising from the lhc experiments and their degree of agreement with the standard model of elementary particle physics @xcite , and can be adopted also to search for couplings between generic scalar fields , not necessarily directly related to the higgs vacuum , and space - time curvature , which may be competitive with bounds arising from the analysis of the cosmic background radiation as reported in @xcite . scalar fields , even if not directly interacting among themselves at level of their classical lagrangian , will have a crosstalk once quantum radiative corrections are considered - the very origin of the hierarchy problem in grand unified theories - so any scalar field will be then coupled with the higgs field and will indirectly affect the mass of elementary particles . bounds based on this analysis could be even more stringent as many of the proposed candidates , for instance scalar fields invoked to accommodate the acceleration of the universe @xcite have a compton wavelength greatly exceeding the one associated to the higgs field , provided that compact astrophysical sources with non - zero ricci scalar may be found . the spectroscopic analysis presented here could be improved in a number of ways in the near future . measurements and calculations of the pressure shifts of c i lines and the @xmath97 swan bands under white dwarf conditions are needed . observations of the ultraviolet carbon and other metal lines which are from lower energy levels would help disentangle the pressure shifts from the gravitational redshift measurement . a detailed scan of the white dwarf s spectral energy distribution combined with an updated atmosphere model would help understanding the details of the molecular and atomic carbon features . additional objects of this type that are in binary pairs would help in the assessment of the local gravity , further diversifying the sample to counterbalance peculiar systematic effects . we are grateful to susanne yelin for a critical reading of the manuscript . some of the observations reported in this paper were obtained with the southern african large telescope ( salt ) under proposals 2013 - 1-dc-001 and 2014 - 1-dc-001 . additional data used observations made with the nasa / esa hubble space telescope , and were obtained from the hubble legacy archive , which is a collaboration between the space telescope science institute ( stsci / nasa ) , the space telescope european coordinating facility ( st - ecf / esa ) and the canadian astronomy data centre ( cadc / nrc / csa ) . this work was also partially funded by the national science foundation through a grant for the institute for theoretical atomic , molecular , and optical physics at harvard university , and the smithsonian astrophysical laboratory . cccc @xmath167 & @xmath168 & @xmath169 & @xmath170 + 0,2 & 6179.90 & 6191.2 & -11.3 + 1,3 & 6114.74 & 6122.1 & -7.4 + 0,1 & 5630.35 & 5635.5 & -5.15 + 1,2 & 5580.07 & 5585.5 & -5.4 + 2,3 & 5536.00 & 5540.7 & -4.7 + 3,4 & 5495.64 & 5501.9 & -6.3 + 4,5 & 5467.73 & 5470.3 & -2.6 + 0,0 & 5159.97 & 5165.2 & -5.2 + 1,1 & 5125.64 & 5129.3 & -3.7 + 2,2 & 5094.45 & 5097.7 & -3.3 + ccccccc line & @xmath101 & @xmath110 & @xmath171 & @xmath172 & @xmath108 & @xmath173 + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) + c@xmath3(0,2 ) & -8.4 @xmath128 0.3 & -12.1 & -11.3 & + 0.18 & -1.0 & 0.0 + c@xmath3(0,1 ) & -3.2 @xmath128 0.5 & -4.6 & -5.15 & + 0.17 & -0.9 & -1.3 + c@xmath3(0,0 ) & -3.2 @xmath128 0.5 & -4.6 & -5.2 & + 0.16 & -0.8 & + 0.0 + c@xmath3(1,0 ) & -3.8 @xmath128 1.0 & -5.5 & -4.0 & + 0.14 & -0.7 & + 0.9 + c@xmath3(2,0 ) & -3.1 @xmath128 1.0 & -4.5 & -3.6 & + 0.13 & -0.7 & + 0.3 + ci(@xmath74771 ) & - & - & -0.65 & + 0.15 & -0.75 & -1.2 + h@xmath127 & - & - & + 0.5 & + 0.20 & -1.0 & -0.3 + cccccccc line & @xmath169 & @xmath168 & @xmath142 & @xmath110 & @xmath108 & @xmath143 & @xmath173 + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) & ( 8) + c@xmath3(0,0 ) & 5165.2 & 5161.8 & -0.15 & + 4.2 & -0.53 & 5165.3 & + 0.1 + c@xmath3(1,1 ) & 5129.3 & 5126.5 & -0.15 & & & & + c@xmath3(1,0 ) & 4737.1 & 4733.1 & -0.14 & + 4.9 & -0.49 & 4737.4 & + 0.3 + c@xmath3(2,1 ) & 4715.2 & 4714.5 & -0.14 & & -0.49 & & + c@xmath3(3,2 ) & 4697.6 & 4696.1 & -0.14 & & -0.48 & & + caii h & 3968.47 & 3969.84 & -0.12 & -1.4 & -0.41 & 3967.9 & -0.6 + caii k & 3933.66 & 3935.18 & -0.11 & -0.4 & -0.41 & 3934.3 & + 0.7 + mgii & 2802.70 & 2803.10 & -0.08 & -0.5 & -0.29 & 2802.2 & -0.5 + mgii & 2795.53 & 2796.85 & -0.08 & -0.6 & -0.29 & 2795.9 & + 0.4 + cccccc system & @xmath174 & @xmath175 & @xmath176 ( ) & @xmath177(@xmath178 m ) & @xmath177(@xmath179 m ) + bpm 27606 & 0.0105 & 0.78 & 1.3 & @xmath180 & @xmath181 + procyon b & 0.0124 & 0.602 & 0.5 & @xmath182 & @xmath183 + table - top experiments & & & & @xmath184 & @xmath185 +

we report on a search for differential shifts between electronic and vibronic transitions in carbon - rich white dwarfs bpm 27606 and procyon b. the absence of differential shifts within the spectral resolution and taking into account systematic effects such as space motion and pressure shifts allows us to set the first upper bound of astrophysical origin on the coupling between the higgs field and the kreschmann curvature invariant . our analysis provides the basis for a more general methodology to derive bounds to the coupling of long - range scalar fields to curvature invariants in an astrophysical setting complementary to the ones available from high - energy physics or table - top experiments .

arxiv

white dwarf ( wd ) stars represent the end points of stellar evolution for all low - mass stars , and are the fate of more than 97% of all stars in our galaxy . roughly 80% of wds belong to the spectral class da , with atmospheres characteristically dominated by hydrogen @xcite . when da wds cool to the appropriate temperature to foster a hydrogen partial ionization zone , they begin their journey through the zz ceti ( or dav ) instability strip , where global pulsations are driven to observable amplitudes and their fundamental parameters can be determined using asteroseismology ( see reviews by @xcite , @xcite and @xcite ) . aside from their variability , the zz ceti stars discovered to date appear to be otherwise normal wds , and are therefore believed to be a natural phase in the evolution of all das . although some das within the empirical instability strip have been observed not to vary to modest limits ( e.g. , @xcite ) , follow - up observations have shown that some of these stars really do pulsate at low amplitude ( e.g. , @xcite ) . higher - quality optical and uv spectra have also moved some of these non - variable interlopers out of the instability strip @xcite . thus , it is currently believed that the zz ceti instability strip is pure , and that all da wds will at some point pass through it and pulsate @xcite . much work has been devoted to observationally mapping the zz ceti instability strip , which runs in temperature from roughly @xmath11 k for standard @xmath4 = 8.0 wds @xcite . there is also a dependence on surface gravity , such that wds with lower @xmath4 pulsate at lower effective temperatures . this trend has been observed for wds with masses from 1.1 @xmath1 down to 0.5 @xmath1 @xcite . the blue edge of the zz ceti instability strip , where pulsations are turning on , has been successfully estimated by both convective period arguments @xcite and full non - adiabatic calculations @xcite . a slightly more efficient prescription for convection has to be assumed , by increasing the value of the mixing - length theory parameter ml2/@xmath12 , to make the theory match the observed blue edge , which was most recently mapped empirically by @xcite . however , estimating the temperature at which pulsations should shut down has remained a challenge . modern non - adiabatic calculations do not predict a red edge until around 5600 k @xcite , more than 5000 k cooler than the empirical red edge @xcite . @xcite argue that a surface reflection criterion can be enforced to limit the maximum mode period , which may push a theoretical red edge to hotter temperatures , nearer what is observed in zz ceti stars @xcite . the recent discovery of pulsating extremely low - mass ( elm , @xmath13 0.25 @xmath1 ) wds provides us with an exciting new opportunity to explore the nature of the physics of wd pulsations at cooler temperatures and much lower masses . since the first discovery by @xcite , more than 160 zz ceti stars have been found , all of which have masses @xmath14 0.5 @xmath1 and thus likely harbor carbon - oxygen ( co ) cores . that changed with the discovery of the first three pulsating elm wds @xcite . these elm wds are likely the product of binary evolution , since the galaxy is not old enough to produce such low - mass wds through single - star evolution @xcite . during a common - envelope phase , the elm wds were most likely stripped of enough mass to prevent helium ignition , suggesting they harbor he cores . the pulsating elm wds will be incredibly useful in constraining the interior composition , hydrogen - layer mass , overall mass , rotation rate , and the behavior of convection in these low - mass wds , which may derive a majority of their luminosities from stable hydrogen burning for the lowest - mass objects @xcite . several groups have recently investigated the pulsation properties of he - core wds , and non - adiabatic calculations have shown that non - radial @xmath15- and @xmath10-modes should be unstable and thus observable in these objects @xcite . pulsating elm wds will also extend our empirical studies of the zz ceti instability strip to significantly lower surface gravities . boosted by the many new elm wds catalogued by the elm survey , a targeted spectroscopic search for elm wds @xcite , we have looked for additional pulsating elm wds throughout a large part of parameter space . the first three pulsating elm wds all have effective temperatures below @xmath16 k , much cooler than any previously known co - core zz ceti star @xcite , which makes up the coolest class of pulsating wds . we now add to that list the two coolest pulsating wds ever found , sdss j161431.28 + 191219.4 ( @xmath17 mag , hereafter j1614 ) and sdss j222859.93 + 362359.6 ( @xmath18 mag , hereafter j2228 ) , bringing to five the number of elm wds known to pulsate . in section [ sec : j1614 ] we detail our discovery of pulsations in j1614 and outline our new spectroscopic observations of this elm wd . in section [ sec : j2228 ] we describe the discovery of multi - periodic variability in the elm wd j2228 and update its determined atmospheric parameters . we conclude with a discussion of these discoveries , and update the observed da wd instability strip . @xcite found that j1614 had @xmath2 @xmath19 k and @xmath4 @xmath20 , based on a single spectrum of this @xmath17 mag wd from the flwo 1.5 m telescope using the fast spectrograph @xcite . we have obtained an additional 51 spectra using the same instrument and setup . we have co - added our spectroscopic observations to determine the atmospheric parameters of the elm wd j1614 ( figure [ fig : j1614spec ] ) . our observations cover a wavelength range from @xmath21 . the model atmospheres used for this analysis are described at length in @xcite and employ the new stark broadening profiles from @xcite . models where convective energy transport becomes important are computed using the ml2/@xmath12 = 0.8 prescription of the mixing - length theory ( see * ? ? ? a discussion of our extension of these models to lower surface gravities and more details of our fitting method can be found in section 2.1.1 of @xcite . our final fit to the phased and co - added spectrum of j1614 is shown in the top panel of figure [ fig : j1614spec ] and yields @xmath2 @xmath3 k and @xmath4 @xmath5 . this corresponds to a mass of @xmath60.20 @xmath1 using the he - core wd models of @xcite , if we assume the wd is in its final cooling stage . the more recent models of @xcite predict a mass of 0.19 @xmath1 given the atmospheric parameters , which we adopt . we have also performed our fit without using the low s / n lines h11@xmath22h12 , but this marginally affects our solution : using only the h@xmath23@xmath22h10 lines of the balmer series , we find @xmath2 @xmath24 k and @xmath4 @xmath25 . to remain consistent with our previous pulsating elm wd atmospheric determinations @xcite , we will include the h11@xmath22h12 lines in our adopted solution for j1614 . @lcccc@ run & ut date & length & seeing & exp . + & & ( hr ) & ( ) & ( s ) + + a2690 & 2012 jun 21 & 2.6 & 1.7 & 5 + a2692 & 2012 jun 22 & 2.0 & 1.8 & 5 + a2695 & 2012 jun 23 & 3.7 & 1.2 & 5 + a2697 & 2012 jun 24 & 3.5 & 1.4 & 5 + a2699 & 2012 jun 25 & 3.6 & 1.3 & 5 + + a2521 & 2011 nov 28 & 3.5 & 1.5 & 10 + a2524 & 2011 nov 29 & 1.6 & 2.5 & 10 + a2528 & 2011 nov 30 & 1.9 & 2.2 & 10 + a2707 & 2012 jul 13 & 2.3 & 1.1 & 5 + a2710 & 2012 sep 17 & 2.8 & 2.4 & 10 + a2719 & 2012 sep 20 & 6.4 & 1.6 & 15 + a2721 & 2012 sep 21 & 7.4 & 1.4 & 10 + elm wds are typically found in close binary systems ; these companions are necessary to strip the progenitor of enough mass to form such a low - mass wd within the age of the universe @xcite . however , using the code of @xcite , we do not detect any significant radial velocity variability in our observations of j1614 . the r.m.s . scatter gives us an upper limit on the rv semi - amplitude : @xmath26 km s@xmath27 . the systemic velocity is @xmath28 km s@xmath27 . we note that this non - detection does not require the lack of a companion to the elm wd in j1614 . rather , the system may be inclined nearly face - on to our line of sight , or the companion may be a much cooler low - mass wd . if the inclination is @xmath29 , which is more than 85% likely if the orientation of the system with respect to the earth is drawn from a random distribution , the companion has @xmath30 @xmath1 if the system has a 7 hr orbital period , the median for elm wd binaries in the elm survey @xcite . empirically , there are similarly low - mass wds in the elm survey with no significant radial velocity variability @xcite . we obtained high - speed photometric observations of j1614 at the mcdonald observatory over five consecutive nights in 2012 june for a total of nearly 15.4 hr of coverage . we used the argos instrument , a frame - transfer ccd mounted at the prime focus of the 2.1 m otto struve telescope @xcite , to obtain @xmath31 s exposures on j1614 . a full journal of observations can be found in table [ tab : jour ] . observations were obtained through a 3 mm bg40 filter to reduce sky noise . we performed weighted , circular , aperture photometry on the calibrated frames using the external iraf package @xmath32 written by antonio kanaan @xcite . we divided the sky - subtracted light curves by the brightest comparison star in the field , sdss j161433.39 + 191058.3 ( @xmath33 mag ) , to correct for transparency variations , and applied a timing correction to each observation to account for the motion of the earth around the barycenter of the solar system @xcite . the top panel of figure [ fig : j1614 ] shows a portion of a typical light curve for j1614 , obtained on 2012 june 23 , and includes the brightest comparison star in the field over the same period . the bottom panel of this figure shows a fourier transform ( ft ) utilizing all @xmath34 light curve points collected thus far . we display the 4@xmath35 significance line at the bottom of figure [ fig : j1614 ] , calculated from the average amplitude , @xmath35 , of an ft within a 1000 @xmath36hz region in steps of 200 @xmath36hz , after pre - whitening by the two highest - amplitude periodicities . .frequency solution for sdss j161431.28 + 191219.4 [ tab : j1614freq ] [ cols="<,^,^,^,^ " , ] we may compare the first five pulsating elm wds to the previously known zz ceti stars by placing them in a @xmath4@xmath2 diagram , shown in figure [ fig : search ] . doing so , we discover there are at least six elm wds with temperatures and surface gravities between the newfound pulsating elm wd j2228 and the other four known pulsating elm wds . these non - variable elm wds have been observed extensively and do not show significant evidence of pulsations to at least 1% relative amplitude . we have excellent limits on the lack of variability in four of these six , ruling out pulsations larger than 0.3% amplitude . we have put limits on three of these new non - detections , detailed in table [ tab : null ] . we note that @xcite previously observed psr 1012 + 5307 , but we have put much more stringent limits on a lack of variability on this faint elm wd with 7 hr of observations in excellent conditions . the other three interlopers have been detailed in previous studies . sdss j0822 + 2753 is a @xmath2 @xmath37 k , @xmath4 @xmath38 wd observed not to vary to 0.2% @xcite . sdss j1443 + 1509 is a @xmath2 @xmath39 k , @xmath4 @xmath40 wd with exquisite limits on lack of variability , to @xmath41% @xcite . finally , nltt 11748 is the @xmath2 @xmath42 k , @xmath4 @xmath43 primary wd in an eclipsing wd+wd binary @xcite . it was shown by @xcite not to vary out of eclipse to above 0.5% . we have obtained an additional 8 hr of photometry of nltt 11748 out of eclipse at mcdonald observatory and can independently rule out variability larger than 0.3% . the discovery of pulsations in j2228 , which is considerably cooler than at least a half - dozen other photometrically constant elm wds , questions the purity of the instability strip for he - core wds and confuses the location of an empirical red edge . however , there is no a priori reason to expect the elm wd instability strip to be pure , or for there to exist a connected low - mass extension of the classical co - core zz ceti instability strip ; evolution through a specific temperature - gravity region is not well established for the elm wds , and they may not all cool through the instability strip in as simple a manner as the co - core zz ceti stars . in fact , some of these elm wds may indeed be in the throws of unstable hydrogen shell burning episodes ; they may not be cooling at all , but rather looping through the hr diagram prior to settling on a final cooling track ( e.g. , @xcite ) . such excursions are not expected for co - core zz ceti stars , which are expected to monotonically cool through an observationally pure instability strip . we have plotted the evolution of theoretical cooling tracks for several different wd masses through the effective temperatures and surface gravities in figure [ fig : search ] . we plot the 0.16 @xmath1 , 0.17 @xmath1 , 0.18 @xmath1 , 0.20 @xmath1 , 0.25 @xmath1 , and 0.35 @xmath1 he - core models of @xcite as dotted magenta lines . we have also used the stellar evolution code mesa @xcite to model the evolution of 0.15 @xmath1 , 0.20 @xmath1 , and 0.25 @xmath1 he - core wds , shown as solid cyan lines in figure [ fig : search ] . for reference , we have also included 0.6 @xmath1 , 0.8 @xmath1 , and 1.0 @xmath1 co - core cooling tracks @xcite . where the lowest - mass wd models enter this diagram depends on how we artificially remove mass from the models , and there is a very noticeable discrepancy between the 0.16 @xmath1 @xcite wd models and our 0.15 @xmath1 wd models using mesa . as an added complication , except for the lowest - mass elm wds ( below roughly 0.18 @xmath1 ) , recurrent hydrogen shell flashes cause the elm wd model to loop many times through this @xmath2-@xmath4 plane , further confusing the picture @xcite . thus , it is not entirely surprising to find non - variable elm wds between j2228 and the four warmer pulsating elm wds . further empirical exploration of the entire elm wd instability strip offers a unique opportunity to constrain physical and evolution models of elm wds , specifically these late thermal pulses and the mass boundary for the occurrence of these episodes . long - term monitoring of the rate of period change of pulsating elm wds also affords an opportunity to constrain the cooling ( or heating ) rate of these objects @xcite . in contrast to the confusion along the red edge of the instability strip , the blue edge is more reliably predicted by theory . the theoretical blue edge ( dotted blue line ) in figure [ fig : search ] has been calculated following @xcite and @xcite . we use the criterion that @xmath44 for the longest period mode that is excited , where @xmath45 is the mode period and the timescale @xmath46 describes the heat capacity of the convection zone as a function of the local photospheric flux , which we compute from a grid of models ( see @xcite for further details ) . we use the criterion @xmath47 s , with the convective prescription ml2/@xmath12=1.5 . we also include the theoretical blue edge of @xcite , which uses a slightly less efficient prescription for convection , ml2/@xmath12=1.0 . we have discovered pulsations in two new extremely low - mass , putatively he - core wds using optical facilities at the mcdonald observatory . spectral fits show that these two elm wds , j1614 and j2228 , are the coolest pulsating wds ever found . this brings to five the total number of pulsating elm wds known , establishing them as a new class of pulsating wd . as with the more than 160 co - core zz ceti stars that have been known for more than four decades , the luminosity variations in these elm wds is so far consistent with surface temperature variations caused by non - radial @xmath10-mode pulsations driven to observability by a hydrogen partial ionization zone . the coolest pulsating elm wd , j2228 , has a considerably lower effective temperature than six similar - gravity elm wds that are photometrically constant to good limits . in contrast to the co - core zz ceti stars , which are believed to represent a stage in the evolution of all such wds , elm wds may not all evolve through an instability strip in the same way , and thus we may not observe their instability strip to be pure . theoretical he - core wd models predict multiple unstable hydrogen - burning episodes , which complicates the evolution of an elm wd through a simple instability strip . empirically discovering elm wds in this space that do or do not pulsate opens the possibility to use the presence of pulsations in elm wds to constrain the binary and stellar evolution models used for low - mass wds , which may better constrain these poorly understood cno - flashing episodes . we acknowledge the anonymous referee for valuable suggestions that greatly improved this manuscript . j.j.h . , m.h.m . and d.e.w . acknowledge the support of the nsf under grant ast-0909107 and the norman hackerman advanced research program under grant 003658 - 0252 - 2009 . m.h.m . additionally acknowledges the support of nasa under grant nnx12ac96 g . b.g.c . thanks the support from cnpq and fapergs - pronex - brazil . the authors are grateful to the essential assistance of the mcdonald observatory support staff , especially dave doss and john kuehne , and to fergal mullally for developing some of the data analysis pipeline used here .

we report the discovery of two new pulsating extremely low - mass ( elm ) white dwarfs ( wds ) , sdss j161431.28 + 191219.4 ( hereafter j1614 ) and sdss j222859.93 + 362359.6 ( hereafter j2228 ) . both wds have masses @xmath0 0.25 @xmath1 and thus likely harbor helium cores . spectral fits indicate these are the two coolest pulsating wds ever found . j1614 has @xmath2 @xmath3 k and @xmath4 @xmath5 , which corresponds to a @xmath60.19 @xmath1 wd . j2228 is considerably cooler , with a @xmath2 @xmath7 k and @xmath4 @xmath8 , which corresponds to a @xmath60.16 @xmath1 wd , making it the coolest and lowest - mass pulsating wd known . there are multiple elm wds with effective temperatures between the warmest and coolest known elm pulsators that do not pulsate to observable amplitudes , which questions the purity of the instability strip for low - mass wds . in contrast to the co - core zz ceti stars , which are believed to represent a stage in the evolution of all such wds , elm wds may not all evolve as a simple cooling sequence through an instability strip . both stars exhibit long - period variability ( @xmath9 s ) consistent with non - radial @xmath10-mode pulsations . although elm wds are preferentially found in close binary systems , both j1614 and j2228 do not exhibit significant radial - velocity variability , and are perhaps in low - inclination systems or have low - mass companions . these are the fourth and fifth pulsating elm wds known , all of which have hydrogen - dominated atmospheres , establishing these objects as a new class of pulsating wd . [ firstpage ] stars : white dwarfs stars : oscillations ( including pulsations ) galaxy : stellar content stars : individual : sdss j161431.28 + 191219.4 , sdss j222859.93 + 362359.6

arxiv

the interstellar medium ( ism ) of a typical disk galaxy is divided into variety of distinct phases , generally classified into hot ionized ( @xmath4 ) gas , the warm neutral medium ( wnm ; @xmath5 ) , and the cold neutral medium ( cnm ; @xmath6 ) . star formation , however , appears to be restricted to cold dense gas with a star formation rate ( sfr ) that strongly correlates with the molecular gas content @xcite . this link between star formation and molecular gas is probably not causal , in the sense that the presence of molecules simply marks those parts of the ism that are cold and dense enough to undergo gravitational collapse @xcite , but the molecular gas nonetheless represents an indispensable tracer of star forming regions in the local and distant universe . understanding these crucial links between star formation , the ism , and galaxy evolution , all intricately coupled through a variety of energetic stellar feedback mechanisms , must rely on identifying the pathways and conditions under which cold , molecular gas is able to develop . furthermore , the gas temperature sets the characteristic size ( @xmath7 ) and mass ( @xmath8 ) of prestellar cores that develop in post - shock regions within magnetized , star - forming clouds ( e.g. , * ? ? ? * ; * ? ? ? since gas cooled by rotational transitions of co is able to reach lower temperature than that cooled by fine - structure lines of atomic carbon , tracking molecular chemistry is crucial to accurately represent small - scale fragmentation in numerical simulations of star - forming clouds . young , massive stars are the main source of far - ultraviolet ( fuv ) photons that permeate the ism and readily photodissociate interstellar molecules , such as molecular hydrogen ( @xmath0 ) and carbon monoxide ( co ) . molecular gas is thus found predominantly in dense , cold regions where dust shielding and self - shielding by the molecules themselves have attenuated the interstellar radiation field ( isrf ) intensity far below its mean . the transition zones separating atomic and molecular gas , so called photodissociation regions ( pdrs ) , have been studied extensively both numerically and theoretically ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? typically , the outer , unshielded layers of a pdr are composed of atomic hydrogen and singly ionized carbon . as column density increases the atomic - to - molecular transition begins to occur , with the c - co transition occurring interior to the @xmath9-@xmath0 one . deeper still , when external fuv photons have been entirely absorbed , the chemistry and thermodynamics becomes dominated by the presence of deep penetrating cosmic rays and the freeze out of molecules onto dust grains . simply understanding the structure of a non - dynamic , one - dimensional pdr is of limited utility . the ism is a dynamic environment where a variety of non - linear , coupled processes determine the global distribution and state of gas . star forming complexes are continually assembled through self - gravity , while stellar feedback acts disperse the structures . turbulent motions , ubiquitous in the ism , dissipate through radiative shocks , but are continually replenished through a combination of supernovae and stellar winds . thermal processes , such as dust grain photoelectric heating and molecular line cooling , further influence the state of the ism and are sensitive functions of ambient radiation fields , column density , and the non - equilibrium chemical state of the gas . implementing these processes in a numerical simulation is highly non - trivial , a challenge further exacerbated by the enormous range of spatial scales involved ; the star forming region of a galactic disk can extend radially for tens of kiloparsecs while individual star forming disks are typically hundreds of aus and smaller , making this a computational _ tour de force _ for even adaptively refined grids . nevertheless , three - dimensional simulations of finite , but representative , portions of galactic disks have attempted to replicate a supernova - driven ism , finding success in reproducing a multiphase ism @xcite , or demonstrating the regulation of the sfr by stellar feedback @xcite . studies that have examined the atomic - to - molecular transition within large - scale , three - dimensional simulations has received far less attention . @xcite and @xcite simulated the conversion of the atomic ism to molecular form in isolated periodic boxes , without explicit feedback but including driven or decaying turbulence . @xcite and @xcite included non - equilibrium @xmath0 formation in simulations of galactic discs , but neglected self - gravity and supernova feedback . these studies did include an @xmath0-dissociating fuv photo - background , though assumed it to be constant in both space and time . @xcite recently conducted a series of galactic scale simulations including supernova feedback , non - equilibrium chemistry , and radiative shielding based on the treecol algorithm @xcite . here , the strength of the fuv photodissociating background scales linearly with the instantaneous sfr , though is spatially uniform . @xcite finds the amount and distribution of molecular gas depends sensitively on a number of parameters , in particular the precise rhythm and spatial distribution of supernovae . while these simulations have provided insight into the relationship between dynamics and chemistry , the price of simulating 3d time - dependent chemistry is that their treatment of photodissociation , the crucial process for regulating the chemical state of the ism , is extremely primitive . the true photodissociation and photoheating rate at any point depends on the flux of fuv radiation integrated over all solid angles and from sources at all distances . in contrast , most of the chemodynamical simulations conducted to date have relied on _ ad hoc _ prescriptions for the dust- and self - shielding - attenuated photodissociation rates in which a single , uniform , permeating radiation field is assumed to exist in all unshielded regions , and the degree of attenuation at a point is characterized by a single characteristic column density . these prescriptions are for the most part untested , and are of unknown accuracy . here our goal is to perform a detailed study on the chemical state of gas in a self - consistently simulated ism , placing special emphasis on accurately computing the molecular content with an rigorous treatment of the generation and attenuation of photodissociating fuv radiation . we do this by post - processing the supernovae driven galactic disk simulations of @xcite with multifrequency ray tracing and a physically motivated spatial distribution of stellar sources . this approach has the price that it discards time - dependent dynamical effects . however , it does not rely on untested approximations for the radiative transfer problem . it therefore provides a useful complement to the more approximate dynamical simulations that have appeared in the literature thus far . in particular , our approach enables tests to evaluate the accuracy of simplified shielding schemes that have been previously used . in this paper , our main goal is to compare a number of commonly used local approximations for the degree of radiative shielding to full solutions of the radiative transfer equation , in order to gauge the effect that different approaches to radiative shielding have on the molecular abundances and temperature . for our tests , we use a density structure obtained from large - scale ism simulations with turbulence self - consistently driven by star formation feedback , which naturally includes vertical stratification and strong density contrasts between warm and cold gas phases . we are also able to test how the distribution of molecular complexes is sensitive to parameters such as the stellar distribution and overall fuv intensity , and the relative importance of dust- and self - shielding . our comparison of different shielding prescriptions will advise us as to the validity and best use of local shielding approximations in upcoming multidimensional simulations of star formation that we shall perform . we organize this paper as follows . in section [ sec : method ] we describe our methodology , including a description of the simulation we apply our post - processing to , the details of our chemical network , and the numerical approach to solving the equation of radiative transfer . in section [ sec : results ] we display our results . finally , we discuss our results and conclude in section [ sec : discussion ] . in this paper we apply radiative and chemical post - processing to the large - scale galactic disk simulations of @xcite to compute the equilibrium chemical and thermal state of the disk given a realistic distribution of stellar sources , a self - consistently derived morphological structure , and multi - frequency , long - characteristics radiative transfer . our methodology broadly consists of two parts . first , we perform frequency dependent radiative transfer through the simulation volume , which we describe in detail in section [ sec : raytrace ] , supplying us with the shielding attenuated chemical photorates . next , these photorates are passed to a chemical network , described in section [ sec : chemistry ] , which is then run to equilibrium , on a cell - by - cell basis , to produce a three - dimensional datacube of chemical abundances and , in a subset of our models , gas temperature ( section [ sec : thermal ] ) . in this procedure , the ray trace and chemical network integration are intricately linked via the photodissociation rates . the chemical abundances have an effect on the global photodissociation rates , which in turn determine the chemical abundances . given the non - local coupling between these two steps , chemical network integration and ray tracing must be carried out iteratively until convergence is obtained in the chemical abundances . the simulations of @xcite ( hereafter k15 ) were run with the athena code @xcite which solves the equations of ideal magnetohydrodynamics on a uniform grid . the models also include self- and external - gravity , heating and cooling , thermal conductivity , coriolis forces , and tidal gravity in the shearing box approximation to model the effect of differential galactic rotation . while k15 ran a suite of simulations with a range of magnetic field strengths and configurations , here we focus exclusively on their mb10 ( solar neighborhood analog ) model that has the following initial properties : gas surface density of @xmath10 , midplane density of stars plus dark matter @xmath11 , galactic rotation angular speed @xmath12 , uniform azimuthal magnetic field of @xmath13 , and box size @xmath14 , @xmath15 , where the @xmath16 direction is perpendicular to the disk . the computational box had a uniform resolution of @xmath17 in each direction such that the number of grid cells along each direction were @xmath18 and @xmath19 . gas cooling , dominated by ly@xmath20 emission at high temperatures and @xmath21 fine structure emission at lower temperatures , is included by use of the fitting formula of @xcite , @xmath22 where @xmath23 is the gas temperature in kelvin . treating radiation from young , massive stars to be the main source of heating via the dust grain photoelectric effect , the gas heating rate is set to be proportional to the recent , global sfr surface density @xmath24 , @xmath25\,{\rm erg}\ , { \rm s}^{-1}\ , , \label{eq : heating}\ ] ] where @xmath26 is the solar neighborhood heating rate and @xmath27 is the sfr surface density in the solar neighborhood . the term @xmath28 accounts for a metagalactic fuv radiation field . radiative shielding of fuv photons , an important effect at high gas column densities , is not included . supernova feedback is included in the form of instantaneous momentum injection at a rate proportional to the local star formation rate , as set by the free - fall time of gas with an efficiency of @xmath29 above a critical density threshold . the momentum from each supernova is set to @xmath30 representing the value at the radiative stage of a single supernova remnant ( see @xcite and references therein ) . this approach does not form or follow the hot ( @xmath31 ) ism phase , but instead focuses on the self - consistent turbulent driving , dissipation , and gravitational collapse of the atomic medium , the main gas reservoir of the ism . the absence of a hot phase will have only a minimal effect on our results , since we are mainly concerned with molecule formation processes that are restricted to much lower temperatures . we utilize the chemical network described in @xcite which we briefly summarize here . the full chemical network consists of 32 atomic and molecular species and @xmath32 chemical and photoreactions . the abundances of only 14 of these species ( h@xmath33 , @xmath0 , he@xmath33 , c@xmath33 , o@xmath33 , oh , h@xmath34o , co , c@xmath34 , o@xmath34 , hco@xmath33 , ch , ch@xmath34 , and ch@xmath35 ) are computed by formal integration , i.e. , via solving a stiff set of coupled ordinary differential equations . the abundances of the remaining species are computed by either assuming instantaneous chemical equilibrium ( as in the case of h@xmath36 , @xmath37 , h@xmath35 , ch@xmath33 , ch@xmath38 , oh@xmath33 , h@xmath34o@xmath33 , and h@xmath39o@xmath33 , species that react so rapidly as to always be close to their equilibrium abundance values ) or by utilizing conservation laws ( for e@xmath36 , h , he , c , and o ) , thus reducing the number of coupled differential equations , and computational cost , that must be integrated to solve the full chemical network . to model photodissociation and photoionization , we take the shape and strength of the interstellar radiation field ( isrf ) to follow the standard @xcite field . in habing units , the strength of the draine field is @xmath40 , where the @xcite radiation field is defined to be @xmath41 where @xmath42 is the angle averaged specific intensity and the integral runs over the fuv range , from @xmath43 . the photodissociation and photoionization rate for any given chemical species can be written as @xmath44 where @xmath45 is the photorate in optically thin gas ( see table b2 in @xcite ) , and @xmath46 is the degree by which the photorate is reduced as compared with the optically thin rate . we defer a discussion of how the shielding factors @xmath46 and unattenuated radiative intensity @xmath47 are determined to section [ sec : raytrace ] . in a subset of our models we evolve the temperature to equilibrium along with the chemical abundances . in this section we describe these heating and cooling processes . in the low temperature , moderate density , molecular ism , line emission originating from the rotational transitions of co is a major source of radiative cooling . the volumetric cooling rate from co can be written as @xmath48 . in general , the co cooling function @xmath49 is a complicated function of temperature , density , column density , and velocity dispersion , and in a multidimensional simulation it is not feasible to self - consistently solve for the level populations in every computational cell . hence , we employ the despotic code @xcite to pre - compute the co cooling function . utilizing the large velocity gradient approximation , we tabulate @xmath49 as a function of @xmath50 , temperature , and , following @xcite , an effective column density per velocity @xmath51 defined such that @xmath52 where @xmath53 is the co number density . the local velocity divergence , @xmath54 , is computed using a three - point stencil around the cell of interest . during the computation we read in the despotic - computed cooling tables and employ tri - linear interpolation to compute the co cooling rate as needed . in addition to co line cooling , we also consider fine - structure line emission from @xmath55 $ ] , @xmath56 $ ] , @xmath57 $ ] . in the case of these species , because their lines remain optically thin under all the conditions found in our calculations , it is straightforward to solve for the level populations and compute the cooling rate directly . we use the collisional rate coefficients , atomic data , and methodology presented in @xcite . the rate of energy exchange between gas and dust is @xmath58 where @xmath59 is the dust - gas coupling coefficient for mliky way dust in @xmath0 dominated regions @xcite and @xmath60 is the dust temperature . in principle @xmath60 can be determined self - consistently by considering dust to be in thermal equilibrium between absorption and emission , but for simplicity here we assume a constant dust temperature of @xmath61 . this should be an acceptable approximation since gas - dust coupling does not typically become important until @xmath62 , a regime not probed here . cooling due to the collisional excitation of atomic hydrogen , or lyman-@xmath63 , is given by @xcite , @xmath64 where @xmath65 , and @xmath66 and @xmath67 are the electron and neutral hydrogen densities , respectively . the photoelectric heating rate , including recombination cooling , is given by @xcite , @xmath68 where @xmath69 is the attenuated radiation field strength described in section [ sec : raytrace ] , @xmath70 , @xmath71 , and @xmath72 is the photoelectric heating efficiency . the heating rate due to cosmic ray ionization is given by @xmath73 where @xmath74 is the cosmic ray ionization rate per hydrogen nucleus which we take to be @xmath75 , a relatively low value that aids in the production of co. the energy added per cosmic ray ionization @xmath76 depends on the chemical composition of the gas . for purely atomic gas we use the recommendation from @xcite , @xmath77 while we use a piecewise fit given in ( * ? ? ? * equation b3 ) of numerical data from @xcite for @xmath78 . note that we do not include any sort of heating due to turbulent dissipation or ambipolar diffusion , two related effects whose combined heating rate can be comparable to the cosmic ray heating rate for the low cosmic ray ionization rate we adopt here ( e.g. , * ? ? ? modeling radiation transport in full generality within a three - dimensional simulation presents a significant numerical challenge . as noted by @xcite , the cost of multifrequency radiation transfer scales as @xmath79 , where @xmath80 is the number of discrete frequency bins and @xmath81 is the number of computational elements . even with moderate resolution , this is orders - of - magnitude larger than the hydrodynamic evolution , which scales as @xmath82 , and generally intractable . while a number of approximate methods exist for solving the equation of radiative transfer in a numerical simulation ( e.g. , * ; * ? ? ? * ; * ? ? ? * ) , these approaches are not suitable for line transfer calculations , which are crucial for determination of the photoreaction rates . here , we elect to solve for the radiation field , and chemical state , by direct ray - tracing applied to static simulation snapshots . -axis with a separation equal to the cell spacing @xmath83 . the next set of rays ( green arrows ) are launched with an angle of @xmath84 with respect to the @xmath85-axis and maintain a perpendicular separation @xmath83 with the first set of rays . this choice guarantees the computational domain is evenly sampled by rays for each angular set of rays . only when the rays are physically located within the computational grid ( solid boxes ) is the ray tracing performed.,scaledwidth=45.0% ] the radiation intensity at any point along a ray is governed by the time - independent radiative transfer equation , @xmath86 where @xmath87 is the specific intensity , @xmath88 is the absorption coefficient , @xmath89 is the emissivity , and @xmath90 is the distance along the ray . we are particularly concerned with fuv photons whose dominant sources are stars with ages of @xmath91 myr or less . since the simulation does not follow young stars explicitly , and since by this age stars are generally no longer predominantly found in clusters ( e.g. , * ? ? ? * ; * ? ? ? * ) , we approximate their distribution as a smooth field that varies with height @xmath16 alone . we therefore assign every grid cell a frequency - dependent fuv emissivity @xmath92 , where the normalization @xmath93 is fixed by the requirement that the unattenuated midplane isrf intensity , @xmath94 , is in accordance with the current sfr surface density : @xmath95 where @xmath24 and @xmath96 retain their definitions from equation [ eq : heating ] . for the vertical distribution of the fuv emissivity we adopt the profile @xmath97 appropriate for an isothermal disc , where @xmath98 is the stellar scale height . for our fiducial model we adopt a scale height of @xmath99 , though we do explore how variations in @xmath98 affect our results . even in a static snapshot , solving the transfer equation in three dimensions with enough frequency resolution to resolve the dominant photodissociation lines of every important species is prohibitively expensive . fortunately , it is also unnecessary for the purpose of computing photoreaction rates , which is our goal . photodissociation for important interstellar species occurs via resonant absorption of an fuv photon in one or more bands that occupy a relatively narrow range of frequencies . one - dimensional simulations that include full frequency - dependent transfer show that the photodissociation rate produced by the isrf decreases as the radiation propagates through increasing columns of material due to two main effects : continuum absorption by dust , and resonant absorption of photons near the centers of the dissociating absorption lines @xcite . for dust , the reduction in the photodissociation rate is given by @xmath100 where @xmath101 is the scaling between the dust opacity at the photodissociation absorption frequency and the v - band . here @xmath102 is the dust opacity evaluated at the absorption band relevant for that species , which is taken to be at a single characteristic frequency , a good approximation since @xmath102 varies slowly with @xmath103 over the band ; @xmath104 is the v - band dust opacity . for the chemical species we consider , @xmath105 ranges from @xmath106 to @xmath107 . the visual extinction in magnitudes @xmath108 is directly proportional to the total column density of h nuclei , @xmath109 , @xmath110 where @xmath111 @xcite is the total dust cross section in the v - band per hydrogen nucleus . to properly model the photodissociation of @xmath0 and co we must also consider self - shielding in which the molecules themselves , in addition to dust grains , contribute to the attenuation of fuv photons . self - shielding of @xmath0 can be well approximated with the following analytic expression from @xcite , @xmath112 where @xmath113 , @xmath114 , and @xmath115 is the doppler broadening parameter for @xmath0 . the total shielding factor for @xmath0 will be @xmath116 . co not only experiences self - shielding and dust shielding , but is additionally shielded by molecular hydrogen . the photodissociation rate for co can be written as @xmath117 , where @xmath118 is the co self - shielding function and @xmath119 accounts for the cross - shielding of co by the overlapping lyman - werner lines of @xmath0 . both @xmath120 and @xmath121 are taken from @xcite , while @xmath122 . a critical point to realize is that , because the co and h@xmath34 self - shielding factors result from a rearrangement of the photon frequency distribution as radiation propagates along a ray , they may be applied independently on every ray . this means that , rather than requiring many frequency bins to resolve the dissociating lines of h@xmath34 and co , we can approximate the effect by considering a single frequency bin for h@xmath34-dissociating photons , and similarly for co - dissociating ones , and use the pre - tabulated shielding functions to model the reduction in photodissociation rate along a ray . the main limitation in this approach is that the shielding functions have been tabulated for gas with a fixed velocity dispersion and zero bulk velocity , whereas in our simulations the velocity fields are significantly more complex . however , we show below that our results are quite insensitive to exactly how we estimate the velocity - dispersion that enters the shielding factors . based on the discussion in the preceding section , we now discretize the equation of radiative transfer ( equation [ eq : rad_transfer ] ) by defining three distinct frequency bins : one describing radiation in the fuv dust continuum , @xmath123 , one that corresponds to the lyman - werner bands for @xmath0 , @xmath124 , and one describing the photodissociation of co , @xmath125 . the three frequency bins can be formally defined through the following filters : @xmath126 defined such that @xmath127 and @xmath128 equal unity in the frequency ranges that overlap with their photodissociation absorption lines , and are zero otherwise . likewise , @xmath129 in the fuv frequency range , excluding the photoabsorption lines of @xmath0 and co , and is zero otherwise . the intensity at any point @xmath90 along a ray is given by the formal solution to equation ( [ eq : rad_transfer ] ) , @xmath130 identifying @xmath131 to be the shielding factor @xmath46 ( equation [ eq : photo_shield ] ) and integrating over the frequency filters defined in equation ( [ eq : filters ] ) gives us the intensity in our three frequency bins : @xmath132 @xmath133 @xmath134 in the above equations , the shielding factors are evaluated using the total column densities between points @xmath90 and @xmath135 , @xmath136 , @xmath137 , and similarly for @xmath138 and @xmath139 we are now in a position to discretize the equations ( [ eq : ih2 ] ) , ( [ eq : ico ] ) , and ( [ eq : icont ] ) in a way suitable for numerical calculation . consider a ray passing through our cartesian grid at some angle , and consider the @xmath140th cell along that ray . let the sequence of cells through which the ray passes on its way to that cell @xmath140 be numbered @xmath141 , and let @xmath142 be the path length of the ray through cell @xmath143 ; let @xmath144 be the corresponding number density of species @xmath145 ( e.g. , h@xmath34 ) , and similarly for all other quantities . in this case equation ( [ eq : ih2 ] ) can be discretized to @xmath146 where the shielding factor , which includes both dust- and self - shielding , is evaluated using the total column @xmath147 between the cells @xmath140 and @xmath143 . similar results hold for equations ( [ eq : ico ] ) and ( [ eq : icont ] ) . we have now written down a discretized version of the radiative transfer equation for a particular direction of propagation . the remaining step is to discretize the problem in angle . we do so by drawing rays in @xmath148 directions ( parametrized by @xmath84 and @xmath149 ) selected via the healpix algorithm for equal area spherical discretization @xcite . all our models use healpix level @xmath150 , corresponding @xmath151 , which we have found provides sufficient resolution to properly sample the radiation field . for each direction , we draw through the computational domain a series of rays separated by one cell spacing @xmath83 in the direction perpendicular to the rays , such that each ray represents a solid angle @xmath152 and area @xmath153 . this choice guarantees that each cell is intersected by at least @xmath148 rays in total . we show a schematic diagram of our ray tracing procedure in figure [ fig : gridfig ] . the radiation intensity in any given cell is then able to be computed through an angle average of all intersecting rays . we are ultimately interested in photodissociation rates which are in turn proportional to the angle - averaged intensity , @xmath154 and similarly for all other species . evaluating this integral requires some subtlety ; although all our rays represent an equal amount of solid angle , not all rays intersecting a cell have the same path length @xmath155 through it , even rays that represent the same direction of radiation propagation . the contribution to the mean intensity in a cell from a given ray @xmath140 will be proportional to the time that the photons propagating along that ray spend in the cell , and thus proportional to @xmath156 . we therefore compute the mean intensity by weighting each ray s contribution by @xmath155 : @xmath157 where @xmath158 is the total number of rays that intersected the cell . the conversion between @xmath159 and @xmath160 is implicitly performed through a proper choice of the normalization of the fuv cell emissivity , @xmath161 . we are interested in comparing the results obtained with the full angle - dependent ray - trace in the previous section to those produced by various local approximations for the degree of radiative shielding that have been proposed . in these approximations , the factor by which a chemical species photodissociation rate is reduced relative to its optically thin rate , @xmath46 , is solely dependent on an effective column density : @xmath162 in the case of dust shielding , and @xmath163 in the case of @xmath0 and co shielding . working under the approximation of uniform density and chemical composition , we can rewrite column density as the product of number density and some shielding length scale , @xmath164 . this significantly reduces the complexity and computational expense of multidimensional , multifrequency radiative transfer to the relatively simple task of computing an appropriate , physically motivated shielding length @xmath165 . various physically motivated expressions for @xmath165 have been proposed and utilized , to varying degrees of success . the sobolev approximation , alternatively known as the large velocity gradient ( lvg ) approach , was originally devised by @xcite to study expanding stellar envelopes , but has been used extensively in simulations to model the photodissociation of @xmath0 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? it relies on the observation that in regions with a constant velocity gradient @xmath166 , a photon will see doppler shifted absorption lines with respect to its emission frame . @xcite proposed that once a photon is doppler shifted by one thermal linewidth , the photon should be free to escape . thus , we can define the sobolev length to be @xmath167 where @xmath168 is the local sound speed and the local velocity divergence @xmath54 serves as a multidimensional analog to @xmath166 . the jeans length @xmath169 is the critical length scale under which the force of gravity overwhelms outward thermal pressure allowing gravitational collapse to proceed . the jeans length is a common approximation for @xmath165 , with the physical justification that at the center of a gravitationally collapsing core , the radiative background will be attenuated by a column of gas with a length scale approximately equal to @xmath170 . we also investigate a shielding length based on the local density and its gradient such that @xmath171 the density gradient approach has been used by @xcite to account for the self - shielding of @xmath0 from photodissociating radiation and provided reasonable accuracy in the column density range @xmath172 . an additional method to account for radiative shielding is to utilize the results of a full ray trace run to calibrate a relationship between density and column density ( or equivalently , visual extinction ) . we we will show in section [ sec : results ] , this relationship can be approximated as @xmath173 which holds above a density of @xmath174 and becomes increasingly accurate with increasing density . this method naturally has the disadvantage that it is perhaps unique to the particular physical system one is examining which detracts form its generality . a highly simplified approach to account for radiative shielding is to assume that all shielding in a given cell is due to matter in the cell in question , such that @xmath175 pc . this single - cell approximation will surely underestimate the degree of radiative shielding , and thus molecular chemical abundances , as compared with the fiducial ray - tracing model , allowing limits to be placed on the effectiveness of radiative shielding . finally , while not a strictly local approach , we perform a model where ray tracing is only performed over the six directions aligned with the cartesian axes , instead of the @xmath176 healpix selected directions in the ray trace models . this approach , known as the six - ray approximation , can be computationally efficient since the transport of rays exploits the alignment of the grid . it is a compromise between a local approach and full - fledged , multi - angle radiative transfer . for this to be a clean comparison , the local and ray - trace models need to utilize the same unattenuated radiation field @xmath47 so that the sole difference between the various methodologies is in the computation of @xmath46 . to achieve this , in the local models we begin by performing the multiangle ray - trace procedure described in section [ sec : raytrace ] , but with all radiative shielding turned off , @xmath177 , providing us with the unattenuated radiation field @xmath47 in each grid cell . this unattenuated field is then used in equation [ eq : photo_shield ] , along with the locally determined @xmath46 , to compute the chemical photodissociation rates . , top - right ) , co fraction versus density ( bottom left ) , and co fraction versus @xmath178 ( bottom - right ) . color corresponds to the mass - fraction that particular region of phase space . for comparison , the overplotted black lines are approximate analytic expressions for the @xmath0 fraction @xcite , mass - weighted in density ( left panel ) or @xmath179 ( right panel ) bins . the shaded region in the top - right panel denotes the analytic prediction from ( * ? ? ? * equation 40 ) for the maximum visual extinction before the h@xmath180-@xmath0 transition , for densities from @xmath181 to @xmath182.,scaledwidth=48.0% ] [ cols="<,^,^,^,^,^,^,^,^ " , ] _ notes _ numbers in parenthesis represent the scientific notation exponent , i.e. , 0.19(-3 ) @xmath183 . ( 1 ) the total gas mass fraction with an @xmath0 ( first column ) or co ( second column ) fraction greater than @xmath106 . the second number , after the slash , is the equivalent quantity when only considering cold , dense gas ( @xmath184 , @xmath185 ) . ( 2 ) the total mass in the fiducial simulation ( f1 ) is @xmath186 . ( 3 ) see text surrounding equation [ eq : err ] for the description of the @xmath0 and co error measurements . to model the spatial distribution of fuv emissivity ( as required by equation [ eq : soln ] ) we treat massive ob stars as the primary emitters of fuv radiation whose vertical density obeys the functional form of equation [ eq : fz ] . the fiducial model f1 utilized a vertical scale height of @xmath187pc , roughly consistent with the measured mean half - width of @xmath0 in the inner galaxy of @xmath188pc @xcite . models v1 and v2 , which use scale heights of @xmath189 and @xmath190pc , respectively , demonstrate the relatively insensitivity of our results to @xmath98 . inspecting table [ tab : results ] , model v1 has @xmath191 more @xmath0 and @xmath192 more co than f1 , reasonable considering that a reduction ( increase ) in @xmath98 lowers ( increases ) the overall global number of fuv photodissociating photons . model v2 has a corresponding decrease in amount of @xmath0 , though less than a @xmath29 reduction in the total co mass as compared with f1 . morphologically , models v1 and v2 are virtually indistinguishable from f1 . models v3 through v6 , in which one or a number of shielding mechanisms are artificially switched off , are designed to assess the relative importance of self- and dust - shielding in determining the molecular abundances . unsurprisingly , each of these models results in overall less @xmath0 and co mass than the comparison model f1 . models v3 , v4 , v5 , v6 have , respectively , @xmath193 , @xmath193 , @xmath194 , and @xmath195% the mass of @xmath0 as model f1 . as for the co mass , these percentages become @xmath196 , @xmath197 , @xmath198 , and @xmath199% ( table [ tab : results ] ) . figures [ fig : proj.shielding.h2 ] and [ fig : proj.shielding.co ] show mass - weighted projections of the @xmath0 and co fractions for models f1 , v3 , v4 , v5 , and v6 . a comparison of models f1 with v3 and v4 in figure [ fig : proj.shielding.h2 ] highlights the importance of @xmath0 self - shielding in attenuating the isrf and reducing the @xmath0 photodissociation rate . the @xmath0 fractions in models v3 and v4 , which do not include @xmath0 self - shielding ( equation [ eq : fh2 ] ; @xmath200 ) , are substantially suppressed ( by nearly a factor of @xmath201 see table [ tab : results ] ) compared with models that do include self - shielding . in these models , @xmath0 fractions of unity are only realized in the highest density gas that approaches the resolution of this study . switching off co self- and cross - shielding , model v5 , unsurprisingly has no effect on the @xmath0 abundance . in model v6 dust shielding is deactivated in an effort to isolate its role in reducing the direct photodissociation rates of @xmath0 and co. importantly , dust shielding remains active for all other species besides @xmath0 and co. this choice removes the indirect effect that increased photodissociation rates for other chemical species would have in altering the chemical formation pathways of co and , to a much lesser extent , @xmath0 . as shown in the far right panel of figure [ fig : proj.shielding.h2 ] , the removal of dust shielding has a fairly small , though non - zero , effect on the @xmath0 fraction . in contrast to @xmath0 , self - shielding , cross - shielding by @xmath0 , though whether this stems from @xmath0 cross - shielding or reduced co formation rates is a subtle issue we explore later in this section ] , and dust shielding are all comparably important in photoshielding co molecules . this can be seen in figure [ fig : proj.shielding.co ] where each model results in a significant decrease in the amount of co as compared with model f1 . model v4 , in which @xmath0 self - shielding is switched - off and the global @xmath0 fractions are significantly reduced , demonstrates the importance of @xmath0 shielding co via their overlapping photodissociation lines in the lyman - werner bands . models v5 and v6 , meanwhile , respectively underscore the importance of self- and dust shielding . however , the inclusion of _ any _ form of radiative shielding , whether from self- , cross- , or dust - shielding , results in the formation of gas with @xmath202 at the highest densities , @xmath203 . model v10 ( not shown ) , in which all radiative shielding is turned off , results in no gas with @xmath204 , a statement that additionally applies to @xmath0 . we can explore this further by examining the cumulative distribution functions ( cdf ) of @xmath205 and @xmath206 for models v3-v6 and f1 , which we plot in figure [ fig : shielding.cdf.cd ] . here , we restrict this analysis to cold , dense gas as this makes for a more potent demonstration of the competing factors at work ; qualitatively identical conclusions would be drawn from the cdfs that included all gas . focusing first on @xmath0 ( left panels ) , we see the removal of dust shielding only reduces the gas fraction with @xmath207 as compared with model f1 . turning off @xmath0 self - shielding ( models v3 and v4 ) , conversely , suppresses the @xmath0 mass @xmath208 for gas with @xmath209 . here it is clear that by and large the dominant mechanism by which @xmath0 is shielded from the fuv isrf is self - shielding , at least for the range of isrf properties probed in these simulations . in the analytic models of @xcite , the relative importance of self - shielding and dust shielding depends on the value of @xmath210 , equation [ eq : chi ] . at milky way values of @xmath210 the two processes are about equally important , but as noted above , the simulations we analyze here have radiation field intensities that are @xmath211 times smaller than those found in the solar neighborhood , so the relative dominance of self - shielding is not surprising . ( left panels ) and co fraction ( right panels ) . this analysis is restricted to cold , dense gas ( @xmath184 , @xmath185 ) . lines of different color denote models with different treatments of dust- and self - shielding as described in the text.,scaledwidth=53.0% ] as for co , a great deal of information can be extracted from a comparison of the various cdfs , shown in the right panels of figure [ fig : shielding.cdf.cd ] . model v4 does not include @xmath0 self - shielding ( @xmath200 ) and as a consequence has a highly suppressed @xmath0 fraction . this choice encapsulates two combined effects , one chemical and one radiative , both of which act to decrease the amount of co. first , @xmath212 ( the cross - shielding factor ) is reduced as the global @xmath0 column densities are significantly reduced . second , the two predominant co formation pathways ( one involving oh , and the other ch ) both rely on the availability of @xmath0 , thus reducing the overall speed of the chemical formation pathways that lead to co. these effects combined result in the co mass being suppressed by a factor of @xmath213 for @xmath214 compared with model f1 the suppression increases with increasing co fraction . from this model alone , however , is it unclear which of these two effects , chemical or radiative , is dominant . model v5 does not include co self - shielding nor cross - shielding from @xmath0 . as can be seen , this has little effect in gas with a low co fraction ( @xmath215 ) . evidently for this diffuse co gas , co self - shielding and @xmath0 cross - shielding is relatively unimportant . for co rich gas ( @xmath216 ) , the mass in co is suppressed by a factor of @xmath217 . chemically , there is no suppression of the @xmath0 abundance in this model so any reduction in @xmath206 is entirely due to the lack of shielding from itself and @xmath0 . the drop in the co fraction between models v5 and v3 is a chemical , not radiative , effect and highlights the role of @xmath0 in the co formation pathways . observe that the sole difference between models v5 and v3 is that the @xmath0 fraction in model v3 is highly suppressed due to the lack of @xmath0 self - shielding . with regards to co , this isolates the effect that the @xmath0 abundance has on the efficiency of the chemical pathways that lead to co. quantitatively , this purely chemical effect reduces the total co mass by @xmath218% ( see table [ tab : results ] ) . furthermore , at the highest co fractions , there is no difference between model v5 and v3 in terms of the cdf , while model v4 still displays a roughly order - of - magnitude drop . this implies that a reduction in the @xmath0 fraction ( via removing @xmath0 self - shielding ) decreases the co fraction predominantly through less @xmath0 cross - shielding and a larger co photodissociation rate , as opposed to a reduction in the co formation rate . the co fractions realized in the absence of dust shielding ( model v6 ) are similar to that of models v3 and v4 for @xmath219 , underscoring the importance of dust shielding for co. in other words , the consequence of removing dust shielding for co is roughly comparable to the combined removal of co self - shielding , @xmath0 cross - shielding , and decreased co formation efficiency due to a suppressed @xmath0 abundance . evidently , dust shielding is critical for the formation of co fractions approaching unity . in the absence of dust shielding , no gas is able to form with @xmath220 . models v7 and v8 consider the scenario where the overall strength of the fuv radiative intensity is , respectively , increased or decreased by a factor of @xmath221 . increasing the isrf by a factor of @xmath221 ( v7 ) results in a @xmath222 decrease in both the total @xmath0 and co mass , while decreasing the radiative intensity tenfold ( v8 ) results in a factor of @xmath223 more @xmath0 and @xmath224 times more co mass . in model v9 , the cosmic - ray ionization rate is increased from @xmath225 to @xmath226s@xmath227 to gauge its impact on molecular chemistry . this has a much more severe effect on the co abundance than the @xmath0 abundance . as seen in table [ tab : results ] , increasing @xmath74 reduced the global @xmath0 mass by roughly a factor of @xmath150 , while the co mass fell to roughly @xmath228 its value in model f1 . this strong dependance of the co abundance on the cosmic ray ionization rate is an indirect effect , stemming mainly from the cosmic ray ionization of neutral helium and the subsequent charge - exchange destructive reaction between he@xmath33 and co. finally , in model v10 , all radiative shielding is switched off , causing a catastrophic decrease in the abundances of @xmath0 and co. as can be seen in table [ tab : results ] , this results in zero gas with an @xmath0 or co fraction higher than @xmath106 . the total molecular mass is also significantly reduced ; the total mass of @xmath0 and co is @xmath229 and @xmath230 , respectively , as compared to model f1 . this result only underscores the importance of radiative shielding in permitting the formation of significant molecular abundances . however , as will be shown in section [ sec : temperature ] , the removal of all radiative shielding has a relatively small effect on the temperature structure of the gas . is entirely due to the vertical dependance of the unattenuated radiation field . the primary effect of changing the fuv intensity ( models t3 and t4 ) is a shift in the density at which gas transitions from the warm to cold neutral medium . the blue shaded region in the top panel denotes this paper s definition of cold , dense gas.,scaledwidth=50.0% ] in models t1-t4 , the gas temperature is evolved to equilibrium along with the chemical abundances , utilizing the thermal processes described in section [ sec : thermal ] . model t1 is identical to model f1 save for the temperature evolution , while model t2 operates under optically thin conditions , i.e. , no radiative shielding . in models t3 and t4 the mean fuv intensity is altered , though are otherwise identical to model t1 . in figure [ fig : dts.compare ] we show density - temperature diagrams for models f1 ( whose density and temperature are unchanged from the values initially supplied by k15 ) , t1 , t2 , t3 , and t4 . at any given density , the spread in temperature for model t1 is significantly reduced as compared with model f1 . in fact , any spread in temperature in model t1 , at a given density , is overwhelmingly due to spatial variations in the photoelectric heating rate ( equation [ eq : photoelectric ] ) which arises due to differences in the degree of radiative shielding ( via @xmath231 ) and the vertical dependance of the fuv emissivity ( equation [ eq : fz ] ) and consequently the unattenuated radiation field @xmath47 . aside from this reduction in the temperature spread , the density - temperature relationship between models f1 and t1 agree remarkably well below a density of @xmath232 . above @xmath233 , the temperature in model t1 begins to drop below that of f1 , reaching @xmath234 by @xmath235 as a result of co line emission , a cooling pathway not included in k15 . the relationship between density and temperature in model t2 , which does not include the effect of radiative shielding , is similar to that of model t1 below @xmath236 . the only significant difference between the two models is the absence of @xmath237 degree gas at the highest densities , due to the complete lack of any significant co cooling , in model t2 . even by increasing the midplane @xmath47 by a factor of @xmath221 to @xmath238 ( model t3 ) , a cold phase still develops , though at a slightly larger density and equilibrium temperature of @xmath239 before co cooling cools the gas below @xmath240 . model t4 , with its tenfold reduction in the strength of the midplane fuv intensity , exhibits the emergence of a molecular cold phase at the relatively low density of @xmath241 . the molecular nature of this cold phase should be viewed as an artifact of our equilibrium assumption , given the chemical time at @xmath242 is of order @xmath221 gyr see section [ sec : discussion ] . the minimum temperature obtained in model t4 is also lower than any other model , nearly reaching the cosmic microwave background ( cmb ) temperature floor ( @xmath243 ) around @xmath244 . while radiative shielding is crucial for the formation of significant molecule fractions ( section [ sec : variations ] ) and for reaching the lowest temperatures observed in the ism , the ubiquitous appearance of a marginally cold phase ( where @xmath6 ) in all these models supports the important point that shielding is not strictly required for the transition to a cold phase , at least in the regime where the background fuv radiation field strength is relatively modest , @xmath245 . the primary objective in this section is to assess the validity and accuracy of utilizing a locally computed shielding length to model radiative attenuation and thus the fuv - radiation regulated chemical state . we will do this by examining how closely the local models l1-l6 reproduce the @xmath0 and co abundances of the fiducial model f1 using a variety of qualitative and quantitative measures . considering first the total mass of @xmath0 ( @xmath246 , table [ tab : results ] column @xmath247 ) , we see all the local models come within a factor of @xmath150 of f1 , with the exception of model l2 ( sobolev approximation ) which results in @xmath248% less @xmath0 . the total mass of co is a bit more scattered than @xmath246 , though the four best performing local models ( l1a , l3 , l4 , and l6 ) have @xmath249 within @xmath250% of model f1 s value . any given local approximation model always produces an increase ( or decrease ) between both the total @xmath0 and co masses , with the exception of model l3 ( density gradient ) which , due to the peculiarity of using density gradients to estimate a global column density , results in @xmath251% less @xmath0 , and nearly @xmath252% _ more _ co , than model f1 . the morphology and distribution of @xmath0 between models l1-l6 and f1 ( figure [ fig : proj.local_compare.h2 ] ) highlights the relative similarity between the distribution of molecular hydrogen in many of the local approximation models compared with the fiducial model . evidently the exact prescription used to account for the attenuation of the isrf is a relatively unimportant factor in modeling the overall distribution of @xmath0 gas . the inclusion of _ some _ form of radiative shielding , however , is a necessary ingredient to form any significant @xmath0 fraction not shown here is model v10 ( no shielding ) in which no cell of gas achieves an @xmath0 ( or co ) fraction greater than @xmath253 and results in a factor @xmath254 less @xmath0 and co mass . it is important , however , to scrutinize any variation between the various local models and model f1 in figure [ fig : proj.local_compare.h2 ] . models in which radiative shielding is based on the local jeans length ( l1 and l1a ) or a power - law fit between @xmath255 and @xmath178 ( l4 ) appear to slightly overestimate the @xmath0 fraction , while a shielding length computed via the velocity divergence ( sobolev length ; l2 ) , the density gradient ( l3 ) , or a single computational cell ( l5 ) , tend to underpredict @xmath205 . by eye , the best level of agreement is achieved by the six - ray approximation ( model l6 ) ; this is not a strictly a local approximation , but is instead effectively the result of reducing the number of ray - tracing propagation directions from @xmath176 to @xmath247 , now all aligned along the cartesian axes . in contrast to @xmath0 , the spatial distribution and amount of co ( figure [ fig : proj.local_compare.co ] ) shows a great deal more sensitivity to the radiative shielding prescription . as compared with model f1 , models l2 and l5 fail to produce any gas with visually discernible co. in model l3 there exists what appears to be an unphysical scattering of cells with @xmath206 approaching unity and a paucity of diffuse co , @xmath256 . as with @xmath0 , the six - ray approximation ( model l6 ) appears to do the best job in reproducing @xmath206 . it should be noted , though , that large concentrations of co can be difficult to discern given the small spatial extent of these concentrations . model l1 , where @xmath257 , seems to perform reasonably well in matching @xmath206 from model f1 , except for the appearance of two large ( @xmath258pc ) structures with @xmath259 that do not exist in model f1 . we attribute this to out - of - equilibrium hot gas with density @xmath260 and temperature @xmath261 . since the jeans length @xmath262 , these cells have an unphysical elevated shielding length which reduces the fuv radiation intensity and boosts the co abundance . this is an undesired effect of modeling @xmath165 with @xmath263 these cells do not represent gravitationally collapsing cores and physically should not experience a higher degree of radiative shielding . this shortcoming can be addressed quite simply , however , by introducing a temperature ceiling into the calculation of @xmath263 . in model l1a the shielding length is given by this temperature capped jeans length with an empirically chosen ceiling temperature of @xmath264 . an inspection of figure [ fig : proj.local_compare.co ] confirms that these unphysical co rich bubbles are strongly suppressed in model l1a . to better understand the relationship amongst the local models , in figure [ fig : local_compare.dens.av.phase ] we plot the mass - weighted effective extinction ( equation [ eq : aveff ] ) as a function of density for each local model as compared with model f1 ( previously plotted in figure [ fig : dens_av_phase ] ) . here it is clear how well models l1 , l1a , l4 , and l6 perform in matching the effective visual extinction as computed from detailed ray - tracing , at least at high densities , @xmath265 . models based on local derivatives of grid - based quantities severely overpredict ( l3 ) or underpredict ( l2 ) the true column density . caution should be used in interpreting figure [ fig : local_compare.dens.av.phase ] ; in doing so , one would be led to conclude that model l4 should be the best performing local model since it was explicitly calibrated to match the @xmath266 relationship from model f1 . as discussed in section [ sec : variations ] , @xmath0 is primarily shielded by itself , the degree of which is a function of the @xmath0 column density @xmath267 , not @xmath162 . thus the @xmath268 relationship is not the ideal proxy for how well a model will predict the @xmath0 abundance . it is important to have a single number that characterizes the level of agreement , with regards to the spatial distribution of @xmath0 and co , between each local approximation model and the fiducial model f1 . many such quantitative metrics could be constructed which effectively compress a three - dimensional error distribution to a single value . for our purposes here , we favor such a metric that represents the ability of a local approximation to accurately model the atomic - to - molecular transition for both @xmath0 and co , rather than one that is sensitive to diffuse molecules and low density gas , since this is most relevant in regards to both observations and molecule - mediated thermo - energetics . to this end , we have found the best metric to be the mass - weighted fractional error of @xmath269 which can be expressed as @xmath270 where @xmath271 , @xmath272 is the density of molecule @xmath85 in model f1 , @xmath273 is the density of molecule @xmath85 in any separate model , @xmath85 denotes either @xmath0 or co , and the sum runs over all cells @xmath274 . equation [ eq : err ] is formulated such that perfect agreement between the molecular abundances would produce @xmath275 , while @xmath276 signifies the local approximation model formed no molecules of species @xmath85 whatsoever . to prevent extremely small , negligible abundances from skewing this error metric and to better quantify the significant differences amongst the local models , we omit cells with concentrations so low where co would not be a significant coolant , adopting a value of @xmath277 for both co and @xmath0 . we compute the above error metric between model f1 and each other model ( see table [ tab : models ] ) except for f2 and f3 , whose different density maps make this error metric meaningless . , equation [ eq : aveff ] ) vs. density . lines of different color denote models with different treatments of radiative shielding ( models l1-l6 ) while the black shaded region represents the @xmath181-@xmath278 dispersion of @xmath178 for model f1.,scaledwidth=42.0% ] an examination of table [ tab : results ] shows that , amongst the local models , the six - ray approximation possesses the smallest @xmath0 weighted error ( l6 , err@xmath279 ) , followed by the temperature capped - jeans length ( l1a , @xmath280 ) and density gradient ( l3 , @xmath281 ) . as for co , model l1a has the lowest error ( err@xmath282 ) followed by the power - law fit ( l4 , @xmath283 ) and single cell shielding ( l5 , @xmath284 ) . this error metric does have limitations , and should be considered along with the other comparison measures . for instance , the relatively low err@xmath285 value of model l5 would suggest it does an excellent job matching the co abundances of f1 , though inspecting its co spatial distribution ( figure [ fig : proj.local_compare.co ] ; see also figure [ fig : xco_fiducial_compare ] ) suggests otherwise . this discrepancy is due to the co error metric ( err@xmath285 , equation [ eq : err ] ) being weighted by @xmath206 , while the logarithmically scaled projections allow the detection of much more diffuse ( @xmath286 ) co. as a final comparison between models l1-l6 and f1 , we show cell - by - cell , mass - weighted abundance comparisons of @xmath0 and co in figures [ fig : xh2_fiducial_compare ] and [ fig : xco_fiducial_compare ] . these plots are generated by comparing the @xmath0 fraction between model f1 ( @xmath287 ) and each local model ( @xmath288 ) on a cell - by - cell basis ( and similarly for co ) . the color scale represents the mass fraction in a particular region of the @xmath289 phase space . perfect agreement between a local model and model f1 would exclusively lie on the diagonal dashed line defined by @xmath290 . local models based on the jeans length ( l1 , l1a ) or a power - law fit ( l4 ) tend to slightly overpredict the @xmath0 abundance by @xmath291 ( see table [ tab : results ] ) . as expected , when all radiative shielding is assumed to take place in the immediate vicinity of a computational cell ( model l5 ) the @xmath0 abundance is slightly underpredicted by roughly a factor of @xmath150 . models in which the local shielding length requires a computation of a discrete gradient or divergence of a grid variable ( l2 and l3 ) tend to exhibit a larger degree of scatter in the @xmath0 abundance as compared to other local models ( figure [ fig : xh2_fiducial_compare ] ) . as before , model l6 performs the best in matching the results of the fiducial model , overpredicting the total @xmath0 mass by only @xmath292 . turning to co ( figure [ fig : xco_fiducial_compare ] ) , we see a much larger scatter and discrepancy amongst the local models . models l1 and l3 tend to significantly overpredict the co abundance by up to four orders of magnitude . comparing models l1 and l1a , however , shows how effectively this disagreement can be greatly reduced by imposing a temperature ceiling of @xmath3 in the calculation of the jeans , and thus shielding , length . model l2 is unique in that effectively no gas exists with @xmath259 . single cell radiative shielding ( model l5 ) results in relatively little scatter but an underprediction of the co fraction in all cells save for those with co fractions approaching unity . model l5 and l1a also have a lower @xmath293 ( 1d error representation ) than model l6 , though the 2d error distribution displayed in figure [ fig : xco_fiducial_compare ] seems to suggest l6 to be better performing than l5 and on - par with model l1a , highlighting the utility of multiple error comparisons to gauge the effectiveness of each model in reproducing the molecular abundances of the fiducial model . finally , it is interesting to note the similarity between models l1a ( jeans with temperature ceiling ) and l4 ( power - law fit ) in their prediction of both the @xmath0 and co abundances . this can be attributed to their respective density scalings : model l4 utilizes a @xmath294 relationship calibrated from model f1 ( @xmath295 ) , while the jeans length approximation can be shown to obey a @xmath296 scaling . the similarity of these exponents should be unsurprising , though , given the physical principle underlying the jeans length : @xmath263 is the appropriate shielding length to utilize when radiative shielding is due to surrounding material in a dense clump undergoing gravitational instability , exactly the physical entities that are able to achieve @xmath297 and @xmath204 in the simulations we analyze here . in this paper we have applied ray - tracing and chemical network integration to static simulation snapshots in an effort to model the distribution , morphology , and amount of @xmath0 and co gas in a finite , but representative , portion of a galactic disk . we have additionally performed calculations where local approximations , instead of ray - tracing , were employed to account for radiative shielding in an effort to assess the validity and accuracy of such radiative transfer alternatives . we find significant concentrations of @xmath0 gas ( @xmath297 ) are vertically restricted to within @xmath298pc of the galactic midplane and possess a filamentary morphology which traces that of the underlying gas distribution . gas with significant co concentrations ( @xmath259 ) is further confined to only the highest density clumps , located interior to @xmath0 rich gas . radiative shielding is crucial for the development of any gas with an @xmath0 or co fraction @xmath299 . when the gas temperature is permitted to reach equilibrium along with the chemical abundances , we find little change from the k15 temperature values ( except for gas at the highest densities , @xmath300 , where co cooling became effective ) , suggesting the bulk of the gas in k15 to exist in thermal equilibrium , and an overall insensitivity of temperature to radiative shielding . different physical mechanisms are responsible for the radiative shielding of @xmath0 and co. for @xmath0 , self - shielding is far dominant over dust shielding in reducing the @xmath0 photodissociation rate in all regimes explored here . the dominant shielding mechanisms for co , on the other hand , are regime dependent . more diffuse co ( @xmath301 ) is shielded primarily by dust and @xmath0 while co rich gas ( @xmath259 ) additionally depends on the presence of self - shielding by the molecule itself . however , this regime dependence for co shielding may be sensitive to the large - scale gas distribution in this particular simulation . we have compared a number of commonly used local approximations for radiative shielding to ray - tracing based solutions of the radiative transfer equation . overall , it is promising how well many of the local models reproduce the @xmath0 and co abundances as compared with a more accurate ray - tracing based approach . one of the main objectives of this work is to inform multidimensional simulations as the validity of these local approximations to account for fuv radiative shielding . based on our analysis , the six - ray approximation ( l6 ) , temperature - capped jeans length ( l1a ) , and power - law ( l4 ) perform the best in matching the @xmath0 and co abundances of model f1 . among the approximations where the effective shielding length for a given cell ( or , potentially , sph particle ) is exclusively computed from local _ quantities ( thus excluding model l6 ) , the temperature - capped ( at @xmath3 ) jeans length performed the best , an assessment based on its superiority in terms of matching the total @xmath0 and co masses of model f1 , and possessing the smallest @xmath0 and co weighted errors ( equation [ eq : err ] ) . we caution , however , that we have only explored one regime , that of a turbulent galactic disk , with relatively coarse ( @xmath302pc ) resolution . while this regime does include both the h i-@xmath0 and , at the highest densities , c ii - co atomic - to - molecular transitions , there are undoubtedly physical environments to which this analysis applies poorly . furthermore , the use of any such an approximation for @xmath165 critically depends on the physics a multidimensional simulation is attempting to probe . if one simply wishes to model the thermal impact of the h i-@xmath0 and c ii - co transitions , then coarse approximations , like those discussed above , are likely physically appropriate . on the other hand , we do not recommend such simple approximations when , for example , attempting to reproduce the detailed distribution and concentration of a large number of chemical species for observational comparison . in these cases , performing detailed , multi - frequency radiative transfer in post - processing , or utilizing advanced radiative transfer modules that operate on - the - fly , is a much more appropriate approach . all the models presented here operate under the assumption of steady - state ( equilibrium ) chemistry , the validity of which can be assessed by comparing the chemical and dynamical timescales . taking @xmath303 to be the rate coefficient for @xmath0 formation on grain surfaces , and @xmath304 to be the gas clumping factor on unresolved scales , the chemical time , the timescale for fully atomic gas to convert to molecular form , is given by @xmath305myr . the dynamical time , roughly the timescale for turbulent motions to displace molecular gas , can be expressed as @xmath306 , which assumes a linewidth - size relation @xmath307 @xcite . the assumption of equilibrium is violated when @xmath308 , or equivalently when @xmath309 . thus the equilibrium chemistry is reasonably justified given we focus our analysis on @xmath0 within the high density h i - h@xmath34 transition . non - equilibrium effects , however , are likely important at lower densities where the @xmath0 formation timescale is much longer . this back - of - the - envelope argument , though , should be compared with actual findings from multidimensional simulations that included non - equilibrium chemistry . the driven turbulence simulations of @xcite identified moderate amounts of molecular hydrogen to be out of chemical equilibrium , a finding they attributed to turbulent transport of @xmath0 from high- to low - density gas that occurs on a shorter timescale than the chemistry can readjust . this same type of mass transport would likely occur within other large - scale simulations . similarly , the colliding flow simulations of @xcite show the assumption of chemical equilibrium can under- or over - predict the true @xmath0 abundance depending on density range and integration time , with the largest discrepancies occurring at early times and , interestingly , high densities . while the assumption of chemical equilibrium likely overestimates the @xmath0 fraction , particularly at low densities , the amount of co is likely _ underestimated _ due to the supernovae feedback prescription in k15 . as discussed in section [ sec : fiducial ] , supernovae events in k15 were modeled by injecting momentum the instant density exceeds the star formation threshold , rather than allowing any time delay . for the original purposes of k15 , which focused on the properties of diffuse atomic gas that is the main ism mass reservoir , the exact timing of supernovae events was unimportant compared to the ability for the supernovae rate to change in time in response to the ism state . however , for the present purposes , this lack of a time delay tends to suppress the formation of large , dense structures where large co abundances would be expected . nonetheless , since our primary objective is to understand the role of radiative shielding and numerical approximations to it in determining the chemical balance of the ism , and not to study the chemical balance itself , this limitation of the simulations does not significantly interfere with our goals . in future work , we plan to utilize our best performing local model , one in which the temperature - capped jeans length is used as a proxy for @xmath165 , to account for radiative shielding in time - dependent radiation - magneto - hydrodynamic amr zoom - in simulations from galactic disk scales down to stellar cluster scales that will achieve significantly higher concentrations of dense , molecule rich gas than is present in the k15 snapshots . post - processing these snapshots with the tools developed here will further validate the utility of such a local approximation to accurately model fuv radiative shielding and photo - regulated chemistry . we additionally plan to extend the radiative transfer post - processing analysis performed here to different physical systems , notably isolated , dwarf galaxies with sub - parsec resolution and densities approaching @xmath310 ( e.g. , * ? ? ? * ) . in these targets of future analyses , the presence of more and much larger co rich structures will permit the measurement of integrated line intensities , the x - factor , and their variation with metallicity and observation line - of - sight . this work was supported by nasa tcan grant nnx-14ab52 g ( for ctss , mrk , ck , eco , jms , sl , cfm , and rik ) . rik , mrk and cfm acknowledge support from nasa atp grant nnx-13ab84 g . cfm and rik acknowledge support from nsf grant ast-1211729 . rik acknowledges support from the us department of energy at the lawrence livermore national laboratory under contract de - ac52 - 07na27344 .

to understand the conditions under which dense , molecular gas is able to form within a galaxy , we post - process a series of three - dimensional galactic - disk - scale simulations with ray - tracing based radiative transfer and chemical network integration to compute the equilibrium chemical and thermal state of the gas . in performing these simulations we vary a number of parameters , such as the isrf strength , vertical scale height of stellar sources , cosmic ray flux , to gauge the sensitivity of our results to these variations . self - shielding permits significant molecular hydrogen ( @xmath0 ) abundances in dense filaments around the disk midplane , accounting for approximately @xmath1% of the total gas mass . significant co fractions only form in the densest , @xmath2 , gas where a combination of dust , @xmath0 , and self - shielding attenuate the fuv background . we additionally compare these ray - tracing based solutions to photochemistry with complementary models where photo - shielding is accounted for with locally computed prescriptions . with some exceptions , these local models for the radiative shielding length perform reasonably well at reproducing the distribution and amount of molecular gas as compared with a detailed , global ray tracing calculation . specifically , an approach based on the jeans length with a @xmath3 temperature cap performs the best in regards to a number of different quantitative measures based on the @xmath0 and co abundances . [ firstpage ] -1 cm stars : formation stars : mass function

arxiv

the recent discovery of an overdensity of stars in the color - magnitude diagram ( cmd ) of the large magellanic cloud ( lmc ) having nearly the same color as the `` red clump '' of core he - burning stars but extending @xmath10.9 mag brighter has been interpreted as an intervening population of stars at @xmath2 kpc that may represent a dwarf galaxy or tidal debris sheared from a small milky way satellite ( zaritsky & lin 1997 , hereafter zl ) . zaritsky & lin label this overdensity the vrc ( vertical extension of the red clump ) , and reject other possible explanations to conclude that the vrc represents a massive foreground population with about 5% of angular surface density of the lmc itself . if true , this conclusion would have profound consequences for the interpretation of galactic microlensing studies ( renault 1997 , alcock 1997a ) since such debris could , in principle , be responsible for a sizable fraction of the microlensing signal toward the lmc ( zhao 1996 , 1998 ) that is generally attributed to microlensing by compact objects in the smoothly - distributed halo of the milky way itself . this particular stellar foreground population as an explanation for the lmc microlensing optical depth has been challenged on several grounds . the macho team find no evidence for a foreground population at @xmath3 kpc in their extensive photometric database , confirming the lmc membership of their cepheids ( alcock 1997b , minniti 1997 ) . they do find an overdensity of stars in a composite macho @xmath4 versus @xmath5 color - magnitude diagram ( cmd ) , but conclude that the _ redder _ color of this feature is incompatible with the hypothesis of a foreground clump population . ( the feature found by macho is unlikely to be the vrc , but rather another stage of stellar evolution associated with the asymptotic giant branch . ) gould ( 1997 ) argues on the basis of surface photometry of lmc performed by devaucouleurs ( 1957 ) that one of the following is true about any luminous foreground population : ( 1 ) it does not extend more than 5 from the lmc center , ( 2 ) is smooth on 15 scales , ( 3 ) has a stellar mass - to - light ratio 10 times that of known populations , or ( 4 ) provides only a small fraction of the microlensing optical depth . using a semi - analytic method to determine the phase space distribution of tidal debris , johnston ( 1998 ) has analyzed the zhao ( 1998 ) proposition , concluding that an ad hoc tidal streamer to explain the microlensing optical depth toward the lmc would cause unobserved overdensities of 10 - 100% in star counts elsewhere in the magellanic plane or would require disruption precisely aligned with the lmc within the last @xmath6 years . bennett ( 1997 ) argues that a recently - determined stellar mass function combined with the assumption that the putative foreground population has a star formation history similar to the lmc results in an implied microlensing optical depth from the vrc that is only a small fraction of that determined by microlensing observations . we will argue that the vrc feature observed by zl in color - magnitude diagrams of the lmc originates in the lmc itself . using bvr ccd photometry of several fields at different locations in the lmc , we confirm the presence of substructure in lmc red clump morphology corresponding to the vrc . in contrast to zl , however , we argue that the origin is likely to be due to stellar evolution , not an intervening population . we begin by illustrating that the vrc is seen in all our fields . because the red clump morphology varies slightly in color and magnitude over the face of the lmc , interpretation of composite cmds is complicated by the superposition of different features . we therefore focus on individual lmc fields , overlaying isochrones and evolutionary tracks of the appropriate metallicity and age in order to demonstrate that the vrc corresponds precisely in magnitude and color to the so called `` blue loops '' experienced by aging intermediate - mass core he - burning stars . we then show that similar red clump morphology is present in the cmd of hipparcos , which probes stellar populations on scales of @xmath7 pc from the sun , where intervening dwarf galaxies or tidal debris can not be invoked . finally , we analyze the argument used by zl to reject stellar evolution as the cause of the vrc , and show that a more realistic model for the star formation history in the lmc is not only consistent with the vrc , but also provides a better fit to the data . in january 1994 , bessel bvr photometry was performed with the danish 1.5 m telescope at eso la silla on the eros#1 , eros#2 and macho#1 microlensing candidates and a fourth field was taken far from the bar ; we will refer to these fields as f1 , f2 , f3 and f4 respectively . the detector was a thinned , back - illuminated , ar - coated tektronix @xmath8 ccd with a nominal gain of 3.47 e-/adu , readout noise of 5.25 e- rms , and pixel size of @xmath9 corresponding to 0.38 on the sky . the detector is linear to better than 1% over the whole dynamic range and is not affected by any large cosmetic defects . observational and field characteristics are listed in table i. the cmd of these fields have been used to calibrate data obtained by the eros microlensing survey , further details can be found in beaulieu ( 1995 ) . we have performed a reanalysis of these bvr data with ( schechter , mateo & saha 1993 ) . typical -reported errors on relative photometry are 0.02 mag at v = 19 ( typical for the clump stars ) for the cosmetically superior ( type 1 ) stars used throughout this analysis . absolute calibration was performed using graham ( 1982 ) and vigneau & azzopardi ( 1982 ) . foreground extinction was estimated using and iras maps ( schwering & israel 1991 ) ; these corrections are listed in table 1 for each field . beginning with this foreground extinction and assuming a metallicity of @xmath10 for the lmc and a helium abundance of @xmath11 , we then vary the internal extinction to achieve the best fit of the main sequence to the isochrones of bertelli ( 1994 ) . in fields f1 and f4 , this produces a total extinction equal to that of the foreground ( no internal reddening ) ; in fields f2 and f3 , the internal extinction results in an additional @xmath12 . we will show that although correcting for extinction is important , the difference between the foreground reddening and the reddening determined from main sequence fitting does not affect our conclusions . unless otherwise stated , we assume a distance modulus of @xmath13 for the lmc . this choice of distance modulus produces the best fits to the isochrones along the main sequence and in the red clump . a calibrated composite @xmath14 cmd for all four fields is shown in fig . 1 both with and without extinction corrections for the individual fields . comparable cmd have been presented and analyzed by vallenari ( 1996 , and references therein ) . the red clump is the most notable feature in the lmc other than the main sequence itself and is clearly visible in the composite cmd at about @xmath15 and @xmath16 ( @xmath17 ) . first identified by cannon ( 1970 ) , red clump stars are the counterpart of the older horizontal branch ( hb ) in globulars , and represent a post helium - flash stage of stellar evolution ( for a review , see chiosi , bertelli & bressan 1992 ) . differential reddening may be responsible for the elongated , tilted red clump in the dustier bar fields f1 and f3 , although differences in the age distribution and contamination by the red giant branch bump ( see 2.2.4 ) are also likely to play a role . the morphology of the red clump itself may also be shaped by mass - loss ( @xcite , @xcite ) . our analysis is aimed at understanding that portion of red clump morphology relevant to testing the hypothesis that the vrc is due to an intervening stellar population . a narrow vertical extension of the clump having nearly the same color as the peak red clump density but extending up to @xmath10.8 mag brighter can also be seen . a second source of substructure , or `` supra - clump , '' is apparent at a position @xmath10.8 mag brighter in @xmath18 and @xmath10.1 mag redder in @xmath19 than the peak of the primary red clump . the positions of these features are marked in fig . 1 . this substructure can be seen not only in the composite cmd , but also in the calibrated , extinction - corrected @xmath14 cmds for each of the individual four lmc fields presented in fig . 2 . fields f1 and f3 are located close to the lmc bar ; fields f2 and f4 further away at about @xmath20 from the center of the lmc . the magnitude and direction of the reddening vector is shown for each field . examination of fig . 2 indicates that although the overall morphology of the cmd is the same in each field , field - to - field variations can be seen in the extent of the red clump and in relative stellar densities along the main sequence and between the main sequence and the red clump . some of this variation is due to differences in crowding ; less severe crowding clearly results in deeper photometry for the f2 field , for example , and thus a larger number of main sequence stars . since histograms reveal a small relative excess of bright main sequence stars in the outer compared to the inner lmc fields , the star formation history of our fields may also be somewhat different , complicating any analysis that rests on a composite cmd drawn from several regions of the lmc . we therefore choose to analyze the fields independently . using contour plots of our four cmd , we determine the position of the red clump at peak density and the extent of other substructures relative to the clump . the results are summarized in table 2 . the _ relative _ position of the vertical extension is remarkably constant in each of our fields and is also consistent with that found by zl : we find that the vertical extension has a @xmath5 color that varies no more than 0.03 mag from that of the primary clump peak and that it extends at least 0.85 mag brighter in @xmath18 than the red clump peak . the second , redder substructure also maintains a constant relative position to the red clump from field to field . to within 0.02 mag in every field , this supra - clump is 0.85 mag brighter in @xmath18 and 0.10 mag redder in @xmath21 than the peak density of the red clump . to test whether the relative stellar density in our cmd within the region of the vrc is consistent with that found by zl , we return to our composite and individual dereddened @xmath22 cmd . in color - magnitude space we define the vertical extension ( @xmath23 , @xmath24 ) , the supra - clump ( @xmath25 , @xmath24 ) and primary clump ( @xmath26 , @xmath27 ) and count the stars within these regions . these counts are summarized in table 3 , indicating that the vrc represents @xmath18% and the supra - clump @xmath114% of the primary clump in the composite cmd . in individual cmds , the vrc to red clump fraction vrc / rc varies from 6% ( in fields f1 and f3 ) to 15% in field f2 . the supra - clump to red clump fraction sc / rc varies from 12% to 16% . slightly different choices for the relevant boxes , for example narrowing them to reduce contamination from stars in other stages of stellar evolution , yield very similar fractions . since our estimate from the composite cmd for the vrc / rc fraction has an uncertainty of @xmath28 from counting statistics alone , it is consistent with the zl estimate of @xmath29 for the relative projected angular surface density of the vrc . to summarize , the characteristics that we measure for vertical extension to the red clump and the redder supra - clump peak are identical within the uncertainties to those found by zl and the macho team respectively . we therefore identify the vertical feature seen in our data with the vrc discussed by zl and the redder supra - clump with the overdensity discussed by alcock in the following section we propose stellar evolutionary origins for each of these features . the position of red clump stars in a color - magnitude diagram depends on their mass , and thus their age , as they pass through this evolutionary stage . using the isochrones of bertelli ( 1994 ) , we plot in fig . 3 the mean locus of the core helium - burning phase for a variety of stellar ages in a @xmath30 cmd . this time - averaged locus marks where stars in core helium - burning phase are likely to be found . we have used tracks with metallicity and helium abundance appropriate to the lmc , namely @xmath31 and @xmath32 . note that clump stars with ages in the range @xmath33 ( 0.4 to 1 gyr ) exhibit color changes smaller than @xmath34mag in @xmath35 while differing in @xmath18 magnitude by 0.79 mag . we now compare these theoretical expectations with the vrc in our own lmc data . we begin by focusing on the field f2 , since the low level of crowding in this outer field has resulted in the best photometry of our four fields . the luminosity function for main sequence stars in this field peaks at about @xmath36 ; in the bar fields this occurs about 0.5 mag sooner . the @xmath30 cmd for field f2 , dereddened by @xmath37 is shown in the upper panel of fig . 4 , for two different choices for the lmc distance modulus . overplotted are theoretical isochrones from bertelli ( 1994 ) computed with new radiative opacities ( opal ) for metallicities appropriate to the lmc ( @xmath10 and @xmath11 ) and ages corresponding to 0.25 , 0.40 , 0.63 , 1.0 and 2.5 gyr ( log(age ) = 8.4 , 8.6 , 8.8 , 9.0 and 9.4 ) . the lower panel enlarges the region of the cmd near the red clump region , which is now plotted as contours under the mean locus of the core he - burning phase from fig . both the fit to the main sequence and the red clump is significantly improved for the smaller distance modulus of @xmath0 . since this improvement was apparent in all our fields , we fixed @xmath38 at this value . using the same isochrones from bertelli ( 1994 ) , zl concluded that stars of age 2.5 gyr provided the best fit to the red clump morphology seen in their lmc data . if , following zl , we do not apply an extinction correction , the 2.5 gyr isochrone does indeed provide a plausible fit to the red giant branch . with our extinction correction , however , this isochrone falls at the very reddest ( or oldest ) edge of the red clump in field f2 . as fig . 5 illustrates , this is true for all of our fields . note that if we had applied only foreground extinction corrections due to the milky way itself , the discrepancy with the 2.5 gyr isochrone would still be present : for two fields the internal extinction ( as determined by our main sequence fitting ) is negligible , for two others @xmath39 is increased by only 25 - 33% . furthermore , younger isochrones fit the red clump of each field similarly despite the fact that different external extinction corrections ( based on estimates from and iras data ) were made . we take this as an indication that these extinction corrections are reasonable and necessary for the proper interpretation of the stellar evolution . the he - core burning phase is associated with the horizontal branch in old , metal - poor globulars , but in systems such as the lmc containing stars with a variety of ages and metallicities the horizontal branch becomes blurred into the red clump in cmd . furthermore , as fig . 3 makes clear , the horizontal branch actually becomes _ vertical _ for core he - burning stars of intermediate masses and ages between @xmath10.4 and @xmath11.0 gyr . stars of these masses experience the well - known `` blue loops '' ( see sweigart 1987 , chiosi 1992 ) that are caused by the increasing temperature of the outward expanding h - burning shell as the he - burning core gains mass . when the hydrogen in the shell is exhausted and helium begins to burn in the shell , the star moves redward again in the cmd to quickly join the asymptotic giant branch ( agb ) . while burning helium in their cores and hydrogen in their envelopes , stars spend most of their time near the bluest end of the blue loops . the position of the most blueward extension of the loops for @xmath40 stars differs substantially in luminosity , but very little in color ( fagotto 1994 ) ; in cmd stars of these masses thus create a vertical extension brightward of the blue end of the red clump . as can be seen in fig . 5 , all of our fields contain main sequence stars as young as 250 myr . since their lifetime on the main sequence is relatively short , intermediate mass stars should also be passing through the core he - burning phase . indeed , stars with ages between @xmath10.4 0.8@xmath41gyr lie on a sequence of blue loops whose densely populated blue edge corresponds to the position of the vrc . this is demonstrated clearly in fig . 6 , where the time - weighted centroid of core he - burning stars from fig . 3 is overplotted on contours of the stellar density for each of our fields . the locus of core he - burning stars agrees with the position of the primary red clump and the vertical extension . the agreement is remarkable , especially considering that we have not adjusted the metallicity of the isochrones to achieve a better fit . we therefore identify the vrc with intermediate mass stars in the lmc currently undergoing core he - burning . in section 2.3 we show that the relative density of stars in the vrc is also consistent with a stellar evolutionary origin . the spatial coincidence of the red and asymptotic giant branches with the supra - clump seen both in our data and in the composite cmd of the macho database ( alcock 1997b ) suggests that this second substructure in the cmd may be associated with giant evolution . indeed , zaritsky & lin ( 1997 ) identify the supra - clump with the so - called `` red giant branch bump '' ( rgbb , see rood 1972 , sweigart , greggio & renzini 1989 , and fusi pecci 1990 ) . during the red giant branch ( rgb ) phase , color and luminosity evolution pauses as the h - burning shell passes through a discontinuity left by the maximum penetration of the convective envelope . this pause in the ascension results in an overdensity along the rgb . using an lmc distance modulus of 18.3 , the apparent magnitude of rgbb stars of 2@xmath42 and lmc metallicities ( sweigart , greggio & renzini 1989 ) agrees with the position and extent of the supra - clump in our data . the same is true using the parameterized models of fusi pecci ( 1990 ) . the position of the rgbb is however quite sensitive to stellar age ; older stars be fainter when in the rgbb phase . the mean age ( and metallicity ) of the stars will thus determine the position of the rgbb for a mixed stellar population . recent simulations based on isochrones from opal opacities indicate that the rgbb may lie at the same magnitude as the red clump itself in the lmc ( gallart & bertelli 1998 ) and thus significantly below the position of the observed supra - clump . an alternative explanation for the supra - clump is offered by these simulations : a stalling of the evolution at the base of the asymptotic giant branch ( agb ) as the star struggles to reach thermal equilibrium after the helium exhaustion in the core ( gallart 1998 ) . this early phase of agb evolution creates an over - density or bump ( denoted the agbb by gallart ) in the cmd since subsequent evolution proceeds much faster along the agb . both secondary features in the cmd are thus explained by stellar evolution : the vrc by young stars of intermediate - mass experiencing core he - burning and the supra - clump by a relatively long - lived phase of he - core exhausted stars at the base of the asymptotic giant branch . the position of the vertical extension to the red clump corresponds to the end of the blue loops experienced by intermediate age stars in their he - core burning phase . this argues against the need for an intervening population to explain the vrc , but is the relative stellar density of the vrc consistent with that expected for a stellar population with the star formation history of the lmc ? comparisons of the stellar density in different parts of the cmd must be done with extreme care due to the selection effects induced by crowding . in addition , comparing regions of the cmd that vary significantly in age increases the chances of error due to uncertainties in the star formation history and the increase of metallicity with decreasing age . for example , comparing the stellar density of the vrc to that of the primary clump not only suffers confusion from the overlapping red giant branch , but also from the tracks of low metallicity old ( low mass ) clump stars that intersect those of higher metallicity younger ( higher mass ) clump stars . for these all reasons , we choose to compare the stellar density of stars in the vrc with stars on the main sequence of comparable age and brightness . our goal is to determine whether simple , reasonable assumptions about the imf and star formation history of the lmc can reproduce not only the position of the vrc , but its relative stellar density as well . detailed models for the stellar density in a particular region of the cmd must combine stellar evolution with assumptions about the star formation history and the initial mass function ( imf ) of the population . the lmc is generally believed to be forming stars continuously and to have undergone more star formation in the last few gyr than previously ( see olszewski , suntzeff & mateo 1996 for a review ) , both on the basis of ground - based studies ( bertelli 1994 , girardi 1995 , and vallenari 1996 ) and deeper studies using hubble space telescope ( hst ) cmd ( gallagher 1996 , holtzman 1997 , elson 1997 , and geha 1997 ) . as fig . 7 demonstrates , in each of our four fields , a constant star formation rate with a salpeter imf ( salpeter 1955 ) described by @xmath43 with @xmath44 provides a good approximation to the relative densities of stars on the main sequence down to the completeness limit of @xmath118.5 19 . field f4 appears to contain a slight excess of very bright main sequence stars ( fig . 5 & 7 ) , which may be indicative of a recent burst of star formation activity . these results are consistent with ground - based and deeper hst studies of the lmc , although most recent work indicates that a slightly steeper imf provides a better match to the luminosity function at the expense of some relative density ratios ( @xcite , @xcite , @xcite ) . for the order of magnitude estimates presented here , we will therefore assume that star formation has proceeded at some relatively constant and arbitrary rate between the epochs @xmath45 ( @xmath46 gyr ) , with @xmath47 . because we will count stars on the main sequence and in the vrc corresponding to the same initial mass , our density analysis is actually independent of the imf as long as it remains unchanged over this same period . only when we examine the effects of crowding will slope @xmath48 of the imf play a role . according to the models of bertelli ( 1994 ) , stars with initial masses between 2.6 and 3.0@xmath42 will be @xmath1500 myr old in the red clump where they will have the colors and magnitudes that place them in the center of the vrc . regardless of its slope , if the imf is constant over the last gyr in the lmc , then the ratio of density of these stars in the vrc to their density on the main sequence will depend only on the ratio of lifetimes in these two phases . here , we define the lifetime in the vrc to be the length of time that these @xmath49 stars burn helium in their cores and have @xmath50 . we use the evolutionary models of fagotto ( 1994 ) to estimate the lifetime on the main sequence , which we define as the time during which the fraction of hydrogen in the core is above 0.1% . this definition is operationally convenient since it allows us to count all stars blueward of the subgiant branch . with these definitions , the ratio of vrc to main sequence lifetimes and thus the ratio of corresponding vrc to main sequence counts is 23% . using these definitions , stars of initial mass in the range @xmath49 have @xmath51 and @xmath52 on the main sequence , and @xmath53 in the vrc where their colors are defined as above . counts in these regions of the cmd yield vrc to main sequence fractions of 98% , 42% , 61% , and 46% for fields f1 through f4 , respectively . note that the bar fields appear to have the highest vrc fraction . however , since all of our fields are complete at the faintest end of the vrc , but none are complete at @xmath54 on the main sequence , these ratios are overestimated and must be corrected for incompleteness . to estimate the completeness of each field , we compare the actual number of stars found in the main sequence bin to that predicted by a salpeter imf normalized to provide good fits to the brighter portion of the main sequence histograms . ( for this purpose , main sequence was defined empirically to mean all stars with @xmath55 . ) the estimates of the completeness _ c@xmath56 _ at magnitudes where @xmath49 main sequence stars would be found are given in fig . 7 for each field ; they range from 42% to 61% . it is difficult to estimate the uncertainty in these completeness estimates , but they are more dominated by counting statistics , which are typically on the order of 15% , than by choice of bins . the resulting vrc to main sequence fractions , corrected for incompleteness , are 59% , 25% , 27% and 19% respectively ; all except bar field f1 are in 1.5@xmath57 agreement with the prediction of 23% based only on the assumption that the star formation rate and imf slope have remained constant over the last 1 gyr in the lmc . the composite completeness - corrected vrc to main sequence ratio is 30% ; if the anomalously high field f1 is excluded , this drops to 25% . the completeness correction assumed a salpeter slope of @xmath44 . a steeper imf would place more stars on the fainter part of the main sequence and would thus result in larger completeness corrections and smaller vrc ratios . the converse is true for shallow imf slopes . how well does stellar evolution fare at predicting the 23% vrc to main sequence ratios for @xmath49 stars if the imf is modified ? redoing the completeness analysis with a shallow imf slope of @xmath58 yields a composite vrc to main sequence fraction of 38% . a steeper slope of @xmath59 , as preferred by many lmc luminosity function studies ( vallenari 1996 , and references therein , holtzman 1997 , and references therein ) , yields a smaller value of 21% , in excellent agreement with the predicted 23% . finally , note that stars with masses of @xmath14 @xmath42 , corresponding to ages of 200 myr in the core he - burning phase evolve so quickly that they would not be expected to be detected in the clump region of the cmd . this may explain the upper magnitude cutoff of the vrc . in conclusion , stellar evolution combined with a salpeter imf and constant star formation over the last 1 gyr can account for @xmath175% of the observed stellar density in the vrc . for the somewhat steeper slopes preferred by most recent lmc studies , the fraction rises to 100% . given the simplicity of the assumptions , counting procedure and incompleteness corrections , this agreement is remarkable and leaves little room for an intervening population of non - lmc stars in this region of the lmc color - magnitude diagram . the parallaxes obtained by the hipparcos mission have allowed the determination of distances to stars brighter @xmath60 with accuracies on the order of 10% ( perryman 1997 ) . this has resulted in a more accurately - determined color - absolute magnitude diagram for stars within @xmath1100 pc than has heretofore been possible , resulting in the first clear detection of `` clump giants '' in the solar neighborhood ( perryman 1995 ) . paczyski & stanek ( 1998 ) conclude from their analysis that the hipparcos red clump has a mean distance of 105 pc and a geometrical mean distance of 98 pc , making it unlikely that extinction significantly affects the magnitude or color of the clump . in fig . 8 , we show the color - absolute magnitude diagram for 16229 stars from the hipparcos catalog with relative distances determined to within 10% and colors determined to within 2.5% ( perryman 1995 , and references therein ) . the red clump is clearly visible as a highly concentrated collection of stars with a density peaking near @xmath61 and @xmath62 . the fainter , redward tail of the clump seen most clearly in the contours may be due to the evolving core he - burning stars overlapping the giant branch ( pre - helium flash giants ) . also visible in the hipparcos cmd is a vertical extension of the red clump toward brighter magnitudes ; its color is indistinguishable from that of the peak of the red clump . centered near @xmath63 , this feature is clearly discernible by eye at @xmath64 and @xmath65 . its presence and location is robust to a variety of different choices for the smoothing and contouring of the cmd . a second overdensity at similar magnitudes ( @xmath66 ) but redder colors ( @xmath67 ) is also apparent , and may correspond to the position of the agbb . the isochrones overplotted in the upper panel of fig . 8 assume a solar metallicity of @xmath68 for stars of ages 0.4 , 1.0 , 2.5 , 6.3 and 10.0 gyr . it thus appears that the solar neighborhood has undergone some star formation in the last 400 myr , but that the bulk of the stars in the red clump in the solar neighborhood are older than those of the lmc . the horizontal extent of the hipparcos red clump is due to older stars that begin burning helium in their cores at fainter , redder positions in the cmd and experience less severe blue loops . in the lower panel of fig . 8 , the area around the clump is enlarged and the stellar density plotted as contours . the vrc in the solar neighborhood can be seen clearly . both the isochrones overplotted in the upper panel of fig . 8 and the theoretical locus of core he - burning stars ( of appropriate metallicity ) overplotted in the lower panel make it clear that the vrc seen in the hipparcos data is due to younger clump stars with ages between 300 myr and 1 gyr . both our cmd and those of zl exhibit the same vertical extension to the red clump , yet we reach different conclusions as to the origin of the vrc despite using the same isochrones from bertelli ( 1994 ) with the same metallicity @xmath10 and helium abundance @xmath11 . why do our conclusions differ ? zl choose an isochrone with an age of log(age)=9.4 as the best match to the red giant morphology of their un - dereddened cmd . they then note that although younger isochrones with log(age)=8.6 can reproduce the increased luminosity of 0.9 mag observed in the vrc , the evolutionary models of bertelli predict a difference in color of the mean locus of the core he - burning phase of @xmath69mag , @xmath70 mag between these two isochrones . since they observe a color difference @xmath71 in @xmath72 between the vrc and the centroid of the red clump , they dismiss stellar evolution as the origin of the vrc . they further note that the maximum change in luminosity predicted by sweigart ( 1987 ) for stars evolving in the clump is 0.6 mag compared with the 0.9 mag that they observe in their cmd . if the cmd are first dereddened before comparing to model isochrones , and the lmc is assumed not to be coeval but to have resulted from star formation that has been relatively constant over the last few gyr , one reaches a different conclusion . fig . 3 shows how the mean @xmath18 magnitude of the core he - burning clump varies as a function of its @xmath73 color for stars of different ages using the models of bertelli ( 1994 ) . indeed , as noted by zl , a measurable color difference is expected between stars of log(age)=8.6 and log(age)=9.4 . however , as demonstrated by our figs . 4 and 5 , stars as old as 2.5 gyr ( log(age)=9.4 ) do not fit the dereddened red clump at all , and for some of our fields appear to be too red to fit the bulk of red giant branch as well . dereddening by even 0.12 mag ( comparable to the average _ foreground _ reddening of our lmc fields ) has a dramatic effect on the conclusions since this shifts the mean age of the clump to younger ages for which the luminosity of the blue loops is a very strong function of age , while the color does not change appreciably . thus , allowing for a small range in the ages of young stars in the red clump is sufficient to reproduce the vertical extent of the vrc without significantly modifying its color . note that this color insensitivity of the blue loops to age for stars between about 400 myr and 1 gyr is also true for the color defined by zl ( @xmath74 , as shown in fig . 9 for regions of the cmd comparable to that shown by zl in their fig . 1 . zaritsky & lin note that if the foreground population hypothesis is correct for the origin of the vrc , then the entire lmc cmd should show traces of a parallel cmd shifted by 0.9 mag . they can find no such traces in their own data , but do point to the hst color - magnitude diagram of holtzman ( 1997 ) as corroborating evidence that such a parallel feature may be present in the lower main sequence of the lmc . they rightly note that binary stars would be a natural explanation for this excess of stars displaced by about @xmath75 mag from the primary main sequence , but dismiss such an explanation as unappealing since it can not simultaneously explain the vrc , where fine - tuning would be required to ensure that both members of the binary arrive in this region of the cmd at the same time . zl conclude that if binaries are invoked as an explanation for the excess population in the lower main sequence , another explanation is necessary for the vrc . while we make no claims as to the origin of a parallel main sequence in the lmc , we do proffer stellar evolution as the required alternate explanation for the vrc . using bvr color magnitude diagrams obtained in four different regions in the lmc , we confirm the presence of a vertical extension of the red clump having the same color as the clump peak , but extending to brighter magnitudes , as first mentioned by zaritsky & lin ( 1997 ) . unlike zl , however , we conclude that this feature is due to stellar evolution , not a foreground population . our argument is based on noting that the feature ( 1 ) is present in all our lmc fields , ( 2 ) is in precise agreement with the time - averaged locus the blue loops that represent a relatively long - lived phase in the stellar evolution of younger core he - burning stars in the clump , ( 3 ) has a relative stellar density consistent with the density on the main sequence and the assumption of continuous and constant star formation history in the lmc over the past 1 gyr , and ( 4 ) is present in the solar neighborhood as demonstrated by the hipparcos color - magnitude diagram . the difference in our conclusions despite using the same model isochrones ( bertelli 1994 ) rests on differences in our reddening corrections and assumptions about the ages of lmc stars . zl perform no reddening correction , whereas we correct for reddening using well - determined foreground estimates with a small correction for internal reddening based on main sequence fitting . this results in an average inferred @xmath76 , only 0.02 mag greater than the average deduced for foreground extinction alone . zl consider a single age of 2.5 gyr for the clump with one possible burst at 400 myr ; we consider a population more distributed in age and weighted toward younger ages than zl . an isochrone of 2.5 gyr does not provide a good fit to the red clump in any of our fields , whereas the presence of younger stars in all our lmc fields is clearly demonstrated by comparison of isochrones with the position of the clump and the stellar density of the upper main sequence . we find that stars of ages @xmath77 gyr are responsible for nearly the full extent of the vrc . the best fit to both the main sequence and red clump of our cmd require a distance modulus to the lmc of @xmath0 , assuming that the agreement of evolutionary models to the cmd of solar neighborhood can be taken to imply that no large systematic offset in luminosity is present in the theory . with our justified assumption of continuous and constant star formation over the last gyr in the lmc , current understanding of stellar evolution predicts with precision not only the position of the vrc seen in both our data and those of zl , but also its relative stellar density . the same stellar evolutionary models ( with appropriate assumptions for solar - metallicities ) reproduce the vrc feature in the solar neighborhood cmd measured by the hipparcos satellite . these models when applied to the lmc with a salpeter imf can account for @xmath175% of stellar density of the vrc compared to that of the upper main sequence . a slightly steeper imf slope , such as preferred by many recent luminosity function studies of the lmc , reproduces the observed vrc stellar density exactly . even with the salpeter imf , only one of our four fields ( f1 ) contains vrc stars 1.5@xmath57 in excess of expectations from stellar evolution and simple assumptions for the star formation history of the lmc . since microlensing events are currently being reported both in the bar and outer regions of the lmc , the 4@xmath57 discrepant bar field alone can not contain a significant fraction of the microlensing optical depth to the lmc in the form of intervening stellar population . indeed three of our four fields ( f1 , f2 and f3 ) contain a known microlensing candidate and thus have non - negligible optical depth to microlensing . we conclude that no foreground population need be invoked to explain the presence of the vertical extension to the red clump in the lmc . if present , such a population is unlikely to account for more than @xmath125% of the vrc since the presence of intermediate mass lmc stars in the vrc are _ required _ by stellar evolutionary models in the observed numbers . whatever the primary source of the measured microlensing optical depth toward the lmc , it is unlikely to be due to a new foreground population that has made its presence evident in this vertical extension of the red clump . we thank konrad kuijken for useful discussion and a prompt yet careful reading of the manuscript . we are grateful to andrew cole for pointing out an error in an early version of this work and to carme gallart and alvio renzini for discussions about the agb . we also thank eric maurice and louis prvot for advice on calibration procedures and richard naber for with subtleties . this work is based on observations carried out at the european southern observatory , la silla . work by pds while at the institute for advanced study in princeton was supported by the nsf grant ast- 92 - 15485 . note added in proof : the red clump is an increasingly popular distance indicator ; two recent studies also find a smaller lmc distance ( udalski 1998 , stanek 1998 ) . alcock ( the macho collaboration ) 1997a , , 486 , 697 alcock ( the macho collaboration ) 1997b , , 490 , l59 beaulieu j.p . 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[ geha 1997]geha97 geha , m.c . 1997 , preprint , astro - ph/9711144 girardi , l. , chiosi , c. , bertelli , g. & bressan , a. 1995 , , 298 , 87 gould , a. 1997 , preprint , astro - ph/9709263 [ graham 1982]gra82 graham j.a . 1982 , , 94 , 244 [ holtzman 1997]hol97 holtzman , j. 1997 , aj , 113 , 656 [ jimenez , flynn & kotoneva 1997]jim97 jimenez , r. , flynn , c. & kotoneva , e. 1997 , submitted to , also astro - ph/9709056 [ johnston 1998]kvj98 johnston , k.v . 1998 , , 495 , 297 minniti d. 1997 , in the proceedings of the uc santa cruz conference on _ galactic halos _ , ed . d. zaritsky , to appear in olszewski , e.w . , suntzeff , n.b . & mateo , m. 1996 , , 24 , 511 paczyski , b. & stanek , k.z . 1998 , , 494l , 219 perryman 1995 , , 304 , 69 perryman 1997 , , 323 , l49 renault c. ( the eros collaboration ) 1997 , , 324 , l69 renzini , a. & fusi pecci , f. 1988 , , 26 , 199 rood , r.t . 1972 , 177 , 681 salpeter , e.e . 1955 , , 121 , 161 schechter p. , mateo m. & saha a , 1993 , , 105 , 1342 schwering , p.b.w . & israel , f. 1991 , , 246 , 231 stanek , k.z . , zaritsky , d. & harris , j. 1998 , submitted to , astro - ph/9803181 . sweigart , a.v . , greggio , l. & renzini , a. 1989 , , 69 , 911 sweigart , a.v . 1987 , , 65 , 95 udalski a. , szymanski , m. , kubiak , m. , pietrzynski , g. , woziak , p. & zebrun , k. 1998 , acta astron . , in press , astro - ph/9803035 . vallenari a. , 1996 , , 309 , 367 [ vigneau & azzopardi 1982]va82 vigneau j. & azzopardi m. 1982 , , 50 , 119 zaritsky , d. & lin , d.n.c . 1997 , , 114 , 2545 ( zl ) zhao , h .- s . 1996 preprint , submitted to , and astro - ph/9606166 zhao , h .- s . 1998 , , 294 , 139 fig . 1 calibrated , composite @xmath22 color - magnitude diagrams for all four lmc fields described in the text . only the 16,139 stars with cosmetically superior point spread functions ( type 1 ) in @xmath18 and @xmath4 are plotted . both uncorrected ( left ) and dereddened ( right ) cmd are shown . positions of the red clump ( rc ) , vertical extension to the red clump ( vrc ) , and the supra - clump are indicated . calibrated , dereddened @xmath22 color - magnitude diagrams for the individual four lmc fields . only stars with cosmetically superior point spread functions in @xmath18 and @xmath4 are plotted . the number of stars plotted is 3614 , 4241 , 5591 , 2693 in @xmath21 for fields f1 through f4 respectively . centroid of the @xmath18-band luminosity of the core he - burning phase as a function of @xmath73 for stars of differing ages with lmc metallicity and helium abundance of z=0.008 and y = 0.25 ( bertelli 1994 ) . note that no significant color change occurs for stars with ages between 400 myr and 600 myr ; these stars correspond to the bulk vrc in the lmc . a similar sequence for z=0.02 and y=0.28 ( appropriate to the solar neighborhood ) yields a @xmath78 color redder by @xmath79mag and is shown in fig . 8 . in the solar neighborhood , older stars are also present , causing the horizontal dispersion in color on the fainter edge of the red clump in the hipparcos data . fig . 4 * top panel : * calibrated , dereddened @xmath30 color - magnitude diagram for our least crowded field f2 . 4666 stars with cosmetically superior point spread functions in @xmath80 and @xmath18 are shown . isochrones with lmc metallicity ( @xmath10 ) and helium abundance ( @xmath11 ) with ages @xmath818.4 , 8.6 , 8.8 , 9.0 , and 9.4 from bertelli ( 1994 ) are shown superposed for two different assumptions for the distance modulus , @xmath38 to the lmc . * bottom panel : * contour representation of the cmd density for field f2 in region of the red clump shown superposed on the mean locus of the core helium burning phase as a function of age as in fig . separate scales for the abscissa of the top and bottom panels are indicated . the region of cmd shown has been divided into 25 bins along both the magnitude and color axes . for this binning , contour levels displayed are 12 , 10 , 8 , 6 , 4 and 2.5 stars . a shorter distance modulus of @xmath0 provides improved fits to the main sequence and red clump . same as the bottom panel of fig 4 , but for all of our lmc fields . the region of cmd shown has been divided into 25 bins along both the magnitude and color axes . for this binning , contour levels displayed are 25 , 20 , 15 , 12 , 10 , 8 , 6 , 4 and 2.5 stars . the highest contour levels are not always present in the outer fields . 7 histograms of main sequence stars for each of our fields fields are show on a logarithmic scale . the number of stars in each magnitude bin expected from a salpeter imf , taking into account the main sequence lifetime and assumed a constant star formation rate over the last 1 gyr , is shown as the solid sloping line . the normalization has been adjusted to achieve a good fit to the bulk of the main sequence stars between @xmath82 . the faintest vrc and main sequence stars used for the counting arguments of 2.3.2 are indicated by the vertical lines at @xmath8318.5 and 19.5 , respectively . the main sequence counts clearly suffer from some incompleteness , whereas the vrc counts do not . comparison with expectations from the salpeter imf produce the main sequence completeness estimates _ c@xmath56 _ indicated for each field . fig . 8 * top panel : * hipparcos @xmath30 color - magnitude diagram containing over 16000 stars with accurate distances and photometry . a vertical extension can be seen clearly on the blueward side of the red clump with nearly the same color as the peak clump density and extending about a magnitude brighter . the redder supra - clump is also visible . isochrones with solar metallicity ( z = 0.02 ) and @xmath84 8.6 , 9.0 , 9.4 , 9.8 and 10.0 are shown overplotted . note that positions of the blue loops again coincide with the vertical extension . * bottom panel : * contour representation of the cmd density in region of the red clump for the solar neighborhood as measured by hipparcos is shown superposed on the mean locus of the core he - burning phase as a function of age for stars with metallicity and helium abundance appropriate to the solar neighborhood ( @xmath85 and @xmath86 ) . separate scales for the abscissa of the top and bottom panels are indicated . the region of cmd shown has been divided into 20 bins along both the magnitude and color axes . for this binning , contour levels displayed are 35 , 25 , 15 , 10 , 5 and 2.5 stars . fig . 9 the same isochrones of lmc metallicity from fig . 3 are plotted in color - magnitude diagrams for the region of the red clump . the color @xmath74 is that used by zaritsky & lin ( 1997 ) as are the bands on the vertical axis : u ( top ) , b ( middle ) , and i ( bottom ) respectively . reddening vectors assuming an average @xmath87 are shown in each panel . the blue loops correspond with the feature seen by zl at a dereddened color of @xmath88 .

we examine the morphology of the color - magnitude diagram ( cmd ) for core helium - burning ( red clump ) stars to test the recent suggestion by zaritsky & lin ( 1997 ) that an extension of the red clump in the large magellanic cloud ( lmc ) toward brighter magnitudes is due to an intervening population of stars that is responsible for a significant fraction of the observed microlensing toward the lmc . using our own ccd photometry of several fields across the lmc , we confirm the presence of this additional red clump feature , but conclude that it is caused by stellar evolution rather than a foreground population . we do this by demonstrating that the feature ( 1 ) is present in all our lmc fields , ( 2 ) is in precise agreement with the location of the blue loops in the isochrones of intermediate age red clump stars with the metallicity and age of the lmc , ( 3 ) has a relative density consistent with stellar evolution and lmc star formation history , and ( 4 ) is present in the hipparcos cmd for the solar neighborhood where an intervening population can not be invoked . assuming there is no systematic shift in the model isochrones , which fit the hipparcos data in detail , a distance modulus of @xmath0 provides the best fit to our dereddened cmd . _ accepted for publication in the astronomical journal _

arxiv

it is well - established that the fraction of star - forming galaxies declines as a function of increasing local galaxy density in the low redshift universe . also known as the star formation - density relation @xcite , this correlation has been confirmed in many studies , primarily using optical and uv data to trace star formation in massive galaxy clusters and field environments . mid - infrared data from the infrared satellite observatory ( iso ) and the multi- band imaging photometer for spitzer ( mips ) have also revealed the presence of highly obscured , dusty star forming galaxies , previously undetected by optical or uv surveys ( e.g. * ? ? ? * ; * ? ? ? * ) . while the sensitivity of mips has enabled detailed studies of obscured star formation in individual local and distant galaxy clusters ( e.g. * * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , there are still only a small number of low redshift clusters that have been systematically surveyed for dusty star - forming galaxies out to the virial radius . there remain many uncertainties in the relationship between star formation in clusters and their global cluster properties . in particular , several studies have tried to understand the correlation between cluster mass and the mass - normalized cluster star formation rate ( sfr ) . while results from @xcite suggest that there is no strong correlation between cluster specific sfr and cluster mass , others such as @xcite , @xcite , and @xcite argue that cluster specific sfr decreases with cluster mass . the large spatial coverage required to observe dusty star - forming galaxies in low redshift clusters out to the cluster infall regions has thus far hindered our ability to understand how star formation is affected by global cluster properties such as cluster mass . in this paper we exploit data from the wide - field infrared survey explorer ( wise ; * ? ? ? * ) to overcome this observational challenge and present results on obscured star formation and how it relates to cluster mass and radius out to 3 in a sample of 69 clusters at @xmath1 . and are commonly used interchangeably with virial radius and total cluster mass , respectively . is the radius within which the average density is 200 times the critical density of the universe and is the mass enclosed within that radius . wise is a medium - class explorer mission funded by nasa and has completed observations of the entire sky in four infrared bands : 3.4 , 4.6 , 12 , and 22 ( w1 to w4 , respectively ) . wise scanned the sky with 8.8 second exposures , each with a 47 arcmin field of view , providing at least eight exposures per position on the ecliptic and increasing depth towards the ecliptic poles . the individual frames were combined into coadded images with a pixel scale of 1.375 arcsec per pixel . cosmic - rays and other transient features were removed via outlier pixel rejection . the photometry used for our analyses is point spread function ( psf ) fitted magnitudes from the `` first - pass operations coadd source working database '' created by the wise data reduction pipeline . galaxies in our cluster sample have a diffraction limited resolution of @xmath2 ( full width half maximum ) in the 22@xmath0 m band . we have confirmed from w4 coadded images that all star - forming galaxies considered in our analyses appear unresolved in the 22 band , and have psf photometry reduced @xmath3 values less than 1.5 . therefore we use the psf magnitudes from the first - pass photometric catalog to obtain estimates of total flux . for the minimum coverage of 8 overlapping frames , the sensitivity for @xmath4 in the w4 band is 6 mjy , including uncertainty due to source confusion @xcite . to ensure an unbiased comparison of global sfrs and total ir luminosities of clusters at different redshifts , we impose a lower limit of sfr=4.6 on our entire cluster sample , which is equivalent to a total ir luminosity of @xmath5 , and corresponds to the 6 mjy flux limit at @xmath6 . we hereafter refer to our sample of star - forming galaxies as demi - lirgs , which have nearly half the total ir luminosity of a luminous infrared galaxy or lirg . however we note for future extragalactic studies using wise data that most coadded observations will have at least 12 overlapping frames and hence better sensitivity than the conservative 6 mjy limit we adopt in this paper . additional information regarding wise data processing is available from the preliminary data release explanatory supplement . we use the cluster infall regions ( cirs ; rines & diaferio 2006 ) sample because it provides high - fidelity mass estimates , is at sufficiently low redshift to enable detection with wise of strongly star - forming galaxies , and has extensive spectroscopy for membership determination . the cirs sample consists of 72 low - redshift x - ray galaxy clusters identified from the rosat all - sky survey that are within the spectroscopic footprint of sdss data release 4 . the redshift range of the cirs clusters is @xmath7 , with a median of @xmath8 . cluster masses are available from @xcite , who utilize the caustics infall pattern to determine total dynamical cluster mass and @xcite . the clusters in this paper consist of the entire cirs sample , excluding three clusters at @xmath9 , which leaves 69 remaining clusters with a minimum redshift of @xmath10 . optical photometric and spectroscopic data are obtained from the sloan digital sky survey data release 7 ( sdss dr7 ) @xcite , which are 90% spectroscopically complete for galaxies with @xmath11 and half - light surface brightness @xmath12 mag arcsec@xmath13 . however , the spectroscopic completeness is lower in high - density regions such as in the core of galaxy clusters , due to fiber collisions . adjacent fibers can not be placed closer than 55 arcsec from each other , which corresponds to a separation of 63 kpc at @xmath8 . to verify that the spectroscopic incompleteness arising from constraints on fiber placement has a negligible effect on radial trends of star formation in clusters , we look at the fraction of w4-bright sources as a function of distance from the cluster center . among the @xmath11 sample , the spectroscopic completeness of w4-bright sources ( w4 @xmath14 ) is @xmath1570% within the central 5 arcmin radial bin , and increases to @xmath1580% in the 5 - 10 arcmin bin . in other words , we are not missing a significant fraction of star - forming galaxies in the cluster core due to fiber collisions . the spectroscopic completeness for all galaxies ( regardless of ir emission ) with @xmath11 shows a similar behavior with distance from the cluster center , and on average reaches at least @xmath1580% completeness beyond the central 5 arcmin radius . since the spectroscopic incompleteness of both w4-bright and optically bright galaxies reaches @xmath1580% beyond the cluster core , our results on radial trends of star formation are negligibly affected by spectroscopic incompleteness . in addition , we are measuring trends on scales of hundreds of kpc , and therefore not significantly affected by small differences in spectroscopic completeness near the cluster core . we use photometric data from sdss dr7 to estimate stellar masses for cluster members , using the tight correlation between stellar mass - to - light ( m / l ) ratio and optical colors , as determined by @xcite . with a sample of more than 10,000 optically bright galaxies ( @xmath16 ) , @xcite construct a grid of stellar population models with a range of metallicities and star formation histories , then compare the best - fit galaxy templates with evolved zero redshift templates to determine present - day m / l ratios . we use the relation between @xmath17 color and @xmath18-band stellar m / l ratio to derive stellar masses for our sample . the estimated uncertainty of the color - based stellar m / l ratios , including random and systematic errors , is @xmath1545% @xcite . spectroscopic redshifts from sdss dr7 are used to determine membership for each of our clusters . figure [ fig : caustic ] illustrates an example of using the caustics infall method to determine cluster membership for abell 1377 , the most massive cluster in our sample . it shows the difference in radial velocity ( @xmath19 ) of galaxies in the cluster field with respect to the cluster systemic velocity , as a function of projected distance from the cluster center . galaxies that are dynamically bound to the cluster form a well - defined region that decreases in velocity offset as a function of projected radius . we define the galaxies that are within the edge of this envelope and within a projected radius of @xmath203 , as cluster members . while the average turnaround radius for the cirs sample is @xmath155 , we restrict our cluster galaxy sample to within 3 , since cluster infall patterns are generally better defined closer in to the cluster center . only galaxies with sdss spectroscopic redshifts are considered in the following analyses . we also limit all cluster and field galaxies to be brighter than @xmath21 , which corresponds to the 90% spectroscopic completeness limit of @xmath22 at @xmath6 . star formation rates and total infrared luminosities ( ) are determined from the wise 22 photometry , using the relations presented in @xcite that are calibrated for mips 24 data . @xcite constructed model average spectral energy distribution ( sed ) templates from a sample of local lirgs and ulirgs and derive correlations between 24 flux , sfr , and . the flux - sfr relation given in their equation 14 can be used for both mips 24 and wise 22 data @xcite . the similarity of the two bandpasses are confirmed by @xcite who quantify the correlation between and the inferred mips 24 and wise 22 luminosities . the mips and wise luminosities are inferred by applying a color - correction to the 18 akari flux , and is derived from fitting the ir sed templates of @xcite to all six akari bandpasses ( 9 , 18 , 65 , 90 , 140 , and 160 ) for @xmath15600 galaxies at @xmath1 . the resulting correlations between the @xmath23 and @xmath24 are nearly identical , with only a @xmath154% offset . therefore we proceed by using the flux - sfr and @xmath23 relations of @xcite , assuming the calibration determined from mips 24 data . the total error attributed to the sfr is @xmath150.2 dex , which is dominated by scatter in the @xmath25- relation , and does not include uncertainties inherent in the assumed stellar initial mass function ( imf ) . the imf adopted by @xcite is similar to the @xcite and @xcite imfs , which have relatively fewer low mass stars compared to a salpeter imf , and is more applicable for extragalactic star - forming regions @xcite . a robust estimate of the cluster star formation rates requires that we first identify and exclude all 22@xmath0 m sources that are dominated by agn rather than star formation . as demonstrated in @xcite , even a few ir - bright agn can significantly bias the inferred global star formation rate of a cluster . there are several methods by which we can identify agn in our data set . the first method relies upon wise colors . similar to the agn wedge in spitzer / irac data @xcite , agn - dominated sources will have red colors in w1-w2 ( [ 3.4]-[4.6 ] ) . specifically , we use the criteria [email protected] ( vega ) to identify candidate agn in the wise data set . this color selection is similar to , though slightly bluer than , the agn selection determined in jarrett et al . 2011 ( submitted ) and stern et al . 2011 ( in prep ) . while this single color cut is in general less robust than the full agn wedge , at the low redshifts that are the focus of this work , star - forming galaxies should have colors uniformly blueward of this threshold . we illustrate in figure [ fig : bpt ] that our [email protected] criterion works well in selecting out agn from star - forming galaxies . profile - fit photometry is used to determine the w1-w2 colors and agn exclusion . while some of the nearest galaxies will be resolved in these bands , we are interested only in the w1-w2 color , rather than in single - band photometry . comparison of agn selection based on w1-w2 color from aperture photometry and profile - fit photometry show that the two methods are similarly successful in isolating agn . however , the color cut based on profile - fit magnitudes has a slightly better overlap with the optically detected agn , discussed below . a second approach to agn identification is use of optical spectroscopy to identify sources that lie in the agn region of the baldwin - phillips - terlevich ( bpt ) diagram ( bpt ; * ? here we use the bpt diagram as a cross - check on the wise color selection for the subset of sources . figure [ fig : bpt ] shows the emission line ratios of [ oiii]/h@xmath27 and [ sii]/h@xmath28 , for a sample of 136 cluster members with @xmath29 , of which 27 have [email protected] . boundaries from @xcite are shown as dotted lines and separate regions where narrow emission lines arise from the presence of seyferts , liners ( low ionization narrow emission line regions ) , and hii regions ( star - forming galaxies ) . galaxies that are selected as agn candidates based purely on having a red wise color ( [email protected] ) are indicated as filled ( red ) circles . figure [ fig : bpt ] shows that out of 22 optically identified seyferts , fourteen ( 65% ) are also identified as agn based on the wise [email protected] selection . sources flagged as quasars ( or qsos ) in the sdss catalog are highlighted with ( purple ) triangle symbols , with all ten quasars being independently identified as agn using the wise color selection . since the bpt diagnostic is meant to classify galaxies based only on _ narrow _ emission line ratios , it is not surprising that nearly all of the sdss quasars are not properly diagnosed in the bpt diagram . the h@xmath30 and h@xmath31 emission lines in eight out of the ten sdss quasars have broad wings relative to their neighboring [ oiii ] or [ sii ] emission lines . overall , we conclude from figure [ fig : bpt ] that the wise color selection is an effective method of excluding agn from our sample of bright w4 sources , identifying 85% of the optically detected seyferts and quasars . the two central questions that we aim to address are how the mean specific star formation rate of cluster galaxies ( mssfr ) depends upon location within a cluster , and how the total integrated star formation rate of a cluster depends upon cluster mass . specifically , the mssfr is defined as the total star formation rate in cluster galaxies _ at _ a given projected radius as inferred from 22@xmath0 m photometry , divided by the total stellar mass of all cluster galaxies _ at _ that radius . for probing the total integrated star formation rate we also define the integrated cluster quantity @xmath32 which is a useful mass - normalized measure of the total star formation rate within the infall region , and referred to as the cluster specific sfr ( cssfr ) . among the 69 clusters , a total of 136 demi - lirgs are detected within 3 , of which 27 are determined to be agn based on their w1-w2 color . in the following sections , all sfr quantities are determined from the remaining 109 star - forming demi - lirgs . eight seyferts identified from the bpt diagram are included in the cluster star - forming galaxy sample because their w1-w2 colors are not indicative of agn activity , and we prefer to maintain a uniform wise selection of agn among all field and cluster galaxies . we note that all results presented in this paper are negligibly affected by the exclusion / inclusion of these eight seyferts . the coordinates , redshift , w4 magnitude , sfr , , projected distance from cluster center , and agn flags are listed for the 136 demi - lirgs in table 1 . there is a long history in the literature demonstrating the existence of a strong radial dependence for star formation ( or color ) in galaxy clusters ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? the main strengths of the current analysis are the uniform 22@xmath0 m wise data , spectroscopic completeness , and existence of measurements for the full sample , which together provide us with an infrared - based view of star formation for a homogeneous sample extending well beyond the virial radius . a common approach in the literature has been to look at the radial dependence of the fraction of star - forming galaxies . here we investigate both the dependence of the star - forming fraction , and also the mssfr out to 3 . for comparison , the identical quantities for a field population are calculated , with the field sample chosen from a catalog of galaxies located within a projected radius of 5 mpc from the cluster center , at the cluster redshift . among these galaxies , we choose those with a radial velocity greater than 5000 away from the cluster systemic velocity , redshift @xmath1 , and absolute r magnitude @xmath33 . these are the same redshift and magnitude limits of the cluster galaxy sample . the highest velocity dispersion of our cluster sample is @xmath15960 , which means that the chosen field galaxies are more than @xmath34 away from the cluster redshift . figure [ fig : caustic ] shows delta - velocity ( @xmath19 ) versus projected distance from the cluster center for galaxies within a 5 mpc projected radius of abell 119 . the field galaxies indicated with ( blue ) cross symbols are clearly not associated with the cluster galaxies , which appear distinctly confined to a trumpet - shaped region . by gathering field galaxies within a projected 5 mpc radius from 69 different regions of the sky , we have compiled a large enough sample to obtain a representative field value of mssfr and the fraction of star - forming galaxies with @xmath29 . there are a total of 11180 field galaxies , of which 566 are demi - lirgs . figure [ fig : ssfr_radius ] shows the mssfr as a function of r/ for our ensemble of 69 clusters , including star formation only for galaxies with @xmath29 l@xmath35 ( sfr@xmath36 m@xmath35 yr@xmath37 ) . we emphasize that by construction our sfr limit therefore means that the observed mssfr is a lower bound but a consistent lower bound across the sample . as expected , the mean specific sfr increases with projected radius , with the most central bin containing the fewest star - forming demi - lirgs twelve out of a total of 109 demi - lirgs found in 69 clusters . the mssfr displays a steep increase from the central bin to , then continues to increase monotonically and nearly flattens out below the field value at larger radii . this is consistent with a low redshift study of h@xmath30 star - forming galaxies by @xcite , who found that the median sfr ( normalized by l@xmath38 ) of cluster galaxies reaches the field value beyond 3 . such a radial trend can be driven by two factors an increase in the ssfr of the sub - population of star - forming galaxies , or an increase in the fraction of star - forming galaxies with radius . we find that there is no statistically significant change in the ssfr of individual star - forming galaxies with radius , implying that the trend is driven primarily by a radial gradient in the star forming fraction . this can be seen in figure [ fig : lirg_fraction ] , where we directly plot the fraction of star - forming galaxies as a function of radius . more than one third of the star - forming galaxies reside at 2@xmath39r@xmath393 . what is striking is that even at these large radii the mssfr and star - forming fraction remain below the field values . in figure [ fig : lirg_fraction ] , the demi - lirg fraction is higher in the field relative to the cluster population at 3 by a factor of @xmath151.5 . one possible interpretation of this result is that these are infalling galaxies that may already be `` pre - processed '' in intermediate - density environments , such as galaxy groups or filaments ( e.g. zabludoff & mulchaey 1998 ) , which would explain why they are suppressed relative to a true field population . there is both observational and theoretical support in favor of the suppressed star - forming galaxy fraction out to 3 . @xcite found that the mean cluster sfr , which they derived from [ oii ] emission of galaxies in 15 clusters , is lower than the mean field sfr by more than a factor of two at 2 . results from @xcite and @xcite similarly conclude that cluster star formation is suppressed at several times the virial radius relative to the field . in addition to galaxies being pre - processed at large radii , another explanation for the suppressed demi - lirg fraction in figure [ fig : lirg_fraction ] could be the contribution of a `` backsplash '' population beyond . simulations from @xcite and @xcite show that up to @xmath1550% of galaxies currently located between and 2 may have previously traveled inward of the virial radius of the cluster . therefore the suppressed demi - lirg fraction at large radii may , to some degree , reflect quenched star formation that occurred within the cluster core . however our galaxy sample , which has an absolute magnitude limit of @xmath33 , is sensitive to massive galaxies brighter than m@xmath40 @xcite , whereas backsplash galaxies are on average significantly less massive than first infall galaxies in the cluster outskirts @xcite . as such , our sample is biased against the lower mass backsplash galaxies relative to first infall galaxies . nonetheless , a contribution of backsplash galaxies may indeed be partially responsible for the suppressed star formation activity at large distances from the cluster core . total mass is , both for astrophysics and cosmology , the most fundamental physical galaxy cluster parameter . there are multiple physical processes that depend on the depth of the cluster potential well ( e.g. ram pressure , harassment , tidal interactions ) which can significantly alter the morphologies , gas content , and star formation rates in cluster galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? furthermore , it has been observed that while the total baryon fraction ( stellar and gas content ) remains roughly constant with cluster mass , the stellar mass decreases and gas content increases in galaxy clusters as a function of cluster mass @xcite . this implies that the integrated star formation efficiency of the history of a cluster is directly tied to cluster mass . in this section we examine the relation between cluster mass and total current cluster star formation to better understand how strongly the present - day star formation rate depends upon cluster mass . our approach is to compute the cluster sfr normalized by cluster mass ( cssfr ; equation 1 ) as a function of cluster mass . for each cluster , the cluster sfr is the sum of sfrs for all member galaxies that have [email protected] and lie within a projected radius of 3 . the left panel of figure [ fig : ssfr_m200 ] shows the cssfr for 62 clusters , which consists of the sample of 69 clusters , excluding seven clusters that have incomplete spatial coverage in the sdss dr7 spectroscopic survey within 3 . the dotted curve shows the limiting detectable cssfr as a function of corresponding to a single demi - lirg with a sfr of 4.6 in a cluster . clusters that have no members above the [email protected] limit are assigned a cssfr of zero . the binned data ( large squares ) indicate the average cssfr in each bin , including clusters with a cssfr of zero . thus the cssfr of the lowest bin is strongly biased by incompleteness , since nearly all of the clusters at low mass fall below the [email protected] limit and have a cluster sfr set to zero . to ensure that our results are not significantly biased by this incompleteness at low cluster mass , we also show the cluster specific sfr versus for a sub - sample of clusters at @xmath41 in the right panel of figure [ fig : ssfr_m200 ] . for these 22 low - z clusters , we are sensitive to star - forming galaxies down to [email protected] , which corresponds to the w4 band 5@xmath42 detection limit of 6 mjy at @xmath8 , and is illustrated with a dotted curve . using the sample of 62 clusters , the left panel of figure [ fig : ssfr_m200 ] shows no significant correlation between cluster specific sfr and cluster mass , over more than one order of magnitude in @xmath43 . the first pass coadd processing used here has the same calibration as in the wise preliminary data release , but was run every other day using only one day of data , creating gaps in coverage at low ecliptic latitude . these gaps should not significantly impact this result , since location of the gaps are random with respect to the cluster centers . we confirm this expectation by examining the cluster specific sfrs of a sub - sample of 24 clusters with ecliptic latitude @xmath44 deg . these clusters are at sufficiently high ecliptic latitude such that they have fairly complete spatial coverage with the first - pass wise coadd data . the average cluster specific sfrs of the high ecliptic latitude sub - sample are overplotted in figure [ fig : ssfr_m200 ] in small purple diamonds . these high ecliptic latitude clusters show no significant correlation between cluster specific sfr and cluster mass , similar to the lack of correlation seen with the full cluster sample . the lack of correlation between cssfr and is also confirmed with the @xmath41 sub - sample , for which we have applied a significantly lower detection limit of [email protected] ( figure [ fig : ssfr_m200 ] , right panel ) . with the lower sfr limit , the clusters in figure [ fig : ssfr_m200 ] have an average of 5.2 star - forming galaxies within r@xmath393 , with a large scatter . five of the low redshift clusters have between 10 to 16 star - forming galaxies . in contrast , for the sample of 62 clusters with the demi - lirg cut ( [email protected] ) , there is an average of 1.6 star - forming galaxies per cluster within r@xmath393 . while the average number of star - forming galaxies per cluster is small in this case , the cssfrs calculated using only the demi - lirgs are well correlated with the cssfrs using the [email protected] cut , with a linear pearson correlation coefficient of 0.9 . therefore , even with an average of only 1.6 star - forming galaxies per cluster , the cssfrs of the full cluster sample are qualitatively representative of cssfrs that sample the fainter end of the infrared luminosity function . the binned data in both panels of figure [ fig : ssfr_m200 ] show cluster specific sfrs that are consistent within 1@xmath42 of each other , across the full range of . the error bars associated with the average cssfr are the poisson uncertainty on the number of star - forming galaxies detected above the sfr limit in each bin . the lack of a significant trend between cssfr and indicates that transformation mechanisms which scale strongly with cluster mass may not play a dominant role in the evolution of star formation in clusters . while figure [ fig : ssfr_m200 ] illustrates the integrated cluster sfr per by taking the sum of all star - forming galaxies within r@xmath393 , our results remain robust when considering only those galaxies within r@xmath39 . since galaxies within are more susceptible to global cluster processes such as ram pressure , the lack of correlation between cssfr ( r@xmath39 ) and cluster mass re - affirms the main conclusion drawn from figure [ fig : ssfr_m200 ] mechanisms that are strongly dependent on cluster mass do not play a significant role in the evolution of star formation in clusters . two cluster processes that strongly scale with cluster mass are galaxy harassment and ram pressure . ram pressure is a mechanism that acts to compress and/or strip cold gas reservoirs in galaxies . the impact of ram pressure is proportional to @xmath45 , where @xmath46 is the density of the intracluster medium ( icm ) and @xmath47 is the velocity of a galaxy with respect to the icm @xcite . galaxy harassment refers to the cumulative effect of high velocity close encounters between galaxies and has the potential to disturb morphologies and quench star formation @xcite . the effects of galaxy harassment , like ram pressure , are expected to scale up with cluster mass , since both cluster velocity dispersion and icm gas density increase with cluster mass . while there has been evidence in support of both ram pressure and harassment having an impact on star formation in individual cluster galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the lack of a correlation between cluster specific sfr and cluster mass in figure [ fig : ssfr_m200 ] indicates that neither of these mechanisms significantly trigger / quench star formation in local galaxy clusters , within the mass range of @xmath48 to @xmath49 . our results are consistent with those of @xcite and @xcite , who use a sample of @xmath15100 clusters at @xmath1 and find no significant correlation between h@xmath30-derived cluster specific sfr and cluster mass . both @xcite and @xcite use sdss dr2 data , with the same limiting r - band magnitude as used in this paper . the limiting h@xmath30 sfr is @xmath153 . similarly , @xcite demonstrate that the passive galaxy fraction is roughly constant as a function of system mass , from @xmath50 to @xmath51 . assuming that the non - passive galaxy fraction is dominated by star - forming galaxies rather than agn , the results of @xcite support our finding that cluster specific sfr is not strongly dependent on cluster mass . in contrast , several other studies have found evidence for an anti - correlation between cluster specific sfr and cluster mass . @xcite and @xcite obtain cluster sfrs by integrating h@xmath30 derived sfrs for members out to 0.5 , and cluster masses from observed velocity dispersions for a sample of eight and nine clusters , respectively . their clusters span an approximate redshift range of 0.2 to 0.8 , and the h@xmath30 sfrs are sensitive to @xmath151 . the anti - correlation observed between cluster specific sfr and cluster mass is significantly weakened when only clusters of the same epoch are considered , indicating that evolutionary effects may be the dominating factor . in addition to interactions between galaxies and the cluster environment , studies have demonstrated that tidal interactions between galaxies or galaxy - galaxy mergers can trigger a burst of star formation ( e.g. * ? ? ? * ; * ? ? ? galaxy - galaxy interactions are generally optimized in intermediate density regions with low velocity dispersions , such as in galaxy groups , whereas the more massive clusters are less likely to host galaxy mergers . figure [ fig : ssfr_m200 ] includes a substantial number of low mass systems ( 57 clusters with @xmath52 ) , yet shows no sign of significantly elevated cluster specific sfr in low mass clusters relative to intermediate mass clusters . any signature of enhanced star formation in low mass clusters due to galaxy - galaxy interactions should be particularly evident in our data , because we integrate the sfr of cluster members out to three times the virial radius , where galaxy - galaxy mergers and tidal interactions are more like to occur . a comparison of the distance to nearest neighbor among the demi - lirg population and the general cluster population as a function of projected radius shows that the two populations have mean nearest neighbor distances that are consistent within 1@xmath42 . this indicates that galaxy - galaxy mergers or close tidal interactions do not play a dominant role in the star formation properties of these clusters . our results imply that even in the low mass @xmath53 groups , star formation has already been quenched to similar levels observed in clusters that are more massive by over an order of magnitude . the lack of correlation between cssfr and ( figure [ fig : ssfr_m200 ] ) , combined with the suppressed demi - lirg fraction at 3 relative to the field ( figure [ fig : lirg_fraction ] , would suggest a scenario in which star formation is quenched in a significant population of galaxies within small groups or filaments . then once the remaining star - forming galaxies are accreted into the cluster environment , a mechanism that is not strongly dependent on cluster mass must operate . we suggest strangulation as a plausible candidate for quenching star formation in cluster star - forming galaxies . as a galaxy enters the cluster icm , its hot halo gas is removed , thereby cutting off its resource for future cold gas supplies , since the halo gas would otherwise eventually cool and settle onto the disk @xcite . the effectiveness of strangulation is less dependent on cluster mass than ram pressure or harassment . simulations from @xcite have shown that strangulation can occur in low mass groups , where the icm - galaxy halo interaction is weak relative to clusters . strangulation has been suggested by several cluster studies as an important cluster mechanism ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , including by @xcite who discovered `` mild gradients '' in the morphological fractions of galaxies in cl0024 + 16 out to large cluster radii , which were best explained by a slow - working gentle mechanism such as strangulation . while strangulation may quench star formation in massive clusters , our results suggest that the dominant process for regulating star formation in dense environments does not depend upon cluster mass , and hence must be efficient in small groups or filaments . we use data from wise and sdss dr7 to study the dependence of star formation on cluster mass and location within a cluster in 69 local ( @xmath1 ) galaxy clusters . cluster membership is determined from sdss dr7 spectroscopic redshifts for galaxies brighter than @xmath21 , and star formation rates are determined with 22 photometry from wise using the relations outlined in @xcite . out of 69 clusters , we find a total of 109 star - forming demi - lirgs with sfr@xmath36 within 3 . both the fraction of demi - lirgs and the mean specific sfr of cluster galaxies increases with projected distance from the cluster center . however , the fraction of demi - lirgs remains below the field value even at 3 times the cluster virial radius . one plausible explanation for the suppressed demi - lirg fraction at 3 is that even galaxies that reside significantly beyond the virial radius have already been quenched in their previous environments , such as small groups or filaments . we also investigate the impact of cluster mass on star formation by presenting the total cluster sfr normalized by as a function of . we find no evidence of a correlation between the cluster specific sfr and cluster mass in this first uniform dataset to detect obscured star formation out to several times the virial radius in a large sample of low redshift clusters . our result indicates that cluster mechanisms which scale with cluster mass , such as ram pressure or harassment , are not likely to play a dominant role in the evolution of star formation in local clusters . this publication makes use of data products from the wide - field infrared survey explorer , which is a joint project of the university of california , los angeles , and the jet propulsion laboratory / california institute of technology , funded by the national aeronautics and space administration . the authors would like to thank emilio donoso for his help and advice on navigating the sdss database . we also thank the anonymous referee for a careful reading and comments which improved the paper . wisepc j235654.30 - 101605.2 & 359.22626 & -10.26812 & 0.074 & 4.12 & 0.025 & 90.3 & 7.2e+11 & 1.1 & agn + wisepc j155850.42 + 272324.5 & 239.71010 & 27.39013 & 0.093 & 4.76 & 0.024 & 86.7 & 7.0e+11 & 1.7 & agn + wisepc j121635.79 + 040709.2 & 184.14912 & 4.11922 & 0.076 & 4.62 & 0.027 & 58.5 & 4.9e+11 & 2.1 & agn + wisepc j130534.25 - 021119.1 & 196.39270 & -2.18864 & 0.088 & 5.41 & 0.039 & 38.8 & 3.4e+11 & 0.2 & agn + wisepc j121742.00 + 034631.2 & 184.42499 & 3.77535 & 0.080 & 5.25 & 0.032 & 35.6 & 3.1e+11 & 1.8 & agn + wisepc j160003.13 + 263707.8 & 240.01305 & 26.61884 & 0.093 & 5.67 & 0.033 & 35.1 & 3.1e+11 & 2.0 & agn + wisepc j111519.95 + 542316.6 & 168.83313 & 54.38794 & 0.070 & 4.98 & 0.026 & 34.2 & 3.0e+11 & 1.5 & agn + wisepc j171031.49 + 643914.6 & 257.63123 & 64.65405 & 0.079 & 5.41 & 0.026 & 29.7 & 2.6e+11 & 0.3 & sf + wisepc j134236.22 + 592324.7 & 205.65091 & 59.39019 & 0.071 & 5.15 & 0.030 & 29.7 & 2.6e+11 & 0.5 & agn + wisepc j170226.20 + 341117.5 & 255.60916 & 34.18821 & 0.105 & 6.16 & 0.035 & 28.9 & 2.6e+11 & 0.1 & agn + wisepc j111448.38 + 401749.3 & 168.70160 & 40.29702 & 0.076 & 5.47 & 0.032 & 25.8 & 2.3e+11 & 1.2 & sf * + wisepc j162021.16 + 301020.7 & 245.08818 & 30.17243 & 0.096 & 6.14 & 0.037 & 23.7 & 2.2e+11 & 2.6 & agn + wisepc j160105.12 + 272539.2 & 240.27135 & 27.42756 & 0.087 & 5.91 & 0.034 & 23.3 & 2.1e+11 & 2.0 & sf + wisepc j170100.39 + 342042.9 & 255.25163 & 34.34524 & 0.101 & 6.27 & 0.039 & 23.1 & 2.1e+11 & 0.2 & sf + wisepc j232514.19 + 151442.1 & 351.30911 & 15.24503 & 0.043 & 4.28 & 0.021 & 21.8 & 2.0e+11 & 0.3 & sf + wisepc j165948.48 + 335944.1 & 254.95201 & 33.99559 & 0.085 & 5.96 & 0.036 & 20.4 & 1.9e+11 & 0.1 & sf + wisepc j152138.79 + 305037.4 & 230.41161 & 30.84372 & 0.081 & 6.07 & 0.040 & 16.2 & 1.5e+11 & 1.0 & sf + wisepc j104319.01 + 050818.0 & 160.82922 & 5.13833 & 0.068 & 5.70 & 0.041 & 15.4 & 1.5e+11 & 0.8 & sf + wisepc j151941.93 + 312905.5 & 229.92471 & 31.48486 & 0.080 & 6.11 & 0.031 & 15.2 & 1.4e+11 & 1.1 & sf + wisepc j221445.86 - 092300.8 & 333.69110 & -9.38356 & 0.082 & 6.24 & 0.051 & 14.2 & 1.4e+11 & 0.4 & agn + wisepc j102200.74 + 382914.4 & 155.50308 & 38.48734 & 0.057 & 5.39 & 0.030 & 14.0 & 1.3e+11 & 0.5 & sf + wisepc j130330.80 - 021400.0 & 195.87834 & -2.23334 & 0.084 & 6.38 & 0.049 & 13.2 & 1.3e+11 & 0.1 & agn + wisepc j152021.87 + 485222.0 & 230.09113 & 48.87279 & 0.074 & 6.09 & 0.033 & 12.8 & 1.2e+11 & 1.5 & sf + wisepc j170957.92 + 335507.3 & 257.49133 & 33.91869 & 0.084 & 6.42 & 0.042 & 12.5 & 1.2e+11 & 0.2 & sf + wisepc j004336.31 - 092547.6 & 10.90130 & -9.42988 & 0.050 & 5.29 & 0.030 & 11.3 & 1.1e+11 & 0.3 & sf + wisepc j155857.04 + 273758.0 & 239.73769 & 27.63278 & 0.090 & 6.70 & 0.056 & 11.2 & 1.1e+11 & 1.7 & sf + wisepc j170231.67 + 335135.6 & 255.63196 & 33.85988 & 0.087 & 6.61 & 0.051 & 11.2 & 1.1e+11 & 2.6 & sf + wisepc j110018.01 + 100256.9 & 165.07504 & 10.04914 & 0.036 & 4.57 & 0.027 & 10.7 & 1.1e+11 & 0.8 & agn + wisepc j101346.84 - 005450.9 & 153.44518 & -0.91414 & 0.042 & 4.96 & 0.028 & 10.6 & 1.0e+11 & 0.7 & agn + wisepc j163032.67 + 392303.2 & 247.63612 & 39.38421 & 0.030 & 4.21 & 0.024 & 10.5 & 1.0e+11 & 2.2 & agn + wisepc j122940.27 + 121743.6 & 187.41780 & 12.29546 & 0.087 & 6.72 & 0.155 & 10.2 & 1.0e+11 & 0.1 & sf + wisepc j114623.86 + 552422.2 & 176.59941 & 55.40616 & 0.053 & 5.53 & 0.024 & 10.0 & 9.9e+10 & 1.7 & sf + wisepc j005656.90 - 011242.4 & 14.23707 & -1.21176 & 0.050 & 5.45 & 0.036 & 9.6 & 9.6e+10 & 0.3 & sf + wisepc j162143.25 + 294332.6 & 245.43019 & 29.72572 & 0.098 & 7.09 & 0.070 & 9.5 & 9.4e+10 & 2.8 & sf + wisepc j102126.42 + 381747.5 & 155.36008 & 38.29652 & 0.055 & 5.72 & 0.038 & 9.1 & 9.1e+10 & 1.2 & sf * + wisepc j165903.80 + 334854.6 & 254.76585 & 33.81516 & 0.086 & 6.81 & 0.062 & 8.9 & 8.8e+10 & 0.2 & sf + wisepc j171447.38 + 643541.1 & 258.69742 & 64.59475 & 0.080 & 6.67 & 0.050 & 8.7 & 8.7e+10 & 0.4 & sf + wisepc j132632.23 + 002800.7 & 201.63428 & 0.46685 & 0.086 & 6.84 & 0.060 & 8.7 & 8.7e+10 & 0.3 & sf + wisepc j075225.73 + 283040.0 & 118.10722 & 28.51112 & 0.062 & 6.08 & 0.044 & 8.5 & 8.5e+10 & 0.3 & agn + wisepc j103342.72 + 392926.9 & 158.42801 & 39.49082 & 0.068 & 6.32 & 0.049 & 8.3 & 8.4e+10 & 1.2 & sf + wisepc j132944.92 - 014239.7 & 202.43716 & -1.71104 & 0.089 & 6.98 & 0.077 & 8.2 & 8.2e+10 & 0.8 & sf + wisepc j162345.89 + 410456.5 & 245.94122 & 41.08235 & 0.034 & 4.70 & 0.016 & 8.1 & 8.1e+10 & 2.1 & agn + wisepc j125830.21 - 015835.4 & 194.62585 & -1.97650 & 0.080 & 6.75 & 0.073 & 8.1 & 8.1e+10 & 0.2 & sf * + wisepc j115720.93 + 051506.0 & 179.33719 & 5.25168 & 0.081 & 6.80 & 0.064 & 8.0 & 8.0e+10 & 1.6 & sf + wisepc j152003.58 + 304350.6 & 230.01491 & 30.73073 & 0.082 & 6.84 & 0.208 & 7.9 & 7.9e+10 & 1.7 & sf + wisepc j155637.07 + 270043.1 & 239.15446 & 27.01196 & 0.091 & 7.11 & 0.079 & 7.7 & 7.7e+10 & 1.6 & sf + wisepc j154351.50 + 363136.9 & 235.96458 & 36.52691 & 0.067 & 6.38 & 0.033 & 7.6 & 7.7e+10 & 1.4 & agn + wisepc j121754.98 + 040117.5 & 184.47910 & 4.02153 & 0.082 & 6.87 & 0.108 & 7.6 & 7.7e+10 & 2.1 & sf + wisepc j102946.81 + 401913.7 & 157.44505 & 40.32048 & 0.067 & 6.39 & 0.047 & 7.6 & 7.6e+10 & 1.3 & agn + wisepc j162637.09 + 390739.3 & 246.65456 & 39.12757 & 0.035 & 4.89 & 0.016 & 7.5 & 7.6e+10 & 2.6 & sf + wisepc j123011.93 + 120632.8 & 187.54971 & 12.10911 & 0.085 & 6.96 & 0.180 & 7.5 & 7.6e+10 & 0.1 & sf + wisepc j112344.58 + 030018.9 & 170.93575 & 3.00525 & 0.051 & 5.75 & 0.036 & 7.5 & 7.5e+10 & 1.8 & sf + wisepc j152255.30 + 305905.4 & 230.73042 & 30.98484 & 0.080 & 6.83 & 0.073 & 7.4 & 7.5e+10 & 1.4 & sf * + wisepc j170102.30 + 340400.7 & 255.25958 & 34.06686 & 0.094 & 7.23 & 0.069 & 7.4 & 7.5e+10 & 0.2 & agn + wisepc j155842.84 + 270736.5 & 239.67848 & 27.12680 & 0.086 & 7.03 & 0.060 & 7.3 & 7.4e+10 & 1.6 & sf + wisepc j115604.23 + 050150.7 & 179.01761 & 5.03075 & 0.079 & 6.81 & 0.075 & 7.3 & 7.4e+10 & 2.1 & sf * + wisepc j152420.43 + 295719.7 & 231.08514 & 29.95546 & 0.076 & 6.72 & 0.049 & 7.3 & 7.3e+10 & 1.5 & sf + wisepc j075213.93 + 292023.5 & 118.05803 & 29.33986 & 0.061 & 6.22 & 0.034 & 7.2 & 7.3e+10 & 0.4 & sf + wisepc j133742.55 + 585209.8 & 204.42731 & 58.86938 & 0.074 & 6.68 & 0.078 & 7.2 & 7.2e+10 & 0.9 & sf + wisepc j130344.14 - 030652.2 & 195.93390 & -3.11451 & 0.081 & 6.91 & 0.081 & 7.1 & 7.2e+10 & 0.4 & sf + wisepc j123018.51 + 113811.4 & 187.57713 & 11.63651 & 0.083 & 6.99 & 0.101 & 7.0 & 7.1e+10 & 2.5 & sf + wisepc j155711.60 + 274455.2 & 239.29832 & 27.74868 & 0.088 & 7.12 & 0.073 & 7.0 & 7.1e+10 & 2.1 & sf + wisepc j011242.10 + 010839.7 & 18.17543 & 1.14435 & 0.044 & 5.51 & 0.033 & 6.9 & 7.0e+10 & 0.5 & sf + wisepc j161241.86 + 484805.8 & 243.17441 & 48.80160 & 0.059 & 6.16 & 0.034 & 6.9 & 7.0e+10 & 2.0 & sf + wisepc j133940.58 + 590307.8 & 204.91910 & 59.05216 & 0.073 & 6.70 & 0.081 & 6.9 & 6.9e+10 & 0.8 & sf + wisepc j165915.56 + 331019.5 & 254.81482 & 33.17208 & 0.087 & 7.13 & 0.072 & 6.8 & 6.9e+10 & 0.0 & sf + wisepc j171424.22 + 633937.8 & 258.60092 & 63.66050 & 0.081 & 6.95 & 0.110 & 6.8 & 6.9e+10 & 0.1 & sf + wisepc j135135.91 + 045100.4 & 207.89963 & 4.85010 & 0.076 & 6.78 & 0.069 & 6.8 & 6.9e+10 & 1.1 & sf + wisepc j132633.84 + 000924.7 & 201.64101 & 0.15686 & 0.086 & 7.09 & 0.061 & 6.8 & 6.9e+10 & 0.5 & sf + wisepc j111459.27 + 402142.0 & 168.74695 & 40.36167 & 0.073 & 6.70 & 0.060 & 6.8 & 6.9e+10 & 0.8 & sf + wisepc j112016.54 + 540624.1 & 170.06892 & 54.10668 & 0.072 & 6.68 & 0.047 & 6.7 & 6.8e+10 & 1.4 & sf + wisepc j170522.43 + 334145.0 & 256.34344 & 33.69582 & 0.090 & 7.23 & 0.077 & 6.6 & 6.7e+10 & 0.0 & sf + wisepc j152043.23 + 304122.7 & 230.18015 & 30.68964 & 0.077 & 6.86 & 0.058 & 6.5 & 6.6e+10 & 0.9 & agn + wisepc j154256.63 + 351602.8 & 235.73595 & 35.26744 & 0.070 & 6.64 & 0.041 & 6.5 & 6.6e+10 & 1.7 & sf + wisepc j170132.94 + 331901.1 & 255.38724 & 33.31697 & 0.088 & 7.18 & 0.069 & 6.4 & 6.6e+10 & 2.5 & agn + wisepc j152227.36 + 303446.0 & 230.61401 & 30.57945 & 0.078 & 6.90 & 0.051 & 6.4 & 6.6e+10 & 1.6 & sf + wisepc j134227.74 + 585930.3 & 205.61560 & 58.99174 & 0.072 & 6.73 & 0.057 & 6.3 & 6.5e+10 & 0.6 & sf + wisepc j235425.88 - 095923.4 & 358.60782 & -9.98984 & 0.076 & 6.88 & 0.067 & 6.2 & 6.4e+10 & 0.9 & sf + wisepc j151933.94 + 303057.6 & 229.89142 & 30.51601 & 0.078 & 6.93 & 0.071 & 6.2 & 6.3e+10 & 1.0 & sf + wisepc j152418.93 + 295638.8 & 231.07889 & 29.94411 & 0.076 & 6.89 & 0.056 & 6.1 & 6.3e+10 & 1.2 & sf + wisepc j111250.48 + 015616.5 & 168.21034 & 1.93791 & 0.076 & 6.91 & 0.077 & 6.1 & 6.2e+10 & 1.5 & sf + wisepc j114951.78 + 560852.8 & 177.46574 & 56.14800 & 0.050 & 5.93 & 0.040 & 6.0 & 6.2e+10 & 1.5 & sf + wisepc j162549.24 + 402042.8 & 246.45515 & 40.34522 & 0.029 & 4.70 & 0.028 & 5.9 & 6.0e+10 & 2.6 & sf + wisepc j111925.05 + 545849.5 & 169.85439 & 54.98043 & 0.071 & 6.78 & 0.056 & 5.9 & 6.0e+10 & 1.6 & sf + wisepc j122747.50 + 120322.3 & 186.94794 & 12.05620 & 0.088 & 7.28 & 0.073 & 5.9 & 6.0e+10 & 0.1 & sf + wisepc j111459.91 + 024551.0 & 168.74963 & 2.76418 & 0.077 & 6.95 & 0.056 & 5.9 & 6.0e+10 & 1.3 & sf + wisepc j162253.90 + 402947.0 & 245.72458 & 40.49639 & 0.031 & 4.85 & 0.026 & 5.9 & 6.0e+10 & 2.6 & sf + wisepc j155849.70 + 272639.2 & 239.70708 & 27.44421 & 0.085 & 7.20 & 0.076 & 5.9 & 6.0e+10 & 1.8 & sf + wisepc j152308.37 + 304847.1 & 230.78488 & 30.81309 & 0.072 & 6.82 & 0.064 & 5.9 & 6.0e+10 & 1.4 & sf + wisepc j102815.22 + 400800.0 & 157.06342 & 40.13334 & 0.067 & 6.65 & 0.064 & 5.8 & 5.9e+10 & 0.8 & sf + wisepc j141309.46 + 442850.4 & 213.28941 & 44.48066 & 0.090 & 7.37 & 0.082 & 5.7 & 5.9e+10 & 1.1 & sf + wisepc j011703.58 + 000027.4 & 19.26492 & 0.00762 & 0.046 & 5.76 & 0.029 & 5.7 & 5.9e+10 & 0.5 & agn + wisepc j111341.36 + 412319.9 & 168.42235 & 41.38885 & 0.073 & 6.88 & 0.062 & 5.7 & 5.9e+10 & 0.8 & sf + wisepc j102134.50 + 410605.2 & 155.39377 & 41.10143 & 0.092 & 7.42 & 0.103 & 5.7 & 5.8e+10 & 0.5 & sf + wisepc j132459.25 + 110431.7 & 201.24689 & 11.07547 & 0.084 & 7.21 & 0.142 & 5.7 & 5.8e+10 & 0.5 & sf + wisepc j111512.08 + 542741.4 & 168.80035 & 54.46151 & 0.066 & 6.64 & 0.057 & 5.7 & 5.8e+10 & 1.5 & sf + wisepc j133450.77 + 593713.3 & 203.71155 & 59.62035 & 0.076 & 6.99 & 0.075 & 5.6 & 5.8e+10 & 0.9 & sf + wisepc j132054.04 + 113912.6 & 200.22517 & 11.65351 & 0.095 & 7.51 & 0.147 & 5.6 & 5.8e+10 & 0.6 & sf + wisepc j103130.85 + 401450.9 & 157.87852 & 40.24746 & 0.066 & 6.66 & 0.058 & 5.6 & 5.7e+10 & 1.3 & sf + wisepc j155844.97 + 270812.4 & 239.68739 & 27.13678 & 0.095 & 7.53 & 0.095 & 5.5 & 5.7e+10 & 1.8 & sf + wisepc j104115.63 + 045313.8 & 160.31512 & 4.88717 & 0.068 & 6.76 & 0.063 & 5.5 & 5.6e+10 & 1.3 & agn + wisepc j162036.15 + 335643.1 & 245.15063 & 33.94529 & 0.031 & 4.97 & 0.026 & 5.4 & 5.6e+10 & 2.1 & sf + wisepc j130306.96 - 031846.9 & 195.77899 & -3.31304 & 0.086 & 7.30 & 0.112 & 5.4 & 5.6e+10 & 0.2 & sf + wisepc j115809.58 + 555140.0 & 179.53992 & 55.86110 & 0.063 & 6.57 & 0.084 & 5.4 & 5.5e+10 & 1.8 & sf + wisepc j171248.41 + 634258.0 & 258.20169 & 63.71612 & 0.075 & 7.01 & 0.067 & 5.4 & 5.5e+10 & 0.3 & sf + wisepc j115938.06 + 561702.5 & 179.90857 & 56.28402 & 0.070 & 6.85 & 0.080 & 5.3 & 5.4e+10 & 2.2 & sf * + wisepc j121734.75 + 035019.8 & 184.39478 & 3.83884 & 0.073 & 6.96 & 0.110 & 5.3 & 5.4e+10 & 1.9 & sf + wisepc j155745.93 + 274043.6 & 239.44136 & 27.67878 & 0.090 & 7.46 & 0.096 & 5.2 & 5.4e+10 & 1.7 & sf + wisepc j101051.19 - 003842.5 & 152.71327 & -0.64514 & 0.043 & 5.75 & 0.076 & 5.2 & 5.4e+10 & 0.8 & sf * + wisepc j130510.25 - 023516.0 & 196.29272 & -2.58778 & 0.088 & 7.41 & 0.133 & 5.2 & 5.4e+10 & 0.4 & sf + wisepc j162858.00 + 391909.3 & 247.24168 & 39.31924 & 0.034 & 5.18 & 0.027 & 5.2 & 5.3e+10 & 2.4 & sf + wisepc j151848.57 + 305449.2 & 229.70239 & 30.91367 & 0.078 & 7.13 & 0.064 & 5.2 & 5.3e+10 & 1.2 & sf + wisepc j123001.03 + 113940.5 & 187.50427 & 11.66124 & 0.082 & 7.24 & 0.124 & 5.2 & 5.3e+10 & 2.2 & sf + wisepc j133105.70 + 584939.1 & 202.77376 & 58.82752 & 0.073 & 6.97 & 0.065 & 5.1 & 5.3e+10 & 0.8 & sf + wisepc j152428.76 + 310750.4 & 231.11983 & 31.13067 & 0.078 & 7.14 & 0.094 & 5.1 & 5.2e+10 & 1.1 & agn + wisepc j103954.63 + 051546.3 & 159.97762 & 5.26285 & 0.064 & 6.69 & 0.069 & 5.1 & 5.2e+10 & 1.7 & sf + wisepc j132422.33 + 105047.7 & 201.09305 & 10.84658 & 0.093 & 7.59 & 0.175 & 5.0 & 5.2e+10 & 0.7 & sf + wisepc j082926.09 + 302528.8 & 127.35872 & 30.42466 & 0.051 & 6.17 & 0.047 & 5.0 & 5.2e+10 & 0.7 & sf + wisepc j111503.18 + 542856.5 & 168.76323 & 54.48237 & 0.067 & 6.80 & 0.076 & 5.0 & 5.2e+10 & 1.5 & sf + wisepc j155957.42 + 272745.9 & 239.98924 & 27.46274 & 0.095 & 7.63 & 0.124 & 5.0 & 5.1e+10 & 1.8 & agn + wisepc j171351.15 + 640012.7 & 258.46313 & 64.00354 & 0.088 & 7.45 & 0.179 & 5.0 & 5.1e+10 & 0.2 & sf + wisepc j155930.98 + 273257.4 & 239.87907 & 27.54927 & 0.091 & 7.53 & 0.132 & 4.9 & 5.1e+10 & 1.9 & sf + wisepc j235345.66 - 103353.6 & 358.44025 & -10.56490 & 0.078 & 7.19 & 0.083 & 4.9 & 5.1e+10 & 0.7 & sf + wisepc j115610.78 + 045943.3 & 179.04491 & 4.99536 & 0.079 & 7.24 & 0.104 & 4.8 & 5.0e+10 & 1.6 & agn + wisepc j122915.88 + 113858.6 & 187.31615 & 11.64962 & 0.088 & 7.49 & 0.160 & 4.8 & 4.9e+10 & 2.5 & sf + wisepc j235514.75 - 102946.9 & 358.81146 & -10.49636 & 0.082 & 7.33 & 0.105 & 4.7 & 4.9e+10 & 0.6 & sf + wisepc j133111.02 - 014338.6 & 202.79590 & -1.72740 & 0.084 & 7.39 & 0.096 & 4.7 & 4.9e+10 & 0.7 & sf + wisepc j235208.99 - 095434.4 & 358.03748 & -9.90957 & 0.077 & 7.19 & 0.082 & 4.7 & 4.9e+10 & 0.6 & sf * + wisepc j134033.01 + 015016.0 & 205.13754 & 1.83778 & 0.078 & 7.24 & 0.094 & 4.7 & 4.9e+10 & 0.5 & sf + wisepc j130335.10 - 025417.0 & 195.89626 & -2.90471 & 0.083 & 7.38 & 0.110 & 4.7 & 4.8e+10 & 0.2 & sf + wisepc j130329.04 - 025337.2 & 195.87102 & -2.89366 & 0.079 & 7.27 & 0.092 & 4.7 & 4.8e+10 & 0.3 & sf + wisepc j122902.48 + 114340.3 & 187.26031 & 11.72786 & 0.086 & 7.49 & 0.243 & 4.6 & 4.8e+10 & 2.3 & sf + wisepc j135143.98 + 045833.5 & 207.93326 & 4.97596 & 0.077 & 7.20 & 0.089 & 4.6 & 4.8e+10 & 0.5 & sf + wisepc j170436.61 + 334454.2 & 256.15256 & 33.74839 & 0.090 & 7.60 & 0.099 & 4.6 & 4.8e+10 & 3.0 & sf + wisepc j121734.57 + 033450.9 & 184.39404 & 3.58081 & 0.074 & 7.13 & 0.089 & 4.6 & 4.8e+10 & 1.8 & sf + wisepc j123102.64 + 113929.5 & 187.76102 & 11.65819 & 0.087 & 7.52 & 0.126 & 4.6 & 4.7e+10 & 2.8 & sf + [ table : demi_lirgs ]

we present results from a systematic study of star formation in local galaxy clusters using 22@xmath0 m data from the wide - field infrared survey explorer ( wise ) . the 69 systems in our sample are drawn from the cluster infall regions survey ( cirs ) , and all have robust mass determinations . the all - sky wise data enables us to quantify the amount of star formation , as traced by 22@xmath0 m , as a function of radius well beyond , and investigate the dependence of total star formation rate upon cluster mass . we find that the fraction of star - forming galaxies increases with cluster radius but remains below the field value even at 3 . we also find that there is no strong correlation between the mass - normalized total specific star formation rate and cluster mass , indicating that the mass of the host cluster does not strongly influence the total star formation rate of cluster members .

arxiv

let @xmath0 and @xmath1 be banach spaces over a real or complex field @xmath6 . we use the traditional notations @xmath7 and @xmath8 for the unit sphere and the closed unit ball of the space @xmath0 , respectively . the banach space of all bounded linear operators @xmath9 will be represented by @xmath10 . in particular , when @xmath11 we denote @xmath12 simply by putting @xmath13 called the dual space of @xmath0 . we say that an operator @xmath14 attains its norm if there exists @xmath15 such that @xmath16 . in this case , we say that @xmath17 is a norm attaining operator and it attains its norm at @xmath18 . the subset of @xmath10 of all norm attaining operators is denoted by @xmath19 . we recall that a bounded linear operator is compact if the closure of the image of the unit ball is compact . for @xmath20 we denote by @xmath21 the euclidean space @xmath22 endowed with the @xmath23-norm @xmath24 with @xmath25 and @xmath26 and @xmath27 endowed with the sup - norm @xmath28 . to simplify the notation we put just @xmath29 when the field that we are working is specified . the bishop - phelps theorem @xcite says that every bounded linear functional can be approximated by norm attaining functionals . in other words , this means that the set of all norm attaining functionals on a banach space @xmath0 is dense in its dual space @xmath13 . it was proved by lindenstrauss @xcite in 1963 that , in general , the same result does not work for bounded linear operators . more precisely , he exhibited a banach space @xmath0 such that the set @xmath30 is not dense in @xmath31 . seven years later , bollobs proved a numerical version of the bishop - phelps theorem which nowdays is known as the bishop - phelps - bollobs theorem @xcite . as a consequence of ( * theorem 2.1 ) we may enunciate this theorem as follows . _ ( bishop - phelps - bollobs theorem , @xcite , @xcite ) _ let @xmath32 and suppose that @xmath33 and @xmath34 satisfy @xmath35 then , there are @xmath36 and @xmath37 such that @xmath38 since we do not have a bishop - phelps version for bounded linear operators , we can not expect a bishop - phelps - bollobs version for this type of functions either . so it is natural to study the conditions that the banach spaces @xmath0 and @xmath1 must satisfy to get a theorem of this nature . in 2008 , acosta , aron , garca and maestre introduced a definition in order to attain this problem . [ bpbp ] ( bishop - phelps - bollobs property , @xcite ) we say that a pair of banach spaces @xmath4 satisfies the _ bishop - phelps - bollobs property _ ( bpbp , for short ) when given @xmath39 , there exists @xmath40 such that whenever @xmath41 and @xmath15 are such that @xmath42 there are @xmath43 and @xmath36 such that @xmath44 there are many classical banach spaces that satisfy the bpbp . for example , when @xmath0 and @xmath1 are finite dimensional spaces , the pair @xmath4 has this property ( * ? ? ? * proposition 2.4 ) . also , if @xmath1 has the property @xmath45 of lindenstrauss , as @xmath46 and @xmath47 do , then the pair @xmath4 satisfies the bpbp for all banach space @xmath0 ( * ? ? ? * theorem 2.2 ) . more positive results appear when we assume that the range space @xmath1 is uniformly convex : the pairs @xmath48 , @xmath49 and @xmath50 all satisfy the bpbp ( see @xcite , @xcite and @xcite , respectively ) . also if @xmath0 is a uniformly convex banach space , then the pair @xmath4 has the bpbp for all banach space @xmath1 ( * ? ? ? * theorem 3.1 ) . just to help make the paper entirely accessible , we remember the concept of uniform convexity . a banach space @xmath0 is uniformly convex if given @xmath39 , there exists @xmath51 such that whenever @xmath52 satisfy @xmath53 , then @xmath54 . we recall that if @xmath55 then @xmath56 is uniformly convex . in 2014 , kim and lee ( * ? ? ? * theorem 2.1 ) given a characterization for uniformly convex banach spaces that associate this type of spaces with a peculiarity on the bishop - phelps - bollobs property . more precisely , they proved that a banach space @xmath0 is uniformly convex if and only if given @xmath39 then we are able to find a positive number @xmath40 such that whenever @xmath57 and @xmath15 satisfy the relation @xmath58 there exists a vector @xmath36 such that @xmath59 note that the theorem says that a banach space @xmath0 is uniformly convex if and only if the pair @xmath60 satisfies the bishop - phelps - bollobs property without changing the initial functional @xmath61 , that is , the functional that almost attains its norm at some point @xmath18 is the same that attains its norm at the new vector that is close to @xmath18 . in this paper , we study this last result for bounded linear operators and we call it as the strong bishop - phels - bollobs property ( sbpbp ) . first , we study it in the case that the real number @xmath62 depends of @xmath39 and also of a fixed operator @xmath17 , and we get some positives results about it . after that , we study the property in the uniform case , that is , when the number @xmath63 depends only of @xmath39 as we are used to work when we work with the bpbp . as we will see in the next section , we get many negatives results about the uniform case and we use these results to prove that there are uniformly convex banach spaces @xmath0 and infinite dimensional banach spaces @xmath1 such that the pair @xmath4 fails the sbpbp . finally , we give a complete characterization to the sbpbp for the pair @xmath5 which describes when these pairs satisfy the property . in this section we study the strong bishop - phelps - bollobs property . namely , we study the conditions that the banach spaces @xmath0 and @xmath1 must have and the hypothesis that we have to add to get a kim - lee type theorem for bounded linear operators . the kim - lee theorem is enunciated as follows . [ kim - lee ] _ ( kim - lee theorem , @xcite ) _ a banach space @xmath0 is uniformly convex if and only if given @xmath39 , there exists @xmath40 such that whenever @xmath57 and @xmath15 satisfy @xmath58 there is @xmath36 such that @xmath64 as we mentioned before , it is like a bishop - phelps - bollobs property without changing the initial functional @xmath61 . a natural question arises : the result is true for other pairs of banach spaces @xmath4 with @xmath0 and @xmath1 having additional hypothesis considering bounded linear operators instead bounded linear functionals ? although it is more natural put the question just like that , it is seems to us to be a strong problem in the sense that it will be hard to find concrete pairs of banach spaces @xmath4 satisfying the kim - lee theorem for bounded linear operators . so we start by considering that the positive real number @xmath65 that appears in theorem [ kim - lee ] does not depend only of @xmath39 but also of a given operator @xmath17 fixed . before we do that , we want to comment that carando , lassalle and mazzitelli @xcite defined a weak bpbp for ideals of multilinear mappings where the real positive number @xmath66 in the definition of the bpbp in this context depends of a given @xmath39 and also of the ideal norm of the operator defined on a normed ideal of @xmath67-linear mappings . in other words , a normed ideal of @xmath67-linear mappings @xmath68 where @xmath69 are banach spaces has the weak bpbp if for each @xmath70 with @xmath71 and @xmath39 , there exists @xmath72 depending also of @xmath73 such that if @xmath74 satisfies @xmath75 , then there exist @xmath76 with @xmath77 and @xmath78 such that @xmath79 , @xmath80 and @xmath81 . they proved that if @xmath82 are uniformly convex banach spaces then @xmath83 has the weak bpbp for ideals of multilinear mappings for all banach space @xmath1 . here we will work on a different context where @xmath62 depends of a fixed operator not of the norm of the operators ideal as we may see in the following definition . [ sbpbp ] a pair of banach spaces @xmath4 has the _ strong bishop - phelps - bollobs property _ ( or sbpbp for short ) if given @xmath39 and @xmath41 , there exists @xmath84 such that whenever @xmath15 satisfies @xmath85 there is @xmath36 such that @xmath86 if @xmath14 is compact and the pair @xmath4 has the sbpbp , then we say that the pair @xmath4 has the sbpbp for compact operators . in the next proposition , we assume that the domain space @xmath0 is finite dimensional to get the sbpbp first result . in fact , as we mentianed in the previous paragraph , we get a positive real number @xmath84 that depends of @xmath39 and also of the operator @xmath41 fixed at the beggining of the proof . the proof is by contradiction . let @xmath41 . if the result is false for some @xmath87 , then for all @xmath88 , there exists @xmath89 such that @xmath90 but @xmath91 , where @xmath92 for @xmath88 . since @xmath0 is finite dimensional , there exists a subsequence @xmath93 of @xmath94 such that @xmath95 for some @xmath96 . this implies that @xmath97 and since @xmath98 we get that @xmath99 and so @xmath100 . then @xmath101 and @xmath102 which is a contradiction . the proof is again by contradiction . let @xmath41 be a compact operator . if the result is false , for some @xmath87 and for all @xmath88 , there exists @xmath89 such that @xmath104 but @xmath91 for all @xmath88 . then @xmath105 as @xmath106 . since @xmath0 is uniformly convex , @xmath0 reflexive and then by the smulian theorem there exists a subsequence @xmath93 of @xmath94 and @xmath96 such that @xmath93 converges weakly to @xmath18 . since @xmath17 is completely continuous , @xmath107 converges in norm to @xmath108 as @xmath109 . therefore @xmath110 and so @xmath100 . thus @xmath101 and @xmath111 this implies that @xmath112 and , using again that @xmath0 is uniformly convex , we get that @xmath113 which is a contradiction because of the following inequalities : @xmath114 consequently we get two more positives results about the sbpbp . in the next corollary we prove that the pair @xmath4 has the sbpbp whenever @xmath0 is uniformly convex and @xmath1 has the schur s property which implies as a particular case that the pair @xmath115 satisfies the property . corollary [ sbpbp3 ] shows that whenever @xmath0 is uniformly convex and @xmath1 has finite dimension , the pair @xmath4 has the sbpbp by using the fact that every bounded linear operator with finite dimensional range is compact . we apply theorem [ sbpbp2 ] . to do this , we prove that every bounded linear operator @xmath103 is compact . indeed , since @xmath17 is continuous , @xmath17 is @xmath116-@xmath116 continuous . let @xmath117 . since @xmath0 is reflexive , by the smulian theorem , there are a subsequence of @xmath118 ( which we denote again by @xmath118 ) and @xmath96 such that @xmath119 . so @xmath120 . now , since @xmath1 has the schur s property , @xmath121 in norm . so @xmath17 is compact . by theorem [ sbpbp2 ] the pair @xmath4 has the sbpbp . [ sbpbp5 ] note that in definition [ sbpbp ] the operator @xmath103 must attains its norm if the pair @xmath4 has the sbpbp . so if @xmath0 is not reflexive , then the pair @xmath4 fails the sbpbp for all banach space @xmath1 . indeed , since @xmath0 is not reflexive , by the james theorem , there is a linear continuous functional @xmath122 such that @xmath123 for all @xmath124 . let @xmath125 and define @xmath103 by @xmath126 . then @xmath127 and @xmath128 for all @xmath124 . this implies that @xmath17 never attains its norm and then the pair @xmath4 can not have the sbpbp . note that in the classic definition of the bishop - phelps - bollobs property the number @xmath62 depends only of @xmath39 . so what happen if we ask for more in the definition of the strong bishop - phelps - bollobs property ? as we will see below , when we put @xmath129 to depends only of @xmath39 we get negative results . however , we use them to get examples of pairs of banach spaces that fail the sbpbp when the domain space is reflexive or when the range space has infinite dimension . just to help to make reference we put a name of it . we say that a pair of banach spaces @xmath4 has the _ uniform strong bishop - phelps - bollobs property _ ( uniform sbpbp , for short ) if given @xmath39 , there exists @xmath40 such that whenever @xmath41 and @xmath15 are such that @xmath42 there is @xmath36 such that @xmath86 we observe that the kim - lee theorem says that a banach space @xmath0 is uniformly convex if and only if the pair @xmath60 has the uniform sbpbp where @xmath130 or @xmath131 . note also that if the pair @xmath4 satisfies the uniform sbpbp then the pair @xmath4 satisfies the bpbp . the first thing that we notice is that if @xmath4 has the uniform sbpbp for some banach space @xmath1 , then @xmath0 must be uniformly convex . given @xmath132 , consider @xmath40 the positive real number that satisfies the uniform sbpbp for the pair @xmath4 . we prove that the pair @xmath60 has the uniform sbpbp with the same @xmath133 . indeed , let @xmath57 and @xmath15 be such that @xmath134 let @xmath125 and define @xmath14 by @xmath135 for all @xmath136 . so @xmath137 and @xmath138 since the pair @xmath4 has the uniform sbpbp with @xmath133 , there exists @xmath36 such that @xmath139 and @xmath140 . since @xmath141 the proof is complete . what about the reciprocal of corollary [ corsbpbp ] ? the first thing that come to mind , since every hilbert space is uniformly convex , is to assume that the domain space @xmath0 is a hilbert space and try to find some banach space @xmath1 such that the pair @xmath4 satisfies the property . but even in the simplest situation the result fails as we may see in the following example . [ exsbpbp ] this example works for both real and complex cases . for a given @xmath39 , suppose that there exists @xmath40 satisfying the uniform sbpbp for the pair @xmath144 . let @xmath145 be defined by @xmath146 for every @xmath147 . for every @xmath148 , we have @xmath149 since @xmath150 , we obtain @xmath151 . moreover , @xmath152 we prove now that every @xmath153 such that @xmath154 assumes the form @xmath155 for @xmath156 . indeed , since @xmath157 and @xmath154 , we have @xmath158 . since @xmath159 , we have @xmath160 and @xmath161 with @xmath156 . in summary , we have a unit operator @xmath17 and a unit vector @xmath162 satisfying @xmath163 but if @xmath17 attains its norm at some point @xmath164 then @xmath165 with @xmath156 . this contradicts the assumption that the pair @xmath144 has the uniform sbpbp since @xmath166 is far from @xmath162 in view of the fact that @xmath167 . this shows that the pair @xmath168 fails to have the uniform sbpbp for @xmath130 or @xmath131 . now what if we add on the hypothesis that both @xmath0 and @xmath1 are hilbert spaces ? the answer for this question is still no as we can see below . [ sbpbp8 ] this example works for both real and complex cases . for a given @xmath39 , suppose that there exists @xmath40 satisfying the uniform sbpbp for the pair @xmath169 . define @xmath170 by @xmath171 for every @xmath147 . then for every @xmath148 , @xmath172 and since @xmath173 , we have @xmath151 . also , we see that @xmath174 . now , if @xmath153 is such that @xmath175 , then @xmath176 and since @xmath177 , we get @xmath178 which implies that @xmath160 and using again that @xmath159 , we obtain @xmath179 with @xmath156 and so @xmath180 which contradicts the hypothesis that the pair @xmath169 has the uniform sbpbp . [ paramiguel1 ] let @xmath181 and @xmath182 be the space @xmath183 endowed with the @xmath23-norm and the @xmath184-norm , respectively , with @xmath185 ( or @xmath186 ) . given @xmath187 , there exists a bounded linear operator @xmath188 with @xmath189 such that let @xmath187 and @xmath185 . define @xmath188 by @xmath194 for every @xmath195 . if @xmath196 , since @xmath197 , we get @xmath198 which implies that @xmath199 . since @xmath200 , we have @xmath189 . now , let @xmath201 be such that @xmath202 . we prove that @xmath179 with @xmath156 . indeed , the equality @xmath203 implies that @xmath204 and since @xmath205 , we do the difference between these two equalities to get @xmath206 since @xmath197 and @xmath207 , @xmath208 and @xmath209 . because of the above equality , we get that @xmath210 . but @xmath211 which implies that @xmath212 thus @xmath160 and then @xmath179 with @xmath156 as desired . so if @xmath191 is such that @xmath192 , then @xmath213 which completes the proof . as a consequence of this last result , we get that all the pairs @xmath214 fail the uniform sbpbp for @xmath185 for @xmath130 or @xmath131 . now we study the pair @xmath215 in the real case when we put the sum norm on the range space . unfortunatly , we can construct a bounded linear operator to get the same contradiction as before . let @xmath216 and @xmath142 be the banach spaces @xmath217 endowed with the @xmath218-norm and the sum - norm , respectively . given @xmath187 , there exists a bounded linear operator @xmath219 with @xmath189 such that let @xmath187 . define @xmath219 by @xmath223 for all @xmath224 . then @xmath225 so @xmath226 and @xmath227 . also , we have @xmath189 . indeed , let @xmath148 . then @xmath228 recall that for every @xmath229 , we may write @xmath230 . with this in mind , if @xmath231 , then @xmath232 and if @xmath233 likewise we have @xmath234 + \frac{1}{2 } |\beta x - y| = \max \ { -\beta x , -y \ } \leq 1.\ ] ] since @xmath235 , we have @xmath189 . now , suppose that @xmath153 is such that @xmath221 . so , @xmath236 again , we have two cases . if @xmath237 , then @xmath238 which implies @xmath239 since @xmath240 , and if @xmath241 , then @xmath242 which implies @xmath243 . since @xmath244 , we obtain @xmath160 and @xmath245 . this means that if @xmath246 attains its norm at some @xmath247 , then @xmath180 . [ paramiguel2 ] let @xmath216 and @xmath182 be the space @xmath217 endowed with the 2-norm and the @xmath184-norm , respectively , with @xmath250 . given @xmath187 , there exists @xmath251 with @xmath189 such that let @xmath250 and define @xmath254 by @xmath255 for every @xmath147 . first of all , note that @xmath256 i.e. , @xmath257 . this shows that @xmath258 . next , we show that @xmath259 and also that the only points which @xmath17 attains its norm are at @xmath260 and @xmath261 . to do so , we study the norm of the operator @xmath17 by using the following compact set : @xmath262 by symmetry , the norm of @xmath17 is the maximum of @xmath263 with @xmath166 in @xmath264 . let @xmath265 a point of @xmath264 such that @xmath17 attains its norm at @xmath266 , that is , @xmath267 . we consider @xmath268 as the segment that connect @xmath269 with @xmath162 , @xmath270 as the segment that connect @xmath269 with @xmath271 and @xmath272 as the arc that connect @xmath162 with @xmath271 . see figure [ fig : maximum ] . it is enough to study the values of @xmath273 on the set @xmath274 since the operator @xmath17 attains its norm at elements of the sphere and @xmath275 . we have @xmath276 with @xmath277 defined as @xmath278 and @xmath279 defined as @xmath280 . since @xmath281 for every @xmath250 , then @xmath282 . ( on the other hand observe that if @xmath283 , then @xmath284 and if @xmath285 , then @xmath286 . ) thus if @xmath287 , then either @xmath288 and then @xmath289 would be a critical point of @xmath290 in @xmath291 , where @xmath292 or @xmath293 and in this case @xmath294 would be a critical point of @xmath295 in @xmath296 , where @xmath297.\ ] ] but , as we will see in the next lines , these can not happen because @xmath298 and @xmath299 for all @xmath300 and then @xmath301 . indeed , we consider first the case that @xmath302 for every @xmath302 , we get @xmath303.\ ] ] for @xmath302 , we have that @xmath304 and since @xmath305 for every @xmath306 on this interval , we obtain that @xmath307 for every @xmath308 . a simply change of the letter @xmath290 by @xmath295 and @xmath306 by @xmath309 implies that @xmath299 for every @xmath310 . everything we did so far was to prove that @xmath311 and that @xmath17 attains its norm on @xmath264 only at @xmath312 and @xmath313 . therefore , we may conclude that @xmath17 attains its maximum at @xmath260 and at @xmath261 . in other words , we proved that @xmath314 . now , for @xmath315 , define @xmath251 by @xmath316 for every @xmath147 . note that @xmath317 and @xmath318 . since @xmath319 , then @xmath199 . also , using that @xmath314 and that @xmath320 , we have that @xmath321 . this implies that if @xmath164 is such that @xmath192 , then @xmath322 and therefore @xmath180 as we wanted . what about the case that @xmath325 and @xmath250 ? we will study the real case of this right now . consider @xmath325 . define @xmath326 by @xmath327 for every @xmath328 . then @xmath329 and @xmath330 . also , if @xmath331 , then @xmath332 and so @xmath333 . this implies that @xmath334 and @xmath335 , and therefore @xmath336 thus @xmath337 . given @xmath315 , let @xmath338 be as in the proposition [ paramiguel2 ] with @xmath250 . now , define @xmath339 by @xmath340 . then @xmath341 . also , @xmath342 and @xmath343 . suppose that there exists @xmath344 such that @xmath345 . then @xmath346 and then , as we can see in the proof of the proposition [ paramiguel2 ] , @xmath166 must be equals to @xmath271 or @xmath347 . in both cases , we have that @xmath348 . we just have proved the following result . suppose that there exists @xmath40 that depends only of a given @xmath39 satisfying the property . let @xmath352 and @xmath52 be such that @xmath353 for @xmath354 . define @xmath355 by @xmath356 for all @xmath136 . then @xmath357 and @xmath358 . moreover , since @xmath359 , we have that @xmath259 . this shows that @xmath151 . therefore , there exists @xmath360 such that @xmath361 and @xmath362 . since @xmath363 and @xmath364 , we have that @xmath365 . on the other hand , since @xmath366 we get a contradiction , since @xmath367 it is a consequence of the fact that the pair @xmath144 fails the uniform sbpbp that the pair @xmath368)$ ] also fails this property . indeed , consider @xmath369 $ ] positive functions defined on @xmath370 $ ] such that @xmath371 , @xmath372 $ ] and @xmath373 $ ] . define @xmath374 $ ] by @xmath375 for all @xmath376 $ ] and @xmath377 . then @xmath378 is linear . since for @xmath379 we have @xmath380 and for @xmath381 $ ] we have @xmath382 , we get that @xmath383 } | \varphi(x , y)(t)| = \max \ { |x| , |y| \ } = \|(x , y)\|_{\infty}\ ] ] for all @xmath377 , using the fact that @xmath371 . this shows that @xmath378 is a linear isometry between @xmath384 and the closed subspace of @xmath385 $ ] generated by @xmath386 and @xmath387 . now suppose by contradiction that the pair @xmath368)$ ] has the uniform sbpbp . then there exists @xmath40 for all @xmath132 , the modulus of the uniform sbpbp for this pair . let @xmath388 with @xmath151 and @xmath389 be such that @xmath390 define @xmath391 $ ] by @xmath392 . then for all @xmath148 , we have that @xmath393 and since @xmath378 is an isometry , we get that @xmath394 which implies that @xmath395 . also , @xmath396 so , there exists @xmath397 such that @xmath398 and @xmath399 . since @xmath400 , we just have proved that the pair @xmath144 also has the uniform sbpbp which contradicts example [ exsbpbp ] and proposition [ sbpbp1 ] . next we show that if @xmath1 is a @xmath218-dimensional banach space , then the pair @xmath401 does not have the uniform sbpbp . to do so , we use the existence of the auerbach base for a finite dimensional banach space ( see , for example , ( * ? ? ? * proposition 20.21 ) ) . let @xmath1 be an @xmath402-dimensional banach space . then there are elements @xmath403 of @xmath1 and @xmath404 of @xmath405 such that @xmath406 for all @xmath407 and @xmath408 for @xmath409 . in fact , @xmath410 is a base of @xmath1 called the auerbach base of @xmath1 . let @xmath411 and @xmath412 satisfying @xmath406 for @xmath413 and @xmath408 for @xmath354 . since @xmath414 is a base for @xmath1 , every @xmath415 has an expression in terms of @xmath416 and @xmath417 given by @xmath418 . given @xmath187 , define the continuous linear operator @xmath419 by @xmath420 for all @xmath421 . then for all @xmath422 , we have that @xmath423 then @xmath199 . also , note that @xmath424 . so @xmath189 . now let @xmath125 be such that @xmath425 . then , using that @xmath426 we get that @xmath427 and therefore @xmath428 . finally , if the pair @xmath401 has the uniform sbpbp , there exists @xmath40 satisfying the property . if we put @xmath429 , there exists an operator @xmath430 such that @xmath151 , @xmath163 and for all @xmath125 which satisfies @xmath431 is such that @xmath432 . this is a contradiction and the pair @xmath401 fails the uniform sbpbp . it is clear but it is worth mentioning that if the pair @xmath4 has the uniform sbpbp , then the pair @xmath433 also has this property for all closed subspace @xmath434 of @xmath1 . this implies , by the above proposition , that the pair @xmath401 fails the uniform sbpbp for all @xmath402-dimensional finite space @xmath1 . anyway , just for curiosity , if @xmath435 , the proof of proposition [ yy ] works as well in this situation . indeed , for @xmath187 we define @xmath436 by @xmath437 for all @xmath415 , where @xmath438 and @xmath439 is given by the auerbach basis . then @xmath440 for @xmath441 and @xmath442 . to prove that @xmath199 , we add and subtract @xmath443 in @xmath444 where @xmath422 , to get @xmath445 . now , it is clear that if @xmath246 attains its norm at some @xmath125 then @xmath446 for all @xmath441 . if @xmath1 is a banach space which contains strictly convex @xmath218-dimensional subspaces , then there exists a uniformly convex banach space @xmath0 such that the pair @xmath4 fails the uniform sbpbp . indeed , let @xmath434 be a subspace of @xmath1 such that @xmath434 is stricly convex and @xmath447 . then @xmath448 is uniformly convex , since @xmath434 is finite dimensional . by proposition [ yy ] , the pair @xmath433 fails the uniform sbpbp and by the above observation the pair @xmath4 can not have this property . next we use the negatives results that we got so far about the uniform sbpbp to get examples of pairs @xmath4 that fail the strong bishop - phelps - bollobs prooperty . in remark [ sbpbp5 ] we noted that if @xmath0 is not reflexive , then the pair @xmath4 does not have the sbpbp . in what follows we get examples of reflexive banach spaces @xmath0 such that the pair @xmath4 fails the sbpbp . we also present a complete characterization for the pairs @xmath5 concerning this property by showing that there are cases that these pairs satisfy the property and other cases not ( see theorem [ sbpbp9 ] ) . first of all we use the fact , which is showed in the next remark , that the pair @xmath449 fails the uniform sbpbp to get examples of banach spaces @xmath1 such that the pair @xmath450 fails the sbpbp ( see corollary [ sbpbp7 ] ) . [ sbpbp4 ] by corollary [ sbpbp3 ] , the pair @xmath451 has the sbpbp if @xmath452 . but in the case of the uniform sbpbp , we get a negative result . we note that the pair @xmath449 fails the uniform sbpbp . indeed , suppose by contradiction that this pair satisfies the property . then given @xmath39 there exists @xmath40 such that whenever @xmath453 and @xmath454 are such that @xmath455 , there is @xmath456 such that @xmath139 and @xmath457 . since the pair @xmath169 fails the uniform sbpbp , there exists some @xmath87 , a norm one linear operator @xmath458 and a norm one vector @xmath459 with @xmath460 such that there is no point @xmath461 such that @xmath462 and @xmath463 . let @xmath464 be the projection on the first two coordinates , i.e. , @xmath465 for all @xmath466 . then @xmath467 . define @xmath468 by @xmath469 . then @xmath470 . let @xmath471 . we have that @xmath472 then there exists @xmath473 such that @xmath474 and @xmath475 . since @xmath476 we get that @xmath477 . on the other hand , @xmath478 . this is a contradiction and then the pair @xmath449 fails the sbpbp as desired . the next theorem connects both sbpbp and uniform sbpbp in order to get examples of pairs of banach spaces @xmath4 that not satisfy the sbpbp by using the examples of the pairs that fail the uniform sbpbp . suppose that the pair @xmath4 fails the uniform sbpbp . then for each @xmath88 , there are @xmath480 , @xmath481 and @xmath89 with @xmath482 such that whenever @xmath124 satisfies @xmath483 we have @xmath484 . define @xmath485 by @xmath486 for every @xmath487 . since @xmath488 for all @xmath88 , we get that @xmath259 . on the other hand , @xmath489 so @xmath151 . we suppose that there is @xmath490 such that the pair @xmath491 has the sbpbp with this constant . denote by @xmath492 the element of @xmath493 such that in the @xmath402-th position is @xmath166 and in the rest is zero for all @xmath494 . we take @xmath88 to be such that @xmath495 observe that @xmath496 thus there is @xmath497 such that @xmath498 by the second inequality , we get that @xmath499 for all @xmath500 and @xmath501 . moreover , by the equality @xmath502 we get that @xmath503 since @xmath504 for all @xmath500 . so @xmath505 . this contradicts the beggining of the proof and we conclude that the pair @xmath491 fails the sbpbp whenever @xmath4 fails the uniform sbpbp . * @xmath144 , * @xmath169 , * @xmath510 for @xmath185 . * @xmath215 , * @xmath511 , for @xmath250 . * @xmath512 for @xmath324 . * @xmath513 for @xmath325 and @xmath350 . * @xmath368)$ ] , * @xmath401 , where @xmath514 . \(i ) by the pitt s theorem every bounded linear operator from @xmath56 into @xmath520 with @xmath519 is compact . by theorem [ sbpbp2 ] the pair @xmath5 has the sbpbp since @xmath56 is uniformly convex for @xmath521 . \(ii ) let @xmath132 and @xmath185 . consider @xmath56 and @xmath520 as the banach spaces @xmath522 and @xmath523 , respectively . for each @xmath88 define @xmath524 by @xmath525 for each @xmath195 . let @xmath526 and let @xmath527 be defined by @xmath528 for each @xmath529 we have that @xmath530 and then @xmath531 so @xmath259 . we consider the vectors @xmath532 thus we get that @xmath533 . so @xmath151 . suppose that there exists @xmath84 such that the pair @xmath5 has the sbpbp . let @xmath88 be such that @xmath534 . so since @xmath535 and @xmath536 there exists @xmath537 such that @xmath538 next we claim that @xmath539 for all @xmath540 . indeed , suppose that there exists some @xmath541 such that @xmath542 . thus @xmath543 which is a contradiction . then @xmath539 for all @xmath540 . because of that , we have @xmath544 this new contradiction shows that the pair @xmath5 fails the sbpbp for @xmath185 . * acknowledgement . * this article is based on parts of my ph.d . thesis at universidad de valencia and i would like to thank domingo garca , manuel maestre and miguel martn for their wonderful supervision on this work .

in this paper we introduce the strong bishop - phelps - bollobs property ( sbpbp ) for bounded linear operators between two banach spaces @xmath0 and @xmath1 . this property is motivated by a kim - lee result which states , under our notation , that a banach space @xmath0 is uniformly convex if and only if the pair @xmath2 satisfies the sbpbp . positive results of pairs of banach spaces @xmath3 satisfying this property are given and concrete pairs of banach spaces @xmath4 failing it are exhibited . a complete characterization of the sbpbp for the pairs @xmath5 is also provided .

arxiv

the analysis of modern accelerator - based neutrino oscillation experiments requires good control over the intermediate - energy neutrino - nucleus scattering cross section @xcite . in particular the importance of multi - nucleon events has been suggested in many calculations of charge - changing quasielastic cross sections @xmath2 , at typical neutrino energies of @xmath3 gev @xcite . the contribution of two - particle - two - hole ( 2p-2h ) excitations is now thought to be essential for a proper description of data @xcite . thus a growing interest has arisen in including 2p-2h models into the monte carlo event generators used by the neutrino collaborations @xcite . the only 2p-2h model implemented up to date in some of the monte carlo neutrino event generators corresponds to the so - called ific valencia model @xcite , which has been incorporated in genie @xcite . there are also plans to incorporate the lyon model @xcite in genie , while phenomenological approaches like the effective transverse enhancement model of @xcite are implemented , for instance , in nuwro generator @xcite . one of the main problems to implementing the 2p-2h models is the high computational time . this is due to the large number of nested integrals involved in the evaluation of the inclusive hadronic tensor with sums over the final 2p-2h states . to speed up the calculations , several approximations can be made , such as choosing an average momentum for the nucleons in the local fermi gas @xcite , neglecting the exchange matrix elements , or reducing the number of integrations to two nested integrals by performing a non - relativistic expansion of the current operators @xcite . the latter approach is only useful for some pieces of the elementary 2p-2h response . in this work we present a fast and very efficient method to calculate the inclusive 2p-2h responses in the relativistic fermi gas model ( rfg ) this approach , denoted as the frozen nucleon approximation , was first explored in @xcite but restricted to the analysis of the 2p-2h phase - space . here it is extended to the evaluation of the full hadronic tensor assuming that the initial momenta of the two struck nucleons can be neglected for high enough energy and momentum transfer , @xmath4 . the frozen nucleon approximation was found to work properly in computing the phase space function for two - particle emission in the range of momentum transfers of interest for neutrino experiments with accelerators . here we investigate the validity of the frozen approximation beyond the phase - space study by including the electroweak meson - exchange current ( mec ) model of @xcite . we find that the presence of virtual delta excitations requires one to introduce a `` frozen '' @xmath1-propagator , designed by a convenient average over the fermi sea . the main advantage of the frozen approximation consists in reducing the number of nested integrals needed to evaluate the inclusive 2p-2h electroweak responses from 7 ( full calculation ) to 1 . thus it is well - suited to computing the 2p-2h neutrino cross sections folded with the neutrino flux , and it can be of great help in order to implement the 2p-2h models in the monte carlo codes currently available . the plan of this work is as follows : in section [ sec_form ] we review the formalism of neutrino scattering and describe mathematically the frozen approximation approach . in section [ sec_results ] we validate the nucleon frozen approximation by computing the 2p-2h response functions and by comparing with the exact calculation . finally , in section [ sec_conclusions ] we summarize our conclusions . the double - differential inclusive @xmath5 or @xmath6 cross section is given by @xmath7 \ , , \end{aligned}\ ] ] where the sign @xmath8 is positive for neutrinos and negative for antineutrinos . the term @xmath9 in eq . ( [ cross ] ) represents the elementary neutrino scattering cross section with a point nucleon , while the @xmath10 are kinematic factors that depend on lepton kinematic variables . their explicit expressions can be found in @xcite . the relevant nuclear physics is contained in the five nuclear response functions @xmath11 , where @xmath12 is the momentum transfer , defining the @xmath13 direction , and @xmath14 is the energy transfer . they are defined as suitable combinations of the hadronic tensor @xmath15 in this work we compute the inclusive hadronic tensor for two - nucleon emission in the relativistic fermi gas , given by @xmath16 where @xmath17 by momentum conservation , @xmath18 is the nucleon mass , @xmath19 is the volume of the system and we have defined the product of step functions @xmath20 with @xmath21 the fermi momentum . finally the function @xmath22 is the elementary hadron tensor for the 2p-2h transition of a nucleon pair with given initial and final momenta , summed up over spin and isospin , @xmath23 which is written in terms of the antisymmetrized two - body current matrix elements @xmath24 the factor @xmath25 in eq . ( [ elementary ] ) accounts for the antisymmetry of the two - body wave function . for the inclusive responses considered in this work there is a global axial symmetry , so we can fix the azimuthal angle of one of the particles . we choose @xmath26 , and consequently the integral over @xmath27 gives a factor @xmath28 . furthermore , the energy delta function enables analytical integration over @xmath29 , and so the integral in eq . ( [ hadronic ] ) can be reduced to 7 dimensions ( 7d ) . in the `` exact '' results shown in the next section , this 7d integral has been computed numerically using the method described in @xcite . the frozen nucleon approximation consists in assuming that the momenta of the initial nucleons can be neglected for high enough values of the momentum transfer . thus , in the integrand of eq . ( [ hadronic ] ) , we set @xmath30 , and @xmath31 . we roughly expect this approximation to become more accurate as the momentum transfer increases . the integration over @xmath32 is trivially performed and the response function @xmath33 , with @xmath34 , is hence approximated by @xmath35 where @xmath36 and @xmath37 are the elementary response functions for a nucleon pair excitation , which are defined similarly to eqs . ( [ rcc][rtprima ] ) . the integral over @xmath38 can be done analytically by using the delta function for energy conservation , and the integral over @xmath39 gives again a factor of @xmath28 . thus only an integral over the polar angle @xmath40 remains : @xmath41 where the sum runs over the , in general two , possible values of the momentum of the first particle for given emission angle @xmath42 . these are obtained as the positive solutions @xmath43 of the energy conservation equation @xmath44 the explicit values of the solutions of the above equation can be found in the appendix of @xcite . care is needed in performing the integral over @xmath42 because the denominator inside the integral can be zero for some kinematics . the quadrature in these cases can be done with the methods explained in @xcite . to investigate the validity of the frozen nucleon approximation , we have to choose a specific model for the two - body current matrix elements @xmath45 entering in the elementary 2p-2h response functions , eqs . ( [ elementary],[anti ] ) . here we use the relativistic model of electroweak mec operators developed in @xcite . the mec model can be summarized by the feynman diagrams depicted in fig . [ fig_feynman ] . it comprises several contributions coming from the pion production amplitudes of @xcite . the seagull current , corresponding to diagrams ( a , b ) , is given by the sum of vector and axial - vector pieces @xmath46_{1'2',12 } \frac{f^2_{\pi nn}}{m^2_\pi } \frac{\bar{u}_{s^\prime_1}({{\bf p}}^\prime_1)\,\gamma_5 { \not{\!k}}_{1 } \ , u_{s_1}({{\bf h}}_1)}{k^2_{1}-m^2_\pi}\ , \nonumber\\ & \times & \bar{u}_{s^\prime_2}({{\bf p}}^\prime_2 ) \left [ f^v_1(q^2)\gamma_5 \gamma^\mu + \frac{f_\rho\left(k_{2}^2\right)}{g_a}\,\gamma^\mu \right ] u_{s_2}({{\bf h}}_2 ) \nonumber\\ & + & ( 1\leftrightarrow2 ) \ , , \label{seacur } \end{aligned}\ ] ] where @xmath47 corresponds to the @xmath8-components of the two - body isovector operator @xmath48 $ ] . the @xmath49 sign refers to neutrino ( antineutrino ) scattering . the four - vector @xmath50 is the momentum carried by the exchanged pion and @xmath51 . the @xmath52 ( @xmath53 ) and axial ( @xmath54 ) couplings , and the form factors ( @xmath55 , @xmath56 ) have been taken from the pion production amplitudes of @xcite . the pion - in - flight current corresponding to diagram ( c ) is purely vector and is given by @xmath57_{1'2',12 } \frac{f^2_{\pi nn}}{m^2_\pi}\ , \frac { f^v_1(q^2 ) \ , \left(k^\mu_{1}-k^\mu_{2}\right)}{\left(k^2_{1}-m^2_\pi\right ) \left(k^2_{2}-m^2_\pi\right)}\ , \nonumber \\ & \times & \bar{u}_{s^\prime_1}({{\bf p}}^\prime_1)\,\gamma_5 { \not{\!k}}_{1 } \ , u_{s_1}({{\bf h}}_1 ) \bar{u}_{s^\prime_2}({{\bf p}}^\prime_2)\,\gamma_5 { \not{\!k}}_{2 } \ , u_{s_2}({{\bf h}}_2 ) \ , , \end{aligned}\ ] ] where @xmath58 is the momentum of the pion absorbed by the second nucleon . the pion - pole current corresponds to diagrams ( d , e ) and is purely axial , given by @xmath59_{1'2',12 } \frac{f^2_{\pi nn}}{m^2_\pi}\ , \frac{f_\rho\left(k_{1}^2\right)}{g_a}\;q^\mu \bar{u}_{s^\prime_1}({{\bf p}}^\prime_1){\not{\!q}}u_{s_1}({{\bf h}}_1 ) \nonumber\\ & \times & \frac { \bar{u}_{s^\prime_2}({{\bf p}}^\prime_2)\,\gamma_5 { \not{\!k}}_{2 } \ , u_{s_2}({{\bf h}}_2 ) } { \left(k^2_{2}-m^2_\pi\right)\left(q^2-m^2_\pi\right ) } + ( 1\leftrightarrow2 ) \,.\end{aligned}\ ] ] finally the @xmath1 current corresponds in fig . [ fig_feynman ] to diagrams ( f , g ) for the forward and ( h , i ) for the backward @xmath1 propagations , respectively . the current matrix elements are given by @xmath60_{1'2',12 } \frac{\bar{u}_{s^\prime_2}({{\bf p}}^\prime_2)\,\gamma_5 { \not{\!k}}_{2 } \ , u_{s_2}({{\bf h}}_2)}{k^2_{2}-m^2_\pi } \nonumber\\ & \times & k^\alpha_{2}\,\bar{u}_{s^\prime_1}({{\bf p}}^\prime_1)g_{\alpha\beta}(h_1+q ) \gamma^{\beta\mu}(h_1,q)u_{s_1}({{\bf h}}_1 ) \nonumber\\ & + & ( 1\leftrightarrow2 ) \label{delta_forward } \\ j^\mu_{\delta,\rm b}&= & \frac{f^ * f_{\pi nn}}{m^2_\pi}\ , \left[u_{\rm b}^{\pm}\right]_{1'2',12}\ ; \frac{\bar{u}_{s^\prime_2}({{\bf p}}^\prime_2)\,\gamma_5 { \not{\!k}}_{2 } \ , u_{s_2}({{\bf h}}_2)}{k^2_{2}-m^2_\pi } \nonumber\\ & \times & k^\beta_{2}\,\bar{u}_{s^\prime_1}({{\bf p}}^\prime_1 ) \hat{\gamma}^{\mu\alpha}(p^\prime_1,q ) g_{\alpha\beta}(p^\prime_1-q)u_{s_1}({{\bf h}}_1 ) \nonumber\\ & + & ( 1\leftrightarrow2)\label{delta_backward } \ , .\end{aligned}\ ] ] the @xmath61 coupling is @xmath62 . the forward , @xmath63 , and backward , @xmath64 , isospin transition operators have the following cartesian components @xmath65 where @xmath66 and @xmath67 are the isovector transition operators from isospin @xmath68 to @xmath69 or vice - versa , respectively . the @xmath70 operator is for neutrino ( antineutrino ) scattering . the @xmath1-propagator , @xmath71 , is given by @xmath72 where @xmath73 is the projector over spin-@xmath68 on - shell particles , @xmath74\end{aligned}\ ] ] and whose denominator has been obtained from the free propagator for stable particles , @xmath75 , with the replacement @xmath76 to take into account the finite decay width of the @xmath77 . the tensor @xmath78 in the forward current is the weak @xmath79 transition vertex a combination of gamma matrices with vector and axial - vector contributions : @xmath80\gamma_5 \\ \gamma^{\beta\mu}_a(p , q)&= & \frac{c^a_3}{m_n}\left(g^{\beta\mu}{\not{\!q}}- q^\beta\gamma^\mu\right)\nonumber\\ & + & \frac{c^a_4}{m^2_n}\left(g^{\beta\mu}q\cdot \left(p+q\right)- q^\beta \left(p+q\right)^\mu\right ) \nonumber\\ & + & c^a_5 g^{\beta\mu}+\frac{c^a_6}{m^2_n } q^\beta q^\mu \ , .\end{aligned}\ ] ] for the backward current , we take @xmath81^{\dagger } \gamma^0 \ , .\ ] ] finally , it is worth noting that the form factors @xmath82 are taken from @xcite . we refer to that work for further details of the model . the evaluation of the relevant elementary responses requires one to contract the electroweak two - body mec with themselves by spin - isospin summation . this leads to the squares of each of the diagrams depicted in fig . [ fig_feynman ] plus all their interferences . the validity of the frozen nucleon approximation relies on the fact that the integrand inside the 2p-2h response is a function that depends slowly on the momenta of the two initial nucleons inside the fermi sea . in that case the mean - value theorem applied to the resolution of the integrals provides very precise results . this is so for all of the diagrams of the mec except for the forward @xmath1 diagram , which shows a sharp maximum for kinematics around the @xmath1 peak for pion emission , located at @xmath83 . this is due to the denominator in the @xmath1 propagator , @xmath84 where @xmath85 is the momentum of the hole that gets excited to a @xmath1 . in these cases the integrand changes very significantly with a small variation of the momentum of the holes and consequently , the frozen approximation can not properly describe the integrand . on the contrary , it only provides a general estimation of the order of magnitude . to get rid of these difficulties we have developed a prescription to deal with the forward @xmath1-propagator appearing in eq . ( [ delta_forward ] ) . this procedure is based on the use of an effective propagator ( `` frozen '' ) for the @xmath1 , conveniently averaged over the fermi gas . this average is an analytical complex function , which is used instead of the `` bare '' propagator inside the frozen approximation , recovering the precision of the rest of diagrams . the `` frozen '' prescription amounts to the replacement : @xmath86 where the frozen denominator is defined by @xmath87 taking the non - relativistic limit for the energies of the holes ( @xmath88 ) , which is justified because hole momenta are below the fermi momentum , itself a value far below the nucleon rest mass , we can write : @xmath89 where @xmath90 assuming the @xmath1 width ( @xmath91 ) to be constant , we can integrate eq . ( [ averaged_prop ] ) over the angles , getting @xmath92 \,.\ ] ] note the complex logarithm inside the integral , which provides the needed kinematical dependence of the averaged propagator , differing from the bare lorentzian shape . finally the integral over the momentum @xmath93 can also be performed , resulting in @xmath94 \right\ } \ , . \nonumber\end{aligned}\ ] ] by comparing the response functions evaluated in the frozen approximation , _ i.e. , _ substituting the denominator of the @xmath1 propagator in eq . ( [ delta_prop ] ) for the frozen expression in eq . ( [ averaged_denom ] ) , with the exact results , we find that the shapes around the @xmath1 peak are similar , but with slightly different width and position of the center of the peak . we have checked that the differences can be minimized by changing the parameters @xmath95 with respect to the `` bare '' ones , given by eqs . ( [ a],[b ] ) . this is because we have computed the averaged denominator without taking into account the current matrix elements appearing in the exact responses , although the functional form and kinematical dependence is the appropriate one . in practice , we adjust @xmath91 and apply a shift in the expression for @xmath96 in eq . ( [ a ] ) in order to obtain the best approximation to the exact results . the effective `` frozen '' parameters we actually introduce in eq . ( [ averaged_denom ] ) , are given by @xmath97 we consider @xmath98 and the frozen shift , @xmath99 , to be tunable parameters depending on the momentum transfer @xmath100 . we have adjusted these parameters for different @xmath100-values and we provide them in table [ table1 ] . .values of the free parameters of the fermi - averaged @xmath1-propagator for different kinematic situations corresponding to different values of the momentum transfer @xmath100 . [ cols="^,^,^",options="header " , ] -propagator in frozen approximation compared to the average propagator . in this evaluation we have taken @xmath101 mev / c and @xmath102 mev.,width=302 ] in this section we validate the frozen approximation by computing the approximate 2p-2h response functions and comparing with the exact results in the rfg . we consider the case of the nucleus @xmath103c with fermi momentum @xmath104 mev / c , and show the different response functions for low to high values of the momentum transfer . for other nuclei with different @xmath21 the frozen parameters of table 1 should be determined again , and we expect their values change slightly . in fig . [ fig_denom ] we show the modulus squared of the @xmath1 propagator , given by the @xmath105 function defined in eq . ( [ denominator ] ) , computed for @xmath106 , as a function of @xmath14 for @xmath107 gev / c . it presents the typical lorentzian shape corresponding to width @xmath102 mev . we observe a narrow peak around @xmath108 mev . this corresponds to the @xmath1-peak position for @xmath107 gev / c . in the same figure we also show the square of the frozen average @xmath109 ( solid line ) . the resulting peak is quenched and broadened as compared to the lorentzian shape , reducing its strength and enlarging its width . this behavior of the averaged @xmath1-propagator drives the actual shape of the exact 2p-2h nuclear responses , being more realistic than the simple lorentzian shape of the frozen approximation without the average , as we will see below . of @xmath103c for different momentum transfers @xmath100 . the exact results are compared to the frozen approximation with and without the averaged @xmath1 propagator . , width=283 ] in fig . [ fig_rt ] we show the weak transverse 2p-2h response function of @xmath103c for four different values of the momentum transfer ranging from 300 to 1500 mev / c . the curves correspond to different calculations or approximations made in the evaluation of the responses , as labeled in the legend . the solid line corresponds to the seven - dimensional calculation with no approximations . the other two curves refer to the different frozen nucleon approximations developed in this work : the dashed line is obtained with eq . ( [ frozen_eq ] ) but performing the replacement expressed in ( [ replacement_rule ] ) for the forward @xmath1-excitation terms in the evaluation of the current matrix elements ; on the contrary , the dotted line corresponds to the same frozen nucleon approximation , eq . ( [ frozen_eq ] ) , but without the fermi - average of the @xmath1-propagator in the forward terms . as it can be seen from fig . [ fig_rt ] , for those values of the momentum transfer for which the @xmath1-peak is not reached ( the panel with @xmath110 mev / c ) , there is really little difference between averaging or not the @xmath1 propagator . this is certainly not the case when the @xmath1-peak is fully reached , as shown in the other panels . in this situation there is a dramatic difference between performing the fermi - average of the @xmath1-propagator or not . this difference is in consonance with the results shown in the previous fig . [ fig_denom ] , and it shows how crucial is the treatment of the @xmath1-propagator to obtain accurate results for the 2p-2h responses in the frozen nucleon approximation , i.e , with only one integration . the results in fig . 3 have been obtained after fitting the parameters @xmath111 for the fermi - averaged @xmath1-propagator at the different values of the momentum transfer quoted in table [ table1 ] . it is also worth noting that there is no way of converting the dotted line into the dashed one by only a suitable fitting of these parameters , _ i.e. , _ without averaging the @xmath1-propagator . c within different models for two values of the momentum transfer . the exact rfg results and the frozen approximation are compared with the shell model ( sm ) results of @xcite . the total shell model results ( 1p-1h ) + ( 2p-2h ) are also shown for comparison . , width=302 ] c. several values of the momentum transfer are displayed : @xmath112 and 2000 mev / c . , width=302 ] in fig . [ fig_shellmodel ] we show results for the transverse electromagnetic 2p-2h response function . the frozen and exact ( 7d ) @xmath113 response of the rfg are compared with the results obtained in the shell model 2p-2h calculation of @xcite . this was one of the first computations of the 2p-2h response within the nuclear shell model . the total nuclear response in the shell model , obtained by adding the 1p-1h to the 2p-2h channel , is also shown to appreciate the relative size of the 2p-2h contribution to the total result . as shown in fig . [ fig_shellmodel ] , the fermi gas results ( either in frozen approximation or not ) are similar to the shell model ones . the small discrepancy between them can not be attributed to relativistic effects because of the low momentum transfer values considered , but to the different coupling constants and form factors used in the model of the @xmath1 meson - exchange current considered in @xcite and the present approach . we can remark the slightly different threshold effects between both calculations . these effects are , as expected , very sensitive to the treatment of the nuclear ground state . note also that the frozen approximation describes reasonably well this low momentum @xmath110 mev / c , considering the simplifications involved . finally in fig . [ fig5 ] we show that the frozen approximation works notably well in a range of momentum transfer from low to high values of @xmath100 . we compare the @xmath113 , @xmath114 , and @xmath115 2p-2h responses in frozen approximation with the exact results obtained computing numerically the 7d integral of the hadronic tensor . the accord is particularly good for the two transverse responses which dominate the cross section . a slight disagreement occurs for very low energy transfer at threshold where the response functions are anyway small . in the case of the cc response function some tiny differences are observed . however , note that this response is small because the dominant @xmath1 current is predominantly transverse . moreover , its global contribution to the cross section is not very significant because it is partially canceled with the contribution of the cl and ll responses . [ [ physical - interpretation - of - the - frozen - approximation ] ] physical interpretation of the frozen approximation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the validity of the frozen approximation led us to conclude that , in the inclusive responses for two - particle emission , the detailed information about the momenta carried out by the two nucleons is lost . this is because the energy and momentum transfer @xmath116 are shared by the two nucleons in multiple ways . this is reminiscent from the phase - space kinematical dependence ( which can be obtained setting the elementary response @xmath117 to unity ) already seen in @xcite . the soft dependence of the elementary response on the initial momenta makes the same argument applicable to the full responses with the exception of the @xmath1 forward current that requires one to soften and average the rapid variations of the @xmath1 propagator . only the low - energy region where the sharing is highly restricted and the cross section is therefore very small , is found to be sensitive to the details of the initial state . this is also supported by the comparison between the shell model and the rfg . on the other hand , in the 2p-2h model of @xcite , an average momentum @xmath118 was determined by imposing quasi - deuteron kinematics . note that this condition is similar to the present frozen approach , but this only guarantees that the total momentum of the two holes is zero , corresponding to selecting back - to - back pair configurations in the ground state only . in this work we have introduced and validated the frozen nucleon approximation for a fast and precise calculation of the inclusive 2p-2h response functions in a relativistic fermi gas model . this approximation neglects the momentum dependence of the two holes in the ground state and requires the use of an effective propagator for the @xmath1 resonance conveniently averaged over the fermi sphere , for which we have provided a simple analytical expression . for momentum transfers above the fermi momentum this approximation makes it possible to compute the responses with only a one - dimensional integral . taking into account all the uncertainties in modeling the two - nucleon emission reactions , this approach can be used instead of the full 7d integral , obtaining very satisfactory results . although we have used a specific model of mec to prove the validity of the approximation , it is reasonable to expect that the frozen approach is also valid for other 2p-2h models . this can be of great interest when implementing 2p-2h models in monte carlo event generators , which up to now have relied on parameterizations from external calculations . in summary , the frozen approximation enables one to make 2p-2h calculations very efficiently and rapidly , instead of interpolating pre - calculated tables , including allowing the parameters of the models to be modified inside the codes , if desired . finally , in the near future this study will be extended to an exploration of how the 2p-2h mec responses depend on nuclear species @xcite . this work has been partially supported by the spanish ministerio de economia y competitividad and erdf ( european regional development fund ) under contracts fis2014 - 59386-p , fis2014 - 53448-c2 - 1 , by the junta de andalucia ( grants no . fqm-225 , fqm160 ) , by the infn under project manybody , and part ( twd ) by the u.s . department of energy under cooperative agreement de - fc02 - 94er40818 . irs acknowledges support from a juan de la cierva fellowship from mineco ( spain ) . gdm acknowledges support from a junta de andalucia fellowship ( fqm7632 , proyectos de excelencia 2011 ) . i. ruiz simo , j. e. amaro , m. b. barbaro , a. de pace , j. a. caballero and t. w. donnelly , arxiv:1604.08423 [ nucl - th ] . j. e. amaro , m. b. barbaro , j. a. caballero , t. w. donnelly and c. maieron , phys . c * 71 * ( 2005 ) 065501 .

we present a fast and efficient method to compute the inclusive two - particle two - hole ( 2p-2h ) electroweak responses in the neutrino and electron quasielastic inclusive cross sections . the method is based on two approximations . the first neglects the motion of the two initial nucleons below the fermi momentum , which are considered to be at rest . this approximation , which is reasonable for high values of the momentum transfer , turns out also to be quite good for moderate values of the momentum transfer @xmath0 . the second approximation involves using in the `` frozen '' meson - exchange currents ( mec ) an effective @xmath1-propagator averaged over the fermi sea . within the resulting `` frozen nucleon approximation '' , the inclusive 2p-2h responses are accurately calculated with only a one - dimensional integral over the emission angle of one of the final nucleons , thus drastically simplifying the calculation and reducing the computational time . the latter makes this method especially well - suited for implementation in monte carlo neutrino event generators . neutrino scattering , meson - exchange currents , 2p-2h . 25.30.pt , 25.40.kv , 24.10.jv

arxiv

accretion of gas onto a super massive black hole ( smbh ) in the nucleus of galaxies is believed to be the source of activity in quasars and seyfert galaxies ( commonly known as active galactic nuclei ( agns ) ; cf . rees 1984 ) . several studies have suggested that the mass of the smbh in these objects is correlated with the luminosity , mass and velocity dispersion of the stellar spheroid of the galaxies ( kormendy & richstone 1995 ; magorrian et al . 1998 ; ferrarese & merritt 2000 ; gebhardt et al . 2000 ; marconi & hunt 2003 ; hring & rix 2004 ) . such correlations may imply an evolutionary relation between the growth of the smbh and the host galaxy itself ( e.g. somerville et al . 2008 ; shankar et al . 2009 ; hopkins & hernquist 2009 ) . in order to study the dependence of the various observed phenomena of agns on the black hole mass and the cosmic evolution of the black holes , independent and reliable estimates of the mass of the black holes are required ( e.g. , goulding et al . 2010 ; rafter , crenshaw & wiita 2009 ) . one independent method to estimate the mass of the black hole is using the reverberation mapping technique ( blandford & mckee 1982 ; peterson 1993 ) . in the optical bands , the continuum flux of some agns , is known to vary on timescales as short as hours ( e.g. , miller , carini & goodrich 1989 ; stalin et al . 2004 ) . if the main source of ionization of the broad line region ( blr ) is the continuum itself , any variation of the continuum emission can also be seen in the broad emission lines . however , the variations in the broad line flux will have a time lag ( @xmath6 ) relative to the continuum variations , which can be interpreted as the light travel time across the blr . as a first approximation , therefore , the size of the blr is @xmath7 , where @xmath8 is the velocity of light . once the @xmath9 is obtained , the mass of the black hole can also be estimated , using the velocity dispersion of the broad component of the emission lines , @xmath10 , and assuming virial equilibrium ( peterson et al . 2004 ; p04 ) ; see peterson 2010 , for a recent review ) . the reverberation mapping technique has been used to make estimates of smbh masses over a large range of redshift . however , because the technique is observationally taxing , as it demands an enormous amount of telescope time , to date the blr radius of only about three dozen agns ( seyfert 1 galaxies and quasars ) have been determined ( p04 ; kaspi et al . 2007 ; bentz et al . 2009a ; denney et al . 2009 , 2010 ) . nevertheless , using these estimates a correlation was found between @xmath9 and the optical continuum luminosity at 5100 ( kaspi et al . 2000 ; kaspi et al . 2007 ; p04 ; denney et al . 2009 ; bentz et al . the r@xmath11@xmath12@xmath13l@xmath14 relation can be considered well constrained between the luminosities 10@xmath15 erg sec@xmath16l@xmath17 erg sec@xmath18 . on the other hand , for luminosities below 10@xmath15 erg sec@xmath18 , only a handful of sources are observed , and the estimated values of @xmath9 could also indicate a flattening of the relation ( see fig . 2 of kaspi et al . this flattening would suggest a lower limit in the possible masses of smbhs in galaxies . although recent revisions of a few sources made by bentz et al . ( 2006 ) and denney et al . ( 2009;2010 ) are consistent with a continuation of the @xmath19l@xmath14 relation to lower luminosities , and consequently with no lower limit in the mass for the smbh , the correlation is still sparsely sampled . moreover , the @xmath19l@xmath14 relation is very useful for estimating the smbh masses from single - epoch spectra and calibrating other surrogate relations used for black hole mass estimates ( vestergaard 2004 ; shen et al . therefore , estimates of @xmath9 for a larger number of sources are required . the extrapolation of the known @xmath19l@xmath14 relation to low luminosities suggests that the time lag between the variations of the broad line and that of the continuum will be of the order of hours to days , as compared to several months for high luminosity sources . thus , monitoring programs of short durations , but fast sampling , are required to estimate the reverberation time lags for low luminosity sources . in this paper , we present the optical spectroscopic and photometric observations of a new low luminosity agn , the x - ray source and seyfert 1.5 galaxy h 0507 + 164 . based on a reverberation mapping campaign that lasted for about a month , during november - december 2007 , we have obtained @xmath9 and estimated the mass of the smbh . in section 2 , the observations and data reductions are described . the results of the analysis are given in section 3 , and the conclusions are presented in section 4 . using the vron - cetty & vron catalogue of quasars and active galactic nuclei ( 12th ed . ; vron - cetty & vron 2006 ) , we have compiled a list of nearby seyfert 1 galaxies , which , based on the available spectra , have a luminosity at @xmath20 of the order of 10@xmath21 erg sec@xmath18 or lower . very few candidates were found ( mostly because of the absence of available spectra ) . the source , h 0507 + 164 , that we selected for our campaign is identified in the catalogue of vron - cetty & vron as an x - ray source , with coordinates @xmath22 , and is classified as a seyfert 1.5 galaxy at a redshift of @xmath0 . optical spectroscopic and photometric observations of h 0507 + 164 were carried out in 2007 between 21 of november and 26 of december at the 2 m himalayan chandra telescope ( hct ) , operated by the indian institute of astrophysics , bangalore . the telescope is equipped with a @xmath23 ccd , coupled to the himalayan faint object spectrograph and camera ( hfosc ) . in imaging mode , only the central @xmath24 pixels region of the ccd is used . the camera has a plate scale of @xmath25 arcsecond / pixel , which yields a field of view of @xmath26 square arcmin . medium resolution spectra of the nucleus were obtained using a 11 arcmin @xmath27 1.92 arcsec wide slit and a grism . the spectra have a spectral range of 3800@xmath126700 with a resolution of @xmath288 . the exposure time varied between 900 and 1000 seconds . the spectra were reduced using standard procedures in iraf . after bias subtraction and flat fielding , one dimensional spectra were extracted and calibrated , in wavelength using an fear lamp , and in flux using various observations of the spectrophotometric standard star feige 34 . since the observed spectra were of low s / n , for further analysis all the spectra were smoothed to a resolution of @xmath2815 . the standard technique of spectral flux calibration is not sufficiently precise to study the variability of agns . since even under good photometric conditions the accuracy of spectrophotometry is not better than 10% ( shapovalova et al . 2008 ) , we used a relative calibration procedure . a first order flux calibration was first obtained in the normal way using the standard star . then all the spectra were inter - calibrated relative to the spectra of one night ( we choose the 21 of november ) , assuming the flux of the narrow line [ o iii]@xmath29 is constant . this is justified , because the narrow line region ( nlr ) is much more extended ( of the order of a few hundred parsecs ) than the blr ( much less than a parsec ) and flux variation can not be observed in this region over short time scales ( cf . osterbrock 1989 ) . each spectra were scaled relative to the reference spectra using the scaling algorithm devised by van groningen & wanders ( 1992 ) . this algorithm uses a chi - square , to minimize the residuals of the [ o iii]@xmath135007 line after subtraction from the reference spectrum . the mean spectrum , averaging the 22 nights of observations , is shown in fig . [ fig : meanspec ] . in parallel to the spectroscopic observations , r - band images were also obtained with the hfosc . unfortunately , some observations turned out to be contaminated by the light of the extremely bright stars located near the source . as a consequence , our r - band photometry covers only 13 of the 22 nights of the campaign . the images were bias subtracted and flat fielded using iraf packages . for the rest of the reduction , calibration and analysis , packages in midas were used . after removing the cosmic rays , profile fitting photometry was done using the daophot and allstar packages . the observed r - band frame is shown in fig . [ fig : image ] . north is up and east is to the left . note the presence of an extremely bright star near h 0507 + 164 . [ fig : image ] , width=377,height=264 ] .continuum and @xmath30 fluxes for the source h 0507 + 164 [ table : fluxes ] [ cols=">,^,^ , > " , ] the lightcurves of the h@xmath2 flux and the continuum at 5100 were obtained using the final inter - calibrated spectra . the continuum flux in the rest - frame of the galaxy at 5100 was obtained using the mean flux within the observed band from 5172 to 5200 . the h@xmath2 emission line fluxes were obtained by integrating the emission profile in the band spanning 4884@xmath125012 , after subtracting a continuum . an average of the mean fluxes in the regions on the blue ( 4808@xmath124852 ) and red ( 5012@xmath125024 ) sides of the h@xmath2 line was used as the continuum below the line . although the measured line fluxes include both the narrow and broad components , any variation observed in the line fluxes can be attributed to the broad component only , since the narrow component is not expected to vary during the period of our observations . the lightcurves for the continuum at 5100 , for the h@xmath2 and for the [ o iii]@xmath135007 lines are shown in fig . [ fig : lightcurve ] . the corresponding fluxes for the continuum and h@xmath2 are listed in table [ table : fluxes ] . as expected from the inter - calibration procedure , the lightcurve for [ o iii]@xmath135007 is nearly constant . on the other hand , both the continuum at 5100 and the h@xmath2 flux are observed to vary . @xmath135007 , h@xmath2 and continuum at 5100 are plotted . the fluxes are in units of 10@xmath31 erg s@xmath18 @xmath32 @xmath18 for the [ o iii]@xmath135007 line , 10@xmath33 erg s@xmath18 @xmath32 @xmath18 for the h@xmath34 line and 10@xmath35 erg s@xmath18 @xmath32 @xmath18 for the continuum . [ fig : lightcurve ] , width=302 ] the observed r - band differential instrumental magnitudes between the galaxy and the comparison star ( marked in fig . [ fig : lightcurve ] ) are also given in table [ table : fluxes ] . the r - band differential lightcurve plotted in fig . [ fig : lightcurve ] ( top panel ) closely follows the lightcurves of the continuum at 5100 and h@xmath2 . the average flux at 5100 is @xmath36 erg @xmath32s@xmath18@xmath18 , which corresponds to @xmath13l@xmath14 of @xmath37 erg s@xmath18 , for the cosmological parameters h@xmath38 km s@xmath18mpc@xmath18 , @xmath39 and @xmath40 . the variability of the light curves are characterised by the parameters , excess variance , f@xmath41 and the ratio between the maximum and minimum flux of the light curves , r@xmath42 ( rodriguez - pascual et al . 1997 ; edelson et al . the continuum and h@xmath2 line have f@xmath41 of 0.16 and 0.18 and r@xmath42 of [email protected] and [email protected] respectively . these values are within the range of values found by other variability studies of agns ( cf . p04 ; denney et al . 2010 ) . the time lag between the variations of the continuum flux and the variations of the h@xmath2 emission can be determined by cross - correlating the two light curves . for cross correlation analysis , the method of interpolated cross - correlation function ( iccf ; gaskell & sparke 1986 ; gaskell & peterson 1987 ) and the method of discrete correlation function ( dcf ; edelson & korlik 1998 ) were used . although both iccf and dcf methods produce similar results ( white & peterson 1994 ) , the interpolation of the light curve during the period of gaps required by the iccf method might not be a reasonable approximation to the behavior of the light curves . thus the dcf method is preferable for data with large gaps ( denney et al . 2009 ) . for comparison , we obtain the results using both the methods . the results of the cross correlation analysis are shown in fig . [ fig : ccf ] . the cross correlation function ( ccf ) obtained using the iccf method is plotted as a thick solid line . the auto correlation functions ( acfs ) of the continuum at 5100 and the @xmath30 line are also shown in fig . [ fig : ccf ] as dashed and dotted lines respectively . for comparison the cross correlation function obtained using the dcf method is also plotted as a thin solid line . as expected , the auto correlations have zero time lags . on the other hand , the time lag in the cross correlation curve is clearly noticeable as an overall shift to the right . the position of the maximum in the ccf provides an estimate of the time lag between the continuum and the h@xmath34 line . the maximum was however determined using the centroid , which gives a better estimate for noisy ccfs , rather than the peak , using the formula : @xmath44 the estimate of the centroid includes all the points that are within 50% of the peak value of the ccf . based on this cross correlation analysis , a statistically significant centroid and the associated uncertainty were obtained using a bootstrap technique that introduces effects of randomness in fluxes and sampling of the light curve ( cf . p04 ) . a method to carry out monte - carlo ( mc ) simulation using the combined effects of flux randomization ( fr ) and the random subset selection ( rss ) procedures is described in peterson et al . additional improvements as suggested by welsh ( 1999 ) are summarised in p04 , which we use for our analysis . lightcurves , as obtained using the iccf method , is plotted as as a thick solid line . the auto correlation functions of the continuum at 5100 and the @xmath30 line are plotted as dashed and dotted lines respectively . the cross correlation function obtained using the dcf method is plotted as a thin solid line . , width=302,height=302 ] first an rss procedure was applied by randomly selecting 22 observations from the light curve . the flux uncertainties of the multiply selected observations were weighted according to welsh ( 1999 ) . this light curve was given as input to the fr procedure , where each measured value of the fluxes are modified by adding the measured flux uncertainties multiplied with a random gaussian value . the modified light curves were then cross - correlated and the centroids were determined as outlined above using ccf values above 50% of the peak value . this procedure was repeated for @xmath45 times , retaining only those ccfs whose maximum cross - correlation coefficient is large enough such that the correlation is significant with a confidence level of 95% or larger . a cross - correlation centroid distribution ( cccd ) was built using the above centroids and is shown in fig . [ fig : ccfdist ] . the average value of cccd was taken to be @xmath46 . since the cccd is non - gaussian ( cf . peterson et al . 1998 ) , the upper and lower uncertainties in @xmath46 were determined such that 15.87% of cccd realizations have @xmath47 and 15.87% realizations have @xmath48 . this error in @xmath46 corresponds to @xmath49 errors for a gaussian distribution . lccccc & + method & & 2 & 2.5 & 3 & 3.5 & 4 + iccf & 3.14@xmath50 & 3.28@xmath51 & 3.27@xmath52 & 3.06@xmath53 & 3.05@xmath54 + + dcf & 3.91@xmath55 & 4.30@xmath56 & 4.19@xmath57 & 4.11@xmath58 & 4.03@xmath59 + the centroid time lags obtained using different cross correlation methods for different bin sizes are given in table [ table : centroid ] . the variations due to different bin sizes are within the error bars . the dcf method gives a mean time lag of about 4 days , whereas the iccf method gives a mean value of about 3 days . this difference is of the order of the estimated errors on the time lags . considering these uncertainties , both the dcf and iccf methods give time delays that are consistent with each other . this suggests that the estimated time lag is not a spurious result for the sampling of the light curves presented here , and the results of the iccf methods are reliable . to be conservative , for further calculations we use the time lag with the largest scatter corresponding to a bin size of 3.5 days , obtained using the iccf method . based on this analysis the average observed frame time lag between the h@xmath60 and the @xmath135100 continuum light - curves was found to be @xmath61 days . after correcting for the time dilation effects using the redshift of the source , we found a time lag of @xmath62 days in the rest frame of the source . the wavelength coverage of our observations , also include the h@xmath63 line , and a time lag using the h@xmath63 line can also be estimated . however , by repeating the analysis procedure presented here , a reliable time lag using h@xmath63 line could not be found , because the correlation curves are too noisy . this may be due to the shorter duration of the observations and the relative flux calibration procedure using the [ o iii ] @xmath29 line situated much farther apart in wavelength from h@xmath63 being unreliable ( grier et al . 2008 ) . unfortunately , the nearby doublet [ s ii ] @xmath64 6716,6731 narrow lines , which could be used for the relative calibration of h@xmath63 , are too weak . thus we do not estimate the time lag using the h@xmath63 line . , width=302,height=302 ] in order to relate the time lag to the mass of the black hole , an estimate of the line width , the line dispersion @xmath10 , of the broad emission component of h@xmath2 , is required . following p04 , it is relatively straight forward and more practical to measure @xmath10 , the second moment of the profile , directly from the root mean square ( rms ) spectrum . indeed , in the rms spectrum the constant components , or those that vary on timescales much longer than the duration of the observation vanish , thus largely obviating the problem of de - blending the lines . to obtain the rms spectrum , all the observed spectra were combined using the formula : @xmath65}^2 \right\}^{1 \over 2}\ ] ] in fig . [ fig : rmsspec ] , we show the root mean square ( rms ) spectrum . it can be seen that the two narrow [ o iii ] lines have almost completely disappeared in the rms spectrum . the mean value of @xmath10 corrected for the instrumental response of the spectrograph and the associated uncertainty were obtained following the bootstrap method described in p04 . from our observed 22 observed spectra , we randomly selected 22 spectra , irrespective of whether a particular spectrum has already been selected or not . since some of the spectra were selected multiple times , the mean value of the resultant number of spectra were smaller by 8 . these randomly selected spectra were then used to construct an rms spectra from which @xmath10 was measured and corrected for the instrumental resolution of the spectrograph . this procedure was repeated 10000 times and the mean and standard deviation of these realisations are taken as @xmath10 and its uncertainty respectively . a distribution of @xmath10 values obtained from the bootstrap method is also shown in fig [ fig : sigmadist ] . we thus estimate a line dispersion of @xmath66 km sec@xmath18 . using the rest frame time delay , the radius of the blr is estimated to be @xmath67 pc . in fig . [ fig : rlum ] we show the measurement of @xmath9 and @xmath13l@xmath14 luminosity of the source presented here along with the most updated data set given by bentz et al . ( 2009b ) and the additional source , mrk 290 , given by denney et al . the solid line is the relation obtained by bentz et al . ( 2009b ) . from this figure it can be seen that our h@xmath60 measurement lags are in agreement with the known @xmath68 relationship . the mass of the black hole was estimated using the formula in p04 : @xmath69 where @xmath70 is the width of the line and g is the gravitational constant . the parameter @xmath71 is a scaling factor , which takes into account the geometry and kinematics of the blr . onken et al . ( 2004 ) found an empirical value of @xmath72 , using a sample of agns having both reverberation based black hole masses and host galaxy bulge velocity dispersion ( @xmath73 ) estimates . this value relies on the assumption that both agns and quiescent galaxies follow the same m@xmath74-@xmath73 relationship ( ferrarese & merritt 2000 ; gebhardt et al . 2000 ) . for this particular scaling , the appropriate velocity width @xmath70 is the line dispersion in the rms spectrum @xmath10 ( bentz et al . 2008 ) . adopting the onken et al . ( 2004 ) scaling factor and the @xmath10 measured from our observations , we estimated the mass of the smbh in h 0507 + 164 to be @xmath75 m@xmath5 . using the bootstrap method described in the text . [ fig : sigmadist ] , width=302,height=302 ] , width=355,height=264 ] we present for the first time monitoring observations of the x - ray source and sy 1.5 galaxy h 0507 + 164 , spanning a time period of about one month . we have obtained 22 nights of spectra during this period , with a mean sampling time of about 1.6 days . we measured an observed frame time lag of about @xmath61 days between the changes in the h@xmath2 emission line flux and the changes in the continuum flux at 5100 . after correcting for the redshift , we find a corresponding time lag of @xmath62 days in the rest frame of the source . from this measured time lag we deduced a size for the blr of @xmath67 parsec and estimated a black hole mass of @xmath76 m@xmath5 . our estimate of r@xmath11 using the measured lag of h@xmath60is in agreement with the r@xmath11@xmath12@xmath13l@xmath14 relationship shown by bentz et al . 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we present the results of our optical monitoring campaign of the x - ray source h 0507 + 164 , a low luminosity seyfert 1.5 galaxy at a redshift , @xmath0 . spectroscopic observations were carried out during 22 nights in 2007 , from the 21 of november to the 26 of december . photometric observations in the r - band for 13 nights were also obtained during the same period . the continuum and broad line fluxes of the galaxy were found to vary during our monitoring period . the r - band differential light curve with respect to a companion star also shows a similar variability . using cross correlation analysis , we estimated a time delay of @xmath1 days ( in the rest frame ) , of the response of the broad h@xmath2 line fluxes to the variations in the optical continuum at 5100 . using this time delay and the width of the h@xmath2 line , we estimated the radius for the broad line region ( blr ) of @xmath3 parsec , and a black hole mass of @xmath4m@xmath5 . [ firstpage ] galaxies : active - galaxies : individual ( h 0507 + 164 ) - galaxies : seyfert

arxiv

the broad emission iron lines are well - known features found in about two dozens of spectra of active galactic nuclei and black hole binaries . they are supposed to originate close to the black hole by the reflection of the primary radiation on the accretion disc . the spin of the black hole plays an important role in the forming of the line shape . especially , it determines the position of the marginally stable orbit which is supposed to confine the inner edge of the accretion disc ( see figure [ intro ] ) . the innermost stable orbit occurs closer to a black hole with a higher spin value . however , the spin affects also the overall shape of the line . over almost two decades the most widely used model of the relativistic disc spectral line has been the one by , which includes the effects of a maximally rotating kerr black hole . in other words , the laor model sets the dimensionless angular momentum @xmath1 to the canonical value of @xmath0 so that it can not be subject of the data fitting procedure . have relaxed this limitation and allowed @xmath1 to be fitted in the suite of ky models . other numerical codes have been developed independently by several groups ( , , ) and equipped with similar functionality . however , the laor model can still be used for evaluation of the spin if one identifies the inner edge of the disc with the marginally stable orbit . in this case the spin is actually estimated from the lower boundary of the broad line . the comparison of the laor and model is shown in the right panel of figure [ intro ] . the other parameters of the relativistic line models are inclination angle @xmath2 , rest energy of the line @xmath3 , inner radius of the disc @xmath4 , outer radius of the disc @xmath5 , emissivity parameters @xmath6 , @xmath7 with the break radius @xmath8 . the emissivity of the line is given by @xmath9 for @xmath10 and @xmath11 for @xmath12 . the angular dependence of the emissivity is characterized by limb darkening profile @xmath13 in the laor model . the model enables to switch between different emission laws . we used further two extreme cases , the with the same limb - darkening law as in the laor model and * with the limb - brightening law @xmath14 . the aim of this paper is to compare the two models applied to the current data provided by the xmm - newton satellite , and to the artificial data generated for the on - coming x - ray mission . for this purpose we have chosen two sources , mcg-6 - 30 - 15 and gx 339 - 4 , which exhibit an extremely skewed iron line according to recently published papers ( ) . and marginally stable orbit @xmath15 . right : comparison of the laor ( black , solid ) and ( red , dashed ) model for two values of the spin @xmath16 ( top ) and @xmath17 ( bottom ) . the other parameters of the line are @xmath18kev , @xmath19 , @xmath20.,scaledwidth=98.0% ] and marginally stable orbit @xmath15 . right : comparison of the laor ( black , solid ) and ( red , dashed ) model for two values of the spin @xmath16 ( top ) and @xmath17 ( bottom ) . the other parameters of the line are @xmath18kev , @xmath19 , @xmath20.,title="fig:",scaledwidth=98.0% ] + and marginally stable orbit @xmath15 . right : comparison of the laor ( black , solid ) and ( red , dashed ) model for two values of the spin @xmath16 ( top ) and @xmath17 ( bottom ) . the other parameters of the line are @xmath18kev , @xmath19 , @xmath20.,title="fig:",scaledwidth=98.0% ] we used the sas software version 7.1.2 ( http://xmm.esac.esa.int/sas ) to reduce the xmm - newton data of the sources . further , we used standard tools for preparing and fitting the data available at http://heasarc.gsfc.nasa.gov ( ftools , xspec ) the galaxy mcg-6 - 30 - 15 is a nearby seyfert 1 galaxy ( @xmath21 ) . the skewed iron line has been revealed in the x - ray spectra by all recent satellites . the xmm - newton observed mcg-6 - 30 - 15 for a long 350ks exposure time during summer 2001 ( revolutions 301 , 302 , 303 ) . the spectral results are described in . we joined the three spectra into one using the ftool mathpha . the black hole binary gx 339 - 4 exhibited a strong broadened line in the 76ks observation in 2002 ( ) when the source was in the very high state ( for a description of the different states see ) . the observation was made in the _ burst mode _ due to a very high source flux . the 97@xmath22 of photons are lost during the reading cycle in this mode , which results into 2.25ks total exposure time . we rebinned all the data channels in order to oversample the instrumental energy resolution maximally by a factor of 3 and to have at least 20 counts per bin . the first condition is much stronger with respect to the total flux of the sources @xmath23erg@xmath24s@xmath25 in 210kev ( @xmath26cts ) for mcg-6 - 30 - 15 and @xmath27erg@xmath24s@xmath25 in 210kev ( @xmath28cts ) for gx 339 - 4 . [ cols="^,^,^ " , ] we investigated the iron line band for two representative sources mcg-6 - 30 - 15 ( active galaxy ) and gx 339 - 4 ( x - ray binary ) . the iron line is statistically better constrained for the active galaxy mcg-6 - 30 - 15 due to a significantly longer exposure time of the available observations for comparison of count rates of the sources see table 3 . the spectra of both sources are well described by a continuum model plus a broad iron line model . we compared modeling of the broad iron line by the two relativistic models , laor and . the model leads to a better defined minimum of @xmath29 for the best fit value . the confidence contour plots for @xmath30 versus other model parameters are more regularly shaped . this indicates that the model has a smoother adjustment between the different points in the parameter space allowing for more reliable constraints on @xmath30 . the laor model has a less accurate grid and is strictly limited to the extreme kerr metric . the discrepancies between the and laor results are within the general uncertainties of the spin determination using the skewed line profile when applied to the current data . however , the results are apparently distinguishable for higher quality data , as those simulated for the xeus mission . we find that the laor model tends to overestimate the spin value and furthermore , it has insufficient energy resolution which affects the correct determination of the high - energy edge of the broad line . the discrepancies in the overall shape of the line are more visible especially for lower values of the spin @xmath30 . as a side - product , we have found that the correct re - binning of the data with respect to the instrumental energy resolution is crucial to obtain statistically the most relevant results . arnaud , m. , barcons , x. , barret , d. , bautz , m. , bellazzini , r. , xeus : the physics of the hot evolving universe , _ experimental astronomy _ , tmp , 24a , 2008 beckwith , k. , done , c. , iron line profiles in strong gravity , _ monthly notices of royal astronomical society _ , 352 , 353 , 2004 brenneman , l. w. , reynolds , c. s. , constraining black hole spin via x - ray spectroscopy , _ astrophysical journal _ , 652 , 1028 , 2006 ade , a. , calvani , m. , relativistic emission lines from accretion discs around black holes , _ monthly notices of royal astronomical society _ , 363 , 177 , 2005 doviak , m. , karas , v. , yaqoob , t. , an extended scheme for fitting x - ray data with accretion disc spectra in the strong gravity regime , _ astrophysical journal supplements _ , 153 , 205 , 2004 fabian , a. c. , vaughan , s. nandra , k. , iwasawa , k. , ballantyne , d. r. et al . , a long hard look at mcg-6 - 30 - 15 with xmm - newton , _ monthly notices of royal astronomical society _ , 335 , l1 , 2002 gallo e. , corbel s. , fender r. p. , a transient large - scale relativistic radio jet from gx 339 - 4 , _ monthly notices of royal astronomical society _ , 347l , 52 g , 2004 laor , a. , line profiles from a disk around a rotating black hole , _ astrophysical journal _ , 376 , 90 , 1991 miller , j. m. , fabian , a. c. , reynolds , c. s. , nowak , m. a. , homan , j. et al . , evidence of black hole spin in gx 339 - 4 : xmm - newton / epic - pn and rxte spectroscopy of the very high state , _ astrophysical journal _ , 606 , l131 , 2004 miller , j. m. , homan , j. , steeghs , d. , rupen , m. , hunstead , r. w. et al . , a long , hard look at the low / hard state in accreting black holes , _ astrophysical journal _ , 653 , 525 , 2006 miller , l. , turner , t. j. , reeves , j. n. , an absorption origin for the x - ray spectral variability of mcg-6 - 30 - 15 , _ astronomy & astrophysics _ , 483 , 437 , 2008 reis , r. c. , fabian , a. c. , ross , r. , miniutti , g. , miller , j. m. , reynolds , c. s. , a systematic look at the very high and low / hard state of gx 339 - 4 : constraining the black hole spin with a new reflection model , _ arxiv:0804.0238 _ , 2008 remillard , r. a. and mcclintock , j. e. , x - ray properties of black hole binaries , _ annual review of astronomy & astrophysics _ , 49 , 2006 vaughan , s. , fabian , a. c. , a long hard look at mcg-6 - 30 - 15 with xmm - newton - ii . detailed epic analysis and modelling , _ monthly notices of royal astronomical society _ , 348 , 1415 , 2004

the analysis of the broad iron line profile in the x - ray spectra of active galactic nuclei and black hole x - ray binaries allows us to constrain the spin parameter of the black hole . we compare the constraints on the spin value for two x - ray sources , mcg-6 - 30 - 15 and gx 339 - 4 , with a broad iron line using present relativistic line models in xspec laor and . the laor model has the spin value set to the extremal value @xmath0 , while the model enables direct fitting of the spin parameter . the spin value is constrained mainly by the lower boundary of the broad line , which depends on the inner boundary of the disc emission where the gravitational redshift is maximal . the position of the inner disc boundary is usually identified with the marginally stable orbit which is related to the spin value . in this way the laor model can be used to estimate the spin value . we investigate the consistency of the laor and models . we find that the spin values evaluated by both models agree within the general uncertainties when applied on the current data . however , the results are apparently distinguishable for higher quality data , such as those simulated for the international x - ray observatory ( ixo ) mission . we find that the laor model tends to overestimate the spin value and furthermore , it has insufficient resolution which affects the correct determination of the high - energy edge of the broad line . +

arxiv

axially symmetric solutions of einstein s field equations corresponding to disklike configurations of matter are of great astrophysical interest , since they can be used as models of galaxies or accretion disks . these solutions can be static or stationary and with or without radial pressure . solutions for static disks without radial pressure were first studied by bonnor and sackfield @xcite , and morgan and morgan @xcite , and with radial pressure by morgan and morgan @xcite . disks with radial tension have been considered in @xcite , and models of disks with electric fields @xcite , magnetic fields @xcite , and both magnetic and electric fields have been introduced recently @xcite . solutions for self - similar static disks were analyzed by lynden - bell and pineault @xcite , and lemos @xcite . the superposition of static disks with black holes were considered by lemos and letelier @xcite , and klein @xcite . also bick , lynden - bell and katz @xcite studied static disks as sources of known vacuum spacetimes and bick , lynden - bell and pichon @xcite found an infinity number of new static solutions . for a recent survey on relativistic gravitating disks , see @xcite . the principal method to generate the above mentioned solution is the `` displace , cut and reflect '' method . one of the main problem of the solutions generated by using this simple method is that usually the matter content of the disk is anisotropic i.e. , the radial pressure is different from the azimuthal pressure . in most of the solutions the radial pressure is null . this made these solutions rather unphysical . even though , one can argue that when no radial pressure is present stability can be achieved if we have two circular streams of particles moving in opposite directions ( counter rotating hypothesis , see for instance @xcite ) . in this article we apply the `` displace , cut and reflect '' method on spherically symmetric solutions of einstein s field equations in isotropic coordinates to generate static disks made of a _ perfect fluid _ , i.e. , with radial pressure equal to tangential pressure and also disks of perfect fluid surrounded by an halo made of perfect fluid matter . the article is divided as follows . section ii gives an overview of the `` displace , cut and reflect '' method . also we present the basic equations used to calculate the main physical variables of the disks . in section iii we apply the formalism to obtain the simplest model of disk , that is based on schwarzschild s vacuum solution in isotropic coordinate . the generated class of disks is made of a perfect fluid with well behaved density and pressure . section iv presents some models of disks with halos obtained from different known exact solutions of einstein s field equations for static spheres of perfect fluid in isotropic coordinates . in section v we give some examples of disks with halo generated from spheres composed of fluid layers . section vi is devoted to discussion of the results . for a static spherically symmetric spacetime the general line element in isotropic spherical coordinates can be cast as , @xmath1 \mbox{.}\ ] ] in cylindrical coordinates @xmath2 the line element ( [ eq_line1 ] ) takes the form , @xmath3 the metric of the disk will be constructed using the well known `` displace , cut and reflect '' method that was used by kuzmin @xcite in newtonian gravity and later in general relativity by many authors @xcite-@xcite . the material content of the disk will be described by functions that are distributions with support on the disk . the method can be divided in the following steps that are illustrated in fig . [ fig_schem1 ] : first , in a space wherein we have a compact source of gravitational field , we choose a surface ( in our case , the plane @xmath4 ) that divides the space in two pieces : one with no singularities or sources and the other with the sources . then we disregard the part of the space with singularities and use the surface to make an inversion of the nonsingular part of the space . this results in a space with a singularity that is a delta function with support on @xmath4 . this procedure is mathematically equivalent to make the transformation @xmath5 , with @xmath0 constant . in the einstein tensor we have first and second derivatives of @xmath6 . since @xmath7 and @xmath8 , where @xmath9 and @xmath10 are , respectively , the heaviside function and the dirac distribution . therefore the einstein field equations will separate in two different pieces @xcite : one valid for @xmath11 ( the usual einstein s equations ) , and other involving distributions with an associated energy - momentum tensor , @xmath12 , with support on @xmath4 . for the metric ( [ eq_line2 ] ) , the non - zero components of @xmath13 are @xmath14 \mbox { , } \label{eq_qtt}\\ q^r_r & = q^{\varphi}_{\varphi } = \frac{1}{16 \pi } \left [ -b^{zz}+g^{zz}(b^t_t+b^r_r+b^z_z ) \right ] \mbox { , } \label{eq_qrr}\end{aligned}\ ] ] where @xmath15 denote the jump of the first derivatives of the metric tensor on the plane @xmath4 , @xmath16 and the other quantities are evaluated at @xmath17 . the `` true '' surface energy - momentum tensor of the disk can be written as @xmath18 , thus the surface energy density @xmath19 and the radial and azimuthal pressures or tensions @xmath20 read : @xmath21 note that when the same procedure is applied to an axially symmetric spacetime in weyl coordinates we have @xmath22 , i.e. , we have no radial pressure or tension . this procedure in principle can be applied to any spacetime solution of the einstein equations with or without source ( stress tensor ) . the application to a static sphere of perfect fluid is schematized in fig . [ fig_schem2 ] . the sphere is displaced and cut by a distance @xmath0 less then its radius @xmath23 . the part of the space that contains the center of the sphere is disregarded . after the inversion of the remaining space , we end up with a disk surrounded by a cap of perfect fluid . the properties of the inner part of the disk will depend on the internal fluid solution , but if the internal spherical fluid solution is joined to the standard external schwarzschild solution , the physical properties of the outer part of the disk will be those originated from schwarzschild s vacuum solution . in isotropic coordinates the matching at the boundary of the fluid sphere leads to four continuity conditions metric functions @xmath24 and @xmath25 together with their first derivatives with respect to the radial coordinate should be continuous across the boundary . in addition , to have a compact body the pressure at the surface of the material sphere has to drop to zero . also to have a meaningfully solution the velocity of sound , @xmath26 should be restricted to the interval @xmath27 . the einstein equations for a static spherically symmetric space time in isotropic coordinates for a perfect fluid source give us that density @xmath28 and pressure @xmath29 are related to the metric functions by @xmath30 \mbox { , } \\ p & = \frac{e^{-\lambda}}{8 \pi}\left [ \frac{1}{4}(\lambda')^2+\frac{1}{2}\lambda'\nu'+\frac{1}{r}(\lambda'+\nu ' ) \right ] \mbox{,}\end{aligned}\ ] ] where primes indicate differentiation with respect to @xmath31 . also static spheres composed of various layers of fluid can be used to generate disks with halos of fluid layers ( see figure [ fig_schem3 ] ) . the disk will then be composed of different axial symmetric `` pieces '' glued together . the matching conditions at the boundary of adjacent spherical fluid layers in isotropic coordinates involves four continuity conditions : the two metric functions @xmath24 and @xmath25 , the first derivative of @xmath32 with respect to the radial coordinate , and the pressure should be continuous across the boundary . at the most external boundary , the metric functions @xmath24 and @xmath25 , and their first derivatives with respect to the radial coordinate should be continuous across the boundary ; also the pressure there should go to zero . we first apply the `` displace , cut and reflect '' method to generate disks discussed in the previous section and depicted in fig.[fig_schem1 ] to the schwarzschild metric in isotropic coordinates @xmath33 , @xmath34 \mbox{. } \label{eq_metrica_sch}\ ] ] expressing solution ( [ eq_metrica_sch ] ) in cylindrical coordinates , and using eq . ( [ eq_qtt ] ) ( [ eq_disk_surf ] ) , we obtain a disk with surface energy density @xmath19 and radial and azimuthal pressures ( or tensions ) @xmath35 given by @xmath36 the total mass of the disk can be calculated with the help of eq.([eq_en_sch ] ) : @xmath37 eq . ( [ eq_en_sch ] ) shows that the disk s surface energy density is always positive ( weak energy condition ) . positive values ( pressure ) for the stresses in azimuthal and radial directions are obtained if @xmath38 . the strong energy condition , @xmath39 is then satisfied . these properties characterize a fluid made of matter with the usual gravitational attractive property . this is not a trivial property of these disks since it is known that the `` displace , cut and reflect '' method sometimes gives disks made of exotic matter like cosmic strings , see for instance @xcite . another useful parameter is the velocity of sound propagation @xmath40 , defined as @xmath41 , which can be calculated using eq . ( [ eq_en_sch ] ) and eq . ( [ eq_p_sch ] ) : @xmath42 the condition @xmath43 ( no tachyonic matter ) imposes the inequalities @xmath44 or @xmath45 . if the pressure condition and the speed of sound less then the speed of light condition are to be simultaneously satisfied , then @xmath44 . this inequality will be valid in all the disk if @xmath46 . , ( b ) pressures @xmath35 , ( c ) sound velocity @xmath40 and ( d ) tangential velocity @xmath47 ( rotation curve or rotation profile ) with @xmath48 and @xmath49 , and @xmath50 as function of @xmath51 . we use geometric units @xmath52.,title="fig : " ] , ( b ) pressures @xmath35 , ( c ) sound velocity @xmath40 and ( d ) tangential velocity @xmath47 ( rotation curve or rotation profile ) with @xmath48 and @xmath49 , and @xmath50 as function of @xmath51 . we use geometric units @xmath52.,title="fig : " ] + , ( b ) pressures @xmath35 , ( c ) sound velocity @xmath40 and ( d ) tangential velocity @xmath47 ( rotation curve or rotation profile ) with @xmath48 and @xmath49 , and @xmath50 as function of @xmath51 . we use geometric units @xmath52.,title="fig : " ] , ( b ) pressures @xmath35 , ( c ) sound velocity @xmath40 and ( d ) tangential velocity @xmath47 ( rotation curve or rotation profile ) with @xmath48 and @xmath49 , and @xmath50 as function of @xmath51 . we use geometric units @xmath52.,title="fig : " ] , ( b ) pressures @xmath35 and ( c ) sound velocity @xmath40 with @xmath53 and @xmath54 , and @xmath55 as function of @xmath56.,title="fig : " ] , ( b ) pressures @xmath35 and ( c ) sound velocity @xmath40 with @xmath53 and @xmath54 , and @xmath55 as function of @xmath56.,title="fig : " ] + , ( b ) pressures @xmath35 and ( c ) sound velocity @xmath40 with @xmath53 and @xmath54 , and @xmath55 as function of @xmath56.,title="fig : " ] with the presence of radial pressure one does not need the assumption of streams of rotating and counter rotating matter usually used to explain the stability of static disk models . however , a tangential velocity ( rotation profile ) can be calculated by assuming a test particle moves in a circular geodesic on the disk . we tacitly assume that this particle only interact gravitationally with the fluid . this assumption is valid for the case of a particle moving in a very diluted gas like the gas made of stars that models a galaxy disk . the geodesic equation for the @xmath57 coordinate obtained from metric ( [ eq_line2 ] ) is @xmath58 for circular motion on the @xmath4 plane , @xmath59 and @xmath60 , then eq . ( [ eq_geod ] ) reduces to @xmath61 the tangential velocity measured by an observer at infinity is then @xmath62 from the metric on the disk , @xmath63 we find that eq . ( [ eq_veloc_tang ] ) can be cast as , @xmath64 } \mbox{.}\ ] ] for @xmath65 , eq . ( [ eq_v_fi_sch ] ) goes as @xmath66 , the newtonian circular velocity . to determine the stability of circular orbits on the disk s plane , we use an extension of rayleigh @xcite criteria of stability of a fluid at rest in a gravitational field @xmath67 where @xmath68 is the specific angular momentum of a particle on the disk s plane : @xmath69 using eq . ( [ eq_phi_t ] ) and the relation @xmath70 one obtains the following expression for @xmath68 : @xmath71 for the functions ( [ eq_sch_coef ] ) , eq . ( [ eq_h ] ) reads @xmath72 the stability criterion is always satisfied for @xmath73 . in figure [ fig_1 ] ( a)(d ) we show , respectively , the surface energy density , pressures , the sound velocity and curves of the tangential velocity ( rotation curves ) eq . ( [ eq_v_fi_sch ] ) with @xmath48 and @xmath49 , and @xmath50 as functions of @xmath51 . figure [ fig_2 ] ( a)(c ) display , respectively , the surface energy density , pressures and sound velocity with parameters @xmath53 and @xmath74 as as functions of @xmath51 . we see that the first three quantities decrease monotonically with the radius of the disk , as can be checked from eq . ( [ eq_en_sch ] ) , ( [ eq_p_sch ] ) and ( [ eq_sound_sch ] ) . energy density decreases rapidly enough to , in principle , define a cut off radius and consider the disk as finite . now we study some disks with halos constructed from several exact solutions of the einstein equations for static spheres of perfect fluid . a survey of these class of solutions is presented in @xcite . the first situation that we shall study is similar to the one depicted in fig . [ fig_schem2 ] wherein we start with a sphere of perfect fluid . this case will not be exactly the same as the one presented in the mentioned figure because the sphere has no boundary . hence the generated disk will be completely immersed in the fluid . an example of exact solution of the einstein equations that represent a fluid sphere with no boundary is the the buchdahl solution that may be regarded as a reasonably close analogue to the classical lane - emden index 5 polytrope @xcite . the metric functions for this solution are : @xmath75 where @xmath76 and @xmath77 are constants . far from the origin the solution goes over into the external schwarzschild metric , when @xmath78 . the density , pressure and sound velocity are given by : @xmath79 the condition @xmath80 is satisfied for @xmath81 . using eq . ( [ eq_sol_buch ] ) and eqs . ( [ eq_qtt])([eq_disk_surf ] ) , we get the following expressions for the energy density , pressure and sound velocity of the disk : @xmath82 ^ 3 } \mbox { , } \label{eq_en_buch}\\ p & = \frac{aka^2}{2 \pi \left[-a+\sqrt{1+k(r^2+a^2 ) } \right ] \left[a+\sqrt{1+k(r^2+a^2 ) } \right]^3 } \mbox { , } \label{eq_p_buch}\\ v^2 & = \frac{a\left[-a+2\sqrt{1+k(r^2+a^2 ) } \right]}{3\left[a-\sqrt{1+k(r^2+a^2 ) } \right]^2 } \mbox{. } \label{eq_v_buch}\end{aligned}\ ] ] the conditions @xmath83 and @xmath84 are both satisfied if @xmath85 . figure [ fig_3 ] ( a)(d ) shows , respectively , @xmath19 , @xmath35 , @xmath40 and rotation curves , eq . ( [ eq_v_fi_buch ] ) , as functions of @xmath51 for the disk calculated from buchdahl s solution . , eq . ( [ eq_en_buch ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_p_buch ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_v_buch ] ) , ( d ) the tangential velocity @xmath47 eq . ( [ eq_v_fi_buch ] ) for the disk with @xmath86 as function of @xmath51.,title="fig : " ] , eq . ( [ eq_en_buch ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_p_buch ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_v_buch ] ) , ( d ) the tangential velocity @xmath47 eq . ( [ eq_v_fi_buch ] ) for the disk with @xmath86 as function of @xmath51.,title="fig : " ] + , eq . ( [ eq_en_buch ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_p_buch ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_v_buch ] ) , ( d ) the tangential velocity @xmath47 eq . ( [ eq_v_fi_buch ] ) for the disk with @xmath86 as function of @xmath51.,title="fig : " ] , eq . ( [ eq_en_buch ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_p_buch ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_v_buch ] ) , ( d ) the tangential velocity @xmath47 eq . ( [ eq_v_fi_buch ] ) for the disk with @xmath86 as function of @xmath51.,title="fig : " ] in figure [ fig_4 ] ( a)(b ) we show , respectively , the density @xmath28 together with pressure @xmath29 , and sound velocity @xmath87 of the halo along the axis @xmath6 for @xmath88 . note that in this solution there is no boundary of the fluid sphere : the disk is completely immersed in the fluid . and pressure @xmath29 eq . ( [ eq_rhop_buch ] ) and ( b ) the velocity of sound v eq . ( [ eq_v_buch ] ) for the halo with @xmath88 along the axis @xmath6.,title="fig : " ] and pressure @xmath29 eq . ( [ eq_rhop_buch ] ) and ( b ) the velocity of sound v eq . ( [ eq_v_buch ] ) for the halo with @xmath88 along the axis @xmath6.,title="fig : " ] the tangential velocity @xmath47 calculated from metric coefficients ( [ eq_sol_buch ] ) is @xmath89^{3/2}+a[1+k(a^2-r^2)]\right\ } } \mbox{.}\ ] ] for @xmath65 , eq . ( [ eq_v_fi_buch ] ) goes as @xmath90 . the specific angular momentum follows from eq . ( [ eq_h ] ) and eq . ( [ eq_sol_buch ] ) : @xmath91^{1/4 } } { \sqrt{[1+k(r^2+a^2)]^2 - 4akr^2\sqrt{1+k(r^2+a^2)}-a^2[1+k(a^2-r^2 ) ] } } \mbox{.}\ ] ] now we shall study the generation of a disk solution with an halo exactly as the one depicted in fig . [ fig_schem2 ] . we start with a solution of the einstein equations in isotropic coordinates that represents a sphere with radius @xmath23 of perfect fluid that on @xmath92 will be continuously matched to the vacuum schwarzschild solution . narlikar , patwardhan and vaidya @xcite gave the following two exact solutions of the einstein equations for a static sphere of perfect fluid characterized by the metric functions @xmath93 and @xmath94 , @xmath95 ^ 2 & \text { for } k=-2+\sqrt{2 } \text { , } \label{eq_nar_sol1c}\end{aligned}\ ] ] where @xmath96 are constants , and @xmath97 . we shall refer to these solutions as npv 1a and npv 1b , respectively . the density , pressure and sound velocity for for the solutions @xmath98 and @xmath94 , will be denoted by @xmath99 and @xmath100 , respectively . we find , @xmath101 \right . \notag \\ & \left.+b_{1a } \left[3k^2 + 8(1+n)+4k(n+3 ) \right ] r^{2n } \right\ } \mbox { , } \label{eq_p1_nar1}\\ p_{1b } & = \frac{a_{1b}+b_{1b } \ln ( r)+2\sqrt{2}b_{1b}}{16 \pi c [ a_{1b}+b_{1b } \ln ( r)]}\mbox { , } \label{eq_p2_nar1 } \\ \mathrm{v}_{1a}^2 & = \frac{1}{k(k+4)(a_{1a}+b_{1a}r^{2n})^2 } \left\ { a_{1a}^2 \left [ -3k^2 + 8(n-1)+4k(n-3 ) \right ] \right . \notag \\ & \left . -b_{1a}^2r^{4n } \left [ 3k^2 + 8(1+n)+4k(n+3 ) \right ] -2a_{1a}b_{1a}r^{2n}\left [ 3k(k+4)+8(1-n^2 ) \right ] \right\ } \mbox{,}\label{eq_v1_nar1}\\ \mathrm{v}_{1b}^2 & = \frac{2b_{1b}^2r^{\sqrt{2}}}{[a_{1b}+b_{1b } \ln ( r)]^2 } \mbox{. } \label{eq_v2_nar1}\end{aligned}\ ] ] the condition of continuity of the metric functions @xmath102 given by ( [ eq_nar_sol1a])([eq_nar_sol1c ] ) and the corresponding functions in eq . ( [ eq_metrica_sch ] ) at the boundary @xmath92 leads to following expressions : @xmath103 @xmath104 has its maximum at @xmath105 , and @xmath106 at @xmath92 . condition @xmath107 is satisfied if @xmath108 . using eq . ( [ eq_nar_sol1a])([eq_nar_sol1c ] ) in eq . ( [ eq_qtt])([eq_disk_surf ] ) , we get the following expressions for the energy density , pressure and sound velocity of the disk : @xmath109}{\left [ a_{1a}+b_{1a}\mathcal{r}^{n}\right ] } \mbox { , } \label{eq_p1_nar1}\\ p_{1b } & = \frac{a}{4 \pi \sqrt{c}\mathcal{r}^{1/2+\sqrt{2}/4}[2a_{1b}+b_{1b } \ln ( \mathcal{r } ) ] } \left [ 2a_{1b}(\sqrt{2}-1)+2b_{1b } \right . \notag \\ & \left . + b_{1b}(\sqrt{2}-1)\ln ( \mathcal{r})\right ] \mbox { , } \label{eq_p2_nar1}\\ v_{1a}^2 & = \frac{1}{k(k+4)\left [ a_{1a}\mathcal{r}^{-n/2}+b_{1a}\mathcal{r}^{n/2}\right]^2 } \left [ -b_{1a}^2\mathcal{r}^n(k^2 + 5k+nk+4n+4 ) \right . \notag \\ & \left . + a_{1a}^2\mathcal{r}^{-n } ( -k^2 - 5k+nk+4n-4 ) + 2a_{1a}b_{1a}(-k^2 - 5k+4n^2 - 4 ) \right ] \mbox{,}\label{eq_v1_nar1}\\ v_{1b}^2 & = \frac{1}{2(2a_{1b}+b_{1b } \ln ( \mathcal{r}))^2 } \left [ b_{1b}^2 \sqrt{2 } \ln^2 ( \mathcal{r } ) + 2b_{1b } ( 2b_{1b}+b_{1b}\sqrt{2}+2a_{1b}\sqrt{2})\right . \notag \\ & \left . \ln ( \mathcal{r } ) + 4a_{1b}b_{1b}(2+\sqrt{2})+4\sqrt{2}a_{1b}^2 + 8b_{1b}^2 \right ] \mbox { , } \label{eq_v2_nar1}\end{aligned}\ ] ] where @xmath110 . @xmath111 have their maximum values at @xmath112 . because the expressions are rather involved , the restrictions on the constants to ensure the velocities are positive and less then one is best done graphically . the curves of @xmath19 , @xmath35 and @xmath40 as function of @xmath51 with parameters @xmath113 are displayed in figure [ fig_5 ] ( a)(c ) , respectively . figure [ fig_6 ] ( a)(b ) show the density @xmath28 , pressure @xmath29 and velocity of sound v for the halo with parameters @xmath114 along the axis @xmath6 . the same physical quantities are shown in figures [ fig_7 ] ( a)(c ) and [ fig_8 ] ( a)(b ) with @xmath115 . we note that @xmath19 and @xmath35 are continuous at the boundary between the internal and external parts of the disk , but the velocity of sound has a discontinuity . ( [ eq_en_nar1 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_p1_nar1 ] ) and ( c ) the velocity of sound @xmath40 eq . ( [ eq_v1_nar1 ] ) for the disk with @xmath116 as function of @xmath51.,title="fig : " ] eq . ( [ eq_en_nar1 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_p1_nar1 ] ) and ( c ) the velocity of sound @xmath40 eq . ( [ eq_v1_nar1 ] ) for the disk with @xmath116 as function of @xmath51.,title="fig : " ] + eq . ( [ eq_en_nar1 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_p1_nar1 ] ) and ( c ) the velocity of sound @xmath40 eq . ( [ eq_v1_nar1 ] ) for the disk with @xmath116 as function of @xmath51.,title="fig : " ] eq . ( [ eq_rho_nar1 ] ) and pressure @xmath29 eq . ( [ eq_p1_nar1 ] ) , ( b ) the velocity of sound v eq . ( [ eq_v1_nar1 ] ) for the halo with @xmath117 along the axis @xmath6.,title="fig : " ] eq . ( [ eq_rho_nar1 ] ) and pressure @xmath29 eq . ( [ eq_p1_nar1 ] ) , ( b ) the velocity of sound v eq . ( [ eq_v1_nar1 ] ) for the halo with @xmath117 along the axis @xmath6.,title="fig : " ] eq . ( [ eq_v_fi_nar1 ] ) and ( b ) the curves of @xmath118 eq . ( [ eq_h_nar1 ] ) with @xmath119 as function of @xmath51 . a region of instability appears on the disk generated with parameter @xmath120.,title="fig : " ] eq . ( [ eq_v_fi_nar1 ] ) and ( b ) the curves of @xmath118 eq . ( [ eq_h_nar1 ] ) with @xmath119 as function of @xmath51 . a region of instability appears on the disk generated with parameter @xmath120.,title="fig : " ] eq . ( [ eq_en_nar1 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_p2_nar1 ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_v2_nar1 ] ) for the disk with @xmath121 as function of @xmath51.,title="fig : " ] eq . ( [ eq_en_nar1 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_p2_nar1 ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_v2_nar1 ] ) for the disk with @xmath121 as function of @xmath51.,title="fig : " ] + eq . ( [ eq_en_nar1 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_p2_nar1 ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_v2_nar1 ] ) for the disk with @xmath121 as function of @xmath51.,title="fig : " ] eq . ( [ eq_rho_nar1 ] ) and pressure @xmath29 eq . ( [ eq_p2_nar1 ] ) , ( b ) the velocity of sound v eq . ( [ eq_v2_nar1 ] ) for the halo with @xmath122 along the axis @xmath6.,title="fig : " ] eq . ( [ eq_rho_nar1 ] ) and pressure @xmath29 eq . ( [ eq_p2_nar1 ] ) , ( b ) the velocity of sound v eq . ( [ eq_v2_nar1 ] ) for the halo with @xmath122 along the axis @xmath6.,title="fig : " ] eq . ( [ eq_v_fi_nar2 ] ) , ( b ) the curves of @xmath118 eq . ( [ eq_h_nar2 ] ) with @xmath123 as function of @xmath51 . as in the previous case , the same region of instability occurs.,title="fig : " ] eq . ( [ eq_v_fi_nar2 ] ) , ( b ) the curves of @xmath118 eq . ( [ eq_h_nar2 ] ) with @xmath123 as function of @xmath51 . as in the previous case , the same region of instability occurs.,title="fig : " ] the tangential velocity @xmath47 is given by @xmath124 [ r^2(k+2)+2a^2 ] } \mbox { , } \label{eq_v_fi_nar1 } \\ \mathrm{v}_{c1b}^2 & = r^2\frac{2\sqrt{2}a_{1b}+b_{1b}[4+\sqrt{2}\ln ( r^2+a^2)]}{[2a_{1b}+b_{1b}\ln ( r^2+a^2 ) ] [ 2a^2+\sqrt{2}r^2 ] } \mbox { , } \label{eq_v_fi_nar2}\end{aligned}\ ] ] and the specific angular momentum @xmath68 , @xmath125 } { 2a^2[a_{1b}+b_{1b}\ln(\sqrt{r^2+a^2})]-2b_{1b}r^2}}\mbox{. } \label{eq_h_nar2 } \end{aligned}\ ] ] in figure [ fig_9 ] ( a)(b ) , the curves of tangential velocity eq . ( [ eq_v_fi_nar1 ] ) and @xmath118 eq . ( [ eq_h_nar1 ] ) , respectively , are displayed as functions of @xmath51 with @xmath126 . the same quantities are shown in figure [ fig_10](a)(b ) with @xmath115 . for @xmath120 the disks have a small region of unstable orbits immediately after the `` boundary radius '' . like in the previous subsections we study the generation of a disk solution with an halo exactly as the one depicted in fig . [ fig_schem2 ] . we also start with a solution of the einstein equations in isotropic coordinates that represents a sphere of radius @xmath23 of perfect fluid that on @xmath92 will the continuously matched to the vacuum schwarzschild solution . we will use another two solutions found by narlikar , patwardhan and vaidya @xcite that we shall refer as npv 2a and npv 2b , respectivelly . that are characterized by the metric functions @xmath127 and @xmath128 , @xmath129 ^ 2}{\left ( a_1r^{1/\sqrt{2}}+a_2r^{-1/\sqrt{2}}\right)^2 } & \text { for } n=\sqrt{2}. \label{eq_nar_sol2c}\end{aligned}\ ] ] where the @xmath76 s and @xmath130 s are constants and @xmath131 . the solution @xmath132 with @xmath133 corresponds to schwarzschild s internal solution in isotropic coordinates ( see for instance ref . this solution has constant density and is conformally flat when @xmath134 . the density , pressure and sound velocity for the solutions @xmath135 and @xmath128 , will be denoted by @xmath136 and @xmath137 , respectively . we find , @xmath138 \mbox { , } \label{eq_rho_nar2}\\ p_{2a } & = \frac{1}{32 \pi } \left [ -12n^2a_1a_2+(3n^2 - 4)\left ( a_1r^{n/2}+a_2r^{-n/2}\right)^2\right . \notag \\ & \left . + \frac{2nx(b_{2a}-b_{1a}r^x)}{b_{2a}+b_{1a}r^x}\left ( a_1 ^ 2r^{n}-a_2 ^ 2r^{-n}\right ) \right ] \mbox { , } \label{eq_pa_nar2}\\ p_{2b } & = \frac{1}{16 \pi ( b_{1b}+b_{2b } \ln ( r ) ) } \left [ ( b_{1b}+b_{2b } \ln(r ) ) \left ( a_1 ^ 2r^{\sqrt{2}}+a_2 ^ 2r^{-\sqrt{2}}-10a_1a_2 \right ) \right . \notag \\ & \left . + 2\sqrt{2}b_{2b } \left ( a_2 ^ 2r^{-\sqrt{2}}-a_1 ^ 2r^{\sqrt{2 } } \right ) \right ] \mbox { , } \label{eq_pb_nar2}\\ \mathrm{v}_{2a}^2 & = \frac{1}{(4-n^2)\left ( a_1 ^ 2r^{n}-a_2 ^ 2r^{-n}\right)(b_{2a}+b_{1a}r^x)^2 } \left\ { b_{2a}^2 \left[a_1 ^ 2r^n(3n^2 + 2nx-4)\right . \right . \notag \\ & \left . + a_2 ^ 2r^{-n}(-3n^2 + 2nx+4 ) \right]+b_{1a}^2r^{2x } \left [ a_1 ^ 2r^n(3n^2 - 2nx-4 ) \right . \right . \notag \\ & \left . + a_2 ^ 2r^{-n}(-3n^2 - 2nx+4 ) \right]+2b_{1a}b_{2a}r^x \left[a_1 ^ 2r^n(3n^2 - 2x^2 - 4)\right . \right . \notag \\ & \left . + a_2 ^ 2r^{-n}(-3n^2 + 2x^2 + 4 ) \right]\right\ } \mbox { , } \label{eq_va_nar2}\\ \mathrm{v}_{2b}^2 & = \frac{1}{[b_{1b}+b_{2b } \ln ( r)]^2\left ( a_1 ^ 2r^{\sqrt{2}}-a_2 ^ 2r^{-\sqrt{2}}\right ) } \left\ { a_1 ^ 2r^{\sqrt{2 } } \left [ ( b_{1b}+b_{2b } \ln ( r ) ) \right . \right . \notag \\ & \left . \times ( b_{1b}+b_{2b } \ln ( r)-2\sqrt{2}b_{2b})+2b_{2b}^2 \right]- a_2 ^ 2r^{-\sqrt{2 } } \right . \notag \\ & \left . \times \left [ ( b_{1b}+b_{2b } \ln ( r))(b_{1b}+b_{2b } \ln ( r)+2\sqrt{2}b_{2b})+2b_{2b}^2 \right]\right\ } \mbox{. } \label{eq_vb_nar2}\end{aligned}\ ] ] the condition of continuity of the metric functions @xmath102 given by ( [ eq_nar_sol2a])([eq_nar_sol2c ] ) and the corresponding functions in eq . ( [ eq_metrica_sch ] ) at the boundary @xmath92 leads to following expressions : @xmath139 \text { , } \label{eq_cond_sol2a}\\ a_2 & = \frac{1}{r_b^{2-n/2}\left ( 1+\frac{m}{2r_b } \right)^3 } \left [ -\frac{m}{2n}+r_b \left ( \frac{1}{2}+\frac{1}{n}+\frac{m}{4r_b } \right ) \right ] \text { , } \label{eq_cond_sol2b}\\ b_{1a } & = \frac{-4r_b^2\left(1-\frac{2m}{r_b}\right)-m^2 + 2xr_b^2 \left(1-\frac{m^2}{4r_b^2}\right)}{4xr_b^{3+x/2}\left(1+\frac{m}{2r_b}\right)^4 } \mbox { , } \label{eq_cond_sol2c}\\ b_{2a } & = \frac{4r_b^2\left(1-\frac{2m}{r_b}\right)+m^2 + 2xr_b^2 \left(1-\frac{m^2}{4r_b^2}\right)}{4xr_b^{3-x/2}\left(1+\frac{m}{2r_b}\right)^4 } \mbox { , } \label{eq_cond_sol2d}\\ b_{1b } & = \frac{1}{4r_b^3\left ( 1+\frac{m}{2r_b } \right)^4 } \left[4r_b^2-m^2+(m^2 - 8mr_b+4r_b^2 ) \ln(r_b ) \right ] \mbox { , } \label{eq_cond_sol2e}\\ b_{2b } & = -\frac{m^2 - 8mr_b+4r_b^2}{4r_b^3\left ( 1+\frac{m}{2r_b } \right)^4 } \mbox{. } \label{eq_cond_sol2f}\end{aligned}\ ] ] @xmath140 has its maximum at @xmath92 , and @xmath141 at @xmath105 . using eq . ( [ eq_nar_sol2a])([eq_nar_sol2c ] ) in eq . ( [ eq_qtt])([eq_disk_surf ] ) , we get the expressions for the energy density , pressure and sound velocity of the disk : @xmath142 \text { , } \label{eq_en_nar2}\\ p_{2a } & = -\frac{a}{8 \pi \left ( b_{1a}\mathcal{r}^{1/2+x/4}+b_{2a}\mathcal{r}^{1/2-x/4 } \right ) } \left [ b_{1a}a_1(2 + 2n - x)\mathcal{r}^{(x+n)/4 } \right . \notag \\ & \left . + b_{1a}a_2(2 - 2n - x)\mathcal{r}^{(x - n)/4}+b_{2a}a_1(2 + 2n+x)\mathcal{r}^{(-x+n)/4 } \right . \notag \\ & \left . + b_{2a}a_2(2 - 2n+x)\mathcal{r}^{-(x+n)/4 } \right ] \mbox { , } \label{eq_pa_nar2}\\ p_{2b } & = -\frac{a}{4 \pi [ 2b_{1b}+b_{2b}\ln ( \mathcal{r})]}\left\ { 2(1+\sqrt{2})b_{1b}a_1\mathcal{r}^{-1/2 + \sqrt{2}/4}\right . \notag \\ & \left . + 2(1-\sqrt{2})b_{1b}a_2\mathcal{r}^{-1/2-\sqrt{2}/4 } + \left [ ( 1+\sqrt{2 } ) \ln ( \mathcal{r})-2 \right ] b_{2b}a_1\mathcal{r}^{-1/2+\sqrt{2}/4}\right . \notag \\ & \left . + \left [ ( 1-\sqrt{2 } ) \ln ( \mathcal{r})-2 \right ] b_{2b}a_2\mathcal{r}^{-1/2-\sqrt{2}/4 } \right\ } \mbox { , } \label{eq_pb_nar2}\\ v_{2a}^2 & = \frac{1}{2(4-n^2)\left[a_2+a_1\mathcal{r}^{n/2}\right ] \left [ b_{2a}+b_{1a}\mathcal{r}^{x/2}\right]^2 } \left\ { a_1\mathcal{r}^{n/2 } \right . \notag \\ & \left . \left [ b_{1a}^2(n-2)(2n - x+2)\mathcal{r}^{x}+b_{2a}^2(n-2)(2n+x+2 ) \right . \right . \notag \\ & \left . \left . -4b_{1a}b_{2a}(-n^2+n+x^2 + 2)\mathcal{r}^{x/2 } \right ] + a_2 \left [ b_{1a}^2(n+2)(2n+x-2)\mathcal{r}^{x } \right . \right . \notag \\ & \left + b_{2a}^2(n+2)(2n - x-2)-4b_{1a}b_{2a}(-n^2-n+x^2 + 2)\mathcal{r}^{x/2 } \right ] \right\ } \mbox { , } \label{eq_va_nar2}\\ v_{2b}^2 & = - \frac{2b_{1b}+2b_{2b}+b_{2b}\ln ( \mathcal{r})}{2[2b_{1b}+b_{2b}\ln ( \mathcal{r})]^2[a_2+a_1\mathcal{r}^{1/\sqrt{2}}]}\left\ { \left [ \sqrt{2}\ln ( \mathcal{r})-4 \right]\right . \notag \\ & \left . \times b_{2b}a_1\mathcal{r}^ { 1/\sqrt{2}}-\left [ \sqrt{2}\ln ( \mathcal{r})+4 \right]b_{2b}a_2 + 2\sqrt{2}b_{1b}\left [ a_1\mathcal{r}^{1/\sqrt{2}}-a_2 \right ] \right\ } \mbox { , } \label{eq_vb_nar2}\end{aligned}\ ] ] where @xmath110 . the curves of @xmath19 , @xmath35 and @xmath40 as function of @xmath51 with parameters @xmath143 are displayed in figure [ fig_11 ] ( a ) ( c ) , respectively . figure [ fig_12 ] ( a ) ( b ) shows the density @xmath28 , pressure @xmath29 and velocity of sound v for the halo with parameters @xmath144 along the axis @xmath6 . the same physical quantities are shown in figures [ fig_13 ] and [ fig_14 ] with @xmath145 . ( [ eq_en_nar2 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_pa_nar2 ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_va_nar2 ] ) for the disk with @xmath146 as function of @xmath51.,title="fig : " ] eq . ( [ eq_en_nar2 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_pa_nar2 ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_va_nar2 ] ) for the disk with @xmath146 as function of @xmath51.,title="fig : " ] + eq . ( [ eq_en_nar2 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_pa_nar2 ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_va_nar2 ] ) for the disk with @xmath146 as function of @xmath51.,title="fig : " ] eq . ( [ eq_rho_nar2 ] ) and pressure @xmath29 eq . ( [ eq_pa_nar2 ] ) , ( b ) the velocity of sound v eq . ( [ eq_va_nar2 ] ) for the halo with @xmath147 along the axis @xmath6.,title="fig : " ] eq . ( [ eq_rho_nar2 ] ) and pressure @xmath29 eq . ( [ eq_pa_nar2 ] ) , ( b ) the velocity of sound v eq . ( [ eq_va_nar2 ] ) for the halo with @xmath147 along the axis @xmath6.,title="fig : " ] eq . ( [ eq_v_fi_nara ] ) , ( b ) the curves of @xmath118 eq . ( [ eq_h_nara ] ) with @xmath148 as function of @xmath51 . the disks have no unstable orbits for these parameters.,title="fig : " ] eq . ( [ eq_v_fi_nara ] ) , ( b ) the curves of @xmath118 eq . ( [ eq_h_nara ] ) with @xmath148 as function of @xmath51 . the disks have no unstable orbits for these parameters.,title="fig : " ] eq . ( [ eq_en_nar2 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_pb_nar2 ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_vb_nar2 ] ) for the disk with @xmath149 as function of @xmath51.,title="fig : " ] eq . ( [ eq_en_nar2 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_pb_nar2 ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_vb_nar2 ] ) for the disk with @xmath149 as function of @xmath51.,title="fig : " ] + eq . ( [ eq_en_nar2 ] ) , ( b ) the pressure @xmath35 eq . ( [ eq_pb_nar2 ] ) , ( c ) the velocity of sound @xmath40 eq . ( [ eq_vb_nar2 ] ) for the disk with @xmath149 as function of @xmath51.,title="fig : " ] eq . ( [ eq_rho_nar2 ] ) and pressure @xmath29 eq . ( [ eq_pb_nar2 ] ) , ( b ) the velocity of sound v eq . ( [ eq_vb_nar2 ] ) for the halo with @xmath150 along the axis @xmath6.,title="fig : " ] eq . ( [ eq_rho_nar2 ] ) and pressure @xmath29 eq . ( [ eq_pb_nar2 ] ) , ( b ) the velocity of sound v eq . ( [ eq_vb_nar2 ] ) for the halo with @xmath150 along the axis @xmath6.,title="fig : " ] eq . ( [ eq_v_fi_narb ] ) , ( b ) the curves of @xmath118 eq . ( [ eq_h_narb ] ) with @xmath151 as function of @xmath51 . the disks have no unstable orbits for these parameters.,title="fig : " ] eq . ( [ eq_v_fi_narb ] ) , ( b ) the curves of @xmath118 eq . ( [ eq_h_narb ] ) with @xmath151 as function of @xmath51 . the disks have no unstable orbits for these parameters.,title="fig : " ] the tangential velocity @xmath152 is given by @xmath153-a_1\mathcal{r}^{n/2}[b_{1a } ( n - x)\mathcal{r}^{x/2}+b_{2a}(n+x)]}{[b_{1a}\mathcal{r}^{x/2}+b_{2a}][a_1\mathcal{r}^{n/2}(2a^2-nr^2 ) + a_2(2a^2+nr^2 ) ] } \mbox { , } \label{eq_v_fi_nara } \\ \mathrm{v}_{c2b}^2 & = r^2\frac{a_2[2\sqrt{2}b_{1b}+b_{2b}(4+\sqrt{2}\ln ( \mathcal{r}))]-a_1\mathcal{r}^{\sqrt{2}/2 } [ 2\sqrt{2}b_{1b}+b_{2b}(-4+\sqrt{2}\ln ( \mathcal{r}))]}{[2b_{1b}+b_{2b}\ln ( \mathcal{r})][a_1\mathcal{r}^{\sqrt{2}/2 } ( 2a^2-\sqrt{2}r^2)+a_2(2a^2+\sqrt{2}r^2)]}\mbox { , } \label{eq_v_fi_narb}\end{aligned}\ ] ] and the specific angular momentum @xmath68 @xmath154 - a_1\mathcal{r}^{n/2}[b_{1a}(n - x)\mathcal{r}^{x/2}+b_{2a}(n+x ) ] \right\}^{1/2 } \mbox { , } \label{eq_h_nara}\\ h_{2b } & = \frac{r^2\mathcal{r}^{-1/2+\sqrt{2}/4}}{[a_1\mathcal{r}^{\sqrt{2}/2}+a_2]^{3/2 } \sqrt{4a^2b_{1b}+2b_{2b}[-2r^2+a^2\ln(\mathcal{r } ) ] } } \times \notag \\ & \left\ { a_2[2\sqrt{2}b_{1b}+b_{2b}(4+\sqrt{2 } \ln ( \mathcal{r } ) ) ] -a_1\mathcal{r}^{\sqrt{2}/2}[2\sqrt{2}b_{1b}+b_{2b}(-4+\sqrt{2}\ln ( \mathcal{r } ) ) ] \right\}^{1/2 } \mbox{. } \label{eq_h_narb}\end{aligned}\ ] ] in figure [ fig_15 ] ( a)(b ) , the curves of tangential velocities eq . ( [ eq_v_fi_nara ] ) and @xmath118 eq . ( [ eq_h_nara ] ) , respectively , are displayed as functions of @xmath51 with @xmath155 . figure [ fig_16 ] shows the same quantities with @xmath145 . unlike solution 2 , no unstable circular orbits are present for the disks constructed with these parameters . we study two examples of disks with halos constructed from spheres of fluids with two layers as the ones depicted in fig . [ fig_schem3 ] . let us consider a fluid sphere is formed by two layers : the internal layer , @xmath156 , will be taken as the internal schwarzschild solution ( solution 2a with n=2 ) , @xmath157 the external layer , @xmath158 is taken as the buchdahl solution , @xmath159 note that the external layer has no boundary , i.e. , this layer has infinite radius . according to the continuity conditions at @xmath160 , the constants are related through : @xmath161 } { \left [ ( 1+kr_1 ^ 2)^2 + 2c\sqrt{1+kr_1 ^ 2}+c^2(1-kr_1 ^ 2)\right ] } \mbox{,}\\ b_2 & = \frac{1-\frac{c}{\sqrt{1+kr_1 ^ 2}}}{\left ( 1+\frac{c}{\sqrt{1+kr_1 ^ 2}}\right)^3 } -b_1r_1 ^ 2 \mbox{. } \label{eq_cond_l2}\end{aligned}\ ] ] with these relations , one verifies that , using eq . ( [ eq_en_buch ] ) and eq . ( [ eq_en_nar2 ] ) , eq . ( [ eq_p_buch ] ) and eq . ( [ eq_pa_nar2 ] ) , both the energy density and the pressure are continuous at the radius @xmath162 of the disk . figure [ fig_17 ] ( a)(c ) shows , respectively , @xmath19 , @xmath35 and @xmath40 for the disk obtained from fluid layers eq . ( [ eq_layer1])([eq_layer2 ] ) with parameters @xmath163 as function of @xmath51 . the density @xmath28 , pressure @xmath29 and velocity of sound v for the halo along the @xmath6 axis with the same parameters for @xmath120 is shown in figure [ fig_18 ] . , ( b ) the pressure @xmath35 , ( c ) the velocity of sound @xmath40 for the disk generated from spherical fluid layers eq . ( [ eq_layer1])([eq_layer2 ] ) with @xmath164 as function of @xmath51.,title="fig : " ] , ( b ) the pressure @xmath35 , ( c ) the velocity of sound @xmath40 for the disk generated from spherical fluid layers eq . ( [ eq_layer1])([eq_layer2 ] ) with @xmath164 as function of @xmath51.,title="fig : " ] + , ( b ) the pressure @xmath35 , ( c ) the velocity of sound @xmath40 for the disk generated from spherical fluid layers eq . ( [ eq_layer1])([eq_layer2 ] ) with @xmath164 as function of @xmath51.,title="fig : " ] and pressure @xmath29 , ( b ) the velocity of sound v for the halo formed by fluid layers eq . ( [ eq_layer1])([eq_layer2 ] ) with @xmath165 along the axis @xmath6.,title="fig : " ] and pressure @xmath29 , ( b ) the velocity of sound v for the halo formed by fluid layers eq . ( [ eq_layer1])([eq_layer2 ] ) with @xmath165 along the axis @xmath6.,title="fig : " ] in figure [ fig_19 ] ( a)(b ) , the curves of tangential velocities and of @xmath118 , respectively , are displayed as functions of @xmath51 . , ( b ) the curves of @xmath118 for the disk generated from fluid layers eq . ( [ eq_layer1])([eq_layer2 ] ) with @xmath164 as function of @xmath51 . the disks obtained with these parameters have no unstable orbits.,title="fig : " ] , ( b ) the curves of @xmath118 for the disk generated from fluid layers eq . ( [ eq_layer1])([eq_layer2 ] ) with @xmath164 as function of @xmath51 . the disks obtained with these parameters have no unstable orbits.,title="fig : " ] now we consider a sphere composed with two finite layers : the internal layer , @xmath166 , is taken as the npv solution 2b with @xmath145 , @xmath167 the external layer , @xmath168 , is taken as the npv solution 1b with @xmath115 , @xmath169 the spacetime outside the sphere , @xmath170 , will be taken as the schwarzschild s vacuum solution is isotropic coordinates . @xmath171 in this case the pressure should be zero at @xmath172 . the continuity conditions at @xmath160 and @xmath172 give the relations : @xmath173 \mbox{,}\\ b_1 & = \frac{r_1^{\sqrt{2}}}{2\sqrt{2c}}\left [ \left(a_3+b_3 \ln ( r_1)\right)\left ( 2\sqrt{2}r_1^{-\sqrt{2 } } -\ln ( r_1)+\ln ( r_1)r_1^{-\sqrt{2}}\right)-2\sqrt{2}b_3 \ln ( r_1 ) \right ] \mbox{. } \label{eq_cond_l4}\end{aligned}\ ] ] using eq . ( [ eq_en_nar1 ] ) and eq . ( [ eq_en_nar2 ] ) , the energy density of the disk at @xmath162 is continuous , but not the pressure . the difference between eq . ( [ eq_pb_nar2 ] ) and eq . ( [ eq_p2_nar1 ] ) is @xmath174 \mbox{.}\ ] ] the pressure is continuous if @xmath175 . figure [ fig_20 ] ( a)(c ) shows , respectively , @xmath19 , @xmath35 and @xmath40 for the disk obtained from fluid layers ( [ eq_layer3])([eq_layer4 ] ) with parameters @xmath176 as function of @xmath51 . the density @xmath28 , pressure @xmath29 and velocity of sound v for the halo along the @xmath6 axis with the same parameters for @xmath177 is shown in figure [ fig_21 ] . , ( b ) the pressure @xmath35 , ( c ) the velocity of sound @xmath40 for the disk generated from spherical fluid layers eq . ( [ eq_layer3])([eq_layer4 ] ) with @xmath178 as function of @xmath51.,title="fig : " ] , ( b ) the pressure @xmath35 , ( c ) the velocity of sound @xmath40 for the disk generated from spherical fluid layers eq . ( [ eq_layer3])([eq_layer4 ] ) with @xmath178 as function of @xmath51.,title="fig : " ] + , ( b ) the pressure @xmath35 , ( c ) the velocity of sound @xmath40 for the disk generated from spherical fluid layers eq . ( [ eq_layer3])([eq_layer4 ] ) with @xmath178 as function of @xmath51.,title="fig : " ] and pressure @xmath29 , ( b ) the velocity of sound v for the halo formed by fluid layers eq . ( [ eq_layer3])([eq_layer4 ] ) with @xmath179 along the axis @xmath6.,title="fig : " ] and pressure @xmath29 , ( b ) the velocity of sound v for the halo formed by fluid layers eq . ( [ eq_layer3])([eq_layer4 ] ) with @xmath179 along the axis @xmath6.,title="fig : " ] in figure [ fig_22 ] ( a)(b ) , the curves of tangential velocities and of @xmath118 , respectively , are displayed as functions of @xmath51 . in this case , regions of unstable orbits exist for parameters @xmath177 and @xmath180 . , ( b ) the curves of @xmath118 for the disk generated from spherical fluid layers eq . ( [ eq_layer3])([eq_layer4 ] ) with @xmath181 as function of @xmath51 . regions of unstable circular orbits appear for the disks obtained with parameters @xmath177 and @xmath182.,title="fig : " ] , ( b ) the curves of @xmath118 for the disk generated from spherical fluid layers eq . ( [ eq_layer3])([eq_layer4 ] ) with @xmath181 as function of @xmath51 . regions of unstable circular orbits appear for the disks obtained with parameters @xmath177 and @xmath182.,title="fig : " ] the `` displace , cut and reflect '' method applied to solutions of einstein field equations in isotropic coordinates can generate disks with positive energy density and equal radial and azimuthal pressures ( perfect fluid ) . with solutions of static spheres of perfect fluid it is possible to construct disks of perfect fluid surrounded also by perfect fluid matter . as far we know these are the first disk models of this kind in the literature . all disks constructed as examples have some common features : surface energy density and pressures decrease monotonically and rapidly with radius . as the `` cut '' parameter @xmath0 decreases , the disks become more relativistic , with surface energy density and pressure more concentrated near the center . also regions of unstable circular orbits are more likely to appear for high relativistic disks . parameters can be chosen so that the sound velocity in the fluid and the tangential velocity of test particles in circular motion are less then the velocity of light . this tangential velocity first increases with radius and reaches a maximum . then , for large radii , it decreases as @xmath183 , in case of disks generated from schwarzschild and buchdahl s solutions . the sound velocity is also a decreasing function of radius , except in solution npv 2a with @xmath184 , where it reaches its maximum value at the boundary . in principle other solutions of static spheres of perfect fluid could be used to generate other disk + halo configurations , but it is not guaranteed that the disks will have the characteristics of normal fluid matter . we believe that the presented disks can be used to describe more realistic model of galaxies than most of the already studied disks since the counter rotation hypothesis is not needed to have a stable configuration . in the table we list the seed metric coefficients , matching conditions at the boundaries and relevant physical quantities of all disks studied in this work . the numbers refer to the equations presented along the paper and npv stands for narlikar , patwardhan and vaidya as before . 99 w. a. bonnor and a. sackfield , _ commun . * 8 * , 338 ( 1968 ) . t. morgan and l. morgan , _ phys . rev . _ * 183 * , 1097 ( 1969 ) . l. morgan and t. morgan , _ phys . d _ * 2 * , 2756 ( 1970 ) . g. gonzlez and p. s. letelier , _ class . quantum grav . _ * 16 * , 479 ( 1999 ) . t. ledvinka , m. zofka , and j. bik , in _ proceedings of the 8th marcel grossman meeting in general relativity _ , edited by t. piran ( world scientific , singapore , 1999 ) , pp . 339 - 341 . p. s. letelier , _ phys . * 60 * , 104042 ( 1999 ) . j. katz , j. bik , and d. lynden - bell , _ class . quantum grav . _ * 16 * , 4023 ( 1999 ) . d. lynden - bell and s. pineault , mon . not . r. astron . 185 , 679 ( 1978 ) j. p. s. lemos , class . grav . 6 , 1219 ( 1989 ) j. p. s. lemos and p. s. letelier , class . 10 , l75 ( 1993 ) j. p. s. lemos and p. s. letelier , phys . rev d49 , 5135 ( 1994 ) j. p. s. lemos and p. s. letelier , int . d5 , 53 ( 1996 ) c. klein , class . 14 , 2267 ( 1997 ) j. bick , d. lynden - bell and j. katz , phys . d47 , 4334 ( 1993 ) j. bick , d. lynden - bell and c. pichon , mon . not . r. astron . 265 , 126 ( 1993 ) o. semerk , _ towards gravitating disks around stationary black holes _ , ( april , 2002 ) , available at http://xxx.lanl.gov/abs/gr-qc/0204025 . g. g. kuzmin , _ astron . _ , * 33 * , 27 ( 1956 ) . a. h. taub , j. math . phys . 21 , 1423 ( 1980 ) j.p.s . lemos and p.s . letelier , phys lett . a 153 , 288 ( 1991 ) . lord rayleigh , proc . ser . a 93 , 148 ( 1916 ) . l. d. landau , e. m. lifshitz , _ fluid mechanics _ ( pergamon press , oxford , 1987 ) , 27 . b. kuchowicz , _ acta phys . _ , * b * 3 , 209 ( 1972 ) . h. a. buchdahl , _ astrophys . j. _ , * 140 * , 1512 ( 1964 ) .

using the well - known `` displace , cut and reflect '' method used to generate disks from given solutions of einstein field equations , we construct static disks made of perfect fluid based on vacuum schwarzschild s solution in isotropic coordinates . the same method is applied to different exact solutions to the einstein s equations that represent static spheres of perfect fluids . we construct several models of disks with axially symmetric perfect fluid halos . all disks have some common features : surface energy density and pressures decrease monotonically and rapidly with radius . as the `` cut '' parameter @xmath0 decreases , the disks become more relativistic , with surface energy density and pressure more concentrated near the center . also regions of unstable circular orbits are more likely to appear for high relativistic disks . parameters can be chosen so that the sound velocity in the fluid and the tangential velocity of test particles in circular motion are less then the velocity of light . this tangential velocity first increases with radius and reaches a maximum .

arxiv

one of the long - sought goals in low - temperature physics is the creation of two interpenetrating superfluids . early efforts were directed at mixtures of helium isotopes . more recently , following the experiments with bose - einstein condensates ( bec ) of atomic gases @xcite , considerable efforts have been made to create systems where two species of atoms condense simultaneously . this goal was partially achieved for two different hyperfine spin states of @xmath1rb , which were condensed in the same trap by the technique of sympathetic cooling @xcite . later the dynamics of the complex relative motion of the condensates has been studied @xcite . the possibility of the measurement of the relative phase between the two condensates has also been demonstrated @xcite . in these experiments the two condensates have a substantial overlap , although they do not completely interpenetrate each other in the stationary state . a similar state called `` spinor condensate '' has been observed for sodium gas @xcite . theoretical investigation of two - component bose systems has started many decades ago with the construction of the phenomenological hydrodynamic equations in the spirit of the landau - khalatnikov two - fluid model for the one - component bec @xcite . later , this construction has been put onto a microscopic basis @xcite . recent experiments with alkali atoms have revived the interest in the theory of such systems . hartree - fock theory has been successfully tested on the two - component @xmath1rb system @xcite . the stability @xcite , ground - state properties @xcite and collective excitations @xcite have been studied theoretically by using the gross - pitaevskii equations . many properties of two - component , or binary , bec can be understood from symmetry arguments . compared to one - component bose superfluids , two - component systems have more interesting pattern of symmetry and symmetry breaking . condensation in binary bose systems corresponds to the spontaneous breaking of _ two _ ( instead of one ) global u(1 ) symmetries . these symmetries are related , by noether s theorem , to the separate conservation of the number of atoms of each of the two species . the quantum state of the binary bose system , therefore , is characterized by two phases of the two condensates . correspondingly , the physics of binary bec is also richer than of usual one - component systems . the effects of a symmetry are often best exposed by violating the symmetry explicitly in a controlled fashion . a very interesting feature , specific to systems consisting of atoms of the same isotope in different spin states , is that it is possible to couple two condensates by a driving electromagnetic field tuned to the transition frequency . in this case atoms can be interconverted between the two spin states and the numbers of atoms of each species are not conserved separately anymore ; only the total number of atoms is constant . this implies that , in the presence of the coupling drive , only one u(1 ) symmetry remains exact , the other one is explicitly violated . the preserved u(1 ) symmetry obviously comes from the conservation of the total number of atoms , and corresponds to changing the phases of the two condensates by the same amount ( i.e. , leaving the relative phase unchanged ) . the violated u(1 ) corresponds to changing the relative phase between the two condensate . the presence of the coupling drive lifts the degeneracy of the ground state with respect to the relative phase . in this work , we show that a sufficiently small violation of the u(1 ) symmetry corresponding to the relative phase leads to the existence of a nontrivial static configuration a domain wall inside which the relative phase changes by @xmath0 . this configuration is a local minimum of the energy . however , the domain wall is _ not _ topologically stable and can `` unwind '' itself . to unwind , however , the system must overcome an energy barrier . thanks to this fact , the rate of the spontaneous decay of the domain wall is exponentially suppressed . our paper is organized as follows . section [ sec : l ] introduces the field - theoretical description of binary bec . in sec.[sec : solution ] we describe the domain wall configuration , whose physical interpretation is given in sec . [ sec : interpretation ] . section [ sec : boundary ] deals with the boundary of finite domain walls and the related phenomenon of `` vortex confinement '' . section [ sec : conclusion ] contains concluding remarks . in appendix [ sec : stability ] we find the metastability condition for the domain wall in the particular case when the densities of the two components are equal , and in appendix [ sec : decay ] two different mechanisms for the decay of the domain wall , operating at different temperature regimes , are considered . in this section , we use field theory to describe general properties of binary bec . our goal is to introduce notations and the formalism to lay the ground for the discussion of the domain walls in the next section . a binary dilute bose system is described by a quantum field theory of two complex scalar fields @xmath2 and @xmath3 . these fields have the meaning of the wave functions of the two condensates . the dynamics of these fields is governed by the following lagrangian , @xmath4 where the hamiltonian @xmath5 has the form @xmath6 in eq . ( [ l ] ) @xmath7 are the chemical potentials of the two species , and @xmath8 are functions of coordinates . we assume here that the trapping potentials are sufficiently wide so that these chemical potentials can be put to constants . ] @xmath9 is the scattering amplitude , in the zero momentum limit , between an atom of the @xmath10-th species and that of the @xmath11-th species , and are proportional to the scattering lengths @xmath12 , @xmath13 and @xmath14 is the rabi frequency arising from the coupling drive . by varying the action @xmath15 with respect to @xmath16 , the familiar gross - pitaevskii equations are directly obtained : @xmath17 let us start by finding the ground state when the coupling drive is off , @xmath18 . in the superfluid ground state , both @xmath2 and @xmath3 have nonzero expectation values . these can be found by minimizing the potential energy part in eq . ( [ l ] ) with respect to @xmath2 and @xmath3 . this minimization procedure gives the equations determining the densities @xmath19 , @xmath20 in terms of the chemical potentials @xmath21 and @xmath8 , @xmath22 more conveniently , one could view eq . ( [ nmu ] ) as the equations fixing the chemical potentials for given values of the densities . strictly speaking , eqs . ( [ nmu ] ) only correspond to an _ extremum _ of the potential energy . for it to be a local minimum , the quadratic form @xmath23 needs to be positive definite : @xmath24 in fact , eq . ( [ stability ] ) is the condition for the mixture of the two bose superfluids to be thermodynamically stable against segregation @xcite . in this paper we shall assume that ( [ stability ] ) is satisfied . in principle , the constants @xmath25 , @xmath26 and @xmath27 can be arbitrary . in this paper we limit ourselves to the regime when all three scattering lengths are close to each other , @xmath28 . in the case when the two species are rubidium atoms in different hyperfine states , these lengths were found experimentally to differ by no more than a few percent . the assumption that @xmath29 also introduces considerable technical simplifications in our treatment . we introduce the `` average '' scattering amplitude @xmath30 and the deviations from the average @xmath31 so that @xmath32 . the stability condition ( [ stability ] ) implies that @xmath33 . analogously , we introduce the average scattering lengths @xmath34 and the deviations @xmath35 and @xmath36 . note that in the limit @xmath37 the hamiltonian has an su(2 ) symmetry which leads to interesting implications @xcite . with the lagrangian at hand , the discussion of symmetry in the introduction can be made concrete . in the absence of the coupling drive , @xmath18 , the lagrangian ( [ l ] ) possesses a u(1 ) @xmath38 u(1 ) symmetry with respect to independent phase rotations of the fields , @xmath39 the corresponding conservation laws are those of the numbers of particles of each species , @xmath40 and @xmath41 . that @xmath42 and @xmath43 at @xmath18 are conserved separately is actually a basic assumption made when we wrote down the lagrangian ( [ l ] ) . this assumption is not automatically satisfied : it requires that only elastic scattering between atoms is allowed ; inelastic scattering is forbidden . for binary bose systems made of rubidium atoms , this appears to be a good approximation @xcite . once the coupling drive is turned on ( @xmath44 ) , the lagrangian is invariant only under a subset of the original u(1 ) @xmath38 u(1 ) rotations ; namely , those which rotate both @xmath2 and @xmath3 by the same angle , @xmath45 therefore one of the u(1 ) symmetries the system enjoyed at @xmath18 is explicitly violated . applying the goldstone theorem , we conclude that , at @xmath18 , there are two gapless excitations and only one of these modes remain gapless at @xmath44 . the gapless modes at @xmath18 are the phonons of the two types of sounds . one corresponds to the ordinary density wave ( b mode in our paper , see below ) , and another to the concentration wave ( a mode ) in which the densities of the two species oscillate relative to each other in such a way that the total density remains constant . if we view the two components as being made of the same atoms , but in different hyperfine levels , then the mode a can be alternatively interpreted as a spin polarization wave , in which the density of nuclear spin is a function of space and time . when the coupling drive is on , only the density wave remains gapless ; the phonon of the concentration ( spin density ) wave is gapped . let us compute the sound speeds at @xmath18 . to this end we write @xmath2 and @xmath3 as @xmath46 and expand eq . ( [ l ] ) to second order of @xmath47 and @xmath48 ( we will see that @xmath49 ) . we find @xmath50 ( we have thrown away total derivatives ) . the density fluctuations @xmath51 can be `` integrated out '' and replaced by the saddle point values @xmath52 . as a result , ( [ lnphi ] ) becomes @xmath53 thus , the dispersion relations for the phonons are linear , @xmath54 , and the sound speed @xmath55 satisfies the characteristic equation @xmath56 when @xmath29 , the solutions are @xmath57 we see that when @xmath58 the speed of the concentration wave ( a ) is much smaller than that of the density wave ( b ) , @xmath59 . the system is `` stiffer '' towards density fluctuations than towards fluctuations of concentration . that modes a and b are indeed concentration and density fluctuations respectively is seen from the corresponding eigenvectors . the a mode corresponds to such fluctuations in which @xmath60 while the b mode corresponds to @xmath61 therefore , in the a sound @xmath62 and @xmath63 fluctuate in such a way that the overall density remains constant ( @xmath64 ) , while the b sound corresponds to density waves in which @xmath65 , or concentration , is unchanged ( @xmath66 ) . the lagrangian ( [ lphi ] ) , in terms of the normal modes @xmath67 has the form @xmath68 + \left [ u_\b^{-2}(\d_t\phib)^2-(\nabla\phib)^2\right ] \biggr\}\,.\ ] ] when the coupling drive is on , @xmath44 , one should add to eq . ( [ lphi ] ) the potential energy term @xmath69 . the b sound is not affected by this term since it corresponds to such fluctuations in which @xmath70 . the phonons of the a sound acquire a gap @xmath71 for small values of @xmath14 . in our further discussion we shall need the formulas for the healing , or correlation , lengths , which are defined via the response of the system to a static source coupled locally to the particle density , @xmath72 . as in the case of the sounds , there are also two healing lengths , @xmath73 as seen from eq . ( [ coh_lengths ] ) , @xmath74 . this is because @xmath75 is the correlation length of fluctuations of the overall density , while @xmath76 is the correlation length of the relative density . if we take the values typical for present day experiments : @xmath77 @xmath78 , @xmath79 , and @xmath80 , then @xmath81 @xmath82 m , and @xmath83 @xmath82 m . these lengths are smaller than the typical system size in experiments . the existence of the domain wall of relative phase in binary bec with a coupling drive can be shown in a rather simple way . let us first focus only on fluctuations of the fields on length scales much larger than the largest healing length @xmath76 . in this case , the amplitudes @xmath62 and @xmath63 of @xmath16 can be regarded as `` frozen '' and the only important degrees of freedom are the phases , i.e. , @xmath84 and @xmath85 . the energy of the system is a functional of @xmath84 and @xmath85 , @xmath86 = \int\ ! d^3\x\,\biggl [ { \hbar^2\over2m}(n_1(\nabla\varphi_1)^2 + n_2(\nabla\varphi_2)^2 ) - \hbar\omega\sqrt{n_1n_2}\cos(\varphi_1-\varphi_2 ) \biggr]\ , . \label{h}\ ] ] the potential energy term @xmath87 has its minimum at @xmath88 . the configuration in which @xmath89 over the whole space is obviously the global minimum of the total energy , and hence is the ground state . the domain wall solution that we shall describe is , in contrast , a _ local _ minimum of the energy . to find the profile of the domain wall , we vary eq . ( [ h ] ) with respect to @xmath84 and @xmath85 and obtain the following equations @xmath90 for domain walls , @xmath84 and @xmath85 are functions of only one coordinate , say , @xmath91 , and at @xmath92 both approach constant values . a nontrivial solution to eq . ( [ walleq ] ) satisfying these conditions is @xmath93 the characteristic width of the domain wall is @xmath94 . the tension of the domain wall is @xmath95 the relative phase @xmath96 changes from 0 to @xmath0 as @xmath91 runs from @xmath97 to @xmath98 . note that @xmath96 is defined modulo @xmath0 so it goes a full circle as one passes through the wall . therefore from the point of view of the energy functional ( [ h ] ) the domain wall ( [ profile ] ) is a topologically nontrivial configuration , which can not be continuously deformed into the @xmath89 configuration . in fact , one can prove that the domain wall is a configuration with minimal energy defined in eq . ( [ h ] ) among those where @xmath96 changes by @xmath0 from @xmath99 to @xmath100 , and hence , can not decay away , _ as long as eq . ( [ h ] ) applies_. the domain wall is similar to the soliton of the sine - gordon model . there is a small difference : the ground states on two sides of the domain wall are _ different _ : at @xmath99 @xmath101 , while the ground state at @xmath100 is @xmath102 , @xmath103 . since @xmath104 are defined mod @xmath0 the latter is equivalent to @xmath105 . in reality , eq . ( [ h ] ) is not the full hamiltonian of the system : it is only an effective description valid at length scales larger than both healing lengths . the full theory ( [ h - full ] ) contains , besides @xmath84 and @xmath85 , also the density fluctuations @xmath106 and @xmath107 . as a consequence , in the full theory , the domain wall ( [ profile ] ) _ can _ be continuously deformed into the trivial configuration @xmath89 . such deformations necessarily pass through field configurations where either @xmath62 or @xmath63 vanishes at some points : at these points @xmath96 is ill - defined . thus , the domain wall is not truly topological and can `` unwind '' , i.e. , decay away . the fact that the ground states on the two sides of the wall are different does not prevent the decay : it is possible to construct a field configuration interpolating between @xmath101 and @xmath105 with arbitrarily small energy per unit area . although the domain wall is not a global minimum of the energy functional , it can still be a _ local _ minimum . in this case , to deform the domain wall into a `` topologically '' trivial configuration with @xmath89 one has to overcome an energy barrier . the wall is in this case metastable . from our previous discussion one can conclude that , roughly speaking , the wall is metastable when eq . ( [ h ] ) applies and is unstable when eq . ( [ h ] ) is not applicable . for eq.([h ] ) to be valid the wall has to be wider than the largest healing length , @xmath76 . since the width of the wall decreases as one increases the rabi frequency @xmath14 , the wall is metastable only when @xmath14 is smaller than some critical value @xmath108 . let us define @xmath109 as the value of the rabi frequency at which the width of the wall @xmath94 , as defined in eq . ( [ profile ] ) , is equal to to the longer healing length @xmath76 in eq.([coh_lengths ] ) : @xmath110 the wall is metastable when @xmath14 is less than some critical value @xmath108 of order @xmath109 , @xmath111 to find the exact value of @xmath108 one needs to perform a more refined stability analysis . we present such an analysis for the special case @xmath112 ( i.e. , when the densities of the two species are equal ) in appendix [ sec : stability ] . parametrically , the result is consistent with eq . ( [ omegac ] ) , and the ratio @xmath113 is found to be @xmath114 . using the numerical values typical for experiments with rb : @xmath115 a few @xmath116 @xmath78 , @xmath117 , one finds @xmath118 hz . therefore to have a stable wall the rabi frequency needs to be smaller than about 100 hz . the domain wall can not be thinner than the longer correlation length @xmath76 , which was estimated above to be a few @xmath82 m . even when @xmath119 , a metastable wall can still spontaneously decay ( burst ) . such a decay , as we have said , requires overcoming a potential barrier . at sufficiently large temperature , the mechanism for the decay is thermal activation ( over the barrier ) . at zero or small temperatures , the mechanism of the decay is quantum tunneling under the energy barrier . the decay through thermal activation , which is relevant for the temperatures achieved in current experiments , is considered in appendix [ sec : decay ] , where it is shown that the decay rate is exponentially suppressed , since a global ( macroscopic ) fluctuation is required , unless @xmath14 is very close to @xmath108 . the domain wall solution found in sec . [ sec : solution ] allows an interesting physical interpretation which suggests a possible way for their creation in experiments @xcite . we first note that in a bec the superfluid velocity is proportional to the gradient of the phase . in a two - component bec , there are two such velocities . @xmath120 equation ( [ walleq ] ) , in particular , implies that the total particle number current vanishes , @xmath121 . individually , however , the particle number current of each species @xmath122 and @xmath123 are nonzero . from eq.([profile ] ) we find @xmath124 thus the domain wall is a configuration where the two components flow in opposite directions . the flow is illustrated in fig.[fig : flow ] . the velocities of the components are largest at the center of the wall ( @xmath125 ) and decrease as one moves toward the edge of the wall . the flow is concentrated on the wall ; outside the wall ( @xmath126 ) there is essentially no flow . equation ( [ walleq ] ) can be rewritten in terms of the currents as @xmath127 for a stationary configuration as the one we are considering , eq.([conversion ] ) means that the number of particles in each species is not conserved . it also implies that there is a _ conversion _ of atoms between the two energy levels due to the coupling drive . in the left half of the wall @xmath128 and there is a conversion of atoms of the second type to atoms of the first type . in the right half @xmath129 and the conversion goes the opposite way ( fig.[fig : flow ] ) . the rate of conversion is @xmath130 as is the flow , the conversion rate is also maximal near the wall . far from the wall ( @xmath126 ) there is essentially no conversion . the conversion rate ( [ conversionrate ] ) changes sign at @xmath131 . since different species correspond to different energy levels of the atom , energy is absorbed in one half of the wall and released in the other . plane accompanied by spin flips on both sides of it . ] the interpretation of the domain walls given above suggests a possible method for their creation in experiments @xcite . one starts with the coupling drive off ( @xmath18 ) and prepares a state where the two condensates flow in opposite directions ( for example , by manipulating the traps ) . in such a state the relative phase @xmath132 is a linear function of the coordinate along the direction of motion ( say , @xmath91 ) . one then slowly increases @xmath14 . the domain walls will be created and the centers of the walls are located at the points where @xmath96 was an odd multiple of @xmath133 ( @xmath134 , @xmath135 , @xmath136 , etc . ) before @xmath14 was turned on . by changing the velocity of the initial relative motion of the condensates and the final values of @xmath14 one can change the separation between the domain walls and their width . such controlled creation of the domain walls , hopefully , can be achieved in future experiments . so far we have always considered infinite domain walls which have no boundary . it is also interesting , and perhaps more realistic , to consider domain walls with a boundary . we shall show that the domain wall can be bounded by a vortex line . suppose we have a finite domain wall whose boundary is a closed contour @xmath137 ( fig . [ fig : boundary ] ) . we shall assume that the length of @xmath137 is much larger than the width of the wall @xmath94 so one can view the domain wall as an infinitely thin membrane stretched on @xmath137 ( we shall call this picture the `` thin - wall approximation '' ) . let us now take another , smaller contour which has a nontrivial linking with @xmath137 ( @xmath138 in fig . [ fig : boundary ] ) . as one goes along @xmath138 , one crosses the membrane once , so the relative phase @xmath96 changes by @xmath0 . this is exactly what one expects from a vortex . therefore , @xmath137 can be a vortex line . we recall that the size of the core of the vortex is the healing length @xmath76 , which is smaller or of the same order as the width of the wall . therefore , in the thin wall approximation , we have an infinitely thin membrane bounded by an infinitely thin vortex line . one should note that such a bounded domain wall will tend to shrink to reduce its energy , which comes from the wall tension and the tension of the boundary vortex . conversely , for small non - zero @xmath14 , a vortex must have a domain wall attached to it to minimize the energy due to the nontrivial phase @xmath96 winding around it . this means that the energy of a single vortex per unit length is increasing linearly with the size of the system in the transverse direction . this is in contrast to the situation at @xmath18 when the vortex tension has only a logarithmic dependence on the size of the system . note that there are two types of vortices in binary bec . those of the first type , which we shall call the @xmath84 vortices , have the condensate @xmath2 vanishing at the vortex center , while @xmath3 is nonzero . analogously the @xmath85 vortices have @xmath139 and @xmath140 at their centers . as one goes around a @xmath84 vortex , @xmath84 changes by @xmath0 , while @xmath85 does not change , and vice versa for a @xmath85 vortex . thus @xmath96 changes by either @xmath0 or @xmath141 for the two types of vortices , so the domain wall can be bounded by a vortex of either type . in contrast to an individual vortex , a pair of @xmath84 and @xmath85 vortices , placed parallel to each other , will have energy per unit length which is only logarithmically divergent . that is because the @xmath96 `` charges '' of the two vortices cancel each other , so @xmath96 is trivial at spatial infinity ( no winding ) . the same situation occurs for a pair of parallel vortices of the same type ( @xmath84 or @xmath85 ) with opposite winding ( however , such vortex - antivortex pair can annihilate , while a @xmath142 pair can not ) . in a certain sense , one can talk about the phenomenon of `` vortex confinement '' : vortices exist only in pairs . this confinement should , in principle , be observable experimentally , in a rotating two - component bec , which can be already created in a laboratory @xcite . the vortices are usually identified by the density depletion at their cores , but can be also seen as dislocations of interference fringes due to phase singularities @xcite . with the coupling drive off ( @xmath18 ) such a system contains an equal number of @xmath84 and @xmath85 vortices which are distributed in space with no particular correlation between @xmath84 and @xmath85 vortices . as one turns on @xmath14 , the vortices will start to pair up and at some point the system will become a collection of composite objects , each being a bound state of a @xmath84 and a @xmath85 vortex ( fig . [ fig : pairs ] ) . the phenomenon of vortex confinement is very similar to that of quark confinement in the theory of strong interaction ( quantum chromodynamics ) . similarly to our vortices , quarks and antiquarks do not exist as individual objects , but are confined into composite objects hadrons . the analogy with quantum chromodynamics actually stretches further . if one places a @xmath84 vortex and a @xmath85 vortex at a distance larger than @xmath94 , then a domain wall that connects these two vortices will be formed . the tension of the domain wall is the force , per unit length , that attracts the two vortices . the attractive force between the two vortices is thus independent of their separation , given that the latter is larger than the width of the domain wall . this is analogous to the confining force between a quark and an antiquark , which is also constant at large distances . a confinement model which resembles most the confinement of the vortices is the three - dimensional compact quantum electrodynamics considered in ref . @xcite . in this theory the worldlines of electrically charged particles are analogous to the vortices in bec . one can also imagine a system of two vortices which rotate around each other so that the confining force ( from the domain wall ) balances the centrifugal force . such a system is analogous to the high - spin meson states in hadronic physics where a quark and an antiquark rotate around each other . we have shown that in a system of two interpenetrating bec with a coupling drive , there exists a domain wall solution . the relative phase between the two condensates changes by @xmath0 as one travels through the wall . the wall solution is formally similar to the kink in the sine - gordon field theory , yet it is not topologically stable and can decay . in this respect , the wall is more similar to a soap film , which can spontaneously burst . from the mathematical point of view , the domain walls discussed in this paper are similar to the ones which have been studied in particle physics . such domain walls appear at least in two contexts : in the theory of the hypothetical axion @xcite and in high - density quark matter @xcite . the similarity is that the domain wall solution arises from the spontaneous breaking of an approximate u(1 ) symmetry . in all cases the domain wall exists only if the explicit violation of the u(1 ) symmetry ( determined by the value of the rabi frequency in our case ) is small enough . the decay of the wall in all the examples occurs via hole nucleation . as to the experimental realization of the domain wall , one could be optimistic since the estimated critical value of the rabi frequency , for densities and scattering lengths typical for the rubidium gas in recent experiments , is of the order 100 hz which is not too small . the width of the wall , which might be as small as a few @xmath82 m if the rabi frequency is not much smaller than critical , can be also accommodated inside condensates of the size characteristic of present - day experiments . the apparent immediate obstacle is still the creation of a system where two bec truly interpenetrate . in the recent experiments with trapped atomic gas the region of overlap between the two bec is still small . one can hope the technical problems of making a genuine two - component bec to be solved in the near future which would enable one to study the domain walls experimentally . the authors thank richard friedberg for discussions . we thank riken , brookhaven national laboratory , and u.s . department of energy [ de - ac02 - 98ch10886 ] for providing the facilities essential for the completion of this work . the work of dts is supported , in part , by a doe oji grant . in the case when the densities of the two components are equal , @xmath112 , the maximal frequency @xmath108 , below which the wall configuration is still locally stable , can be found analytically . for illustrative purposes we shall consider this particular case in details . let us recall that when the scattering lengths @xmath144 , @xmath145 , and @xmath146 are approximately equal ( assuming @xmath147 ) there are two healing lengths @xmath76 and @xmath75 . the healing length related to fluctuations of the overall density @xmath75 is much smaller than the one related to fluctuations of the relative density @xmath76 : @xmath148 . we shall be interested in the case when the width of the wall @xmath94 is much larger than @xmath75 ( but we shall not presume any relation between @xmath94 and @xmath76 ) . in this case , the total density @xmath149 can be considered as frozen , and the system can be described in terms of three variables : @xmath150 , @xmath84 and @xmath85 , @xmath151 where @xmath150 runs from @xmath152 and @xmath133 . in terms of these three variables , the lagrangian has the form @xmath153-h\ , , \label{lthetaphi}\ ] ] where @xmath154 is the following hamiltonian @xmath155 + { 1\over2}(\delta\mu n - \delta g ' n^2)\cos\theta \nonumber\\ & & - { 1\over4}\delta gn^2\sin^2\theta - { 1\over2}\hbar\omega n\sin\theta\cos(\varphi_1-\varphi_2)\ , . \label{hthetaphi}\end{aligned}\ ] ] in eq . ( [ hthetaphi ] ) @xmath156 . the ground state is found by minimizing the potential term in eq . ( [ hthetaphi ] ) . from here on we consider the special case when in the ground state the density of atoms of the two species are equal , @xmath157 ( or equivalently @xmath158 . ) this requires @xmath159 . in this case @xmath154 possesses a discrete symmetry with respect to replacing @xmath160 , @xmath161 . this symmetry is what makes it possible to find @xmath108 analytically . it is more convenient to use , instead of @xmath104 , the normal modes @xmath162 defined in eqs . ( [ phiab ] ) , which in the case @xmath157 have the form @xmath163 in terms of these variables the hamiltonian becomes @xmath164 \nonumber \\ & & - { 1\over4}\delta gn^2\sin^2\theta - { 1\over2}\hbar\omega n\sin\theta\cos\phia\ , . \label{hphiab}\end{aligned}\ ] ] in order to find the domain wall configuration , we need to extremize the energy with respect to @xmath150 and @xmath165 . varying with respect to @xmath166 , one finds @xmath167 this equation determines @xmath166 for given @xmath150 and @xmath96 . the task of solving eq . ( [ phibthetaphia ] ) becomes much simpler if one assumes that all variables depend only one coordinate @xmath91 . in this case @xmath168 which can be trivially solved : @xmath169 . after eliminating @xmath166 , the energy functional one has to minimize is @xmath170 -{1\over4}\delta gn^2\sin^2\theta - { 1\over2}\hbar\omega n\sin\theta\cos\phia\ , . \label{hphia}\ ] ] it is easy to check that the following configuration is always a local extremum of eq . ( [ hphia ] ) : @xmath171 equation ( [ extremum1 ] ) can be guessed from the symmetry of eq.([hphia ] ) under @xmath160 . to see if the domain wall solution is a local minimum , one needs to expand @xmath154 in the vicinity of ( [ extremum1],[extremum ] ) . one writes @xmath172 to the second order in @xmath173 and @xmath174 the hamiltonian ( [ hphia ] ) is @xmath175 + { \hbar^2n\over8 m } \left[(\nabla\tilde\phia)^2 + k^2\cos\bar\phia\cdot\tilde\phia^2\right]\ , . \label{h2ndorder}\ ] ] one has to find the eigenmodes of eq . ( [ h2ndorder ] ) : if there are no negative modes then the domain wall is a local minimum of the energy ; if there exist a negative mode then the domain wall is unstable . the second , @xmath150-independent , term in eq . ( [ h2ndorder ] ) does not have negative modes ( it has only one zero mode corresponding to the translation of the wall along the @xmath91 direction ) and does not lead to instability , and hence can be ignored . taking into account the explicit solution @xmath176 , the first term in eq.([h2ndorder ] ) is @xmath177\,.\ ] ] the well - known operator @xmath178 has the lowest eigenvalue equal to @xmath179 , corresponding to the eigenfunction @xmath180 , which implies that @xmath154 does not have a negative mode if @xmath119 , where @xmath181 when @xmath182 , the configuration ( [ extremum ] ) is not a local minimum of the energy functional : the domain wall does not exist . the value ( [ omegac_exact ] ) is of the same order as @xmath109 in eq . ( [ omega0 ] ) : @xmath183 . the numerical value @xmath184 is specific for the case @xmath112 ; if @xmath185 then @xmath113 is , in general , different . as we have seen above , as long as @xmath119 , the wall minimizes the energy of the system with respect to small _ local _ variations of the condensates . however , the global minimum of the energy is achieved when the phases of the condensates are constant in space . since , as discussed above , the wall configuration can be continuously deformed into the global minimum energy configuration , i.e. , the wall can decay . here we shall estimate the lifetime of the wall due to such a decay . since the wall minimizes the energy locally , such a deformation necessarily goes through a potential barrier . thus the decay can only occur by a quantum tunneling through this barrier or , at finite temperature , by a thermal fluctuation over the barrier . in both cases , the decay rate is exponentially suppressed : in the first case by the wkb factor @xmath186 , and in the second case by the boltzmann factor @xmath187 , where @xmath188 is the height of the barrier . the first formula applies at sufficiently low temperature , while the second one applies at higher temperatures . a crude estimate ( see below ) suggests that the corresponding crossover temperature is quite small ( nanokelvins for parameters typical for present - day experiments ) , and is lower than temperatures achieved in present - day experiments . thus we shall limit our discussion to the decay by thermal fluctuation . this case is also simpler theoretically . the quantum mechanical decay , relevant for very low temperatures , will be left beyond the scope of this paper . assuming the temperature is much smaller than the critical temperature at which one of the condensates melts ( so that most atoms are still in the condensates ) , the rate of thermal activation across the barrier is @xmath189 , where @xmath188 is the height of the barrier at zero temperature . since we are dealing with an infinite - dimensional configuration space , @xmath188 should be understood as the energy at the lowest point of the barrier . this point is a saddle point ( the energy has a single negative curvature direction ) . some information about the exponent @xmath190 can be obtained by a simple scaling argument without a detailed calculation . in terms of the dimensionless variables @xmath191 defined as @xmath192 the energy becomes @xmath193 = { \hbar^3n^{1/2}\over m^{3/2}\delta g^{1/2}}\int\!d^3\widetilde \x\ , & & \biggl\ { { 1\over2 } \biggl [ { 1\over4}(\tilde\nabla\theta)^2 + \cos^2{\theta\over2}(\tilde\nabla\varphi_1)^2 + \sin^2{\theta\over2}(\tilde\nabla\varphi_2)^2\biggr ] \nonumber\\ & & -{1\over4 } ( \sin^2\theta+2\cos\theta_0\cos\theta ) -{1\over2 } { \hbar\omega\over\delta gn } \sin\theta\cos(\varphi_1-\varphi_2)\biggr\}\ , , \label{hrescale}\end{aligned}\ ] ] where @xmath194 determines the relative condensate densities in the ground state at @xmath18 : @xmath195 . consider the dependence on @xmath14 at a given @xmath194 . the energy functional eq.([hrescale ] ) is equal to a dimensionful constant times a dimensionless functional which depends on @xmath14 only via the ratio @xmath196 , where @xmath197 , same as in ( [ omega0 ] ) . thus the saddle point of the energy is also a function of this dimensionless ratio : @xmath198 if , moreover , we define a temperature @xmath199 as @xmath200 which is of the same order as the critical temperature of the bose - einstein condensation , then the decay rate for a given @xmath65 ( or @xmath194 ) can be written as @xmath201 = \exp\biggl[-{1\over\sqrt{4\pi \delta an^{1/3 } } } { t_0\over t } f\biggl({\omega\over\omega_0}\biggr)\biggr]\ , . \label{decayscaling}\ ] ] the form of the function @xmath202 can not be found from scaling arguments alone . however , when @xmath203 one can expect @xmath204 , and then the exponent @xmath205 is large , since @xmath206 , and @xmath207 . the function @xmath202 can be computed in the regime @xmath208 , where the saddle point configuration can be found in the `` thin - wall '' approximation . in this approximation field configurations are described at length scales much larger than the width of the domain wall . from this point of view the domain wall is an infinitely thin membrane . the saddle point configuration is a membrane with a round hole in it ( fig . [ fig : hole ] ) . the radius of the hole @xmath209 must be much larger than the width of the wall for the thin wall approximation to be valid . as discussed in sec . [ sec : boundary ] , the rim of the hole must be a vortex , since it is the boundary of the domain wall . there are two contributions to the energy difference @xmath188 between the saddle point and the domain wall configurations . one contribution is negative and comes from the hole ( since the hole is the absence of the wall ) . another contribution is positive and comes from the rim . therefore , @xmath210 where @xmath211 is the energy per unit length of the vortex ( the vortex tension ) and @xmath212 is the domain wall tension . the energy ( [ enusigma ] ) has a maximum when the radius of the hole is @xmath213 this is the radius of the critical hole . indeed , if a hole with a radius @xmath214 is nucleated , then it will expand and eventually eat up the whole wall . if , in contrast , the radius of the nucleated hole is less than critical , then the hole will shrink and disappear . substituting @xmath215 in eq . ( [ enusigma ] ) , we find the height of the energy barrier @xmath216 as there are two types of vortices , in eq . ( [ barrier ] ) @xmath211 refers to the vortex with the smaller tension . the tension of a straight vortex is logarithmically divergent : @xmath217 where the index @xmath218 refers to the two types of vortices , and @xmath209 is the long - distance cutoff . the role of @xmath209 is played by either the size of the critical hole @xmath219 or the width of the wall @xmath94 . we shall see at the end of this section that the two lengths differ only by a logarithm , which does not affect eq . ( [ nu_i ] ) . therefore , to logarithmic accuracy , the vortex tension is @xmath220 the argument of the logarithm is @xmath221 , and is large when @xmath208 , which justifies the use of the logarithmic approximation . without loss of generality , we can assume that @xmath222 . then the vortex of the first type has the smallest @xmath211 ( [ nu_i ] ) . substituting eqs.([nui ] ) and ( [ tension ] ) into eq . ( [ barrier ] ) , one finds that the barrier height has the form of eq . ( [ escaling ] ) , where @xmath223 the decay rate ( [ decayscaling ] ) can be rewritten in the following form @xmath224^{2/3 } \biggl({n\over n_2}\biggr)^{3/2 } { 1\over\sqrt{4\pi\delta an_1^{1/3}}}{t_{c1}\over t } \biggl({\omega_0\over\omega}\biggr)^{1/2 } \ln^2{\omega_0\over\omega}\biggr\}\ , , \label{gamma}\ ] ] where we introduced @xmath225 the critical temperature for the smallest of the two condensates ( @xmath62 by our choice ) . the rate ( [ gamma ] ) is exponentially suppressed when @xmath208 and @xmath226 . to check the consistency of our assumptions , we note that the radius of the critical hole is @xmath227 comparing with the width of the wall in eq . ( [ profile ] ) , we find that @xmath219 is larger than @xmath94 by a factor @xmath228 , which is assumed to be parametrically large . thus , the use of the thin wall approximation is justified . to make a good estimate of the crossover temperature , below which crossover proceeds via quantum tunneling , rather than via thermal activation , one needs to estimate the action on the tunneling trajectory and compare it to the exponent in the thermal activation rate @xmath190 . for a very crude estimate we can take @xmath229 the exponents @xmath230 and @xmath190 become comparable at a temperature of order few nanokelvin , given typical parameters of present day experiments .

we consider a system of two interpenetrating bose - einstein condensates of atoms in two different hyperfine spin states . we show that in the presence of a small coupling drive between the two spin levels , there exist domain walls across which the relative phase of the two condensates changes by @xmath0 . we give the physical interpretation of such walls . we show that the wall tension determines the force between certain pairs of vortices at large distances . we also show that the probability of the spontaneous decay of the domain wall is exponentially suppressed , both at finite and at zero temperature , and determine the exponents in the regime of small rabi frequency . we briefly discuss how such domain walls could be created in future experiments .

arxiv

an anomaly in field theory occurs if a symmetry of the action or the corresponding conservation law , valid in the classical theory , is violated in the quantized version . this surprising feature of quantum theory discovered by adler @xcite , bell and jackiw @xcite , and by bardeen @xcite in 1969 plays a fundamental role in physics ( for details see refs.@xcite @xcite ) . physically there is a difference between external and internal symmetries . the breakdown of an external symmetry is not dangerous for the consistency of the theory , on the contrary , it provides for instance the physical explanation for the @xmath1-decay @xcite , @xcite or the solution to the @xmath2 problem in qcd @xcite . on the other hand , the breakdown of an internal symmetry ( i.e. gauge symmetry ) leads to an inconsistency of the quantum theory , the anomalous ward identities destroy the renormalizability of the theory @xcite , and also the unitarity of the @xmath3-matrix may be lost @xcite . to avoid such anomalies imposes severe restrictions to the physical content of a theory . for instance , in the famous @xmath4 standard theory for electroweak interactions one had to demand the existence of the top quark long before it was discovered . gravitation regarded as a gauge theory also suffers from anomalies . the gauges are the general coordinate transformations ( diffeomorphisms ) or the rotations in the tangent frame ( lorentz transformations ) or the conformal transformations ( weyl transformations ) . then in the quantum case the classical conservation law of the energy - momentum tensor can be broken an einstein anomaly occurs or an antisymmetric part of the energy - momentum tensor can exist a lorentz anomaly occurs or the trace of the tensor is nonvanishing a weyl anomaly arises ( for details see e.g. refs.@xcite,@xcite , @xcite ) . whereas the anomaly in the tensor trace has been found already in the seventies @xcite @xcite the study of the gravitational anomalies started with the pioneering work of alvarez - gaum and witten @xcite in the eighties . the anomalies have been first found within perturbation theory , they are local polynomials in the connection and curvature . the authors @xcite @xcite have calculated ( ultraviolet divergent ) feynman diagrams where the external gravitational field couples to a fermion loop via the energy - momentum tensor . of course , the anomaly reflecting the deep laws of quantum physics must show up within other approaches too . so they have been calculated by the heat kernel method @xcite , @xcite , by fujikawa s path integral approach @xcite , @xcite , and by modern mathematical techniques such as differential geometry and cohomology @xcite @xcite and topology ( index theorems ) @xcite , @xcite , @xcite , ( for an overview see ref.@xcite ) . deeply related to anomalies are the socalled schwinger terms @xcite @xcite ( for an introduction see e.g. refs . @xcite , @xcite ) . in a yang - mills gauge theory schwinger terms ( st ) show up as additional terms ( extensions ) in the canonical algebra of the equal time commutators ( etc ) of the gauss law operators ( see e.g. @xcite cohomologically they are described by the faddeev - mickelsson cocycle @xcite , @xcite , and geometrically they can be related to a berry phase in the vacuum functional @xcite @xcite . st are frequently calculated within perturbation theory where the bjorken - johnson - low limit @xcite , @xcite works very well . however , the definition of a point - splitting method turns out to be more subtle and might not lead to the correct result @xcite , @xcite . in gravitation schwinger terms occur in the etc of the energy - momentum tensors , c - number terms that are proportional to derivatives of the @xmath0-function . they can be related to the gravitational anomalies @xcite , and they have been calculated explictly via the invariant spectral function and via cohomological techniques @xcite . furthermore there exists an interesting relation of the st to the curvature of the determinant line bundle @xcite , @xcite . our work deals with the calculation of the gravitational anomalies , specifically the einstein anomaly and the weyl anomaly . the lorentz anomaly is not independent of the einstein anomaly , both types of anomalies can be shifted into each other by a suitable counterterm @xcite . for convenience we choose the case where the lorentz anomaly is vanishing . we also calculate the gravitational schwinger terms . the purpose of our work is to show that all these anomalous features are easily obtained by the method of dispersion relations , a less familiar but very useful approach . some of our results we have briefly presented in refs . @xcite , @xcite . already since their first introduction into quantum field theory @xcite dispersion relations ( dr ) proved to be a very valuable tool . in connection with anomalies dr have been fomulated by dolgov and zakharov @xcite and also by kummer @xcite . in the following several authors @xcite @xcite used successfully dr to determine the anomalies in the chiral current . recently hoej ' i and schnabl @xcite have applied the method to the well - known trace anomaly @xcite , @xcite which is related to the broken dilatation ( or scale ) invariance . we extend in our work the method of dr to the case of pure gravitation . so we consider chiral fermion loops coupled to gravitation for their evaluation it is enough to use gravitation as an external or nonquantized field and we show that the gravitational anomalies and the schwinger terms are , in fact , completely determined by dispersion relations . all calculations are performed in two dimensions . conceptuelly the dr approach is an independent and complementary view of the anomaly phenomenon as compared to the ultraviolet regularization procedures . within dr the anomaly manifests itself as a very peculiar infrared feature of the imaginary part of the amplitude . but as we shall show there is a link between the two approaches , the dr method and the n - dimensional regularization precedure . our paper is organized as follows . in section 2 we present the general structure of the considered ( pseudo- ) tensor amplitude and we discuss the ward identities which we have to study . in section 3 we introduce the dispersion relations for the relevant formfactors of the amplitude and calculate their imaginary parts via the cutkosky rule . in order to reproduce the dr results in a definite ultraviolet regularization scheme we have worked out in detail the t hooft - veltman regularization procedure in section 4 and we have compared the several amplitudes with the results of tomiya @xcite and alvarez - gaum and witten @xcite . the equivalence between the dispersive approach and the dimensional regularization procedure is given in section 5 . in section 6 we derive the anomalous ward identities and explain the source of the anomaly in the dr approach . from the ward identities we deduce the linearized gravitational anomalies the einstein- and the weyl anomaly and we also determine their covariant versions , a comparison with the exact results is given . the gravitational schwinger terms occuring in the etc of the energy - momentum tensors we calculate in section 7 , where we adapt the dispersive approach to a method proposed by klln @xcite . finally we summarize our main results in section 8 . in two dimensions the lagrangian describing a weyl fermion in a gravitational background field can be written as @xmath5 where @xmath6 is the zweibein and @xmath7 its inverse @xmath8 . the determinant of the zweibein is @xmath9 and @xmath10 is the covariant derivative with the spin connection @xmath11 . we use the following conventions in 2 dimensions : * for the flat metric @xmath12 * for the epsilon tensor @xmath13 * for the dirac matrices @xmath14 the einstein and the weyl anomaly are determined by the one - loop diagram in fig.[loop ] and it is sufficient to use the linearized gravitational field @xmath15 since in two dimensions the spin connection @xmath11 does not contribute ( see e.g. ref.@xcite ) we find the following linearized interaction lagrangian ( for convenience @xmath16 is absorbed into @xmath17 , @xmath18 acts only on @xmath19 ) @xmath20 and @xmath21 from this expression follow the feynman rules for the vertices in the loop diagram @xmath22 and the explicit form of the ( symmetric ) energy - momentum tensor @xmath23 where we have dropped the terms proportional to @xmath24 as they do not contribute to the amplitude . + then the whole amplitude is given by the two - point function @xmath25 due to lorentz covariance and symmetry the general structure of the amplitude can be written in the following way @xmath26 @xmath27 @xmath28 t_{8}(p^{2 } ) \ , , \label{formfactors3}\end{aligned}\ ] ] where we have separated the amplitude into its pure tensor part @xmath29 ( coming from the vector piece of the chirality projection in eq.([energy - momentum - tensor ] ) ) and into its pseudo - tensor part @xmath30 ( coming from the axial piece in eq.([energy - momentum - tensor ] ) ) . the functions @xmath31 are the formfactors that are to be evaluated . + classically the energy - momentum tensor has the following properties : 1 . @xmath32 , symmetric 2 . @xmath33 , conserved 3 . @xmath34 , traceless which lead to the canonical ( naive ) ward identities : 1 . @xmath35 2 . @xmath36 3 . @xmath37 . we are interested in the pure einstein anomaly therefore we demand the quantized energy - momentum tensor to be symmetric . this is always possible due to the bardeen zumino theorem @xcite which states that the gravitational anomaly can be shifted from pure lorentz type to pure einstein type , and vice versa . ( we disregard here leutwyler s point of view @xcite who emphasizes his preference for the lorentz anomaly ) . thus the symmetry property 1 . ) of the amplitude is fulfilled and an other symmetry @xmath38 is trivially satisfied . however , the naive ward identities 2 . ) and 3 . ) need not be satisfied , they can be broken by the einstein and weyl anomalies respectively . the canonical ward identities we re - express by the formfactors @xmath39 { } \nonumber\\{}&&+ \varepsilon_{\rho \tau } p^{\tau}\left[p_{\nu}p_{\sigma } ( p^{2 } t_{6 } + t_{7 } + t_{8 } ) + g_{\nu \sigma } p^{2 } t_{8}\right ] { } \nonumber\\{}&&+ \varepsilon_{\sigma \tau } p^{\tau}\left[p_{\nu}p_{\rho } ( p^{2 } t_{6 } + t_{7 } + t_{8 } ) + g_{\nu \rho } p^{2 } t_{8}\right ] \label{pta}\end{aligned}\ ] ] and @xmath40 where @xmath41 is the dimension . we call from now on ward identity ( wi ) the property 2 . ) and trace identity ( ti ) the property 3 . ) . for the pure tensor part of the amplitude the wi may be written as @xmath42 and the ti as @xmath43 of course , relations ( [ ti2 ] ) and ( [ ti1 ] ) are not independent of each other , ( [ ti2 ] ) follows from ( [ ti1 ] ) and ( [ vwi1 ] ) ( [ vwi3 ] ) . in the following we shall use a renormalization procedure which keeps the wi in the pure tensor part ( [ vwi1 ] ) ( [ vwi3 ] ) so that the anomaly occurs only in the axial pieces representing the pseudotensor part of the amplitude . + calculating the amplitude with massive fermions ( the loop in fig . [ loop ] ) with help of the feynman rules gives @xmath44 \frac{1\pm\gamma_{5}}{2}{}\nonumber\\&&{}\times \frac{/\hspace{-6.5pt}p + /\hspace{-6.5pt}k+m}{(p+k)^{2}-m^{2}+i \varepsilon } \left [ \gamma_{\rho } ( p+2k)_{\sigma } + \gamma_{\sigma } ( p+2k)_{\rho}\right ] \frac{1\pm\gamma_{5}}{2 } \frac{/\hspace{-6pt}k+m}{k^{2}-m^{2}+i \varepsilon } \biggr\}.\end{aligned}\ ] ] for convenience we split the pure tensor piece of the loop in the following way @xmath45 where @xmath46 represents the loop with the identity instead of the chirality projectors and @xmath47 denotes the part proportional to @xmath48 . we separate the no interchange amplitudes as @xmath49 finally , the axial part of the amplitude is connected to the vector part due to relation @xmath50 ( valid only in 2 dimensions for our conventions ( [ definition of the flat metric ] ) - ( [ definition of the gamma matrices ] ) ) . a symmetric decomposition turns out to be useful @xmath51 the formfactors of the amplitude ( [ formfactors1 ] ) ( [ formfactors3 ] ) are analytic functions in the complex @xmath52 plane except a cut on the real axis starting at @xmath53 . due to cauchy s theorem they can be expressed by dispersion relations which relate the real part of the amplitude to its imaginary part . the imaginary parts of the amplitude can be easily calculated via cutkosky s rule @xcite . it states to replace in the amplitude each propagator by its discontinuity on mass shell @xmath54 { } \nonumber\\ { } & & \!\!\!\times \delta(k^{2}-m^{2 } ) \delta\bigl((p+k)^{2}-m^{2}\bigr ) \theta(-k^{0 } ) \theta(k^{0}+p^{0 } ) \label{abstract i m tni}\\ i m t^{dv , ni}_{\mu \nu \rho \sigma}(p ) & = & \frac{1}{32 } m^{2 } g_{\mu\rho } \int \!d^{2}k ( p+2k)_{\nu } ( p+2k)_{\sigma}{}\nonumber\\ { } & & \!\!\!\times \delta(k^{2}-m^{2 } ) \delta\bigl((p+k)^{2}-m^{2}\bigr ) \theta(-k^{0 } ) \theta(k^{0}+p^{0 } ) \ , . \label{abstract i m tdvni}\end{aligned}\ ] ] the explicit integration gives @xmath55 p_{\mu}p_{\nu}p_{\rho}p_{\sigma}{}\nonumber\\ { } & & \qquad + \left [ -\frac{1}{6}p^{2}+\frac{4}{3}m^{2}-\frac{8}{3}\frac{m^{4}}{p^{2}}\right ] \left(p_{\mu}p_{\nu } g_{\rho \sigma } + p_{\mu}p_{\sigma } g_{\nu \rho } + p_{\nu}p_{\rho } g_{\mu \sigma } + p_{\rho}p_{\sigma } g_{\mu \nu}\right ) { } \nonumber\\ { } & & \qquad + \left [ \frac{1}{3}p^{2}-\frac{2}{3}m^{2}-\frac{8}{3}\frac{m^{4}}{p^{2}}\right ] p_{\mu}p_{\rho } g_{\nu \sigma } + \left[\frac{1}{3}p^{2}-\frac{5}{3}m^{2}+\frac{4}{3}\frac{m^{4}}{p^{2}}\right ] p_{\nu}p_{\sigma } g_{\mu \rho } { } \nonumber\\ { } & & \qquad + \left[\frac{1}{6}(p^{2})^{2}-\frac{4}{3}p^{2}m^{2}+\frac{8}{3}m^{4}\right ] \left(g_{\mu \nu } g_{\rho \sigma } + g_{\mu \sigma } g_{\nu \rho}\right ) { } \nonumber\\{}&&\qquad + \left[-\frac{1}{3}(p^{2})^{2}+\frac{5}{3}p^{2}m^{2}-\frac{4}{3}m^{4}\right ] g_{\mu \rho } g_{\nu \sigma } \biggr\ } \\ & & i m t^{dv , ni}_{\mu \nu \rho \sigma}(p ) = \frac{1}{32 } j_{0 } m^{2 } g_{\mu\rho } \biggl\ { \left ( -p^{2}+4m^{2 } \right ) g_{\nu\sigma } + \left(1 - 4\frac{m^{2}}{p^{2 } } \right ) p_{\nu}p_{\sigma } \biggr\},\end{aligned}\ ] ] with the threshold function @xmath56 from which we quickly find all imaginary parts of the formfactors in the total amplitude @xmath57 : @xmath58 considering in addition the amplitude @xmath46 we have the following imaginary parts of the formfactors which we denote by : @xmath59 clearly , the imaginary parts ( [ i m a1 ] ) ( [ i m a5 ] ) of the amplitude @xmath46 satiesfy the wi ( [ vwi1])([vwi3 ] ) with @xmath60 , and the subtraction procedure we choose in the following keeps this property for the entire formfactors @xmath61 . + now we start with an unsubtracted dispersion relation ( dr ) for the formfactors @xmath62 and we observe that , for instance , the integral for @xmath63 is convergent whereas for @xmath64 it is logarithmically divergent and needs to be subtracted once , and for @xmath65 it is linearly divergent and needs to be subtracted twice . we can infer already from the @xmath52 behaviour of the imaginary parts which kind of dispersion relation we have to use , see table [ subtraktionen ] . .the formfactors and the used type of dispersion relations [ cols= " < , > " , ] so for the formfactors @xmath66 an unsubtracted dr is sufficient and we get @xmath67 \label{t1 in dr}\end{aligned}\ ] ] with @xmath68 a once subtracted dr defined by @xmath69 we use for the following formfactors ( see table 1 ) @xmath70 for the remaining formfactors a twice subtracted dr defined by @xmath71 is necessary and we find @xmath72\\ t^{r}_{5}(p^{2 } ) & = & \frac{p^{4}}{\pi } \int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t - p^{2 } } \frac{1}{t^{3}}\left(1-\frac{4m^{2}}{t}\right)^{\hspace{-3pt}-\frac{1}{2 } } \hspace{-3pt } \left ( -\frac{1}{96}t^{2 } + \frac{1}{48}t m^{2}+\frac{1}{12 } m^{4 } \right ) { } \nonumber\\{}&= & p^{2}\left[-\frac{5}{288\pi } - \frac{1}{24\pi } \frac{m^{2}}{p^{2 } } + \frac{1}{48\pi } \bigl(1 + 2 \frac{m^{2}}{p^{2 } } \bigr ) a(p^{2})\right]\\ a^{r}_{5}(p^{2 } ) & = & \frac{p^{4}}{\pi } \int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t - p^{2 } } \frac{1}{t^{3}}\left(1-\frac{4m^{2}}{t}\right)^{\hspace{-3pt}-\frac{1}{2 } } \hspace{-3pt } \left ( -\frac{1}{48}t^{2 } - \frac{1}{12}t m^{2}+\frac{2}{3}m^{4}\right ) { } \nonumber\\{}&= & p^{2}\left[-\frac{1}{72\pi } - \frac{1}{3\pi } \frac{m^{2}}{p^{2 } } + \frac{1}{24\pi } \bigl(1 + 8 \frac{m^{2}}{p^{2 } } \bigr ) a(p^{2})\right ] . \label{a5r in dr}\end{aligned}\ ] ] with these explicit expressions for the formfactors we have determined the whole amplitude @xmath57 , eqs.([amplitude expressed by t - product])([formfactors3 ] ) , from which the correct ward identities will follow . although the method of dispersion relations appears quite different to the methods of regularizations we can reproduce its results in a definite regularization scheme , namely in the n - dimensional regularization procedure of t hooft veltman @xcite . it is instructive to work it out in more detail . we start with amplitude ( [ amplitude by dimensional regularization with massive fermions ] ) , calculate the @xmath73-matrices and follow the standard procedure by inserting the feynman parameter integral @xmath74^{2 } } \ , \ , .\ ] ] then we obtain for the ` no interchange ' amplitudes of the pure tensor pieces @xmath75^{2 } } { } \nonumber\\{}&&\quad \times \bigl [ ( l+px)_{\mu}(l - p(1-x))_{\rho } + ( l+px)_{\rho}(l - p(1-x))_{\mu } - g_{\mu\rho } ( l+px)^{\lambda}(l - p(1-x))_{\lambda } \bigr]\end{aligned}\ ] ] and @xmath76^{2 } } \ , \ , , \ ] ] with @xmath77 calculating the t hooft veltman integrals @xmath78^{\alpha } } = \frac{(-1)^{\alpha}i } { ( 4\pi)^{\omega } } \frac{\gamma(\alpha - \omega)}{\gamma(\alpha ) } \delta^{\omega-\alpha } \\ p_{1}^{\mu\nu } & = & \int\!\ ! \frac{d^{2\omega}l}{(2\pi)^{2\omega } } \frac{l^{\mu}l^{\nu}}{[l^{2}-\delta]^{\alpha } } = \frac{\delta}{2(\omega - \alpha + 1 ) } g^{\mu\nu } p_{0 } \label{integral p1}\\ p_{2}^{\mu\nu\rho\sigma } & = & \int\!\ ! \frac{d^{2\omega}l}{(2\pi)^{2\omega } } \frac{l^{\mu } l^{\nu } l^{\rho } l^{\sigma } } { [ l^{2}-\delta]^{\alpha } } { } \nonumber\\{}&= & \frac{\delta^{2}}{4(\omega - \alpha + 1)(\omega - \alpha + 2 ) } ( g^{\mu\nu } g^{\rho\sigma } + g^{\mu\rho } g^{\nu\sigma } + g^{\mu\sigma } g^{\nu\rho } ) p_{0 } \label{integral p2}\end{aligned}\ ] ] with @xmath79 provides the following expressions @xmath80 p_{\nu}p_{\sigma } g_{\mu \rho } { } \nonumber\\ { } & & \qquad + 2 \frac{\delta^{2 } } { \omega(\omega-1 ) } ( g_{\mu \nu } g_{\rho \sigma } + g_{\mu \sigma } g_{\nu \rho } ) + \left [ -2\omega \frac{\delta^{2 } } { \omega(\omega-1 ) } + 2x(1-x)p^{2 } \frac{\delta}{\omega-1 } \right ] g_{\mu \rho } g_{\nu \sigma } \biggr\}\label{explicit tni in massive dim reg}\\ & & t^{dv , ni}_{\mu \nu \rho \sigma}(p ) = - i \frac{2^{\omega}}{32 } m^{2 } g_{\mu\rho } \int\limits_{0}^{1 } \!dx p_{0 } \biggl\ { 2 \frac{\delta}{\omega-1 } g_{\nu\sigma } + ( 1 - 2x)^{2 } p_{\nu}p_{\sigma } \biggr\}\label{explicit tdvni in massive dim reg}.\end{aligned}\ ] ] these expressions we compare now with the formfactor decompositions ( [ formfactors2 ] ) and ( [ formfactors3 ] ) ( recall eqs.([without interchange ] ) and ( [ axial part with gamma5 symmetrically distributed ] ) ) then we obtain all formfactors of the amplitude @xmath57 explicitly @xmath81 the formfactors of the amplitude @xmath46 follow via eq.([pure vector amplitude ] ) @xmath82 as we can see the formfactors @xmath83 and @xmath84 ( eqs.([t1 in massive dim reg])and ( [ a1 in massive dim reg ] ) ) are finite in the limit @xmath85 and explicitly we find @xmath86 \label{t1r in massive dim reg}\end{aligned}\ ] ] with @xmath87 given by eq.([definition of a ] ) . however , the remaining formfactors are divergent in the limit @xmath85 and we have to renormalize them in an appropriate way . we separate the formfactors into a divergent part and a finite part @xmath88 so that we can extract the finite result by a suitable prescription . the formula we need for this procedure is @xmath89 - \frac{1}{2\pi}\left.\frac{df}{d\omega}\right|_{\omega=1}+ o(1-\omega ) \ , .\end{aligned}\ ] ] it is the pole part @xmath90 of the formfactor which tells us which kind of renormalization we have to choose , for instance , for a constant a simple subtraction is sufficient , for a term proportional to @xmath91 a double subtraction . in this way the following formfactors are determined by a simple subtraction @xmath92 whereas for the remaining formfactors a double subtraction is needed @xmath93 { } \nonumber\\ { } & = & p^{2}\left [ \frac{1}{18\pi } - \frac{1}{6\pi } \frac{m^{2}}{p^{2 } } - \frac{1}{24\pi } \bigl(1 - 4 \frac{m^{2}}{p^{2 } } \bigr ) a(p^{2})\right]\label{t4r in massive dim reg}\\ t^{r}_{5}(p^{2 } ) & = & t_{5}(p^{2 } ) - t_{5}(0 ) - p^{2 } \left.\frac{d}{dp^{2}}t_{5}(p^{2})\right|_{p^{2}=0 } { } \nonumber\\ { } & = & \frac{1}{16\pi } \int\limits^{1}_{0}\!dx \ p^{2}x(1-x ) \ln \frac{m^{2}-p^{2}x(1-x)}{m^{2 } } { } \nonumber\\{}&= & p^{2}\left[-\frac{5}{288\pi } - \frac{1}{24\pi } \frac{m^{2}}{p^{2 } } + \frac{1}{48\pi } \bigl(1 + 2 \frac{m^{2}}{p^{2 } } \bigr ) a(p^{2})\right]\\ a^{r}_{5}(p^{2 } ) & = & a_{5}(p^{2 } ) - a_{5}(0 ) - p^{2 } \left.\frac{d}{dp^{2}}a_{5}(p^{2})\right|_{p^{2}=0 } { } \nonumber\\ { } & = & \frac{1}{8\pi } \int\limits^{1}_{0}\!dx \biggl[\bigl ( m^{2 } + p^{2}x(1-x ) \bigr ) \ln \frac{m^{2}-p^{2}x(1-x)}{m^{2 } } + p^{2}x(1-x)\biggr ] { } \nonumber\\{}&= & p^{2}\left[-\frac{1}{72\pi } - \frac{1}{3\pi } \frac{m^{2}}{p^{2 } } + \frac{1}{24\pi } \bigl(1 + 8 \frac{m^{2}}{p^{2 } } \bigr ) a(p^{2})\right ] . \label{a5r in massive dim reg}\end{aligned}\ ] ] of course , the formfactors @xmath94 of the tensor amplitude @xmath46 satisfy the wi ( [ vwi1 ] ) ( [ vwi3 ] ) and as the difference @xmath47 to the pure tensor amplitude is proportional to @xmath48 ( recall eqs.([pure vector amplitude]),([explicit tdvni in massive dim reg ] ) ) the amplitude @xmath29 will also satisfy the wi ( [ vwi1 ] ) ( [ vwi3 ] ) in the limit @xmath95 . this procedure works in complete analogy to the dispersion relation approach ( see table 1 ) and we clearly obtain the same results for the formfactors as one can see by comparing eqs.([t1r in massive dim reg])([a5r in massive dim reg ] ) with ( [ t1 in dr])([a5r in dr ] ) . even more , as we shall show in the next chapter , the two approaches are equivalent . but before we want to consider the case of massless fermions , the limit @xmath95 , in order to compare our results with the ones of other authors . taking the limit @xmath95 in the @xmath96 expansion ( eq.([pole terms ] ) ) of the formfactors we find the following results @xmath97 \label{t2 for wi } \\ t_{3}(p^{2 } ) & = & -\frac{1}{p^{2}}t_{5}(p^{2 } ) = \mp 4\ t_{8}(p^{2 } ) { } \nonumber\\{}&= & -\frac{1}{96\pi } \biggl [ \frac{1}{\omega-1 } + \gamma - \frac{2}{3 } + \ln\left|\frac{p^{2}}{2\pi\mu^{2}}\right| - i\pi \biggr ] \label{t3 for wi}.\end{aligned}\ ] ] which agree with the expressions we obtain when starting with @xmath98 from the very beginning ( we also introduced here the mass @xmath99 to keep the correct dimensionality , see ref.@xcite ) . now , for comparison tomiya @xcite in his work on the gravitational anomaly defines the amplitude with the covariant @xmath100-product and calculates the following formfactors @xmath101 they also satiesfy the wi ( [ vwi1 ] ) ( [ vwi3 ] ) , however , they ( some of them ) differ from ours ( [ t1 for wi ] ) ( [ t3 for wi ] ) , which is not surprising as the covariant @xmath100-product differs from the ordinary @xmath102-product by suitable seagull terms . but as we shall show below his and our results lead to the same amplitudes and consequently to the same anomaly expression . using on one hand the wi ( [ vwi1 ] ) ( [ vwi3 ] ) for the formfactors @xmath103 in the massless limit , @xmath95 , and on the other the fact that @xmath104 we find for the several amplitudes ( [ formfactors1 ] ) ( [ formfactors3 ] ) the following expressions @xmath105 as we can see all amplitudes depend only on the ( convergent ) formfactor @xmath63 , explicitly given by eq.([t1 for wi ] ) , and are independent of a second ( divergent ) formfactor , say @xmath64 , which one might have expected to contribute too ( since we have six restrictive relations between the eight formfactors @xmath103 ) . so the amplitudes do not determine the formfactors uniquely ! therefore @xmath64 can be chosen at will and tomiya @xcite in his work renormalized the formfactors conveniently such that @xmath64 came out to be zero . this is in accordance with the amplitude result of deser and schwimmer @xcite , @xcite who consider fields without chirality . on the other hand , alvarez - gaum and witten @xcite consider the gravitational anomaly within light - cone coordinates and use a specific regularization prescription to calculate the following amplitude @xmath106 their result is consistent with our calculations as we have @xmath107 { } \nonumber\\{}&= & \frac{1\mp1}{24\pi p^{2}}p_{+}^{4 } \ , \ , .\end{aligned}\ ] ] the dimensional regularization procedure is a method which satisfies the wi ( [ vwi1 ] ) ( [ vwi3 ] ) . on the other hand , within the dispersion relation approach we also have effectively renormalized in a way which keeps this wi property ( [ vwi1 ] ) ( [ vwi3 ] ) . if we had subtracted more often or at some other point than @xmath108 we would have lost this property . the dispersive approach and the dimensional regularization procedure are equivalent in the following sense . for the formfactors that were convergent or logarithmically divergent the corresponding integrals in both approaches do not have just identical values but we even can transform them into each other by a suitable substitution . more precise , we transform the feynman parameter integrals over x in the dimensional regularization procedure into the corresponding dispersion integrals over t. for the formfactors that were linearly divergent the situation is as follows . accidentially for @xmath109 there exists such a substitution , however , not for @xmath110 and @xmath111 . but anyway , for these formfactors the integrals are identical , what is actually sufficient . the substitutions which link the two approaches are @xmath112 to show the equivalence we need the following relations which are valid for any function @xmath113 with suitable differentiation properties @xmath114 now we turn to the calculation of the ward identities and gravitational anomalies . we consider the massless limit , @xmath95 , where the formfactors @xmath115 ( i = 1, ... ,5 ) fulfill the wi ( [ vwi1 ] ) ( [ vwi3 ] ) . this means that the wi for the pure tensor part ( [ ptv ] ) is satiesfied @xmath116 calculating next the wi for the pseudo tensor part ( [ pta ] ) we use the formfactor identities ( [ axial vector ] ) and we find @xmath117 in the flat space - time limit @xmath118 we have the relation @xmath119 and we finally obtain the anomalous result @xmath120 as we discovered already before when calculating the amplitudes , the anomalous wi depends only on the finite formfactor @xmath121 with its explicit result ( [ t1 for wi ] ) . so the anomaly is independent of a specific renormalization procedure ( as long as it preserves the wi ( [ vwi1 ] ) ( [ vwi3 ] ) ) and we clearly agree with the anomaly results of tomiya @xcite and alvarez - gaum and witten @xcite . we want to emphasize that our subtraction procedure is _ the _ ` natural ' choice dictated by the @xmath122behaviour of the imaginary parts @xmath123 of the formfactors . since on general grounds the imaginary parts of the amplitudes fulfill the wi ( [ vwi1 ] ) ( [ vwi3 ] ) ( in the limit @xmath95 ) the naturally chosen dispersion relations @xmath124=0 \\ & & \frac{1}{\pi}\int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t - p^{2}}\frac{p^{4}}{t^2}\left[t\ , im\,t_{2}(t)+im\,t_{4}(t)\right]=0 \\ & & \frac{1}{\pi}\int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t - p^{2}}\frac{p^{4}}{t^2}\left[t\ , im\,t_{3}(t)+im\,t_{5}(t)\right]=0\end{aligned}\ ] ] imply the pure tensor wi ( [ vwi1 ] ) ( [ vwi3 ] ) for the renormalized formfactors ( in the limit @xmath95 ) @xmath125 therefore our subtraction procedure automatically shifts the total anomaly into the pseudotensor part of the wi ( [ pta ] ) . + what is the origin of the anomaly in this dispersive approach ? the source of the anomaly is the existence of a superconvergence sum rule for the imaginary part of the formfactor @xmath63 @xmath126 the anomaly originates from a @xmath0-function singularity of @xmath127 when the threshold @xmath128 approaches zero ( the infrared region ) @xmath129 the limit must be performed in a distributional sense . + then the unsubtracted dispersion relation for @xmath63 , eq.([unsubtracted dr ] ) , provides the result ( [ t1 for wi ] ) . this threshold singularity of the imaginary part of the relevant formfactor is a typical feature of the dispersion relation approach for calculating the anomaly ( see e.g. refs.@xcite , @xcite @xcite , and the appendix b in ref . + next we turn to the energy - momentum tensor . from the anomalous wi @xmath130 we can deduce the linearized consistent einstein ( or diffeomorphism ) anomaly @xmath131 for comparison we demonstrate that result ( [ linearized consistent einstein anomaly ] ) is indeed the linearization of the exact result that follows from differential geometry and topology ( see for instance ref.@xcite ) . the exact einstein anomaly in two dimensions is given by @xmath132 where @xmath133 therefore we get @xmath134 considering the linearized gravitational field , eq.([linearization of the vielbein with upper indices ] ) , the christoffel symbols become ( with explicit @xmath16-dependence ) @xmath135 so that we find as linearization of the exact result ( [ exact consistent einstein anomaly ] ) @xmath136 this agrees with our result ( [ linearized consistent einstein anomaly ] ) since in two dimensions we have the identity @xmath137 now what about the covariant einstein anomaly ? it arises when considering the covariantly transforming energy - momentum tensor @xmath138 which is related to our tensor definition ( [ energy - momentum - tensor ] ) by the bardeen - zumino polynomial @xmath139 @xcite @xmath140 this polynomial is calculable and explicitly we find ( in two dimensions ) @xcite @xmath141 ( @xmath142 denotes the ricci scalar ) which leads to the covariant einstein anomaly @xmath143 linearizing the ricci scalar ( with explicit @xmath16-dependence ) @xmath144 and using ( [ linearization of the christoffel symbols ] ) , ( [ epsilon identity ] ) we get @xmath145 so that we find for the linearized covariant einstein anomaly @xmath146 it is twice the linearized consistent result ( [ linearized consistent einstein anomaly ] ) as it should be . + finally we also calculate the trace identity , eqs.([gtv ] ) and ( [ gta ] ) . using again relations ( [ axial vector ] ) we find @xmath147,\ ] ] and taking into account the wi ( [ vwi1 ] ) provides us the anomalous result @xmath148.\ ] ] the anomalous ti depends only on the finite formfactor @xmath121 what we know already from the previous discussion so that it is independent of a specific renormalization procedure ( as long as it preserves the wi ( [ vwi1 ] ) ( [ vwi3 ] ) ) . inserting the formfactor , eq.([t1 for wi ] ) , gives @xmath149,\ ] ] which implies the following linearization of the weyl ( or trace ) anomaly @xmath150.\ ] ] again , we compare our calculation with the exact result for the weyl anomaly which is given by ( see for instance ref.@xcite ) @xmath151 and consequently we have @xmath152 considering the linearizations of the spin connection ( with explicit @xmath16-dependence ) @xmath153 and of the ricci scalar eq.([linearization of r ] ) , we see that our result ( [ linearized weyl anomaly ] ) is indeed the linearization of the exact expression ( [ exact consistent weyl anomaly ] ) . adding last but not least the bardeen - zumino polynomial @xmath139 @xmath154 with its linearization @xmath155 we find for the covariant trace anomaly @xmath156 and @xmath157 for its linearized version . clearly these results are in agreement with ref . in quantum field theory the energy - momentum tensors form an algebra which is generally not closed but has central extensions , socalled schwinger terms , for example @xmath158 the schwinger terms , the c - number terms @xmath159 , can be determined by considering the vacuum expectation value of the etc . to evaluate the st we work with a technique that has been introduced by klln @xcite and is closely related to the dispersive approach used before . this technique has been applied already by skora @xcite to compute the st for currents in yang - mills theories . our aim is to generalize this procedure to the case of gravitation , where the current is replaced by the energy - momentum tensor . from the lagrangian ( [ lagrangian ] ) describing a weyl fermion in a gravitational background field in two dimensions we get the following classical energy - momentum tensor @xmath160 where @xmath161 means normal ordering . using relation ( [ elimination of gamma5 ] ) and the equations of motions we can express the pseudo tensor part of the enery - momentum tensor by the pure tensor part ( recall that the tensor is symmetric ) @xmath162 in two dimensions we have the identity @xmath163 so that we find @xmath164\vert 0\rangle = \frac{1}{4 } \biggl\ { \langle 0\vert [ t^{v}_{\mu\nu}(x),t^{v}_{\rho\sigma}(0)]\vert 0\rangle + \langle 0\vert [ t^{v}_{\rho\nu}(x),t^{v}_{\mu\sigma}(0)]\vert 0\rangle { } \nonumber\\ { } & & \qquad- g_{\mu\rho } \langle 0\vert [ t^{v\lambda}_{\nu}(x),t^{v}_{\lambda\sigma}(0)]\vert 0\rangle \mp \varepsilon_{\mu}^{\ \lambda}\langle 0\vert [ t^{v}_{\lambda\nu}(x),t^{v}_{\rho\sigma}(0)]\vert 0\rangle \mp \varepsilon_{\rho}^{\ \lambda } \langle 0\vert [ t^{v}_{\mu\nu}(x),t^{v}_{\lambda\sigma}(0)]\vert 0\rangle \biggr\}.\end{aligned}\ ] ] let us define @xmath165 by inserting the completeness relations @xmath166 and using the translation invariance @xmath167 we obtain @xmath168 where the sum runs over many - particle states @xmath169 with positive energy and momentum @xmath170 . + we may write @xmath171 where @xmath172 from lorentz covariance and symmetry we get the same decompositon into formfactors as for @xmath29 ( see eq . ( [ formfactors2 ] ) @xmath173 making use of @xmath174 provides the ward identity @xmath175 that can be expressed by the formfactors @xmath176 now let us explicitly evaluate @xmath177 . as we are considering the energy - momentum tensor as a free ( non interacting ) tensor analogously to the case of free currents we only need to sum up states that consist of one fermion - antifermion pair in eq . ( [ symbolic gmunurhosigma ] ) . we get @xmath178 where @xmath179 let us assume that the fermion fields are canonically quantized @xmath180\\ \bar\psi(x)&= & \frac{1}{(2\pi)^{1/2}}\int\!dp\sum_{s=1}^{2}\sqrt{\frac{m}{e_{p}}}\left [ b^{\dagger(s)}(p)\bar u^{(s)}(p)e^{ipx}+d^{(s)}(p)\bar v^{(s)}(p)e^{-ipx}\right],\end{aligned}\ ] ] then we find @xmath181 this provides @xmath182 without the interchanges we call this @xmath183 . + if we use the completeness relations @xmath184 and the integral equation @xmath185 we obtain @xmath186.\end{aligned}\ ] ] integrating next over the first @xmath0-function and evaluating the trace gives @xmath187.\end{aligned}\ ] ] if we compare this with ( [ abstract i m tni ] ) and ( [ abstract i m tdvni ] ) ( @xmath188 ) , we see that @xmath177 is given by the imaginary part of the amplitude of the pure vector loop @xmath189 this is the important relation which links the schwinger terms to the gravitational anomalies . from eq . ( [ i m a1 ] ) ( [ i m a5 ] ) we read off the formfactors @xmath190 now we consider the commutator @xmath191 in terms of g } \langle 0\vert [ t^{v}_{\mu\nu}(x ) , t^{v}_{\rho\sigma}(0 ) ] \vert 0\rangle = f_{\mu\nu\rho\sigma}(x)-f_{\rho\sigma\mu\nu}(-x ) = \int\!d^{2}p\ e^{-ipx } \varepsilon(p^{0 } ) g_{\mu\nu\rho\sigma}(p).\ ] ] if we remove the mass , @xmath192 , that acted as an infrared cutoff , we get @xmath193 from eq . ( [ [ t , t ] in terms of g ] ) we explicitly find @xmath194 \vert 0\rangle_{et } & = & \lim_{x_{0}\rightarrow 0 } -\frac{1}{24\pi^{2}}\int\!d^{2}p\ e^{-ipx}p_{1}^{4}\ \varepsilon(p^{0 } ) \delta(p^{2 } ) = 0 \\ \langle 0\vert [ t^{v}_{11}(x ) , t^{v}_{11}(0 ) ] \vert 0\rangle_{et } & = & \lim_{x_{0}\rightarrow 0 } -\frac{1}{24\pi^{2}}\int\!d^{2}p\ e^{-ipx}p_{0}^{4}\ \varepsilon(p^{0 } ) \delta(p^{2 } ) = 0 \\ \langle 0\vert [ t^{v}_{00}(x ) , t^{v}_{11}(0 ) ] \vert 0\rangle_{et } & = & \langle 0\vert [ t^{v}_{01}(x ) , t^{v}_{01}(0 ) ] \vert 0\rangle_{et } { } \nonumber\\{}&= & \lim_{x_{0}\rightarrow 0 } -\frac{1}{24\pi^{2}}\int\!d^{2}p\ e^{-ipx}p_{0}^{2}p_{1}^{2}\ \varepsilon(p^{0 } ) \delta(p^{2 } ) = 0 \\ \langle 0\vert [ t^{v}_{00}(x ) , t^{v}_{01}(0 ) ] \vert 0\rangle_{et } & = & \lim_{x_{0}\rightarrow 0 } -\frac{1}{24\pi^{2}}\int\!d^{2}p\ e^{-ipx}p_{0}p_{1}^{3}\ \varepsilon(p^{0 } ) \delta(p^{2 } ) \\ \langle 0\vert [ t^{v}_{01}(x ) , t^{v}_{11}(0 ) ] \vert 0\rangle_{et } & = & \lim_{x_{0}\rightarrow 0 } -\frac{1}{24\pi^{2}}\int\!d^{2}p\ e^{-ipx}p_{0}^{3}p_{1}\ \varepsilon(p^{0 } ) \delta(p^{2 } ) \ , .\label{[t01,t11 ] no interchange } \end{aligned}\ ] ] the first three expressions vanish because @xmath195 is antisymmetric . to evaluate the next two we use the pauli - jordan function @xmath196 with the properties @xmath197 and we find @xmath198 so we conclude @xmath199 \vert 0\rangle_{et } & = & \langle 0\vert [ t^{v}_{01}(x ) , t^{v}_{11}(0 ) ] \vert 0\rangle_{et } { } \nonumber\\{}&= & \lim_{x_{0}\rightarrow 0}-\frac{1}{12\pi}\partial_{1}^{3}\partial_{0}\triangle(x ) = \frac{i}{12\pi}\partial_{1}^{3}\delta(x^{1 } ) \ , .\end{aligned}\ ] ] with relation ( [ symbolic vev of commutator ] ) we finally obtain the schwinger terms in the etc of the energy - momentum tensors @xmath200 \vert 0\rangle_{et } & = & \langle 0\vert [ t_{11}(x ) , t_{11}(0 ) ] \vert 0\rangle_{et } = \mp\frac{i}{24\pi}(\partial_{1})^{3}\delta(x^{1})\\ \langle 0\vert [ t_{00}(x ) , t_{11}(0 ) ] \vert 0\rangle_{et } & = & \langle 0\vert [ t_{01}(x ) , t_{01}(0 ) ] \vert 0\rangle_{et } = \mp\frac{i}{24\pi}(\partial_{1})^{3}\delta(x^{1})\\ \langle 0\vert [ t_{00}(x ) , t_{01}(0 ) ] \vert 0\rangle_{et } & = & \langle 0\vert [ t_{01}(x ) , t_{11}(0 ) ] \vert 0\rangle_{et } = \frac{i}{24\pi}(\partial_{1})^{3}\delta(x^{1 } ) \ , .\end{aligned}\ ] ] our result agrees with the one of tomiya @xcite who works with an invariant spectral function method and in addition uses a cohomological approach . it also coincides with the result of ebner , heid and lopes - cardoso @xcite who derive the schwinger terms directly from the gravitational anomaly . in our approach ( [ g and tpv ] ) is the basic relation . it connects the schwinger terms , determined by @xmath201 , with the ( linearized ) gravitational anomalies given by @xmath202 in our dispersion relations procedure . we have investigated the gravitational anomalies , specifically the pure einstein anomaly and the weyl anomaly . so we demanded the quantized energy - momentum tensor to be symmetric no lorentz anomaly occurs which is a possible choice . the relevant amplitude , the two - point function of the energy - momentum tensors @xmath57 , we have separated into its pure tensor part @xmath29 and into its pseudo - tensor part @xmath30 , eqs.([amplitude expressed by t - product ] ) ( [ formfactors3 ] ) , and we have decomposed the amplitudes into a general structure of tensors containing 8 formfactors @xmath31 . these formfactors we have expressed by dispersion relations where we had to calculate only the imaginary parts via the cutkosky rules . our subtraction procedure for the formfactors no subtraction for @xmath83 , one subtraction for @xmath203 and two subtractions for @xmath204 is _ the _ natural choice dictated by the @xmath122behaviour of the imaginary parts @xmath123 . it implies that the pure tensor wi ( [ vwi1 ] ) ( [ vwi3 ] ) for the renormalized formfactors is satisfied ( in the limit @xmath95 ) , so that the total anomaly is automatically shifted into the pseudotensor part of the wi ( [ pta ] ) . it turns out that the anomalous ward identity and the anomalous trace identity depend only on the finite formfactor @xmath121 , with its explicit result ( [ t1 for wi ] ) , demonstrating such the independence of a special renormalization procedure . from the anomalous ward identity and the anomalous trace identity we could deduce the linearized gravitational anomalies the linearized einstein- and weyl anomaly and determine their covariant versions . the origin of the anomaly is the existence of a superconvergence sum rule for the imaginary part of the formfactor @xmath63 . in the zero mass limit the imaginary part of the formfactor approaches a @xmath0-function singularity at zero momentum squared , exhibiting in this way the infrared feature of the gravitational anomalies this is an independent and complementary view of the anomalies as compared to the ultraviolet regularization procedures . if we compare , however , the dr approach with the n - dimensional regularization procedure of t hooft veltman we find an equivalence . the two approaches are linked by the substitutions ( [ substitution dis rel - dim reg ] ) . we have also calculated the gravitational schwinger terms which occur in the etc of the energy - momentum tensors . we have adapted our dispersive approach to the method of klln . as a result all gravitational schwinger terms are determined by the formfactor @xmath205 which in the zero mass limit approaches a @xmath0-function singularity at zero momentum squared as in the case of anomalies . so also the schwinger terms show this peculiar infrared feature of the anomalies . we have performed all calculations in two dimensions , where already all essential features of the dr approach show up , analogously to the chiral current case . from a practical point of view the method appears quite appealing . all one has to calculate is the imaginary part of an amplitude ( formfactor ) , which is an easy task . however , this computational simplicity is a special ( and convenient ) feature of the two space - time dimensions . in higher dimensions the amplitude will contain more formfactors and we have to calculate dispersion relations for higher loop diagrams , which is a much more delicate task , but nevertheless we expect the method to work here too . r. jackiw , _ field theoretic investigations in current algebra , topological investigations of quantized gauge theories _ , in : _ current algebra and anomalies _ , treiman , r. jackiw , b. zumino and e. witten ( eds . ) , p.81 , and p.211 , world scientific , singapore ( 1985 ) . p. van nieuwenhuizen , _ anomalies in quantum field theory : cancellation of anomalies in @xmath206 supergravity . leuven notes in mathematical and theoretical physics _ , vol . 3 , leuven university press ( 1988 ) .

we are dealing with two - dimensional gravitational anomalies , specifically with the einstein anomaly and the weyl anomaly , and we show that they are fully determined by dispersion relations independent of any renormalization procedure ( or ultraviolet regularization ) . the origin of the anomalies is the existence of a superconvergence sum rule for the imaginary part of the relevant formfactor . in the zero mass limit the imaginary part of the formfactor approaches a @xmath0-function singularity at zero momentum squared , exhibiting in this way the infrared feature of the gravitational anomalies . we find an equivalence between the dispersive approach and the dimensional regularization procedure . the schwinger terms appearing in the equal time commutators of the energy momentum tensors can be calculated by the same dispersive method . although all computations are performed in two dimensions the method is expected to work in higher dimensions too .

arxiv

space time explorer & quantum equivalence space test ( ste - quest ) is a medium - sized mission candidate for launch in 2022/2024 in the cosmic vision programme of the european space agency . after recommendation by the space science advisory committee , it was selected to be first studied by esa , followed by two parallel industrial assessment studies . this paper gives a brief summary of the assessment activities by astrium which build on and extend the preceding esa study as described in@xcite . ste - quest aims to study the cornerstones of einstein s equivalence principle ( eep ) , pushing the limits of measurement accuracy by several orders of magnitude compared to what is currently achievable in ground based experiments@xcite . on the one hand , experiments are performed to measure the gravitational red - shift experienced by highly accurate clocks in the gravitational fields of earth or sun ( space time explorer ) . on the other hand , differential accelerations of microscopic quantum particles are measured to test the universality of free fall , also referred to as weak equivalence principle ( quantum equivalence space test ) . these measurements aim at finding possible deviations from predictions of general relativity ( gr ) , as postulated by many theories trying to combine gr with quantum theory . examples include deviations predicted by string theory @xcite , loop quantum gravity@xcite , standard model extension@xcite , anomalous spin - coupling@xcite , and space - time - fluctuations@xcite , among others . the ste - quest mission goal is summarized by the four primary science objectives@xcite which are listed in tab.[tab : mission_objectives ] together with the 4 measurement types geared at achieving them . lll primary mission objective & measurement accuracy & measurement strategy + measurement of & to a fractional frequency & ( a ) space - clock comparison + earth gravitational red - shift & uncertainty better than & to ground clock at apogee + & @xmath0 & ( b ) space - clock comparison + & & between apogee and perigee + measurement of & to a fractional frequency & ( c ) comparison between + sun gravitational red - shift & uncertainty better than & two ground clocks via + & @xmath1 , goal : @xmath2 & spacecraft + measurement of & to a fractional frequency & ( c ) comparison between + moon gravitational red - shift & uncertainty better than & two ground clocks via + & @xmath3 , goal : @xmath4 & spacecraft + measurement of & to an uncertainty in the & ( d ) atom interferometer + weak equivalence principle & etvs param . smaller & measurements at perigee + & @xmath5 & + the ste - quest mission is complex and comprises a space - segment as well as a ground segment , with both contributing to the science performance . highly stable bi - directional microwave ( x / ka - band ) and optical links ( based on laser communication terminals ) connect the two segments and allow precise time - and - frequency transfer . the space - segment encompasses the satellite , the two instruments , the science link equipment , and the precise orbit determination equipment . the ground - segment is composed of 3 ground terminals that are connected to highly accurate ground - clocks . in order to fulfil the mission objectives , various measurement types are used that are shown in fig.[fig : measurement_principle ] . we shall briefly discuss them . _ earth gravitational red - shift measurements : _ the frequency output of the on - board clock is compared to that of the ground clocks . in order to maximize the signal , i.e. the relativistic frequency offset between the two clocks , a highly elliptical orbit ( heo ) is chosen . when the spacecraft is close to earth during perigee passage , there are large frequency shifts of the space - clock due to the strong gravitational field . when it is far from earth during apogee passage , there are only small gravitational fields and therefore small frequency shifts . whilst measurement type ( a ) compares the space - clock at apogee to the ground - clock , relying on space - clock accuracy , measurement type ( b ) compares the frequency variation of the space - clock over the elliptical orbit , which requires clock stability in addition to visibility of a ground terminal at perigee ( see also section [ sec : orbit_ground ] ) . , scaledwidth=100.0% ] + _ sun gravitational red - shift measurements : _ the relativistic frequency offset between a pair of ground clocks is compared while the spacecraft merely serves as a link node between the ground clocks for a period of typically 5 to 7 hours , as described by measurement type ( c ) . in that case , the accuracy of the space - clock is of minor importance whereas link stability and performance over a long period are essential . as this requirement is also hard to fulfil by the microwave links alone , the optical links play an important role . however , availability and performance of optical links are strongly affected by weather conditions which in turn depend on the location and altitude of the ground terminals . + _ moon gravitational red - shift measurements : _ as in the preceding case , the relativistic frequency offset between a pair of ground clocks is compared while the spacecraft merely serves as a link node between the ground clocks . the potential signals for a violation of the eep with the moon as the source mass can be easily distinguished from those with the sun as the source mass due to the difference in frequency and phase . it is important to point out that unless the eep is violated the measured frequency difference between two distant ground clocks ( due to the sun or the moon ) is expected to be zero up to small tidal corrections and a constant offset term from the earth field@xcite . + _ atom interferometer measurements : _ these measurements do not require contact to ground terminals but must be performed in close proximity to earth where the gravity gradients are large and the etvs parameter ( see eq.[equ : otvos ] below ) becomes small . the latter defines the magnitude of a possible violation of the weak equivalence principle ( wep ) , as required for mission objective 4 . for this reason , the atom interferometer operates during perigee passage at altitudes below 3000 km , where gravity accelerations exceed @xmath6 . the ste - quest payload primarily consists of two instruments , the atomic clock and the atom interferometer , as well as equipment for the science links ( microwave and optical ) and for precise orbit determination ( pod ) . whilst a detailed description of the proposed payload elements is found in @xcite , here we shall only give a brief overview over the payload as summarized in table [ tab : payload ] . note that some payload elements , such as the optical links and the onboard clock , may become optional in a future implementation of ste - quest , which is discussed in the official esa assessment study report ( yellow book)@xcite . lll payload element , & subsystem , & components + instrument & unit & + & pharao ng & caesium tube , laser source + * atomic clock * & & + & microwave - optical local & laser head , frequency comb , + & oscillator ( molo ) & highly - stable cavity + & & + & mw synthesis & frequency & ultra - stable oscillator , frequency gen . , + & distribution ( msd ) & comparison and distribution ( fgcd ) + & & + & instr . control unit ( icu ) & control electronics , harness + & physics package & vacuum chamber , atom source , + * atom interf . * & & magnetic coils and chip , laser + & & interfaces , mu - metal shields + & & + & laser subsystem & cooling , repumping , raman , + & & detection , optical trapping lasers + & & + & electronics & ion pump control , dmu , magnetic coil + & & drive , laser control electronics + & microwave links & mw electronics , x-/ka - band antennas + * equipment * & & + & optical links & 2 laser communication terminals + & & ( lcts ) , synchronization units + & gnss equipment & gnss receiver and antennas + * equipment * & & + & laser ranging equipment & corner cube reflectors ( optional ) + + _ the atomic clock _ is essentially a rebuilt of the high - precision pharao clock used in the aces mission@xcite , with the additional provision of an optically derived microwave signal ( mw ) for superb short term stability which is generated by the molo . in pharao , a cloud of cold caesium atoms is prepared in a specific hyperfine state and launched across a vacuum tube . there , the cloud is interrogated by microwave radiation in two spatially separated ramsey zones before state transition probabilities are obtained from fluorescence detection . repeating the cycle while scanning the mw fields yields a ramsey fringe pattern whose period scales inversely proportional to the flight time t between ramsey interactions . in the molo , a reference laser is locked to an ultra - stable cavity , which provides superb short - term stability in addition to the long - term stability and accuracy obtained from pharao by locking the reference signal to the hyperfine transition of the caesium atoms . the reference laser output is frequency - shifted to correct for long term drifts and then fed to the femtosecond frequency comb@xcite to convert the signal from the optical to the microwave - domain . core components of the molo , such as the highly stable cavity@xcite , the frequency comb@xcite , and optical fibres @xcite are currently being developed or tested in dedicated national programs . additionally , other components of optical clocks have been investigated in the ongoing space - optical - clock program ( soc)@xcite of the european space agency . + _ the atom interferometer_@xcite uses interference between matter waves to generate an interference pattern that depends on the external forces acting on the atoms under test . in order to reject unwanted external noise sources as much as possible , differential measurements are performed simultaneously with two isotopes of rubidium atoms , namely @xmath7 and @xmath8 . the measured phases for both isotopes are subtracted to reveal any possible differences in acceleration under free fall conditions of the particles . this way , the common mode noise which is present on both signals is rejected to a high degree@xcite . in the proposed instrument @xcite a number of the order of @xmath9 atoms of both isotopes are trapped and cooled by means of sophisticated magnetic traps in combination with laser cooling . thus , the residual temperature and consequently the residual motion and expansion of the atomic clouds is reduced to the nk regime , where the cloud of individual atoms makes the transition to the bose - einstein condensed state . the atoms are then released from the trap into free fall before being prepared in superpositions of internal states and split into partial waves by pulses of lasers in a well defined time sequence . by using a sequence of 3 laser pulses separated by a period of @xmath10 , a mach - zehnder interferometer in space - time is formed . the distribution of the internal atomic states is then detected either by fluorescence spectroscopy or by absorption imaging . the distribution ratio of the internal atomic states provides the information about the phase shift which is proportional to the acceleration acting on the atoms . the etvs parameter @xmath11 ( see mission objective 4 ) depends on the ratio of inertial mass @xmath12 to gravitational mass @xmath13 for the two isotopes and can be found from the measured accelerations @xmath14 as follows : @xmath15 where subscript 1 refers to physical parameters of the first isotope ( @xmath7 ) and subscript 2 to those of the second isotope ( @xmath8 ) . + the programmatics of european space atom interferometers , as proposed for ste - quest strongly builds on the different national and european - wide activities in this direction , in particular the space atom interferometer ( sai)@xcite and the nationally funded projects quantus@xcite and ice@xcite . in these activities innovative technologies are developed and tested in parabolic flights , drop tower campaigns and sounding rocket missions . + _ the science links _ comprise optical and microwave time & frequency links to compare the on - board clock with distributed ground clocks . similarly , the links allow to compare distant ground clocks to one another using the on - board segment as a relay . the microwave link ( mwl ) is designed based on re - use and further development of existing mwl technology for the aces mission@xcite , while implementing lessons learned from aces . this applies to the flight segment and the ground terminals , as well as for science data post - processing issues down to the principles and concept for deploying and operating a network of ground terminals and associated ground clocks . digital tracking loops with digital phase - time readout are used to avoid beat - note concepts . different to aces and given the highly eccentric orbit , ka - band is used instead of ku - band , x - band instead of s - band for ionosphere mitigation , and the modulation rate is changed from 100 mchip / s to 250 mchip / s . the mwl flight segment comprises four receive channels to simultaneously support comparisons with 3 ground terminals and on - board calibration . the optical link baseline design relies on two tesat laser communication terminals ( lcts)@xcite on - board the satellite , both referenced to the space - clock molo signal , and the ground lcts to perform two time bi - directional comparisons . the lcts operate at 1064 nm with a 5 w fiber laser , similar to the edrs lct@xcite , and a 250 mhz rf modulation . a telescope aperture of 150 mm in space and 300 mm on ground was chosen . a preliminary performance estimate has been made by extrapolation from existing measurements with more detailed investigations following now in dedicated link studies . the optical links are found to be particularly sensitive to atmospheric disturbances and local weather conditions . these particularly affect common view comparisons between two ground terminals which are typically located in less favorable positions concerning local cloud coverage . + _ the precise - orbit - determination ( pod ) equipment _ is required to determine the spacecraft position and velocity , which allows calculating theoretical predictions of the expected gravitational red - shifts and the relativistic doppler , and compare them to the measured ones . our studies found that a gnss - receiver with a nadir - pointing antenna suffices to achieve the required accuracy@xcite . additional corner - cube reflectors to support laser ranging measurements , such as also used for the giove - a satellite@xcite , may optionally be included as a backup and for validation of gnss - derived results . _ the baseline orbit _ is highly elliptical with a typical perigee altitude of 700 km and an apogee altitude of 50000 km , which maximizes gravity gradients between apogee and perigee in order to achieve mission objective 1 ( earth gravitational red - shift ) . the corresponding orbit period is 16h , leading to a 3:2 resonance of the ground track as depicted in fig.[fig : orbit ] left . some useful orbit parameters are summarized in tab . [ tab : baseline_orbit ] . llllll perigee alt . & apogee alt . & orbit period & inclination & arg . perigee & raan drift + 700 - 2200 km & @xmath16 50 000 km&16 h & @xmath17 & @xmath18 & @xmath19 + contrary to the old baseline orbit@xcite , where the argument of perigee was drifting from the north to the south during the mission , the argument of perigee for the new baseline orbit is frozen in the south . therefore , contact to the ground terminals located in the northern hemisphere can only be established for altitudes higher than 6000 km , which considerably impairs the modulation measurements of the earth gravitational red - shift ( measurement type b in table[tab : mission_objectives ] ) . on the pro side , for higher altitudes and in particular around apogee , common view links to several terminals and over many hours can be achieved which greatly benefits the sun gravitational red - shift measurements ( type c ) . the baseline orbit also features a slight drift of the right - ascension of the ascending node ( raan ) due to the @xmath20 term of the non - spherical earth gravitational potential@xcite , which leads to a total drift of approximately 30@xmath21 per year . this drift being quite small , the orbit can be considered as nearly - inertial during one year so that seasonal variations of the incident solar flux are expected ( see also fig.[fig : solar_flux ] ) . the major advantage of the new baseline orbit over the old one is a greatly reduced @xmath22v for de - orbiting , which allows reducing the corresponding propellant mass from 190 kg to 60 kg . the reason for that is the evolution of the perigee altitude due to the 3-body interaction with the sun and the moon@xcite . the perigee altitude starts low at around 700 km , goes through a maximum of around 2200 km at half time , and then drops back down below 700 km in the final phase , as depicted in fig.[fig : orbit ] right . the low altitude at the end of mission facilitates de - orbiting using the 6 x 22-n oct thrusters during an apogee maneuver that is fully compliant with safety regulations . + _ the ground segment _ comprises 3 ground terminals , each equipped with microwave antennas for x- and ka - band communication as well as ( optional ) optical terminals based on the laser communication terminals ( lct ) of tesat@xcite . the terminals must be distributed over the earth surface with a required minimum separation of five thousand kilometers between them . this maximizes their relative potential difference in the gravitational field of the sun in pursuit of mission objective 2 . additionally , the terminals must be in the vicinity of high - performance ground clocks ( distance @xmath23 km ) to which they may be linked via a calibrated glass - fibre network . this narrows the choice of terminal sites to the locations of leading institutes in the field of time - and frequency metrology . factoring in future improvements in clock technology and design , those institutes are likely able to provide clocks with the required performance of a fractional frequency uncertainty on the order of @xmath24 ) at the start of the mission , considering that current clock performance comes already quite close to the requirement@xcite . accounting for all the aspects mentioned above , boulder ( usa ) , torino ( italy ) , and tokyo ( japan ) were chosen as the baseline terminal locations . the spacecraft has an octagonal shape with a diameter of 2.5 m and a height of 2.7 . it consists of a payload module ( plm ) , accommodating the instruments as well as science link equipment , and a service module ( svm ) for the spacecraft equipment and propellant tanks ( see fig . [ fig : spacecraft_overview ] ) . the laser communication terminals ( lcts ) are accommodated towards the nadir end of opposite skew panels . the nadir panel ( + z - face ) accommodates a gnss antenna and a pair of x - band / ka - band antennas for the science links . the telemetry & tele - commanding ( tt&c ) x - band low gain and medium gain antennas are accommodated on y - panels of the service module . the two solar array wings are canted at @xmath25 with respect to the spacecraft y - faces and allow rotations around the y - axis . radiators are primarily placed on the + x/-x faces of the spacecraft and good thermal conductivity between dissipating units and radiators is provided through the use of heat - pipes which are embedded in the instrument base - plates . a set of 2 x 4 reaction control thrusters ( rcts ) for orbit maintenance and slew maneuvers after the perigee measurement phase is used as part of the chemical propulsion system ( cps ) together with 1 x 6 orbit control thrusters ( octs ) for de - orbiting . the latter is becoming a stringent requirement for future mission designs to avoid uncontrolled growth of space debris and reduce the risk of potential collisions between debris and intact spacecraft@xcite . the micro - propulsion system ( mps ) is based on gaia heritage@xcite and primarily used for attitude control , employing a set of 2x8 micro - proportional thrusters to this purpose . as primary sensors of the attitude and orbit control subsystem ( aocs ) , the spacecraft accommodates 3 star trackers , 1 inertial measurement unit ( imu ) , and 6 coarse sun sensors . the payload module of fig . [ fig : payload_module ] shows how the two instruments are accommodated in the well protected central region of the spacecraft , which places them close to the center - of - mass ( com ) and therefore minimizes rotational accelerations whilst also providing optimal shielding against the massive doses of radiation accumulated over the five years of mission duration . the pharao tube of the atomic clock is aligned with the y - axis . this arrangement minimizes the coriolis force which acts on the cold atomic cloud propagating along the vacuum tube axis when the spacecraft performs rotations around the y - axis . these rotations allow preserving a nadir - pointing attitude of the spacecraft as required for most of the orbit ( see also section [ sec : aocs ] ) . the sensitive axis of the atom interferometer must point in the direction of the external gravity gradient . it is therefore aligned with the z - axis ( nadir ) and suspended from the spacecraft middle plate into the structural cylinder of the service module which provides additional shielding against radiation and temperature variations . the total wet mass of the spacecraft with the launch adapter is approximately 2300 kg , including unit maturity margins and a 20% system margin . the total propellant mass ( cps+mps ) is 260 kg , out of which 60 kg are reserved for de - orbiting . the total power dissipation for the spacecraft is 2230 w , including all required margins , if all instruments and payload units operate at full power . the spacecraft is designed to be compatible with all requirements ( including launch mass , envelope , loads , and structural frequencies ) for launch with a soyuz - fregat from the main esa spaceport in kourou . the two instruments pose stringent requirements on attitude stability and non - gravitational accelerations which drive the design of the attitude and orbit control system ( aocs ) and the related spacecraft pointing strategy . to ensure full measurement performance , the atom interferometer requires non- gravitational center - of - mass accelerations to be below a level of @xmath26 . among the external perturbations acting on the spacecraft and causing such accelerations , air drag is by far the dominant one . it scales proportional to the atmospheric density and therefore decreases with increasing altitudes . figure [ fig : drag_and_rotation ] ( left ) displays the level of drag accelerations for various altitudes and under worst case assumptions , i.e. maximal level of solar activity and maximal cross section offered by the spacecraft along its trajectory . the results indicate that below altitudes of approximately 700 km drag accelerations become intolerably large . we conclude that , for the baseline orbit , drag forces are generally below the required maximum level and therefore need not be compensated with the on - board micro - propulsion system , although such an option is feasible in principle . another requirement is given by the maximal rotation rate of @xmath27 during perigee passage when sensitive atom interferometry experiments are performed . figure [ fig : drag_and_rotation ] ( right ) plots the spacecraft rotation rate ( black solid line ) against the time for one entire orbit , assuming a nadir - pointing strategy . the plot demonstrates that the requirement ( red broken line ) would not only be violated by the fast rotation rates encountered at perigee , even for the comparatively slow dynamics at apogee the rotation rates would exceed the requirement by more than one order of magnitude . we therefore conclude that atom interferometer measurements are incompatible with a nadir - pointing strategy throughout the orbit , requiring the spacecraft to remain fully inertial during such experiments . considering that atom interferometry is generally performed at altitudes below 3000 km , the following spacecraft pointing strategy was introduced for ste - quest ( see figure [ fig : pointing_strategy ] ) : * * perigee passage : * for altitudes below 3000 km , lasting about @xmath28 each orbit , the atom interferometer is operating and the spacecraft remains inertial . the atom interferometer axis is aligned with nadir at perigee and rotated away from nadir by 64 degrees at 3000 km altitude ( see figure [ fig : pointing_strategy ] ) . the solar array driving mechanism is frozen to avoid additional micro - vibrations . * * transition into apogee : * for altitudes between 3000 km and 7000 km , right after the perigee passage , the spacecraft rotates such that it is nadir pointed again ( slew maneuver ) . the solar arrays are unfrozen and rotated into optimal view of the sun . * * apogee passage : * for altitudes above 7000 km the spacecraft is nadir pointed and rotational accelerations are generally much slower than during perigee passage . the solar array is allowed to rotate continuously for optimal alignment with the sun . atom interferometer measurements are generally not performed . * * transition into perigee : * for altitudes between 7000 km and 3000 km , right after the apogee passage , the spacecraft rotates such that it is aligned with nadir at perigee . the solar arrays are frozen . during the science operations at perigee and apogee , the attitude control relies solely on the mps . the slew maneuvers , i.e. @xmath29 pitch rotations , are performed by means of the cps in @xmath30 at a rate of @xmath31 . the cps is also used to damp out residual rotational rates which are caused by solar array bending modes and fuel sloshing due to the fast maneuvers . when the perigee passage is entered , the mps is activated to further attenuate the residual rate errors coming from the minimum impulse bit and sensor bias ( ca . @xmath32 ) , which is shown in fig.[fig : rate_errors]b . closed - loop simulations , based on a dynamical computer model of the spacecraft in - orbit and simulated aocs equipment , demonstrated that the damping requires about 500 s. the complete transition , including maneuvers , lasts about @xmath33 . in order to mitigate the effects of external disturbances during the perigee passage , an aocs closed - loop control has been designed that is based on cold - gas thrusters and includes roll - off filtering to suppress sensor noise while maintaining sufficient stability margins . this allows compensating external torques from drag forces , solar radiation pressure and the earth magnetic field ( among others ) , yielding rotation rates well below the required @xmath27 after the initial damping time of approximately @xmath34 s , as can be seen in in fig.[fig : rate_errors]b . the associated mean relative pointing error ( rpe ) is found to be smaller than @xmath35 in a 15 s time window corresponding to one experimental cycle . note that the spacecraft rotation rates grow to intolerably large levels of several hundred @xmath36 , if the spacecraft is freely floating without control loops to compensate the environmental torques ( see fig.[fig : rate_errors]a ) . for orbit maintenance and de - orbiting , use of the cps is foreseen . is outside the plot scale.,scaledwidth=100.0% ] the problems associated with micro - vibrations lie at the heart of two key trades for our current spacecraft configuration . one major cause of micro - vibrations are reaction wheels . to investigate the option of using them instead of the mps for attitude control , extensive simulations were performed with a finite element model of the spacecraft which provided the transfer functions from the reaction wheels to the atom interferometer interface . based on this model and available measurement data of low - noise reaction wheels ( bradford w18e ) , the level of micro - vibrations seen at instrument interface could be compared against the requirements . the results of the analyses are summarized in fig . [ fig : micro_vibrations ] left which clearly shows a violation of the requirement for frequencies above 20 hz within the measurement band [ 1 mhz , 100 hz ] . the major contribution to the micro - vibrations comes from the h1 harmonic when the disturbance frequency corresponds to the wheel rotation rate up to the maximum wheel rate of 67 hz . further micro - vibrations are induced by the amplification of solar array flexible modes through the wheel rotation . the violation of the requirement for the maximum level of tolerable micro - vibrations when using reaction - wheels , as found from our simulations , lead us to use a micro - propulsion system for attitude control in our baseline configuration . } $ ] . note that for ease of comparison , requirements for the spectral density have been transformed into rms requirements by integrating over the measurement bandwidth.,scaledwidth=100.0% ] similar analyses were performed to determine the micro - vibrations induced by the solar array driving mechanism ( sadm ) on the molo interface , where a complex model of the sadm stepper motor together with the spacecraft finite - element - model ( fem ) with deployed solar arrays was combined in a matlab simulink environment . the stepper motor generates two kinds of torque perturbations , one relating to the magnetic detent effect and the other relating to the commanded micro - stepping of the motor . one particular resonance of the spacecraft structural model , corresponding to a torsional mode of the solar arrays around their rotation axis ( y - axis ) , was found to be the most critical one . when this resonance was hit by a harmonic disturbance originating from the detent torque of the solar arrays driving mechanism , the resulting response displayed a low - frequency isolated resonance peak at the excitation frequency of 1.3 hz , with a few more peaks at higher frequencies due to harmonic excitation from the micro - stepping ( see fig . [ fig : micro_vibrations ] right ) . restricting the solar array rotation rates to below 0.13 deg / s , the level of micro - vibrations is found to be fully compliant throughout the specified frequency range , as can be seen in figure [ fig : micro_vibrations ] . allowing for higher rates up to 0.08 deg / s , which are required to follow the fast spacecraft rotations with the solar array at low altitudes below 10000 km , we find that only one specific rotation rate at 0.044 deg / s leads to a violation of the requirement . however , this is easily mitigated by quickly moving over the resonance , therefore avoiding to rotate the solar arrays at this particular rate . for earth gravitational red - shift measurements of type ( a ) , the measured frequency shift of the space - clock with respect to a ground clock is compared to the predicted frequency - shift . in order to compute the theoretical prediction , it is necessary to know the precise position and velocity of the space - clock ( and therefore the precise orbit ) , as both quantities enter the equation describing the de - phasing of the space - clock proper time @xmath37 with respect to the coordinate time @xmath38 , where the latter is referenced to an inertial earth - centered coordinate system . this dependence is expressed in eq.[equ : proper_time ] , which is derived from a post - newtonian expansion to second order in @xmath39 @xcite . @xmath40 where @xmath41 denotes the gravitational potential , @xmath42 the spacecraft position , and @xmath43 its velocity . note that the first term in brackets corresponds to the gravitational time dilation whereas the second term corresponds to the special relativistic time dilation . + _ clock and link uncertainty : _ considering only the gravitational time dilation , the frequency shift @xmath44 between a space clock at apogee and a ground clock may then be written as @xmath45/c^2=6.2\times10^{-10}$ ] , where @xmath46 and @xmath47 are the earth radius and the radial distance to apogee , respectively . it follows that the space - clock and the science links must have a fractional frequency uncertainty of better than @xmath48 to allow measuring the gravitational red - shift of the earth to the specified accuracy . from similar deliberations one finds that the ground clock and science link performance must be even better , on the order of @xmath24 , to measure the gravitational red - shift of the sun to the specified accuracy . this requires very long contact times between the spacecraft and ground terminals , on the order of @xmath49 , which is only achieved after integration over many orbits . + the above requirements can be directly translated into the required stability for the link , expressed in modified allan deviation@xcite , and into the frequency uncertainty for a space - to - ground or ground - to - ground comparison . a detailed performance error budget was established to assess all contributions to potential instabilities . for the microwave links we found that , provided the challenging requirements for high thermal stability are met , the aces design with the suggested improvements can meet the specifications . the assessment of the optical links showed that a significantly better performance than the state - of - the - art is required . a major issue for further investigation is the validity of extrapolating noise power spectral densities over long integration times in presence of atmospheric turbulence , which is required for comparing the estimated link performance to the specified link performance expressed in modified allan deviation . + _ orbit uncertainty : _ as the fractional frequency uncertainty is specified to be around @xmath48 for the space - clock , a target value of @xmath50 for the orbit uncertainty does not compromise the overall measurement accuracy . although preliminarily a constant target value of 2 m in position and 0.2 mm / s in velocity accuracy has been derived in @xcite , these values can be significantly relaxed at apogee@xcite . a final analysis will have to build on a relativistic framework for clock comparison as outlined in @xcite . + _ gnss tracking : _ the proposed method of precise orbit determination ( pod ) relies on tracking gps and galileo satellites with a gnss receiver supporting all available modulation schemes , e.g. bpsk5 , bpsk10 and mboc@xcite , and an antenna accommodated on the nadir panel . considering that gnss satellites broadcast their signal towards the earth and not into space , they become increasingly difficult to track at altitudes above their constellation ( 20200 km altitude ) , when the spacecraft can only receive signals from gnss satellites behind the earth . in order to increase the angular range of signal reception , the receiver is required to support signal side - lobe tracking . this way , a good signal visibility for six hours during each orbit and for altitudes up to 34400 km can be achieved , with a corresponding pvt ( position - velocity - time ) error of 1.7 m at 1 hz output rate@xcite . the transition maneuvers split the visibility region ( green arch in fig . [ fig : gnss_visibility ] ) into separate segments , which must be considered separately in the orbit determination . for both regions we find orbit errors less than 10 cm in position and less than 0.2 mm / s in velocity . extrapolation of the spacecraft trajectory from the end of the visibility arch as far as apogee introduces large orbit errors ( @xmath51 ) due to large uncertainties in parameters describing the solar radiation pressure . nonetheless , these errors are sufficiently small to achieve the specified frequency uncertainty for the space - clock measurement if one considers the arguments for relativistic clock comparison derived in @xcite . a brief discussion of the possibility to relax the ste - quest orbit accuracy close to apogee is also given in @xcite . finally , it shall be pointed out , that the accuracies obtained from gnss - based pod are sufficient to probe the flyby anomaly , an inexplicable momentum transfer which has been observed for several spacecraft careening around the earth . these investigations can be performed without further modifications to the baseline hardware or mission concept , as described in ref . the major focus of the assessment study was to define preliminary designs of the spacecraft subsystems in compliance with all mission requirements . to this end , key aspects such as pointing strategy of spacecraft and solar arrays , variability of solar flux and temperature stability , measurement performance and pointing stability , instrument accommodation and radiative shielding , and launcher specifications , among others , must be reflected in the custom - tailored design features . additional provisions for ease of access during integration and support of ground - based instrument testing on spacecraft level are also important considerations . as we intend to give a concise overview in this paper , we will limit our discussion on a few distinctive design features . + _ thermal control subsystem ( tcs ) : _ the ste - quest payload dissipates a large amount of power which may total up to more than 1.7 kw ( including maturity and system margins ) if all payload equipment is active . this represents a major challenge for the thermal system which can only be met through dedicated heat - pipes transporting the heat from the protected accommodation region in the spacecraft center to the radiator panels . also problematic is the fact that the baseline orbit is not sun - synchronous and features a drift of the right - ascension of the ascending node ( raan ) , which leads to seasonally strongly variable thermal fluxes incident on the spacecraft from all sides . this is mitigated by optimal placement of the radiators in combination with a seasonal yaw - rotation by @xmath52 to minimize the variability of the external flux on the main radiators . the variation of the thermal flux during the 6 years of mission is plotted in fig.[fig : solar_flux ] ( left ) . the time when the largest temperature fluctuations occur during one orbit , defined as the maximum temperature swing , is close to the start of the mission and coincides with the time of the longest eclipse period . the additional use of heaters in proximity to the various payload units allows us to achieve temperature fluctuations of the payload units of less than @xmath53c , whilst average operating points consistently lie within the required range from @xmath54c to @xmath55c . during periods when only a minimum of solar flux is absorbed by the spacecraft , e.g. cold case of fig.[fig : solar_flux ] , the heaters of the thermal control system may additionally contribute up to 150 w to the power dissipation . these results were found on the basis of a detailed thermal model of the spacecraft implemented with esatan - tms , a standard european thermal analysis tool for space systems@xcite . + _ electrical subsystem & communications : _ similar to the tcs , the design of the electrical subsystem is driven by the highly variable orbit featuring a large number of eclipses ( more than 1800 during the mission with a maximum period of 66 minutes ) , the high power demand of the instruments , and the satellite pointing strategy in addition to the required stability of the spacecraft power bus . in order to meet the power requirements , two deployable solar arrays at a ca nt angle of 45 degrees are rotated around the spacecraft y - axis . this configuration , in combination with two yaw - flips per year , ensures a minimum solar flux of approximately @xmath56 on the solar panels ( see fig.[fig : solar_flux ] right ) , which generates a total power of 2.4 kw , including margins . the option of using a 2-axis sadm has also been investigated in detail and could be easily implemented . however , it is currently considered too risky as such mechanisms have not yet been developed and qualified in europe and they might cause problems in relation to micro - vibrations . as far as telemetry and tele - commanding ( tt&c ) is concerned , contact times to a northern ground station ( cebreros ) are close to 15 h per day on average for the baseline orbit , which is easily compliant with the required minimum of 2 h per day . this provides plenty of margin to download an estimated 4.4 gbit of data using a switched medium gain antenna ( mga ) and low gain antenna ( lga ) x - band architecture which is based on heritage from the lisa pathfinder mission@xcite . the switched tt&c architecture allows achieving the required data volume at apogee on the one hand , whilst avoiding excessive power flux at perigee to remain compliant with itu regulations on the other hand . + _ mechanical subsystem , assembly integration and test ( ait ) and radiation aspects : _ the spacecraft structure is made almost entirely from panels with an aluminum honeycomb structure ( thickness 40 mm ) and carbon - fibre reinforced polymer ( cfrp ) face sheets , which is favorable from a mechanical and mass - savings perspective . a detailed analysis based on a finite - element - model ( fem ) of the spacecraft structure revealed a minimum eigenfrequency of 24 hz in transverse and 56 hz in axial direction , well above the launcher requirements of 15 hz and 35 hz , respectively . the first eigenmode of the atom interferometer physics package , corresponding to a pendulum mode in radial direction , was found at 67 hz . the spacecraft structure is designed to optimize ait and aspects of parallel integration . to this end , the two instruments can be integrated and tested separately before being joined on payload module level without the need to subsequently remove any unit or harness connection . the heat - pipe routing through the spacecraft structure allows functional testing of both instruments on ground by operating the heat - pipes either in nominal mode or re - flux mode ( opposite to the direction of gravity ) . as an important feature in the integration process , payload and service module components are completely separated in their respective modules up to the last integration step , when they are finally joined and their respective harness connected on easily accessible interface brackets . the second major aspect in structural design and instrument accommodation has been to minimize radiation doses of sensitive instrument components . as the spacecraft crosses the van allen belt twice per orbit and more than 5000 times during the mission , it sustains large radiation doses from trapped electrons and solar protons . dedicated design provisions and protected harness routing ensure that the total ionizing doses ( tids ) are below 30 krad for all instrument units and below 5 krad for the sensitive optical fibres . the non - ionizing energy loss , which is primarily caused by low - energy protons , is found to be well below @xmath57 , expressed in terms of 10 mev equivalent proton fluence . these results were obtained from the total dose curves accumulated over the orbit during the mission in combination with a dedicated sectoring analysis performed with a finite - element - model of the spacecraft . ste - quest aims to probe the foundations of einstein s equivalence principle by performing measurements to test its three cornerstones , i.e. the local position - invariance , the local lorentz - invariance and the weak equivalence principle , in a combined mission . it complements other missions which were devised to explore the realm of gravity in different ways , including the soon - to - be - launched aces@xcite and microscope@xcite missions and the proposed step@xcite mission . the ste - quest measurements are supported by two instruments . the first instrument , the atomic clock , benefits from extensive heritage from the aces mission@xcite , which therefore reduces associated implementation risks . the other instrument , the atom interferometer , would be the first instrument of this type in space and poses a major challenge which is currently met by dedicated development and qualification programs . the mission assessment activities summarized in this paper yielded a spacecraft and mission design that is compliant with the challenging demands made by the payload on performance and resources . several critical issues have been identified , including low maturity of certain payload components , potential unavailability of sufficiently accurate ground clocks , high sensitivity of the optical links to atmospheric distortions and associated performance degradation , and high energy dissipation of the instruments in addition to challenging temperature stability requirements . however , none of these problems seems unsurmountable , and appropriate mitigation actions are already in place . the work underlying this paper was performed during the mission assessment and definition activities ( phase 0/a ) for the european space agency ( esa ) under contract number 4000105368/12/nl / hb . the authors gratefully acknowledge fruitful discussions and inputs from the esa study team members , in particular martin gehler ( study manager ) , luigi cacciapuoti ( project scientist ) , robin biesbroek ( system engineer ) , astrid heske , ( payload manager ) , florian renk ( mission analyst ) , pierre waller , ( atomic clock support ) , and eric wille ( atom interferometer support ) . we also thank the astrium team members for their contributions to the study , in particular felix beck , marcel berger , christopher chetwood , albert falke , jens - oliver fischer , jean - jacques floch , sren hennecke , fabian hufgard , gnter hummel , christian jentsch , andreas karl , johannes kehrer , arnd kolkmeier , michael g. lang , johannes loehr , marc maschmann , mark millinger , dirk papendorf , raphael naire , tanja nemetzade , bernhard specht , francis soualle and michael williams . the authors are grateful for important contributions from our project partners mathias lezius ( menlo systems ) , wolfgang schfer , thorsten feldmann ( timetech ) , and sven schff ( astos solutions ) . finally , we are indebted to rdiger gerndt and ulrich johann ( astrium ) for regular support and many useful discussions . martin gehler , luigi cacciapuoti , astrid heske , robin biesbroek , pierre waller , and eric wille . the esa ste - quest mission study - space mission design to test einstein s equivalence principle . in _ proceedings of the aiaa space 2013 conference , paper number : 2013.5464 , doi : 10.2514/6.2013 - 5464_. american institute of aeronautics and astronautics , 2013 . s. schiller and et al . pace time explorer and quantum equivalence principle space test ( ste - quest ) : proposal in response to esa m3 call 2010 for the cosmic vision programme , 2010 . available at _ http://www.exphy.uni - duesseldorf.de / publikationen/2010/ste - quest_final.pdf_. mp plattner , u hugentobler , d voithenleitner , m hseinze , v klein , k kemmerle , and s bedrich . optical clock technology for optimized satellite navigation . in _ eftf 2010 , the 24th european frequency and time forum , 13 - 16 april 2010 _ , volume 1 , page 2 , 2010 . m lezius , k predehl , w stower , a turler , m greiter , ch hoeschen , p thirolf , w assmann , d habs , alexander prokofiev , et al . radiation induced absorption in rare earth doped optical fibers . , 59(2):425433 , 2012 . c schubert , j hartwig , h ahlers , k posso - trujillo , n gaaloul , u velte , a landragin , a bertoldi , b battelier , p bouyer , et al . differential atom interferometry with rb87 rb85 for testing the uff in ste - quest . , 2013 . d aguilera , h ahlers , b battelier , a bawamia , a bertoldi , r bondarescu , k bongs , p bouyer , c braxmaier , l cacciapuoti , et al . -test of the universality of free fall using cold atom interferometry . , 2013 . fiodor sorrentino , kai bongs , philippe bouyer , luigi cacciapuoti , marella de angelis , hansjoerg dittus , wolfgang ertmer , antonio giorgini , jonas hartwig , matthias hauth , et al . a compact atom interferometer for future space missions . , 22(4):551561 , 2010 . guillaume stern , baptiste battelier , rmi geiger , gal varoquaux , andr villing , frdric moron , olivier carraz , nassim zahzam , yannick bidel , w chaibi , et al . light - pulse atom interferometry in microgravity . , 53(3):353357 , 2009 . mp hess , j kehrer , m kufner , s durand , g hejc , h fruhauf , l cacciapuoti , r much , and r nasca . status and test results . in _ frequency control and the european frequency and time forum ( fcs ) , 2011 joint conference of the ieee international _ , pages 18 . ieee , 2011 . a seidel , mp hess , j kehrer , w schafer , m kufner , m siccardi , l cacciapuoti , i aguilar sanches , and s feltham . the aces microwave link : instrument design and test results . in _ frequency control symposium , 2007 joint with the 21st european frequency and time forum . ieee international _ , pages 12951298 . ieee , 2007 . m gregory , f heine , h kmpfner , r meyer , r fields , and c lunde . laser communication terminal performance results on 5.6 gbit coherent inter satellite and satellite to ground links . in _ international conference on space optics _ , volume 4 , page 8 , 2010 . gerald hechenblaikner , jean - jacques floch , francis soualle , and marc - peter hess . -based precise orbit determination for a highly eccentric orbit in the ste - quest mission . in _ proceedings of the 26th international technical meeting of the satellite division of the institute of navigation ( ion gnss 2013 ) , nashville , september 2013_. institute of navigation , 2013 . c urschl , g beutler , w gurtner , u hugentober , and m ploner . orbit determination for giove - a using slr tracking data . in _ extending the range . proceedings of the 15th international workshop on laser ranging _ , pages 4046 , 2008 . till rosenband , db hume , po schmidt , cw chou , a brusch , l lorini , wh oskay , re drullinger , tm fortier , je stalnaker , et al . frequency ratio of al+ and hg+ single - ion optical clocks ; metrology at the 17th decimal place . , 319(5871):18081812 , 2008 . g matticari , g noci , l ceruti , l fallerini , a atzei , c edwards , g morris , and k pfaab . old gas micro propulsion for european science missions : status on development and realization activities for lisa pathfinder , microscope and forthcoming euclid , following the successful delivery of the first fm for gaia program . 49th aiaa / asme / sae / asee joint propulsion conference & exhibit . t. j. sumner , j. anderson , j .- blaser , a. m. cruise , t. damour , h. dittus , c. w. f. everitt , b. foulon , y. jafry , b. j. kent , n. lockerbie , f. loeffler , g. mann , j. mester , c. pegrum , r. reinhardt , m. sandford , a. scheicher , c. c. speake , r. torii , s. theil , p. touboul , s. vitale , w. vodel , and p. w. worden . . , 39:254258 , 2007 .

ste - quest is a fundamental science mission which is considered for launch within the cosmic vision programme of the european space agency ( esa ) . its main scientific objectives relate to probing various aspects of einstein s theory of general relativity by measuring the gravitational red - shift of the earth , the moon and the sun as well as testing the weak equivalence principle to unprecedented accuracy . in order to perform the measurements , the system features a spacecraft equipped with two complex instruments , an atomic clock and an atom interferometer , a ground - segment encompassing several ground - terminals collocated with the best available ground atomic clocks , and clock comparison between space and ground via microwave and optical links . the baseline orbit is highly eccentric and exhibits strong variations of incident solar flux , which poses challenges for thermal and power subsystems in addition to the difficulties encountered by precise - orbit - determination at high altitudes . the mission assessment and definition phase ( phase - a ) has recently been completed and this paper gives a concise overview over some system level results .

arxiv

pulsed high - frequency ( hf ) electromagnetic ( em ) waves from transmitters on the ground are regularly used for sounding the density profile and drift velocity of the overehead ionosphere [ _ hunsucker _ , 1991 ; _ reinisch et al . _ , 1995 , _ reinisch _ , 1996 ] . in 1971 , it was shown theoretically by _ perkins and kaw _ [ 1971 ] that if the injected hf radio beams are strong enough , weak - turbulence parametric instabilities in the ionospheric plasma of the type predicted by _ silin _ [ 1965 ] and _ dubois and goldman _ [ 1965 ] would be excited . ionospheric modification experiments by a high - power hf radio wave at platteville in colorado [ _ utlaut _ , 1970 ] , using ionosonde recordings and photometric measurements of artificial airglow , demonstrated the heating of electrons , the deformation in the traces on ionosonde records , the excitation of spread @xmath0 , etc . , after the hf transmitter was turned on . the triggering of weak - turbulence parametric instabilities in the ionosphere was first observed in 1970 in experiments on the interaction between powerful hf radio beams and the ionospheric plasma , conducted at arecibo , puerto rico , using a scatter radar diagnostic technique [ _ wong and taylor _ , 1971 ; _ carlson et al . _ , 1972 ] . a decade later it was found experimentally in troms that , under similar experimental conditions as in arecibo , strong , systematic , structured , wide - band secondary hf radiation escapes from the interaction region [ _ thid et al . _ , 1982 ] . this and other observations demonstrated that complex interactions , including weak and strong em turbulence , [ _ leyser _ , 2001 ; _ thid et al . _ , 2005 ] and harmonic generation [ _ derblom et al . _ , 1989 ; _ blagoveshchenskaya et al . _ , 1998 ] are excited in these experiments . numerical simulations have become an important tool to understand the complex behavior of plasma turbulence . examples include analytical and numerical studies of langmuir turbulence [ _ robinson _ , 1997 ] , and of upper - hybrid / lower - hybrid turbulence in magnetized plasmas [ _ goodman et al . _ , 1994 ; _ xi _ , 2004 ] . in this letter , we present a full - scale simulation study of the propagation of an hf em wave into the ionosphere , with ionospheric parameters typical for the high - latitude eiscat heating facility in troms , norway . to our knowledge , this is the first simulation involving realistic scale sizes of the ionosphere and the wavelength of the em waves . our results suggest that such simulations , which are possible with today s computers , will become a powerful tool to study hf - induced ionospheric turbulence and secondary radiation on a quantitative level for direct comparison with experimental data . we use the mks system ( si units ) in the mathematical expressions throughout the manuscript , unless otherwise stated . we assume a vertically stratified ion number density profile @xmath1 with a constant geomagnetic field @xmath2 directed obliquely to the density gradient . the em wave is injected vertically into the ionosphere , with spatial variations only in the @xmath3 direction . our simple one - dimensional model neglects the em field @xmath4 falloff ( @xmath5 is the distance from the transmitter ) , the fresnel pattern created obliquely to the @xmath3 direction by the incident and reflected wave , and the the influence on the radio wave propagation due to field aligned irregularities in the ionosphere . for the em wave , the maxwell equations give @xmath6 @xmath7 where the electron fluid velocity is obtained from the momentum equation @xmath8\ ] ] and the electron density is obtained from the poisson equation @xmath9 . here , @xmath10 is the unit vector in the @xmath3 direction , @xmath11 is the speed of light in vacuum , @xmath12 is the magnitude of the electron charge , @xmath13 is the vacuum permittivity , and @xmath14 is the electron mass . ms.,scaledwidth=48.0% ] the number density profile of the immobile ions , @xmath15 $ ] ( @xmath3 in kilometers ) is shown in the leftmost panel of fig . [ fig1 ] . instead of modeling a transmitting antenna via a time - dependent boundary condition at @xmath16 km , we assume that the em pulse has reached the altitude @xmath17 km when we start our simulation , and we give the pulse as an initial condition at time @xmath18 s. in the initial condition , we use a linearly polarized em pulse where the carrier wave has the wavelength @xmath19 ( wavenumber @xmath20 ) corresponding to a carrier frequency of @xmath21 ( @xmath22 ) . the em pulse is amplitude modulated in the form of a gaussian pulse with a maximum amplitude of @xmath23 v / m , with the @xmath24-component of the electric field set to @xmath25\sin(0.1047\times 10^{3 } z)$ ] ( @xmath3 in kilometers ) and the @xmath26 component of the magnetic field set to @xmath27 at @xmath18 . the other electric and magnetic field components are set to zero ; see fig . [ the spatial width of the pulse is approximately 30 km , corresponding to a temporal width of 0.1 milliseconds as the pulse propagates with the speed of light in the neutral atmosphere . it follows from eq . ( 1 ) that @xmath28 is time - independent ; hence we do not show @xmath28 in the figures . the geomagnetic field is set to @xmath29 tesla , corresponding to an electron cyclotron frequency of 1.4 mhz , directed downward and tilted in the @xmath30-plane with an angle of @xmath31 degrees ( @xmath32 rad ) to the @xmath3-axis , i.e. , @xmath33 . in our numerical simulation , we use @xmath34 spatial grid points to resolve the plasma for @xmath35 km . the spatial derivatives are approximated with centered second - order difference approximations , and the time - stepping is performed with a leap - frog scheme with a time step of @xmath36 s. the splitting of the wave is due to faraday rotation.,scaledwidth=48.0% ] ms . b ) a closeup of the region of the turning points of the r - x and l - o modes . we see that the wave - energy of the l - o mode is concentrated into one single half - wave envelop at @xmath37 km , while the turning point of the less localized r - x mode is at @xmath38 km.,scaledwidth=48.0% ] ms . b ) a closeup of the region of the turning points of the r - x and l - o modes . here , the l - o mode oscillations at @xmath37 km are radiating em waves with perpendicular ( to the @xmath3 axis ) electric field components.,scaledwidth=48.0% ] in the simulation , the em pulse propagates without changing shape through the neutral atmosphere , until it reaches the ionospheric layer . at time @xmath39 ms , shown in fig . [ fig2 ] , the em pulse has reached the lower part of the ionosphere . the initially linearly polarized em wave undergoes faraday rotation due to the different dispersion properties of the l - o and r - x modes ( we have adopted the notation `` l - o mode '' and `` r - x mode '' for the two high - frequency em modes , similarly as , e.g. , _ goertz and strangeway _ [ 1995 ] ) in the magnetized plasma , and the @xmath40 and @xmath41 components are excited . at @xmath42 ms , shown in fig . [ fig3 - 4 ] , the l - o and r - x mode pulses are in the vicinity of their respective turning points , the turning point of the l - o mode being at a higher altitude than that of the r - x mode ; see panel a ) of fig . [ fig3 - 4 ] . a closeup of this region , displayed in panel b ) , shows that the first maximum of the r - x mode is at @xmath38 km , and the one of the l - o mode is at @xmath37 km . the maximum amplitude of the r - x mode is @xmath43 v / m while that of the l - o mode is @xmath44 v / m ; the latter amplitude maximum is in agreement with those obtained by _ thid and lundborg _ , [ 1986 ] , for a similar set of parameters as used here . the electric field components of the l - o mode , which at this stage are concentrated into a pulse with a single maximum with a width of @xmath45 m , are primarily directed along the geomagnetic field lines , and hence only the @xmath46 and @xmath47 components are excited , while the magnetic field components of the l - o mode are very small . at @xmath48 ms , shown in panel a ) of fig . [ fig5 - 6 ] , both the r - x and l - o mode wave packets have widened in space , and the em wave has started turning back towards lower altitudes . in the closeup of the em wave in panel b ) of fig . [ fig5 - 6 ] , one sees that the l - o mode oscillations at @xmath37 km are now radiating em waves with significant magnetic field components . finally , shown in fig . [ fig7 ] at @xmath49 , the em pulse has returned to the initial location at @xmath17 km . due to the different reflection heights of the l - o and r - x modes , the leading ( lower altitude ) part of the pulse is primarily r - x mode polarized while its trailing ( higher altitude ) part is l - o mode polarized . in the center of the pulse , where we have a superposition of the r - x and l - o mode , the wave is almost linearly polarized with the electric field along the @xmath26 axis and the magnetic field along the @xmath24 axis . the direction of the electric and magnetic fields here depends on the relative phase between the r - x and l - o mode . ms.,scaledwidth=48.0% ] at @xmath50 km , near the turning point of the r - x mode , and b ) the amplitude of the electric field component @xmath46 at @xmath51 km , near the turning point of the l - o mode . c ) a snapshot of low - amplitude electrostatic waves of wavelength @xmath52 m ( wavenumber @xmath53 ) , observed at time @xmath54 ms , and d ) dispersion curves ( lower panel ) obtained from the appleton - hartree dispersion relation with parameters @xmath55 ( 5 mhz ) , @xmath56 ( 1.4 mhz ) and @xmath57 rad . we identify the high - frequency r - x and l - o modes , as well as the z - mode which extends to the electrostatic langmuir / upper hybrid branch for large wavenumbers ; the circles indicate the approximate locations on the dispersion curve for the electrostatic oscillations shown in panel c ) . for completeness we also show the low - frequency electron whistler branch in panel d).,scaledwidth=50.0% ] at the altitude @xmath50 km , and b ) of @xmath46 at the altitude @xmath51 km.,scaledwidth=48.0% ] in fig . [ fig8 - 9 ] , panel a ) , we have plotted the electric field component @xmath47 at @xmath50 km , near the turning point of the r - x mode and in panel b ) we have plotted the @xmath46 component at @xmath51 km , near the turning point of the l - o mode . we see that the maximum amplitude of @xmath47 reaches @xmath58 v / m at @xmath59 ms , and that of @xmath46 reaches @xmath60 v / m at @xmath61 ms . the electric field amplitude at @xmath50 km has two maxima , due to the l - o mode part of the pulse , which is reflected at the higher altitude @xmath51 km and passes twice over the altitude @xmath50 km . we also observe weakly damped oscillations of @xmath46 at @xmath51 km for times @xmath62 ms , which decrease exponentially in time between @xmath63 ms and @xmath64 ms as @xmath65 with @xmath66 s@xmath67 . we found from the numerical values that @xmath68 , where @xmath69 is the inverse ion density scale length at @xmath70 km , but we are not certain how general this result is . no detectable magnetic field fluctuations are associated with these weakly damped oscillations , and we interpret them as electrostatic waves that have been produced by mode conversion of the l - o mode . the amplitudes of the @xmath47 and @xmath40 components are also much weaker than that of the @xmath46 component for these oscillations . a closeup of these electrostatic oscillations at @xmath54 ms is displayed in panel c ) of fig . [ fig8 - 9 ] , where we see that they have a wavelength of approximately 33 m ( wavenumber @xmath71 ) . in panel d ) of fig . [ fig8 - 9 ] , we have plotted the frequency @xmath72 as a function of the wavenumber @xmath73 , where @xmath74 is obtained from the appleton - hartree dispersion relation [ _ stix _ , 1992 ] @xmath75 here @xmath76^{1/2}$ ] , @xmath77 ( @xmath78 ) is the electron plasma ( cyclotron ) frequency , and @xmath79 is the angle between the geomagnetic field and the wave vector @xmath80 , which in our case is directed along the @xmath3-axis , @xmath81 . we use @xmath82 ( corresponding to @xmath83 mhz ) , @xmath84 ( corresponding to @xmath85 mhz ) and @xmath57 rad . the location of the electrostatic waves whose wavelength is approximately 33 m and frequency 5 mhz is indicated with circles in the diagram ; they are on the same dispersion surface as the langmuir waves and the upper hybrid waves / slow z mode waves with propagation parallel and perpendicular to the geomagnetic field lines , respectively . the mode conversion of the l - o mode into electrostatic oscillations are relatively weak in our simulation of vertically incident em waves , and theory shows that the most efficient linear mode conversion of the l - o mode occurs at two angles of incidence in the magnetic meridian plane , given by , e.g. , eq . ( 17 ) in [ _ mjlhus _ , 1990 ] . the nonlinear effects at the turning point of the l - o and r - x modes are investigated in fig . [ fig10 ] which displays the frequency spectrum of the electric field component @xmath47 at the altitude @xmath86 km and of @xmath46 at the altitude @xmath51 km . the spectrum shows the large - amplitude pump wave at 5 mhz and the relatively weak second harmonics of the pump wave at 10 mhz at both altitudes ( the slight downshift is due to numerical errors produced by the difference approximations used in space and time ) . visible are also low - frequency oscillations ( zeroth harmonic ) due to the nonlinear down - shifting / mixing of the high - frequency wave field . in conclusion , we have presented a full - scale numerical study of the propagation of an em wave and its linear and nonlinear interactions with an ionospheric layer . we observe the reflection of the l - o and r - x modes at different altitudes , the mode conversion of the l - o mode into electrostatic langmuir / upper hybrid waves as well as nonlinear harmonic generation of the high - frequency waves . second harmonic generation have been observed in ionospheric heating experiments [ _ derblom et al . _ , 1989 ; _ blagoveshchenskaya et al . _ , 1998 ] and may be partially explained by the cold plasma model presented here . blagoveshchenskaya , n. f. , v. a. kornienko , m. t. rietveld , b. thid , a. brekke , i. v. moskvin , and s. nozdrachev ( 1998 ) , stimulated emissions around second harmonic of troms heater frequency observed by long - distance diagnostic hf tools . _ geophys . 25_(6 ) , 863876 . derblom , h. , b. thid , t. b. leyser , j. a. nordling , . hedberg , p. stubbe , h. kopka , and m. rietveld ( 1989 ) , troms heating experiments : stimulated emission at hf pump harmonic and subharmonic frequencies , _ j. geophys . 94_(a8 ) , 1011110120 . thid , b. , e. n. sergeev , s. m. grach , t. b. leyser , and t. d. carozzi ( 2005 ) , competition between langmuir and upper - hybrid turbulence in a high - frequency - pumped ionosphere , _ _ 95 _ , 255002 .

the time evolution of a large - amplitude electromagnetic ( em ) wave injected vertically into the overhead ionosphere is studied numerically . the em wave has a carrier frequency of 5 mhz and is modulated as a gaussian pulse with a width of approximately 0.1 milliseconds and a vacuum amplitude of 1.5 v / m at 50 km . this is a fair representation of a modulated radio wave transmitted from a typical high - power hf broadcast station on the ground . the pulse is propagated through the neutral atmosphere to the critical points of the ionosphere , where the l - o and r - x modes are reflected , and back to the neutral atmosphere . we observe mode conversion of the l - o mode to electrostatic waves , as well as harmonic generation at the turning points of both the r - x and l - o modes , where their amplitudes rise to several times the original ones . the study has relevance for ionospheric interaction experiments in combination with ground - based and satellite or rocket observations .

arxiv

the mott - hubbard model of interacting bosons on a lattice has been used to describe superfluid mott - insulator transitions in a variety of systems , e.g. , josephson arrays and granular superconductors @xcite . the recent suggestion @xcite to experimentally observe this transition in a system of cold bosonic atoms in an optical lattice and its successful experimental demonstration @xcite has rekindled the interest in the mott - insulator transition and triggered a great deal of theoretical @xcite and experimental @xcite activity . the possibility to directly manipulate and test the many - body behavior of a system of trapped bosonic atoms in an optical lattice @xcite is very attractive . possible applications include the use of a mott state of bosonic atoms in an optical lattice as a starting point to create controlled multiparticle entanglement as an essential ingredient for quantum computation @xcite the mott - insulator quantum phase transition is driven by the interplay of the repulsive interaction of bosons on the same lattice site and the kinetic energy . hence the ratio of the onsite energy and the bandwidth forms the key parameter in the system . in optical lattices , this parameter can be easily controlled and varied by several orders of magnitude , enabling detailed studies of the quantum phase transition . probing the system by taking absorption pictures to image the expansion patterns after a reasonable expansion time yields information about the momentum distribution of the state . this procedure was used to experimentally confirm the mott transition in an optical lattice @xcite . the essential physics of cold bosonic atoms in an optical lattice is captured by a bosonic mott - hubbard model describing the competition between hopping and on - site interaction . a number of approximation schemes have been used to study this model analytically @xcite as well as numerically , using approaches like the gutzwiller mean - field ansatz @xcite , density - matrix renormalization group ( dmrg ) @xcite , exact diagonalization ( ed)@xcite and quantum monte carlo ( qmc ) @xcite . in this article , we study the short - range correlations , not included by the gutzwiller ansatz , by using perturbation theory . the main purpose is to find corrections to the short - range behavior of the one - particle density matrix , which is directly relevant to experimentally observed expansion patterns . these patterns are important for determining the location of the insulator - superfluid transition . we note that in the insulating state our perturbative approach is identical to the one used in @xcite ( see also @xcite ) , although there the goal was different , viz . , studying corrections to the phase diagram . the remainder of the article is organized as follows : in section [ modsec ] , we will introduce the model and its mean - field solution . the general perturbative approach is briefly outlined in section [ secpa ] , while details may be found in the appendix . numerical results are presented and discussed in section [ secnr ] , first for local observables ( [ secslso ] ) and then for the density matrix ( [ secrho ] ) . implications for expansion patterns both for bulk systems and a harmonic confining potential are discussed in section [ harmotrap ] . the cold bosonic gas in the optical lattice can be described by a mott - hubbard model @xcite @xmath0 here , @xmath1 is the total number of lattice sites , @xmath2 ( @xmath3 ) creates ( annihilates ) a boson on site @xmath4 , @xmath5 , @xmath6 is the on - site repulsion describing the interaction between bosons on the same lattice site , and @xmath7 denotes the chemical potential . the kinetic term includes only hopping between nearest - neighbor sites , this is denoted by the summation index @xmath8 ; @xmath9 is the hopping matrix element that we will assume to be lattice - site independent . finally , @xmath10 describes an external on - site potential that is commonly present in experiments . the gutzwiller ( gw ) approach is based on an ansatz for the many - body ground state that factorizes into single lattice - site wavefunctions @xmath11 the gutzwiller wavefunction represents the ground state of the following mean - field version of the mott - hubbard hamiltonian , eq . ( [ motthubb ] ) : @xmath12 here @xmath13 is the mean - field potential on the @xmath4-th lattice site , which is self - consistently defined as the expectation value of @xmath3 in terms of the gutzwiller wavefunction , @xmath14 @xcite . using the gutzwiller ansatz to obtain an approximate variational solution for the mott - hubbard hamiltonian ( [ motthubb ] ) corresponds , however , to restricting the hilbert space to the subset of product states . consequently , even in higher dimensions , this ansatz fails to describe the correct behavior of short - range correlations between different lattice sites , which are important for experimentally measurable observables , such as expansion patterns ( momentum distributions ) . nevertheless , in the thermodynamic limit and higher dimensions , the gutzwiller wavefunction provides a good approximation in the limits of @xmath15 and @xmath16 ( i.e. , deep in the mott insulator ( mi ) and superfluid ( sf ) phases ) . to get a satisfactory description of the short - range correlations we will now derive perturbative corrections to the gutzwiller mean - field result . our aim is to start from the gutzwiller approximation and improve it by perturbatively including the short - range correlations between lattice sites . we re - express the mott - hubbard hamiltonian ( [ motthubb ] ) by adding the appropriate perturbation to the mean - field hamiltonian , eq . ( [ mf ] ) : @xmath17 with @xmath18 as the mean - field hamiltonian represents a sum of single lattice - site hamiltonians , the excited states @xmath19 and the excitation spectrum @xmath20 can be obtained numerically for each lattice site @xmath4 separately . hence we can write the excitations of @xmath21 as product states of single lattice - site excitations , @xmath22 and the excitation spectrum as a sum over the single lattice - site excitation energies , @xmath23 where @xmath24 describes the set of quantum numbers characterizing the given many - body energy eigenstate . having obtained the mean - field solution from the gutzwiller ansatz , we can now proceed to improve our wavefunction performing rayleigh - schrdinger perturbation theory @xcite in @xmath25 : @xmath26 where @xmath27 and @xmath28 are the @xmath29-th order corrections to the wavefunction and grand - canonical energy @xmath30 , respectively . knowing the excited wavefunctions , eq . ( [ es ] ) , and the excitation spectrum , eq . ( [ esp ] ) , perturbative corrections to observables can be calculated explicitly . the resulting expressions up to second order in @xmath25 , which we use in the following , are derived and illustrated diagrammatically in appendix [ appendixa ] . we start by computing the gutzwiller wavefunction numerically using a conjugate - gradient descent method . propagation steps of the time - dependent gutzwiller equations @xcite in imaginary time are performed to test whether the minimum found by the conjugate - gradient descent method is indeed the ground state . afterwards , we calculate eigenenergies and -vectors of each site , with respect to the mean - field hamiltonian . as explained above , this forms the basis for perturbation theory , which we use to calculate the corrections to the observables up to second order . the results obtained from perturbation theory show important modifications to single - site observables as well as to the correlation function . these observables are composed of operators acting only on single lattice sites . hence , they describe local properties and will be less sensitive to the correlations between lattice sites . thus , values for single - site observables obtained from the gutzwiller wavefunction already provide a good approximation in most cases ( unless fluctuations are concerned ) . ( diamonds ) , @xmath31 ( triangles ) , and @xmath32 ( squares ) dimensions . the dashed - dotted lines in fig . [ slso]a - c are the gutzwiller results . ( a ) number fluctuations @xmath33 calculated for a commensurate filling of one boson per lattice site . the dashed line shows the result from the exact diagonalization for 7 lattice sites and @xmath34 bosons . ( b ) order parameter @xmath35 and ( c ) compressibility @xmath36 both computed at a fixed chemical potential @xmath37 . [ slso],width=302 ] the leading corrections to the mean values of single lattice - site observables ( slso ) are of second order in @xmath25 . the results for the order parameter @xmath38 , the compressibility @xmath36 and the number fluctuations @xmath39 are shown in fig . [ slso ] , where the ration @xmath40 has been scaled by the dimension , to keep the same mean - field transition point . the solid lines in fig . [ slso ] show the results from perturbation theory , as compared to the gutzwiller result ( shown as the dashed - dotted line ) . all three quantities show a vanishing perturbative correction both for small and large @xmath40 , as the gutzwiller wavefunction becomes a good approximation in these regimes ( for lattice dimensions @xmath41 ) . as expected , deviations from the mean - field picture are strongest near the mi - sf transition , where higher - order corrections will become more and more important . the order parameter @xmath42 shown in fig . [ slso]b gets suppressed in the sf . perturbative corrections to the gutzwiller result are particularly large in 1d ( where the order parameter vanishes in reality ) , but get smaller with increasing dimension . this is not surprising , as gutzwiller is a mean - field approach and hence a better approximation for higher - dimensional systems . the critical value @xmath43 is not modified within the present perturbative approach . figure [ slso]c shows the results for the compressibility @xmath44 the results of perturbation theory ( pt ) show a decrease of the compressibility , pointing to an increasing stiffness of the sf phase induced by the short - range interaction . finally , we computed the local particle number fluctuations @xmath45 . exact diagonalization calculations have shown that the number fluctuations are changing smoothly at the mi - sf transition @xcite , whereas the gutzwiller result predicts vanishing fluctuations in the mi phase , @xmath46 ( see dashed - dotted line in fig . [ our perturbative results reproduce the non - vanishing part of @xmath39 in the mi regime and agree well with our result obtained from exact diagonalization of a one - dimensional system of @xmath47 lattice sites . significant deviations from the exact 1d result are seen in the mi - sf transition regions , starting from the mean - field critical value , @xmath48 , up to values of the order of the critical values of @xmath49 usually obtained from dmrg @xcite and qmc @xcite calculations . nevertheless , based on the good agreement of our perturbative result with the exact diagonalization for a large region in the mi , we conclude that the number fluctuations are mainly produced by next - neighbor particle - hole fluctuations included in perturbation theory . the single - particle density matrix @xmath50 is of particular importance as it describes the correlation between the different lattice sites . the correlation function @xmath51 shows off - diagonal long - range order in the sf state ( in dimensions @xmath41 ) , in contrast to the mi phase where @xmath51 decays exponentially . the experimental observation @xcite of the mi transition relies on the different behavior of the density matrix in the mi and sf regimes , which can be visualized by taking absorption pictures of the freely expanding atomic cloud . assuming that the expansion time is long enough and that the gas is dilute enough ( such that atom - atom interactions can be neglected during the expansion ) , the shape of the cloud reflects the initial momentum distribution @xmath52 , which is directly given by the fourier - transform of the density matrix @xmath51 : @xmath53 as a function of site distance @xmath54 . results for a homogeneous lattice of 7 sites with @xmath34 bosons have been obtained from exact diagonalization ( ed ) , from second - order perturbation theory ( pt ) , and from the gutzwiller mean - field ansatz ( gw ) . ( a ) @xmath55 ( mi regime ) , ( b ) @xmath56 , ( c ) @xmath57 , ( d ) @xmath58 ( deep sf regime ) . all calculations use periodic boundary conditions . the mean - field value @xmath43 for the mi transition in the commensurate case with one boson per lattice site is @xmath48 in 1d ( see fig . [ slso ] ) . ( the mf value differs strongly from @xmath59 derived from qmc calculations @xcite or @xmath60 for dmrg calculations @xcite ) . [ vgled],width=302 ] here , @xmath61 is the fourier transform of the wannier functions @xmath62 describing the wavefunction of a single lattice site . the presence of the factor @xmath61 in eq . ( [ fourier ] ) provides a cutoff at high momenta . the mean - field results for @xmath51 only describe the different long - range behavior in the mi and sf . for a homogeneous lattice the correlation function calculated from the gutzwiller wavefunction gives @xmath63 for the diagonal elements and then drops instantly to @xmath64 for all off - diagonal elements @xmath65 . short - range correlations are not reproduced by the gutzwiller approach . this deviation is particularly severe in the mi , where the mean - field result predicts a completely flat momentum distribution , whereas the short - range correlations ( i.e. the exponential decay of @xmath51 ) yield smooth bumps in the expansion pattern . these can be distinguished from the @xmath66-peaks of the sf only after a sufficiently long expansion time . applying perturbation theory to the gw wavefunction improves the structureless gw correlation function . in fig . [ vgled ] , we have compared the results of gw mean - field theory , of pt , and of exact diagonalization ( ed ) . the diagonalization has been carried out for a small 1d lattice , where it is easily feasible . although there is no long - range order in the sf phase for the 1d case , where the density matrix exhibits a power - law decay towards zero , it is still reasonable to compare the short - range correlations . indeed , we find a nice agreement between perturbation theory and exact results , not only for the mi ( see fig . [ vgled]a ) , but also for the short - range behavior in the sf . this agreement is made possible by the fact that @xmath51 decays only slowly and higher - order corrections show only negligible corrections for small lattices . an example for the sf case is shown in fig . [ vgled]d . however , there is still a considerable difference for @xmath67 in fig . [ vgled]d . this is not surprising as we do pt up to second order . hence correlations over a distance of three and more lattice sites are only corrected by the global mean - field correction for the infinite lattice ( see eq . ( [ cpsione ] ) , eq . ( [ cpsitwo ] ) , and fig . [ 2ndcpsi ] in appendix [ appendixa ] ) . we expect better agreement for larger lattice sizes as finite - size effects , arising in small lattices , are still considerable for @xmath68 sites used in our ed calculations . finally , for intermediate values of @xmath40 , shown in fig . [ vgled]b , c , we observe a faster drop in the off - diagonal correlations , such that higher - order contributions in the pt become more important . in any dimension , the perturbation @xmath25 is no longer small at the tip of the mi - sf transition lobe ( fig . [ vgled]c ) , and the pt breaks down . nevertheless , comparing with exact diagonalization results , fig . [ vgled]b shows still good agreement with ed , in contrast to fig . [ vgled]c , which shows clear disagreement . even though the parameters @xmath57 chosen for fig . [ vgled]c are close to the mi - sf transition for 1d - lattices ( as predicted by dmrg @xcite and qmc @xcite calculations ) , pt reproduces the correct slope for the off - diagonal decay and lacks only the wrong offset from the mean field . thus , even for this case , pt represents a qualitative improvement on the gutzwiller result . the results for both approximations , gutzwiller and pt , are expected to become better in higher dimensions ( with the perturbative corrections diminishing in size ) . in second - order pt as a function of @xmath69 and @xmath40 . the order parameter @xmath70 vanishes inside the mott - insulating lobes , whose mean - field boundaries are given by the white dashed line . plots ( a ) ( f ) display the resulting momentum distribution without the wannier form factor , @xmath71 , calculated for a 2d lattice with @xmath72 lattice sites . ( a)-(c ) are the gutzwiller mean - field results , and ( d)-(f ) are calculated using second - order pt . the inset in ( c),(f ) shows a cut of one peak taken along @xmath73 ; dashed line for gw and solid line for the pt result . arrows indicate the position of the respective plots in the @xmath74 phase diagram . the parameters used are : @xmath75 , @xmath76 for ( a ) and ( d ) ; @xmath77 , @xmath78 for ( b ) and ( e ) ; and @xmath79 , @xmath80 for ( c ) and ( f ) . the gray - scales of plots belonging to the same parameter set are identical . expansion patterns ( a),(b),(d),and ( e ) are normalized to the peak maximum ; ( c ) and ( f ) are normalized to @xmath81 of the peak maximum.[expans ] , width=302 ] as discussed before , the perturbative enhancement of the description of short - range correlations is expected to lead to strong consequences for the momentum distributions . as an example we discuss a set of momentum distributions @xmath71 for a homogeneous 2d lattice , fig . [ expans]a - f , and compare the gutzwiller results to those improved by pt . the improved pt versions , figs . [ expans]d - f , show much finer structures than the mean - field results , fig . [ expans]a - c . pt predicts broad peaks in the mi regions down to very small values of @xmath40 , fig . [ expans]e , whereas the gutzwiller result without pt shows a structureless flat distribution for the whole mi region , fig . [ expans]a , b . naturally , the modifications of @xmath71 are strongest near the phase transition , figs . [ expans]a and [ expans]d . going towards larger values @xmath40 into the sf phase , pt gives rise to a suppression of the peaks ( inset in fig . [ expans]c , f ) . this suppression can be larger than @xmath82 of the original peak height and stems from the corrections to the mean field . additionally , fig . [ expans]f shows broad peaks induced by the inclusion of short - range correlations . however , for large lattices , these broad peaks are small compared to the ( finite - size broadened ) sf @xmath66-peaks . on a plane through the trap center for a 3d lattice with @xmath83 lattice sites . ( a ) @xmath84 , @xmath85 , and @xmath86 . ( b ) @xmath75 , @xmath87 , and @xmath88 . ( c ) @xmath84 , @xmath87 , and @xmath86.[spd],width=302 ] -sites in the presence of a harmonic potential . ( a ) momentum distribution without the wannier form factor , @xmath71 , calculated along the @xmath89 direction . filled symbols are the pt results and open symbols are the gw results . the graphs for the mi surrounded by a sf shell ( diamonds ) are rescaled by a factor @xmath90 . the results denoted by circles correspond to the occupation distribution of fig . [ spd]a , diamonds to fig . [ spd]c , and squares to fig . ( b ) difference between pt and gw results . [ expans3d],title="fig:",width=264 ] -sites in the presence of a harmonic potential . ( a ) momentum distribution without the wannier form factor , @xmath71 , calculated along the @xmath89 direction . filled symbols are the pt results and open symbols are the gw results . the graphs for the mi surrounded by a sf shell ( diamonds ) are rescaled by a factor @xmath90 . the results denoted by circles correspond to the occupation distribution of fig . [ spd]a , diamonds to fig . [ spd]c , and squares to fig . ( b ) difference between pt and gw results . [ expans3d],title="fig:",width=264 ] in contrast to what was assumed in the last section , optical lattices used in experiments are not homogeneous . magnetic or optical trapping potentials are used @xcite to confine the atomic gas to a finite volume . the inhomogeneity caused by the trapping potential leads to slowly varying on - site energies , @xmath10 , in the mott - hubbard model eq . ( [ motthubb ] ) , that can be interpreted as a spatially varying chemical potential @xmath91 . consequently the lattice is in general not in a pure mi or sf phase , but shows alternating shells of sf and mi regions . an example for a sf region surrounded by a mi shell is shown in fig . [ spd]b . considering the slowly varying on - site energy as a spatially varying chemical potential gives a qualitative understanding of the shell structures . the spatial variation of the chemical potential corresponds to a path parallel to the @xmath69 axis in the @xmath69-@xmath40-diagram . starting with the potential minimum , in the trap center , and then moving off the center , decreases the effective local chemical potential . whenever a mi - sf ( sf - mi ) phase boundary is hit along the path in the @xmath69-@xmath40-diagram , a change from a mi to a sf ( sf to mi ) shell appears . the inclusion of short - range correlations gives rise to considerable modifications also in the presence of an inhomogeneous trapping potential . examples , calculated for a 3d - lattice , are shown in fig . [ expans3d ] , where an underlying harmonic potential @xmath92 was chosen . the three situations considered here correspond to a case with a large mi fraction ( fig . [ spd]a ) , a sf island surrounded by a mi shell ( fig . [ spd]b ) , and a mi island surrounded by a sf phase ( fig . [ spd]c ) . calculating the expansion patterns for these situations , fig . [ expans3d ] , shows that the perturbative corrections arising from the short - range correlations lead to substantially different behavior in the different cases . for the almost complete mi state we get a correction to all wavevectors @xmath93 , with a fast drop at values close to the peak center @xmath94 , which leads to a peak broadening in the expansion picture ( see circles in fig . [ expans3d ] ) . particularly large changes were found for the case of a sf island surrounded by a mi phase , fig . [ expans3d ] ( squares ) . again , corrections arise for all wavevectors , but , in contrast to the almost homogeneous case , the largest increase is now found for @xmath94 , with changes of about @xmath82 of the peak - maximum . finally , the reversed situation , a mi island surrounded by a sf phase ( diamonds in fig . [ expans3d ] ) does not show an increase of its maximum peak height but a considerable reduction ( over 5% of the peak maximum ) . this is not surprising , as the majority of the lattice sites are now contributing to the sf phase , and a peak reduction was also observed for the bulk sf phase . we would now like to compare our approach with the results obtained by kashurnikov _ @xcite , who used qmc calculations to calculate expansion patterns for a small 3d lattice with harmonic confinement . there is good qualitative agreement in all cases with high superfluid fraction ( compare for example fig . [ kash]a(b ) and fig . 2b(c ) in ref . @xcite ) . however , the features of these expansion patterns are already well reproduced using the gutzwiller mean - field ansatz alone , in particular , the satellite peak , which was discussed as a signature of the mi - sf shell structure in @xcite . corrections arising from pt show a suppression of the sf peak , as was discussed above . however , there are also considerable discrepancies to the qmc results for situations with a large mi fraction , even after implementing second - order pt . in these cases ( fig . [ kash]c(d ) and fig . 2d(e ) in ref . @xcite ) , the influence of the mi phase on the expansion picture broadens the peak and leads to a homogeneous background . the discrepancies to the qmc are clearly visible in fig . [ kash]c showing no peak broadening and a sattelite peak in contrast to the qmc results ( fig.2d in @xcite ) . including the short range correlations perturbatively corrects the expansion pattern in the right direction , giving rise to a suppression of the sf - peak . considering the expansion pattern with the clearest mi features ( fig [ kash]d and fig . 2e in ref . @xcite ) , we obtain the correct peak broadening from gw / pt calculation , but a larger ratio of mi background to sf peak . concerning these discrepancies , we note that the expansion pattern is highly sensitive to the value of the mean field . even small deviations can lead to a change in the sf - peak height sufficient to mask the flat distribution of the mi phase . we checked , however , that the observed discrepancy is not due to a lack in accuracy of our numerical calculations . we therefore believe that the discrepancies between qmc and gw / pt for situations with a large mi fraction can be attributed to the insufficiency of gw and low - order pt in describing the long - range correlations in this inhomogeneous situation . in addition , we note that the lattice employed in @xcite is comparatively small for the given harmonic confinement potential ( with no complete shell of empty sites at the perimeter , see insets of fig . [ kash ] ) , i.e. , the choice of boundary conditions ( periodic in the case of our numerical calculations ) may have non - negligible effects on the outer lattice sites . ) for a system of @xmath95 lattice sites considered in @xcite . all plots are without the wannier form factor , @xmath96 . the insets show the occupation number @xmath97 for a cut along the @xmath98-direction . ( a ) @xmath99 , @xmath100 , @xmath101 . ( b ) @xmath102 , @xmath103 , @xmath104 . ( c ) @xmath102 , @xmath105 , @xmath106 . ( d ) @xmath102 , @xmath107 , @xmath108 . ( note however , that the hamiltonian used in @xcite differs from eq . ( [ motthubb ] ) and hence the parameters are converted to the corresponding quantities used in our definitions . ) [ kash],width=321 ] we have used perturbation theory to incorporate the effects of short - range correlations on top of the mean field solution of the bosonic mott hubbard model . we derived corrections to local quantities , as well as to observable expansion patterns . we numerically calculated the corrections to the mf - result , using pt up to second order , thus including correlations between next and next - nearest lattice sites . modifications to the particle number fluctuations @xmath39 , arising from the pt , gave rise to the expected smooth transition of @xmath39 at the mi - sf transition . moreover , comparing the pt results to the ed results for @xmath39 in 1d lattices showed good agreement for small values of @xmath40 . of particular importance are the corrections to the correlation function , and thus to the expansion patterns . we compared the correlation function obtained from pt with calculations from ed for small 1d - lattices and found good agreement . studying the expansion patterns showed that the inclusion of the short - range correlations to the mf - ansatz gives rise to distinct modifications . a broad peak can be seen in the pt results for the mi regime , visible even down to small values of @xmath40 . comparing pt and mf expansion patterns obtained for parameters in the sf region displayed a considerable suppression of the sf peak in the pt results . additionally , on approaching the sf - mi transition from the sf side , broad peaks underlying the sf peaks were found in the pt - expansion patterns . including a harmonic confinement potential leads to situations where sf and mi regions coexist . hence the perturbative corrections to the expansion pattern become more complex . we studied lattices with different underlying harmonic traps giving rise to different constellations of sf - mi regions , finding modifications of up to @xmath82 of the peak maximum in the expansion patterns . we would like to thank krishnendu sengupta for discussions . our work was supported by the swiss nsf and the nccr nanoscience , as well as a dfg research fellowship ( f.m . ) . schematic diagram illustrating the term @xmath109 appearing in the first - order correction @xmath110 of the density matrix , where @xmath111 is the perturbation connecting sites @xmath4 and @xmath112 . the diagram shows the lattice sites in the horizontal direction . the different steps needed to obtain the matrix element are shown vertically . open circles denote the ground state ( gs ) of the given lattice site , while filled circles are excited states ( es ) of this site , for the mean - field hamiltonian @xmath21.,width=264 ] we use standard stationary perturbation theory @xcite to calculate the corrections to the mean - field results induced by the perturbation @xmath113 where we introduce the new operators @xmath114 the expectation value @xmath115 is taken with respect to the mf wavefunction @xmath116 . defining the operator for the energy denominator as @xmath117 the expectation value of an observable @xmath118 including all corrections up to second order is @xmath119 the first line in eq . ( [ perta ] ) is the mf - result @xmath120 , followed by two contributions which are the first - order corrections . lines two and three in eq . ( [ perta ] ) are the second - order corrections to the mean value . as an example we will discuss the corrections to the density matrix @xmath121 . the first - order correction to the density matrix is : @xmath122 for two different lattice sites @xmath123 , we find @xmath124 since the gutzwiller ground state is a product state . as a consequence , the contributions @xmath125 and @xmath126 vanish . the only remaining contributions to @xmath127 stem from @xmath128 and @xmath129 . the graph shows a decomposition of the matrix element , with each row showing the wavefunction at an intermediate step in the evaluation of the matrix element . as we deal with product states @xmath130 we represent the wavefunction by a row of circles , where each circle denotes the state @xmath131 of a particular lattice site @xmath4 . open circles in fig . [ diagramm ] denote a lattice site in its ground state ( gs ) , filled circles refer to an excited state ( es ) of this particular lattice site , with respect to the local mean - field hamiltonian . note that , in general , this can be an arbitrarily highly excited state ( although higher contributions are suppressed by the energy denominator , and a cutoff is used in practice ) . starting with a row of open circles , denoting the gs , @xmath116 , each following row corresponds to the state after the action of @xmath25 or @xmath132 , as indicated on the left side of the graph . as all matrix elements in eq . ( [ perta ] ) can be expressed in terms of a gs expectation value @xmath115 and a sequence of @xmath25 and @xmath132 operators , the first and last row must always be a line of open circles . let us consider for instance the second term in eq . ( [ firstorder ] ) @xmath133 reading the graph in fig . [ diagramm ] from top to bottom corresponds to reading the matrix element from right to left . starting with the gs , @xmath134 , the first row consists of open circles . the second row shows the state after the action of the perturbation @xmath25 . acting with @xmath135 to the right onto the gs results in a state latexmath:[\[v_{ij } |g_0\rangle = \sum_{\alpha,\beta } f_{\alpha,\beta } @xmath112 are neighboring lattice sites and @xmath137 denotes the state with lattice site @xmath4 ( @xmath112 ) in the exited state @xmath138 ( @xmath139 ) and all other sites in their gs . thus the second row shows the lattice sites @xmath4 and @xmath140 in an excited state ( filled circle ) , as the perturbation , eq . ( [ pert ] ) , allows only next neighbor interactions . finally , the action of @xmath132 has to bring the excited states back to the gs , in order to get a non - vanishing contribution . therefore , in first order pt , only next neighbor corrections to the correlation function arise , as the final row must represent the ground state @xmath141 again . the graph representing the remaining first term in eq . ( [ firstorder ] ) is obtained by rotating the graph in fig . [ diagramm ] by @xmath142 . and @xmath143 to the second - order correction of the density matrix . ( a ) terms representing contributions of the form eq . ( [ cpsione ] ) . ( b ) contributions of the form eq . ( [ cpsitwo ] ) . all notations are the same as in fig . [ diagramm ] . [ 2ndcpsi],width=302 ] rewriting the second - order corrections to the density matrix , @xmath144 , in terms of the operators @xmath145 and @xmath146 gives eq . ( [ atoc ] ) , but with @xmath147 replaced by @xmath148 . in contrast to the first - order corrections , now the terms proportional to @xmath149 and @xmath150 also give non - vanishing contributions . using eq . ( [ perta ] ) we obtain : @xmath151 here , the primed sums run over all neighbors @xmath93 to site @xmath112 . the corresponding subset of graphs for @xmath152 and @xmath153 are given by fig . [ 2ndcpsi]a and fig . [ 2ndcpsi]b for eq . ( [ cpsione ] ) and eq . ( [ cpsitwo ] ) respectively . note that all terms of eq . ( [ cpsione ] ) and eq . ( [ cpsitwo ] ) give a correction to all matrix elements of the density matrix independent of the distance between the lattice sites . we can understand these contributions as a modification to the mf value of the density matrix . graphs showing second - order corrections arising from @xmath154 . diagrams ( a ) and ( b ) are coming from direct - neighbor contributions as given by eq . ( [ 2ndccone ] ) and eq . ( [ 2ndcctwo ] ) respectively . diagrams ( c ) and ( d ) are next - nearest - neighbor contributions : ( c ) corresponds to eq . ( [ 2ndccthree ] ) and eq . ( [ 2ndccfour ] ) ; ( d ) corresponds to eq . ( [ 2ndccfive]).,width=264 ] * lattice site @xmath4 and @xmath112 being direct neighbors . in this case we get @xmath155 corrections for eq . ( [ 2ndccone ] ) and eq . ( [ 2ndcctwo ] ) are shown in fig . [ 2ndcc]a and fig . [ 2ndcc]b , respectively . * configurations corresponding to two lattice sites @xmath156 connected by two successive hopping steps via site @xmath93 . this gives rise to six contributions : + @xmath157 + an example for the contributions arising from eq . ( [ 2ndccthree ] ) and eq . ( [ 2ndccfour ] ) is shown in fig . [ 2ndcc]c . the last term , eq . ( [ 2ndccfive ] ) , has the representation shown in fig . [ 2ndcc]d . note that for lattices with dimensions @xmath41 , the sites @xmath4,@xmath112 and @xmath93 need not necessarily form a straight line but can form a chevron .

we study the mott - insulator transition of bosonic atoms in optical lattices . using perturbation theory , we analyze the deviations from the mean - field gutzwiller ansatz , which become appreciable for intermediate values of the ratio between hopping amplitude and interaction energy . we discuss corrections to number fluctuations , order parameter , and compressibility . in particular , we improve the description of the short - range correlations in the one - particle density matrix . these corrections are important for experimentally observed expansion patterns , both for bulk lattices and in a confining trap potential .

arxiv

cosmological models with scalar field matter have been much studied in the context of inflation and , more recently , in the context of the late - time acceleration that is indicated by current astronomical observations ( see @xcite for a recent review ) . one theoretical motivation for these studies is that scalar fields arise naturally from the compactification of higher - dimensional theories , such as string or m - theory . however , the type of scalar field potential obtained in these compactifications is sufficiently restrictive that until recently it was considered to be difficult to get accelerating cosmologies in this way , although the existence of an accelerating phase in a hyperbolic ( @xmath13 ) universe obtained by compactification had been noted @xcite , and non - perturbative effects in m - theory have since been shown to allow unstable de sitter vacua @xcite . in an earlier paper , we pointed out that compactification on a compact hyperbolic manifold with a time - dependent volume modulus naturally leads to a flat ( @xmath14 ) expanding universe that undergoes a transient period of accelerating expansion @xcite . numerous subsequent studies have shown that such cosmological solutions are typical to all compactifications that involve compact hyperbolic spaces or non - vanishing @xmath15-form field strengths ( flux ) @xcite , and this was additionally confirmed in a systematic study @xcite . furthermore , the transient acceleration in these models is easily understood @xcite in terms of the positive scalar field potential that both hyperbolic and flux compactifications produce in the effective , lower - dimensional , action . this perspective also makes clear the generic nature of transient acceleration . for any realistic application one would want the lower - dimensional spacetime to be four - dimensional , but for theoretical studies it is useful to consider a general @xmath16-dimensional spacetime . assuming that we have gravity , described by a metric @xmath17 , coupled to @xmath18 scalar fields @xmath19 taking values in a riemannian target space with metric @xmath20 and with potential energy @xmath1 , the effective action must take the form @xmath21 where @xmath22 is the ( spacetime ) ricci scalar . we are interested in solutions of the field equations of the action ( [ action ] ) for which the line element has the friedmann - lematre - robertson - walker ( flrw ) form for a homogeneous and isotropic spacetime . in standard coordinates , @xmath23 where the function @xmath24 is the scale factor , and @xmath25 represents the @xmath26-dimensional spatial sections of constant curvature @xmath27 . we normalise @xmath27 such that it may take values @xmath28 for a riemann tensor @xmath29 with metric @xmath30 on @xmath31 . the scalar fields are taken to depend only on time , which is the only choice compatible with the symmetries of flrw spacetimes . the universe is expanding if @xmath32 and accelerating if @xmath33 . we need only discuss expanding cosmologies because contracting cosmologies are obtained by time - reversal . in some simple cases , the target space has a one - dimensional factor parametrised by a dilaton field @xmath34 , and the potential takes the form @xmath35 for some ` dilaton coupling ' constant @xmath36 and constant @xmath37 . this model is of special interest , in part because of its amenability to analysis . the special case for which the dilaton is the _ only _ scalar field was analysed many years ago ( for @xmath38 ) using the observation that , for an appropriate choice of time variable , cosmological solutions correspond to trajectories in the ` phase - plane ' parametrised by the first time - derivatives of the dilaton and the scale factor @xcite . this method ( which has recently been extended to potentials that are the sum of two exponentials @xcite ) allows a simple visualisation of all possible cosmological trajectories . moreover , all trajectories for flat cosmologies can be found explicitly @xcite ( see also @xcite ) , and a related method allows a visualisation of their global nature @xcite . it was noted in @xcite that there is both a ` critical ' and a ` hypercritical ' value of the dilaton coupling constant @xmath36 , at which the set of trajectories undergoes a qualitative change . in spacetime dimension @xmath16 , these values are @xcite @xmath39 below the critical value ( @xmath40 ) there exists a late - time attractor universe undergoing accelerating expansion , whereas only transient acceleration is possible above it . the hypercritical coupling ( @xmath41 ) separates ` steep ' exponential potentials ( which arise in flux compactifications ) and ` gentle ' exponential potentials ( which arise from hyperbolic compactification , in which case @xmath42 so the potential is still too steep to allow eternal acceleration ) . one aim of this paper is to generalise this type of analysis to the multi - scalar case . for @xmath14 , this has been done already for what could be called a ` multi - dilaton ' model @xcite , and for the multi - scalar model with exponential potential ( [ eq.exppot ] ) @xcite . here we consider _ all _ cosmological trajectories ( arbitrary @xmath27 ) for an exponential potential of either sign , and for any spacetime dimension @xmath16 . in particular , we find exact solutions for all flat cosmologies when @xmath43 , following the method used in @xcite for @xmath44 , and the exact phase - plane trajectories for all @xmath27 when @xmath45 . a more ambitious aim of this paper is to determine what can be said about cosmologies with more general scalar potentials . what kind of model - independent behaviour can one expect , and how generic is the phenomenon of transient acceleration ? exponential potentials are simple partly because of the power - law attractor cosmologies that they permit , but such simple solutions do not occur for other potentials so other methods are needed . in this paper , we develop an alternative method of visualising cosmological solutions that applies to _ any _ potential , and we illustrate it by an application to exponential potentials . as we explain , exponential potentials serve as ` reference potentials ' in determining the late - time behaviour of cosmologies arising in a large class of models with other potentials . our starting point for the new formalism is the observation @xcite that flat cosmological solutions of gravity coupled to @xmath18 scalar fields with @xmath5 can be viewed as null geodesics in an ` augmented ' target space of dimension @xmath46 with a metric of lorentzian signature ( see @xcite for related results ) . trajectories in this space corresponding to non - flat cosmologies are neither null nor geodesic . however , we will see that they are projections of null geodesics in a ` doubly - augmented ' target space with a metric of signature @xmath47 for @xmath13 and signature @xmath48 for @xmath49 . it turns out that the extension of these results to @xmath50 is very simple . flat cosmologies are again geodesics in the augmented target space but with respect to a conformally rescaled metric ; the geodesic is timelike if @xmath44 and spacelike if @xmath43 . null geodesics are unaffected by the conformal rescaling and hence continue to correspond to ( @xmath14 ) cosmologies for @xmath5 . the analysis of acceleration is also simple in this framework : within the lightcone is an ` acceleration cone ' , and if a geodesic enters its acceleration cone then the corresponding universe will accelerate . in the case of a flat target space with an exponential potential , the fixed point geodesics are just straight lines , and the universe is accelerating if the straight line lies within the acceleration cone . the further extension to @xmath51 cosmologies is achieved exactly as in the @xmath5 case . cosmological trajectories are projections of geodesics in the doubly - augmented target space with respect to a conformally rescaled metric of signature @xmath47 for @xmath13 and @xmath48 for @xmath49 . the geodesic is timelike if @xmath44 , null if @xmath5 and spacelike if @xmath43 . the plan of this paper is as follows . section [ sec.mscalar ] reviews , in our conventions , the equations of motion in multi - scalar cosmologies , and we derive a useful alternative criterion for the acceleration of the scale factor . in section [ sec.fixedpoints ] , we consider possible fixed points , concluding that these occur only for exponential potentials , and we determine the nature of these fixed points for arbitrary @xmath16 , @xmath27 and sign of the potential . in section [ sec.phase ] we discuss the phase - plane trajectories and present some exact @xmath43 solutions . in section [ sec.manifold ] we develop the interpretation of multi - scalar cosmologies as geodesic motion . in section [ sec.applications ] we introduce the notion of the acceleration cone , and we show how the cosmologies arising in exponential - potential models fit into the new framework ; other potentials of current interest are also considered . we summarise in section [ sec.discussion ] . we begin by briefly reviewing the equations of motion for multi - scalar cosmologies that follow from the action ( [ action ] ) . these equations will be given in two different forms , distinguished by the choice of time coordinate . the second form will be useful to our analysis in the next section of possible ` fixed points ' , which correspond to power - law cosmologies in the first form . to simplify the equations we define @xmath52 where @xmath53 is the levi - civit connection for the target space metric @xmath54 , and we introduce the following ` characteristic functions ' of the potential v : @xmath55 these are the components of a 1-form on the target space dual to a vector field @xmath56 . in a vector notation , the scalar field equation can now be written as @xmath57 where @xmath58 is the hubble function . the friedmann constraint is @xmath59 = \left(d-1\right)\alpha_c^{-2}k,\ ] ] where the norm @xmath60 ( and hence the inner product ) is the one induced by the target space metric : @xmath61 . differentiating the friedmann constraint , and using the scalar field equation ( [ eq.sca ] ) , we deduce the ` acceleration equation ' @xmath62 is the ` critical ' constant given in ( [ exponents ] ) . clearly , acceleration is possible only if @xmath44 . note too that @xmath63 can not vanish unless either @xmath43 or @xmath64 ( or @xmath5 , @xmath14 and @xmath65 ) so only in these cases can an expanding universe recollapse . for the following section , it will prove useful to rewrite the above equations in terms of a new time coordinate @xmath66 , defined as a function of @xmath67 by the relation @xmath68 note that we have allowed for the possibility that @xmath6 , although @xmath5 is excluded . we will also set @xmath69 in terms of the variable @xmath70 , the condition @xmath71 for expansion is @xmath72 , while the condition @xmath73 for acceleration is @xmath74 where the overdot denotes differentiation with respect to @xmath75 . in the new time coordinate , the scalar field equation becomes @xmath76 and the friedmann constraint becomes @xmath77 = \left(d-1\right)\alpha_c^{-2 } k\,.\ ] ] the acceleration equation is now @xmath78 and the condition ( [ accel1 ] ) for acceleration is therefore equivalent to @xmath79 , and hence valid for all @xmath80 . as expected , it can be satisfied only if @xmath44 . we now wish to determine whether the system of equations ( [ newscalar ] , [ accelerationeq ] ) subject to the constraint ( [ newfried ] ) , admits any fixed point solutions for which @xmath81 we shall see that these conditions are consistent only for exponential potentials , for which the equations ( [ newscalar ] , [ accelerationeq ] ) become those of an autonomous dynamical system , and we determine the fixed points and their type for all @xmath16 , @xmath27 and for either sign of the potential . for this analysis , it is convenient to introduce the quantity @xmath82 recall that @xmath83 is the ` hypercritical ' constant given in ( [ exponents ] ) . given @xmath84 , equation ( [ newscalar ] ) yields @xmath85 where @xmath86 solves the quadratic equation @xmath87 given @xmath88 , equation ( [ accelerationeq ] ) yields @xmath89 \nonumber\\ & = & \left(d-2\right)c^2a^2 + { 1\over2}\left(d-1\right ) \left(d-2\right)\left[\dot\beta^2 - ca^2 \dot\beta \right ] \end{aligned}\ ] ] where we have used ( [ cterms ] ) to get to the second line . using this equation to eliminate the @xmath90 term from ( [ quadc ] ) we arrive at the quadratic equation @xmath91 there are therefore two types of fixed point : * @xmath92 . using this in ( [ quadc ] ) we deduce that @xmath93 and that @xmath94 this requires @xmath95 and @xmath96 it then follows that @xmath97 this is just the friedmann constraint for @xmath14 , so this type of fixed point occurs only for @xmath14 . * @xmath98 . using this in ( [ quadc ] ) we deduce that @xmath99 it follows that this type of fixed point can occur only for @xmath44 , and that @xmath100 using this in the friedmann constraint we deduce that @xmath101 we may exclude the possibility that @xmath102 because this yields a @xmath14 fixed point of the type already considered , so @xmath51 and we need @xmath64 for @xmath40 and @xmath103 for @xmath104 . note that @xmath105 for this type of fixed point , so the fixed point cosmology is neither accelerating nor decelerating . the above results are in complete analogy with those of @xcite for the one - scalar case . to see why , it should first be appreciated that the fixed point conditions have been derived on the assumption that @xmath106 is constant , and that @xmath107 is covariantly constant . it then follows from ( [ cterms ] , [ quadc ] ) that @xmath56 must be covariantly constant too . however , for any space that admits a non - zero covariantly constant vector , there exist coordinates in which this vector is _ constant _ , not just covariantly constant . for coordinates in which @xmath56 is constant the potential takes the form @xmath108 which is of the form ( [ eq.exppot ] ) with @xmath109 . moreover , for any target space on which this potential is globally defined , one can find new coordinates such that @xmath110 where @xmath111 is a metric on a ` reduced ' target space with @xmath112 coordinates @xmath113 . in these target space coordinates , the scalar field equations are @xmath114 and the friedmann constraint is @xmath115 - \left(d-1\right)\dot\beta^2 = k\left(d-1\right)e^{2\left(a\varphi-\beta\right ) } |v_0|^{-1}\ , .\ ] ] these imply the acceleration equation for @xmath116 . if we suppose that the reduced target space is flat then the scalar field equations , together with the acceleration equation , define an autonomous dynamical system : @xmath117 - \dot\beta^2 + a\dot\beta \dot\varphi\ , , \nonumber\\ \ddot\chi & = & a\dot\varphi\ , \dot\chi - \left(d-1\right)\dot\beta\ , \dot\chi\ , . \end{aligned}\ ] ] note that we may consistently set @xmath118 to recover the equations of the one - scalar model . as there are no fixed points with @xmath119 , the fixed points of the multi - scalar model are the same as those of the one - scalar model . if the reduced target space is not flat then a connection term must be added to the last equation in ( [ autonomous ] ) . as this introduces dependence on @xmath113 , the equations no longer define an autonomous dynamical system in just @xmath46 variables . it is not clear how much difference this makes to the results : note that the connection term has no effect on the stability of fixed points with @xmath118 because it is quadratic in @xmath120 . however , we will suppose for the analysis to follow that the reduced target space _ is _ flat . we may then assume with no essential loss of generality that it is also one - dimensional ( i.e. @xmath121 ) so we have an autonomous dynamical system for just three variables . we linearise about a fixed point with @xmath122 and @xmath123 by writing @xmath124 the linearised equations for @xmath125 take the form @xmath126 , where @xmath127 is the @xmath128 matrix @xmath129 the eigenvalues of this matrix determine the nature of the fixed point . we consider the two types of fixed point in turn : * @xmath14 . in this case @xmath130 the eigenvalues of @xmath127 are @xmath131 for an expanding universe we must take the top sign ( corresponding , for @xmath44 , to the upper branch of the @xmath14 hyperboloid ) . then we have a stable node ( all eigenvalues real and negative ) for @xmath40 . for @xmath132 we have a saddle , with one real - negative and two real - positive eigenvalues ; the instability due to the positive eigenvalues is what leads to the ` flatness problem ' of standard big - bang cosmology . + for @xmath43 we may have @xmath133 , in which case the fixed point is an unstable node ( all eigenvalues real and positive ) . all @xmath13 trajectories start at this unstable fixed point and end at the stable fixed point on the other branch of the @xmath14 hyperboloid ; as @xmath134 at this other fixed point , all @xmath13 universes recollapse to a big crunch singularity . however , for @xmath135 there is a long quasi - static period on some of these trajectories . * @xmath51 . in this case @xmath136 the eigenvalues of @xmath127 are @xmath137 , \qquad \mp 2a/\alpha_c\,,\ ] ] where @xmath138 for one eigenvalue and @xmath139 for another . take the top sign again . for @xmath140 all eigenvalues are real but one is positive and the others negative , so we have a saddle point . for @xmath42 all eigenvalues are real and negative provided _ either _ @xmath141 _ or _ ( if @xmath142 ) @xmath143 otherwise ( i.e. @xmath142 and @xmath144 ) we have one real negative eigenvalue and two complex eigenvalues with negative real parts , and hence a stable ` spiral - node ' ( a spiral in the @xmath145 plane and a node in the @xmath146 direction ) . note that @xmath147 where the equality occurs for @xmath148 . thus , for any @xmath16 the @xmath51 fixed point is an unstable saddle if @xmath40 , and is stable for @xmath42 but can be either a node or a spiral - node . for @xmath141 it is always a node , but for @xmath148 it becomes a spiral - node at @xmath41 and for @xmath149 it becomes a spiral - node at @xmath150 . in the supergravity context we are restricted to @xmath151 , but there is no scalar potential possible in @xmath152 and the only possible potential in @xmath153 is a positive exponential ( with @xmath154 so there is no fixed - point solution ) . we now have all the information needed to determine the qualitative behaviour of all phase - space trajectories . we begin with the one - scalar model , obtained by setting @xmath155 . the trajectories for @xmath44 are sketched in @xcite but , for the reader s convenience , we give a brief summary in words . all @xmath14 trajectories begin , after a big bang singularity , with a period of kinetic energy domination with effectively vanishing potential . for @xmath133 they end up approaching a similar late - time phase , but for @xmath156 they approach the @xmath14 fixed - point solution , which is accelerating if @xmath40 and decelerating if @xmath42 . for @xmath40 , all @xmath51 trajectories sufficiently close to a @xmath14 trajectory also approach the @xmath14 fixed point ; in fact , all @xmath13 trajectories have this property . the behaviour of a @xmath49 trajectory is determined by its relation to the separatrix of the unstable @xmath49 fixed point ( which is the einstein static universe for @xmath157 ) ; those on one side approach the @xmath14 fixed point while those on the other side represent universes that recollapse to a big crunch . for @xmath42 _ all _ @xmath49 universes recollapse to a big crunch whereas all @xmath13 universes now approach a zero - acceleration milne universe ( the @xmath14 fixed point now being a saddle point ) . for @xmath144 the @xmath13 fixed point is an attractor spiral so that all @xmath13 trajectories spiral around a point of zero - acceleration ; as observed in @xcite , this leads to eternal oscillation between acceleration and deceleration . for @xmath43 the @xmath14 trajectories were studied in @xcite but there seems to be no complete study of all trajectories for this case . we will therefore present some results of the one - scalar model for @xmath43 before moving on to discuss modifications that arise in the multi - scalar case . it is well known that anti - de sitter space is a solution of einstein s equations with a negative cosmological constant . this space must correspond to one of the cosmological trajectories in a model with @xmath45 ( which implies that @xmath157 ) . set @xmath158 to get the equations @xmath159\ ] ] where @xmath160 these equations imply that @xmath161 = cx^{2/(d-1)}\ ] ] for constant @xmath162 . the curves with @xmath163 are the @xmath49 trajectories and those with @xmath164 are the @xmath13 trajectories . the two branches of the @xmath165 curve , a hyperbola , are the @xmath14 trajectories . the ads solution corresponds to @xmath166 , and hence @xmath167 ; for this special case we have @xmath168\ ] ] which has the solution @xmath169 , and hence @xmath170 the friedmann constraint implies that @xmath13 , as expected , and @xmath171,\ ] ] and hence @xmath172 this is anti - de sitter space . in contrast to de sitter space , which can be realised ( for @xmath173 ) as a flrw universe for _ any _ @xmath27 ( in particular as the @xmath14 fixed - point solution ) , anti - de sitter space is realised as an flrw cosmology only for @xmath13 , and it does not correspond to a fixed point cosmology ( at least in the sense of this paper ) . we now turn to the @xmath14 trajectories for arbitrary @xmath36 . in this case we have to solve @xmath174 subject to the constraint @xmath175 the constraint is solved by setting @xmath176 and the equation of motion becomes @xmath177.\ ] ] this equation is immediately integrated if @xmath41 . having found @xmath178 we integrate ( [ xydef ] ) to deduce that @xmath179^{1\over 2\left(d-1\right)}\,,\ ] ] for integration constant @xmath180 ( we have set to zero the other integration constant as it can be absorbed into @xmath37 ) . if @xmath93 then we introduce the new variable @xmath181 the two branches of the @xmath14 hyperbola correspond to positive and negative @xmath182 ; we take @xmath183 . we now consider separately the cases @xmath133 and @xmath156 : * @xmath156 . in this case the equation for @xmath182 is @xmath184,\ ] ] where we recall that @xmath185 is the function of @xmath36 given in ( [ omegadef ] ) . integration of ( [ xydef ] ) now leads to @xmath186^{{\alpha_c^2\over 2\alpha_h\omega^2}}\ , . \end{aligned}\ ] ] * @xmath154 . in this case the equation for @xmath182 is @xmath187 there is a fixed point at @xmath188 , so we must distinguish between @xmath189 and @xmath190 . one finds that @xmath191 for @xmath189 integration of ( [ xydef ] ) yields @xmath192^{\alpha_c^2\over 2\alpha_h\omega^2}\ , . \end{aligned}\ ] ] for @xmath190 integration of ( [ xydef ] ) yields @xmath193^{\alpha_c^2\over 2\alpha_h\omega^2}\ , . \end{aligned}\ ] ] in the multi - scalar case , the ` spectator ' fields @xmath113 have an effect on the scale factor due to the @xmath194 term in the acceleration equation , and in the friedmann constraint ( [ friedchi ] ) . phase space trajectories with non - zero @xmath120 will depend on the reduced target space metric but , as long as there are no new fixed points , one may expect the qualitative behaviour to be independent of this metric . let us therefore continue to suppose that the target space ( and hence the reduced target space ) is flat . in this case , all @xmath14 trajectories lie in a hyperboloid in phase space that separates the @xmath13 and @xmath49 trajectories . this is true for either sign of @xmath195 but we will now suppose that @xmath44 ; in this case , we see from ( [ eq.accpot ] ) that acceleration occurs whenever the phase - plane trajectory enters the region with hyperspherical boundary @xmath196 . this was observed in @xcite in the context of an analysis of @xmath14 trajectories , but it is also valid for non - zero @xmath27 . the only exponential models with eternally accelerating cosmological solutions are those for which a fixed point lies inside or on the sphere of acceleration , as happens when @xmath140 . for all other values of @xmath197 , any acceleration can be at most transient , and only a subset of the trajectories undergo even transient acceleration ; these are the trajectories that pass through the sphere of acceleration . the more scalar fields there are , the more freedom there is to avoid the sphere of acceleration , so transient acceleration in multi - scalar models is less generic than it is in the single - scalar model . to make this statement quantitative would require an understanding , along the lines of @xcite , of what the appropriate measure might be on the space of trajectories . the most significant fact about a @xmath14 fixed point is whether it lies inside or outside the sphere of acceleration . figure [ fig.k0psdia ] illustrates a few examples in a model with two scalar fields and a positive exponential , scalar potential . we now develop a method to describe the generic evolution of multi - scalar cosmologies . it will be convenient to define a new independent variable @xmath198 by @xmath199 although we will still use @xmath70 where this is more convenient . the friedmann constraint can now be written as @xmath200 and the scalar field equation as @xmath201 these two equations imply the acceleration equation @xmath202 = 0\,.\ ] ] using the friedmann constraint to eliminate @xmath203 from the scalar and acceleration equations we have @xmath204 & = & k\,\alpha_c^{-2}\left(d-1\right)e^{-2\beta}\ , \mathbf{a}\ , , \nonumber\\ \partial_t^2\gamma + \alpha_h \left|\partial_t{{\bm{\phi}}}\right|^2 & = & k\,\alpha_c^{-2}\alpha_h e^{-2\beta}\ , . \label{accgamma}\end{aligned}\ ] ] before considering the general case ( arbitrary @xmath27 and arbitrary @xmath195 ) , we first discuss the special cases of @xmath5 ( but arbitrary @xmath27 ) and @xmath14 ( but arbitrary @xmath195 ) . we may consider the @xmath205 variables @xmath206 to be maps from the cosmological trajectory to an ` augmented ' target space . in this notation , the friedmann constraint for @xmath5 can be written as @xmath207 where @xmath208 is a _ lorentz - signature _ metric on the augmented target space . using the friedmann constraint to eliminate the @xmath209 term in the second equation in ( [ accgamma ] ) yields @xmath210 if this is taken together with the scalar field equation then the two combine into the single ( albeit coordinate - dependent ) equation @xmath211 for @xmath14 this is the equation for a geodesic in a non - affine parametrisation . in terms of the the new time variable @xmath212 defined by @xmath213 we have @xmath214 for @xmath14 this is the equation of an affinely parametrised geodesic . although the cosmological trajectory in the augmented target space is not a geodesic for @xmath51 , it can be viewed as the projection of a geodesic in a ` doubly - augmented ' target space of dimension @xmath215 that is foliated by hypersurfaces isometric to the augmented target space . let @xmath216 be the @xmath215 coordinates . we take the metric to be @xmath217 where @xmath218 as we now have one more variable we also need another equation . we take this to be the ` projection equation ' @xmath219 with this choice , the friedmann constraint becomes @xmath220 and the combined scalar field and acceleration equations ( [ knonzero ] ) are equivalent to the single geodesic equation @xmath221 thus , cosmological trajectories for @xmath5 are null geodesics in the doubly - augmented target space . note that the signature of this space is lorentzian if @xmath103 but non - lorentzian ( with two ` times ' ) if @xmath64 . because @xmath222 is not constant , the motion is not restricted to a single hypersurface of constant @xmath222 and this accounts for the fact that the projection of the motion onto one such hypersurface is not geodesic . this argument fails for @xmath14 , for which the projected motion _ is _ geodesic , because the metric on the doubly - augmented target space is degenerate when @xmath14 . the friedmann constraint for @xmath14 , but arbitrary @xmath195 , can be written as @xmath223 thus , the cosmological trajectory in this space is timelike for @xmath44 , null for @xmath5 and spacelike for @xmath43 . as we have seen , it is a null _ geodesic _ when @xmath5 . when @xmath50 the cosmological trajectories are no longer geodesic with respect to the metric @xmath224 , but they _ are _ geodesics with respect to the conformally - rescaled metric @xmath225 where the conformal factor is @xmath226 in fact , one finds that the equations ( [ basiceqs ] ) are equivalent , for @xmath14 , to the equation @xmath227 where @xmath228 is the levi - civit connection for @xmath229 , and when @xmath5 as @xmath230 is undefined in this case . ] @xmath231 this is the equation for a geodesic in a non - affine parametrisation ; noting that @xmath232 we deduce that the geodesics are affinely parametrised by a new time - coordinate @xmath233 for which @xmath234 in other words , @xmath235 the friedmann constraint now takes the form @xmath236 notice that @xmath233 differs from @xmath212 as defined ( for @xmath5 ) by ( [ tprime ] ) . in fact , @xmath237 this difference occurs because the affine parameter of a null geodesic is affected by a conformal rescaling of the metric , and the metric we are now considering is the conformally rescaled one @xmath238 ; clearly a null curve that is geodesic with respect to @xmath239 is also geodesic with respect to any other conformally equivalent metric , such as @xmath238 , but the affine parametrisation will differ in general . in contrast , when @xmath50 the cosmological trajectories in the augmented target space are geodesics with respect to a unique metric ( up to homothety ) in the class of metrics that are conformally equivalent to @xmath239 , and this metric is @xmath238 . the steps that led to ( [ affinegeo ] ) for @xmath14 lead , for general @xmath27 , to the equation @xmath240 although this is not the equation of a geodesic when @xmath51 , it is the projection of a geodesic in the doubly - augmented target space with respect to the conformally rescaled metric @xmath241 where the conformal factor @xmath242 is as given in ( [ conformalfac ] ) , and @xmath243 is as given in ( [ doubleaug ] ) . from ( [ phistar ] ) and ( [ ttprime ] ) we deduce that @xmath244 given this , one then finds that ( [ general ] ) is equivalent to the geodesic equation @xmath245 and that the friedmann constraint is @xmath246 thus , the general cosmological trajectory is a geodesic in the doubly - augmented target space , one that is timelike for @xmath44 , null for @xmath5 and spacelike for @xmath43 . we now consider some applications of the formalism just developed . as we are particularly interested in accelerating cosmologies we first consider the implications of the acceleration condition . we will then see how to interpret the fixed point cosmologies that arise for exponential potentials . this turns out to be useful when considering the asymptotic behaviour of more general potentials . we first show how the condition for acceleration ( [ eq.accpot ] ) acquires a geometrical meaning in the new framework . we will now assume that @xmath44 , as acceleration can not otherwise occur , and before proceeding we note that the friedmann constraint ( [ newfried ] ) can be written as @xmath247 we now introduce the following ` acceleration ' metric on the augmented target space , @xmath248 and the corresponding conformally rescaled metric @xmath249 where the conformal factor is the same as before . noting that @xmath250 one finds that @xmath251 using the friedmann constraint in the form ( [ friedform ] ) to eliminate the @xmath252 term , we deduce that @xmath253 + { k\over 2\alpha_c^{2}v}\ , e^{-2\beta}\,.\ ] ] recalling that acceleration occurs when @xmath254 , we see that the condition for acceleration is equivalent , for @xmath14 , to @xmath255 geometrically , this states that a universe corresponding to a timelike geodesic trajectory is accelerating when its tangent vector lies within a subcone of the lightcone defined by the acceleration metric on the augmented target space . as one might expect , an analogous result holds for @xmath51 in terms of the doubly - augmented target space . in this case , the acceleration metric is @xmath256 . \end{aligned}\ ] ] using the projection equation ( [ phistarhat ] ) one finds that @xmath257 , \ ] ] where @xmath258 is the conformally rescaled acceleration metric . the condition for acceleration is therefore equivalent to @xmath259 this states that a universe is accelerating when the tangent to its geodesic trajectory in the doubly - augmented target space lies with an acceleration subcone of the lightcone . we now discuss how the cosmologies arising for an exponential potential of the form ( [ exppot2 ] ) fit into the new geometrical framework . for reasons given earlier we assume in this case that the target space is flat , in which case the augmented target space metric @xmath239 is also flat , and hence @xmath238 is conformally flat , with conformal factor @xmath260.\ ] ] our first task is to determine the trajectories in the augmented target space that are associated with the fixed point solutions . consider first the @xmath14 fixed point . this solution has the property that the linear function @xmath261 is time - independent : @xmath262 . in other words , @xmath263 is a vector tangent to the hyperplanes of constant @xmath264 . for @xmath265 there is only one such direction , which is determined by the gradient of @xmath242 . thus , @xmath266 for @xmath265 . in fact , this remains true for arbitrary @xmath18 as a direct calculation shows : @xmath267 a fixed point solution for @xmath51 has the property that the linear function @xmath268 is time - independent . as surfaces of constant @xmath269 are hyperplanes , a @xmath51 fixed point trajectory is _ also _ a straight line , as a direct computation confirms : @xmath270 thus , fixed point solutions are ( particular ) straight - line trajectories in the augmented target space . in addition , as follows from our earlier results , the @xmath14 straight - line trajectories are geodesics with respect to the metric @xmath271 , whereas this is not true for @xmath51 . this difference can be understood as follows : as we have assumed a flat target space , the metric @xmath239 is flat and _ all _ straight lines are geodesics with respect to it . however , the relevant metric is @xmath238 , and the only geodesic of @xmath239 that is also a geodesic of @xmath238 is the line of steepest descent of the function @xmath242 ; as we have seen , this is precisely the direction of the @xmath14 fixed point trajectory . note that the line determined by ( [ direction.nonzerok ] ) is a generator of the acceleration cone ( as expected from the zero - acceleration property of the @xmath51 fixed point ) . more generally , if a straight line trajectory lies within the acceleration cone then it corresponds to an eternally accelerating universe , and if it lies outside the acceleration cone ( @xmath42 ) then it corresponds to an eternally decelerating universe . in the latter case it might also lie outside the ` lightcone ' ( @xmath133 ) , in which case the geodesic corresponds to the @xmath14 fixed point for @xmath43 . note that a change in the value of the constant vector @xmath56 effects a lorentz transformation of the @xmath14 straight - line trajectory in the augmented target space . such a transformation can take any timelike line into any other timelike line , and any spacelike line into any other spacelike line , but it can not take a spacelike line to a timelike one or vice versa ( as expected from the fact that this requires a change of sign of the potential ) . also , it can not take a timelike or spacelike line into a null line , a fact that is consistent with the absence of a fixed - point solution for @xmath41 . generic geodesics , corresponding to generic @xmath14 cosmologies , are not straight lines in the extended target space , but they still have a simple description . the regime where the solutions are dominated by kinetic energy is given by geodesics with tangent vectors having large @xmath272 . this means that the generic geodesics start out in null directions , as could be anticipated from the fact that @xmath14 cosmologies are null geodesics when @xmath5 . let us now follow the subsequent evolution for the @xmath44 case . as the potential becomes more important , the geodesics bend into the timelike cone , ultimately approaching the timelike straight line corresponding to the fixed point solution if @xmath156 . on the other hand , if @xmath273 , then the geodesic ultimately approaches a null straight - line geodesic . for the single - scalar case , this behaviour is shown in figure [ fig.k0geodesics ] . cosmologies accelerate precisely when their corresponding geodesics bend their tangent vectors into the acceleration cone . for @xmath51 , cosmological trajectories are no longer geodesics in the augmented target space but they are projections of geodesics in the doubly - augmented target space . the vector tangent to the geodesic corresponding to a @xmath51 fixed point is @xmath274 where we have used ( [ phistarhat ] ) to get the last entry . note that this is _ not _ a constant @xmath215-vector ( even allowing for the fact that @xmath275 is constant at the fixed point ) so the fixed point solution is not a straight line in the doubly - augmented target space , as was to be expected because its metric is neither flat nor conformally flat . however , using ( [ fixedcurve ] ) , and then ( [ keq ] ) , one finds that @xmath276 = 0\,.\ ] ] this again confirms the zero acceleration of the @xmath51 fixed point cosmologies . the asymptotic ( late - time ) behaviour of a large class of cosmologies in models with rather general potentials can be determined by comparison with the asymptotic behaviour of cosmologies in models with exponential potentials . to demonstrate this , we discuss the case of flat cosmologies with positive potentials . in this case , the scalar fields must approach @xmath277 at late times , where some components of the constant @xmath18-vector @xmath278 may be infinite . the characteristic functions of the potential in this limit are also constant : @xmath279 the absolute value @xmath280 determines the late - time behaviour , by comparison with the critical and hypercritical exponents : if @xmath281 , then there will be late - time acceleration , with the trajectory approaching an approximate accelerating attractor ; obvious examples are models for which @xmath195 has a strictly positive lower bound , so most models of interest will be those for which @xmath195 either tends to zero or has a minimum at zero . for @xmath282 , there is late - time deceleration corresponding to the existence of an approximate late - time decelerating attractor . the asymptotic geodesics in both these cases are timelike . in contrast , for @xmath283 the timelike geodesic asymptotes a null ( and hence decelerating ) trajectory . to illustrate the above observations , we consider the following single - scalar examples . * inverse power law potentials : @xmath284 these potentials have been much studied in quintessence models ( see e.g. @xcite ) . they also arise from dynamical breaking of supersymmetry @xcite . with @xmath285 , one finds @xmath286 . this implies late - time acceleration , _ even though the potential tends to zero at late - times_. * that any other value of @xmath287 may occur is illustrated by the potential @xmath288 for which @xmath289 if we suppose that @xmath290 at late times . for such cosmologies the asymptotic behaviour is the same as it would be in the exponential potential model with dilaton coupling constant @xmath36 . * it may happen that at late times the scalar fields become trapped near a minimum of the potential . near such a minimum , which we may assume to occur at the origin of field space , the potential takes the form @xmath291 for integer @xmath292 . for positive @xmath293 one finds @xmath294 which tends to zero ( i.e. @xmath286 ) as @xmath295 for any @xmath296 . this implies late - time acceleration , due to the effective cosmological constant @xmath293 . however , we may now consider the limit @xmath297 , in which case @xmath298 , implying that the late - time trajectory is null , and hence decelerating . in this paper we have developed a geometrical method for the classification and visualisation of homogeneous isotropic cosmologies in multi - scalar models with an arbitrary scalar potential @xmath195 . the method involves two steps . in the first step , the target space parametrised by the @xmath18 scalar fields is augmented to a larger @xmath205-dimensional space of lorentzian signature , the logarithm of the scale factor playing the role of time . this is reminiscent of the role of the scale factor in mini - superspace models of quantum cosmology , especially if one views the scalar fields of our model as moduli - fields for extra dimensions . in this case , the lorentzian metric on the augmented target space is the one induced from the wheeler - dewitt metric on the space of higher - dimensional metrics . this was the point of view adopted in @xcite , where it was also observed that when @xmath5 all flat ( @xmath14 ) cosmologies are null geodesics in this metric ( an observation that goes back to work of dewitt on kasner metrics @xcite ) . when @xmath50 , flat cosmologies again correspond to trajectories in the augmented target space but they are neither null nor geodesic . however , as we have shown , _ all _ flat cosmologies are geodesics with respect to a _ conformally rescaled _ metric on the augmented target space , the conformal factor depending both on the scale factor and the potential . the conformal rescaling has no effect on null geodesics , of course , so flat cosmologies are still null geodesics whenever @xmath5 , but now the trajectories of flat cosmologies are timelike geodesics when @xmath44 and spacelike geodesics when @xmath43 . this is true not only for a potential of fixed sign but also for one that changes sign ; in this case the tangent to the geodesic becomes null just as the scalar fields take values for which @xmath5 . the second step , which is needed for non - flat ( @xmath299 ) cosmologies , is to further enlarge the target space to a ` doubly - augmented ' target space foliated by copies of the augmented target space . one can choose the metric on this space , of signature @xmath47 for @xmath13 and @xmath48 for @xmath300 and degenerate for @xmath14 , such that geodesics yield ( on projection ) cosmological trajectories for _ any _ @xmath27 when projected onto a given hypersurface , and the geodesics are again timelike , null or spacelike according to whether @xmath195 is positive , zero or negative . this general construction thus includes all the previous ones as special cases . accelerating cosmologies have a simple interpretation in this new framework . within the lightcone ( defined by the metric with respect to which the trajectories are geodesics ) there is an acceleration subcone . a given trajectory corresponds to an accelerating universe whenever its tangent vector lies within the acceleration cone . in particular , a flat cosmology undergoes acceleration whenever its geodesic trajectory bends into the acceleration cone within the lightcone of the augmented target space . this can happen only if the trajectory is timelike , which it will be if the potential is positive . we have also presented in this paper a complete treatment ( complementing many previous studies ) of cosmological trajectories , for any @xmath27 and any spacetime dimension @xmath16 , for the special case of simple exponential potentials . these potentials are characterised by a sign ( the potential may be positive or negative ) and a ( dilaton ) coupling constant @xmath36 ( the magnitude of an @xmath18-vector coupling constant @xmath56 ) . as has long been appreciated @xcite , cosmological solutions in such models correspond , for an appropriate choice of time parameter , to trajectories of an autonomous dynamical system in a ` phase space ' parametrised by the time - derivatives of the scale factor and scalar fields . the qualitative features of these trajectories ( which should not be confused with trajectories in the ` augmented target space ' parametrized by the fields themselves ) are determined by the position and nature of any fixed points , which are of two types . there is always a @xmath14 fixed point unless @xmath41 ( the ` hypercritical ' value of @xmath36 defined by this property ) but it occurs only for @xmath44 if @xmath133 and only for @xmath43 if @xmath133 . there is also a @xmath51 fixed point if @xmath44 , which coincides with the @xmath14 fixed point when @xmath102 , where @xmath301 is the ` critical ' value of @xmath36 ( defined as the value at which fixed point cosmologies have zero acceleration ) . these fixed point cosmologies have a very simple interpretation as trajectories in the augmented target space : they are straight lines . in the @xmath14 case these lines are also geodesics . for @xmath43 these geodesic lines lie outside the lightcone . for @xmath44 they lie inside the lightcone , but may lie either inside or outside the acceleration cone . for models obtained by ( classical ) compactification from a higher dimensional theory without a scalar field potential , the @xmath14 geodesic line always lies outside the acceleration cone , so only transient acceleration is possible in these models . for a single scalar field the transient acceleration is generic in the sense that it is a feature of all @xmath14 trajectories when @xmath133 and of some when @xmath156 . in contrast , in the multi - scalar case , there are trajectories that correspond to eternally decelerating universes even when @xmath133 , so transient acceleration is less generic for more than one scalar field . although exponential potentials are of limited phenomenological value , they are important as ` reference potentials ' in determining the asymptotic behaviour of cosmologies in models with other potentials , such as inverse power potentials . essentially , any potential that falls to zero slower than the critical exponential potential ( as do inverse power potentials ) will lead to late - time eternal acceleration . any potential that falls to zero faster than the hypercritical exponential potential will have a late - time behaviour that is approximately that of a model with zero potential . of course , there is no evidence for the existence of cosmological scalar fields , but there _ is _ evidence that the expansion of the universe is accelerating and hence for dark energy . whatever produces this energy , it seems reasonable to suppose that it can be modeled by scalar fields . if so , it is possible that future observations may be interpreted as telling us something about the potential energy of these fields . in view of our current complete ignorance of what this potential might be , we have tried , as much as possible , to understand generic properties , and we hope that the methods developed here will be of further use in this respect . the interpretation of solutions of gravitational theories as geodesics in a suitable metric space has a considerable history of which we were mostly unaware at the time of writing this paper . the geodesic interpretation of @xmath14 cosmologies that we have presented is closely related to the maupertuis - jacobi principle of classical mechanics . relatively recent work on this topic includes @xcite . a particle mechanics formulation of the geodesic interpretation described here , including our proposed extension to @xmath51 , has recently been found , and applied to a model in which cosmological singularities correspond to horizons in the augmented target space @xcite .

for gravity coupled to @xmath0 scalar fields , with arbitrary potential @xmath1 , it is shown that all flat ( homogeneous and isotropic ) cosmologies correspond to geodesics in an @xmath2-dimensional ` augmented ' target space of lorentzian signature @xmath3 , timelike if @xmath4 , null if @xmath5 and spacelike if @xmath6 . accelerating cosmologies correspond to timelike geodesics that lie within an ` acceleration subcone ' of the ` lightcone ' . non - flat ( @xmath7 ) cosmologies are shown to evolve as projections of geodesic motion in a space of dimension @xmath8 , of signature @xmath9 for @xmath10 and signature @xmath11 for @xmath12 . this formalism is illustrated by cosmological solutions of models with an exponential potential , which are comprehensively analysed ; the late - time behaviour for other potentials of current interest is deduced by comparison .

arxiv

in the course of an extensive search for very wide , common proper motion companions to nearby hipparcos stars from the nomad catalog @xcite , we came across an optically faint and red star at an angular separation of @xmath5 , position angle @xmath6 from the active and rapidly rotating g9v star hip 115147 ( bd + 78 826 , hd 220140 , v368 cep ) . this faint star was subsequently identified with lspm j2322 + 7847 , a candidate nearby low - mass dwarf detected by @xcite from the lspm - north catalog @xcite . the original identification of this star as one with significant proper motion traces to luyten ( 1967 ) where it was reported as a magnitude 17 object with @xmath7 mas yr@xmath8 at a position angle of @xmath9 and assigned the name lp12 - 90 . in the present paper we present new @xmath2 photometry of hip 115147 , its known visual companion hip 115147b and lspm j2322 + 7847 , and obtain preliminary trigonometric parallax astrometry for the latter companion . then we discuss the possible young age and origin of this interesting triple system . following the identification of lspm j2322 + 7847 as a potential widely separated common proper motion companion to hip 115147 , @xmath2 photometry on the cousins system was obtained for the fainter star on ut 30 august 2005 using the 1.0-meter reflector at the flagstaff station . the photometry was calibrated with an instrumental zero point term and a first order airmass term . the calibration field was the standard field pg2213 - 006 from @xcite . additional photometric observations were subsequently obtained on ut 16 and 17 june 2007 when the individual components of the brighter system hip 115147ab ( sep . 11 " ) were also measured . the photometric results are presented in table 1 along with @xmath10 photometry extracted from 2mass , proper motions from nomad and parallax determinations . the estimated uncertainties in the @xmath2 measures are @xmath11 mag in the case of lspm j2322 + 7847 and @xmath12 - 0.04 mag for the hip 115147 components where the short exposure times introduced additional error from scintillation and shutter timing . since the august 2005 photometry indicated that lspm j2322 + 7847 was most likely an m - dwarf at approximately the same distance as hip 115147 , it was added to the trigonometric parallax program at usno s flagstaff station . through june 2007 , a total of 66 acceptable ccd observations have been accumulated on this field , covering an epoch range of 1.65 years . the same tek2k ccd , observational procedures , and reduction algorithms have been employed as summarized in @xcite . using a total of 29 reference stars , the current preliminary solution yields @xmath13 mas . this solution appears to be very robust , with the separate solutions for parallax in the ra and dec directions in very satisfactory agreement ( @xmath14 mas versus @xmath15 mas , respectively ) . correction to absolute parallax was performed using usno @xmath16 photometry for the individual reference stars along with a calibrated @xmath17 versus @xmath18 relationship to derive a mean photometric parallax of @xmath19 mas for the 29 star ensemble . together this then translates to @xmath20 mas for lspm j2322 + 7847 . the star hip 115147 was identified with the bright x - ray source h2311 + 77 , which led @xcite to suggest rs cvn - type activity . it was shown later on that the star is not evolved , and that it is not a short - period spectroscopic binary , justifying the currently accepted classification as a very young , naked " post - t tauri dwarf @xcite . being one of the most powerful x - ray emitters in the solar neighborhood @xcite at @xmath21 ergs s@xmath8 @xcite , the star has the same space velocity as the local young stream ( or local association , @xcite ) . this association limits its age to @xmath22 myr . this stream includes isolated stars , groups and associations of diverse ages , some as young as 1 myr ( e.g. , in lupus and ophiuchus ) . therefore , the assignment to this stream by itself does not lend a more precise estimation of age . an x - ray luminosity of @xmath23 is typical of weak - lined tt stars in taurus - auriga - perseus , but significantly larger than that of classical tt stars ; if anything , it points at an age older than a few myr . hip 115147 is listed as variable star v368 cep @xcite . the slight variability allowed @xcite to determine the period of rotation of this star , 2.74 d. the fast rotation is responsible for the high degree of chromospheric and coronal activity . the primary star is identified as the extreme ultraviolet source euve j@xmath24 with strong detections at @xmath25 band , as well as at 250 ev in x - rays @xcite . hip 115147 is one of the 181 extreme - ultraviolet sources in the rosat wfc all - sky bright source survey identified with late - type main - sequence stars @xcite , with high signal - to noise detections in both @xmath26-@xmath27 and @xmath28-@xmath29 passbands . an unusually high level of chromospheric activity of @xmath30 was determined by @xcite ; a spectral type k2v is also specified in the latter paper as opposed to g9v given in the simbad database . since the rate of rotation diminishes fairly quickly with age in single stars , so does x - ray luminosity , and open clusters older than @xmath31 persei ( 50 myr ) are usually more quiescent than the youngest ones ( ic 2602 and ic 2391 ) . the high degree of chromospheric and extreme ultraviolet activity suggests a very young age , possibly less than 20 myr . v368 cep is more powerful in x - rays than an average k - type pleiades member by a factor of 10 , indicating an age less than 100 myr . finally , the equivalent width of at @xmath32 is smaller than the upper limit for the pleiades by @xmath33 according to @xcite , which points at an age similar to the pleiades or older . however , the lithium surface content is a poor estimator of age for young stars because of the large intrinsic spread of this parameter in stars of the same mass and age . we are left with the most reliable and frequently used method of age estimation by model isochrones . using the photometric data from the literature and our own ( table 1 ) , we plot hr diagrams in the 2mass @xmath34 and @xmath3 , and @xmath35 passbands in figs . [ jk.fig ] and [ vk.fig ] . for reference , the theoretical isochrones at @xmath36 , @xmath37 and @xmath38 myr are drawn from @xcite . according to the data in @xcite derived from strmgren _ uvby@xmath39 _ photometry , the star hip 115147 has a markedly subsolar metallicity at [ fe / h]@xmath40 . this determination has to be taken with caution , because the photometrically derived metallicities are sensitive to the color index @xmath41 ( b. nordstrm 2007 , priv . comm . ) , which may be abnormally high for chromospherically active stars @xcite . a wider range of metal abundances is supplied in the evolution models of @xcite , and we present also a 25 myr isochrone for @xmath42 from their models in fig . [ jk.fig ] with a dashed line . we find that the siess et al . 50 myr isochrone ( @xmath43 ) provides the best fit for all three alleged comoving components ( marked with crossed circles ) , including the secondary m dwarf hip 115147 b , separated by @xmath44 from the primary @xcite . this pair is listed in the wds @xcite as wds 23194@xmath45 ( lds 2035 , discovered by luyten in 1969 ) , at separation @xmath46 and position angle @xmath47 . the relative position did not change significantly between the first ac2000.2 measures taken in 1901 and the latest epoch 2000 in 2mass ( w. hartkopf & g. wycoff 2007 , priv . comm . ) , confirming that the inner pair is physical . the small deviation of the faintest component lspm j2322 + 7847 from the siess 50 myr isochrone appears to arise mostly from a sudden twist of the isochrone at the latest data point corresponding to mass @xmath48 . we are not in a position to discuss if this twist has a certain physical meaning , but the overall match is good , and the estimated mass of the star is roughly @xmath48 . the younger isochrones for 16 and 7 myr from @xcite lie significantly higher in fig . [ jk.fig ] than the matching 50 myr isochrone . the 25 myr isochrone from @xcite at a lower metallicity provides a poor fit to our stars , predicting much bluer colors , but these models may be strongly biased for young ages as was found by @xcite for solar - type stars in the alpha persei open cluster at 52 myr . the members of the very young twa association are shown in fig . [ jk.fig ] with open circles . these stars are certainly much younger than hip 115147 and its companions . they appear to be brighter and redder than the 7 myr isochrone corresponding to the probable age of this group ( see also * ? ? ? * ) , but this deviation may be the result of the near - infrared @xmath49 and @xmath34 excess commonly observed in classical t tauri stars and attributed to a hot inner rim in their dusty accretion disks @xcite . since our stars under investigation have no accretion disks and their metal abundance may be lower , we need to find a young star with similar parameters . it was pointed out by @xcite that a few young stars currently in the solar neighborhood , that traveled from the vicinity of the ophiuchus star forming region , share the moderately poor metal abundance of hip 115147 as determined from the _ uvby@xmath39 _ photometry , probably affected by the high degree of chromospheric activity . one of these stars is the extremely active dwarf pz tel , a likely member of the @xmath39 pic associations estimated to be 12 - 20 myr old . this star is indicated with an open square in fig . [ jk.fig ] . its position matches the 16 myr , @xmath43 isochrone quite well . pz tel is significantly brighter than hip 115147 in @xmath3 having approximately the same color ; thus , the latter star and its companions are probably older than 16 myr . it is verified in fig . [ vk.fig ] that this difference in @xmath50 between the two stars is not originating in a @xmath3-band excess , since both hip 115147 and lspm j2322 + 7847 match best the 52 myr , @xmath43 isochrone in a @xmath50 versus @xmath51 diagram as well . the latter star is brighter in @xmath3 than the empirically determined main sequence for field dwarfs from @xcite by 1 mag . if this star were an unrelated old m6 dwarf , its distance would be only 12 pc . our trigonometric parallax ( table 1 ) yields a distance @xmath52 pc . a number of stars with outstanding signs of activity and young age scattered in the solar vicinity can be traced back to their places of origin in , or close to , the ob associations in the sco - cen complex @xcite . the star hip 115147 has come from the vicinity of the molecular cloud ldn 1709 in the ophiuchus star forming region , flanking the sco - cen complex at @xmath53 @xcite . the closest approach to the estimated center of that association is 10.7 pc , while the separation today is 170 pc . the closest approach 16 myr ago saw the star flying by at a relative velocity of 10.8 km s@xmath8 . if hip 115147 was born in the ophiuchus association , its probable age should be 16 myr , considerably younger than the previous isochrone analysis and the lithium abundance suggest . because of the high departing velocity , it is more likely that this close fly - by is a chance occurrence . finally , we note that the difference in proper motion between hip 115147 and lspm j2322 + 7847 , which appears to be statistically significant , may be caused by a systematic error of 5 to 10 mas yr@xmath8 in the nomad proper motions for very faint stars . the formal errors specified in nomad may be strongly underestimated for this optically dim star , whose position at the mean epoch 1979 is based on schmidt plates . the proper motion difference , if it is true , implies that the two stars are not gravitationally bound as a truly multiple system . this is commonly observed among very young stars in the solar neighborhood ; for example , the common proper motion pair at mic and au mic ( approximately 10 myr old ) are separated by at least 0.23 pc and have proper motions different by @xmath54% , but their physical association is not in doubt . at the suggested age ( 20 - 50 myr ) , the hip 115147 companions are likely to stay close to each other as a moving group . more accurate absolute proper motions for both companions and spectroscopic radial velocities are required to clarify the dynamical status of the system . the currently available evidence is consistent with lspm j2322 + 7847 being one of the youngest , later - type stars in the near solar neighborhood . we thank t. tilleman and c. dahn for taking some of the parallax images . the research described in this paper was in part carried out at the jet propulsion laboratory , california institute of technology , under a contract with the national aeronautics and space administration . this research has made use of the simbad database , operated at cds , strasbourg , france ; and data products from the 2mass , which is a joint project of the university of massachusetts and the infrared processing and analysis center , california technology institute , funded by nasa and the nsf . ra j2000 & 23 19 26.632 & 23 19 24.53 & 23 22 53.873 + dec . j2000 & + 79 00 12.67 & + 79 00 03.8 & + 78 47 38.81 + @xmath55 & @xmath1 & & @xmath0 + proper motion & @xmath56 & @xmath57 & @xmath58 + @xmath59 & 8.60 & 13.75 & 17.99 + @xmath35 & 7.73 & 12.24 & 16.16 + @xmath60 & 7.17 & 11.04 & 14.61 + @xmath61 & 6.76 & 9.54 & 12.54 + @xmath34 & 5.90 & 8.04 & 10.42 + @xmath62 & 5.51 & 7.39 & 9.84 + @xmath3 & 5.40 & 7.20 & 9.52 +

we report a late m - type , common proper motion companion to a nearby young visual binary hip 115147 ( v368 cep ) , separated by 963 arcseconds from the primary k0 dwarf . this optically dim star has been identified as a candidate high proper motion , nearby dwarf lspm j2322 + 7847 by lpine in 2005 . the wide companion is one of the latest post - t tauri low mass stars found within 20 pc . we obtain a trigonometric parallax of @xmath0 mas , in good agreement with the hipparcos parallax of the primary star ( @xmath1 mas ) . our @xmath2 photometric data and near - infrared data from 2mass are consistent with lspm j2322 + 7847 being brighter by 1 magnitude in @xmath3 than field m dwarfs at @xmath4 , which indicates its pre - main sequence status . we conclude that the most likely age of the primary hip 115147 and its 11-arcsecond companion hip 115147b is 20 - 50 myr . the primary appears to be older than its close analog pz tel ( age 12 - 20 myr ) and members of the twa association ( 7 myr ) . [ firstpage ]

arxiv

in recent years , many cosmological observations have provided the strong evidence that the universe is flat and its energy density is dominated by the dark energy component whose negative pressure causes the cosmic expansion to accelerate @xcite@xmath0@xcite . in order to clarify the origin of the dark energy , one has tried to understand the connection of the dark energy with particle physics . in a dynamical model proposed by fardon , nelson and weiner ( mavans ) , relic neutrinos could form a negative pressure fluid and cause the cosmic acceleration @xcite . in this model , an unknown scalar field which is called `` acceleron '' is introduced and neutrinos are assumed to interact through a new scalar force . the acceleron sits at the instantaneous minimum of its potential , and the cosmic expansion only modulates this minimum through changes in the neutrino density . therefore , the neutrino mass is given by the acceleron , in other words , it depends on its number density and changes with the evolution of the universe . the equation of state parameter @xmath1 and the dark energy density also evolve with the neutrino mass . those evolutions depend on a model of the scalar potential and the relation between the acceleron and the neutrino mass strongly . typical examples of the potential have been discussed in ref . @xcite . the variable neutrino mass was considered at first in ref . @xcite , and was discussed for neutrino clouds @xcite . ref . @xcite considered coupling a sterile neutrino to a slowly rolling scalar field which was responsible for the dark energy . @xcite considered coupling of the dark energy scalar , such as the quintessence to neutrinos and discuss its impact on the neutrino mass limits from baryogenesis . in the context of the mavans scenario , there have been a lot of works @xcite@xmath0@xcite . the origin of the scalar potential for the acceleron was not discussed in many literatures , however , that is clear in the supersymmetric mavans scenario @xcite . in this work , we present a model including the supersymmetry breaking effect mediated by the gravity . then we show evolutions of the neutrino mass and the equation of state parameter in the model . the paper is organized as follows : in section ii , we summarize the supersymmetric mavans scenario and present a model . sec . iii is devoted to a discussion of the supersymmetry breaking effect mediated by the gravity in the dark sector . in sec . iv , the summary is given . in this section , we discuss the supersymmetric mass varying neutrinos scenario and present a model . the basic assumption of the mavans with supersymmetry is to introduce a chiral superfield @xmath2 in the dark sector , which is assumed to be a singlet under the gauge group of the standard model . it is difficult to build a viable mavans model without fine - tunings in some parameters when one assumes one chiral superfield in the dark sector , which couples to only the left - handed lepton doublet superfield @xcite . therefore , we assume that the superfield @xmath2 couples to both the left - handed lepton doublet superfield @xmath3 and the right - handed neutrino superfield @xmath4 . in this framework , we suppose the superpotential @xmath5 where @xmath6 are coupling constant of @xmath7 and @xmath8 , @xmath9 , @xmath10 and @xmath11 are mass parameters . the scalar and spinor component of @xmath2 are @xmath12 , and the scalar component is assumed to be the acceleron which cause the present cosmic acceleration . the spinor component is a sterile neutrino . the helium-4 abundancy gives the most accurate determination of a cosmological number of neutrinos and does not exclude a fourth thermalized neutrino at @xmath13 @xcite . the third term of the right - hand side in eq . ( [ w ] ) is derived from the yukawa coupling such as @xmath14 with @xmath15 , where @xmath16 is the higgs doublet . in the mavans scenario , the dark energy is assumed to be the sum of the neutrino energy density and the scalar potential for the acceleron : @xmath17 since only the acceleron potential contributes to the dark energy , we assume the vanishing vacuum expectation values of sleptons , and thus the effective scalar potential is given as @xmath18 we can write down a lagrangian density from eq . ( [ w ] ) , @xmath19 it is noticed that the lepton number conservation in the dark sector is violated because this lagrangian includes both @xmath20 and @xmath21 . after integrating out the right - handed neutrino , the effective neutrino mass matrix is given by @xmath22 in the basis of @xmath23 , where we assume @xmath24 . the first term of the @xmath25 element of this matrix corresponds to the usual term given by the seesaw mechanism @xcite in the absence of the acceleron . the second term is derived from the coupling between the acceleron and the right - handed neutrino but the magnitude of this term is negligible small because of the suppression of @xmath26 . therefore , we can rewrite the neutrino mass matrix as @xmath27 where @xmath28 . it is remarked that only the mass of a sterile neutrino is variable in the case of the vanishing mixing ( @xmath29 ) between the left - handed and a sterile neutrino on cosmological time scale . the finite mixing ( @xmath30 ) makes the mass of the left - handed neutrino variable . we will consider these two cases of @xmath29 and @xmath30 later . in the mavans scenario , there are two constraints on the scalar potential . the first one comes from observations of the universe , which is that the magnitude of the present dark energy density is about @xmath31 , @xmath32 being the critical density . thus , the first constraint turns to @xmath33 where `` @xmath34 '' represents a value at the present epoch . the second one is the stationary condition . in this scenario , the neutrino mass is assumed to be a dynamical field which is a function of the acceleron . therefore , the dark energy density should be stationary with respect to the variation of the neutrino mass : @xmath35 if @xmath36 , this condition is equivalent to the usual stationary condition stabilized by an ordinary scalar field . ( [ stationary ] ) is rewritten by using the cosmic temperature @xmath37 : @xmath38 where @xmath39 , @xmath40 and @xmath41 we can get the time evolution of the neutrino mass from eq . ( [ stationary1 ] ) . since the stationary condition should be always satisfied in the evolution of the universe , this one at the present epoch is the second constraint on the scalar potential : @xmath42 in addition to two constraints for the potential , we also have two relations between the acceleron and the neutrino mass at the present epoch : @xmath43 ^ 2 + 4m_d^2 } } { 2},\\ m_\psi ^0&=&\frac{c+m_a+\lambda _ 1\phi ^0}{2}\nonumber\\ & & -\frac{\sqrt{[c-(m_a+\lambda _ 1\phi ^0)]^2 + 4m_d^2}}{2}. \end{aligned}\ ] ] next , we will consider the dynamics of the acceleron field . in order that the acceleron does not vary significantly on distance of inter - neutrino spacing , the acceleron mass at the present epoch must be less than @xmath44 @xcite . here and below , we fix the present acceleron mass as @xmath45 once we adjust parameters which satisfy five equations ( [ v ] ) and ( [ stationary2])@xmath0([amass ] ) , we can have evolutions of the neutrino mass by using the eq . ( [ stationary1 ] ) . the dark energy is characterized by the evolution of the equation of state parameter @xmath1 . the equation of state in this scenario is derived from the energy conservation equation in the robertson - walker background and the stationary condition eq . ( [ stationary1 ] ) : @xmath46\rho _ \nu}{3\rho _ { \mbox{{\scriptsize de } } } } , \end{aligned}\ ] ] where @xmath47 it seems that @xmath1 in this scenario depend on the neutrino mass and the cosmic temperature . this means that @xmath1 varies with the evolution of the universe unlike the cosmological constant . in the last of this section , it is important to discuss the hydrodynamic stability of the dark energy from mavans . the speed of sound squared in the neutrino - acceleron fluid is given by @xmath48 where @xmath49 is the pressure of the dark energy . recently , it was argued that when neutrinos are non - relativistic , this speed of sound squared becomes negative in this scenario @xcite . the emergence of an imaginary speed of sound shows that the mavans scenario with non - relativistic neutrinos is unstable , and thus the fluid in this scenario can not acts as the dark energy . however , finite temperature effects provide a positive contribution to the speed of sound squared and avoid this instability @xcite . then , a model should satisfy the following condition , @xmath50 where @xmath51 is the redshift parameter , @xmath52 , and @xmath53 actually , some models satisfy this condition @xcite . in order to consider the effect of supersymmetry breaking in the dark sector , we assume a superfield @xmath54 , which breaks supersymmetry , in the hidden sector , and the chiral superfield @xmath2 in the dark sector is assumed to interact with the hidden sector only through the gravity . this framework is shown graphically in fig . [ fig:0 ] . once supersymmetry is broken at tev scale , its effect is transmitted to the dark sector through the following operators : @xmath55 where @xmath56 is the planck mass . then , the scale of the soft terms @xmath57-@xmath58 is expected . such a framework was discussed in the `` acceleressence '' scenario @xcite . now , we consider only one superfield which breaks supersymmetry for simplicity . if one extend the hidden sector , one can consider a different mediation mechanism between the standard model and the hidden sector from one between the dark and the hidden sector . we will return to this point later . in this framework , taking supersymmetry breaking effect into account , the scalar potential is given by @xmath59 where @xmath60 and @xmath61 are supersymmetry breaking parameters , and @xmath62 is a constant determined by the condition that the cosmological constant is vanishing at the true minimum of the acceleron potential . this scalar potential is the same one presented in @xcite . we consider two types of the neutrino mass matrix in this scalar potential . they are the cases of the vanishing and the finite mixing between the left - handed and a sterile neutrino . when the mixing between the left - handed and a sterile neutrino is vanishing , @xmath29 in the neutrino mass matrix ( [ mm ] ) . then we have the mass of the left - handed and a sterile neutrino as @xmath63 in this case , we find that only the mass of a sterile neutrino is variable on cosmological time scale due to the second term in eq . ( [ sterilem ] ) . let us adjust parameters which satisfy eqs . ( [ v ] ) and ( [ stationary2])@xmath0([amass ] ) . ( [ v ] ) , the scalar potential eq . ( [ v1 ] ) is used . putting typical values for four parameters by hand as follows : @xmath64 we have @xmath65 there is a tuning between @xmath8 and @xmath61 in order to satisfy the constraint on the present accerelon mass of eq . ( [ amass ] ) . ) ] now , we can calculate evolutions of the mass of a sterile neutrino and the equation of state parameter @xmath1 by using these values . the numerical results are shown in figs . [ fig:2 ] , [ fig:3 ] and [ fig:4 ] . especially , the behavior of the mass of a neutrino near the present epoch is shown in fig . [ fig:3 ] . we find that the mass of a sterile neutrino have varied slowly in this epoch . this means that the first term of the left hand side in eq . ( [ constraint ] ) , which is a negative contribution to the speed of sound squared , is tiny . we can also check the positive speed of sound squared in a numerical calculation . therefore , the neutrino - acceleron fluid is hydrodynamically stable and acts as the dark energy . ) ] ( @xmath66 ) ] next , we consider the case of the finite mixing between the left - handed and a sterile neutrino ( @xmath30 ) . in this case , the left - handed and a sterile neutrino mass are given by @xmath67 ^ 2 + 4m_d^2 } } { 2},\\ m_\psi&=&\frac{c+m_a+\lambda _ 1\phi } { 2}\nonumber\\ & & -\frac{\sqrt{[c-(m_a+\lambda _ 1\phi ) ] ^2 + 4m_d^2}}{2}. \end{aligned}\ ] ] we can expect that both the mass of the left - handed and a sterile neutrino have varied on cosmological time scale due to the term of the acceleron dependence . taking typical values for four parameters as @xmath68 we have @xmath69 where we required that the mixing between the active and a sterile neutrino is tiny . in our model , the small present value of the acceleron is needed to satisfy the constraints on the scalar potential in eqs . ( [ v ] ) and ( [ stationary2 ] ) . ) ] ) ] ( @xmath66 ) ] when we add following terms of the supersymmetry breaking effects to the scalar potential , @xmath70 a small gravitino mass ( @xmath71 ) is favored . such a small gravitino mass has been given in the gauge mediation model of ref.@xcite . therefore , a mediation mechanism between the standard and the hidden sector which leads to a small gravitino mass is suitable for our framework . values of parameters in ( [ value1 ] ) are almost same as the case of the vanishing mixing ( [ value ] ) . however , the mass of the left - handed neutrino is variable unlike the vanishing mixing case . the time evolution of the left - handed neutrino mass is shown in fig . . the mixing does not affect the evolution of a sterile neutrino mass and the equation of state parameter , which are shown in figs . [ fig:6 ] , [ fig:7 ] . since the variation in the mass of the left - handed neutrino is not vanishing but extremely small , the model can also avoid the instability of speed of sound . the value of the sum of the left - handed and a sterile neutrino mass is within the limit provided by analyses from wmap three - years date @xcite . finally , we comment the smallness of the evolution of the neutrino mass at the present epoch . in our model , the mass of the left - handed and a sterile neutrino include the constant part . a variable part is a function of the acceleron . in the present epoch , the constant part dominates the neutrino mass because the present value of the acceleron should be small . this smallness of the value of the acceleron is required from the cosmological observation and the stationary condition in eqs . ( [ v ] ) and ( [ stationary2 ] ) . we presented a supersymmetric mavans model including the effects of the supersymmetry breaking mediated by the gravity . evolutions of the neutrino mass and the equation of state parameter have been calculated in the model . our model has a chiral superfield in the dark sector , whose scalar component causes the present cosmic acceleration , and the right - handed neutrino superfield . in our framework , supersymmetry is broken in the hidden sector at tev scale and the effect is assumed to be transmitted to the dark sector only through the gravity . then , the scale of soft parameters are @xmath72-@xmath73 is expected . we considered two types of model . one is the case of the vanishing mixing between the left - handed and a sterile neutrino . another one is the finite mixing case . in the case of the vanishing mixing , only the mass of a sterile neutrino had varied on cosmological time scale . in the epoch of @xmath74 , the sterile neutrino mass had varied slowly . this means that the speed of sound squared in the neutrino acceleron fluid is positive , and thus this fluid can act as the dark energy . in the finite mixing case , the mass of the left - handed neutrino had also varied . however , the variation is extremely small and the effect of the mixing does not almost affect the evolution of the sterile neutrino mass and the equation of state . therefore , this model can also avoid the instability . t. yanagida , in proceedings of the _ `` workshop on the unified theory and the baryon number in the universe '' _ , tsukuba , japan , 1979 , edited by o. sawada and a. sugamoto , kek report no . kek-79 - 18 , p. 95 ; prog . * 64 * ( 1980 ) 1103 ;

we discuss the effect of the supersymmetry breaking on the mass varying neutrinos(mavans ) scenario . especially , the effect mediated by the gravitational interaction between the hidden sector and the dark energy sector is studied . a model including a chiral superfield in the dark sector and the right handed neutrino superfield is proposed . evolutions of the neutrino mass and the equation of state parameter are presented in the model . it is remarked that only the mass of a sterile neutrino is variable in the case of the vanishing mixing between the left - handed and a sterile neutrino on cosmological time scale . the finite mixing makes the mass of the left - handed neutrino variable .

arxiv

the _ quark gluon plasma _ ( @xmath0 ) usually is defined as the phase of _ quantum chromodynamics _ ( @xmath2 ) in which the quarks and gluons degrees of freedom , that is normally confined within the hadrons , are mostly liberated . the possible phases of the @xmath2 and the precise locations of critical boundaries or points are currently being actively studied . in fact , revealing the @xmath2 phase transition structure is one of the central aims of the ongoing and the future theoretical and the experimental research in the field of the hot and/or the dense @xmath2 @xcite . it is about thirty years since the study of the hot and the dense nuclear matter in the form of the @xmath0 has been started . the experiments at the @xmath3 s _ super proton synchrotron _ ( @xmath4 ) first tried to create the @xmath0 in the 1980s and 1990s : the first hints of the formation of a new state of matter was obtained from the @xmath4 data in terms of the global observable , the event - by - event fluctuations , the direct photons and the di - leptons . the current experiments at the @xmath5 national laboratory s _ relativistic heavy ion collider _ ( @xmath6 ) are still continuing these efforts . in the april 2005 , the formation of the quark matter was tentatively confirmed by the results obtained at the @xmath6 . the consensus of the four @xmath6 research groups was in favor of the creation of the quark - gluon liquid at the very low viscosity @xcite . since , in this new phase of matter , the quarks and the gluons are in the asymptotic freedom region , one expects that they interact weakly . so , the _ perturbative _ methods can be used for such a system . the asymptotic freedom suggests two procedures for the creation of the @xmath0 : @xmath7i@xmath8 the recipe for the @xmath0 at high temperature . if one treats the quarks and the gluons as the massless noninteracting gas of molecules , such that the baryon density vanishes , the critical temperature which above that , the hadronic system dissolves into a system of quarks and gluons ( @xmath0 ) is @xmath9 @xcite . however , the modern lattice @xmath2 calculation estimates the critical temperature , @xmath10 , to be about @xmath11 @xmath12 @xcite @xmath7ii@xmath8 the recipe for the @xmath0 at high baryon density . at zero temperature , the critical baryon density required the transition to take place in @xmath13 @xcite i.e. four times the empirical nuclear matter density . on these grounds , one should expect to find the @xmath0 in two places in the nature : firstly , in the early universe , about @xmath14 after the _ cosmic big bang _ or secondly , at the core of _ super - dense stars _ such as the neutron and the quark stars . this new phase of matter can also be created in the initial stage of the _ little big bang _ by means of the relativistic nucleus - nucleus collisions in the heavy ion accelerators @xcite . the critical temperature ( critical baryon density ) at the zero baryon density ( zero temperature ) has been obtained simply for the non - interacting , massless up and down quarks and gluons in the reference @xcite . some primary works in the zero baryon density and in the framework of the bag model have been also presented in the reference @xcite ( and the reference therein ) . for the more complicated field theoretical approaches see the references @xcite and the references therein . so it would be interesting to perform a similar calculation to reference @xcite , but with the non zero interaction , for the region in which both @xmath15 and @xmath16 . so , in this work we generalize above calculations for the region with the finite temperature and the baryon density by considering the weakly interacting quarks in the framework of the _ one gluon exchange _ scheme . since it is intended to use the @xmath17 method in our calculation , it is assumed that @xmath18 . the dependence of @xmath19 on the temperature and the baryon density is ignored @xcite . we perform calculations beyond the zero hadronic pressure approximation of reference @xcite and the pressure of the excluded volume hadron @xcite model is also taken into account to find the corresponding @xmath0 transition phase diagram . so , the paper is organized as follows . in the section @xmath20 we review the derivation of the _ one gluon exchange _ interaction formulas @xcite based on the landau fermi liquid picture @xcite . the section @xmath21 is devoted to the calculation of the thermodynamics properties of the @xmath0 . the result and its comparison with the strange quark matter @xcite are given in the section @xmath22 . finally a conclusion and summary are presented in the section @xmath23 . the landau fermi liquid theory describes the relativistic systems such as : the nuclear matter under the extreme conditions , the quark matter , the quark gluon plasma , and other relativistic plasmas . the basic framework of the landau theory of relativistic fermi liquids is given by @xmath24 and @xmath25 @xcite . in this framework , one can evaluate the energy density of a weakly interacting quarks in @xmath0 by the following formulas : @xmath26 with @xmath27 where @xmath28 , is the degeneracy of quarks . @xmath29 , @xmath30 and @xmath31 are the kinetic and the potential energies and the familiar fermi - dirac liquid distributions of our quasi - particles , respectively . in the equation ( [ tatsumi ] ) , @xmath32 is the landau - fermi interaction function , which is a criterion for the interaction between the two quarks ( the `` @xmath33 '' superscript refers to the unpolarized quark matter @xcite ) . it is related to the two - particle forward scattering amplitude i.e. : @xmath34 @xmath35 is the commonly defined lorentz invariant matrix element . now , using the _ feynman rules _ for the @xmath2 , @xmath36 can be calculated . since , the direct term is proportional to the trace of the _ gelman matrices _ , the color symmetric matrix element is only given by exchange contribution : @xmath37 where @xmath38 ( our choice is @xmath39 ) . since our system is unpolarized , it is possible to sum over all the spin states and get the average of this quantity as : @xmath40 then for massless up and down quarks , @xmath41 so , the total energy of interacting quarks can be evaluated by using the equations ( 1 ) , ( 2 ) , ( 3 ) and ( 7 ) . in the next section , the various thermodynamic quantities of the interacting @xmath0 like the internal energy , the pressure , the entropy , etc are calculated . we begin with calculation of the free ultra - relativistic massless quarks partition function density , which follows by its non - interacting gluon contribution s . since we are interested in the heavy ion processes , it is assumed that the initial state to be the nuclear matter with the equal neutron and proton densities i.e. @xmath42 . so , @xmath43 and the contributions of @xmath44 and @xmath45 quarks are equal . then write the @xmath46 partition function for each quarks is written as ( @xmath47 is the volume ) , @xmath48\frac{4\pi g_{q}}{(2\pi)^{3 } } k^{2 } dk.\ ] ] this integral can be evaluated analytically i.e. , @xmath49 where @xmath50 and @xmath51 ( in the @xmath52 factor unit ) . and the strange quark matter , respectively.,width=415 ] and the strange quark matter , respectively.,width=415 ] and the strange quark matter , respectively.,width=415 ] and the strange quark matter , respectively.,width=415 ] equation of state as a function of baryon density at two fixed temperatures . the solid and dash curves are for the weakly interacting @xmath0 and the strange quark matter , respectively , but without the bag constant.,width=415 ] .,width=415 ] by taking into account the nuclear matter excluded volume pressure at four different bag pressures ( @xmath12).,width=415 ] now we are in the position to evaluate the various thermodynamic quantities , such as the energy density , the entropy and the free energy from the above partition function , especially we have , @xmath53_{\cal t } & & \nonumber\\ & & \hspace{-44 mm } = \frac{4\pi g_{q}}{3(2\pi)^{3 } } ( \pi^{2 } { \cal t}^{2 } \mu+\mu^{3}),\end{aligned}\ ] ] where @xmath54 is the quark degeneracy for two , up and down quarks , flavors . all of the thermodynamic quantities are obtained as a function of chemical potential ( @xmath55 ) and temperature ( @xmath56 ) . the temperature is a suitable experimental quantity but the chemical potential is not . so , it is better to rewrite the thermodynamic quantities as a function of the temperature and the baryon density , instead of the chemical potential . since the baryon number of a quark is @xmath57 , we have , @xmath58 , and : @xmath59 then , the chemical potential can be found as a function of @xmath60 and @xmath61 , from the above equation i.e. , @xmath62 so our thermodynamic quantities become a function of baryon density and temperature . for the energy density of gluons we have , @xmath63 where again @xmath64 is the degeneracy of gluons . our system is ultra - relativistic , so , there is a simple relation between the pressure and the energy density : @xmath65 the bag pressure and the vacuum energy contributions should be included , in addition to the quark and the gluon contribution energies , @xmath66 having the pressure , the entropy density of system is evaluated , @xmath67 to study the phases of the @xmath0 , one should concentrate on the @xmath0 and the hadronic pressures , @xmath68 and @xmath69 @xcite , respectively , i.e. for @xmath70 , the system is in the confinement phase and the quarks and the gluons are inside the bag , but for @xmath71 , the quarks and the gluons pressures can overcome to the bag and the hadronic pressures and the system is in the de - confinement phase , i.e. the @xmath0 phase is created . so , the phase diagram can be extracted by solving the following equation : @xmath72 where in this work , both @xmath73 @xcite and @xmath74 @xcite cases are considered . for @xmath74 , the excluded volume effect for the nuclear matter equation of state is used @xcite to calculate the hadronic pressure ( equation ( 29 ) of reference @xcite ) i.e. , @xmath75},\label{n2}\ ] ] where @xmath76 is the pressure of free ideal nucleonic ( fermion ) matter @xcite ( here , degeneracy is 4 , @xmath77 , @xmath78 and @xmath79 , e.g see the equation ( 9 ) of the reference @xcite ) . now , the interaction energy between the quarks can be calculated by using the landau fermi liquid model and the results derived by us in the previous section . the potential energy density of the interacting massless up or down quarks is found by using the equations ( 3 ) and ( 7 ) , @xmath80 where @xmath81 is the quark degeneracy of one flavor . by using the fermi - dirac distribution and the value of each quark flavor potential energy i.e. the equation ( 21 ) , an analytical formula for the total potential energy density of the two quark flavours is written as follows : @xmath82 note that , the potential energy for massless quarks is always positive . so , the interaction between quarks inside the bag is repulsive and it helps the interacting quarks and the gluons to penetrate from the bag more easily , rather than the noninteracting case , and further more , the _ one gluon exchange _ interaction , because of its repulsive properties , makes the conditions easier for the system to make the transition to the @xmath0 phase . the internal energy density of the @xmath0 is evaluated by performing the summation over the interacting and the non - interacting parts of the energy density of quarks , the vacuum energy and the gluon energy density which were calculated before , and having that , the other thermodynamic quantities of the @xmath0 are found . as it was pointed out before , it is assumed that the bag ( hadronic ) pressure is the one has been used in the reference @xcite i.e. @xmath83@xmath84 ( zero ) , in order to compare our phase diagram with this reference , and since it is intended to compare our results with the reference @xcite , the @xmath2 coupling constant os chosen to be @xmath85 . on the other hand , for the none zero hadronic pressure , the baryonic chemical potential and the bag pressure are varied to reach to the critical temperature of the lattice @xmath2 predictions @xcite , i.e. @xmath86 @xmath12 . we begin by presenting the calculated free energy per baryon for both the @xmath0 and the strange quark matter @xcite in the figure 1 ( 2 ) , as a function of baryon density ( temperature ) at the two different temperatures ( baryon densities ) . the free energy for the @xmath0 is larger than those of strange quark matter , since we know that the strange quark matter should be more stable than the @xmath0 and therefore the strange quark matter free energy should be smaller than that of the @xmath0 . as one should expect , the free energy increases ( decreases ) by increasing the baryon density ( temperature ) . while , the @xmath0 has less temperature dependent with respect to the strange quark matter , they have similar density dependent at fixed temperature . similar comparisons are made for the entropies in the figures 3 and 4 . the entropy per baryon for the @xmath0 is an increasing ( decreasing ) function of temperature ( baryon density ) and it is smaller than that of the strange quark matter @xcite . again their dependence on the density is the same , but they behave especially differently at larger temperatures . the plots of the equation of states of both the @xmath0 ( without the effect of constant bag pressure ) and the strange quark matter as a function of baryon density at the two different temperatures are given in the figure 5 . the @xmath0 equation of state is much harder than that of strange quark matter at the same baryon density and temperature . the pressure of weakly interacting @xmath0 for the two different @xmath2 coupling constants , and the noninteracting @xmath0 ( without the effects of constant bag pressure ) at zero temperature as a function of baryon density is shown in the figure 6 . the increase in the interaction strength makes the pressure to rise , and therefore at the smaller baryon densities , the pressure of quarks becomes equal to the bag pressure . so , the interaction facilitates the quarks transition to the deconfined phase at lower density . the @xmath2 coupling constant also plays the same role , i.e it will reduce the transition density . finally , the phase diagrams for both the interacting and the noninteracting @xmath0 are shown in the figure 7 for @xmath87 . the _ one gluon exchange _ interaction which is repulsive , causes to get the @xmath0 at the smaller baryon densities and temperatures . as it was pointed out before , the reason is very simple , the repulsive interaction between quarks helps them to escape from the bags . so the formation of the @xmath0 happens much easier for the interacting quarks than the noninteracting one . but , the critical temperature is about @xmath88 , which is much less than the lattice @xmath2 suggestion of @xmath89 . in the figure 8 , the hadronic pressure has been also taken into the account ( see the equations ( [ n1 ] ) and the reference [ n2 ] ) . the slashed area is the forbidden region , i.e. for the bag pressure approximately larger than @xmath90 there is no critical temperature with the zero baryonic chemical potential ( density ) . with the bag pressure about @xmath91 ( note that the bag pressure estimated to as large as @xmath92 @xmath12 @xcite ) and the low baryonic chemical potential ( density ) it is possible to get results near to the lattice @xmath2 prediction , i.e. @xmath86 @xmath12 . in conclusion , the _ one gluon exchange _ interaction was used to evaluate the strength of potential energy of the @xmath0 in the fermi liquid model . by calculating the @xmath0 partition function , the different thermodynamic properties of the @xmath0 as a function of baryon density and temperature for the both interacting and the non - interacting cases were discussed . it was found that the @xmath0 internal and free energies are much larger than those of strange quark matter . on the other side , if we consider the massive quarks like the strange quarks in our the @xmath0 , the potential energy becomes a negative quantity , but for the massless quarks it is always a positive quantity , therefore the internal and the free energy densities of strange quark matter become smaller than the @xmath0 ones . we have seen how the _ one gluon exchange _ interaction for the massless quarks affects the phase diagram of the @xmath0 and causes the system to reach to the deconfined phase at the smaller baryon densities and temperatures . our results depend on the values of bag pressure and the @xmath2 coupling constant which the latter does not have a dramatic effect on our results . the increase of the hadronic and the bag pressure can improve our results toward the lattice @xmath2 calculations . in the future works , we could adjust our phase diagram to get the relation between the bag constant and the @xmath2 coupling constant . on the other hand it is possible to generalize our method for the non - constant @xmath2 coupling and the bag pressure . finally , we can also add the interaction between the gluons to our present calculations .

the thermodynamic properties of the _ quark gluon plasma _ ( @xmath0 ) as well as its phase diagram are calculated as a function of baryon density ( chemical potential ) and temperature . the @xmath0 is assumed to be composed of the light quarks only , i.e. the up and the down quarks , which interact weakly and the gluons which are treated as they are free . the interaction between quarks is considered in the framework of the _ one gluon exchange _ model which is obtained from the fermi liquid picture . the bag model is used , with fixed bag pressure ( @xmath1 ) for the _ nonperturbative _ part and the _ quantum chromodynamics _ ( @xmath2 ) coupling is assumed to be constant i.e. no dependence on the temperature or the baryon density . the effect of weakly interacting quarks on the @xmath0 phase diagram are shown and discussed . it is demonstrated that the _ one gluon exchange _ interaction for the massless quarks has considerable effect on the @xmath0 phase diagram and it causes the system to reach to the confined phase at the smaller baryon densities and temperatures . the pressure of excluded volume hadron gas model is also used to find the transition phase - diagram . our results depend on the values of bag pressure and the @xmath2 coupling constant which the latter does not have a dramatic effect on our calculations . finally , we compare our results with the thermodynamic properties of strange quark matter and the lattice @xmath2 prediction for the @xmath0 transition critical temperature .

arxiv

in ref . @xcite an @xmath7 conformally symmetric model was proposed for strong interactions at low energies , based on the observation , published in 1919 by h. weyl in ref . @xcite , that the dynamical equations of gauge theories retain their flat - space - time form when subject to a conformally - flat metrical field , instead of the usual minkowski background . confinement of quarks and gluons is then described through the introduction of two scalar fields which spontaneously break the @xmath7 symmetry down to @xmath8 and @xmath9 symmetry , respectively . moreover , a symmetric second - order tensor field is defined that serves as the metric for flat space - time , coupling to electromagnetism . quarks and gluons , which to lowest order do not couple to this tensor field , are confined to an anti - de - sitter ( ads ) universe @xcite , having a finite radius in the flat space - time . this way , the model describes quarks and gluons that oscillate with a universal frequency , independent of the flavor mass , inside a closed universe , as well as photons which freely travel through flat space - time . the fields in the model of ref . @xcite comprise one real scalar field @xmath10 and one complex scalar field @xmath11 . their dynamical equations were solved in ref . @xcite for the case that the respective vacuum expectation values , given by @xmath12 and @xmath13 , satisfy the relation @xmath14 a solution for @xmath12 of particular interest leads to ads confinement , via the associated conformally flat metric given by @xmath15 . the only quadratic term in the lagrangian of ref . @xcite is proportional to @xmath16 hence , under the condition of relation ( [ slvacua ] ) , one obtains , after choosing vacuum expectation values , a light @xmath10 field , associated with confinement , and a very heavy complex @xmath11 field , associated with electromagnetism . weak interactions were not contemplated in ref . @xcite , but one may read electroweak for electromagnetism . here , we will study the supposedly light mass of the scalar field that gives rise to confinement . the conformally symmetric model of ref . @xcite in itself does not easily allow for interactions between hadrons , as each hadron is described by a closed universe . hence , in order to compare the properties of this model with the actually measured cross sections and branching ratios , the model has been further simplified , such that only its main property survives , namely its flavor - independent oscillations . this way the full ads spectrum is , via light - quark - pair creation , coupled to the channels of two or more hadronic decay products for which scattering amplitudes can be measured . the ads spectrum reveals itself through the structures observed in hadronic mass distributions . however , as we have shown in the past ( see ref . @xcite and references therein ) , there exists no simple relation between enhancements in the experimental cross sections and the ads spectrum . it had been studied in parallel , for mesons , in a coupled - channel model in which quarks are confined by a flavor - independent harmonic oscillator @xcite . empirically , based on numerous data on mesonic resonances measured by a large variety of experimental collaborations , it was found @xcite that an ads oscillation frequency of @xmath17 agrees well with the observed results for meson - meson scattering and meson - pair production in the light @xcite , heavy - light @xcite , and heavy @xcite flavor sectors , thus reinforcing the strategy proposed in ref . @xcite . another ingredient of the model for the description of non - exotic quarkonia , namely the coupling of quark - antiquark components to real and virtual two - meson decay channels @xcite via @xmath18 quark - pair creation , gives us a clue about the size of the mass of the @xmath10 field . for such a coupling it was found that the average radius @xmath19 for light - quark - pair creation in quarkonia could be described by an flavor - independent mass scale , given by @xmath20 where @xmath21 is the effective reduced quarkonium mass . in earlier work , the value @xmath22 @xcite was used , which results in @xmath23 mev for the corresponding mass scale . however , the quarkonium spectrum is not very sensitive to the precise value of the radius @xmath19 , in contrast with the resonance widths . in more recent work @xcite , slightly larger transition radii have been applied , corresponding to values around 40 mev for @xmath24 . nevertheless , values of 3040 mev for the flavor - independent mass @xmath24 do not seem to bear any relation to an observed quantity for strong interactions . however , we will next present experimental evidence for the possible existence of a quantum with a mass of about 38 mev , which in the light of its relation to the @xmath18 mechanism we suppose to mediate quark - pair creation . moreover , its scalar properties make it a perfect candidate for the quantum associated with the above - discussed scalar field for confinement . @xcite , we made notice of an apparent interference effect around the @xmath25 threshold in the invariant - mass distribution of @xmath26 events , which we observed in preliminary radiation data of the babar collaboration @xcite . the effect , with a periodicity of about 74 mev , could be due to interference between the typical oscillation frequency of 190 mev of the @xmath27 pair and that of the gluon cloud . [ cols="^ " , ] thus , a signal with the shape of a narrow breit - wigner resonance seems to be visible on the slope of the @xmath28 resonance , though with little more than 2@xmath10 relevance . nevertheless , by coincidence or not , it comes out exactly in the expected place , namely at @xmath29 mev . unfortunately , the data @xcite do not have enough statistics to pinpoint possible higher excitations as well . so we can not relate , to a minimum degree of accuracy , the other observed deficit enhancements to the possible existence of hybrid states . from the fact that the @xmath30 has not been observed before , one must conclude that it probably does not interact at least to leading order through electroweak forces , but instead couples exclusively to quarks and gluons . the interference effect we discussed in sec . [ oscillations ] might well be explained by @xmath30 exchange between the quarks , which interferes with the natural quark oscillations . moreover , since the interference effect is smaller for light quarks than for heavier ones , it is likely that their coupling to the @xmath30 is proportional to flavor mass , as one expects from theory . the @xmath30 could very well be just a light scalar glueball , albeit much lighter than found in refs . its low mass precludes decay into hadrons , while the absence of electroweak couplings does not allow it to decay into leptons either , at least to lowest order . it may decay , though , into photons via virtual quark loops , and through photons , eventually , into @xmath0 pairs . however , the probability for such reactions to occur is extremely remote , since the coupling between the quarks and the light scalar is proportional to the quark mass @xcite . in the case of bottom quarks , we found events of the order of or less than one percent of the total . for light quarks , their mass ratio with respect to bottom quarks reduces this rate to @xmath31@xmath32 . hadrons will certainly interact with a light scalar ball of glue . for example , a proton struck by such a scalar particle may absorp it and then emit photons , or decay into a neutron and a lepton - neutrino pair . yet another possibility is that , being closed universes themselves , these scalar particles mainly collide elastically with hadronic matter . in that case , depending on their linear momentum , they may remove light nuclei from atoms . nevertheless , we do not see such processes happening around us . therefore , these light scalars are probably not abundantly present near us . however , in the early universe they may have existed , most probably inflated to hadrons under collisions . on the other hand , interactions with hadrons might have been observed in bubble - chamber experiments , where isolated protons could be the result of collisions with a light scalar particle emerging from one of the interaction vertices . light higgs fields have been considered in supersymmetric extensions of the standard model @xcite . furthermore , axions , which appear in models motivated by astrophysical observations , are assumed to have higgs - like couplings @xcite . model predictions for the branching fraction of @xmath33higgs decays , for higgs masses below @xmath34 @xcite , range from @xmath32 @xcite to @xmath35 @xcite . furthermore , the three anomalous events observed in the hypercp experiment @xcite were interpreted as the production of a scalar boson with a mass of 214.3 mev , decaying into a pair of muons @xcite . however , in ref . @xcite the babar collaboration found no evidence for dimuon decays of a light scalar particle in radiative decays of @xmath28 and @xmath36 mesons . the babar limits for dimuon decays of a light scalar particle rule out much of the parameter space allowed by the light - higgs @xcite and axion @xcite models . nonetheless , in ref . @xcite y .- j . zhang and h .- s . shao , pointed out that the transitions @xmath37higgs are not yet excluded by the lepton - universality test in @xmath38 decays studied by babar in refs . @xcite . the light scalar glueball we have discussed here seems to correspond to the lowest - order empty - universe solution of ref . @xcite for strong interactions . it has similar properties as the electroweak higgs , but now for strong interactions . quarks couple to it with an intensity which is proportional to their mass , in the same way that mass couples to gravity @xcite . in the standard model @xcite , the higgs boson and the graviton are the only particles yet to be observed , and no higgs particle for strong interactions is anticipated . however , n. trnqvist recently proposed @xcite the light scalar - meson nonet @xcite as the higgs bosons of strong interactions , while in ref . @xcite he obtained a nonzero pion mass by means of a small breaking of a relative symmetry between the electroweak and the strong interactions . a relation between the lightest scalar - meson nonet and glueballs has often been advocated by p. minkowski and w. ochs ( see e.g. ref . @xcite ) . in our view though , the light scalar mesons are dynamically generated through @xmath39 pair creation / annihilation , which mixes the quark - antiquark and dimeson sectors @xcite . in sec . [ intro ] we discussed why we expect an additional scalar particle to exist , besides the higgs boson for the electroweak sector . furthermore , we have estimated its mass based on the average radius for @xmath18 quark - pair creation , which had been extracted over the past three decades from numerous data on mesonic resonances ( see ref . @xcite and references therein ) . in sec . [ oscillations ] we recalled our results on an apparent interference effect in annihilation data , and stressed the possibility that it may stem from some internal oscillation with a frequency of about 38 mev . in sec . [ phenomenon ] we showed that missing data in the reactions @xmath0 @xmath1 @xmath40 @xmath1 @xmath3 and @xmath0 @xmath1 @xmath40 @xmath1 @xmath4 exhibit maxima at @xmath24 , @xmath41 , and @xmath42 , for @xmath43 mev . each of the results , viz.the interference effect observed in ref . @xcite , the small flavor - independent oscillations in electron - positron and proton - antiproton annihilation data , observed in ref . @xcite and summarized in fig . [ interference ] , the excess signals visible in the @xmath5 mass distributions of @xmath28 @xmath1 @xmath40 @xmath1 @xmath4 ( fig . [ mumuall2s ] ) , in @xmath36 @xmath1 @xmath40 @xmath1 @xmath4 ( fig . [ alldiff]a ) , in @xmath36 @xmath1 @xmath44 @xmath1 @xmath4 ( fig . [ alldiff]b ) , and in the @xmath0 mass distributions of @xmath0 @xmath1 @xmath3 ( fig . [ alldiff]c ) , and finally the resonance signal shown in fig . [ hybrid ] , is much too small to make firm claims . however , we observe here that all points in the same direction . indeed , the probability must be close to zero that one accidentally finds the same oscillations in four different sets of data ( refs . @xcite ) involving different flavors , statistical fluctuations at @xmath45 mev in yet another four sets of different data ( figs . [ mumuall2s ] and [ alldiff ] ) , and moreover a resonance - like fluctuation at @xmath46 mev in a further set of data ( fig . [ hybrid ] ) . furthermore , the related mass comes where predicted by our analyses in mesonic spectroscopy ( see ref . @xcite and references therein ) . in sec . [ gluons ] we discussed that , most probably , the missing signal is due to the emission of an as yet unobserved light scalar particle , while part of the excess data corroborates such an interpretation . since the corresponding particle has all the right properties , we conclude that we found first indications , of the possible existence of a higgs - like particle , namely the scalar boson related to confinement . furthermore , the data also suggest the existence of two replicas of the @xmath30 with masses that are two and three times heavier than the @xmath30 . in addition , we believe that this 38 mev boson , which we designate by @xmath30 , consists of a mini - universe filled with glue , thus forming a very light scalar glueball . furthermore , we have pinpointed the masses of possible @xmath6 hybrids , one of which shows up as an enhancement in the invariant - mass distribution of babar data , albeit with a 2@xmath10 significance at most . finally , we urge the babar collaboration to inspect their larger data set in order to settle , with higher statistics , the possible existence of the @xmath30 and the related @xmath6 hybrid spectrum . we are grateful for the precise measurements and data analyses of the babar , cdf , and cmd-2 collaborations , which made the present analysis possible . one of us ( evb ) wishes to thank drs . b. hiller , a. h. blin , and a. a. osipov for useful discussions . this work was supported in part by the _ fundao para a cincia e a tecnologia _ of the _ ministrio da cincia , tecnologia e ensino superior _ of portugal , under contract cern / fp/ 109307/2009 . m. dohse , arxiv:0706.1887 . e. van beveren and g. rupp , pos hql2010:003 ( 2010 ) [ arxiv:1011.2360 ] . e. van beveren , c. dullemond , and g. rupp , , 772 ( 1980 ) [ erratum - 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we present evidence for the existence of a light scalar particle that most probably couples exclusively to gluons and quarks . theoretical and phenomenological arguments are presented to support the existence of a light scalar boson for confinement and quark - pair creation . previously observed interference effects allow to set a narrow window for the scalar s mass and also for its flavor - mass - dependent coupling to quarks . here , in order to find a direct signal indicating its production , we study published babar data on leptonic bottomonium decays , viz.the reactions @xmath0 @xmath1 @xmath2 @xmath1 @xmath3 ( and @xmath4 ) . we observe a clear excess signal in the invariant - mass projections of @xmath0 and @xmath5 , which may be due to the emission of a so far unobserved scalar particle with a mass of about 38 mev . in the process of our analysis , we also find an indication of the existence of a @xmath6 hybrid state at about 10.061 gev . further signals could be interpreted as replicas with masses two and three times as large as the lightest scalar particle .

arxiv

the term ` stochastic resonance ' was introduced in the early 80s ( see @xcite and @xcite ) in the study of periodic advance of glaciers on earth . the stochastic resonance is the effect of nonmonotone dependence of the response of a system on the noise when this noise ( for instance the temperature ) is added to a periodic input signal ( see e.g. @xcite , in which the author explains also differences and similarities with the notion of stochastic filtering ) . an extensive review on stochastic resonance and its presence in different fields of applications can be found in @xcite . following @xcite , as stochastic resonance we intend the phenomenon in which the transmission of a signal can be improved ( in terms of statistical quantities ) by the addition of noise . from the statistical point of view the problem is to estimate a signal @xmath0 transmitted through a channel . this signal has to be detected by a receiver that can reveal signals louder than a threshold @xmath1 . if @xmath2 is bounded from above by @xmath1 , the signal is not observable and the problem has not a solution . but , if some noise @xmath3 is added to the signal , the perturbed signal @xmath4 may be observable and inference can be done on @xmath2 . too few noise is not sufficient to give good estimates and too much noise deteriorates excessively the signal . the optimal in some sense level of the noise will be called stochastic resonance in this framework . usually ( see @xcite ) the criterion applied to measure optimality of estimators are the shannon mutual information or the kullback divergence . more recently the fisher information quantity have been also proposed ( see @xcite and @xcite ) . here we are concerned with the fisher information quantity . it happens that this quantity , as a function of the noise , can be maximized for certain noise structures . if there is only one global maximum , the corresponding noise level is the value for which we have stochastic resonance , if several local maxima are present , the phenomenon is called stochastic multi - resonance . in this paper we study the problem of estimation and hypotheses testing for the following model : we suppose to have a threshold @xmath5 and a subthreshold constant and non negative signal @xmath6 , @xmath7 . we add , in continuous time , a noise that is a trajectory of a diffusion process @xmath8 and we observe the perturbed signal @xmath9 where @xmath10 is the level of the noise . we propose two schemes of observations : _ i ) _ we observe only the proportion of time spent by the perturbed signal over the threshold @xmath1 and _ ii ) _ we measure the energy of the perturbed signal when it is above the threshold . the asymptotic is considered as time goes to infinity . this approach differs from the ones in the current statistical literature basically for two reasons : the noise structure is an ergodic diffusion process and not a sequence of independent and identically distributed random variables and data are collected in continuous time . this second aspect is a substantial difference but it is not a problem from the point of view of applications for the two schemes of observations proposed if one thinks at analogical devices . we propose two different estimators for the schemes and we study their asymptotic properties . we present an example where , in both cases , it emerges the phenomenon of stochastic resonance . for the same model we also solve the problem of testing the simple hypothesis @xmath11 against the simple alternative @xmath12 by applying the bayesian maximum a posterior probability criterion . it emerges that the overall probability of error is nonmonotonically dependent on @xmath13 . we show again that there exists a non trivial local minimum of this probability that is again the effect of stochastic resonance . the presence of stochastic resonance in this context is noted for the first time here . the paper is organized as follows . in section [ sec : model ] we set up the regularity assumptions of the model . in sections [ sec : time ] and [ sec : energy ] we prove some asymptotic properties estimators for the two schemes and we calculate numerically the points where the fisher information quantity attains its maximum for both models . it turns out that the estimators proposed are asymptotically equivalent to the maximum likelihood estimators . section [ sec : test ] is devoted to the problem of hypotheses testing . all the figures are collected at the end of the paper . let @xmath1 be the threshold and @xmath6 a constant signal . taking @xmath14 will not influence the calculations that follows but may improve the exposition , so we use this assumption . let @xmath15 be a given diffusion process solution to the following stochastic differential equation @xmath16 with non random initial value @xmath17 . the process @xmath15 is supposed to have the ergodic property with invariant measure @xmath18 and invariant distribution function @xmath19)$ ] as @xmath20 . the functions @xmath21 and @xmath22 satisfy the global lipschitz condition @xmath23 where @xmath24 is the lipschitz constant . under condition [ cond : c1 ] , equation has a unique strong solution ( see e.g. @xcite ) but any equivalent condition to [ cond : c1 ] can be assumed because we do not use explicitly it in the sequel . the following conditions are needed to ensure the ergodicity of the process @xmath15 . if @xmath25 and @xmath26 then there exists the stationary distribution function @xmath27 and it takes the following form @xmath28 again , any other couple of conditions that imply the existence of @xmath27 can be used instead of [ cond : c3 ] and [ cond : c3 ] . we perturb the signal @xmath6 by adding , proportionally to some level @xmath10 , the trajectory diffusion process @xmath15 into the channel . the result will be the perturbed signal @xmath29 . this new signal will be detectable only when it is above the threshold @xmath1 . moreover , @xmath30 is still ergodic with trend and diffusion coefficients respectively @xmath31 and @xmath32 and initial value @xmath33 , but we will not use directly this process . we denote by @xmath34 the observable part of the trajectory of @xmath30 , being @xmath35 the indicator function of the set @xmath36 . we consider two possibile schemes of observation : * we observe only the proportion of time spent by @xmath30 over the threshold @xmath1 @xmath37 * we measure the energy of the signal @xmath38 @xmath39 in the next sections , for the two models we establish asymptotic properties of estimators given by the generalized method of moments . in @xcite different properties of the generalized method of moments for ergodic diffusion processes are studied . in this note we follows the lines given in the paper of@xcite for the i.i.d . setting . these results are interesting in themselves independently from the problem of stochastic resonance . we give an example of stochastic resonance based on the process where the phenomenon of stochastic resonance appears pronounced and in which results in a closed form can be written down . the random variable @xmath40 can be rewritten in terms of the process @xmath15 as @xmath41 by the ergodic property of @xmath15 we have that @xmath42 where @xmath43 has @xmath27 as distribution function . from it derives that @xmath44 so that @xmath6 is a one - to - one continuous function of @xmath45 . from the glivenko - cantelli theorem ( see e.g. @xcite ) for the empirical distribution function ( edf ) defined by @xmath46 follows directly that @xmath40 is a @xmath47-consistent estimator of @xmath45 thus also @xmath48 is a @xmath49-consistent estimator for @xmath6 . we can calculate the asymptotic variance of this estimator . it is known that ( see @xcite and @xcite ) the edf is asymptotically gaussian and in particular @xmath50 where @xmath51 is the inverse of the analogue of the fisher information quantity in the problem of distribution function estimation : @xmath52 where @xmath53 and @xmath54 . the quantity @xmath55 is also the minimax asymptotic lower bound for the quadratic risk associated to the estimation of @xmath56 , so that @xmath57 is asymptotically efficient in this sense . the asymptotic variance @xmath58 of @xmath59 can be derived by means of the so - called @xmath60-method ( see e.g. @xcite ) : @xmath61 thus @xmath62 where @xmath2 is the density of @xmath27 . the quantity @xmath58 can also be derived from the asymptotic minimal variance @xmath63 of the edf estimator . in fact , with little abuse of notation , by putting @xmath64 and @xmath65 we have that @xmath66 we now show that @xmath40 also maximizes the approximate likelihood of the model . in fact , for the central limit theorem for the edf we have @xmath67 where @xmath68 is a standard gaussian random variable . thus @xmath69 where @xmath70 is the distribution function of @xmath68 . we approximate the likelihood function of @xmath40 by @xmath71 that is maximal when @xmath72 thus , the maximum likelihood estimator of @xmath6 ( constructed on the approximated likelihood ) reads @xmath73 so if the approximation above is acceptable , one can infer the optimality property of @xmath40 of having minimum variance from being also the maximum likelihood estimator . to view the effect of stochastic resonance on the fisher information we consider a particular example . by setting @xmath74 and @xmath75 the noise become a standard process solution to the stochastic differential equation @xmath76 in such a case , the ergodic distribution function @xmath27 is the gaussian law with zero mean and variance 1/2 . the asymptotic variance @xmath58 assumes the following form @xmath77 where @xmath78 is the classical error function . in figure [ fig1 ] it is shown that for this model there exists the phenomenon of stochastic resonance . for a fixed level of noise @xmath13 the fisher information increases as the signal @xmath6 is closer to the threshold @xmath1 . for a fixed value of the signal @xmath6 , the fisher information , as a function of @xmath13 , has a single maximum , that is the optimal level of noise . for example , if @xmath79 then the optimal level is @xmath80 and for @xmath81 it is @xmath82 . suppose that it is possibile to observe not only the time when the perturbed process is over the threshold but also its trajectory above @xmath1 , say @xmath83 , @xmath84 . we now show how it is possibile to estimate the unknown signal @xmath6 from the equivalent of the energy of the signal for @xmath85 : literally from the quantity @xmath86 we use the following general result from @xcite on the estimation of functionals of the invariant distribution functions for ergodic diffusion processes . _ let @xmath87 and @xmath88 be such that @xmath89 . then @xmath90 is a @xmath49-consistent estimator for @xmath91 where @xmath43 is distributed according to @xmath27_. in our case @xmath92 and @xmath93 . the estimator @xmath94 can be rewritten as @xmath95 and it converges to the quantity @xmath96 that is a continuous and increasing function of @xmath6 , @xmath97 . its inverse @xmath98 allows us to have again @xmath99 . by applying the @xmath60-method once again , we can obtain the asymptotic variance of @xmath100 from the asymptotic variance of @xmath94 . @xmath101 where @xmath102 the asymptotic variance of @xmath94 is given by ( see @xcite ) @xmath103 where @xmath104 and its inverse is also the minimal asymptotic variance in the problem of estimation of functionals for ergodic diffusion . thus , the asymptotic variance of @xmath100 is given by @xmath105 * remark : * _ by the asymptotic normality of @xmath94 follows that @xmath100 is also the value that maximizes the approximate likelihood function . in fact , as in the previous example , if we approximate the density function of @xmath94 with @xmath106 it is clear that @xmath100 is its maximum . _ as before , we put in evidence the phenomenon of stochastic resonance by using the process as noise . the quantities involved ( @xmath107 and @xmath108 ) transform into the following @xmath109 ( from which it appears that @xmath107 is an increasing function of @xmath6 ) and @xmath110 in figure [ fig1 ] it is plotted the fisher information of the model as a function of @xmath6 and @xmath13 . also in this case there is evidence of stochastic resonance . for a fixed value of @xmath6 is then possibile to find the optimal noise level @xmath13 . for example , taking @xmath111 then we have stochastic resonance at @xmath112 and for @xmath113 , @xmath114 . we now study a problem of testing two simple hypotheses for the model discussed in the previous section . as in @xcite , we apply the maximum _ a posteriori _ probability ( map ) criterion . we will see that the decision rules for our model are similar to the one proposed by chapeau - blondeau in the i.i.d . setting . given the observation @xmath40 we want to verify the null hypothesis that the unknown constant signal is @xmath115 against the simple alternative @xmath116 , with @xmath117 : @xmath118 suppose that , before observing @xmath40 , we have a _ prior _ information on the parameter , that is @xmath119 and @xmath120 . the map criterion uses the following likelihood ratio @xmath121 and the decision rule is to accept @xmath122 whenever @xmath123 ( decision @xmath124 ) or refuse it otherwise ( decision @xmath125 ) . the overall probability of error is @xmath126 let now be @xmath127 then , the likelihood @xmath128 appears as @xmath129 to write explicitly the decision rule and then study the effect of stochastic resonance we have to distinguish three cases : @xmath130 , @xmath131 and @xmath132 . 1 . let it be @xmath130 , then put @xmath133 then , if @xmath134 accept @xmath122 and @xmath135 . if @xmath136 , then if @xmath137 reject @xmath122 otherwise accept it . in both cases @xmath138 2 . let it be @xmath131 , then put @xmath139 then , if @xmath134 reject @xmath122 and @xmath140 . if @xmath136 , then if @xmath137 accept @xmath122 otherwise reject it . in both cases @xmath141 3 . let it be @xmath132 , then put @xmath142 then , if @xmath143 reject @xmath122 otherwise accept it . in both cases @xmath144 as before , we apply this method to the model . in this case the variance @xmath145 , for a fixed threshold @xmath1 and noise level @xmath13 , is a non decreasing function of @xmath6 being @xmath146 only a scale factor ( figure [ fig2 ] gives a numerical representation of this statement ) . thus , the @xmath147 is , in general , given by formula . what is amazing is the behavior of @xmath147 . in figure [ fig3 ] it is reported the graph of @xmath147 as a function of @xmath13 and @xmath116 given @xmath148 and @xmath149 . for @xmath116 around 1/2 the @xmath147 shows the effect of stochastic resonance . so it appears that in some cases the noise level @xmath13 can reduce sensibly the overall probability of making the wrong decision . this kind of behavior is non outlined in the work of @xcite . * remark : * _ following the same scheme , similar results can be obtained for the model _ ii ) _ when we observe the energy @xmath94 . in this case it is sufficient to replace in the values of @xmath150 and @xmath151 with the quantities @xmath152 and @xmath40 with @xmath94 in the decision rule . _ the use of ergodic diffusions as noise in the problem of stochastic resonance seems quite powerful . characterizations of classes of ergodic process that enhance the stochastic resonance can be done ( see e.g. @xcite ) but not in a simple way as in the i.i.d . case as calculations are always cumbersome . the problem of a parametric non constant signal can also be treated while the full nonparametric non constant signal requires more attention and will be object for further investigations . for i.i.d . observations , @xcite and @xcite considered the problem of non parametric estimation for regression models of the form @xmath153 , @xmath154 . their approach can be applied in this context . other criterion of optimality than the fisher information quantity can be used as it is usually done in information theory ( e.g. shannon mutual information or kullback divergence ) . the analysis of the overall probability of error seems to put in evidence something new with respect to the current literature ( see e.g. @xcite ) . it is worth noting that in a recent paper @xcite models driven by ergodic diffusions have also been used but the effect of stochastic resonance is not used to estimate parameters . barndorff - nielsen , o. , cox , d.r . , _ asymptotic techniques for use in statistics_. chapman & hall , 1989 , bristol . benzi , r. , sutera , a. , vulpiani , a. , the mechanism of stochastic resonance . _ j. phys . a _ , v * 14 * , ( 1981 ) , l453-l458 . berglund , n. , gentz , b. , a sample - path approach to noise - induced synchronization : stochastic resonance in a double - well potential . wias preprint no . 627 , ( 2000 ) . chapeau - blondeau , f. , nonlinear test statistic to improve signal detection in non - gaussian noise . _ ieee sig . _ , v * 7 * , n 7 , ( 2000 ) , 205 - 207 . chapeau - blondeau , f. , rojas - varela , j. , information - theoretic measures improved by noise in nonlinear systems . 14th int . conf . on math . theory of networks and systems _ , perpignane , france , ( 2000 ) , 79 - 82 . greenwood , p.e . , ward , l.m . , wefelmeyer , w. , statistical analysis of stochastic resonance in a simple setting . _ phys . rev . , v * 60 * , n 4 , ( 2000 ) , 4687 - 4695 . gammaitoni , l. , hnggi , p. , jung , p. , marchesoni , f. , stochastic resonance . _ rev . of modern phys . _ , 70 , ( 1998 ) , 223 - 288 . klimontovich , yu.l . , what are stochastic filtering and stochastic resonance ? _ physics - uspecky _ , v * 42 * , n 1 , ( 1999 ) , 37 - 44 . kutoyants , yu.a . , efficiency of the empirical distribution function for ergodic diffusion . _ bernoulli _ , v * 3 * , n 4 , ( 1997a ) , 445 - 456 . kutoyants , yu.a . , on semiparametric estimation for ergodic diffusion . a. razmadze math . _ , 115 , ( 1997b ) , 45 - 58 . kutoyants , yu.a . , _ statistical inference for ergodic diffusion processes_. monography to appear ( 2000 ) . ibragimov , i.a . , khasminskii , r.z . , _ statistical estimation ( asymptotic theory)_. springer , 1981 , new york . liptser , r.s . , shiryayev , a.n . , _ statistics of random processes _ , part . springer - verlag , 1977 , new york . mller , u. u. , nonparametric regression for threshold data . _ canadian j. statist _ , * 28 * , ( 2000 ) , to appear . mller , u. u. , ward , l. m. , stochastic resonance in a statistical model of a time - integrating detector _ e ( 3 ) _ , * 61 * , ( 2000 ) , to appear . negri , i. ( 1998 ) . stationary distribution function estimation for ergodic diffusion processes . _ statistical inference for stochastic processes _ , v * 1 * , n 1 , 61 - 84 . nicolis , c. , stochastic aspects of climatic transitions response to a periodic forcing . _ tellus _ , * 34 * , n 1 , ( 1982 ) , 1 - 9 . [ cols="^ " , ] as a function of @xmath13 for different values of @xmath116 given @xmath148 and @xmath149 . from left - top to right - bottom : @xmath116 = 0.1 , 0.25 , 0.5 , 0.75 , 0.9 , 0.95 . the bottom plot is to show the effect of stochastic resonance for @xmath116 = 0.5 . ]

a subthreshold signal is transmitted through a channel and may be detected when some noise with known structure and proportional to some level is added to the data . there is an optimal noise level , called stochastic resonance , that corresponds to the highest fisher information in the problem of estimation of the signal . as noise we consider an ergodic diffusion process and the asymptotic is considered as time goes to infinity . we propose consistent estimators of the subthreshold signal and we solve further a problem of hypotheses testing . we also discuss evidence of stochastic resonance for both estimation and hypotheses testing problems via examples . * keywords : * stochastic resonance , diffusion processes , unobservable signal detection , maximum a posterior probability . * msc : * 93e10 , 62m99 .

arxiv

the order parameter in high - temperature superconductors ( hts ) has been proved to have mainly a @xmath4 pairing symmetry.@xcite in particular , try - crystal phase - sensitive measurements @xcite have determined the presence of a predominant @xmath5gap symmetry up to the superconducting critical temperature @xmath1 over , if any , other minor components smaller than 5% of that with the @xmath5symmetry . thermal transport measurements have also shown the predominance of this symmetry for the gap function . in particular , when a magnetic field is rotated parallel to the cuo@xmath6 planes , the longitudinal thermal conductivity shows a fourfold oscillation which can be explained in terms of both andreev scattering of quasiparticles by vortices ( as ) and doppler shift ( ds ) in the energy spectrum of the quasiparticles if one takes a @xmath5gap into account.@xcite nevertheless , the fourfold oscillation in the thermal conductivity has been resolved up to @xmath7 k. above this temperature , no direct evidence for this type of symmetry from this kind of measurements has been published . furthermore , deviations from the expected angular pattern within a pure @xmath5symmetry have been attributed to the effect of pinning of vortices.@xcite the variation of the thermal conductivity as a function of angle @xmath8 between the heat current and the magnetic field applied parallel to the cuo@xmath6 planes depends mainly on the heat transport by quasiparticles , their interaction with the supercurrents ( vortices ) and the symmetry of the order parameter.@xcite an angular pattern showing properties of the order parameter symmetry can only be achieved when the temperature is low enough so that the quasiparticle momentum is close to the nodal directions of the gap . otherwise , thermal activation would also induce quasiparticles at different orientations from those of the nodes and hence , the sensitivity of the probe to measure gap characteristics will be reduced . the first question we address in this paper is related to the temperature range at which the nodal characteristics of the gap are directly observable by thermal conductivity . the second question we would like to clarify in this paper is whether the thermal activation of quasiparticles with increasing temperature , treated phenomenologically within a fermi liquid approximation for thermal transport including the @xmath5gap function , can explain the experimental results in the whole temperature range , in particular the change of symmetry of @xmath0 as a function of temperature . in this paper , we calculate numerically the thermal conductivity at different angles and temperatures at fixed magnetic field , assuming an andreev reflection model for the scattering of quasiparticles by supercurrents , originally proposed by yu et al . @xcite within the two dimensional brt expression @xcite for the thermal conductivity , and compare it to the experimental data . angle scans in a magnetic field applied parallel to the cuo@xmath6 planes were performed in order to measure the angular variation of the longitudinal thermal conductivity @xmath0 in two single crystals of yba@xmath6cu@xmath9o@xmath10 high - temperature superconductor . the overall results agree with the theoretical model and confirm the predominance of the @xmath5gap up to @xmath1 . the measurements provide also new results that improve our knowledge of the thermal transport at temperatures at which the nodal properties of the gap are not directly observable . following the experimental and sample details of the next section , we present in sec . [ res ] the main experimental results . in sec . [ comparison ] we describe the used model and compare it with the experimental data . a brief summary is given in sec . in order to rule out effects concerning shape and structure characteristics of the crystal we have used two different samples of yba@xmath6cu@xmath9o@xmath10 ( ybco ) : a twinned single crystal with dimensions ( length @xmath11 width @xmath11 thickness ) @xmath12mm@xmath13 and critical temperature @xmath14k previously studied in refs . @xcite and an untwinned single crystal with dimensions @xmath15mm@xmath13 and @xmath16k.@xcite for the measurement of the thermal conductivity , a heat current @xmath17 was applied along the longest axis of the crystal studied . in the untwinned sample , j was parallel to the @xmath18-axis and in the twinned crystal was parallel to the @xmath19-axes ( twin planes oriented along ( 110 ) ) . in both cases the position of the lattice axis with respect to the crystal axis was determined using polarized light microscopy and x - ray diffraction . the longitudinal temperature gradient @xmath20 was measured using previously calibrated chromel - constantan ( type e ) thermocouples @xcite and a dc picovoltmeter . special efforts were made in order to minimize the misalignment of the plane of rotation of the magnetic field applied perpendicular to the c - axis with the cuo@xmath6 planes of the sample . this misalignment was minimized step by step , measuring the angle dependence of @xmath0 until a satisfactory symmetrical curve was obtained . in this way , we estimate a misalignment smaller than @xmath21 . an in - situ rotation system enabled measurement of the thermal conductivity as a function of the angle @xmath8 defined between the applied field and the heat flow direction along @xmath22 , see fig . [ esquema ] . for more details on the experimental arrangement see ref . @xcite . as pointed out by aubin et al . @xcite and observed in refs . @xcite , the effect of the pinning of vortices plays an important role in determining the correct angle pattern in this kind of measurement . in fact , when the angle of the magnetic field is changed , a non uniform vortex distribution due to pinning forces may appear . as argued in ref . @xcite , the pinning of the josephson - like vortices parallel to the planes is strongly affected by vortices perpendicular to the planes which may appear due to the misalignment of the crystal axes with respect to the applied magnetic field . in this situation , even hysteresis in the angular patterns of the thermal conductivity can be measured . @xcite we note that pinning of vortices is influenced by the distribution of the oxygen vacancies in the sample as well as by defects and impurity centers . thus , in order to rule out the influence of this effect in the measurements , a field - cooled procedure have to be used . this procedure consists of two steps . in the first step the sample is driven into the normal state by heating to a few kelvin above @xmath1 and the angle is changed . secondly , it is cooled down to the desired temperature at constant field . in fig . [ fig1 ] we show the longitudinal thermal conductivity as a function of the temperature in both crystals at zero magnetic field . as argued by many authors ( see for example refs . @xcite ) , the observed behavior in fig . [ fig1 ] provides qualitative information about the quality of the sample . the height of the peak observed in the temperature dependence of the thermal conductivity is related to the relative contributions between the density of impurity scattering centers and the strength of the inelastic electron - electron scattering . crystals showing large peaks may have a small amount of impurity scattering centers @xcite and/or larger quasiparticle - related inelastic contribution . the origin of the peak is explained in terms of a competition between the decrease of the inelastic scattering rate @xcite and the decrease of the population of quasiparticles as temperature decreases.@xcite taking this into account we might conclude that the untwinned crystal has a larger impurity scattering and a smaller quasiparticle - related inelastic scattering since its peak is considerably smaller than that of the twinned sample . this is supported by its reduced t@xmath23 with respect to the twinned sample ( which indicates a larger density of oxygen vacancies ) . since the thermal conductivity at @xmath1 appears to be larger for the untwinned sample , we conclude that there should be a substantial reduction of the inelastic scattering . as we show below , this makes the measurement of electronic properties related to the gap symmetry more difficult . the angular patterns of the longitudinal thermal conductivity @xmath24 at different temperatures for the twinned and untwinned crystals are shown in figs . [ fig2 ] and [ fig3 ] respectively . at low enough temperatures ( @xmath25 ) , a magnetic field parallel to the cuo@xmath6 planes of the sample produces a fourfold oscillation in the longitudinal thermal conductivity as the field is rotated from @xmath26 to @xmath27 . we note that at both @xmath26 and @xmath28 the magnetic field is perpendicular to the heat flow . at @xmath29 the field is parallel to the heat flow ( see fig . [ esquema ] ) . as noted in refs . @xcite and @xcite the angular patterns are not free of the phononic contribution to the total thermal conductivity . however , this contribution seems to be constant under variations of the magnetic field orientations for two reasons . 1 ) for fields parallel to the cuo@xmath6 planes , vortices are unlikely to have a normal core ( josephson vortices ) , making a change of the phonon attenuation with field and angle unlikely . 2 ) there is no experimental evidence that the phononic contribution does change substantially with field . the similarity between the temperature dependencies of @xmath24 and @xmath30 below @xmath1 where a large change in the quasiparticle density ocurrs@xcite , indicates that the phonon - electron interaction does not play a main role in the temperature dependence of @xmath31 . the electronic contribution is entirely responsible for the magnetic field dependence , revealed in both thermal conductivity @xcite and in microwave measurements @xcite , and indicates against any large phonon - electron scattering . in fact , no remarkable phonon - electron scattering have been addressed in those experiments where the measured magnetic field dependencies are larger than the oscillation amplitudes measured in this paper . therefore , measurable contributions of the phonons via phonon - electron scattering are more unlikely to occur in the angular patterns . this characteristic makes the angular profiles of the thermal conductivity suitable to be compared with electronic models by using the quantity @xmath32 where the phononic contribution is subtracted@xcite . as pointed out in ref . @xcite , the variation of the thermal conductivity in figs . [ fig2 ] and [ fig3 ] can be explained with a model involving a doppler shift ( ds ) in the energy spectrum of the quasiparticles along with an accurate inclusion of the impurity scattering @xcite and/or assuming andreev scattering ( as ) of quasiparticles by vortices@xcite . briefly , in the mixed state the quasiparticles are in the presence of a phase gradient produced by the superfluid flow of the vortices . thus , as viewed from the laboratory frame they experience a doppler shift in their energy spectrum given by the scalar product of the momentum and superfluid velocity , * * p**@xmath33**v**@xmath34 . when a quasiparticle of momentum * p * is moving parallel to the magnetic field , the product is zero and hence , no ds occurs . thus , the angular characteristic of this effect is to produce an excess of quasiparticles in the direction perpendicular to the field and thereby , reducing the local " thermal resistance at this orientation @xcite . when the field is placed parallel to the heat current , the doppler shift is the same for both nodal directions at @xmath35 and @xmath36 . therefore and since thermal resistances must be added in parallel , a simple picture in which there are only quasiparticles at the nodes would produce a fourfold oscillation in the thermal conductivity with opposite sign to that observed in the measurements . however , as pointed out in refs . @xcite , the ds affects both the carrier density as well as the scattering rate of quasiparticles , and since at high enough temperatures the latter dominates , the quasiparticles move more easily parallel to the magnetic field . therefore , the sign of the fourfold oscillation observed in the experiments is also recovered within this picture . in the as mechanism a phase gradient , namely , the superfluid flow surrounding the vortices may induce andreev reflection of the quasiparticles @xcite . thus , the ds in the energy spectrum * * p**@xmath33**v**@xmath34 is implicitly taken into account in the as picture . as viewed from the laboratory frame , when the quasiparticle energy equals the value of the gap , then the quasiparticle is transformed into a quasihole reversing its velocity and hence , decreasing its contribution to the thermal conductivity . thus , a quasiparticle with momentum * p * parallel to the magnetic field does not experience a ds and hence , no andreev reflection can take place . the field acts as a filter @xcite for the quasiparticle that contributes to reduce the total temperature gradient . note that the as picture gives rise to a fourfold oscillation in the thermal conductivity as the magnetic field is rotated parallel to the cuo@xmath6 planes without more considerations . then , qualitatively both as and ds can explain in principle the angle profiles observed at low temperatures . however as pointed out in ref . @xcite , neither as nor ds alone can explain the magnetic field dependence of the oscillation amplitude in the whole field range @xmath37t@xmath38 t . therefore , a more realistic picture of the thermal transport should take both into account . as argued in ref . @xcite , this scenario produces different regimes influenced by the predominance of either the ds or the as in the magnetic field dependence . thus , at constant temperature the strength of the magnetic field becomes the parameter that changes the regime . at low magnetic fields the ds dominates and the as dominates at high fields @xcite . in both cases , an increase of the oscillation amplitudes with increasing magnetic field strength is predicted . in fig . [ fig2 ] the oscillation amplitudes of the thermal conductivity at 3 t and 8 t for different temperatures in the twinned crystal are shown . as the temperature increases the fourfold oscillation is no longer observable , see figs . [ fig2 ] and [ fig3 ] . this fact can be understood if one takes into account the thermal activation of the quasiparticles at different orientations from those of the nodes of the order parameter . in other words , if we suppose that the same processes that govern the thermal transport at low temperatures ( as and ds ) are responsible for the high temperature behavior as well , we have to conclude that an increasing number of carriers appears in the direction of the heat current as the temperature is raised . as we shall see below from the numerical results using the as model , this is , in fact , what takes place as the temperature is increased . however , although the observed angle profiles can be well described by the numerical analysis , the amplitude of the oscillations are smaller than the simulation results at @xmath39k . as argued for the peak of the curves in fig . [ fig1 ] and shown in the following section , this discrepancy can be solved by the inclusion of an inelastic scattering into the calculations.@xcite in fig . [ fig3 ] we show the angle dependence of the untwinned crystal at 8 t. similar angle profiles as for the twinned sample have been found , but the amplitude of the oscillation is considerably smaller . this can be explained in terms of a larger concentration of impurity scatterers in the untwinned sample , as discussed above and in ref . in fact , if the impurity scattering rate is sufficiently large , the relative weight of the directionality of the as and ds mechanisms to @xmath0 weakens . the influence of the impurity scattering has been considered in the original models by inclusion of an impurity rate.@xcite we note also that the symmetry of the @xmath40curves differs slightly for the twinned and untwinned samples at the same absolute or reduced temperature . the effect of impurity scatterers can be accounted for by an impurity dependent gap parameter as done in ref . @xcite . in general , however , it can be viewed as an effect related to the ratio of the modulus of the gap and the density of states of the quasiparticles at a temperature t and momentum * p*. therefore , since our twinned sample has a larger critical temperature @xmath1 than our untwinned sample , the oxygen deficiency in the latter could be also responsible for a reduced density of states , and therefore , for a different angle patterns respect to the sample with a higher @xmath1 . figure [ fig4 ] shows the complete angle patterns measured in the twinned crystal at 8 t. at @xmath41 k no clear change in the thermal conductivity when the field is rotated parallel to the cuo@xmath6 planes is observed . this can be explained by taking into account the temperature dependence of the gap , which vanishes at @xmath1 , and the relative increase of the inelastic scattering rate . to compare the experimental results with theory we take recently published results of the longitudinal and transverse thermal conductivity@xcite into account that indicate that at high fields ( @xmath42 t ) , applied parallel to the planes , the main scattering mechanism for the quasiparticles appears to be the andreev reflection by vortex supercurrents.@xcite therefore , we use the formulation proposed by yu et al.@xcite using a two dimensional model of the brt expression for the thermal conductivity,@xcite which has been useful in the interpretation and discussion of previous results.@xcite under this model the longitudinal thermal conductivity can be written as @xmath43 where @xmath44 is the the x - axis component of the group velocity , and @xmath45 is the quasiparticle energy . for this energy we use a free quasiparticle model @xmath46 where @xmath47 is the effective mass of the quasiparticle . @xmath48 may be taken as a relaxation rate given by the sum of the following scattering mechanisms acting in series : scattering of qp by impurities @xmath49 , by phonons @xmath50,@xcite by quasiparticles @xmath51@xcite and as by vortex supercurrents @xmath52.@xcite the expression for this scattering rate according to the model for the as , proposed by yu et al . @xcite , is given by @xmath53 ^2}{p_f^2\hbar ^2\ln ( a_\upsilon /\xi _ 0)\sin ^2\psi ( { \bf p})}\right\}\,,\ ] ] where @xmath54 is the intervortex spacing given in this model by @xmath55 we use the bcs - like parameterization for the temperature dependence of the gap amplitude @xmath56 and the @xmath4-pairing symmetry in the resulting gap @xmath57 enters as follows @xmath58\,.\ ] ] we showed recently@xcite that at low temperatures and high fields , a quantitative agreement between this model and the oscillation amplitudes of the thermal conductivity tensor is achieved only if the intervortex spacing is increased by about five times the value defined in eq.([spacing ] ) if we use the parameters from ref . @xcite ( @xmath59 mev , ginzburg - landau parameter @xmath60 , anisotropy @xmath61 , @xmath62 t ) . furthermore , from those fits@xcite for the twinned crystal we obtained a momentum independent impurity scattering @xmath63 ps and @xmath64 . assuming a total scattering rate given by the sum of the as and impurity rates , i.e. @xmath65 , the model given by eqs . ( 1 ) to ( 5 ) reproduces remarkably well the symmetry of the curves in fig . [ fig4 ] in the whole measured temperature range . because in this calculation we do not take explicitly into account the inelastic scattering rate @xmath66 given by phonons @xmath67 and by quasiparticles @xmath68 , the calculated oscillation amplitude above 20 k increases up to @xmath69 times that observed in the experiment . in fig . [ fig5 ] we show the results of the numerical simulation . each of the curves in this figure has been multiplied by a factor @xmath70 in order to fit the experimental oscillation amplitude ( fig . [ fig4 ] ) . we note that the symmetry of the results can be explained quite satisfactorily by this model . this result , on one hand , supports the predominance of @xmath4-pairing symmetry of the order parameter up to temperatures close to @xmath1 and , on the other hand , confirms the idea of a competition between the thermal activation of quasiparticles in the direction of the thermal current and the gap structure . we note that the @xmath4-gap symmetry is the only gap function that can explain the whole angular patterns up to t@xmath23 . of course , a @xmath71-gap also gives a onefold oscillation similar to the experiments above @xmath72 50 k , but it does produce neither the fourfold oscillation below @xmath72 15 k nor the curves between 15 k and 50 k. thus , the set of curves in fig . [ fig4 ] can be only explained if one uses a main @xmath4-gap into the calculations . furthermore , the anisotropy of @xmath73-wave gap proposed for bscco,@xcite does not seem to occur in ybco since in the latter the nodes are observed at both field orientations @xmath8= 45@xmath74 and @xmath8= -45@xmath74(see also refs . @xcite ) . the factor @xmath70 is related to the inelastic scattering rate @xmath66 , which was not taken explicitly into account in the previous calculations . however , an approximation can be carried out to get roughly the temperature dependence of the total scattering rate , which could be in part associated to the temperature dependence of @xmath66 . since the as mechanism is weakly temperature dependent below @xmath75 , we may approximate the total scattering rate as @xmath76 the temperature dependence of the inelastic scattering rate is given then by the factor @xmath70 in this approximation as @xmath77 in fig . [ fig6 ] we plot @xmath78 . as expected , we recover the overall behavior of the inelastic scattering rate already described by many authors@xcite . below @xmath1 it decreases rapidly and becomes negligible in comparison with the impurity and as scattering rates below @xmath79 k. although the approximation given by ( [ aprox ] ) is too rough to obtain the true temperature dependence of the inelastic scattering rate , it is instructive to compare the obtained power dependence with results from literature . thermal hall angle measurements performed in the same ybco twinned crystal show that @xmath80 down to @xmath81k ( @xmath82 is the effective mass and @xmath83 is the hall scattering time of the quasiparticles responsible for the hall signal).@xcite the power law dependence obtained for @xmath84 is similar to that obtained for the longitudinal scattering rate assuming a @xmath3wave pairing.@xcite since quasiparticle - quasiparticle scattering mechanism should be the dominant temperature dependent inelastic scattering below @xmath1 , we expect a rate proportional to the density of quasiparticles . interestingly , within the simple two - fluid model we expect a density of quasiparticles proportional to @xmath85 . on the other hand , a @xmath86 dependence is expected within the spin - fluctuation scattering picture@xcite . in summary , we have measured the longitudinal thermal conductivity @xmath24 in two single crystals of ybco in the presence of a planar magnetic field which was rotated parallel to the cuo@xmath6 planes , from @xmath87 k up to a few kelvins below @xmath1 . fourfold oscillations were recorded below @xmath88 k. above this temperature the angle dependence of the longitudinal thermal conductivity changes ; from the minimum at @xmath89 ( field parallel to the heat current ) a maximum develops at high @xmath90 . the observed behavior in the whole temperature range can be very well reproduced by a model involving andreev scattering of quasiparticles by vortices and the two - dimensional brt expression for the thermal conductivity assuming a @xmath3wave pairing . the overall results agree with the @xmath4-pairing symmetry of the order parameter . this agreement suggests that the mechanisms that influence the behavior of the quasiparticles below @xmath1 and above @xmath91k are well described by the fermi liquid theory at nearly optimal doping . the small differences found in the angle patters between the twinned and untwinned samples can be explained in terms of the different impurity concentration as well as oxygen deficiency and are accounted for by the phenomenology exposed in this work .

the angle dependence at different temperatures of the longitudinal thermal conductivity @xmath0 in the presence of a planar magnetic field is presented . in order to study the influence of the gap symmetry on the thermal transport , angular scans were measured up to a few kelvin below the critical temperature @xmath1 . we found that the four - fold oscillation of @xmath0 vanishes at @xmath2k and transforms into a one - fold oscillation with maximum conductivity for a field of 8 t applied parallel to the heat current . nevertheless , the results indicate that the @xmath3wave pairing symmetry remains the main pairing symmetry of the order parameter up to @xmath1 . numerical results of the thermal conductivity using an andreev reflection model for the scattering of quasiparticles by supercurrents under the assumption of @xmath3wave symmetry provide a semiquantitative description of the overall results .

arxiv

_ dehn surgery _ is an operation to modify a three - manifold by drilling and then regluing a solid torus . denote by @xmath6 the resulting three - manifold via dehn surgery on a knot @xmath0 in @xmath7 along a slope @xmath8 . two dehn surgeries along @xmath0 with distinct slopes @xmath8 and @xmath9 are called _ purely cosmetic _ if @xmath10 as oriented manifolds . in gordon s 1990 icm talk ( * conjecture 6.1 ) and kirby s problem list ( * ? ? ? * problem 1.81 a ) , it is conjectured that two surgeries on inequivalent slopes are never purely cosmetic . we shall refer to this as the _ cosmetic surgery conjecture_. in the present paper we study purely cosmetic surgeries along knots in the three - sphere @xmath11 . we show that for most knots @xmath0 in @xmath11 , @xmath12 as oriented manifolds for distinct slopes @xmath8 , @xmath9 . more precisely , our main result gives a sufficient condition for a knot @xmath0 that admits no purely cosmetic surgery in terms of its jones polynomial @xmath13 . [ main1 ] if a knot @xmath0 has either @xmath1 or @xmath2 , then @xmath12 for any two distinct slopes @xmath8 and @xmath9 . here , @xmath14 and @xmath15 denote the second and third order derivative of the jones polynomial of @xmath0 evaluated at @xmath16 , respectively . note that in ( * ? ? ? * proposition 5.1 ) , boyer and lines obtained a similar result for knots @xmath0 with @xmath17 , where @xmath18 is the normalized alexander polynomial . we shall see that @xmath19 ( lemma [ a2 ] ) . hence , our result can be viewed as an improvement of their result ( * ? ? ? * proposition 5.1 ) . previously , other known classes of knots that are shown not to admit purely cosmetic surgeries include the genus @xmath20 knots @xcite and the knots with @xmath21 @xcite , where @xmath22 is the concordance invariant defined by ozsvth - szab @xcite and rasmussen @xcite using floer homology . theorem [ main1 ] along with the condition @xmath21 give an effective obstruction to the existence of purely cosmetic surgery . for example , we used knotinfo @xcite , knot atlas @xcite and baldwin - gillam s table in @xcite to list all knots that have simultaneous vanishing @xmath14 , @xmath15 and @xmath22 invariant . we get the following result : the cosmetic surgery conjecture is true for all knots with no more than @xmath3 crossings , except possibly @xmath23 @xmath24 in @xcite , ozsvth and szab gave the example of @xmath25 , which is a genus two knot with @xmath26 and @xmath27 . @xmath28 and @xmath29 have the same heegaard floer homology , so no heegaard floer type invariant can distinguish these two surgeries . this example shows that theorem [ main1 ] and those criteria from heegaard floer theory are independent and complementary . the essential new ingredient in this paper is a surgery formula by lescop , which involves a knot invariant @xmath30 that satisfies a crossing change formula ( * ? ? ? * section 7 ) . we will show that @xmath30 is actually the same as @xmath31 . meanwhile , we also observe that @xmath30 is a finite type invariant of order @xmath5 . this enables us to reformulate theorem [ main1 ] in term of the finite type invariants of the knot ( theorem [ cormain ] ) . as another application of theorem [ main1 ] , we prove the nonexistence of purely cosmetic surgery on certain families of two - bridge knots and whitehead doubles . along the way , an explicit closed formula for the canonically normalized finite type knot invariant of order 3 @xmath32 is derived for two - bridge knots in conway forms @xmath33 in proposition [ v3formula ] , which could be of independent interest . the remaining part of this paper is organized as follows . in section 2 , we review background and properties of jones polynomial , and prove crossing change formulae for derivatives of jones polynomial . in section 3 , we define an invariant @xmath34 for rational homology spheres and then use lescop s surgery formula to prove theorem [ main1 ] . in section 4 and section 5 , we study in more detail cosmetic surgeries along two - bridge knots and whitehead doubles . * acknowledgements . * the authors would like to thank tomotada ohtsuki and ryo nikkuni for stimulating discussions and drawing their attention to the reference @xcite@xcite . the first named author is partially supported by jsps kakenhi grant number 26400100 . the second named author is partially supported by grant from the research grants council of hong kong special administrative region , china ( project no . 14301215 ) . suppose @xmath35 , @xmath36 , @xmath37 is a skein triple of links as depicted in figure [ crossings ] . , @xmath36 , @xmath38 are identical except at one crossing.,title="fig : " ] ( -188,-10)@xmath39 ( -109,-10)@xmath36 ( -30 , -10)@xmath38 recall that the _ jones polynomial _ satisfies the skein relation @xmath40 and the _ conway polynomial _ satisfies the skein relation @xmath41 the _ normalized alexander polynomial _ @xmath42 is obtained by substituting @xmath43 into the conway polynomial . for a knot @xmath0 , denote @xmath44 the @xmath45-term of the conway polynomial @xmath46 . it is not hard to see that @xmath47 . if one differentiates equations ( [ jones ] ) and ( [ conway ] ) twice and compares the corresponding terms , one can also show that @xmath48 . see @xcite for details . in summary , we have : [ a2]for all knots @xmath49 , @xmath50 in @xcite , lescop defined an invariant @xmath30 for a knot @xmath0 in a homology sphere @xmath7 . when @xmath51 , the knot invariant @xmath30 satisfies a crossing change formula @xmath52 where @xmath53 is a skein triple consisting of two knots @xmath54 and a two - component link @xmath55 ( * ? ? ? * proposition 7.2 ) . clearly , the values of @xmath56 are uniquely determined by this crossing change formula once we fix @xmath30(@xmath57 ) for the unknot . this gives an alternative characterization of the invariant @xmath30 for knots in @xmath11 . the next lemma relates it to the derivatives of jones polynomial . [ w3 ] for all knots @xmath49 , @xmath58 the main argument essentially follows from nikkuni ( * ? ? ? * proposition 4.2 ) . we prove the lemma by showing that @xmath31 satisfies an identical crossing change formula as equation ( [ w3def ] ) . to this end , we differentiate the skein formula for the jones polynomial ( [ jones ] ) three times and evaluate at @xmath16 . abbreviating the jones polynomial of the skein triple @xmath59 , @xmath60 and @xmath61 by @xmath62 , @xmath63 and @xmath64 , respectively , we obtain @xmath65 the terms on the right hand side can be expressed as * @xmath66 * @xmath67 * @xmath68 , @xmath69 * @xmath70 * @xmath71 * @xmath72 here , ( a ) and ( d ) are well - known ; ( b),(c),(e ) and ( f ) are proved by murakami @xcite . after doing substitution and simplification , we have @xmath73 meanwhile , it follows from ( [ conway ] ) and hoste ( * ? ? ? * theorem 1 ) that @xmath74 this enables us to further simplify @xmath75 and reduce it to the same expression as the right hand side of ( [ w3def ] ) . as @xmath76 also equals @xmath77 when @xmath0 is the unknot , @xmath31 must equal @xmath56 for all @xmath49 . we conclude the section by remarking that both lemma [ a2 ] and lemma [ w3 ] can be seen in a simpler way from a more natural perspective . a knot invariant @xmath78 is called a _ finite type invariant _ of order @xmath79 if it can be extended to an invariant of singular knots via a skein relation @xmath80 where @xmath81 is the knot with a transverse double point ( see figure [ crossings2 ] ) , while @xmath78 vanishes for all singular knots with @xmath82 singularities . with a transverse double point ] it follows readily from the definition that the set of finite type invariant of order @xmath77 consists of all constant functions . one can also show that @xmath44 and @xmath14 are finite type invariants of order @xmath83 , while @xmath56 and @xmath15 are finite type invariants of order @xmath5 . as the dimension of the set of all finite type invariants of order @xmath84 and @xmath85 are two and three , respectively ( see , e.g. , @xcite ) , there has to be a linear dependence among the above knot invariants , from which one can easily deduce lemma [ a2 ] and lemma [ w3 ] . in fact , if we denote @xmath86 and @xmath87 the finite type invariants of order @xmath83 and @xmath5 respectively normalized by the conditions that @xmath88 and @xmath89 for any knot @xmath0 and its mirror image @xmath90 and that @xmath91 for the right hand trefoil @xmath92 , then it is not difficult to see that @xmath93 and @xmath94 the goal of this section is to prove theorem [ main1 ] . recall the following results about purely cosmetic surgery from ( * ? ? ? * proposition 5.1 ) and ( * ? ? ? * theorem 1.2 ) . [ mainlemma ] suppose @xmath0 is a nontrivial knot in @xmath11 , @xmath95 are two distinct slopes such that @xmath96 as oriented manifolds . then the following assertions are true : * @xmath27 . * @xmath97 . * if @xmath98 , where @xmath99 are coprime integers , then @xmath100 . * @xmath26 , where @xmath22 is the concordance invariant defined by ozsvth - szab @xcite and rasmussen @xcite . our new input for the cosmetic surgery problem is lescop s @xmath34 invariant which , roughly speaking , is the degree @xmath83 part of the kontsevich - kuperberg - thurston invariant of rational homology spheres @xcite . like the famous le - murakami - ohtsuki invariant , the kontsevich - kuperberg - thurston invariant is universal among finite type invariants for homology spheres @xcite@xcite@xcite . see also ohtsuki @xcite for the connection to perturbative and quantum invariants of three - manifolds . we briefly review the construction . jacobi diagram _ is a graph without simple loop whose vertices all have valency @xmath5 . the degree of a jacobi diagram is defined to be half of the total number of vertices of the diagram . if we denote by @xmath101 the vector space generated by degree @xmath79 jacobi diagrams subject to certain equivalent relations as and ihx , then the degree @xmath79 part @xmath102 of the kontsevich - kuperberg - thurston invariant takes its value in @xmath101 . simple argument in combinatorics implies that * @xmath103 is an @xmath20-dimensional vector space generated by the jacobi diagram @xmath104{tata.pdf } } $ ] * @xmath105 is a @xmath83-dimensional vector space generated by the jacobi diagrams @xmath104{tata.pdf } } \ ; { \includegraphics[bb= 0 50 163 160 , scale=.09]{tata.pdf } } $ ] and @xmath106{tetra.pdf } } $ ] many interesting real invariants of rational homology spheres can be recovered from the kontsevich - kuperberg - thurston invariant @xmath107 by composing a linear form on the space of jacobi diagrams . in the simplest case , the casson - walker invariant @xmath108 is @xmath109 , where @xmath110{tata.pdf } } ) = 2 $ ] . we shall concentrate on the case of the degree @xmath83 invariant @xmath111 , where @xmath112{tetra.pdf } } ) = 1 $ ] and @xmath113{tata.pdf } } \ ; { \includegraphics[bb= 0 50 163 160 , scale=.09]{tata.pdf } } ) = 0 $ ] . the following surgery formula for @xmath34 is proved by lescop and will play a central role in the proof of our main result . * theorem 7.1 ) [ lescopsurgery ] the invariant @xmath34 satisfies the surgery formula @xmath114 for all knots @xmath115 . here , @xmath44 is the @xmath45-coefficient of @xmath46 , and @xmath116 is the lens space obtained by @xmath117 surgery on the unknot . then @xmath56 is a knot invariant , which was shown earlier in lemma [ w3 ] to be equal to @xmath31 for @xmath49 . the terms @xmath118 and @xmath119 are both explicitly defined in @xcite , but they will not be needed for our purpose . for the moment , we make the following simple observation . [ w3cosmetic ] suppose @xmath0 is a knot in @xmath11 with @xmath120 , and @xmath121 are nonzero integers satisfying @xmath122 . then @xmath123 if and only if @xmath124 . we apply the surgery formula in theorem [ lescopsurgery ] . note that the first and third terms of the right hand side are clearly equal for @xmath117 and @xmath125 surgery . next , recall the well - known theorem that two lens spaces @xmath126 and @xmath127 are equivalent up to orientation - preserving homeomorphisms if and only if @xmath128 . in particular , this implies the lens spaces @xmath129 as oriented manifolds if @xmath130 , so their @xmath34 invariants are obviously the same . consequently , @xmath131 and the statement follows readily . in light of theorem [ mainlemma ] , we only need to consider the case when @xmath27 and @xmath132 , for otherwise , the pair of manifolds @xmath133 and @xmath134 will be non - homeomorphic as oriented manifolds . thus @xmath135 . if we now assume @xmath136 , then lemma [ w3 ] implies that @xmath137 . we can then apply proposition [ w3cosmetic ] and conclude that @xmath138 . consequently , @xmath139 . given ( [ v2 ] ) and ( [ v3 ] ) , theorem [ main1 ] can be stated in the following equivalent way , which is particularly useful in the case where it is easier to calculate the finite type invariant @xmath87 ( or equivalently @xmath30 ) than the jones polynomial . [ cormain ] if a knot @xmath0 has the finite type invariant @xmath140 or @xmath141 , then @xmath12 for any two distinct slopes @xmath8 and @xmath9 . in this section , we derive an explicit formula for @xmath87 and use it to study the cosmetic surgery problem for two - bridge knots . following the presentation of ( * ? ? ? * section 2.1 ) , we sketch the basic properties and notations for two - bridge knots . every two - bridge knot can be represented by a rational number @xmath142 for some odd integer @xmath143 and even integer @xmath144 . if we write this number as a continued fraction with even entries and of even length @xmath145=2b_1 + \cfrac{1}{2c_1 + \cfrac{1}{\cdots + \cfrac{1}{2b_m + \cfrac{1}{2c_m } } } } \ ] ] for some nonzero integers @xmath146 s and @xmath147 s , then we obtain the _ conway form _ @xmath148 of the two - bridge knot , which is a special knot diagram as depicted in figure [ conwayform ] . we will write @xmath149 for the knot of conway form @xmath148 . the genus of @xmath150 is @xmath151 ; and conversely , every two - bridge knot of genus @xmath151 has such a representation . of a two bridge knot . in the figure , there are @xmath152 positive ( resp . negative ) full - twists if @xmath153 ( resp @xmath154 ) , and there are @xmath155 negative ( resp . positive ) full - twists if @xmath156 ( resp . @xmath157 ) for @xmath158 . ] burde obtained the following formula for @xmath159 , the @xmath45-coefficient of the conway polynomial of @xmath150 . * proposition 5.1)[burde ] for the two - bridge knot @xmath149 , the @xmath45-coefficient of the conway polynomial is given by @xmath160 the above formula can be proved by recursively applying equation ( [ a2crossingchange ] ) . the similar idea can be used to find an analogous formula for @xmath30 , which is the main task of the next few lemmas . [ w3recursivelemma ] the invariant @xmath30 satisfies the recursive formula @xmath161 this follows from a direct application of the crossing change formula ( [ w3def ] ) at the rightmost crossing in figure [ conwayform ] , and the observation that both @xmath162 and @xmath163 are the unknot with @xmath164 . [ w3lemma2 ] the invariant @xmath30 satisfies the recursive formula @xmath165 we first prove the lemma for @xmath166 . we repeatedly apply lemma [ w3recursivelemma ] until @xmath167 is reduced to @xmath77 . note that the knot @xmath168 can be isotoped to @xmath169 by untwisting the far - right @xmath170 full twists . therefore , @xmath171 now , the lemma follows from substituting @xmath172 which is an immediate corollary of proposition [ burde ] . the case when @xmath173 is proved analogously . finally , applying lemma [ w3lemma2 ] and induction on @xmath151 , we obtain an explicit formula for @xmath30 , and consequently also for @xmath87 . [ v3formula ] @xmath174 we use induction on @xmath151 . for the base case @xmath175 , lemma [ w3lemma2 ] readily implies that @xmath176 so @xmath177 satisfies the formula . next we prove that if the formula holds for @xmath178 , then it also holds for @xmath33 . it suffices to show that @xmath179 where @xmath180 the above identity can be verified from tedious yet elementary algebra . we omit the computation here . for the rest of the section , we apply theorem [ cormain ] and proposition [ v3formula ] to study the cosmetic surgery problems for the two - bridge knots of genus @xmath83 and @xmath5 , which correspond to the conway form @xmath181 and @xmath182 , respectively . note that the cosmetic surgery conjecture for genus one knot is already settled by wang @xcite . if a genus @xmath83 two - bridge knot @xmath181 is not of the form @xmath183 for some integers @xmath184 , then it does not admit purely cosmetic surgeries . suppose there are purely cosmetic surgeries for the knot @xmath181 . theorem [ cormain ] implies that @xmath185 and @xmath186 where the formula for @xmath187 and @xmath87 follows from proposition [ burde ] and proposition [ v3formula ] , respectively . from equation ( [ v2genus2 ] ) , we see @xmath188 and @xmath189 , which was then substituted into the second and the third terms of equation ( [ v3genus2 ] ) , and gives @xmath190 hence , @xmath191 . plugging this identity back to equation ( [ v2genus2 ] ) , we see @xmath192 . as a result , the two - bridge knot @xmath181 can be written as @xmath183 for some integers @xmath167 and @xmath193 . we can perform a similar computation for a genus @xmath5 two - bridge knot @xmath182 . by proposition [ v3formula ] , @xmath194 in particular , we see @xmath195 consequently , theorem [ cormain ] implies the family of two - bridge knots @xmath196 does not admit purely cosmetic surgeries . as explained in @xcite , both @xmath197 and @xmath198 are @xmath77 for the knot @xmath196 . hence , purely cosmetic surgery could not be ruled out by previously known results from theorem [ mainlemma ] . we are devoted to @xmath199 in this section , where @xmath199 denotes the satellite of @xmath0 for which the pattern is a positive - clasped twist knot with @xmath79 twists . the knot @xmath199 is called the _ positive @xmath79-twisted whitehead double _ of a knot @xmath0 . see figure [ whitehead ] for an illustration . we apply the crossing change formula ( [ w3def ] ) at either one of the crossings of the clasps . note that @xmath204 , @xmath205 is the unknot , and @xmath206 . the classical formula for the alexander polynomial of a satellite knot implies that @xmath207 , from which we compute @xmath208 also observe that @xmath209 . therefore , @xmath210 and so @xmath211 since the invariant @xmath212 , the whitehead double @xmath213 does not admit purely cosmetic surgeries if @xmath214 . when @xmath215 , proposition [ v3whitehead ] gives @xmath216 . hence , theorem [ cormain ] immediately implies the following corollary . there is no purely cosmetic surgery for the positive @xmath79-twisted whitehead double @xmath213 for @xmath214 . moreover , if @xmath217 , then there is no purely cosmetic surgery for the untwisted whitehead double @xmath218 .

we show that two dehn surgeries on a knot @xmath0 never yield manifolds that are homeomorphic as oriented manifolds if @xmath1 or @xmath2 . as an application , we verify the cosmetic surgery conjecture for all knots with no more than @xmath3 crossings except for three @xmath4-crossing knots and five @xmath3-crossing knots . we also compute the finite type invariant of order @xmath5 for two - bridge knots and whitehead doubles , from which we prove several nonexistence results of purely cosmetic surgery .

arxiv

anomaly - mediated supersymmetry breaking ( amsb ) models have received much attention in the literature due to their attractive properties@xcite : the soft supersymmetry ( susy ) breaking terms are completely calculable in terms of just one free parameter ( the gravitino mass , @xmath11 ) , the soft terms are real and flavor invariant , thus solving the susy flavor and @xmath12 problems , the soft terms are actually renormalization group invariant@xcite , and can be calculated at any convenient scale choice . in order to realize the amsb set - up , the hidden sector must be `` sequestered '' on a separate brane from the observable sector in an extra - dimensional universe , so that tree - level supergravity breaking terms do not dominate the soft term contributions . such a set - up can be realized in brane - worlds , where susy breaking takes place on one brane , with the visible sector residing on a separate brane . the soft susy breaking ( ssb ) terms arise from the rescaling anomaly . in spite of its attractive features , amsb models suffer from the well - known problem that slepton mass - squared parameters are found to be negative , giving rise to tachyonic states . the original solution to this problem is to suppose that scalars acquire as well a universal mass @xmath13 , which when added to the amsb ssb terms , renders them positive . thus , the parameter space of the `` minimal '' amsb model ( mamsb ) is given by m_0 , m_3/2 , , sign ( ) . an alternative set - up for amsb has been advocated in ref . @xcite , known as hypercharged anomaly - mediation ( hcamsb ) . it is a string motivated scenario which uses a similar setup as the one envisioned for amsb . in hcamsb , susy breaking is localized at the bottom of a strongly warped hidden region , geometrically separated from the visible region where the mssm resides . the warping suppresses contributions due to tree - level gravity mediation@xcite and the anomaly mediation@xcite can become the dominant source of susy breaking in the visible sector . possible exceptions to this sequestering mechanism are gaugino masses of @xmath14 gauge symmetries @xcite . thus , in the mssm , the mass of the bino the gaugino of @xmath15 can be the only soft susy breaking parameter not determined by anomaly mediation@xcite . depending on its size , the bino mass @xmath16 can lead to a small perturbation to the spectrum of anomaly mediation , or it can be the largest soft susy breaking parameter in the visible sector : as a result of rg evolution its effect on other soft susy breaking parameters can dominate the contribution from anomaly mediation . in extensions of the mssm , additional @xmath17s can also communicate susy breaking to the mssm sector @xcite . besides sharing the same theoretical setup , anomaly mediation and hypercharge mediation cure phenomenological shortcomings of each other . the minimal amsb model predicts a negative mass squared for the sleptons ( and features relatively heavy squarks ) . on the other hand , the pure hypercharge mediation suffers from negative squared masses for stops and sbottoms ( and features relatively heavy sleptons ) : see sec . [ sec : pspace ] . as a result , the combination of hypercharge and anomaly mediation leads to phenomenologically viable spectra for a sizable range of relative contributions @xcite . we parametrize the hcamsb ssb contribution @xmath18 using a dimensionless quantity @xmath2 such that @xmath19 so that @xmath2 governs the size of the hypercharge contribution to soft terms relative to the amsb contribution . then the parameter space of hcamsb models is given by , m_3/2 , , sign ( ) . in the hcamsb model , we assume as usual that electroweak symmetry is broken radiatively by the large top - quark yukawa coupling . then the ssb @xmath20 term and the superpotential @xmath21 term are given as usual by the scalar potential minimization conditions which emerge from requiring an appropriate breakdown of electroweak symmetry . in hcamsb , we take the ssb terms to be of the form : m_1 & = & _ 1+m_3/2,m_a & = & m_3/2 , a=2 , 3 m_i^2 & = & -14\ { _ g+_f}m_3/2 ^ 2 a_f & = & m_3/2 , where @xmath22 , @xmath23 is the beta function for the corresponding superpotential coupling , and @xmath24 with @xmath25 the wave function renormalization constant . the wino and gluino masses ( @xmath26 and @xmath27 ) receive a contribution from the bino mass at the two loop level . thus , in pure hypercharge mediation , they are one loop suppressed compared to the scalar masses . for convenience , we assume the above ssb mass parameters are input at the gut scale , and all weak scale ssb parameters are determined by renormalization group evolution . we have included the above hcamsb model into the isasugra subprogram of the event generator isajet v7.79@xcite . after input of the above parameter set , isasugra then implements an iterative procedure of solving the mssm rges for the 26 coupled renormalization group equations , taking the weak scale measured gauge couplings and third generation yukawa couplings as inputs , as well as the above - listed gut scale ssb terms . isasugra implements full 2-loop rg running in the @xmath28 scheme , and minimizes the rg - improved 1-loop effective potential at an optimized scale choice @xmath29@xcite to determine the magnitude of @xmath21 and @xmath30 . all physical sparticle masses are computed with complete 1-loop corrections , and 1-loop weak scale threshold corrections are implemented for the @xmath31 , @xmath32 and @xmath33 yukawa couplings@xcite . the off - set of the weak scale boundary conditions due to threshold corrections ( which depend on the entire superparticle mass spectrum ) , necessitates an iterative up - down rg running solution . the resulting superparticle mass spectrum is typically in close accord with other sparticle spectrum generators@xcite . once the weak scale sparticle mass spectrum is known , then sparticle production cross sections and branching fractions may be computed , and collider events may be generated . then , signatures for hcamsb at the cern lhc may be computed and compared against standard model ( sm ) backgrounds . our goal in this paper is to characterize the hcamsb parameter space and sparticle mass spectrum , and derive consequences for the cern lhc @xmath34 collider , which is expected to begin operation in fall , 2009 . some previous investigations of mamsb at lhc have been reported in ref . @xcite . the remainder of this paper is organized as follows . in sec . [ sec : pspace ] , we calculate the allowed parameter space of hcamsb models , imposing various experimental and theoretical constraints . we also show sample mass spectra from hcamsb models , and show their variation with @xmath2 and @xmath11 . we show typical values of @xmath35 and @xmath36 that result . in sec . [ sec : lhc ] , we explore consequences of the hcamsb model for lhc sparticle searches . typically , collider events are characterized by production of high @xmath37 @xmath32 and @xmath31 quarks , along with @xmath38 and observable tracks from late decaying charginos @xmath39 . for small @xmath2 , slepton pair production may be visible , while for large @xmath2 , direct @xmath40 and @xmath41 production may be visible . the lhc reach for 100 fb@xmath3 should extend up to @xmath42 ( 105 ) tev , corresponding to a reach in @xmath43 ( 2.2 ) tev , for small ( large ) values of @xmath2 . the hcamsb model should be easily distinguishable from the mamsb model at the lhc if @xmath6 is not too large , due to the presence of @xmath44 candidates in cascade decay events . the presence of these reflects the mass ordering @xmath0 in the hcamsb model , while @xmath45 in the mamsb model . in sec . [ sec : conclude ] , we present our conclusions and outlook for hcamsb models . we begin our discussion by plotting out in fig . [ fig : m10 ] the mass spectra of various sparticles versus _ a _ ) . @xmath46 in mamsb and _ b _ ) . @xmath2 in the hcamsb model , for @xmath11 fixed at 50 tev , while taking @xmath47 , @xmath48 and @xmath49 gev . for @xmath13 and @xmath50 , the yellow - shaded region yields the well - known tachyonic slepton mass - squared values , which could lead to electric charge non - conservation in the scalar potential . in mamsb , as @xmath13 increases , all the scalars increase in mass , while @xmath51 , @xmath52 and @xmath53 remain roughly constant , and the superpotential @xmath21 term decreases . the large @xmath13 limit of parameter space is reached around @xmath54 , where ewsb is no longer properly broken ( signaled by @xmath55 ) . we also see the well - known property of mamsb models that @xmath56 . in addition , an important distinction between the two models is the mass ordering which enters into the neutralino mass matrix : we find typically that @xmath0 in the hcamsb model , while @xmath1 in mamsb . thus , both models will have a wino - like @xmath57 state . however , in the hcamsb model , the @xmath58 are dominantly higgsino - like states , with @xmath59 being bino - like , while in the mamsb model , we expect @xmath60 to be bino - like with @xmath61 being higgsino - like . this mass ordering difference will give rise to a crucial distinction in lhc susy cascade decay events ( see sec . [ sec : lhc ] ) which may serve to distinguish the two models . in the hcamsb case , as @xmath2 increases , the gut scale gaugino mass @xmath16 increases . thus , the bino mass increases with @xmath2 , while the light charginos @xmath39 and neutralino @xmath57 remain wino - like with mass fixed near @xmath26 , and the gluino remains with mass fixed at nearly @xmath62 . many of the scalar masses also vary with @xmath2 . the reason is that as @xmath2 increases , so does the gut scale value of @xmath16 . the large value of @xmath16 feeds into the scalar masses via their renormalization group equations , causing many of them to increase with @xmath2 , with the largest increases occurring for the scalars with the largest weak hypercharge assignments @xmath63 . thus , we see strong increases in the @xmath64 , @xmath65 and especially the @xmath66 masses with increasing @xmath67 . the @xmath68 squark only receives a small increase in mass , since its hypercharge value is quite small : @xmath69 . from fig . [ fig : m10]_b _ ) . , we already see an important distinction between mamsb and hcamsb models : in the former case , the @xmath65 and @xmath66 states are nearly mass degenerate , while in the latter case these states are highly split , with @xmath70 . an exception to the mass increase with @xmath2 in fig . [ fig : m10]_b _ ) . occurs in the values of @xmath71 and @xmath72 . in these cases , the large increase in @xmath73 feeds into the rge @xmath74 term@xcite , and _ amplifies _ the top - quark yukawa coupling suppression of the @xmath75 term . since the doublet @xmath76 contains both the @xmath77 and @xmath78 states , both of these actually suffer a _ decrease _ in mass with increasing @xmath2 . thus , we expect in hcamsb models with moderate to large @xmath2 that the third generation squark states will be highly split . for large @xmath67 , we expect the light third generation squarks @xmath79 and @xmath80 to be quite light , with a dominantly left- squark composition . the heavier squarks @xmath81 and @xmath82 will be quite heavy , and dominantly right - squark states . in addition , we see from fig . [ fig : m10]_b _ ) . that the superpotential @xmath21 term _ decreases _ with increasing @xmath2 . at moderate - to - large @xmath6 , the @xmath21 term is from the tree - level scalar potential minimization conditions @xmath83 . the running of @xmath84 versus energy scale @xmath85 is shown in fig . [ fig : mhu ] for @xmath86 and 0.195 . we see that as @xmath2 increases , the value of @xmath87 actually decreases , leading to a small @xmath88 value . the relevant rge reads = ( - g_1 ^ 2m_1 ^ 2 -3g_2 ^ 2m_2 ^ 2 + g_1 ^ 2 s+3f_t^2 x_t ) . a large value of @xmath16 thus leads to an _ upwards _ push to @xmath84 in its early running from @xmath89 , which is only later compensated by the downward push of the yukawa - coupling term involving the top yukawa coupling @xmath90 . in the figure , for the case of @xmath91 , the weak scale value of @xmath84 is actually positive . upon adding the large 1-loop corrections to the effective potential ( due to the light top - squark ) , the rg - improved scalar potential yields a positive value of @xmath88 . thus , in the region of large @xmath2 , where @xmath21 becomes small and comparable to @xmath26 , we expect the neutralino @xmath57 to become a mixed wino - higgsino particle , and the corresponding @xmath92 mass gap to increase beyond the value @xmath93 mev which is expected in amsb models@xcite . parameter as a function of energy scale @xmath85 for @xmath94 , @xmath95 and @xmath96 for @xmath97 tev and @xmath47 , @xmath48 in the hcamsb model . , scaledwidth=50.0% ] an interesting coincidence related to the rg evolution of @xmath84 in the limit where hypercharge mediation dominates is that the electroweak symmetry breaking _ requires _ the electroweak scale to be @xmath98 orders of magnitude below the scale @xmath99 ( @xmath99 may be of order the gut scale or string scale ) at which the bino mass @xmath16 is generated . if the hierarchy between the electroweak scale and @xmath99 was smaller , then a susy breaking scenario in which hypercharge mediation dominates would not be capable of triggering ewsb ( the energy interval for rg evolution would not be large enough to drive the @xmath84 parameter to negative values ) . this is a very uncommon feature among susy breaking scenarios . for a more detailed comparison , we list in table [ tab : cases ] the sparticle mass spectrum for a mamsb point with @xmath100 gev , @xmath97 tev , @xmath47 and @xmath48 , and two hcamsb points with small and large @xmath2 values equal to @xmath101 and @xmath96 . while all three cases have a comparable gluino mass , we see that the rather small splitting amongst @xmath102 and also @xmath103 states in mamsb is turned to large left - right splitting in the hcamsb cases . we also see that the @xmath104 mev mass gap in amsb and hcamsb1 which leads to long - lived and possibly observable @xmath105 tracks in collider detectors opens up to a few gev in the hcamsb2 case . the latter mass gap is large enough to make the @xmath105 state less long lived , although still maintaining possibly measureable tracks in collider scattering events . the value of @xmath106 versus @xmath2 is shown in fig . [ fig : ctau ] , where we usually get @xmath107 mm for most @xmath2 values . the value drops to shorter lengths for large @xmath2 . the shorter travel time of the @xmath105 would distinguish the large @xmath2 hcamsb case with a mixed higgsino - wino @xmath57 state from the low @xmath2 hcamsb case where @xmath57 is instead nearly pure wino - like . .masses and parameters in gev units for three case study points amsb , hcamsb1 and hcamsb2 using isajet 7.79 with @xmath49 gev and @xmath48 . we also list the total tree level sparticle production cross section in fb at the lhc . [ cols="<,^,^,^",options="header " , ] in this paper , we have examined some phenomenological consequences of hypercharged anomaly - mediated susy breaking models at the lhc . we have computed the expected sparticle mass spectrum , and mapped out the relevant parameter space of the hcamsb model . we have computed sparticle branching fractions , production cross sections and expected lhc collider events , and compared against expectations for sm backgrounds . our main result was to compute the reach of the lhc for hcamsb models assuming 100 fb@xmath3 of integrated luminosity . we find an lhc reach to @xmath42 tev ( corresponding to @xmath43 tev ) for low values of @xmath2 , and a reach to @xmath108 tev ( corresponding to @xmath109 tev ) for large @xmath2 . we expect the reach for @xmath110 to be similar to the reach for @xmath48 , due to similarities in the spectra for the two cases ( see fig . [ fig : m10 ] . ) we also expect the reach for large @xmath6 to be similar to the reach for low @xmath6 in the @xmath111 and @xmath112 channels ( differences in the multi - lepton channels can occur due to enhanced -ino decays to taus and @xmath32s at large @xmath6 ) . the lhc reach for hcamsb models tends to be somewhat lower than the reach for mamsb models , where ref . @xcite finds a 100 fb@xmath3 reach of @xmath113 tev for low values of @xmath13 . this is due in part because , in mamsb , the various squark states are more clustered about a common mass scale @xmath13 , while in hcamsb the squark states are highly split , with @xmath114 . the hcamsb lhc event characteristics suffer similarities and differences with generic mamsb models . both hcamsb and mamsb give rise to multi - jet plus multi - lepton plus @xmath38 event topologies , and within these event classes , it is expected that occasional hits of length a few _ cm _ will be found , arising from production of the long - lived wino - like chargino states . some of the major differences between the models include the following . a severe left - right splitting of scalar masses is expected in hcamsb , while left - right scalar degeneracy tends to occur in mamsb . this may be testable if some of the slepton states are accessible to lhc searches . it is well known that in mamsb , @xmath56 , while in hcamsb , @xmath115 , since the @xmath66 state has a large weak hypercharge quantum number . in addition , the lightest stau state , @xmath116 , is expected to be mainly a left- state in hcamsb , while it is mixed , but mainly a right- state in mamsb . while it is conceivable that the left - right mixing might be determined at lhc ( using branching fractions or tau energy distributions ) , such measurements would be easily performed at a linear @xmath117 collider , especially using polarized beams@xcite . in hcamsb models , the light third generation squarks @xmath79 and @xmath80 are expected to be generically lighter than the gluino mass , and frequently much lighter . this leads to cascade decays which produce large multiplicities of @xmath32 and @xmath31 quarks in the final state . thus , in hcamsb models , a rather high multiplicity of @xmath32 jets is expected . in mamsb , a much lower mutiplicity of @xmath32-jets is expected , although this depends also on the value of @xmath6 which is chosen . in hcamsb models , the @xmath14 gaugino mass @xmath16 is expected to be the largest of the gaugino masses , with a mass hierarchy of @xmath0 . this usually implies that the @xmath59 neutralino is mainly bino - like , while @xmath60 and @xmath118 are higgsino - like , and @xmath57 is wino - like . in contrast , in the mamsb model , usually the ordering is that @xmath1 , so that while @xmath57 is again wino - like , the @xmath60 state is bino - like , and @xmath118 and @xmath59 are higgsino - like . the compositions of the @xmath119 for @xmath120 will not be easy to determine at lhc , but will be more easily determined at a linear @xmath117 collider . however , the mass ordering gives rise to os dilepton distributions with a prominent @xmath121 peak in hcamsb , while such a peak should be largely absent in mamsb models ( except at large @xmath6 where there is greater mixing in the neutralino sector ) . thus , cascade decay events containing hits along with a @xmath121 peak in the os dilepton invariant mass distribution may be a smoking gun signature for hcamsb models at the lhc , at least within the lower range of @xmath6 . this work was supported in part by the u.s . department of energy . 99 l. randall and r. sundrum , _ nucl . * b 557 * ( 1999 ) 79 ; 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we investigate the phenomenological consequences of string models wherein the mssm resides on a d - brane , and the hypercharge gaugino mass is generated in a geometrically separated hidden sector . this hypercharged anomaly - mediated susy breaking ( hcamsb ) model naturally solves the tachyonic slepton mass problem endemic to pure amsb scenarios . in hcamsb , one obtains a mass ordering @xmath0 with split left- and right- scalars , whereas in mamsb models , one obtains @xmath1 with nearly degenerate left- and right- scalars . we compute the allowed parameter space and expected superparticle mass spectrum in the hcamsb model . for low values of the hc and amsb mixing parameter @xmath2 , the spectra is characterized by light left - sleptons , while the spectra for large @xmath2 is characterized by light top- and bottom- squarks . we map out the approximate reach of lhc for hcamsb , and find that with 100 fb@xmath3 of integrated luminosity , a gravitino mass of @xmath4 ( 105 ) tev can be probed for low ( high ) values of @xmath2 , corresponding to a gluino mass reach of @xmath5 ( 2.2 ) tev . both cases contain as is typical in amsb models long lived charginos that should yield visible highly ionizing tracks in the lhc detector . also , in the lower @xmath6 range , hcamsb models give rise to reconstructable @xmath7 candidates in susy cascade decay events , while mamsb models should do so only rarely . * prospects for hypercharged anomaly mediated + susy breaking at the lhc * + howard baer@xmath8 , radovan derm ' iek@xmath9 , shibi rajagopalan@xmath10 , heaya summy@xmath10 + _ 1 . dept . of physics and astronomy , university of oklahoma , norman , ok 73019 , usa + 2 . dept . of physics , indiana university , bloomington in 47405 , usa + _

arxiv

the ideas of zero range potential ( zrp ) approach were recently developed to widen limits of the traditional treatment by demkov and ostrovsky @xmath4 and albeverio et al . @xmath5 . the advantage of the theory is the possibility of obtaining an exact solution of scattering problem . the zrp is conventionally represented as the boundary condition on the matrix wavefunction at some point . alternatively , the zrp can be represented as pseudopotential ( breit @xmath6 ) . on the other hand , darboux transformation ( dt ) allows to construct in natural way exactly solvable potentials . general starting point of the theory goes up to matveev theorem ( see @xcite ) . the transformation can be also defined on the base of covariance property of the schrdinger equation with respect to a transformation of wavefunction and potential ( matveev and salle @xmath7 ) . darboux formulas in multi - dimensional space could be applied in the sense of andrianov , borisov and ioffe ideas @xcite . in the circumstances , dt technique can be used so as to correct zrp model . we attempt to dress the zrp in order to improve the possibilities of the zrp model . we use notations and some results from @xcite . dt modifies the generalized zrp ( gzrp ) boundary condition ( section @xmath8 ) and creates a potential with arbitrarily disposed discrete spectrum levels for any angular momentum @xmath9 . in the section @xmath10 we consider @xmath11-representation for a non - spherical potential so as to dress a multi - centered potential , which includes @xmath0 zrps . as an important example , we consider electron scattering by the @xmath1 and @xmath2 structures within the framework of the zrp model ( section @xmath12 ) . in section @xmath13 we present the our calculations for the electron-@xmath3 scattering and discuss them . let us start from the simplest case of a central field . then angular momentum operator commutates with hamiltonian and therefore wavefunction @xmath14 can be expanded in the spherical waves @xmath15 where @xmath16 , @xmath17 is initial particle direction , @xmath18 are partial waves , and @xmath19 are phase shifts . consider the radial schrdinger equation for partial wave with angular momentum @xmath9 . the atomic units are used throughout the present paper , i.e. @xmath20 and born radius @xmath21 . @xmath22 @xmath23 @xmath24 denotes differential operator , and @xmath25 are hamiltonian operators of the partial waves . this equations describe scattering of a particle with energy @xmath26 . the wavefunctions @xmath18 at infinity have the form @xmath27 let us consider gzrp in coordinate origin . this potential is conventionally represented as boundary condition on the wavefunction ( see @xmath28 ) @xmath29 where @xmath30 are inverse scattering lengths . the potential @xmath31 and therefore wavefunctions @xmath18 can be expressed in terms of the spherical functions @xmath32 where spherical functions @xmath33 are related to usual bessel functions as @xmath34 , @xmath35 . in the vicinity of zero they have the asymptotic behavior @xmath36 , and @xmath37 . to substituting the equation @xmath38 into the boundary condition we obtain the elements of @xmath11-matrix @xmath39 the bound states correspond to the poles of the @xmath11-matrix ( i.e the zeros of the denominator @xmath40 ) , which lie on the imaginary positive semi - axis of the complex @xmath41-plane . it is obvious that bound state , with orbital momentum @xmath9 , exists only if @xmath42 ( elsewise an antibound state exists ) and has the energy @xmath43 . thus , spectral problem for gzrp is solved for any value @xmath41 . on the other hand , the equations ( [ e ] ) are covariant with respect to dt that yields the following transformations of the potentials ( coefficients of the operator @xmath25 ) @xmath44 and the wavefunctions @xmath18 @xmath45 where @xmath46 are some solutions of the equations @xmath47 at @xmath48 , and @xmath49 are real parameters , which can be both positive or negative . the dt @xmath50 combines the solutions @xmath18 and a solution @xmath46 that corresponds to another eigen value @xmath51 . repeating the procedure we obtain a chain of the integrable potentials @xmath52 . in general , dressed potential @xmath53 is real for real function @xmath46 . the next step in the dressing procedure of the zero - range potential ( @xmath31 ) is a definition of the free parameters of the solutions @xmath46 . suppose the prop functions @xmath46 satisfy the boundary conditions @xmath54 with @xmath55 . in the simplest case of @xmath56 we have @xmath57 and @xmath58 the dt @xmath50 gives rise to the following requirement on dressed wavefunction @xmath59 the dressed potential @xmath60 is given by @xmath61 it is regular on semiaxis only if @xmath62 . in the limiting case at @xmath63 we obtain long - range interaction @xmath64 , which can be regular on semiaxis only if @xmath65 . assuming @xmath66 we get @xmath67 ( trivial transformation ) , and boundary condition can be obtained by the substitution : @xmath68 to dress free wave @xmath69 we obtain zrp at the coordinate origin . thus , zrp can be also introduced in terms of dt . to consider transformation with parameter @xmath70 we obtain regular solution @xmath71 and tangent of phase shift is @xmath72 in the other cases asymptotic of the functions @xmath73 at zero is given by @xmath74 it is clear that the each dt introduces short - range core of centrifugal type ( which depends on angular momentum @xmath9 ) in the potential . in this situation the boundary conditions on the dressed wavefunctions @xmath75 $ ] require modification . thus , in the case @xmath76 the boundary conditions become @xmath77 and in the case @xmath78 we obtain @xmath79 in the generalized case , zrp with angular momentum @xmath9 generates also @xmath80 complex poles of the @xmath11-matrix , which correspond the quasi - stationary states ( resonances ) . the dts @xmath50 with the parameters @xmath49 results in the @xmath11-matrix elements for dressed gzrp @xmath81 we can use darboux transformation in order to add ( or remove ) poles of the @xmath11-matrix . the principal observation allows to built a zero - range potential eigen function in the multi - center problem . let us consider @xmath0 zrps at the points @xmath82 and interaction @xmath83 . the wavefunction @xmath14 can be expressed in terms of the ( outgoing - wave ) green function , defined by the equation @xmath84 the second ( ingoing - wave ) green function is defined by @xmath85 . the partial waves @xmath86 , defined by @xmath87 for multi - centered target which includes @xmath0 zrps and interaction @xmath83 can be expressed as a superposition of the green functions @xmath88 in which @xmath89 are phase shifts , @xmath90 denote @xmath11-matrix orthonormal eigenfunctions , @xmath91 are real numbers . they naturally generalize the spherical partial waves @xmath92 for a non - spherical potential @xmath93 . expanding the partial waves @xmath86 at the infinity we obtain the expressions for @xmath90 @xmath94 where @xmath95 is wavefunction for potential @xmath83 , and @xmath96 is scattering amplitude . scattering amplitude for potential @xmath83 and @xmath0 zrp is given by @xmath97 by the imposition of the boundary conditions @xmath54 the calculation of the partial waves is reduced to the solution of the following system @xmath98 in which we use @xmath99 where @xmath100 , @xmath101 . the tangents @xmath102 can be found from compatibility condition of this system . in simplest case @xmath103 @xmath104 the system @xmath105 is reduced to the usual equations of zrp theory @xmath93 . in order to construct functions @xmath106 and @xmath107 for dressed potential , i.e. for @xmath83 , we need to write down green function as single - center expansion over spherical harmonics . in the simplest case , when the prop function is @xmath108 , the green function is given by @xmath109 where @xmath110 the function @xmath111 has the following asymptotic at infinity @xmath112 the integral cross section can be readily derived using the optical theorem : @xmath113 thus , averaged integral cross section is given by @xmath114 for purpose of illustration we consider scattering problem for a dressed multi - center potential . the multi - center scattering within the framework of the zrp model was investigated by demkov and rudakov @xmath93 ( 8 centers , cube ) , drukarev and yurova @xmath116 ( 3 centers in line ) , szmytkowski @xmath117 ( 4 centers , regular tetrahedron ) . let structure @xmath1 contains @xmath0 identical scatterers , which involve only @xmath119 waves . denote position vector of the scatterer @xmath120 by the @xmath121 . suppose , for the sake of simplicity , the distance between any two scatterers @xmath122 and @xmath120 is @xmath123 . there are three such structures in three - dimensional space - dome @xmath124 , regular trihedron @xmath125 , regular tetrahedron @xmath126 . the partial waves @xmath127 and phases @xmath89 can be classified with respect to symmetry group representation , degeneracy being defined by the dimension of the representation @xmath93 . the structures @xmath124 , @xmath125 , @xmath126 belong to the @xmath128 point groups respectively . the equation @xmath105 leads to algebraical problem @xmath129 the phases can be readily found from compatibility condition of this system . to factorize the determinant we derive the expressions for the phases @xmath130 @xmath131 thus , assuming @xmath132 we obtain the phases of a regular tetrahedron @xmath117 . the partial and integral cross sections can be expressed as @xmath133 the structures @xmath2 can be used , for instance , to study a slow electron scattering by the polyatomic molecules like @xmath135 , @xmath136 , @xmath137 , etc . let @xmath138 denote the position vectors of the scatterers @xmath139 and @xmath140 denotes the position vector of the scatterer @xmath141 . the scatterers @xmath139 are situated in vertices of a regular structure @xmath1 , i.e. @xmath123 . suppose , for the sake of simplicity , the distance between the scatterer @xmath141 and any scatterer @xmath139 is @xmath142 . therefore the position of the scatterer @xmath141 perfectly fixed only if @xmath132 ( geometric center of the tetrahedron , @xmath143 ) . the partial waves are given by the equation @xmath127 , where the summation should be performed over the @xmath144 . the constants @xmath91 and phases can be derived analytically . thus , we obtain the phases @xmath145 where @xmath146 is inverse scattering length of the scatterers @xmath139 . the @xmath147 obey the quadratic equation @xmath148 where @xmath149 indicates inverse scattering length of the scatterer @xmath141 . in the limiting case when the distance between @xmath139 and @xmath141 scatterers is very large , i.e. @xmath150 , the expression for @xmath151 becomes @xmath152 and @xmath153 . this situation corresponds to independent scattering on the structure @xmath1 and scatterer @xmath141 . the tangent of @xmath154 also reduces to @xmath155 for structure @xmath1 in the limit @xmath156 . the integral cross sections for @xmath3 ( closed - shell ground - state @xmath157 ) are plotted in fig . @xmath158 for a number of values of @xmath159 , which are regarded as constant in the range of interest . the present calculations were carried out with @xmath160 . our calculations were made within the framework of the zrp model and hence they are not expected to be correct for low impact energies ( i.e for energies @xmath161 ev ) where polarization effects ( @xmath3 spherical polarizability @xmath162 ) are known to be important . because induced polarization potentials are always attractive , including polarization algebraically increases the computed phases ( @xmath163 at @xmath164 ) and therefore decreases the integral cross section . thus , fig . @xmath158 clearly shows that polarization effects reproduce the deep minimum ( ramsauer - townsend minimum near @xmath165 ev ) seen in the experimental data @xmath166 and numerical calculations @xmath167 , which are incomparably smaller in the our calculations . the ics can be corrected by the zrps dressing . fig . @xmath168 show ics for dressed @xmath169 structure . in our calculation , the parameters @xmath170 , @xmath171 were used . in the higher collision energies our cross section ( fig . @xmath158 ) differs from the experimental results in size but coincides in shape . it is known that role of the higher partial waves is important at the higher collision energies . thus , taking into consideration the @xmath172 waves for the scatterers @xmath139 and @xmath141 we can add in partial waves and improve the agreement both in size at the higher energies and in the position of shape resonance . we demonstrate the posibilities of dt in multi - center scattering problem . thus , these transformations allow to correct the zrp model at low energies . in the limiting case @xmath63 dt induces the long - range forces @xmath173 depending on angular momentum and leads to singularity in the cross section at zero energy . albeverio s , gesztesy f , hegh - krohn r and holden h ( 1988 ) solvable models in quantum mechanics , springer - verlag , new york breit g ( 1947 ) phys . 71:215 matveev v b ( 1979 ) lett . 3:213 matveev v b , salle m a ( 1991 ) darboux transformations and solitons , springer , new york andrianov a , borisov n , ioffe m ( 1984 ) phys . a 105:19 - 22 baltenkov a s ( 2000 ) phys . a 286:92 - 99 szmytkowski r , szmytkowski c ( 1999 ) j. math . chem . 26:243 - 254 wan h x , moore j h , tossel j a ( 1989 ) j. chem . 91:1340 zaitsev a , leble s ( 1999 ) preprint 12.01.1999 math - ph/9903005 ; ( 2000 ) romp 46:155

a dressing of a nonspherical potential , which includes @xmath0 zero range potentials , is considered . the dressing technique is used to improve zrp model . concepts of the partial waves and partial phases for non - spherical potential are used in order to perform darboux transformation . the problem of scattering on the regular @xmath1 and @xmath2 structures is studied . the possibilities of dressed zrp are illustrated by model calculation of the low - energy electron - silane ( @xmath3 ) scattering . the results are discussed .

arxiv

the convection - diffusion model can be expressed mathematically , which is a semi linear parabolic partial differential equation . specially , we consider an initial value system of _ convection - diffusion _ equation in @xmath3 dimension as : @xmath4,\ ] ] together with the dirichlet boundary conditions : @xmath5,\ ] ] or neumann boundary conditions : @xmath6.\ ] ] where @xmath7 is the boundary of computational domain @xmath8 \times [ c , d]\subset { \mathbb{r}}^2 $ ] , @xmath9 $ ] is time interval , and @xmath10 @xmath11 and @xmath12 are known smooth functions , and @xmath13 denote heat or vorticity . the parameters : @xmath14 and @xmath15 are constant convective velocities while the constants @xmath16 are diffusion coefficients in the direction of @xmath17 and @xmath18 , respectively . the convection - diffusion models have remarkable applications in various branches of science and engineering , for instance , fluid motion , heat transfer , astrophysics , oceanography , meteorology , semiconductors , hydraulics , pollutant and sediment transport , and chemical engineering . specially , in computational hydraulics and fluid dynamics to model convection - diffusion of quantities such as mass , heat , energy , vorticity @xcite . many researchers have paid their attention to develop some schemes which could produce accurate , stable and efficient solutions behavior of convection - diffusion problems , see @xcite and the references therein . in the last years , the convection - diffusion equation has been solved numerically using various techniques : namely- finite element method @xcite , lattice boltzmann method @xcite , finite - difference scheme and higher - order compact finite difference schemes @xcite . a nine - point high - order compact implicit scheme proposed by noye and tan @xcite is third - order accurate in space and second - order accurate in time , and has a large zone of stability . an extension of higher order compact difference techniques for steady - state @xcite to the time - dependent problems have been presented by spotz and carey @xcite , are fourth - order accurate in space and second or lower order accurate in time but conditionally stable . the fourth - order compact finite difference unconditionally stable scheme due to dehghan and mohebbi @xcite have the accuracy of order @xmath19 . a family of unconditionally stable finite difference schemes presented in @xcite have the accuracy of order @xmath20 . the schemes presented in @xcite are based on high - order compact scheme and weighted time discretization , are second or lower order accurate in time and fourth - order accurate in space . the high - order alternating direction implicit ( adi ) scheme with accuracy of order @xmath21 proposed by karaa and zhang @xcite , is unconditionally stable . a high - order unconditionally stable exponential scheme for unsteady @xmath22d convection - diffusion equation by tian and yua @xcite have the accuracy of order @xmath21 . a rational high - order compact alternating direction implicit ( adi ) method have been developed for solving @xmath3d unsteady convection - diffusion problems @xcite is unconditionally stable and have the accuracy of order @xmath21 . a unconditionally stable fourth - order compact finite difference approximation for discretizing spatial derivatives and the cubic @xmath23- spline collocation method in time , proposed by mohebbi and dehghan @xcite , have the accuracy of order @xmath19 . an unconditionally stable , semi - discrete based on pade approximation , by ding and zhang @xcite , is fourth - order accurate in space and in time both . the most of schemes are based on the two - level finite difference approximations with dirichlet conditions , and very few schemes have been developed to solve the convection - diffusion equation with neumann s boundary conditions , see @xcite and references therein . the fourth - order compact finite difference scheme by cao et al . @xcite is of @xmath24th - order accurate in time and 4th - order in the space . a high - order alternating direction implicit scheme based on fourth - order pade approximation developed by you @xcite is unconditionally stable with the accuracy of order @xmath21 . the differential quadrature method ( dqm ) dates back to bellman et al . @xcite . after the seminal paper of bellman , various test functions have been proposed , among others , spline functions , sinc function , lagrange interpolation polynomials , radial base functions , modified cubic b - splines , see @xcite , etc . shu and richards @xcite have generalized approach of dqm for numerical simulation of incompressible navier - stokes equation . the main goal of this paper is to find numerical solution of initial value system of @xmath3d _ convection - diffusion _ equation with both kinds of boundary conditions ( dirichlet boundary conditions and neumann boundary conditions ) , approximated by dqm with new sets of modified cubic b - splines ( modified extended cubic b - splines , modified exponential cubic b - splines , modified trigonometric cubic b - splines ) as base functions , and so called modified trigonometric cubic - b - spline differential quadrature method ( mtb - dqm ) , modified exponential cubic - b - spline differential quadrature method ( mexp - dqm ) and third modified extended cubic - b - spline differential quadrature method ( mecdq ) . these methods are used to transform the convection diffusion problem into a system of first order odes , in time . the resulting system of odes can be solved by using various time integration algorithm , among them , we prefer ssp - rk54 scheme @xcite due to its reduce storage space , which results in less accumulation errors . the accuracy and adaptability of the method is illustrated by three test problems of two dimensional convection diffusion equations . the rest of the paper is organized into five more sections , which follow this introduction . specifically , section [ sec - metho - decr - tem ] deals with the description of the methods : namely- mtb - dqm , mexp - dqm and mecdq . section [ sec - impli ] is devoted to the procedure for the implementation of describe above these methods for the system together with the boundary conditions as in and . section [ sec - stab ] deals with the stability analysis of the methods . section [ sec - num ] deals with the main goal of the paper is the numerical computation of three test problems . finally , section [ sec - conclu ] concludes the results . the differential quadrature method is an approximation to derivatives of a function is the weighted sum of the functional values at certain nodes @xcite . the weighting coefficients of the derivatives is depend only on grids @xcite . this is the reason for taking the partitions @xmath25 $ ] of the problem domain @xmath26 distributed uniformly as follows : @xmath27 = \ { ( x_i , y_j)\in \omega : h_x = x_{i+1}-x_{i } , h_y = y_{j+1}-y_{j } , i\in \delta_x , j\in \delta_y\},\ ] ] where @xmath28 , and @xmath29 are the discretization steps in both @xmath17 and @xmath18 directions , respectively . that is , a uniform partition in each @xmath30-direction with the following grid points : @xmath31 @xmath32 let @xmath33 be the generic grid point and @xmath34 the @xmath35-th order derivative of @xmath36 , for @xmath37 , with respect to @xmath30 at @xmath33 for @xmath38 is approximated as follows : @xmath39 where the coefficients @xmath40 and @xmath41 , the time dependent unknown quantities , are termed as the weighting functions of the @xmath35th - order derivative , to be computed using various type of base functions . the trigonometric cubic b - spline function @xmath42 at node @xmath43 in @xmath17 direction , read as @xcite : @xmath44 where @xmath45 , @xmath46 . then the set @xmath47 forms a base over the interval @xmath48 $ ] . setting @xmath49 the values of @xmath50 and its first and second derivatives in the grid point @xmath51 , denoted by @xmath52 , @xmath53 and @xmath54 , respectively , read : @xmath55 the modified trigonometric cubic b - splines base functions are defined as follows @xcite : @xmath56 now , the set @xmath57 is the base over @xmath48 $ ] . the procedure to define modified trigonometric cubic b - splines in @xmath18 direction , is followed analogously . in order to evaluate the weighting coefficients @xmath60 of first order partial derivative in eq . , we use the modified trigonometric cubic b - spline @xmath61 , @xmath62 in dq method as base functions . setting @xmath63 and @xmath64 . using mtb - dqm , the approximate values of first order derivative is obtained as follows : @xmath65 setting @xmath66 $ ] , @xmath67 $ ] , and @xmath68 $ ] . eq . reduced to compact matrix form : @xmath69 the coefficient matrix @xmath70 of order @xmath71 can be read from and as : @xmath72\ ] ] and the columns of the matrix @xmath73 read as : @xmath74 = \left [ \begin{array}{c } 2 a_4\\ a_3-a_4\\ 0 \\ \vdots \\ \\ 0 \\ 0 \\ \end{array } \right ] , \im'[2 ] = \left[\begin{array}{c } a_4 \\ $ 0 $ \\ a_3 \\ $ 0 $ \\ \vdots \\ \\ $ 0 $ \\ \end{array}\right ] , \ldots , \im'[n_x-1 ] = \left [ \begin{array}{c } $ 0 $ \\ \vdots \\ \\ $ 0 $ \\ a_4 \\ $ 0 $ \\ a_3 \\ \end{array } \right ] , \mbox { and } \im'[n_x ] = \left [ \begin{array}{c } $ 0 $ \\ \\ \vdots \\ \\ $ 0 $ \\ 2a_4 \\ a_3- a_4 \\ \end{array } \right].\ ] ] the exponential cubic b - splines function @xmath75 at node @xmath43 in @xmath17 direction , reads @xcite : @xmath76 where @xmath77 the set @xmath78 forms a base over @xmath48 $ ] . setting the values of @xmath79 and its first and second derivatives at @xmath51 by @xmath80 , @xmath81 and @xmath82 , respectively . then @xmath83 the modified exponential cubic b - splines base functions are read as : @xmath84 the set @xmath85 is a base over @xmath48 $ ] . the procedure to define modified trigonometric cubic b - splines in @xmath18 direction , is followed analogously . setting @xmath86 and @xmath87 for all @xmath88 . using mexp - dqm , the approximate values of the first - order derivative is given by @xmath89 setting @xmath90 $ ] , @xmath67 $ ] , and @xmath91 $ ] , then eq . can be reduced to compact matrix form : @xmath92 let @xmath93 and @xmath94 . using eqns . and , the coefficient matrix @xmath95 of order @xmath71 , read as : @xmath96\ ] ] and the columns of the matrix @xmath97 read : @xmath98 = \left [ \begin{array}{c } \omega / h_x\\ -\omega / h_x\\ 0 \\ \vdots \\ \\ 0 \\ 0 \\ \end{array } \right ] , \beth'[2 ] = \left[\begin{array}{c } \omega/2h_x \\ $ 0 $ \\ -\omega/2h_x \\ $ 0 $ \\ \vdots \\ \\ $ 0 $ \\ \end{array}\right ] , \ldots , \beth'[n_x-1 ] = \left [ \begin{array}{c } $ 0 $ \\ \vdots \\ \\ $ 0 $ \\ \omega/2h_x \\ $ 0 $ \\ \\ \end{array } \right ] , \mbox { and } \beth'[n_x ] = \left [ \begin{array}{c } $ 0 $ \\ \\ \vdots \\ \\ $ 0 $ \\ \omega / h_x \\ -\omega / h_x \\ \end{array } \right].\ ] ] the extended cubic b - splines function @xmath99 , in the @xmath17 direction and at the knots , reads @xcite : @xmath100 where @xmath101 and @xmath102 , and @xmath103 is a free parameter @xcite . the set @xmath104 forms a base over @xmath48 $ ] . let @xmath105 and @xmath106 , then the values of @xmath107 and its first and second derivatives in the grid point @xmath51 , denoted by @xmath108 , @xmath109 and @xmath110 , respectively , read : @xmath111 the modified extended cubic b - splines base functions are defined as follows @xcite : @xmath112 the set @xmath113 is a base over @xmath48 $ ] . setting @xmath114 and @xmath115 . using mecdq method , the approximate values of the first - order derivative is given by @xmath116 setting @xmath117 $ ] , @xmath67 $ ] , and @xmath118 $ ] . eq . can be re - written in compact matrix form as : @xmath119 using eqns . and , the matrix @xmath120 of order @xmath71 read as : @xmath121\ ] ] and the columns of the matrix @xmath122 read : @xmath123 = \left [ \begin{array}{c } -1/h_x\\ 1/h_x\\ 0 \\ \vdots \\ \\ 0 \\ 0 \\ \end{array } \right ] , \phi'[2 ] = \left[\begin{array}{c } -1/2h_x \\ $ 0 $ \\ 1/2h_x \\ $ 0 $ \\ \vdots \\ \\ $ 0 $ \\ \end{array}\right ] , \ldots , \phi'[n_x-1 ] = \left [ \begin{array}{c } $ 0 $ \\ \vdots \\ \\ $ 0 $ \\ -1/2h_x \\ $ 0 $ \\ 1/2h_x \\ \end{array } \right ] , \mbox { and } \phi'[n_x ] = \left [ \begin{array}{c } $ 0 $ \\ \\ \vdots \\ \\ $ 0 $ \\ -1/h_x \\ 1/h_x \\ \end{array } \right].\ ] ] using thomas algorithm " the system , and have been solved for the weighting coefficients @xmath124 , for all @xmath125 . similarly , the weighting coefficients @xmath126 , in either case , can be computed by employing these modified cubic b - splines in the @xmath18 direction . using @xmath127 and @xmath126 , the weighting coefficients , @xmath128 and @xmath129 ( for @xmath130 ) can be computed using the shu s recursive formulae @xcite : @xmath131 where @xmath35 denote @xmath35-th order spatial derivative . in particular , the weighting coefficients @xmath132 of order @xmath3 can be obtained by taking @xmath133 in . after computing the approximate values of first and second order spatial partial derivatives from one of the above three methods , one can re - write eq as follows : @xmath134 in case of the dirichlet conditions , the solutions on boundaries can directly read from as : @xmath135.\ ] ] on the other hand , if the boundary conditions are neumann or mixed type , then the solutions at the boundary are obtained by using any above methods ( mtb - dqm , mexp - dqm or mecdq method ) on the boundary , which gives a system of two equations . on solving it we get the desired solution on the boundary as follows : from eq . with @xmath136 and the neumann boundary conditions at @xmath137 and @xmath138 , we get @xmath139 in terms of matrix system for @xmath140 , the above equation can be rewritten as @xmath141 \left [ \begin{array}{c } u_{1j } \\ u_{n_x j } \\ \end{array } \right ] = \left [ \begin{array}{c } s_j^a \\ s_j^b \\ \end{array } \right],\ ] ] where @xmath142 and @xmath143 . on solving , for the boundary values @xmath144 and @xmath145 , we get @xmath146 analogously , for the neumann boundary conditions at @xmath147 and @xmath148 , the solutions for the boundary values @xmath149 and @xmath150 can be obtained as : @xmath151 where @xmath152 and @xmath153 . after implementing the boundary values , eq can be written in compact matrix form as follows : @xmath154 where 1 . @xmath155 $ ] is the solution vector : + @xmath156 , and @xmath157 represents the initial solution vector . 2 . @xmath158 $ ] is the vector of order @xmath159 containing the boundary values , i.e. , + @xmath160 3 . @xmath161 be a square matrix of order @xmath162 given as @xmath163 where @xmath164 and @xmath165 @xmath166 are the block diagonal matrices of the weighting coefficients @xmath58 and @xmath167 , respectively as given below @xmath168 , \mbox { and } & ~ b_r = \left [ \begin{array}{cccc } m_r & o & \ldots & o \\ o & m_r & \ldots & o \\ \vdots & \vdots & \ddots & \vdots \\ o & o & \ldots & m_r \\ \end{array } \right ] , \end{array}\ ] ] where @xmath169 and @xmath170 are the matrices of order @xmath171 , and the sub - matrix @xmath172 of the block diagonal matrix @xmath173 is given by @xmath174 . \end{array}\ ] ] finally , we adopted ssp - rk54 scheme @xcite to solve the initial value system as : @xmath175 where @xmath176 the stability of the method mtb - dqm for @xmath3d convection - diffusion equation depends on the stability of the initial value system of odes as defined in . noticed that whenever the system of odes is unstable , the proposed method for temporal discretization may not converge to the exact solution . moreover , being the exact solution can directly obtained by means of the eigenvalues method , the stability of depends on the eigenvalues of the coefficient matrix @xmath161 @xcite . in fact , the stability region is the set @xmath177 , where @xmath178 is the stability function and @xmath179 is the eigenvalue of the coefficient matrix @xmath161 . the stability region for ssp - rk54 scheme is depicted in ( * ? ? ? * fig.1 ) , from which one can clam that for the stability of the system it is sufficient that @xmath180 for each eigenvalue @xmath179 of the coefficient matrix b. hence , the real part of each eigenvalue is necessarily either zero or negative . it is seen that the eigenvalues of the matrices @xmath181 and @xmath173 @xmath182 have identical nature . therefore , it is sufficient to compute the eigenvalues , @xmath183 and @xmath184 , of the matrices @xmath185 and @xmath186 for different values of grid sizes @xmath187 . the eigenvalues @xmath183 and @xmath184 for @xmath188 has been depicted in figure [ eq - eignv ] . analogously , one can compute the eigenvalues @xmath183 and @xmath184 using mecdq method @xcite or mexp - dqm . it is seen that in either case , @xmath183 and @xmath184 have same nature as in figure [ eq - eignv ] . further , we can get from figure [ eq - eignv ] that each eigenvalue @xmath179 of the matrix @xmath161 as defined in eq . is real and negative . this confirms that the proposed methods produces stable solutions for two dimensional convection - diffusion equations . this section deals with the main goal , the numerical study of three test problems of the initial value system of convection - diffusion equations with both kinds of the boundary conditions has been done by adopting the methods mtb - dqm , mexp - dqm and mecdq method along with the integration ssp - rk54 scheme . the accuracy and the efficiency of the methods have been measured in terms of the discrete error norms : namely- average @xmath189 norm ( @xmath190-error norm ) and the maximum error ( @xmath191 error norm ) . [ ex1 ] consider the initial value system of @xmath3d convection - diffusion equation with @xmath192 , while values of @xmath193 for @xmath194 can be extracted from the exact solution @xmath195 where initial condition is a gaussian pulse with unit hight centered at @xmath196 . we have computed the numerical solution of problem [ ex1 ] for @xmath197 and different values of @xmath198 . for @xmath199 ^ 2 $ ] : @xmath200 errors and the rate of convergence ( roc ) @xcite at @xmath201 for different values of @xmath202 ( @xmath203 ) has been reported in table [ tab1.1 ] for @xmath204 . table [ tab1.1 ] confirms that the proposed solutions obtained by these methods are more accurate , and approaching towards exact solutions . the behavior of the approximate solution at time @xmath201 taking @xmath205 is depicted in figure [ ex1-fig1.1 ] for @xmath206 and in figure [ ex1-fig1.2 ] for @xmath207 . for @xmath208 ^ 2 $ ] , time @xmath209 , and grid space step size @xmath210 : the @xmath190 and @xmath200 error norms in the proposed solutions have been compared with the errors in the solutions by various schemes in @xcite in table [ tab1.2 ] for @xmath211 and in table [ tab1.3 ] for @xmath212 . the initial solution and mtb - dqm solutions in @xmath213 ^ 2 $ ] with the parameter values @xmath214 have been depicted in figure [ ex1-fig1.3 ] , and similar behavior is seen from the other two methods . it is evident from the above reports that the proposed results are more accurate as compared to @xcite and approaching towards the exact solutions . l*8l + @xmath71 & & & & & + & @xmath200 @xmath215 & roc @xmath215 & & @xmath200 @xmath215 & roc @xmath215 & & @xmath200 @xmath215 & roc + 5 & 2.91e-03 & & & 2.91e-03 & & & 1.94e-03 & + 10&6.41e-04 & 2.18 & & 6.42e-04&2.181 & & 4.93e-04 & 1.98 + 20&6.50e-05 & 3.30 & & 6.51e-05&3.303 & & 6.05e-05 & 3.03 + 40&7.19e-06 & 3.18 & & 7.18e-06&3.180 & & 6.91e-06 & 3.13 + 80@xmath216 & 7.22e-07 @xmath217 & 3.31 & @xmath218 & 7.00e-07@xmath217&3.358 & @xmath218 & 6.99e-07 @xmath219&3.31 + + 5 & 4.93e-04 & & & 4.97e-04 & & & 3.20e-04 & + 10&4.86e-05 & 3.34 & & 4.87e-05 & 3.35 & & 4.39e-05&2.87 + 20&4.70e-06 & 3.37 & & 4.70e-06 & 3.37 & & 4.35e-06&3.34 + 40&4.51e-07 & 3.38 & & 4.46e-07 & 3.40 & & 4.50e-07&3.27 + 80@xmath216&4.58e-08@xmath216 & 3.30 @xmath216 & & 3.49e-08 @xmath216&3.67 & & 4.37e-08@xmath216 & 3.36 + * 3l schemes & @xmath190 & @xmath200 + p - r - adi @xcite & 3.109e-04 @xmath220 @xmath220 & 7.778e-03 + noye @xmath221 tang @xcite & 1.971e-05 & 6.509e-04 + kalila @xcite & 1.597e-05 & 4.447e-04 + kara @xmath221 zhang adi @xcite & 9.218e-06 & 2.500e-04 + tang @xmath221 ge adi@xcite & 9.663e-06 & 2.664e-04 + dehghan @xmath221 mohebbi @xcite@xmath220 @xmath220 & 9.431e-06 & 2.477e-04 + mtb - dqm & 8.026e-12 & 4.327e-08 + mexp - dqm(p=0.0001 ) & 7.030e-12 & 4.154e-08 + mecdq ( @xmath222 ) & 4.504e-12 & 3.343e-08 + mcb - dqm ( @xmath223 ) & 8.269e-12 & 4.388e-08 + * 3l schemes & @xmath190 & @xmath200 + noye @xmath221 tang @xcite & 1.430e-05 & 4.840e-04 + kalila et al.@xcite & 1.590e-05 & 4.480e-04 + dehghan @xmath221 mohebbi @xcite @xmath220 @xmath220 & 9.480e-06 @xmath220 @xmath220 @xmath220 & 2.469e-04 + mtb - dqm & 8.900e-12 & 4.501e-08 + mexp - dqm(p=0.0001 ) & 9.147e-12 & 4.315e-08 + mecdq ( @xmath224 ) & 4.646e-12 & 3.852e-08 + mcb - dqm ( @xmath223 ) & 9.147e-12 & 4.561e-08 + [ ex2 ] consider the initial value system of @xmath3d convection - diffusion equation with @xmath225 ^ 2 $ ] with @xmath226 , where @xmath227 and the neumann boundary condition @xmath228,\]]or the dirichlet s conditions can be extracted from the exact solution @xcite : @xmath229 the computation by the proposed methods has been done for different values of @xmath230 and @xmath231 . for @xmath232 , we take @xmath233 for the solutions at @xmath234 . the rate of convergence and @xmath190 error norms in the proposed solutions has been compared with that of due to fourth - order compact finite difference scheme @xcite for @xmath235 , in table [ tab2.1 ] . it is found that the proposed solutions from either method are more accurate in comparison to @xcite , and are in good agreement with the exact solutions . for @xmath236 , we take @xmath237 for the solution at @xmath234 . in table [ tab2.2 ] , the @xmath190 and @xmath200 error norms are compared with that obtained by fourth - order compact finite difference scheme @xcite for @xmath236 , and @xmath238 . rate of convergence have been reported in table [ tab2.3 ] . from tables [ tab2.2 ] and [ tab2.3 ] , we found that the proposed solutions are more accurate as compared to the results in @xcite , while the rate of convergence is linear . the behavior of solutions is depicted in figure [ ex2-fig2.2 ] and [ ex2-fig2.3 ] with @xmath239 for @xmath240 and @xmath241 , respectively . for the same parameter , mentioned above , the numerical solution is obtained by using neumann conditions and reported in table [ tab2.4 ] . the obtained results are in good agreement with the exact solutions , the rate of convergence for @xmath242 is quadratic for each method . * 2lclcllclcllclcllc + & & & & & & & + @xmath243 & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc & & @xmath190 & roc + 0.2 & 4.231e-06 & & 4.969e-10 & & & 1.714e-05 & & 3.584e-09 & & & 4.231e-06 & & 4.999e-10 & & & & + 0.1 & 5.155e-07 & 3.0 & 2.472e-11 & 4.3 & & 5.168e-07 & 5.1 & 2.473e-11 & 7.2 & & 5.153e-07 & 3.0 & 2.478e-11 & 4.3 & & 2.289e-10 & + 0.05 & 5.108e-08 & 3.3 & 9.041e-13 & 4.8 & & 5.105e-08 & 3.3 & 9.044e-13 & 4.8 & & 5.107e-08 & 3.3 & 9.049e-13 & 4.8&&1.621e-11 & 3.82 + 0.025 & 1.147e-08 & 2.2 & 1.922e-14 & 5.6 & & 1.147e-08 & 2.2 & 1.922e-14 & 5.6 & & 1.147e08 & 2.4 & 1.923e-14 & 5.6&&8.652e-13 & 4.23 + + 0.2 & 7.4031e-06 & & 1.4907e-09 & & & 3.5634e-05 & & 1.5314e-08 & & & 2.0585e-06 & & 6.7290e-11 & & & & + 0.1 & 1.1460e-06 & 2.7 & 1.3103e-10 & 3.5 & & 1.1421e-06 & 5.0 & 1.3148e-10 & 6.9 & & 1.5919e-07 & 3.7 & 1.1624e-12 & 5.9 & & 2.749e-09 & + 0.05 & 1.4578e-07 & 3.0 & 7.5377e-12 & 4.1 & & 1.4582e-07 & 3.0 & 7.5396e-12 & 4.1 & & 1.9012e-08 & 3.1 & 5.4939e-14 & 4.4 & & 2.394e-10 & 3.52 + 0.025 & 1.3902e-08 & 3.4 & 2.7129e-13 & 4.8 & & 1.3904e-08 & 3.4 & 2.7132e-13 & 4.8 & & 6.2220e-09 & 1.6 & 9.0456e-15 & 2.6 & & 1.658e-11 & 3.85 + * 6l*6l*6l & & & & & & & + @xmath243 & @xmath200 & & @xmath190 & & @xmath200 & & @xmath190 & & @xmath200 & & @xmath190 & & @xmath200 & & @xmath190 + 0.04 @xmath215 & 4.1718e-03@xmath215 & & 1.7833e-03@xmath215 & & 1.7475e-03@xmath215 & & 7.9168e-04 & & 1.2939e-03@xmath215 & & 3.6957e-04 & & 1.1826e-01 @xmath215 & & 4.9331e-03 + 0.02 & 5.2095e-04 & & 6.8410e-05 & & 5.2097e-04 & & 6.8445e-05 & & 6.3735e-04 & & 6.8622e-05 & & 1.5310e-02 & & 4.1351e-04 + 0.01 & 3.2496e-04 & & 2.1508e-05 & & 3.2484e-04 & & 2.1493e-05 & & 2.5394e-04 & & 1.8260e-05 & & 9.4696e-04 & & 2.8405e-05 + * 5l*5l*5l & + & & & & & + @xmath243 & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc + 0.1 & 2.6263e-02 & & 3.6277e-02 & & & 2.6280e-02 & & 3.6670e-02 & & & 2.4464e-02 & & 4.0175e-02 & + 0.05 & 6.8144e-03 & 1.9 & 4.1447e-03 & 3.1 & & 6.8144e-03 & 1.9 & 4.1339e-03 & 3.1 & & 3.3580e-03 & 2.9 & 1.6264e-03 & 4.6 + 0.025 & 1.1634e-03 & 2.6 & 2.7780e-04 & 3.9 & & 1.0719e-03 & 2.7 & 1.9697e-04 & 4.4 & & 8.1776e-04 & 2.0 & 1.0090e-04 & 4.0 + & + & & & & & + 0.01 & 5.0998e-04 & & 3.5191e-03 & & & 5.0738e-04 & & 1.0116e-05 & & & 7.4180e-04 & & 2.5558e-05 & + 0.05 & 1.8776e-04 & 1.4 & 1.8193e-03 & 1.0 & & 1.8776e-04 & 1.4 & 2.9979e-06 & 1.8 & & 2.0972e-04 & 1.8 & 5.1381e-06 & 2.3 + 0.025 & 9.9342e-05 & 0.9 & 9.1372e-04 & 1.0 & & 9.9342e-05 & 0.9 & 7.9462e-07 & 1.9 & & 9.5677e-05 & 1.1 & 8.1188e-07 & 2.7 + l*4c*5c*5c & + & & & & & + @xmath243 & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc & & @xmath242 & roc & @xmath190 & roc + 0.05 & 1.0176e-02 & & 6.7504e-03 & & & 1.0177e-02 & & 6.7513e-03 & & & 9.1549e-03 & & 5.7412e-03 & + 0.025 & 2.2164e-03 & 2.2 & 1.0428e-03 & 2.7 & & 2.2165e-03 & 2.2 & 1.0429e-03 & 2.7 & & 1.9834e-03 & 2.2 & 8.7214e-04 & 2.7 + 0.0125 & 2.9460e-04 & 2.9 & 7.2888e-05 & 3.8 & & 2.9460e-04 & 2.9 & 7.2889e-05 & 3.8 & & 2.5565e-04&3.0 & 5.6518e-05 & 3.9 + & + & & & & & + 0.05 & 1.2070e-01 & & 6.3285e-01 & & & 1.2070e-01 & & 6.3293e-01 & & & 1.0921e-01 & & 6.0454e-01 & + 0.025 & 2.7808e-02 & 2.1 & 6.2302e-02 & 3.3 & & 2.7809e-02 & 2.1 & 6.2304e-02 & 3.3 & & 2.6004e-02 & 2.1 & 5.5004e-02 & 3.5 + 0.0125&6.0348e-03 & 2.2 & 6.4433e-03 & 3.3 & & 6.0348e-03 & 2.2 & 6.4433e-03 & 3.3 & & 5.4733e-03 & 2.2 & 5.5668e-03 & 3.3 + & + & & & & & + 0.05 & 1.4931e-05 & & 5.8044e-08 & & & 1.4932e-05 & & 5.8056e-08 & & & 1.3598e-05 & & 4.8332e-08 & + 0.025 & 3.3009e-06 & 2.2 & 1.0664e-08 & 2.4 & & 3.3009e-06 & 2.2 & 1.0665e-08 & 2.4 & & 2.9824e-06 & 2.2 & 8.7286e-09 & 2.5 + 0.0125 & 4.5947e-07 & 2.8 & 8.0780e-10 & 3.7 & & 4.5947e-07 & 2.8 & 8.0780e-10 & 3.7 & & 3.9840e-07 & 2.9 & 6.0761e-10 & 3.8 + & + & & & & & + 0.05 & 1.7132e-05 & & 6.9905e-08 & & & 1.7132e-05 & & 6.9910e-08 & & & 1.6511e-05 & & 6.5219e-08 & + 0.025 & 3.9931e-06 & 2.1 & 1.4135e-08 & 2.3 & & 3.9932e-06 & 2.1 & 1.4136e-08 & 2.3 & & 3.7892e-06 & 2.1 & 1.2756e-08 & 2.4 + 0.0125 & 8.8931e-07 & 2.2 & 2.7052e-09 & 2.4 & & 8.8931e-07 & 2.2 & 2.7052e-09 & 2.4 & & 8.2511e-07 & 2.2 & 2.3316e-09 & 2.5 + [ ex3 ] the initial value system of @xmath3d convection - diffusion equation with @xmath244 ^ 2 $ ] and @xmath245 , and @xmath246 where @xmath247 the distribution of the initial solution is depicted in figure [ ex3-fig3.1 ] . the solutions behavior is obtained for the parameter values : @xmath248 and @xmath249 , and is depicted in figure [ ex3-fig3.2 ] due to mtb - dqm , also we noticed the similar characteristics obtained using mexp - dqm and mecdq method . the obtained characteristics agreed well as obtained in @xcite . in this paper , the numerical computations of initial value system of two dimensional _ convection - diffusion _ equations with both kinds of boundary conditions has been done by adopting three methods : modified exponential cubic b - splines dqm , modified trigonometric cubic b - splines dqm , and mecdq method @xcite , which transforms the _ convection - diffusion _ equation into a system of first order ordinary differential equations ( odes ) , in time , which is solved by using ssp - rk54 scheme . the methods are found stable for two space convection - diffusion equation by employing matrix stability analysis method . section [ sec - num ] shows that the proposed solutions are more accurate in comparison to the solutions by various existing schemes , and are in good agreement with the exact solutions . the order of accuracy of the proposed methods for the convection - diffusion problem with dirichlet s boundary conditions is cubic whenever @xmath250 and otherwise it is super linear , in space . on the other hand , the order of accuracy of the proposed methods for the convection - diffusion problem with neumann boundary condition is quadratic with respect to @xmath251 error norms , see table [ tab2.4 ] . huai - huo cao , li - bin liu , yong zhang and sheng - mao fu , a fourth - order method of the convection - diffusion equations with neumann boundary conditions , applied mathematics and computation 217 ( 2011 ) 9133 - 9141 . hengfei ding and yuxin zhang , a new difference scheme with high accuracy and absolute stability for solving convection - diffusion equations , journal of computational and applied mathematics 230 ( 2009 ) 600 - 606 . kalita , d.c . dalal and a.k . dass , a class of higher order compact schemes for the unsteady two - dimensional convection - diffusion equation with variable convection coefficients , int . methods fluids 38 ( 2002 ) 1111 - 1131 . m. zerroukat , k. djidjeli and a. charafi , explicit and implicite messless method for linear advection - diffusion - type partial differential eqations , international journal numerical method in engineering 48 ( 2000 ) 19 - 35 . g. arora , v. joshi , a computational approach for solution of one dimensional parabolic partial differential equation with application in biological processes , ain shams eng j ( 2016 ) , http://dx.doi.org/10.1016/j.asej.2016.06.013 . singh , p. kumar , an algorithm based on a new dqm with modified extended cubic b - splines for numerical study of two dimensional hyperbolic telegraph equation , alexandria eng . j. ( 2016 ) , http://dx.doi.org/10.1016/j.aej.2016.11.009 . g. arora , rc mittal and b. k. singh , numerical solution of bbm - burger equation with quartic b - spline collocation method , journal of engineering science and technology , special issue 1 , 12/2014 , 104 - 116 . v. k. srivastava and b. k. singh , a robust finite difference scheme for the numerical solutions of two dimensional time - dependent coupled nonlinear burgers equations , international journal of applied mathematics and mechanics 10(7 ) ( 2014 ) 28 - 39 . a. korkmaz and h.k . akmaz , extended b - spline differential quadrature method for nonlinear viscous burgers equation , proceedings of international conference on mathematics and mathematics education , pp 323 - 323 , elazi , turkey 12 - 14 may , 2016 . muhammad abbas , ahmad abd . majid , ahmad izani md . ismail , abdur rashid , the application of cubic trigonometric b - spline to the numerical solution of the hyperbolic problems , applied mathematics and computation 239 ( 2014 ) 74 - 88 . ( left ) of 2d convection - diffusion equation with @xmath238 , @xmath254 for @xmath255 , title="fig:",width=236,height=170 ] ( left ) of 2d convection - diffusion equation with @xmath238 , @xmath254 for @xmath255 , title="fig:",width=379,height=170 ] ( left ) of 2d convection - diffusion equation with @xmath238 , @xmath254 for @xmath255 , title="fig:",width=283,height=194 ] ( left ) of 2d convection - diffusion equation with @xmath238 , @xmath254 for @xmath255 , title="fig:",width=283,height=194 ] d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=283,height=194 ] d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=283,height=194 ] d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=283,height=194 ] d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=283,height=194 ] d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=292,height=194 ] d convection - diffusion equation in example [ ex3 ] at @xmath256 ( left ) and @xmath257 ( right),title="fig:",width=292,height=194 ]

this paper deals with the numerical computations of two space dimensional time dependent _ parabolic partial differential equations _ by adopting adopting an optimal five stage fourth - order strong stability preserving runge - kutta ( ssp - rk54 ) scheme for time discretization , and three methods of differential quadrature with different sets of modified b - splines as base functions , for space discretization : namely- @xmath0 mecdqm : ( dqm with modified extended cubic b - splines ) ; @xmath1 mexp - dqm : dqm with modified exponential cubic b - splines , and @xmath2 mtb - dqm : dqm with modified trigonometric cubic b splines . specially , we implement these methods on _ convection - diffusion _ equation to convert them into a system of first order ordinary differential equations ( odes ) , in time . the resulting system of odes can be solved using any time integration method , while we prefer ssp - rk54 scheme . all the three methods are found stable for two space convection - diffusion equation by employing matrix stability analysis method . the accuracy and validity of the methods are confirmed by three test problems of two dimensional _ convection - diffusion _ equation , which shows that the proposed approximate solutions by any of the method are in good agreement with the exact solutions . convection - diffusion equation , modified trigonometric cubic - b - splines , modified exponential cubic - b - splines , modified extended cubic - b - splines , differential quadrature method , ssp - rk54 scheme , thomas algorithm

arxiv

surveys with bolometer arrays at millimetre ( mm ) and submillimetre ( submm ) wavelengths are potentially sensitive to dusty objects at extreme redshifts , galaxies that drop out of surveys at shorter and longer wavelengths due to obscuration and unfavourable @xmath11 corrections . the first cosmological surveys using scuba ( holland et al . 1999 ) and mambo ( kreysa et al . 1998 ) quickly and radically changed the accepted picture of galaxy formation and evolution , moving away from the optocentric view of the last century . the discovery of so - called ` scuba galaxies ' ( smail , ivison & blain 1997 ) was greeted with surprise due to the remarkable evolution in the dusty , starburst galaxy population implied by such a large source density at the flux levels accessible to the first generation of bolometer arrays ( blain et al . excitement was replaced by pessimism with the first efforts to study smgs at optical and infrared ( ir ) wavelengths : early reports , backed up with a study in the hubble deep field north by hughes et al . ( 1998 ) , suggested that the majority of the submm population had no plausible optical counterparts . attention was diverted to various redshift engines and broadband photometric techniques ( e.g.townsend et al . 2001 ; aretxaga et al . 2003 ; wiklind 2003 ) . as a result , only a handful of detailed studies were attempted , often for extreme and possibly unrepresentative galaxies ( e.g. knudsen et al . 2004 ) . recent progress has largely been the result of radio imaging of submm survey fields . early radio follow - up detected roughly half of the submm sources observed ( smail et al . 2000 ; ivison et al . 2002 hereafter i02 ) , with an astrometric precision of @xmath30.3@xmath12 and , combined with the submm flux density , provide a rough estimate of redshift ( carilli & yun 1999 ) . radio data also enabled some refinement of submm samples ( i02 ) , increasing the detection fraction to two thirds of smgs at 0.85-mm flux density levels in excess of @xmath35mjy . with positions in hand , these bright smgs were found to be a diverse population some quasar - like , with broad lines and x - ray detections ( e.g. ivison et al . 1998 ) , some morphologically complex ( e.g. ivison et al . 2000 downes & solomon 2003 ; smail , smith & ivison 2005 ) , some extremely red ( e.g. smail et al.1999 ; gear et al . 2000 ; i02 ; webb et al . 2003b ; dunlop et al.2004 ) , some with the unmistakable signatures of obscured active nuclei and/or superwinds ( e.g. smail et al . 2003 ) . spectroscopic redshifts have been difficult to determine . the first survey based on a submm / radio sample was undertaken by chapman et al.(2003 , 2005 hereafter c03 , c05 ) : the median redshift was found to be @xmath32.2 for @xmath13-mjy galaxies selected using scuba and pinpointed at 1.4ghz . the accurate redshifts reported by c03 and c05 facilitated the first systematic measurements of molecular gas mass for smgs ( @xmath310@xmath14m@xmath15 ) via observations of co ( neri et al . 2003 ; greve et al . 2005 ) , as well as constraints on gas reservoir size and dynamical mass ( tacconi et al.2005 ) . the data suggest smgs are massive systems and provide some of the strongest tests of galaxy - formation models to date ( greve et al.2005 ) . in spite of this progress , a detailed understanding of smgs remains a distant goal . confusion currently limits our investigations to the brightest smgs ( although surveys through lensing clusters have provided a handful of sources more typical of the faint population that dominates the cosmic background smail et al . 2002 ; kneib et al . 2004 ; borys et al . we must also recall that selection biases have potentially skewed our understanding : around half of all known smgs remain undetected in the radio ( due simply to the lack of sufficiently deep radio data , which do not benefit from the same @xmath11 correction as submm data ) and the radio - undetected fraction remains largely untargeted by existing spectroscopic campaigns . these is also only limited coverage of red and ir wavelengths in spectroscopic surveys . here , we present a robust sample of bright smgs selected using scuba and mambo in one of the ` 8-mjy survey ' regions : the lockman hole ( see scott et al . 2002 ; fox et al . 2002 ; i02 ; greve et al . 2004 ; mortier et al . our goal is to provide a bright sample which we would expect to detect in well - matched radio imaging ( @xmath16 ) whilst minimising , so far as is practicable , the possibility that sources are spurious or anamalously bright . we may thus determine the true fraction of radio drop - outs amongst smgs ( potentially lying at very high redshift , @xmath7 ) , as well as practical information such as the intrinsic positional uncertainty for smgs in the absence of radio / ir counterparts . throughout we adopt a cosmology , with @xmath17 , @xmath18 and @xmath19km@xmath20mpc@xmath5 . existing surveys have typically employed a snr threshold of 3.03.5 . at these snrs , false detections are dominated by ` flux boosting ' ( 2.2 ) , possibly at the 1040 per cent level ( scott et al . 2002 ; laurent et al . 2005 ) . our goal is to provide a highly reliable submm source catalogue , free from concerns about contamination by spurious or artificially bright sources . this issue has limited our ability to address the true recovery fraction in the radio , and hence the corrections that must be made to the redshift distributions that are used to determine star - formation histories and galaxy - formation models . to achieve this we have combined independent submm and mm maps of the lockman hole , constructing a single , reliable catalogue that is several times larger than would have been realised by simply adopting a high snr threshold in the individual submm and mm maps . greve et al . ( 2004 ) argued that several maps with low signal - to - noise ratio ( snr ) of the same region , with only marginal differences in frequency , produce several visualisations of essentially the same sky , tracing the same population of luminous , dust - enshrouded galaxies and we have adopted the same philosophy in the present work . our maps came from the survey of greve et al . ( 2004 ) who presented a 1.2-mm map of the lockman hole region ( as well as data on the elais n2 region ) , centred on the coordinates mapped at 0.85 mm by scuba in the ` 8-mjy survey ' ( scott et al . 2002 , 2004 ) . mortier et al . ( 2005 ) have recently present a refined analysis of scuba data which lies within the lockman hole mambo map ; we began with a 2-@xmath1 mambo catalogue , extracted as described by greve et al . ( 2004 ) , and looked for 0.85-mm sources within 14arcsec ( roughly the area of five beams ) in the mortier et al . ( 2005 ) sample , applying a correction for separation and checking for a combined significance above our initial threshold , 4.5@xmath1 . we included sources that exceed 4.5@xmath1 in either dataset , as long as there is a valid reason why the source is not seen in the other image . smgs from scott et al . ( 2002 ) were substituted where blends were evident in the mortier et al.catalogue . [ cols="<,^,^,^,^,^,^,^,^,^,^ " , ] notes : @xmath21 sources in parentheses lack robust radio identifications . + @xmath22 values in bold are for sources identified in the radio ( @xmath23 ) . + @xmath24 values in bold are for redshifts that we consider to be most robust . [ speccat ] table 3 lists the mag_best magnitudes of our sample in @xmath25 , measured using sextractor . fig . 7 shows a histogram of @xmath25 magnitudes compared with those from the surveys by smail et al.(2002 ) , i02 , clements et al . ( 2004 ) and pope et al . the various surveys broadly agree , with median @xmath25-band magnitudes ranging between @xmath2626 and a wide range in magnitude within each sample . the clements et al . ( 2004 ) sample has a bright tail which is less evident in the other surveys . our sample appears to show the smallest dispersion , although the presence of upper - limits in all five samples make this statement hard to quantify . nevertheless , from this comparison we can state that the bright , radio - identified smg population , spanning barely a magnitude in submm flux density , covers over 10 orders of magnitude in rest - frame ultraviolet flux . the optical magnitudes of the radio - identified smgs discussed in this paper span around @xmath2726 , with a faint tail . the median magnitude is @xmath28 , comparable to the sample analysed by c05 , @xmath29 , and @xmath30.6mag fainter than the spectroscopically - identified sample of c05 . as expected , a similar bias in the spectroscopically identified subset exists within our own sample , which is @xmath30.9mag brighter than our complete catalogue . the redshift distribution determined here inevitably suffers from spectroscopic incompleteness , but our sample should be otherwise unbiased . concentrating on cases where our spectroscopy failed to secure a robust redshift the failures tend to coincide with the faintest optical counterparts , @xmath30 , as one might expect . looking at the long - wavelength flux ratios ( fig . 5 ) for those targets where we failed to obtain identifications or redshifts , e.g.lh1200.002 or .004 , we see no indication that their ( sub)mm / radio photometric properties differ from those of the sample as a whole . it seems unlikely therefore that they lie at substantially higher redshifts than the subset of smgs for which we have obtained redshifts . the exceptions are lh1200.022 , and especially lh1200.007 , which have relatively low 0.85-/1.2-mm flux density ratios and fairly high 1.2-mm/1.4-ghz flux density ratios . this suggests they may lie at high redshift , or be particularly cold . however , as discussed in 3.2 , these two sources are not particularly secure ( although that may simply reflect their general faintness in all bands due to their high redshifts ) . reliable identification of these two sources is therefore only achievable through higher resolution ( interferometric ) ( sub)mm observations ( e.g. lutz et al . 2001 ) . for the six smgs where we have a single radio identification ( and hence an unambigious counterpart ) and for which we have measured robust redshifts ( table 3 ) , we determine a median of @xmath31 ( where the scatter is estimated from bootstrap re - sampling ) . this rises slightly to @xmath10 if we include the four sources with robust redshifts for at least one radio counterpart and the five smgs where we have less secure redshifts . these figures are slightly lower than , but entirely consistent with , the spectroscopic redshift distribution determined by c05 ( _ z _ = 2.2 ) . this suggests that the statistical properties of the c05 sample has not been strongly biased by the modest significance @xmath323@xmath1 ) of some of the submm data used in their analysis . the median redshift we derive is in reasonable agreement with that predicted by the galform semi - analytic model ( baugh et al.2005 ) , which gives a median redshift of @xmath33 at our flux limit , with a quartile range of @xmath340.9 and only 1020 per cent of the population at @xmath35 . this again suggests that much of the activity in the bright smg population is amenable to study via the precise positions derived from their radio counterparts . this provides a much - needed route to identify the true far - ir - luminous source within these frequently morphologically complex and crowded fields . we illustrate in fig . 8 the distribution of 1.2-mm/1.4-ghz and 0.85-/1.2-mm flux ratios for our sample versus our spectroscopic redshifts . the former show a trend to higher flux ratios at higher redshifts , in line with predictions from sed modelling ( e.g. carilli & yun 1999 ) , although with a large scatter ; the latter flux ratio , however , is essentially a scatter plot . we conclude that significantly more reliable flux measurements ( in terms of both absolute calibration and overall snr ) will be needed to use the 0.85-/1.2-mm flux ratio for astrophysical analysis . at face value , both our spectroscopic redshift distribution and that of c05 are inconsistent with the redshift distribution ( _ z _ @xmath32 3 ) claimed by eales et al . ( 2003 ) using the @xmath36 flux density ratio ( fig . the eales et al.sample contained 23 sources with @xmath37 = 3.16.5mjy ( cf . 19 sources , @xmath37 = 1.65.7mjy here ) , with a radio - detected fraction of 73 per cent ( cf . 79 per cent here ) . thus , the two samples differ only in that the eales et al . photometry targets were selected above 4@xmath1 at 1200@xmath38 m rather than above 5@xmath1 at a combination of 850 and 1200@xmath38 m . we believe the disagreement in the mean redshifts derived by these studies most likely results from the fact that current photometric measurements in the mm and submm wavebands are insufficiently precise for typical smgs to allow them to be used as a reliable redshift indicator , particularly when some data do not comprise fully sampled images . the disagreement may have been compounded by the lower significance of the eales et al . targets , and the dual - wavelength extraction performed here could exclude the most distant starbursts ( though we note that the radio - detection of a substantial number of the eales et al.targets is hard to reconcile with the high median redshift claimed for that sample ; indeed , the radio - based estimates given in eales et al . , with _ z _ = 2.35 , are more consistent with the spectroscopic results ) . we have developed and applied a dual - survey extraction technique to scuba and mambo images of the lockman hole , resulting in a robust sample of 19 smgs . of these , 15 are detected securely by our deep radio imaging . those undetected at 1.4ghz can be explained by a combination of contamination by spurious sources ( 10 per cent ) and the large observed scatter in radio flux densities , which is probably due to a significant range in dust temperature . we determine 15 spectroscopic redshifts , of which we consider ten to be secure . the resulting redshift distribution ( _ z _ = 2.14 for the full spectroscopic sample ) is consistent with that determined for a much larger sample by c05 ( _ z _ = 2.2 ) . our results thus support the conclusions of c05 , who modelled their incompleteness and estimated only a small shift ( @xmath39 ) in the median redshift as a result . those galaxies for which our spectroscopy failed to determine redshifts are usually optically faint , @xmath30 , where the sample ranges from @xmath27 to @xmath40 with a median of 25.0 . from the radio detections , and the spectroscopy , is seems unlikely that a significant fraction of bright smgs lie at very high redshift ( @xmath41 ) and we conclude that the bright smg population is readily amenable to study via radio - selected samples , down to a 850-@xmath38 m flux density limit of @xmath37mjy . " we note , however , that the dual - wavelength extraction performed here could potentially bias our sample against very - high - redshift sources which would only be detected at 1.2 mm . an analysis of separations between smgs and their radio counterparts has allowed us to re - calibrate the rule - of - thumb relationship between positional accuracy , beam size and significance . the most secure smgs , at @xmath310@xmath1 , representative of those expected in upcoming , confusion - limited , wide - field ( tens of square degrees ) 0.85-mm surveys using scuba-2 ( audley et al . 2004 ) , with lower significance , simultaneous detections at 0.45 mm , will be located with a uncertainty of @xmath42arcsec , at which level the precision of the telescope pointing and scuba-2 flat - field may become important contributors to the positional error budget . assuming these sources of uncertainty can be minimised , the current requirement for deep radio coverage to identify counterparts and enable follow - up spectroscopy may not be as urgent , particularly if 3@xmath433-arcsec deployable integral - field units are employed for spectroscopic follow up ( e.g. kmos sharples et al . 2003 , 2004 ) . is acknowledges support from the royal society . awb acknowledges support from nsf grant ast-0205937 , the research corporation and the alfred p. sloan foundation . 99 aretxaga i. , hughes d.h . , chapin e.l . , gaztaaga e. , dunlop j.s . , ivison r.j . , 2003 , mnras , 342 , 759 audley m.d . et al . , 2004 , spie , 5498 , 63 baugh c.m . , lacey c.g . , frenk c.s . , granato g.l . , silva l. , bressan a. , benson a.j . , cole s. , 2005 , mnras , 356 , 1191 blain a.w . , smail i. , ivison r.j . , kneib j .- , 1999 , mnras , 302 , 632 blain a.w . , chapman s.c . , smail i. , ivison r.j . , 2004 , apj , 611 , 52 borys c. et al . , 2004 , mnras , 352 , 759 borys c. , scott d. , chapman s. , halpern m. , nandra k. , pope a. , 2005 , mnras , 355 , 485 carilli c.l . , yun m.s . , 1999 , apj , 513 , l13 chapman s.c . , smail i. , ivison r.j . , blain a.w . , 2002 , mnras , 335 , l17 chapman s.c . , blain a.w . , ivison r.j . , smail i. , 2003 , nature , 422 , 695 ( c03 ) chapman s.c . , smail i. , blain a.w . , ivison r.j . , 2004a , apj , 614 , 671 chapman s.c . , smail i. , windhorst r. , muxlow t. , ivison r.j . , 2004b , apj , 611 , 732 chapman s.c . , blain a.w . , smail i. , ivison r.j . , 2005 , apj , 622 , 772 ( c05 ) clements d. et al . , 2004 , mnras , 351 , 611 downes a.j.b . , peacock j.a . , savage a. , carrie d.r . , 1986 , mnras , 218 , 31 downes d. , solomon p.m. , 2003 , apj , 582 , 37 dunlop j.s . et al . , 2004 , mnras , 350 , 769 dunlop j.s . , 2005 , in : ` starburst from 30 doradus to lyman break galaxies ' , eds de grijs r. , gonzalez - 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ph/0506176 ) swinbank a.m. , smail i. , chapman s.c . , blain a.w . , ivison r.j . , keel w.c . , 2004 , apj , 617 , 64 townsend r.h.d . , ivison r.j . , smail i. , blain a.w . , frayer d.t . , 2001 , mnras , 328 , l17 webb t.m . et al . , 2003a , apj , 587 , 41 webb t.m.a . , lilly s.j . , clements d.l . , eales s. , yun m. , brodwin m. , dunne l. , gear w.k . , 2003b , apj , 597 , 680 wiklind t. , 2004 , apj , 588 , 736 notes on our spectroscopic observations of the individual sources , referring to the slit positions shown in fig . 3 . * lh1200.001 : * a complex field . lris slits on several masks had been placed on a faint @xmath25-band source coincident with the most probable radio counterpart ( @xmath44 ) . its spectrum , shown in fig . a1 , is consistent with a redshift of 3.036 , with ly@xmath45 and hints of weak nv and several other lines . we searched for [ oii ] and [ oiii ] using nirspec ( 4.1 ) , yet we were not able to confirm or rule out @xmath46 with confidence . a search for co(32 ) at iram was unsuccessful ( greve et al . 2005 ) , though this could have been due to insufficient velocity coverage . the second radio counterpart , 4arcsec to the north and also with @xmath23 , is associated with an ero ( fig . the nirspec longslit was positioned to cover this galaxy ; no lines were evident . its closest neighbour was observed with gmos ( fig . a2 ; [ 28 ] ) and found to have emission lines ( ly@xmath45 , possibly nv and siii ) consistent with @xmath47 ( the apparent continuum blueward of ly@xmath45 is merely an artifact of the reduction process ) . at the edge of the same slit , near the northern radio position , there is tentative evidence for line emission from ly@xmath45 , nv and possibly civ at @xmath48 . the optically bright radio source 11arcsec se of the smg centroid ( fig . a2 ; [ 26 ] ) is an agn , with broad ciii ] and mgii emission lines , this time at @xmath49 . the optical galaxy 8arcsec sse of the smg centroid ( fig . a2 ; [ 27 ] ) has [ oii ] at @xmath50 . a further ero , overlooked by i02 , and detected at 3.5@xmath1 in our smoothed radio image , lies 7.5 arcsec nne of the smg centroid ( fig . we conclude that the submm emission likely originates from the two central radio components , which most probably lie at @xmath51 . * lh1200.002 : * two galaxies in this field were targeted spectroscopically with gmos . one target , slit [ 25 ] , lies close to the faint radio emission near the smg centroid , but we were unable to identify its redshift . the other target probably unassociated with the smg has a faint emission line which , if [ oii ] , indicates @xmath52 ( fig . a2 ; [ 24 ] ) . * lh1200.003 : * another complex field . lh1200.003 may be a blend of 23 smgs , with mambo resolving the source into two , and the vla resolving three faint radio sources ( one marginally outside our radio search region ) . slits had been placed on all likely counterparts . the radio source to the se , targeted by slits on five lris masks at various position angles , is a starburst at @xmath53 . the most probable ( central ) radio identification is associated with a galaxy at @xmath54 ( fig . a1 ) . the galaxy to the nnw possibly associated with faint mm emission , lh1200.213 in greve et al . ( 2004 ) was targeted spectroscopically using lris and gmos and is an agn at @xmath55 , with broad civ emission , as well as narrower ly@xmath45 , heii , niv and nv ( fig . a2 ; [ 15 ] ) . * lh1200.004 : * from two possibilities , an lris slit had been placed on the galaxy with the marginally higher @xmath56 value ( 0.044 versus 0.033 ) . the galaxy has a weak line in its spectrum at 558 nm , possibly ly@xmath45 , but we were unable to determine the redshift reliably . * lh1200.005 : * lris slits were placed on a faint optical galaxy @xmath33 arcsec west of the obvious radio counterpart . the galaxy is estimated to lie at @xmath57 from absorption lines in its optical spectrum ( c05 ) , a value tentatively confirmed via h@xmath45 in the @xmath11 band , although nirspec slit - rotation problems mean the line can not be recovered in the final spectrum or image . although optically faint ( fig . 3 , @xmath58 ) , the extremely red object ( ero ) described by lutz et al.(2001 ) , seen in fig . 4 at the radio position , was targeted using lris and gmos . no redshift was forthcoming . the likelihood of finding a @xmath59 galaxy so close to the smg centroid is slim , so the two may well be associated , but we regard the redshift as tentative until h@xmath45 is detected unambiguously . * lh1200.006 : * lris slits had been placed on by far the most probable radio counterpart in the region ( @xmath60 ) . its optical spectrum was identified by c05 as that of a starburst at @xmath61 ( fig . this position was also observed by swinbank et al . ( 2004 ) using nirspec , although only [ nii ] would have been accessible . since the one - dimensional optical spectrum is not wholly convincing , this redshift can not be relied upon absolutely . the optically bright object to the nw was targeted by gmos : a foreground galaxy ( @xmath62 ) with h@xmath63 and [ oiii ] emission lines evident ( fig . a2 ; [ 220 ] ) . * lh1200.007 : * there is no secure radio counterpart . an lris slit was placed on a nearby _ xmm - newton _ x - ray source , which is marginally detected in our smoothed radio image . it lies at @xmath64 ( fig . a1 ) , and may contribute submm flux to what could well be a blended submm source . the faint , red galaxy to the north , barely seen in a noisy part of our @xmath11 image ( fig . 4 ) , was not targeted spectroscopically . * lh1200.008 : * an lris slit had been placed on by far the most probable radio counterpart in the region ( @xmath65 ) , an @xmath66 galaxy amongst a dense ensemble of fainter objects ( fig . 3 ) , with absorption lines in its spectrum corresponding to @xmath67 ( fig . a1 ) , with ciii ] and heii weakly in emission . * lh1200.009 : * gmos slits were placed on the radio counterpart and on an optically bright galaxy to the nnw . an unambiguous redshift could not be determined for either , although the nnw galaxy may show faint [ oii ] at @xmath68 ( fig . a2 ; [ 231 ] ) . lris slits had been placed on a faint @xmath25-band galaxy @xmath36arcsec ssw of the obvious radio counterpart . this was due to an positional offset in an earlier version of the greve et al . mambo catalogue . the galaxy 3arcsec nw of the slit centre , 4arcsec sww of the radio source , appears to be a ly@xmath45 emitter at @xmath69 ( fig . a1 ) , though no other lines are seen . as with lh1200.005 , the likelihood of finding a @xmath59 galaxy so close to the submm centroid is low , so it may be associated with the radio source and the smg . the position of the radio source is blank to @xmath70 in our deep gemini / niri imaging ( fig . the ly@xmath45 emitter is detected , barely , in @xmath11 . we note that at least one similarly faint smg ( smmj14009 + 0252 , @xmath71 ) has been found at @xmath59 ( smith et al . , in preparation ) . * lh1200.010 : * lris slits had been placed on both of the radio - identified galaxies in this region with one of them ( and several other radio - quiet galaxies ) targeted by gmos . by far the most robust counterpart ( @xmath72 ) , morphologically complex in @xmath25 ( fig . 3 ) but relatively uncluttered in @xmath11 ( fig . 4 ) , has the spectrum of a starburst at @xmath73 ( c05 , smmj105230.73 + 572209.5 ) , confirmed in h@xmath45 by swinbank et al . ( 2005 ) and in nv by gmos ( ly@xmath45 falling between chips fig . a2 ; [ 210 ] ) , although undetected in co(32 ) at iram ( greve et al . the lris spectrum of the radio - bright disk - like galaxy 10arcsec to the sw is also consistent with this redshift . nothing was seen at the remaining gmos slit positions , [ 211 ] and [ 212 ] . * lh1200.011 : * an lris slit had been placed on by far the most probable radio counterpart in the region ( @xmath74 ) , a compact source with a 5-arcsec - long tail visible in @xmath25 and @xmath11 ( figs 3 & 4 ) . it has the spectrum of a starburst at @xmath75 ( c05 , smmj105158.02 + 571800.2 ) , confirmed convincingly in h@xmath45 and [ nii ] by swinbank et al . the fainter radio source , 14arcsec to the nee , lies at @xmath76 . simpson et al . ( 2004 ) identified a @xmath77-band feature , presumably a noise spike , as [ oii ] at @xmath78 ( rest - frame 359 nm for @xmath79 ) , as well as a continuum break at 1.2@xmath38 m ( consistent with the balmer break at either redshift ) . * lh1200.012 : * a complex field , with the robust radio identification ( @xmath80 ) lying between 34 galaxies visible in @xmath25 , and on top of a galaxy seen in the _ spitzer _ 3.6-@xmath38 m imaging of huang et al . a ns - oriented lris slit had been placed on the brightest of the @xmath25-band galaxies , @xmath32arcsec sse , a starburst at @xmath81 ( fig . a1 ; c05 , smmj105155.47 + 572312.7 ) . the nirspec slit , centred on the same point , yielded continuum but no strong lines . however , 2arcsec nw along the slit where fuzz is visible in @xmath25 , 1arcsec sw of the radio centroid there is more red continuum emission a line at 2.405@xmath38 m , corresponding to h@xmath45 at @xmath82 ( fig . we view the likelihood of its association with the smg as high . another lris slit was placed on the radio source to the nw , associated with another _ spitzer _ galaxy , this time at @xmath83 ( @xmath84 in h@xmath45 according to swinbank et al.2005 ) . gmos slits [ 129 ] through [ 132 ] were placed on all of the brightest optical knots , but nothing was seen in the spectra . * lh1200.014 : * an interesting case , with strong similarities to lh1200.010 and .012 . the brightest of the two radio counterparts associated with faint @xmath11-band emission ( fig . 4 ) is the least likely identification , although by a small margin ( @xmath85 versus 0.022 ) . lris slits were placed on the brightest @xmath25-band galaxy ( fig . 3 ) , just to the wsw , which displays [ oii ] in emission as well as ca h / k and several balmer lines in absorption at @xmath86 ( fig . a1 lh1200.014a ; c05 , smmj105200.22 + 572420.2 ) . its association with the smg , and with the radio - identified galaxies , is plausible , as is the possibility that it acts as a lens ( chapman et al . the slit passed over the radio centroid which appears to share the same broad spectral features ( fig . a1 lh1200.014b ) . a gmos slit , [ 123 ] , was placed directly on the radio centroid , but the spectral coverage ( @xmath87550 nm in this case ) meant we were unable to confirm the presence of [ oii ] . * lh1200.042 : * lris slits had been placed on by far the most probable radio counterpart in the region ( @xmath88 ) . although optically faint , it has an absorption - line spectrum consistent with @xmath89 ( fig . a1 lh1200.042a ) . the bright optical galaxy 3arcsec to the sse , targeted by both gmos [ 225 ] and lris , is associated with very faint radio emission and is tentatively consistent with @xmath90 ( fig . a1 lh1200.042b ) . * lh1200.096 : * again , lris and gmos slits had been placed on the most probable radio counterpart ( @xmath91 ) , an ero ( i02 ) visible out to 24@xmath38 m ( egami et al . 2004 ) , ignoring several brighter optical galaxies . the lris spectrum was classified as a starburst at @xmath92 by c05 ( fig . a1 smmj105151.69 + 572636.0 ) ; gmos saw nothing at this position . this is the curious smg discussed by i02 , apparently associated with the steep - spectrum lobe of a radio galaxy , the flat - spectrum core of which lies to the west , with [ oii ] evident at @xmath93 in its spectrum ( fig . a2 , [ 120 ] ) . is this system a jet - triggered burst , a galaxy projected onto an unrelated radio lobe , or a faint , dusty , ultra - steep - spectrum radio galaxy ? our picture of this system is muddled and contradictory . to confuse matters further , our niri @xmath11-band imaging ( fig . 4 ) reveals another ero , wsw of the radio emission , visible out to 8@xmath38 m in the _ spitzer _ imaging , and just missed by gmos slit [ 119 ] on a nearby , bluer galaxy . * lh1200.104 : * despite the lack of radio detections in the vicinity , a bright _ spitzer _ counterpart described by ivison et al . ( in preparation ) suggests this smg is not spurious . a diffraction spike from the nearby star makes identification impossible in our optical imaging .

the modest significance of most sources detected in current ( sub)millimetre surveys can potentially compromise some analyses due to the inclusion of spurious sources in catalogues typically selected at @xmath03.03.5@xmath1 . here , we develop and apply a dual - survey extraction technique to scuba and mambo images of the lockman hole . cut above 5@xmath1 , our catalogue of submillimetre galaxies ( smgs ) is more robust than previous samples , with a reduced likelihood of real , but faint smgs ( beneath and around the confusion limit ) entering via superposition with noise . our selection technique yields 19 smgs in an effective area of 165arcmin@xmath2 , of which we expect at most two to be due to chance superposition of scuba and mambo noise peaks . the effective flux limit of the survey ( @xmath34mjy at @xmath31 mm ) is well matched to our deep 1.4-ghz image ( @xmath4jy beam@xmath5 ) . the former is sensitive to luminous , dusty galaxies at extreme redshifts whilst the latter probes the @xmath6 regime . a high fraction of our robust smgs ( @xmath380 per cent ) have radio counterparts which , given the @xmath310-per - cent contamination by spurious sources , suggests that very distant smgs ( @xmath7 ) are unlikely to make up more than @xmath310 per cent of the bright smg population . this implies that almost all of the @xmath8mjy smg population is amenable to study via the deepest current radio imaging . we use these radio counterparts to provide an empirical calibration of the positional uncertainty in smg catalogues . we then go on to outline the acquisition of redshifts for radio - identified smgs , from sample selection in the submillimetre , to counterpart selection in the radio and optical / infrared , to slit placement on spectrograph masks . we determine a median of @xmath9 from a sample of six secure redshifts for unambigious radio - identified submillimetre sources and @xmath10 when we include submillimetre sources with multiple radio counterparts and/or less reliable redshifts . these figures are consistent with previous estimates , suggesting that our knowledge of the median redshift of bright smgs population has not been biased by the low significance of the source catalogues employed . galaxies : starburst galaxies : formation cosmology : observations cosmology : early universe

arxiv

with the introduction of efficient multi - object spectrographs on 4m - class telescopes , it has become possible to construct large samples of faint galaxies with measured redshifts . with such a sample , one can compute the luminosity function ( lf ) of galaxies as a function of redshift and thereby directly observe the evolution ( or lack thereof ) of the galaxy population . several groups have now presented the results of deep , faint galaxy redshift surveys ( @xcite , cfrs ; @xcite , autofib ; @xcite ; @xcite , cnoc ) . the conclusions from these surveys are in broad agreement : the population of blue , star - forming galaxies has evolved strongly since @xmath16 while the population of red galaxies shows at most modest signs of evolution ( although , see kauffmann , charlot , & white ( 1996 ) for an alternative analysis of the red galaxies ) . however , there are important differences as well . lin et al . ( 1996a ) demonstrate that the lfs from the various groups are formally inconsistent with each other . since there are many selection effects involved with the construction and analysis of faint redshift surveys , it is difficult to pinpoint the reasons for the disagreement between the various groups . while it is likely that the small numbers of galaxies in each survey and the small areas covered are partly responsible , it is also likely that systematic errors are in important contributor to the differences in detail . quantitative estimates of the evolution are , of course , dependent upon having a reliable measurement of the local lf , and it is , therefore , of concern that there remain considerable uncertainties about the _ local _ lf . the lfs derived from large - area local redshifts survey ( e.g. , the stromlo / apm survey , loveday et al . 1992 ; the cfa survey , marzke , huchra , & geller 1994a ; the las campanas redshift survey , lin et al . 1996b ) all have similar shapes , but there are still substantial differences over the overall normalization , the characteristic luminosity , and the slope at low luminosities . the rapid evolution at @xmath17 required to match steep @xmath18-band counts at intermediate magnitudes @xmath19 ( maddox et al . 1990 ) could be reduced if the normalization or the faint - end slope have been underestimated . the results of the largest of the local surveys , the las campanas redshift survey ( lcrs ) with 18678 galaxies used in the lf analysis and a median redshift of @xmath20 , are seemingly consistent with both a low normalization and a flat faint - end slope . the lcrs is selected from ccd drift scans rather than photographic plates and surveys what should be a fair volume of the universe ( shectman et al . 1996 , davis 1996 ) . it also probes both the southern and northern galactic caps . accordingly , the local luminosity function computed from their data should be free from systematic photometric errors and fluctuations in large - scale structure in the distribution of galaxies . however , both the cfa survey and the autofib survey find a normalization which is a factor of 2 higher than that obtained from the lcrs . while the normalization of the cfa survey can be questioned on the grounds that it does not sample a fair volume , the autofib survey is the concatenation of many fields distributed across the sky . the autofib survey is particularly important because the galaxy sample was selected with a much fainter surface brightness threshold than any of the other local surveys . mcgaugh ( 1994 ) emphasizes that a large population of intrinsically luminous but low surface brightness galaxies may be missed in the shallow photometry on which all the local surveys , except autofib , are based . a steep faint - end slope of the lf , with a power law exponent of @xmath21 , is a natural prediction of galaxy formation theories based on hierarchical structure formation models ( kauffmann , guiderdoni , & white 1994 ) . there is only weak evidence for a steep faint - end slope in the local field galaxy lf . marzke et al . ( 1994b ) report an upturn in the luminosity function of late - type galaxies with @xmath22 , but lcrs , autofib , and cowie et al . ( 1996 ) all derive a flat faint - end slope . there is , however , evidence for a steep faint - end slope in galaxy clusters ( e.g. , de propris et al . 1995 , bernstein et al . environmental influences on galaxy evolution may be reflected in variations of the lf for galaxies in different environments , and it is therefore important to measure the lf in a variety of environments . in this paper , we investigate the evolution and environmental dependence of the galaxy lf based on data obtained during the course of our redshift survey of the corona borealis supercluster . the primary motivation for the survey was to study the dynamics of the supercluster . however , the majority of galaxies for which we measured redshifts actually lie behind the corona borealis supercluster , thus providing a sample suitable for study of the evolution of the lf . the galaxies were originally selected from plates taken as part of the second palomar observatory sky survey ( poss - ii ; @xcite ) and have been calibrated in the gunn @xmath23 and @xmath7 bands , which correspond roughly to the photographic @xmath24 and @xmath25 bands . previous redshift surveys have generally been either selected in bluer bands ( @xmath18 ) , for sensitivity to changes in star - formation rates , or redder bands ( @xmath26 and @xmath27 ) , for sensitivity to old stellar populations which more reliably trace stellar mass . although we had no option but to use the @xmath23 and @xmath7 bands , the two bands turn out fortuitously to have the virtue that corrections to the rest @xmath18 band , where lfs are traditionally computed and compared , are small since the @xmath23 band matches the rest @xmath18 band at @xmath28 and the @xmath7 band matches the rest @xmath18 band at @xmath29 . the cnoc survey used photometry in @xmath23 and @xmath7 as well , and so it is particularly interesting to compare our results to that survey since there should be no systematic effects due to using different passbands for galaxy selection . finally , with over 400 redshifts in the corona borealis supercluster , and roughly 300 in a background supercluster , we can explore the variation of the lf from the field to the supercluster environment . the paper , the second in the series presenting results from the norris survey of the corona borealis supercluster , is organized as follows . in 2 , we summarize our survey , particularly emphasizing those features that are directly relevant to the computation of the lf . we discuss the details of the computation of the lf in 3 . the results are given in 4 for both field galaxies and for the two superclusters individually and are discussed in 5 . finally , we summarize our conclusions in 6 . we use a hubble constant @xmath30 km s@xmath31 mpc@xmath32 and a deceleration parameter @xmath33 . for comparison to the most recent work in the field ( e.g. , cfrs and cnoc ) , we use the ab - normalized @xmath18 band , @xmath6 ( oke 1974 ) . the offsets from @xmath6 to @xmath8 and @xmath18 are @xmath34 and @xmath35 ( fukugita , shimasaku , & ichikawa 1995 ) . the norris survey of the corona borealis supercluster has been described in detail in small et al . ( 1996 ) , paper i of the current series , and will be only briefly reviewed here . the core of the supercluster covers a @xmath36 region of the sky centered at right ascension @xmath37 , declination @xmath38 and consists of 7 rich abell clusters at @xmath11 . since the field - of - view of the 176-fiber norris spectrograph is only 400 arcmin@xmath39 , we planned to observe 36 fields arranged in a rectangular grid with a grid spacing of 1 . as it turned out , we successfully observed 23 of the fields and 9 additional fields along the ridge of galaxies between abell 2061 and abell 2067 , yielding redshifts for 1491 extragalactic objects . we have extended our survey with 163 redshifts from the literature , resulting in 1654 redshifts in the entire survey . 1022 of these galaxies lie beyond the corona borealis supercluster , although of these 1022 , 325 ( 318 with @xmath4 ) galaxies are in a background supercluster ( @xmath13 ) which we have dubbed the `` abell 2069 supercluster . '' the survey fields are distributed across an area of 25 deg@xmath39 . the total area covered by the 32 observed fields , albeit sparsely sampled , is 2.99 deg@xmath39 . as noted above and described in detail in paper i , the objects have been selected from poss - ii photographic plates of poss - ii field 449 , which neatly covers the entire core of the supercluster . we have both a @xmath24 ( kodak iii - aj emulsion with a gg395 filter ) plate and an @xmath25 ( kodak iii - af emulsion with a rg610 filter ) plate . the plates were digitized with 1 arcsec@xmath39 pixels at the space telescope science institute and then processed using the sky image cataloging and analysis tool ( skicat , @xcite ) . the instrumental intensities recorded by skicat were calibrated with ccd sequences in the @xmath23 and @xmath7 bands of galaxies in abell 2069 . the random magnitude errors are 0.25@xmath40 for @xmath23 and @xmath7 brighter than 21@xmath40 and become substantially worse at fainter magnitudes . ( we describe how we correct our computed lfs for these magnitude errors in 3.2 . ) with the skicat system , the star - galaxy separation is 90% accurate to @xmath41 . our lf analysis is limited to galaxies with @xmath42 . since our original motivation for the survey was to study the dynamics of the corona borealis supercluster , we chose a comparatively high spectral resolution for a faint galaxy redshift survey . a third of the objects were observed with @xmath43spectral resolution during the period when the largest ccd available at palomar was a 1024@xmath39 device with 24 pixels ; the rest were observed with @xmath44 resolution with a very efficient 2048@xmath39 ccd ( also with 24 pixels ) . since the operation of retrieving the fibers for one set - up and redeploying the fibers for another takes roughly an hour with the norris spectrograph , we decided to observe only two fields per night in order to minimize the amount of time lost due to changing fields . thus , our exposures were 2 - 4 hours long , and we generally obtained high quality spectra on @xmath45 galaxies . in figure [ figures : success ] , we plot our success rate , the fraction of objects ( i.e. , including stars and quasars ) on which fibers were deployed for which we successfully measured redshifts , as a function of magnitude . figure [ figures : success ] shows that our success rate falls substantially below unity beyond @xmath46 . therefore , we have computed weights for each galaxy to correct for a our incomplete sampling . the weight for a particular object is defined to be simply the ratio of the total number of objects in the photometric catalog to the number of objects with redshifts in an interval of @xmath47 centered on the magnitude of the object . this prescription assumes that the redshift distribution of the objects for which we failed to measure redshifts is identical to the redshift distribution of the objects for which we successfully measured redshifts . the color distribution of the objects that we observed but failed to identify is similar to that of the objects that we successfully observed , leading us to conclude that we do not suffer biases against particular types of galaxies ( paper i ) . moreover , we do not believe that we should have redshift - dependent biases in our success rate . since we limit the computation of the lf to @xmath48 , we are unlikely to be affected by a bias in redshift . the 4000 break and ca h , ca k lines of old stellar populations and the [ ] line of star - forming galaxies are all within our spectral range out to @xmath1 . we plot the calculated weights as a function of @xmath7 magnitude for all galaxies with @xmath42 and which satisfy our surface brightness threshold ( see below ) in figure [ figures : weights ] . the weights are greater than unity even for bright galaxies because of the sparse sampling of our survey area . ( since the very brightest galaxies ( @xmath49 ) produce scattered light contamination of nearby spectra on the ccd , we usually did not place fibers on galaxies with @xmath49 , and thus the weights increase for the very brightest galaxies . ) in paper i , we carefully studied the surface brightness selection effects present in our sample . we found that by restricting our sample to objects with @xmath42 and with core magnitudes @xmath50 ( where the core magnitude is the integrated magnitude within the central 9 arcsec@xmath39 ) , we are free from surface brightness selection effects . for comparison , @xmath51 corresponds to a central surface brightness of @xmath52 @xmath7 mag arcsec@xmath53 for a galaxy with an @xmath54 ( @xmath55 mag in the @xmath7 band ) disk . we compute galaxy lfs in the rest - frame @xmath6 band and , for the local lf , in the rest - frame gunn @xmath7 band as well . rest - frame colors and @xmath56-corrections are computed from the spectral energy distributions compiled by coleman , wu , & weedman ( 1980 , hereafter ccw ) . we assign each galaxy a spectral type based on its @xmath57 color and its redshift . following lilly et al . ( 1995 ) , the spectral type is a real number which takes the values 0 for an elliptical galaxy , 2 for an sbc galaxy , 3 for an scd galaxy , and 4 for an i m galaxy . we then interpolate between the ccw spectral energy distributions to construct the spectral energy distribution appropriate for the given spectral type . galaxies whose colors are redder than a ccw e galaxy or bluer than a ccw i m galaxy are simply assigned the spectral energy distribution of an e galaxy or an i m galaxy , respectively . the number of galaxies with colors outside the limits defined by the ccw e and i m types is known to be small even to large redshifts ( e.g. , crampton et al . the fact that many of our galaxies lie outside the ccw limits ( see figure 17 , paper i ) is due to the large errors in our colors ( @xmath58 ) . we compute the absolute rest - frame @xmath6-band magnitude as follows : @xmath59 where @xmath60 incorporates the corrections based on the spectral energy distribution and @xmath61 is the luminosity distance in mpc . the @xmath62 term represents the change in the bandwidth with redshift and is included in the traditional @xmath56-correction . again following lilly et al . ( 1995 ) , we separate the bandwidth stretching term , which has negligible error since it depends only on the accurately measured redshift , from the terms which depend on the spectral energy distribution and are therefore much more uncertain . we plot @xmath60 for the @xmath23 and @xmath7 bands in figure [ figures : keff ] . by converting from @xmath63 for objects with @xmath64 and from @xmath65 for objects with @xmath66 , @xmath60 may be kept less than @xmath67 for @xmath68 for all spectral types . we use the step - wise maximum - likelihood ( swml ) method of efstathiou , ellis , & peterson ( 1988 ) to estimate the lf . the probability of observing a galaxy of absolute magnitude @xmath69 at redshift @xmath70 in a flux - limited catalog is given by , @xmath71 where @xmath72 is the lf and @xmath73 is the intrinsically faintest galaxy observable at @xmath70 in the flux - limited catalog . the lf is parameterized as a set of @xmath74 numbers @xmath75 such that @xmath76 and then the likelihood , @xmath77 where @xmath78 is the number of galaxies in the sample , is maximized with respect to the @xmath75 . we constrain the values of @xmath75 to satisfy @xmath79 the virtues of the swml method are that it is not biased by the presence of clustering since the normalization of the lf cancels out of the expression for the probability @xmath80 and also that one does not have to assume a particular functional form for the lf . in order to include the weights , we make the substitution @xmath81 where @xmath82 is the weight of galaxy @xmath83 ( zucca , pozzetti , & zamorani 1994 , lin et al . one must then estimate the mean galaxy density separately . we use a standard technique which we describe below . since the weights are greater than one , their use will increase the calculated likelihood for the sample and thus lead to artificially small error estimates . by renormalizing the weights so that @xmath84 , the error estimates are appropriate for the true sample size ( lin et al . 1996b ) . note that the normalization constraint on the @xmath75 reduces the estimated errors . we do not use the traditional @xmath85 method ( @xcite ) employed by ellis et al . ( 1996 ) and lilly et al . ( 1995 ) since the method is sensitive to clustering . we have , however , compared the results of the two techniques for samples with @xmath86 , where the clustering in our survey is not pronounced , and found that they agree satisfactorily . for @xmath87 , we can construct volume - limited sub - samples with @xmath4 in which any galaxy with @xmath88 is visible in the entire volume . of course , the value of the lf in a given magnitude bin for a volume - limited sample is estimated by counting the number galaxies with absolute magnitudes in the bin and then dividing by the volume of the sample and the width of the bin . the swml lfs for @xmath87 agree well with the lfs estimated from the volume - limited samples . we compute the mean density @xmath89 of a magnitude - limited sub - sample using the following estimator : @xmath90 where @xmath91 is the volume of the sample , @xmath78 is the number of objects in the sample , and @xmath92 is the selection function . the selection function , @xmath93 gives the fraction of the lf observable at a given redshift . here , @xmath94 is the lf , @xmath95 is the maximum absolute magnitude that an object can have at redshift @xmath96 and still be included in the sample , and @xmath97 is the absolute magnitude of the most instrinsically faint galaxy in the sample . in practice , one does not begin evaluating the integrals at @xmath98 , but rather at the absolute magnitude of the most instrinsically bright galaxy in the sample . the estimator in equation [ equations : mean_density ] is almost identical to the minimum variance estimator derived by davis & huchra ( 1982 ) for @xmath99 and is unbiased by density inhomogeneities . an additional complication of computing a lf in the @xmath6 band where the objects have been selected in the @xmath7 band is that one must ensure that any object , regardless of its color , would have been detectable in both bands . if one ignores this complication , then the faintest objects at a given redshift will be biased in color . in our survey , since the @xmath7 band is centered at a longer wavelength than the @xmath18 band , the faintest objects would be biased to the red . in order to avoid such a bias , we adjust our absolute @xmath6 magnitude limits as a function of redshift so that the bluest galaxy at any @xmath6 magnitude limit would be observable in the @xmath7 band . for the local lf computed in the @xmath7 band , the bias works in the opposite sense , and so we adjust our rest - frame absolute @xmath7 magnitude limits to ensure that the reddest galaxy a given limit would be detected in the observed @xmath7 band . for each lf , we estimate the parameters of the best - fitting schechter ( 1976 ) function , @xmath100 where @xmath101 is the normalization , @xmath102 determines the location of the bright - end exponential cutoff , and @xmath103 is the faint - end slope . the fitting was performed using a standard @xmath104-minimization algorithm ( press et al . 1992 ) with the schechter function integrated over the width of the adopted magnitude bin . we intend these fits to be useful for comparisons with other work . usually , there are too few points for the fits to be well defined . our random magnitude errors ( @xmath105 ) will artificially brighten the characteristic magnitude @xmath102 of the lf and steepen the faint end . we correct for the magnitude errors by fitting to the data points a schechter function ( equation [ equations : schechter ] ) convolved with a gaussian of dispersion @xmath106 ( efstathiou et al . in fact , however , the corrections to the schechter function parameters are substantially less than the 1@xmath107 statistical errors . the error in the normalization from the @xmath104-fitting only includes the uncertainties due to @xmath102 and @xmath103 . the error due to large - scale structure fluctuations is @xmath108 ( davis & huchra 1982 ) , where @xmath109 is the second moment of the two - point spatial correlation function ( peebles 1980 ) and @xmath91 is the appropriate volume . we use @xmath110 ( @xmath111 mpc)@xmath112 ( tucker et al . 1997 ) and record the error from density fluctuations alongside the uncertainty in @xmath101 due to @xmath102 and @xmath103 . in the following subsections , we report our results for the local lf , the evolution of the lf , and the lfs of the corona borealis and abell 2069 superclusters . all samples are magnitude - limited at @xmath113 . the parameters of the best fitting schechter functions are summarized in table 1 , where the sample is given in the first column , the number of galaxies used to compute the lf in the second column , the absolute range over which the fit is valid in the third column , @xmath101 in the fourth column , @xmath102 in the fifth column , @xmath103 in the sixth column , the reduced @xmath104 in the seventh column , and the estimate of the variance due to density fluctuations in the eighth column . the numbers of galaxies listed in the second column are slightly smaller than the total number of galaxies satisfying the sample listed in the first column because a few galaxies have been trimmed from each sample , as described above , to ensure that there are no color biases in the faintest bins . corrections to @xmath101 to match galaxy counts , as discussed below , have _ not _ been applied to the values listed in table 1 . we wish to emphasize that the fitted schechter functions are intended only to guide the eye and that comparisons of the various lfs in this paper are best done by comparing the individual data points in the plots . the @xmath6-band local galaxy lf is plotted in figure [ figures : local_lf ] . the unfilled circles show the lf for @xmath114 with the superclusters removed . the filled circles show the lf for @xmath114 with the the superclusters included . in order to remove the superclusters , we simply delete all objects with @xmath115 . the median redshift of our local sample with the superclusters removed is @xmath116 . the stromlo / apm lf is plotted with the solid line . we also plot the autofid local lf with the dashed line . in figure [ figures : local_lf ] and all subsequent figures where appropriate , we convolve the schechter function fits to the lfs from other surveys with a gaussian of dispersion @xmath106 in order to facilitate comparisons with our lf data points , which are constructed with galaxies whose photometry suffers from random magnitude errors of @xmath106 . all of the local luminosity functions have similar shapes , but the normalizations and low luminosity ends vary significantly . in order to investigate further the normalization of the local luminosity function , we compute the _ shape _ of local luminosity in the rest - frame @xmath7 band and then normalize this lf to the @xmath7-band counts of weir , djorgovski , & fayyad ( 1995 ) . we plot our @xmath7-band local lf , normalized to the counts of weir et al . ( 1995 ) , in figure [ figures : local_lf_r ] , along with the @xmath7-band local lf from lcrs . to convert from isophotal @xmath117 magnitudes to total gunn @xmath7 magnitudes , we apply a 25% isophotal - to - total light correction and then use @xmath118 ( shectman et al . thus , the corrections compensate for each other and @xmath119 . the counts of weir et al . ( 1995 ) are based on 4 overlapping , high galactic latitude plates taken as part of the poss - ii survey . knowing the shape of the lf , we can estimate the differential number counts as @xmath120 { dv \over dz } dz \nonumber \\ & = & \phi^\ast \int_0^\infty \phi^\prime[m(m , z ) ] { dv \over dz } dz \\ & = & \phi^\ast { di \over dm } \nonumber , \label{equations : diff_counts}\end{aligned}\ ] ] where @xmath121 is @xmath72 with @xmath101 set equal to 1 , @xmath122 is the absolute magnitude of an object at redshift @xmath96 with apparent magnitude @xmath123 , and the @xmath124 are the magnitude intervals . we estimate @xmath101 by minimizing the quantity @xmath125 ^ 2 } \over { \phi^\ast di(m_i)}}\ ] ] with respect to @xmath101 ( efstathiou et al . 1988 ) , which yields @xmath126 ^ 2 / di(m_i ) } \over { \sum_i di(m_i)}}.\ ] ] we find @xmath127 mpc@xmath32 where the errors reflect the changes due to varying jointly @xmath102 and @xmath103 by @xmath128 . this is a 21% reduction from the normalization determined in the norris field itself . although the median redshift of our local sample is @xmath116 , the agreement with lcrs ( @xmath129 ) in the @xmath7 band leads us to believe that we have computed a fair estimate of the local lf . in figure [ figures : r_counts ] , we plot the @xmath7-band differential number counts . the histogram shows the counts from the poss - ii plate from which we selected our objects . the thick solid line is the counts from weir et al . ( 1995 ) , and the triangles are ccd counts from metcalfe et al . the dashed line represents the predicted counts based on our field galaxy lf , including evolution at @xmath86 ( see 4.2 below ) , with the superclusters removed . even with the superclusters removed , our local lf appears to be normalized too high . the solid line , however , gives the predicted counts with the normalization reduced by 21% . with the reduced normalization , the predicted counts agree quite well with the observed counts to @xmath113 . for comparison , we also show , as the dotted line , the predicted counts from the lcrs . these counts fall below the weir et al . ( 1995 ) counts for @xmath130 because they do not include the evolution of the lf beyond @xmath131 . since the volume of the norris region , with the superclusters removed , is @xmath132 mpc@xmath112 , we would expect from equation [ equations : delta_n ] @xmath133 . it is therefore not cause for concern that this region is overdense by 21% . we compute the field galaxy lf in two redshift intervals : @xmath134 and @xmath135 . the results are plotted in figure [ figures : field_lf ] . the normalization of the local lf ( unfilled circles ) has been reduced by 21% to match the @xmath7-band counts . the filled circles are the lf of the high redshift interval . we also plot the lfs of the stromlo / apm survey ( solid line ) , the cnoc survey ( @xmath136 , dashed line ) , and the cfrs survey ( @xmath135 , dotted line ) , all of which have been convolved with a gaussian with @xmath137 to account for random photometry errors . since the lf for the autofib survey was divided into redshift intervals which do not neatly match our redshift intervals and , more importantly , since lin et al . ( 1996a ) have already performed a detailed comparison with the autofib survey , we do not plot the autofib lfs . the @xmath135 lf has clearly evolved with respect to the local lf . we create two sub - samples of galaxies according to the rest - frame equivalent width of [ ] @xmath93727 . the division is made at a rest - frame equivalent width of 10 , which roughly corresponds to dividing the sample into types earlier and later than sbc ( kennicutt 1992 ) and thus allows direct comparison with the results of cnoc and cfrs . for galaxies with @xmath139 , [ ] is not redshifted into our observed wavelength range , and so we use the strength of h@xmath140 to divide our sample . after correcting for stellar absorption , we have 7 galaxies with @xmath139 and @xmath138(h@xmath140 ) @xmath141 5 , which we include in the @xmath138 ( [ ] ) @xmath141 10 sample . we have also investigated separating our sample by color , but we have found that the large errors on our colors tend to dilute trends which are seen clearly in samples defined by @xmath138 ( [ ] ) . the lfs for galaxies with @xmath138 ( [ ] ) @xmath142 10 and @xmath138 ( [ ] ) @xmath141 10 are shown in figure [ figures : no_o2_lf ] and figure [ figures : o2_lf ] , respectively . we also plot in each figure the lfs for the corresponding color - selected samples from cnoc and cfrs . it is important to remember when considering possible detailed discrepancies between our lfs and the cnoc and cfrs lfs that our sample is divided by @xmath138 ( [ ] ) , which , while roughly equivalent to color selection , is not identical . there is no significant indication that the population of early - type galaxies ( i.e. , @xmath138 ( [ ] ) @xmath142 10 ) has evolved since @xmath1 ; this result is not surprising given that the light of early - type galaxies is dominated by red , long - lived stellar populations . in contrast , the lfs of the late - type galaxies ( i.e. , those with @xmath138 ( [ ] ) @xmath141 10 ) show striking evidence for evolution , even though the sample sizes are small and the error bars are large . the lfs of the corona borealis supercluster and the abell 2069 supercluster are given in figure [ figures : clusters_lf ] . we take the redshift range of the corona borealis supercluster to be @xmath143 and that of the abell 2069 supercluster to be @xmath144 . the normalization of the corona borealis supercluster function is a factor of 2 greater than that of the abell 2069 supercluster , but the shapes of the lfs of the two superclusters are similar . note that the volumes used to normalize the superclusters lfs are in redshift space ; the real - space volumes may be quite different ( see 5.3 below for a detailed discussion ) . since the bright ends of the supercluster lfs are clearly not well - described by a schechter function , we restrict our fit to @xmath145 . in addition , the fit to the corona borealis supercluster is limited to @xmath146 since the faintest two data points appear to describe a sharp upturn in the lf . prior evidence for rapid evolution of the galaxy lf to @xmath17 from galaxy counts was based crucially on normalizing the local lf to the bright ( @xmath147 ) galaxy counts from schmidt - telescope photographic surveys ( e.g. , maddox et al . . with this low normalization , predicted counts from the no - evolution model fall well short of the observed counts for @xmath148 . however , the evolution of the lf which we and others observe is not enough to make up for this shortfall . in figure [ figures : b_counts ] , we plot the observed counts from the apm survey ( maddox et al . 1990 ) and from the ccd survey of metcalfe et al . ( 1991 ) , along with various predicted counts . the dotted line shows the expected counts using the loveday et al . ( 1992 ) lf for @xmath134 and the cnoc @xmath6 lf for @xmath86 . despite including the observed evolution of the lf , the predicted counts only begin to agree with the observed counts for @xmath149 , by which point the predicted counts are dominated by galaxies with @xmath86 . in contrast , the predicted counts computed using the evolving lfs measured for our survey and for the autofib survey , while substantially overpredicting the counts for @xmath150 , match the observed counts for @xmath151 . in order to help unravel this confusing situation , we plot in figure [ figures : b_and_r_lf ] various @xmath6- and @xmath7-band local lfs on the same diagram . @xmath6-band lfs are plotted with respect to the bottom axis , while @xmath7-band lfs are plotted with respect to the top axis . the two axes are offset by @xmath152 , which is the median color we measure for field galaxies with @xmath87 . our @xmath6- and @xmath7-band local lfs , both of which have been reduced by 21% following the discussion in 4.1 , are plotted with filled and unfilled circles , respectively . with the color offset , they agree extremely well . the stromlo / apm lf , represented by the solid line , lies consistently below our @xmath6-band lf . the lcrs @xmath7-band lf is consistent with our @xmath7-band lf . unlike lin et al . ( 1996b ) , we do _ not _ conclude that the lcrs @xmath7-band lf matches the stromlo / apm lf . the reason for the disagreement lies in the different measurements of the median color of local galaxies . our median color is that of an sb galaxy , whereas the lcrs median color is that of a much redder e galaxy . the median color of galaxies in the _ third reference catalogue of bright galaxies _ ( de vaucouleurs et al . 1991 ) is @xmath153 ( see table 2 of fukugita et al . 1995 ) , which is roughly that of an sb galaxy . sebok ( 1986 ) also concludes that the typical local galaxy has the color of an sb galaxy . it is thus surprising that the mean color of the lcrs galaxies is so red . since our @xmath7-band lf agrees with the lcrs @xmath7-band lf and since the lcrs computed the colors of their galaxies by matching directly to the apm catalog , the most natural explanation for the anomalous red colors of the lcrs is a systematic error in the bright apm magnitudes . we note that weir et al . ( 1995 ) conclude that magnitudes derived from @xmath24 plates are only reliable for @xmath154 or , equivalently , @xmath155 . galaxies brighter than @xmath156 are saturated on the photographic plates . a systematic error in the bright apm counts would remove the need for rapid , and otherwise unsubstantiated , galaxy evolution at @xmath17 . we note also that there is possible evidence for a rise in the local lf above an @xmath157 for the least luminous galaxies in our survey . such a rise is evident in the data of marzke et al . ( 1994b ) for irregular galaxies , the cfrs , and the low surface brightness galaxy redshift survey of sprayberry et al . similar behavior is also perhaps visible in the local lf of the autofib survey . although ellis et al . ( 1996 ) argue against a rise in the lf for @xmath158 , the three faintest points in the their local lf ( their figure 8) all lie above their preferred schechter function fit . however , since such faint galaxies are only visible in our survey in a quite small volume , we refrain from attempting to make a definitive statement . we asserted in 4.2 that the lf of star - forming galaxies ( @xmath138 ( [ ] ) @xmath141 10 ) evolved from @xmath0 to @xmath1 and that the luminosity function of galaxies with weak [ ] emission ( @xmath142 10 ) did not . a powerful method to verify this result is to compute @xmath159 for appropriate samples ( @xcite ) . if there is no evolution in the number density of objects , @xmath160 ; if the number density declines , @xmath161 ; and if the number density increases , @xmath162 . our @xmath159 analysis is complicated by the need to remove the superclusters and to account for the 21% overdensity of our local field . in order to excise the superclusters , we simply remove all galaxies with @xmath115 . if the maximum redshift @xmath163 at which a galaxy could be observed in our survey lies in the range 0.06 to 0.13 , we set @xmath164 . we correct for the overdensity of our local field by reducing the weights of galaxies with @xmath114 by 21% . the values of @xmath159 for various samples of galaxies are given table 2 , both with and without the 21% correction to the weights of the galaxies with @xmath114 . for samples selected by color , we compute the @xmath159 statistic for galaxies in the redshift range @xmath2 ( with the supercluster region excluded ) . for samples selected by the strength of [ ] , we use the redshift range @xmath165 ( with the supercluster region excluded ) since [ ] from objects with @xmath139 is not redshifted into our observed wavelength range . the differences between our weighted and unweighted statistics are small . the @xmath159 test supports our claim , at the @xmath166 level for the weighted statistic , that the population of star - forming galaxies is evolving . the rate of evolution increases with the strength of [ ] . the population of galaxies with @xmath138 ( [ ] ) @xmath141 20 has @xmath167 . this result is analogous to the results from cfrs in which the rate of evolution is the strongest for the bluest population of galaxies . @xmath159 for the population of red galaxies is consistent with no evolution , in agreement with the lf analysis . broadly speaking , our results are in accord with the results of cfrs , autofib , cowie et al ( 1996 ) , and cnoc . since the cnoc survey , like our survey , uses photometry in the @xmath23 and @xmath7 bands , it is particularly interesting to compare our results in detail to theirs since many of the systematic effects associated with @xmath56- and color - corrections ought to be the same . it is encouraging to see ( figure [ figures : no_o2_lf ] and [ figures : o2_lf ] ) that , given the small samples , our lfs agree well with those of cnoc . the agreement is significantly improved if one reduces the normalization of the cnoc lfs by 20% , as , in fact , is recommended by lin et al . ( 1996a ) . now that we have confirmed that the population of blue galaxies is evolving with redshift , we wish to investigate whether we can detect differences in the colors and the spectral properties of the evolving population with redshift . first , we reiterate that the color distribution of objects with measured redshifts is similar to the color distribution of unidentified objects , leading us to believe that the type distribution of the identified objects is not strongly biased ( paper i ) . although emission lines are generally easier to detect than absorption lines , the difficulty of identifying emission lines at observed wavelengths longer than 5577 , where there are many strong night sky features , combined with the strength of ca h , ca k , and the 4000 break in absorption line objects at @xmath168 mitigate the bias in favor of emission line objects . a sample of the spectra of 8 absorption line objects in this redshift range is shown in figure [ figures : ab_line_objs ] to illustrate the ease of detection of their characteristic absorption features . in figure [ figures : obs_g - r ] , we plot the observed @xmath5 color as a function of redshift of all the objects in our survey along with the tracks of five representative model galaxy spectra . the bluest spectrum is simply a flat - spectrum object , @xmath169 the four other spectra are typical of the hubble types e , sbc , scd , and i m and are taken from ccw . the large , solid diamonds mark the observed median color in the redshift ranges @xmath170 ( arranged to exclude the superclusters ) , @xmath171 , @xmath172 , @xmath173 , and @xmath174 . perhaps counter - intuitively , we see that the observed median color does not become progressively bluer with respect to the model spectra with increasing redshift . as discussed by lilly et al . ( 1995 ) and illustrated in our figure [ figures : rest - frame_g - r ] , the median color does not become bluer because the color - magnitude relation in the local universe ( i.e. , the fact that more luminous galaxies are redder ) breaks down for @xmath131 as the population of blue galaxies brightens while the population of red galaxies does not evolve significantly . we plot in figure [ figures : rest - frame_g - r ] the absolute @xmath6-band magnitudes versus rest - frame @xmath5 for our galaxies divided into four intervals in redshift . in the low - redshift interval , the color - magnitude relation is apparent , but it disappears in the higher redshift intervals . at @xmath175 , we observe far down the lf to absolute magnitudes where blue galaxies are dominant . at @xmath131 , we do not observe as far down the lf , but the population of blue galaxies has brightened so as to be included in the samples . the increase in the luminosity of the population of blue galaxies is presumably associated with a change in the star formation activity at earlier times . in our spectra , there are two convenient star formation indicators , [ ] @xmath93727 and h@xmath10 @xmath94101 . [ ] emission is found in galaxies with ongoing star formation , and its strength is proportional to the strength of h@xmath103 ( @xcite ) . strong h@xmath10 absorption is a signature of the presence of a population of a - stars , which are visible @xmath1761 gyr after a burst of star formation . these lines are reliably measured by automated programs ( see paper i ) since both occur in regions of the spectrum where the continuum is featureless and there is little crowding from other lines . an important virtue of the h@xmath10 line is that , since it appears in absorption , a galaxy with detectable h@xmath10 would have been identified no matter what its spectral characteristics , which implies that there is no bias towards detecting objects with h@xmath10 absorption . a galaxy with [ ] emission is , of course , easier to identify than if it had had only absorption lines . however , as we discussed above , the combination of the difficulty of identifying weak emission lines in the face of strong sky subtraction residuals and of the ease of identifying the strong features characteristic of absorption - line galaxies at moderate redshifts suggests that our survey is not strongly biased towards detecting objects with [ ] . by adding together spectra of late - type galaxies in two redshift intervals in order to obtain two spectra with very high signal - to - noise ratios , heyl et al . ( 1996 ) find that the two star - formation indicators [ ] @xmath93727 and h@xmath10 have both become stronger in the higher redshift mean spectrum . those authors interpreted the increase in the strength of [ ] and h@xmath10 in the @xmath86 spectra to imply that not only were the higher - redshift galaxies forming stars more rapidly , but that also the nature of the star formation was changing with redshift . specifically , they asserted that the strong h@xmath10 absorption in the higher redshift sample is evidence that the star formation in that sample is dominated by bursts . with our high - quality spectra , we can reliably measure [ ] and h@xmath10 without adding together the spectra of many galaxies . in figure [ figures : o2_and_hd ] , we plot as a function of redshift the fraction of galaxies with [ ] emission and the fraction of galaxies with h@xmath10 absorption . the vertical dashed line marks the redshift beyond which h@xmath10 is shifted into the region of the spectrum in which sky subtraction becomes increasingly difficult . while the fraction of galaxies with [ ] emission increases with redshift , the fraction of galaxies with strong h@xmath10 absorption shows no significant variation with redshift , suggesting that the star formation is _ not _ occurring in short - lived bursts ( c.f . , hammer et al . 1997 ) in order to investigate this disagreement more carefully , we repeat the analysis of heyl et al . ( 1996 ) and construct high signal - to - noise ratio composite spectra . the individual spectra have been median - combined after scaling by the median count level . the magnitude weights , which correct for our incompleteness at faint magnitudes , have not been applied since scaling by the median count level in each spectrum effectively incorporates the magnitude weights . in figure [ figures : coadd ] , we plot the composite spectra of galaxies with @xmath138 ( [ ] ) @xmath141 20 , @xmath177 , and either @xmath178 ( thick line , @xmath179 ) or @xmath180 ( thin line , @xmath181 ) . @xmath138 ( [ ] ) @xmath141 20 is typical for the late - type galaxies in which heyl et al . ( 1996 ) observe spectral changes with redshift . while the [ ] line is modestly stronger in the higher redshift composite spectrum , there is no difference in the strength of h@xmath10 ( which is partially filled - in by emission in both spectra ) , which indicates that the nature of the star formation has not changed from @xmath1 to @xmath182 in contrast to heyl et al . ( 1996 ) , we again do not conclude that the spectra of intermediate - redshift blue galaxies show spectral signs of short bursts of star formation . ( we do not believe that the small differences in the rest - equivalent widths of the ca h and k lines are significant since it is difficult to fit a reliable continuum in the vicinity of the 4000 break . ) the modest increase in the strength of the [ ] line with redshift for galaxies with @xmath183 is due , as previously discussed by cowie et al . ( 1996 ) , to the increase in the intrinsic luminosity of star - forming galaxies with redshift . in figure [ figures : o2_ew ] , we plot absolute magnitude versus the rest equivalent width of [ ] for the galaxies in our sample divided into four redshift intervals . as the redshift increases , more and more luminous galaxies exhibit strong [ ] . notice , however , that the range of rest equivalent widths does not vary significantly with redshift . it is difficult to draw firm conclusions about the nature of the evolving galaxy population from the lfs alone . the lfs provide only a statistical view of the entire population and include information about the evolution of individual galaxies only indirectly . furthermore , the luminosity of a galaxy is an unreliable tracer of the physical state of the galaxy . the luminosity of a galaxy , especially in the rest - frame ultraviolet , can change dramatically on short timescales , making the identification of the descendants of distant galaxies in the local population very difficult . in order to make further progress , additional data on the nature of distant galaxies is required . the morphologies of distant galaxies , measured with the _ hubble space telescope _ ( hst ) , are already providing crucial clues ( @xcite , @xcite , @xcite ) . the counts of faint elliptical and early - type spiral galaxies match predictions based on counts in the local universe , provided that the local lf is normalized , as advocated here , a factor of @xmath1761.5 - 2 higher than found by loveday et al . in contrast , the hst number counts of late - type and irregular galaxies are far in excess of the counts expected from observations of nearby galaxies , even with a high normalization of the local lf . an alternative method for selecting distant galaxies is by their gas absorption cross section . steidel , dickinson , & persson ( 1994 ) study the evolution of 58 galaxies selected by @xmath1842796 , 2803 absorption seen in the spectra of high - redshift quasars . these authors found that the galaxies responsible for quasar absorption lines in the range @xmath185 , which typically have luminosities near @xmath54 , do not evolve over the redshift range . however , their sample is small and , when divided by color , is not inconsistent with results from surveys of galaxies selected by apparent magnitude ( lilly et al . in addition , steidel et al . ( 1994 ) note that intrinsically faint blue galaxies do not appear in their sample , and so the population of galaxies which is observed to be evolving most rapidly is not included in their survey . it is now possible , especially with 10m - class telescopes , to measure the masses ( e.g. , vogt et al . 1993 , 1996 ; rix et al . 1996 , guzmn et al . 1996 ) and chemical abundances of distant galaxies . since the masses and metallicities of galaxies evolve more smoothly than the luminosities , measurements of these two quantities will allow more easily interpretable comparisons of distant and local populations ( guzmn et al . 1996 ) . in addition , the mass and metallicity are , compared to morphology and even gas absorption cross section ( churchill , steidel , & vogt 1996 ) , straightforward to define and interpret . a survey to measure the masses and chemical abundances of the faint blue galaxies ought to yield important insights into the nature of this rapidly evolution galaxy population and aid in the identification of their present - day counterparts . the lfs of the two superclusters have quite similar shapes and generally resemble the field galaxy lf . since the normalizations of the superclusters lfs are computed in redshift space , in which distances along the line - of - sight may be substantially altered with respect to real space , it is not straighforward to compare the normalization of the supercluster lfs with that of the local field galaxy lf . for the purpose of the discussion here , we introduce a factor @xmath186 , the ratio of the redshift - space volume to the real - space volume . we expect @xmath186 to be in the range @xmath187 . the lower limit corresponds to assuming that the peculiar velocities in the supercluster regions are small ; the upper limit corresponds to assuming that the depth of the superclusters along the line - of - sight is similar to the linear sizes of the superclusters on the plane of the sky ( @xmath188 mpc ) . the real - space mean density of galaxies in the corona borealis supercluster , obtained simply by integrating the measured lf data points ( @xmath189 ) , is @xmath190 mpc@xmath32 . the mean density of field galaxies in the same range of absolute magnitude is @xmath191 mpc@xmath32 . similarly , the real - space mean density of galaxies in the a2069 supercluster ( @xmath192 ) is @xmath193 mpc@xmath32 , while the mean density of field galaxies in the same range of absolute magnitude is @xmath194 mpc@xmath32 . thus , the overdensities are @xmath195 and @xmath196 for the corona borealis and a2069 superclusters , respectively . it is also important to know whether our sampling of the superclusters is biased towards either the abell clusters within the superclusters or the `` field '' of the superclusters . owing to the difficulties of converting redshift - space volumes to real - space volumes , we assess our sampling of the superclusters using projected surface densities , which are not affected by redshift - space distortions . for the corona borealis supercluster , the project surface density @xmath197 is the mean redshift - space galaxy volume density multiplied by the depth in redshift space of the supercluster along the line - of - sight , @xmath198 mpc . thus , @xmath199 mpc@xmath53 @xmath200 mpc@xmath53 . we compare this value with the median surface density of the regions surrounding successfully observed corona borealis supercluster galaxies . we compute the surface density around a given corona borealis supercluster galaxy by counting the number of supercluster galaxies in a 6 arcmin diameter circle surrounding the chosen galaxy . we either simply count the number of galaxies with measured redshifts within the corona borealis supercluster , or we count the total number of galaxies on the original poss - ii @xmath25 plate , weighted by the empirically - determined fraction of galaxies at a given magnitude which are in the supercluster . the results of these two methods agree well : the median values of the raw and weighted surface densities are @xmath201 mpc@xmath53 and @xmath202 mpc@xmath53 . these two values bracket the projected surface density measured by multiplying the mean galaxy density ( computed from the lf ) by the line - of - sight depth of the supercluster . we conclude , therefore , that we have fairly sampled the corona borealis supercluster . we perform an identical analysis for the background a2069 supercluster . the redshift - space depth of the a2069 supercluster is also @xmath203 mpc . given a mean galaxy volume density of @xmath204 mpc@xmath32 , the projected surface density is @xmath205 mpc@xmath53 . the median surface density of the regions surrounding the successfully observed a2069 supercluster galaxies , computed in the same fashion as for the corona borealis supercluster , is @xmath206 mpc@xmath53 ( unweighted ) or @xmath207 mpc@xmath53 ( weighted ) . thus , we are slightly biased to the denser regions of the a2069 supercluster . the overall resemblance between the supercluster lfs and the field galaxy lf suggests that the fundamental physical processes which drive galaxy formation and evolution must not depend strongly on environment . there are , however , important differences between the lfs in the field and in the superclusters that must ultimately be due to environmental effects . the most striking difference between the field and supercluster lfs is that the supercluster lfs to do not continue the exponential decline for galaxies brighter than @xmath208 . both superclusters evidently contain a population of very luminous galaxies . despite the fact that there are only 6 galaxies with @xmath209 in the two superclusters combined , it is clear that these galaxies are giant ellipticals found in the densest regions of the superclusters . the 4 of the 6 for which we have spectra ( the other 2 were taken from the literature ) are dominated by the light of an old , red stellar population . all of the galaxies are found in regions in the upper 27th percentile of local surace density , with half of them found in regions in the upper 10th percentile . in fact , the four galaxies in the corona borealis supercluster are all found in the dense ridge of galaxies between the abell clusters a2061 and a2067 . for the corona borealis supercluster , the characteristic magnitude @xmath102 is @xmath210 brighter than in the field and is quite close to the value measured by colless ( 1989 ) for rich clusters . there is no significant difference between @xmath102 for the abell 2069 supercluster and that of the field . for the corona borealis supercluster lf , the data points for the two faintest magnitude bins hint that the lf may steepen significantly for galaxies fainter than @xmath211 . since the hint is based on only two data points , which are themselves based on only 29 galaxies , we must be cautious in our interpretation . however , a steepening of the supercluster lf fainter than @xmath211 would be in accord with observations of the faint end of the lf in galaxy clusters and groups , in which a number of workers report steep ( @xmath212 ) lfs ( impey , bothun & malin 1988 ; ferguson & sandage 1991 ; biviano et al . 1995 ; de propris et al . 1995 ; driver & phillipps 1996 ) . with the exception of the study of the coma cluster by biviano et al . ( 1995 ) , the observations of steep faint ends in cluster and group lfs have depended , since redshifts were not available , on the subtraction of a background component from the foreground cluster , a procedure which is prone to systematic errors . although it would be unwise to draw strong conclusions from our two data points , they do have the virtue of being based on galaxies with measured redshifts . we have presented an analysis of the lf of galaxies in the norris survey of the corona borealis supercluster . our @xmath7-band lf of local field galaxies , when normalized to counts in high galactic latitude fields , agrees well with the lcrs . however , the normalization of our @xmath6 local lf is roughly a factor of 1.6 higher than that of the stromlo / apm survey . since lin et al . ( 1996b ) claim that the lcrs local lf agrees well with the stromlo / apm survey , the difference must lie in a systematic photometry error in one ( or more ) of the three surveys . a clue to the nature of this error is provided by examining the mean colors of norris and lcrs galaxies . the mean color of local norris galaxies is that of an sb galaxy , whereas the mean color of lcrs galaxies , computed by matching lcrs galaxies directly with galaxies in the apm catalog , is that of an e galaxy . given the agreement of the norris and lcrs @xmath7-band lfs , we therefore believe that the error is most likely in the apm catalog . indeed , brightening the magnitudes of apm galaxies with @xmath213 by @xmath1760.25@xmath40 would bring all of the local lfs into agreement . a ccd - based local redshift survey ( e.g. , the sloan survey , gunn & weinberg 1995 ) will certainly resolve any remaining questions about the local lf . we have observed evolution of the field galaxy lf within our sample , thereby confirming the conclusions drawn from several previous redshift surveys . the evolution is limited to the population of blue , star - forming galaxies . the population of blue galaxies becomes more luminous with increasing redshift , and thus the median color of the field galaxy population does not change . the evolution of the population of blue galaxies is reflected in the larger fraction of galaxies at higher redshift exhibiting spectral signatures of ongoing star formation . in contrast to the results of heyl et al . ( 1996 ) , but in agreement with hammer et al . ( 1996 ) , we find that the star formation is long - lived . we do not see evidence for short - term bursts of star formation . we are unable to detect any evolution of the population of galaxies with @xmath138 ( [ ] ) @xmath142 10 . the fact that the evolution which we observe in our @xmath23- and @xmath7-band selected survey is consistent with the results of the surveys of lilly et al . ( 1995 ) , ellis et al . ( 1996 ) , cowie et al . ( 1996 ) , and lin et al . ( 1996 ) adds to the already strong evidence that a consistent picture of the evolution of the galaxy lf is emerging . in particular , it is quite reassuring that our lfs agree well with those of lin et al . ( 1996a ) since both used the @xmath23 and @xmath7 bands and should therefore have very similar systematic effects . the lfs of the two superclusters show significant differences from the field galaxy lf , despite considerable overall similarity . since the superclusters are @xmath214 denser than the field , we are likely to be observing the influence of the environment on galaxy formation and evolution . the most prominent difference is an excess of very bright galaxies ( @xmath208 ) relative to the best - fitting schechter function , which accurately describes the field lf over the observed absolute magnitude range . these very bright galaxies are found in very dense regions of the superclusters and have spectra dominated by an old , red stellar population . in the corona borealis supercluster , the characteristic magnitude @xmath102 is @xmath210 brighter than in the field . @xmath102 for the abell 2069 supercluster is , however , very close to the value in the field . we have also presented suggestive evidence that there is a sharp upturn in the supercluster lf for @xmath215 . while there is also a suggestion of an upturn in the local field galaxy lf for the least luminous galaxies in our survey , it does not appear as dramatic as the upturn seen in the supercluster lf , but more data are needed before the possible difference can be quantified . we are grateful to the kenneth t. and eileen l. norris foundation for their generous grant for construction of the norris spectrograph . we thank the staff of the palomar observatory for the expert assistance we have received during the course of the survey , david hogg for many enlightening discussions , and the referee for a careful reading of this paper and helpful suggestions . this work has been supported by an nsf graduate fellowship ( tas ) and nsf grant ast-92213165 ( wlws ) .

we measure the field galaxy luminosity function ( lf ) as a function of color and redshift from @xmath0 to @xmath1 using galaxies from the norris survey of the corona borealis supercluster . the data set consists of 603 field galaxies with @xmath2 and spans a wide range in apparent magnitude ( @xmath3 ) , although our field galaxy lf analysis is limited to 493 galaxies with @xmath4 . we use the observed @xmath5 colors of the galaxies to compute accurate corrections to the rest @xmath6 and @xmath7 bands . we find that our local @xmath7-band lf , when normalized to counts in high galactic latitude fields , agrees well with the local lf measured in the las campanas redshift survey . our @xmath6-band local lf , however , does not match the @xmath8-band lf from the stromlo / apm survey , having a normalization 1.6 times higher . we see compelling evidence that the @xmath6-band field galaxy lf evolves with redshift . the evolution is strongest for the population of star - forming galaxies with [ ] @xmath93727 rest - frame equivalent widths greater than 10 . the population of red , quiescent galaxies shows no sign of evolution to @xmath1 . the evolution of the lf which we observe is consistent with the findings of other faint galaxy redshift surveys . the fraction of galaxies with [ ] emission increases rapidly with redshift , but the fraction of galaxies with strong h@xmath10 absorption , a signature of a burst of star - formation , does not . we thus conclude that the star formation in distant galaxies is primarily long - lived . we also compute the lfs of the corona borealis supercluster ( @xmath11 , 419 galaxies with @xmath12 ) and the abell 2069 supercluster ( @xmath13 , 318 galaxies with @xmath14 ) . the shapes of the two supercluster luminosity functions are broadly similar to the shape of the local luminosity function . however , there are important differences . both supercluster lfs have an excess of very bright galaxies . in addition , the characteristic magnitude of the corona borealis supercluster lf is roughly half a magnitude brighter than that of the local field galaxy lf , and there is a suggestion of an upturn in the lf for galaxies fainter than @xmath15 .

arxiv

over the past few years rapidity gaps , i.e. pseudorapidity regions without hadronic activity , have been observed in hadronic collisions at both the hera @xmath7 collider @xcite and in @xmath8 collisions at the fermilab tevatron @xcite . such rapidity gaps are widely attributed to the exchange of color singlet quanta between incident partons @xcite , the exchange of two gluons in a color singlet state being the simplest such model @xcite . at the tevatron , a fraction @xmath9 of all dijet events with jet transverse energies @xmath10 gev and jet separations of more than three units of pseudorapidity exhibit rapidity gaps between the jets . this observation is particularly striking since it demonstrates that color singlet exchange effects in qcd events are relevant at momentum transfers of order 1,000 gev@xmath6 , raising the hope that perturbative methods can be used for quantitative descriptions . a gap fraction of order one percent was in fact predicted by bjorken @xcite , in terms of a fraction @xmath11 of dijet events which are due to @xmath0-channel color - singlet exchange and a survival probability @xmath12 of rapidity gaps of order 10% @xcite , [ eq : ps ] f_gap = f_s p_s . here the survival probability estimates the fraction of hard dijet events without an underlying event , i.e. without soft interactions between the other partons in the scattering hadrons . such multiple interactions would fill the rapidity gap produced in the hard scattering process . for @xmath13 elastic scattering , bjorken estimated the color - singlet fraction @xmath14 in terms of the imaginary part of the two - gluon @xmath0-channel exchange amplitude , which is known to dominate the forward scattering amplitude for @xmath0-channel color - singlet exchange . in impact parameter space , at impact parameters small compared to @xmath15 , the result is f_s^impact & = & 29 & & 12 |433 - 2n_f|^2 = 0.15 . [ eq : fbjorken ] here 2/9 is the relative color factor of the two - gluon color - singlet to the one - gluon color - octet exchange cross section and @xmath16 is an infrared cutoff parameter which regularizes the two - gluon loop - integral . this model for the calulation of the color singlet fraction @xmath14 , with the two gluon - exchange amplitude replaced by its imaginary part , will be called the two - gluon exchange model in the following . in this model , the color singlet fraction grows with the color charge of the scattered partons . for @xmath2 and @xmath3 elastic scattering @xmath14 would be larger by factors @xmath17 and @xmath18 , respectively @xcite . this results in a substantial decrease of the observable gap fraction as the contribution from gluon induced dijet events is reduced , _ e.g. _ by increasing the average transverse momentum of the observed jets and thereby the feynman-@xmath19 values of the incident partons . such measurements have recently been reported by both the cdf @xcite and the d0 @xcite collaborations , and no such effect is observed . in fact , the d0 data are compatible with a slight increase of the gap fraction with increasing jet @xmath20 , casting doubt on the validity of the two - gluon exchange model @xcite . in this paper we reconsider the basic ideas behind the two - gluon exchange model . we demonstrate its limitations and show that , even when starting from this perturbative picture of rapidity gap formation , the determination of the color singlet exchange fraction @xmath14 is essentially nonperturbative . we start from a basic feature of the two - gluon exchange model : unitarity fixes the imaginary part of the @xmath0-channel two - gluon exchange amplitude in terms of the born amplitude and this imaginary part dominates @xmath0-channel color singlet exchange @xcite . rewriting this relationship in terms of phase shifts , the one- and two - gluon exchange amplitudes are found to be too large to be compatible with unitarity . phase shift unitarization leads to a more realistic description , in which the total differential cross section remains unchanged compared to the born result , but with @xmath0-channel color singlet exchange fractions which differ substantially from the expectations of the two - gluon exchange model . these features are demonstrated analytically for fixed values of the strong coupling constant , @xmath21 , in section [ sec2 ] . in section [ sec3 ] we then perform a numerical analysis for running @xmath21 , showing that the key properties of the fixed-@xmath21 results remain unchanged . the predicted color singlet fractions are found to very strongly depend on the regularization of gluon exchange at small momentum transfer , however , and thus can not be reliably calculated within perturbation theory . within our unitarized model the non - perturbative effects can be summarized in terms of two parameters , the survival probability of gaps , @xmath22 , and a universal coulomb phase shift , @xmath23 . implications for the formation of gaps at the tevatron are analyzed in section [ sec4 ] . in particular we calculate how the gap fraction between two hard jets varies with jet transverse energies and jet pseudorapidity separation and then compare predicted fractions with tevatron data @xcite . our conclusions are given in section [ sec5 ] . consider the elastic scattering of two arbitrary partons , @xmath24 and @xmath25 , p(i_1)+p(j_1)p(i_2)+p(j_2 ) , at momentum transfer @xmath26 . here @xmath27 denote the colors of the initial and final state partons . the cross section and the partial wave amplitudes are completely dominated by the forward region , @xmath28 , where the rutherford scattering amplitude , [ eq : mborn ] m = -8_s t^at^a = 8_s f_c = m_0f_c , provides an excellent approximation . note that helicity is conserved in forward scattering , hence spin need not be considered in the following . the only process dependence arises from the color factor @xmath29 . in order to study unitarity constraints , we need to diagonalize the amplitude in both momentum / coordinate space and in color space . the first step is most easily achieved by transforming to impact parameter space , [ eq : fourier ] t(*b * ) = ( * q * ) e^-i*q * . neglecting multi - parton production processes , _ i.e. _ inelastic channels , unitarity of the @xmath30-matrix implies the relation [ eq : unitarity1 ] t(*b * ) = @xmath31 . ( [ eq : unitarity1 ] ) represents a matrix relation in color space . more fully it can be written as t(*b*)_i_2 j_2 , i_1 j_1 = _ i , j t(*b*)_i_2 j_2 , i j t^(*b*)_i j , i_1 j_1 , [ eq : unitarity2 ] where the sum runs over the dimension of the color space , @xmath32 for @xmath33 and @xmath13 scattering and @xmath34 ( 64 ) for @xmath2 ( @xmath3 ) elastic scattering . since the color factors can be written as hermitian matrices , the right - hand side of eq . ( [ eq : unitarity2 ] ) represents a simple matrix product of the color matrices . this product is easily diagonalized by decomposing the color factors @xmath35 into a linear combination of projection operators onto the irreducible color representations which are accessible in the @xmath36-channel , [ eq : col.factor ] f_c = ( f_c)_i_2j_2,i_1j_1 = _ k f_k ( p_k)_i_2j_2,i_1j_1 = _ k f_k p_k . for the case of quark - antiquark elastic scattering , for example , with color decomposition @xmath37 , the color factor can be written in terms of gell - mann matrices as ( f_c)_i_2j_2,i_1j_1 = ( ^a2 ) _ i_2i_1 ( ^a2 ) _ j_1j_2 = 49_j_1i_1_i_2j_2- 13(^a2 ) _ j_1i_1 ( ^a2 ) _ i_2j_2 = 43p_1 - 16 p_8 . for all cases , @xmath13 , @xmath33 , @xmath2 and @xmath3 elastic scattering , the decomposition into @xmath36-channel projectors is summarized in table [ tab : colorop ] . this color decomposition , combined with the transformation to impact parameter space , diagonalizes the unitarity relation for elastic scattering amplitudes . .representations and color operators for qcd elastic scattering . the indices of the projection operators @xmath38 in the last column represent the dimensionalities of the irreducible color representations in the @xmath36-channel . results for the @xmath39 decomposition are taken from ref . @xcite . [ cols="^,<,<,^",options="header " , ] both the d0 @xcite and the cdf @xcite collaborations at the tevatron have analyzed the fraction of dijet events with rapidity gaps , as a function of both the transverse energy , @xmath20 , and the pseudorapidity separation , @xmath40 , of the two jets . as these phase space variables change , the composition of dijet events varies , from mostly gluon initiated processes at small @xmath20 and @xmath41 ( and , hence , small feynman-@xmath19 , @xmath42 ) to @xmath13 scattering at large values . a dependence of the gap fraction on the color structure of the scattering partons would thus be reflected in a variation with @xmath20 and @xmath41 . bjorken s two gluon exchange model , which is equivalent to the small @xmath43 region in our analysis , predicts a larger fraction of color singlet exchange events for gluon initiated processes @xcite ( see fig . [ fig : fs.vs.psi ] for @xmath44 ) . the gap fraction should thus decrease with increasing @xmath20 or @xmath41 . the opposite behavior is expected in statistical models of color rearrangement @xcite . here the eight color degrees of freedom for gluons , as compared to three for quarks , make it less likely for gluon initiated processes that @xmath0-channel color singlet exchange is achieved by random color rearrangement . this would lead to a smaller gap fraction at small @xmath42 and therefore small @xmath20 or @xmath41 . in the unitarized rpwe framework , the dependence on the regularization parameters is sufficiently strong to encompass both scenarios . this is demonstrated in figs . [ fig : fgap.exp.d0 ] and [ fig : fgap.exp.cdf ] , where the results of the running coupling analysis for three choices of the regularization parameters are compared with tevatron data , taken at @xmath45 gev . the data correspond to dijet events with two opposite hemisphere jets of @xmath46 gev , @xmath47 ( cdf ) or @xmath48 gev , @xmath49 ( d0 ) . d0 data are taken from ref . @xcite and show the fraction of dijet events with rapidity gaps . cdf @xcite shows the ratio of gap fractions in individual @xmath41 and @xmath20 bins to the overall gap fraction in the acceptance region . for comparing our calculation with the data we fix the survival probability @xmath22 in eq . ( [ eq : ps ] ) to reproduce the overall gap fraction in the acceptance region , which was measured as @xmath50 for the d0 sample and @xmath51 for the cdf sample . required survival probabilities strongly depend on regularization parameters and vary between 1.9% and 5.5% , which is on the low side of previous estimates @xcite . for a given choice of regularization parameters , predictions for the @xmath20 or @xmath41 dependence of the gap fraction are quite similar for the cdf and d0 cuts . to the extent that the two data sets are consistent within errors , it is not yet possible to discriminate between different choices of regularization parameters , i.e. to obtain sensitivity to the non - perturbative dynamics . d0 data somewhat favor color singlet fractions which grow with @xmath42 and which are more in line with expectations from color evaporation models @xcite . note that our unitarized gluon exchange model , with @xmath52 gev and @xmath53 gev is able to describe this trend , even though it is an extension of the two - gluon exchange model . cdf data slightly prefer a gap fraction which decreases with increasing @xmath41 and , hence , with larger @xmath42 . the unitarized gluon exchange model , with @xmath54 gev and @xmath55 gev describes such a situation . comparison with fig . [ fig : fs.reg.dep ] shows that this set of parameters predicts a much smaller gap fraction for @xmath13 scattering than for gluon initiated processes , which is qualitatively similar to the two - gluon exchange model @xcite . indeed , the shape of the gap fraction for the two gluon exchange model is very similar to the long - dashed curves in fig . [ fig : fgap.exp.cdf ] . clearly , the data are not yet precise enough to unambiguously distinguish between these different scenarios . on the theoretical side , the variation of the rpwe predictions with model parameters highlights the limitations of a perturbative approach to the color singlet exchange probability . taking the phase @xmath23 and the survival probability @xmath22 as free parameters , the unitarized two - gluon exchange model is clearly capable of fitting the present tevatron data , however . the formation of rapidity gaps in hadronic scattering events is a common occurrence , and its ubiquity asks for a theoretical explanation within qcd . the formation of gaps between two hard jets at the tevatron is particularly intriguing and is commonly being explained in terms of color singlet exchange in the @xmath0-channel , be it via an effective color singlet object like the `` pomeron '' or via a statistical color rearrangement , in terms of multiple soft gluon exchange . `` pomeron '' exchange models build on the observation that color singlet exchange in the @xmath0-channel can be achieved in qcd via the exchange of two gluons , with compensating colors @xcite . when trying to build a quantitative model for the formation of rapidity gaps @xcite , one encounters infrared divergences in the color singlet hard scattering amplitude , which in a full treatment would be regularized by the finite size and the color singlet nature of physical hadrons @xcite . in turn , this indicates that non - perturbative information may be indispensable for a quantitative understanding of the hard color singlet exchange process . we have analyzed this question within a particular model , based on the unitarization of single gluon exchange in the @xmath0-channel . the low - nussinov model @xcite corresponds to a truncation of the unitarization at order @xmath56 . we find that , for any reasonable range of regularization parameters , the two - gluon exchange approximation violates partial wave unitarity , and thus a fully unitarized amplitude is needed for phenomenological applications . the unitarization of hard elastic quark and gluon scattering is not unique , of course , but any acceptable method must preserve the successful description of hard dijet events by perturbative qcd . the phase shift approach used here fulfills this requirement : the unitarization does not change the born - level predictions for the color averaged differential cross sections . as a corollary , the color - inclusive dijet cross section is independent of the regularization parameters which need to be introduced for the full phase shift analysis . the situation is entirely different when considering the @xmath0-channel color singlet exchange component which is introduced by the exchange of two or more gluons or by unitarization . the @xmath0-channel color singlet exchange fraction , @xmath14 , is strongly affected by the full unitarization and deviates from the expectations of the two - gluon exchange approximation , changing even the qualitative predictions of the low - nussinov model . these strong unitarization effects are reflected by a strong dependence on the precise regularization procedure . this cutoff dependence , which parameterizes non - perturbative effects , does not allow to make quantitative predictions for the color singlet exchange fractions in particular partonic subprocesses . these limitations , which have been demonstrated here for the two - gluon exchange approximation to the pomeron , may be generic to pomeron exchange models , and should be analyzed more generally . in spite of these limitations , we find some intriguing features of the unitarized gluon exchange amplitudes . for all partonic subprocesses and for all regularization parameters , the color singlet exchange fractions can be described in terms of a single universal phase , @xmath57 , which absorbs all non - perturbative effects . this suggests a unified phenomenological description of the rapidity gap data , via the gap survival probability @xmath22 and the phase @xmath57 . such an analysis goes beyond the transverse momentum and pseudorapidity dependence of rapidity gap fractions which have just become available , and should best be performed directly by the experimental collaborations . we would like to thank f. halzen and j. pumplin for useful discussions on the physics of rapidity gaps . this research was supported in part by the u.s . department of energy under grant no . de - fg02 - 95er40896 , and in part by the university of wisconsin research committee with funds granted by the wisconsin alumni research foundation . e. berger , j. collins , g. sterman , and d.soper , nucl . * b286 * , 704 ( 1987 ) ; n. nikolaev and b. zakharov , z. phys . * c53 * , 331 ( 1992 ) ; v. del duca , scientifica acta * 10 * , 91 ( 1995 ) [ hep - ph/9503226 ] ; e. levin , proceedings of the _ gleb wataghin school on high energy phenomenology _ , campinas , brazil , july 1116 , 1994 , [ hep - ph/9503399 ] ; and references therein . d. zeppenfeld , in _ particles and fields _ , proceedings of the _ viiith j. a. swieca summer school _ , rio de janeiro , brazil , february 618 , 1995 , ed . by j. barcelos - neto , s. f. novaes , and v. o. rivelles ( world scientific , singapore , 1996 ) , p. 78 [ hep - ph/9603315 ] .

rapidity gaps between two hard jets at the tevatron have been interpreted as being due to the exchange of two gluons which are in an overall color - singlet state . we show that this simple picture involves unitarity violating amplitudes . unitarizing the gluon exchange amplitude leads to qualitatively different predictions for the fraction of @xmath0-channel color singlet exchange events in forward @xmath1 , @xmath2 or @xmath3 scattering , which better fit tevatron data . = cmssbx10 scaled 2 to + r. oeckl@xmath4 and d. zeppenfeld@xmath5 + .5 cm @xmath5_department of physics , university of wisconsin , madison , wi 53706 , usa _ + @xmath6_damtp , university of cambridge , cambridge cb3 9ew , uk _

arxiv

numerous techniques for forecasting electric energy consumption have been proposed in the last few decades . for operators , energy consumption ( load ) forecast is useful in effectively managing power systems . consumers can also benefit from the forecasted information in order to yield maximum satisfaction . in addition to these economic reasons , load forecasting has also been used for system security purposes . when deployed to handle system security problems , it provides expedient information for detecting vulnerabilities in advance . forecasting energy consumed within a particular geographical area greatly depends on several factors , such as , historical load , mean atmospheric temperature , mean relative humidity , population , gdp per capita . over the years , there has been rapid growth annually of about 10% from year 1999 to 2005 for energy demand in the gaza strip . with about 75% of energy demands from service and household sectors , these demands are barely met @xcite . in order to meet these demands and efficiently utilize the limited energy , it is imperative to observe historic trends and make futuristic plans based on past data . in the past , computationally easier approaches like regression and interpolation , have been used , however , this methods may not give sufficiently accurate results . as advances in technology and sophisticated tools are made , complex algorithmic approaches are introduced and more accuracy at the expense of heavy computational burden can be observed . several algorithms have been proposed by several researchers to tackle electric energy consumption forecasting problem . previous works can be grouped into three@xcite : _ * time series approach : * _ : : in this approach , the trend for electric energy consumption is handled as a time series signal . future consumption is usually predicted based on various time series analysis techniques . however , time series approach is characterized with prediction inaccuracies of prediction and numerical instability . this inaccurate results is due to the fact the approach does not utilize weather information . studies have shown that there is a strong correlation between the behavior of energy consumed and weather variables . zhou r. _ et al_. @xcite proposed a data driven modeling method using time series analysis to predict energy consumed within a building . the model in @xcite was applied on two commercial building and is limited to energy prediction within a building . basu k. _ et al_. @xcite also used the time series approach to predict appliance usage in a building for just an hour . + simmhan y. _ et al_. @xcite used an incremental time series clustering approach to predict energy consumption . this method in @xcite was able to minimize the prediction error , however , very large number of data points were required . autoregressive integrated moving average ( arima ) is a vastly used time series approach . arima model was used by chen j. _ et al_. @xcite to predict energy consumption in jiangsu province in china based on data collected from year 1985 to 2007 . the model @xcite was able to accurately predict the energy consumption , however it was limited to that environment . the previous works on time series usually use computationally complex matrix - oriented adaptive algorithms which , in most scenarios , may become unstable . _ * functional based approach : * _ : : here , a functional relationship between a load dependent variable ( usually weather ) and the system load is modelled . future load is then predicted by inserting the predicted weather information into the pre - defined functional relationship . most regression methods use functional relationships between weather variables and up - to - date load demands . linear representations are used as forecasting functions in conventional regression methods and this method finds an appropriate functional relationship between selected weather variables and load demand . liu d. _ et al_. @xcite proposed a support vector regression with radial basis function to predict energy consumption in a building . the approach in @xcite was only able to forecast the energy consumed due to lighting for some few hours . + in @xcite , a grey model , multiple regression model and a hybrid of both were used to forecast energy consumption in zhejiang province of china . yi w. _ et al_. @xcite proposed an ls - svm regression model to also forecast energy consumption . however , these models were limited to a specific geographic area . _ * soft computing based approach : * _ : : this is a more intelligent approach that is extensively being used for demand side management . it includes techniques such as fuzzy logic , genetic algorithm and artificial neural networks ( ann ) ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the ann approach is based on examining the relationship that exist between input and output variables . ann approach was used in @xcite to forecast regional load in taiwan . empirical data was used to effectively develop an ann model which was able to predict the regional peak load . catalo j. p. s. _ et al_. @xcite used the ann approach to forecast short - term electricity prices . levenberg - marquardt s algorithm was used to train data and the resulting model @xcite was able to accurately forecast electricity prices . however , it was only able to predict electricity prices for about 168 hours . + pinto t. _ et al_. @xcite also worked on developing an ann model to forecast electricity market prices with a special feature of dynamism . this model @xcite performs well when a small set of data is trained , however , it is likely to perform poorly with large number of data due to the computational complexities involved . load data from year 2006 to 2009 were gathered and used to develop an ann model for short - term load forecast in @xcite . in @xcite , ann hybrid with invasive weed optimization ( iwo ) was employed to forecast the electricity prices in the australian market . the hybrid model @xcite showed good performance , however , the focus was on predicting electricity prices in australia . most of the ann models developed in existing work considered some specific geographic area@xcite , some models were able to forecast energy consumption for buildings ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and only for few hours@xcite . this study used available historical data from year 1994 to 2013 ( but trained data from year 1994 to 2011 ) in determining a suitable model . the resulting model from training will be used to predict electric energy consumption for future years ] , while the error criteria such as mean squared error ( mse ) , root mean squared error ( rmse ) , mean absolute error ( mae ) and mean absolute percentage error ( mape ) are used as measures to justify the appropriate model @xcite . the model was used to predict the behavior for year 2012 and 2013 . the remainder of this paper is organized as follows : section [ sec:2 ] gives a brief description of the ann concept . section [ sec:3 ] presents the ann approach used to analyse our data . section [ sec:4 ] evaluates the performance of the ann model . section [ sec:5 ] draws conclusions . ann is a system based on the working principles of biological neural networks , and is defined as a mimicry of biological neural systems . ann s are at the vanguard of computational systems designed to create or mimic intelligent behavior . unlike classical artificial intelligence ( ai ) systems that are aimed at directly emulating rational and logical reasoning , ann s are targeted at reproducing the causal processing mechanisms that give results to intelligence as a developing property of complex systems . ann systems have been designed for fields such as capacity planning , business intelligence , pattern recognition , robotics . . in computer science and engineering , ann techniques has gained a lot of grounds and is vastly deployed in areas such as forecasting , data analytics and even data mining . the science of raw data examination with the aim of deriving useful conclusions can simply be defined as data analytics . on the other hand , data mining describes the process of determining new patterns from large data sets , by applying a vast set of approaches from statistics , artificial intelligence , or database management . forecasting is useful in predicting future trends with reliance on past data . the focus on this paper will be on using the ann approach in forecasting energy consumption . practically , ann provides accurate approximation of both linear and non - linear functions . a mathematical abstraction of the internal structure of a neuron is shown in fig . [ fig:1 ] . despite the presence of noise or incomplete information , it is possible for neurons to learn the behaviour or trends and consequently , make useful conclusions . usually , a neural network is trained to perform a specific function by adjusting weight values between elements as seen in fig . [ fig:2](a ) . the neural network function is mainly determined by the connections between the elements @xcite . by observing the data , it is possible for the ann to make accurate predictions . unlike other forecasting approach , the ann technique has the ability to predict future trends with theoretically poor , but rich data set . in this section , we present the overall modeling process . the implementation of the ann model can be described using the flow chart in fig . [ fig:2](b ) . historical monthly data ( historical energy consumption , mean atmospheric temperature , mean relative humidity , population , gdp per capita ) has been gathered from years 1994 to 2013 from gaza region as shown in fig . [ fig:3 ] . based on the gathered data , this study will develop a forecast model that predicts energy consumption for year 2012 and 2013 . using the ann technique , training and learning procedures are fundamental in forecasting future events . the training of feed - forward networks is usually carried out in a supervised manner @xcite . with a set of data to be trained ( usually extracted from the historical data ) , it is possible to derive an efficient forecast model . the proper selection of inputs for ann training plays a vital role to the success of the training process . on the other hand , the learning process involves providing both input and output data , the network processes the input and compares the resulting output with desired result . the system then adjusts the weights which acts as a control for error minimization . in order to minimize error , the process is repeated until a satisfactory criterion for convergence is attained . the knowledge acquired by the ann via the learning process is tested by applying it to a new data set that has not been used before , called the testing set . it should now be possible that the network is able to make generalizations and provide accurate result for new data . due to insufficient information , some networks do not converge . it is also noteworthy that over - training the ann can seriously deteriorate forecasts . also , if the ann is fed with redundant or inaccurate information , it may destabilize the system . training and learning process should be thorough in order to achieve good results . to accurately forecast , it is imperative to consider all possible factors that influence electricity energy consumption , which is not feasible in reality . electric energy consumption is influenced by a number of factors , which includes : historical energy consumption , mean atmospheric temperature , mean relative humidity , population , gdp per capita , ppp , etc . in this paper , different criteria were used to evaluate the accuracy of the ann approach in forecasting electric energy consumption in gaza . they include : mean squared error ( mse ) , root mean squared error ( rmse ) , mean absolute error ( mae ) and mean absolute percentage error ( mape ) . a popular and important criterion used for performance analysis is the mse . it is used to relay concepts of bias , precision and accuracy in statistical estimation . here , the difference between the estimated and the actual value is used to get the error , the average of the square of the error gives an expression for mse . the mse criterion is expressed in equation ( [ eqn:1 ] ) . @xmath0 where @xmath1 is the actual data and @xmath2 is the forecasted data . the rmse is a quadratic scoring rule which measures the average magnitude of the error . the rmse criterion is expressed in equation ( [ eqn:2 ] ) . rmse usually provides a relatively high weight to large errors due to the fact that averaging is carried out after errors are squared . this makes this criterion an important tool when large errors are specifically undesired . @xmath3 the mae measures the average error function for forecasts and actual data with polarity elimination . equation ( [ eqn:3 ] ) gives the expression for the mae criterion used . the mae is a linear score which implies that all the individual differences are weighted equally in the average . @xmath4 mape , on the other hand , measures the size of the error in percentage ( % ) terms . it is calculated as the average of the unsigned percentage error . equation ( [ eqn:4 ] ) gives the expression for the mape criterion used . @xmath5 validation techniques are employed to tackle fundamental problems in pattern recognition ( model selection and performance estimation ) . in this study , 2-fold and k - fold cross validation techniques will be employed and the validation set will only be used as part of training and not part of the test set . the test set will be used to evaluate how well the learning algorithm works as a whole . the forecast model was simulated to obtain results of the energy consumed for year 2012 and 2013 in gaza . table 1 compares the actual and the forecasted energy consumption for year 2012 . 2-fold and k - fold cross validation techniques were used and the performance of the forecast model based on different error criteria is shown in table 2 . similarly , table 3 and 4 shows the results for year 2013 . the results obtained have good accuracy and shows that the proposed ann model can be used to predict future trends of electric energy consumption in gaza . .2-fold and k - fold cross validation for year 2012 [ cols="^,^,^,^",options="header " , ] [ tab:4 ] in this paper , an ann model to forecast electric energy consumption in the gaza strip was presented . to the best of our knowledge , this is the first of it s kind in existing literature . based on the performance evaluation , the error criteria were within tolerable bounds . empirical results presented in this paper indicates the relevance of the proposed ann approach in forecasting electric energy consumption . future works will consider other forecasting techniques . park d. c. , el - sharkawi m. a. , marks ii r. j. , atlas l. e. and damborg m. j.,(1991 ) electric load forecasting using an artificial neural network , _ ieee transactions on power engineering _ , vol.6 , pp . 442 - 449 zhou r. , pan y. , huand z. and wang q. , ( 2013 ) building energy use prediction using time series analysis , _ ieee 6th international conference on service - oriented computing and applications _ , pp . 309 - 313 . basu k. , debusschere v. and bacha s. , ( 2012 ) appliance usage prediction using a time series based classification approach , _ ieee 38th annual conference on ieee industrial electronics society _ , pp . 1217 - 1222 . catalo j. p. s. , mariano s. j. p. s. , mendes v. m. f. and ferreira l. a. f. m. , ( 2007 ) an artificial neural network approach for short - term electricity prices forecasting , _ ieee intelligent systems applications to power systems _ , pp . 1 - 7 . pinto t. , sousa t. m. and vale z. , ( 2012 ) dynamic artificial neural network for electricity market prices forecast , _ ieee 16th international conference on intelligent engineering systems _ , pp . 311 - 316 . khamis m. f. i. , baharudin z. , hamid n. h. , abdullah m. f. and solahuddin s. , ( 2011 ) electricity forecasting for small scale power system using artificial neural network , _ ieee 5th international power engineering and optimization conference _ , pp . 54 - 59 . safari m. , dahlan y. n. , razali n. s. and rahman t. k. , ( 2013 ) electricity prices forecasting using ann hybrid with invasive weed optimization ( iwo ) , _ ieee 3rd international conference on system engineering and technology _ , pp . 275 - 280 .

due to imprecision and uncertainties in predicting real world problems , artificial neural network ( ann ) techniques have become increasingly useful for modeling and optimization . this paper presents an artificial neural network approach for forecasting electric energy consumption . for effective planning and operation of power systems , optimal forecasting tools are needed for energy operators to maximize profit and also to provide maximum satisfaction to energy consumers . monthly data for electric energy consumed in the gaza strip was collected from year 1994 to 2013 . data was trained and the proposed model was validated using 2-fold and k - fold cross validation techniques . the model has been tested with actual energy consumption data and yields satisfactory performance . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

arxiv

low mass x - ray binaries ( lmxbs ) are compact binaries where the primary is a compact object and the secondary a low mass star ( @xmath81@xmath7 ) . the secondary is transferring mass via roche - lobe overflow , forming an accretion disk around the compact object that gives rise to the observed x - rays . by far , most of the persistently bright lmxbs are neutron star systems that can be divided into two groups , the z - sources and atoll sources ( hasinger & van der klis 1989 ) . z - sources are usually the brightest lmxbs in x - rays ( they are thought to have mass accretion rates that reach the eddington limit ) and trace a z - like shape in their x - ray colour - colour diagrams . atoll sources on the other hand have lower accretion rates ( @xmath91 - 2 orders of magnitude lower ) and their colour - colour diagrams usually consists of fragmented island - like regions . apart from the difference in accretion rates , the main physical difference between z - sources and atoll sources are thought to be the strength of the neutron star magnetic field and their evolutionary history ( hasinger & van der klis 1989 ) . lmcx-2 is a persistent lmxb that shows the properties of a z - source ( smale et al . 2003 ) , and it is therefore thought to be a neutron star system that has an accretion rate around the eddington limit . it is one of the most x - ray luminous lmxbs known ( @xmath10@xmath910@xmath11 erg s@xmath3 ) , but due to its extra - galactic nature ( it is located in the large magellanic cloud at a distance of @xmath948 kpc ) , its x - ray flux is rather low . its optical counterpart was identified by pakull ( 1978 ) as a @[email protected] blue star . despite being x - ray luminous and having a known optical counterpart , thus far little is known about the system parameters of lmcx-2 . even the estimates for the orbital period range from 6.4 hrs ( motch et al . 1985 ) or 8.2 hrs ( callanan et al . 1990 , smale & kuulkers 2000 ) up to 12 days by crampton et al . ( 1990 ) . to make things even more complicated , no periodic variability was detected in 6 years of macho data ( alcock et al . 2000 ) . in recent years steeghs & casares ( 2002 ) developed a new technique to detect a signature of the donor star in persistent lmxbs . using phase - resolved spectroscopy they detected narrow emission lines in scox-1 , especially in the bowen blend ( a blend of niii and ciii lines between 4630 - 4650 ) , that were interpreted as coming from the irradiated side of the donor star . this discovery in scox-1 was followed by a survey of other lmxbs that are optically bright enough to also resolve these narrow components . thus far these narrow emission lines have been detected in x1822@xmath14371 ( casares et al . 2003 ) , gx339@xmath144 ( hynes et al . 2003 ) , v801ara and v926sco ( casares et al . 2006 ) , grmus ( barnes et al . 2007 ) , aqlx-1 and gx9@xmath159 ( cornelisse et al 2007a , b ) , leading to constraints on their system parameters . in this paper we apply the technique of bowen fluorescence to lmcx-2 . we will show that it is possible to detect a periodic signal in our spectroscopic dataset that we identify as the orbital period . furthermore , similar to the other x - ray binaries thus far , the bowen region shows the presence of narrow emission lines that we identify as coming from the irradiated side of the companion , giving the first ever constraints on the system parameters of lmcx-2 . on november 21 and 22 2004 we obtained a total of 77 spectra of lmcx-2 with an integration time of 600s each , using the fors2 spectrograph attached to the vlt unit 4 ( yepun telescope ) at paranal observatory ( eso ) . each spectrum was taken with the 1400v volume - phased holographic grism using a slit width of 0.7@xmath16 , giving a wavelength coverage of @xmath1@xmath14514 - 5815 and a resolution of 70 km s@xmath3 ( fwhm ) . the seeing during the first night was between 0.4 and 0.7 arcsec , while on the second night it varied between 0.5 and 2.7 arcs . the slit was orientated at a position angle of 7@xmath17 to include a comparison star in order to correct for slit losses . during daytime he , ne , hg and cd arc lamp exposures were taken for the wavelength calibration scale . we de - biased and flat - fielded all the images and used optimal extraction techniques to maximise the signal - to - noise ratio of the extracted spectra ( horne 1986 ) . we determined the pixel - to - wavelength scale using a 4th order polynomial fit to 20 reference lines giving a dispersion of 0.64 pixel@xmath3 and rms scatter @xmath180.05 . we also corrected for any velocity drifts due to instrumental flexure by cross - correlating the sky spectra . finally , we divided all spectra of lmcx-2 by a corresponding low order spline - fit of the comparison star to get the final fluxed spectra . since we did not observe a spectro - photometric standard star , we were not able to correct for instrumental response , and all spectra are therefore in relative fluxes . we created average spectra for each individual night . since we do not have a flux standard to derive an absolute flux for lmcx-2 , we decided to normalise the continuum flux to one by dividing each average spectrum by a low order spline fit . in fig.[spectrum ] we show the results . both spectra are dominated by the very narrow high excitation heii @xmath14686 emission line , while also bowen emission is present in both spectra . however , compared to other x - ray binaries , such as scox-1 , x1822@xmath14371 , v801ara and v926sco the bowen emission is much weaker compared to heii in lmcx-2 ( steeghs & casares 2002 , casares et al . 2003 , casares et al . this might be due to the much lower metal abundances in the large magellanic cloud ( motch & pakull 1979 ) . the most striking difference between the spectra is the dramatic change of h@xmath19 ( see fig.[spectrum ] ) . during the first night ( 21 november ) h@xmath19 is dominated by a weak emission feature superposed on a broad absorption feature , while in the second night ( 22 november ) the emission feature has become almost as strong as heii @xmath14686 . furthermore , also the hei @xmath14922@xmath205016 lines have become more prominent during the second night ( although they might be present during the first night ) . in order to quantify the change in the most prominent emission lines we estimated the equivalent widths and their line fluxes ( in arbitrary units ) for the two nights , and show them in table[width ] . for h@xmath19 we decided to also include the absorption component , giving negative values for the first night . table[width ] shows that there is no change in equivalent width of heii and the bowen region , and the line fluxes have dropped by @xmath940% during the second night . on the other hand the h@xmath19 and hei lines have all increased significantly in both equivalent width and line flux . unfortunately , due to the faintness of these lines ( or the presence of an absorption feature ) , it is not possible to say if they have all changed by the same amount , but it is likely that the same process is responsible for this change in line intensity . finally , in order to see if these changes could be related to a change in brightness we have also created a lightcurve of the continuum flux for lmcx-2 . we have normalised the flux of the first night around unity , and show the result in fig.[light ] . we note that during the second night the continuum flux was @xmath940% lower compared to the first night , a similar fraction as was observed for the line fluxes of the heii lines and bowen region , keeping their equivalent widths the same and suggesting a common origin . we determined phase - resolved radial velocities by cross - correlating each spectrum with a gaussian of width 150 km s@xmath3 centered on the core of the heii @xmath14686 line . since the conditions during the second night were much worse , together with the fact that the source was @xmath940% fainter , we have binned these spectra together in groups of three , and then determined the radial velocity . in fig.[radial ] we show the results . the first thing to note in fig.[radial ] is that during both nights the radial velocity shows a sine - like variation , which we interpret as orbital motion of a region that is co - rotating with the binary , and perhaps is connected to the dynamical properties of the primary . however , during the second night it appears that either the semi - amplitude of the radial velocity or the off - set compared to the rest wavelength has increased . this last behaviour was noted by crampton et al . ( 1990 ) , who observed a long - term variation in the radial velocity of heii . we searched the radial velocity curve for any periodic signal with a duration between 1 hr and 2 days using the lomb - scargle technique ( scargle 1982 ) . apart from the 24 hr alias due to the separation of our 2 observing nights , only two significant peaks with comparable strength were present in the power spectrum ; one peak is at a period of [email protected] days and another at [email protected] day . interestingly , the 0.32 day period is similar to the photometric period detected by callanan et al . ( 1990 ) , suggesting that this is the orbital period . although we can not exclude the possibility that the 0.45 day period is real ( but see sect.3.3 ) , we tentatively interpret the 0.32 day period as the orbital period and use it in the rest of this paper . fitting a sine curve with a period of 0.32 day to the radial velocity curve gives a phase zero at hjd 2,453,[email protected] , an off - set of 344@xmath011 km s@xmath3 , and the semi - amplitude of our sine fit is 41@xmath04 km s@xmath3 . note that we did not attempt to account for the seemingly variable systemic velocity from night to night , but only did a single sine fit to both nights . .lmc x-2 equivalent widths and spectral line fluxes . [ width ] [ cols="<,^,^,^,^ " , ] we used doppler tomography on the most prominent emission lines in order to probe the structure of the accretion disk ( marsh & horne 1988 ) . in order to create the maps , we used the orbital period of [email protected] hrs as determined by callanan et al . ( 1990 ) and a systemic velocity derived from the radial velocity curve in sect.3.2 . to comply with the standard definition of orbital phase 0 in doppler maps ( when the donor star is at inferior conjunction ) we used the phasing derived from the bowen map ( see below ) and applied a shift of @xmath90.5 orbital phase compared to the value derived in sect.3.2 . since heii @xmath14686 is by far the strongest emission line in lmcx-2 we decided to create doppler maps for each individual night , and we show the result in fig.[hedopp ] . both maps show a ring - like structure , but there are some minor differences . during the second night it appears as if the outer edge of the structure is at higher velocities than during the first night ( 218@xmath010 km s@xmath3 compared to 180@xmath010 km s@xmath3 ) . furthermore , during the first night there is a clear emission feature in the lower part of the map , that appears to have shifted toward the lower - right quadrant during the second night . since we only have 2 nights of observations , it is not clear if these changes are real , but it suggests that there is some change in accretion disk structure from night to night . we also created a doppler map of the bowen region by simultaneous fitting all the major niii ( @xmath14634/4640 ) and ciii ( @xmath14647/4650 ) lines using the relative strengths as given by mcclintock et al . the map is dominated by a bright emission feature , that we used to rotate the map ( by @xmath90.5 orbital phase ) until it was located in the top . fig.[bowdopp ] shows the resulting map . the bright emission feature is at a velocity of @xmath21=351@xmath028 km s@xmath3 , and another ( much fainter ) spot is also present in the map . if we interpret the bright spot as arising on the surface of the donor star ( see sect.4.2 ) , the fainter spot is in a region where we could expect an interaction between the accretion disk and the accretion flow . we do note that the velocity of the bright spot is much higher than the disk velocities in the heii doppler map ( fig.[hedopp ] ) , and we will discuss this in sect.4.2 . furthermore , since the radial velocity curve in sect.3.2 could not exclude a 0.45 days orbital period , we also created a bowen map for this period . although there are several spots present in this map , none is as sharp and significant as those in fig.[bowdopp ] . this is further support that the 0.32 day period is the correct orbital period . we have presented phase - resolved spectroscopy of lmcx-2 , one of the brightest x - ray sources in the large magellanic cloud , and also one of the most luminous lmxbs known . this enables us to derive the first constraints on its system parameters , and in particular give new insights on the previously reported orbital periods that ranged from @xmath96 - 8 hrs by motch et al . ( 1985 ) or callanan et al . ( 1990 ) up to @xmath912 days by crampton et al . ( 1990 ) . callanan et al . ( 1990 ) based their claim of a short orbital period on an extended @xmath92 week photometric campaign . a clear 8.15 hr modulation was present in their data that was interpreted as the orbital period , although there is indication of a long term ( @xmath910 day ) variability that was also observed by crampton et al . ( 1990 ) . on the other hand , crampton et al . did not detect the @xmath98 hr period in a @xmath91 week photometric campaign . however , there was a variation in the heii radial velocity curve over a period of 4 nights in their spectroscopic data that they interpreted as a @xmath912 day orbital period . interestingly , they noticed that the h@xmath19 emission lines also changed in strength over those 4 nights ( even going into absorption ) , with maximum line strength occurring at minimum ( continuum ) light . although crampton et al . ( 1990 ) did speculate that the @xmath912 day period is a precession or beat period , they discarded this due to the absence of shorter periods in their data set . in sect.3.2 we have shown that there is a @xmath98 hr period present in the radial velocity of heii @xmath14686 that is similar to the period detected by callanan et al . ( 1990 ) . since the semi - amplitude of this radial velocity curve is rather small ( @xmath941 km s@xmath3 ) , it might have been difficult to detect with the 4 m class telescope and an instrument with a resolution of @xmath134 used by crampton et al . we , therefore , tentatively identify this period with the orbital period . however , that does leave the question of the long term variation observed by both crampton et al . ( 1990 ) and callanan et al . similarly to crampton et al . ( 1990 ) , we also observe a large change in the emission line strength of h@xmath19 , with a stronger line occurring at lower continuum flux levels . unfortunately we only have two nights of observations , and are therefore not able to check if there is also a long term variation in our heii radial velocity curve . however , fig.[radial ] does suggest that during the second night ( when the continuum flux was lower ) either the amplitude or the mean velocity of the radial velocity has increased compared to the first night ( when the continuum flux was higher ) , similar to what crampton et al . observed . one explanation for the observed properties could be the presence of an inclined precessing , warped , accretion disk in lmcx-2 , as is also observed in herculesx-1 and ss433 ( katz 1973 , margon 1984 ) . we will discuss this suggestion in detail in a forthcoming paper by shih et al . ( in preparation ) , but here we will briefly highlight the spectroscopic evidence . during the first night we could be observing the accretion disk more edge on compared to the second night , and this could explain the much lower h@xmath19 and hei line intensities observed . this is further strengthened by the fact that the doppler maps suggest that heii is extending to higher radial velocities during the second night . if true , this could also explain the change in continuum flux observed in fig.[light ] . callanan et al . ( 1990 ) already suggested that a significant contribution to the optical light is either coming from the heated surface of the secondary or the outer disk bulge . if the fraction of the secondary or disk bulge that is in the shadow of the accretion disk changes as a function of precession period , this would lead to a change in the optical , with maximum light occurring when the accretion disk is most edge on . we can compare the characteristics of lmcx-2 to those of xtej1118@xmath15480 , an x - ray transient that is known for having a precessing ( although not necessarily inclined ) accretion disk ( uemura et al . 2000 ) . zurita et al . ( 2002 ) showed that the nightly average h@xmath22 lines changed in both velocity and width that is consistent with a periodic variation on the precession period ( torres et al . 2004 ) . estimating the average wavelength of heii @xmath14686 for the two nights in lmcx-2 gives [email protected] and [email protected] , respectively , suggesting a slight velocity shift . however , we must be careful with this slight shift , since we do not have a full orbital coverage each night and this could lead to a systematic off - set to the average wavelength . a better way to find out if this slight shift in velocity is real , is by examining the two heii doppler maps . they show a bright spot that appears to have moved over night , suggesting the presence of an irradiated region that shows movement on a much longer time - scale than the orbital period , such as the warped and irradiated part of the accretion disk . unfortunately , we only have two nights of data and can therefore not follow the long - term evolution of this bright spot to unambiguously claim that it moves periodically on a longer timescale . a spectroscopic campaign with a vlt - class telescope would be needed to follow the evolution of the emission lines in lmcx-2 over a full expected precession cycle ( of @xmath91 week ) and show that this bright spot in the heii doppler maps is long - lived and connected to a precessing disk . the radial velocity curve of the heii @xmath14686 emission line shows a periodic variability that we have interpreted as the orbital period . this could suggest that it also traces the primary and that we have an estimate for both orbital phase zero and @xmath23 . however , the doppler maps in fig.[hedopp ] show that the heii emission is dominated by the bright spot due to the warped accretion disk . since this spot does not have a similar phasing as the primary ( and even moves due to precession ) , we can not use the radial velocity curve to determine the orbital phasing of the primary . furthermore , if lmcx-2 harbours an inclined precessing accretion disk it is also not likely that semi - amplitude of the heii radial velocity curve traces @xmath23 . in this case the accretion disk , or at least the irradiated side that produced the bright spot in the heii doppler map , is tilted out of the orbital plane thereby changing its radial velocity . this is clear from fig.[radial ] , where it appears that either the average velocity or the semi - amplitude of the radial velocity curve has changed . only from long - term spectroscopic monitoring of lmcx-2 might it be possible to determine the @xmath23 velocity , but currently we can not constrain this value . this also means that currently we can not be certain that the average velocity that we determined corresponds to the systemic velocity @xmath24 . we have detected narrow emission lines in the bowen region that dominate the bowen doppler map . although these lines are not visible in the individual spectra , they become prominent when we create an average spectrum that is shifted into the rest - frame of these narrow lines . as fig.[average ] shows , all important bowen lines are present , and especially the niii @xmath14640 line is very strong , suggesting that this spot is real and not just a noise feature in the doppler map . these narrow lines have been detected in many other x - ray binaries thus far , such as scox-1 , x1822@xmath14371 , gx339@xmath144 , v801ara , v926sco , aqlx-1 , gx9@xmath159 and grmus ( steeghs & casares 2002 , casares et al . 2003 , hynes et al . 2003 , casares et al . 2006 , cornelisse et al . 2007a , b , barnes et al . since there are few compact regions that could produce such narrow emission lines , it was proposed that they arise on the irradiated surface of the donor star . especially in x1822@xmath14371 , but also in v801ara this connection could unambiguously be made , strengthening the claim in all other sources . furthermore , the width of these emission lines in lmcx-2 suggests that they come from a very compact region in the system , and apart from the donor star surface not many other regions in the binary could produce such narrow lines . therefore , following the other systems and given the narrowness of these emission lines we tentatively identify them as coming from the donor star of lmcx-2 , despite the fact that the absolute phasing of the system is unknown . in lmcx-2 there is another problem with identifying the compact spot in the bowen doppler map with the secondary , namely the fact that all emission in the heii doppler maps is at much lower velocities than the compact bowen spot . this would suggest that all emission in the heii map is at sub - keplerian velocities , and not related to the accretion disk . however , such behaviour is not unique to lmcx-2 . also in the heii @xmath14686 doppler maps of many other lmxbs ( such as scox-1 , x1822@xmath14371 , v801ara and v926sco amongst others ) is most , if not all , emission at sub - keplerian velocities ( steeghs & casares 2002 , casares et al . 2003 , 2006 ) . however , if the compact spot in the bowen map is produced on the donor star , lmcx-2 would be the most extreme system in showing too low disk velocities . we do note that one of the assumptions of the technique of doppler tomography is that all motion occurs in the orbital plane . if the accretion disk in lmcx-2 is inclined , as suggested above , these assumptions are no longer fulfilled . this would make the interpretation of the accretion disk structure using the doppler maps more difficult , and could be a reason why we observe these sub - keplerian velocities . for example , in ss433 the precession angle of the jets ( and most likely also the accretion disk ) is thought to be @xmath920@xmath25 ( margon 1984 ) , and if lmcx-2 has a similar precession angle ( and a relatively low inclination ) this could easily account for observed accretion disk velocities that are a factor 2 too low . again , long term spectroscopic monitoring would be needed to give more insight into the real accretion disk velocities . however , we do note that this should not impact our interpretation of the compact spot arising on the secondary star , and we remain confident that we have detected a signature of the donor star . assuming that we have detected the donor star in lmcx-2 we can use these data to constrain the system parameters . firstly , the narrow lines must arise on the irradiated surface , therefore the determined @xmath2 ( 351@xmath028 km s@xmath3 ) must be a lower limit to the center of mass velocity of the secondary ( @xmath4 ) . however , since @xmath2 must be smaller than @xmath4 this already gives a lower limit to the mass function of @xmath26=@xmath27@xmath28@xmath29/(1+@xmath30)@[email protected] @xmath7 ( at 95% confidence ) , where @xmath30 is the binary mass ratio @xmath32/@xmath27 and @xmath29 the inclination of the system . in order to further constrain the mass function the @xmath33-correction must be determined . unfortunately , this depends on @xmath30 , @xmath29 and the disk flaring angle @xmath22 , all of which are unknown for lmcx-2 ( muoz - darias et al . furthermore , the @xmath33-correction by muoz - darias et al . ( 2005 ) assumes an idealised accretion disk that is located in the orbital plane of the system , both of which are most likely not true for lmcx-2 . therefore , we must be very careful with any constraints we will derive in the rest of this section . we can set a strict lower - limit to @xmath4 by assuming that it is equal to @xmath2 , i.e that all emission is coming from the poles . furthermore , we can use the 4th order polynomials for the @xmath33-correction by muoz - darias et al . ( 2005 ) to determine @xmath4 as a function of @xmath30 in the case of @xmath22=0@xmath25 and @xmath29=40@xmath25 ( since lmcx-2 does not show dips or eclipses its inclination must be lower than @xmath970@xmath25 , and therefore the polynomials for the @xmath33-correction when @xmath29=40@xmath25 are a better approximation than in the case of @xmath29=90@xmath25 ) . we show these limits on @xmath4 in fig.[limits ] . note that these limits are still true even if the accretion disk is severely warped or out of the orbital plane . assuming that the width of the narrow emission lines is mainly due to rotational broadening we can derive a lower - limit to @xmath30 using @xmath34@xmath35@xmath29=0.462@xmath4@xmath36(1+@xmath30)@xmath37 ( wade & horne 1988 ) . from fig.[average ] we derive a _ fwhm _ of the narrow lines of [email protected] km s@xmath3 . since these emission lines are expected to arise from only part of the secondary , this value must be a strict lower - limit and the true _ fwhm _ must be higher . furthermore , our estimate still includes the effect of the intrinsic instrumental resolution of 70 km s@xmath3 . following casares et al . ( 2006 ) we accounted for this effect by broadening a strong line in our arc spectrum using a gray rotational profile without limb darkening ( since the fluorescence lines occur in optically thin conditions ) until we reached the observed _ fwhm _ ( gray 1992 ) . we found that a rotational broadening of @xmath38@xmath35@xmath29@xmath660@xmath018 km s@xmath3 reproduced our results . since this is a strict lower - limit on the true rotational broadening this will also give a lower limit on both @xmath30 and @xmath4 that is shown in fig.[limits ] . casares et al . ( 2006 ) derived an upper - limit to @xmath30 by assuming that the secondary is the largest possible zero - age main sequence star fitting in its roche - lobe . however , lmcx-2 shows the x - ray properties of a z - source ( smale et al . z - sources trace a z - like shape in their x - ray colour - colour diagrams and are thought to have more evolved secondaries ( see e.g. hasinger & van der klis 1989 ) . although the evolved nature of the secondary in z - sources is not unambiguously confirmed , it is still reasonable that the assumption of a zero - age main sequence donor is most likely not true for lmcx-2 . we therefore can not use this assumption to derive an upper - limit on the mass ratio , since the evolved donor star could be more massive than a main sequence star ( schenker & king 2002 ) . however , we can use an alternative way to set an upper - limit to @xmath30 , namely using the fact that lmcx-2 has a precessing accretion disk . one of the main criteria to produce a precessing accretion disk is to have an extreme mass ratio ( e.g. whitehurst 1988 ) . from an overview by patterson et al . ( 2005 ) of compact binaries that have a precessing accretion disk and for which the mass ratio is known , we can use the system with the largest measured @xmath30 thus far and assume that it must be smaller for lmcx-2 . this leads to a conservative upper - limit of @[email protected] , and our final limit for the system parameters shown in fig.[limits ] . all these limits constrain an area in fig.[limits ] that is still quite large , and at the moment it is still not possible to constrain any of the system parameters tightly enough in order to say anything about the component masses or the evolution of lmcx-2 . however , we can make two more assumptions , although more speculative , for lmcx-2 to further constrain its system parameters . given that lmcx-2 shows the x - ray properties of a z - source ( smale et al . 2003 ) , we can speculate that its compact object is a neutron star . in this case the maximum mass for the compact object would be @xmath93.2@xmath7 , and we could use this to set tighter upper - limits to @xmath4 than derived from the assumptions we have made thus far . in fig.[limits ] we have therefore also indicated the maximum limit where the compact object can still be a neutron star , i.e. where @xmath27@xmath35@xmath39=3.2@xmath7 , as a solid dark line . however since we can not completely rule out the possibility that the compact object in lmcx-2 is a low mass black hole , we have decided not to use this limit to constrain @xmath4 . finally , we can speculate that the semi - amplitude that we have derived from the heii @xmath14686 radial velocity curve is @xmath23 . in this case we set an upper - limit to @xmath30=@xmath23/@[email protected] . note that we have used this value for @xmath30 and @xmath4=@xmath2 to draw the roche lobe in fig.[bowdopp ] to illustrate that it encompasses the bright spot . however , this set of system parameters is just one possible solution within the allowed region in fig.[limits ] and we do not imply that it represents the true or even preferred system parameters of lmcx-2 . we note that this value of @xmath30=0.12 is similar to the mass ratios determined from other x - ray binaries that show a precessing accretion disk ( odonoghue & charles 1996 ) , suggesting that it might be close to the true mass ratio of lmcx-2 . also , if we assume that the @xmath912 day variation observed by crampton et al . ( 1990 ) is the precession period , and that the 8.2 hr photometric period is a superhump period , we can estimate the fractional period excess of the superhump over the orbital period @xmath40 . we find that @[email protected] and with the @xmath40(@xmath30)-@xmath30 relation by patterson et al . ( 2005 ) we estimate that @[email protected] . this is again close to the mass ratio derived above . although this would set tight constraints to @xmath30 , and makes a black hole nature for the compact object very unlikely , we have argued in sect.4.2 that we can not make this assumption and have therefore not plotted this in fig.[limits ] . finally , we do want to point out that , despite the large range of system parameters possible , lmcx-2 most likely harbours a neutron star that is more massive than the canonical 1.4@xmath7 . since lmcx-2 does not show eclipses or dips we can estimate an upper - limit on the inclination of 65@xmath25 ( see e.g. paczynski 1974 ) . combined with the lower limit on the mass function already gives @xmath41(@xmath27)/@xmath28@[email protected]@xmath7 . for the compact object to be 1.4@xmath7 , this only leaves @[email protected] ( and @[email protected] ) . furthermore , already for @xmath4=309 km s@xmath3 ( and @xmath29@xmath865@xmath25 ) the mass of the compact object will exceed 1.4m@xmath42 . this only leaves a very small corner in the bottom - left part of our allowed system parameters , and makes lmcx-2 another strong candidate to harbour a massive neutron star . we have detected for the first time a spectroscopic period in lmcx-2 that is close to previously published photometric period of 8.15 hrs by callanan et al . we interpret this as the orbital period , while the 12 day period detected by crampton et al . ( 1990 ) is most likely the super - orbital period due to an inclined precessing accretion disk . the signature of such a precessing accretion disk is also present in our data , mainly in the heii doppler maps , but also in the form of a nightly change in semi - amplitude and/or mean velocity of the radial velocity curve of heii @xmath14686 . in a forthcoming paper by shih et al . ( in prep . ) we will explore the consequences of such a precessing accretion disk in more detail . the main result of our spectroscopic data - set is the detection of narrow emission lines in the bowen region . following previous detections of such lines in other lmxbs , we tentatively identify these as arising from the surface of the secondary . this gives us for the first time the possibility to derive the mass function of lmcx-2 and constrain its system parameters . although they point toward a massive neutron star , the constraints on the system parameters are currently not very tight . however , this could change with a better determination of both the spectroscopic and precession period during a long term spectroscopic campaign to study the emission line kinematics . using the relation between the mass ratio @xmath30 and the fractional period excess of the spectroscopic and the photometric period @xmath40 could give strong constraints on @xmath30 and thereby the other system parameters . in particular this would be an excellent way to identify the nature of the donor star to find out if z - sources really have more evolved secondaries . this work is based on data collected at the european southern observatory paranal , chile ( obs.id . 074.d-0657(a ) ) . we acknowledge the use of the molly and doppler software packages developed by t.r . marsh . rc acknowledges financial support from a european union marie curie intra - european fellowship ( meift - ct-2005 - 024685 ) . jc acknowledges support from the spanish ministry of science and technology through project aya2002 - 03570 . ds acknowledges a smithsonian astrophysical observatory clay fellowship as well as support from nasa through its guest observer program . ds acknowledges a pparc / stfc advanced fellowship . 99 alcock , c. , allsman , r.a . , alves , d.r . , axelrod , t.s . , becker , a.c . , bennett , d.p . , charles , p.a . , cook , k.h . , drake , a.j . , freeman , k.c . , and 16 coauthors 2000 , mnras ; 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two nights of phase - resolved medium resolution vlt spectroscopy of the extra - galactic low mass x - ray binary lmcx-2 have revealed a [email protected] day spectroscopic period in the radial velocity curve of the heii @xmath14686 emission line that we interpret as the orbital period . however , similar to previous findings , this radial velocity curve shows a longer term variation that is most likely due to the presence of a precessing accretion disk in lmcx-2 . this is strengthened by heii @xmath14686 doppler maps that show a bright spot that is moving from night to night . furthermore , we detect narrow emission lines in the bowen region of lmcx-2,with a velocity of @xmath2=351@xmath028 km s@xmath3 , that we tentatively interpret as coming from the irradiated side of the donor star . since @xmath2 must be smaller than @xmath4 , this leads to the first upper - limit on the mass function of lmcx-2 of @[email protected]@xmath7 ( 95% confidence ) , and the first constraints on its system parameters . [ firstpage ] accretion , accretion disks stars : individual ( lmcx-2 ) x - rays : binaries .

arxiv

undoubtedly the axial anomaly represents a fundamental issue for understanding the basic aspects of quantum field theory . this issue has been analysed deeply over the years . + the anomaly problem has been treated by means of renormalization procedure , giving the interpretation of its origin in terms of ultraviolet divergences @xcite . a more formal analysis of the axial anomaly can be made by using the path integral formalism @xcite . + dolgov and zakharov @xcite have shown an alternative approach to the axial anomaly , by studying the @xmath2 triangular diagram through dispersion relations . from this approach follows the interpretation of the axial anomaly as an infrared phenomenon . it appears as due to a singularity present in the chiral limit in the absorbitive part of the triangular diagram . + the infrared aspect of the axial anomaly , rised in this paper , is complementary to the more familiar ultraviolet one , which emerges from the renormalization procedure . a particularly interesting feature of this approach is that it allows to shed light upon the physical meaning of the anomalous chiral symmetry breaking , which is connected to a non conservation of helicity . + the connection between the anomaly and the breaking of a given symmetry has received a lot of attention in the literature and this subject has been discussed and developed in several papers . gribov @xcite , in a seminal work , has described the source of the anomalies as a collective motion of particles with arbitrarily large momenta in the vacuum . related to this work , in ref . @xcite , mueller has discussed the manifestation of the axial anomaly as a flow of landau levels . in the papers of refs . @xcite the origin of the axial anomaly has been studied in two dimensions and , again , as a level crossing phenomenon . the infrared interpretation of the axial anomaly , according to the dolgov and zakharov approach , has been possibly advanced in @xcite . in a series of papers @xcite the leading terms in the chiral limit have been correctly evaluated . furthermore the dispersive analysis of the triangular @xmath2 diagram is fundamental in the formulation of the t hooft consistency condition @xcite . the axial anomaly plays an essential role also in the interpretation of spin dependent parton distribution ( see @xcite ) . + in this work we attempt to relate the dolgov and zakharov approach to the axial anomaly to some effects in the dynamics of physical reactions , as the radiative decays of @xmath0 and @xmath1 . we will try to show that the axial anomaly can be related to polarized radiative decays , as in the usual ultraviolet interpretation it is connected to the @xmath3 decay . we calculate the corresponding decay rates for the cases where the outgoing leptons are in a definite helicity state and we examine in some detail the structure of the mass singularities and their cancellation . we study how the kinoshita - lee - nauenberg ( kln ) theorem @xcite applies and we consider the analogies and the differences with respect to the corresponding unpolarized decay rates . + the paper is organized as follows . in the section [ anomaly ] we briefly reconsider the dispersive approach to the axial anomaly . in particular we concentrate on its physical origin . + in section [ helicity ] we extend the dolgov and zakharov approach to the study of the radiative pion decay . we calculate the differential decay rate for the process , where the outgoing lepton undergoes an helicity flip and we interpret its behaviour in the chiral limit , as a manifestation of the axial anomaly . + in section [ mass ] we study the behaviour of the inner bremsstrahlung contribution to the pion decay rate in the collinear and infrared limits . we consider separately the unpolarized decay rate and both the cases of right - handed and of left - handed outgoing lepton . we find that the mass singularities cancellation mechanism occurs in different ways , according to the polarization of the outgoing lepton . we discuss the various realizations of the kln theorem . + we also consider a more general process , i.e. the radiative @xmath1 decay in a lepton - antilepton pair , with a right - handed polarized lepton . we examine how the mass singularities cancel in this case and discuss the differences with respect to the pion decay . + finally , in the conclusions , we summarize our arguments . + a short discussion of the main results obtained has been already given in @xcite . the dolgov and zakharov approach @xcite to derive the axial anomaly is based on the dispersion relation method . in this framework , the triangular diagram with two vector and one axial vertices is seen as the lowest order contribution to the process : @xmath4 as described by the diagrams of fig . 1 . [ cols="^ " , ] [ fig2 ] in terms of dispersion relations , the functions @xmath5 , @xmath6 can be expressed as follows : @xmath7 we can use unsubtracted dispersion relations , because the integrals contained in @xmath5 , @xmath6 are convergent , since these functions are multiplied by three and two powers of momentum , respectively . + the imaginary part of the invariant scalar functions can be derived from the absorbitive part of the triangular diagram , calculated by cutting the diagram as shown in fig . [ fig2 ] and by using the cutkosky rules or the perturbative unitarity relation . we obtain @xcite : @xmath8 using eqs . ( [ def1 ] ) , ( [ def2 ] ) , ( [ dr ] ) and the above expressions , we derive the complete triangular diagram contribution : @xmath9 and the anomalous axial - vector ward - takahashi identity : @xmath10 thus , in the dispersive approach we find the same result obtained by using the renormalization procedure @xcite . the method above allows a more direct treatment , since we avoid evaluating divergent integrals and introducing regularization schemes . + the fact that the axial anomaly can be derived without using the renormalization procedure , suggests that this should not be considered as the only origin of the anomalous breaking of the chiral symmetry . moreover , by studying the axial anomaly with the renormalization procedure , that is by considering its ultraviolet interpretation , an important aspect of this phenomenon remains obscure and we are bound by a formal derivation only . as stressed in @xcite and @xcite , the dispersive approach shows that the anomaly is related to the chiral limit and therefore , it can be interpreted as an infrared effect . in this work we are close to this infrared interpretation , which , as we will see , can help to shed light on some aspects of the physics connected to the axial anomaly . + let us briefly discuss the physics involved in the amplitudes contributing to the absorbitive part of the triangular diagram . the two born diagrams , obtained from the cut of the triangular diagram ( see fig . [ fig1 ] ) , describe the following two processes : 1 . the production of a fermion - antifermion pair ( for example @xmath11 ) by an axial - vector source ; 2 . the subsequent annihilation of the pair into two real photons . in both these processes there occur helicity flips , thus the chirality is not conserved in the zero mass limit . + let us go to the center of mass frame of the two final photons . in the first process the axial - vector source produces an @xmath11 pair of total spin zero , since a spin 1 state can not annihilate into a two real photons state . thus @xmath12 and @xmath13 must have the same helicity and hence opposite chirality in the massless limit . in the process b ) the @xmath11 pair annihilates into two real photons by going through an intermediate virtual state . there are four possible virtual states @xcite : one is drawn in fig . 3 c [ fig3 ] ( @xmath14 and @xmath15 are the @xmath13 and @xmath12 linear momenta ) , a second one is obtained by reversing the virtual state helicity ; the remaining two are obtained replacing the outgoing virtual fermion with an incoming virtual antifermion . let us study the case shown in fig . 3 and assume that @xmath12 and @xmath13 are both right - handed . in the vertex b the chirality is conserved in the massless limit , while in the vertex a there is an helicity flip , thus the chiral symmetry is broken . as can be easily checked , for all the remaining virtual states we always have an allowed vertex and a forbidden one . + at the born level , these reactions are described by the classical qed lagrangian , which , in the @xmath16 limit , is invariant under chiral transformations . thus it seems at first sight that the absorbitive part of the triangular diagram , being proportional to the product of amplitudes relative to processes forbidden by chiral invariance , vanishes . on the contrary , one sees that taking the limit @xmath16 in ( [ g2 ] ) gives a finite result @xcite : @xmath17 therefore , by studying the absorbitive part of the triangular diagram , one establishes that a non - conservation of helicity occurs , becoming , in the massless limit , a non - conservation of chirality . we interpret this as related to the presence of the anomalous term in the divergence of the axial - vector current . thus the axial anomaly can be derived by studying the properties of the amplitude in the infrared region . + even if in this work we will analyse the cancellation of mass singularities in polarized processes , we will not discuss the physical implication of the zero fermion mass limit . + as stated in refs . @xcite , the result in eq . ( [ limit ] ) indicates that there occurs a cancellation of the suppression factor @xmath18 , due to the terms coming from the vertices with helicity flip . in ( [ g1 ] ) the logarithmic factor conspires to give a finite result . this logarithm is a collinear one ; we shall discuss about this kind of logarithms in section [ mass ] . its presence is a manifestation of the singularity occurring in the fermion propagator as @xmath16 , which exactly cancels the suppression factor @xmath18 in the numerator . + we observe that the behaviour of the absorbitive part of the triangular diagram given in eq . ( [ limit ] ) shows that the @xmath16 limit is not smooth ; if we evaluate the amplitudes with the massless theory , they identically vanish . if , on the contrary , we take the chiral limit after summing their product over the intermediate states , we obtain a result different from zero . in the process b ) contributing to the absorbitive part of the triangular diagram a charged polarized fermion , changes helicity by emitting a photon . thanks to the collinear singularity within the propagator relative to this fermion , the chiral suppression factor @xmath18 is cancelled and the absorbitive part of the triangular diagram does nt vanish in the chiral limit . + let us now extend these remarks to physical processes having the same features as the ones characterizing the absorbitive part of the triangular diagram . this means that we want to investigate about a manifestation of the axial anomaly in reactions where a fermion changes helicity by emitting a photon . we call these processes helicity changing processes . they have been considered by lee and nauenberg @xcite , who observed that states with opposite helicity do nt decouple in the massless limit , provided that this limit is taken after having summed the transition probability over the final phase space . in the process b ) the incoming @xmath12 and @xmath13 are in a definite helicity state , as discussed above . in the reactions we want to study , we calculate the probability that an outgoing fermion assumes an helicity opposite to the one required by the interaction before the emission of the photon . in other words , we evaluate the probability that in the fermion - photon vertex occurs an helicity flip . according to the dolgov and zakharov analysis , we interpret the presence of a term independent of the fermion mass in the corresponding cross sections as a manifestation of the axial anomaly . due to this term , the probability for a process with helicity flip does nt vanish in the chiral limit . + we first examine the non radiative pion decay @xmath19 where @xmath20 is an antilepton ( @xmath21 or @xmath22 ) and @xmath23 is the associated neutrino . at the born level the total decay rate is given by : @xmath24 where @xmath25 is the fermi coupling constant , @xmath26 is the pion decay constant , @xmath27 is the ckm matrix element , @xmath28 and @xmath29 are the lepton and pion masses , respectively . the decay rate ( [ born ] ) is proportional to @xmath30 , since , due to angular momentum conservation , the pion produces a left - handed lepton , while the structure of the weak coupling requires the @xmath31 to be right - handed for @xmath32 . + this situation is confirmed by the expression of the total decay rate for the process in which the antilepton is polarized : @xmath33 where @xmath34 and @xmath35 are the linear momentum and the spin vector of the lepton , respectively , giving @xmath36 , for right - handed and left - handed lepton , respectively . ( [ bornpol ] ) indicates that the lepton is mandatory left - handed , as requested by angular momentum conservation . + we now consider the radiative correction , due to the emission of a real photon , to the pion decay , @xmath37 when the outgoing antilepton is polarized , i.e. when it is in a definite helicity state . as we have seen in the non - radiative case , due to angular momentum conservation , the @xmath0 is coupled to a left - handed lepton . we calculate the probability that the lepton flips its helicity and becomes right - handed , by emitting a real photon . + the amplitude describing the radiative pion decay can be divided into two parts , the inner bremsstrahlung and the structure dependent amplitudes @xcite : @xmath38 the inner bremsstrahlung amplitude , where the photon is radiated from the external charged particles , can be calculated using the rules of qed , with a point like pion coupling ; the structure - dependent amplitude is governed by the strong interactions . + clearly , the relevant part for the problem we are considering is the inner bremsstrahlung contribution described by the diagrams of fig . c [ innerb ] the term associated to the @xmath39 diagram is the so called contact term and it is introduced to ensure gauge invariance ( see , for example , @xcite ) . + we consider only tree level diagrams , since , to reveal the effects of the axial anomaly , it is sufficient to take into account the contribution of the vertex with the helicity flip . for the moment , we may neglect the corrections due to the emission of virtual photons ; these will be discussed in section [ mass ] . finally , at this order in perturbation theory , we retain terms of all powers in the lepton mass . we will argue that the manifestation of the axial anomaly is strictly connected to these terms . + the inner bremsstrahlung amplitude is given by @xcite : @xmath40 v(p_l , s_{lr } ) , \label{e:4.5}\end{aligned}\ ] ] where @xmath41 and @xmath42 are the dirac spinors for the neutrino and the lepton , respectively , @xmath43 is the pion momentum , @xmath44 and @xmath45 are the momentum and the polarization vector of the lepton , @xmath46 and @xmath47 are the momentum and the polarization vector of the photon . + we see that @xmath48 is proportional to the lepton mass @xmath28 , thus the decay rate is proportional to @xmath30 . as we have said above , this factor is a consequence of the structure of the weak coupling . thus , if we remove it by normalizing the radiative decay rate with respect to the non radiative one , we can emphasize the mass dependence of the radiation emission process . + the differential inner bremsstrahlung contribution for right - handed lepton is : @xmath49 \nonumber \\ & & + \,\big[(a+2r)(1+r)^2 + ay(y-4r ) + 2ry(1+y)-y(1+y^2 + 5r^2)\big]\,\ln{\frac{y+a}{y - a } } \nonumber \\ & & + \,(1-y+r)^2(y-2r - a ) \ln{\frac{y+a-2}{y - a-2}}\bigg\}. \label{ibr}\end{aligned}\ ] ] the dimensionless variable @xmath50 is defined as @xmath51 and @xmath52 the physical region for @xmath50 is : @xmath53 we see that , in eq . ( [ ibr ] ) , there is a term independent of the lepton mass , the one in the first square brackets . owing to this term , the differential decay rate does nt vanish in the limit @xmath54 . we have : @xmath55 this indicates that there occurs an helicity flip and thus a chirality non conservation in the limit of zero lepton mass . according to the interpretation given above , this term can be interpreted as connected to the axial anomaly . it corresponds to the anomalous term present in the divergence of the axial current . + since the polarized radiative process with the right - handed @xmath31 is forbidden , in the limit @xmath54 , by the chiral invariance of the massless qed lagrangian , the appearance of a term different from zero , in this limit , indicates the action of a cancellation mechanism , analogous to the one acting in the absorbitive part of the triangular diagram . + to see how this mechanism acts , let us examine the decay rate differential with respect to the lepton energy and to the emission angle : @xmath56 \ , \frac{1}{(y - a\cos{\theta } ) } \nonumber \\ & & + ( 1-y+r)^2(y - a-2r)\ , \frac{1}{(y - a\cos{\theta } -2 ) } \bigg\}. \label{integrale}\end{aligned}\ ] ] the first term is proportional to the lepton propagator squared . thanks to this term , when we carry out the final integration over the emission angle , we obtain , besides logarithmic collinear divergences , also power collinear divergences . these power terms are essential for the convergent behaviour of the distribution . the cancellation of the chiral suppression would not take place were these terms absent . the term related to the axial anomaly ( [ massa0 ] ) ) originates from this cancellation . to obtain the differential decay rate , we have to evaluate integrals of the form : @xmath57 where the propagator has a power mass singularity that exactly cancels the factor @xmath30 , coming from the vertex lepton - photon . + the integration of the terms in equation ( [ integrale ] ) containing the lepton propagator gives rise to the collinear logarithms : @xmath58 in the differential decay rate ( [ ibr ] ) there are also terms proportional to the logarithm @xmath59 this one is another collinear logarithm ; it diverges in the limit @xmath60 and @xmath54 and corresponds to the possibility that the photon is emitted parallel to the pion . + finally , we point out that the chiral limit ( @xmath54 ) is not smooth . in fact we get different results depending upon whether we describe an helicity changing process using the massless theory or we take the @xmath54 limit , after carrying out the integration over the final phase space . the radiative pion decay is not a good process to see this , since , owing to the angular momentum conservation in the pion vertex , this process can not take place within a massless theory , given the structure of the @xmath61 coupling of the electroweak theory . to see that the chiral limit is not a smooth one , we consider the scattering of a polarized electron with a proton ( treated as a point like particle ) of initial and final momenta @xmath62 and @xmath63 respectively , accompanied by the emission of a real photon . we consider a left - handed incoming electron with momentum @xmath43 and spin @xmath64 : we calculate the probability that the electron makes an helicity flip , emitting a real photon with momentum @xmath46 and polarization @xmath47 and thus becoming right - handed electron with momentum @xmath65 and spin @xmath66 : @xmath67 we study this process with the massless qed . the left - handed and right - handed spinors are given respectively by : @xmath68 the corresponding transition amplitude identically vanishes : @xmath69 ( l^2 + i \epsilon ) } \nonumber \\ & & \bar{u}(p')\frac{1-\gamma_5}{2 } { \not\!}\epsilon({\not\!}p ' + { \not\!}k ) \gamma_{\rho } \frac{1-\gamma_5}{2}u(p ) \bar{u}(q')\gamma^{\rho}u(q ) = 0,\end{aligned}\ ] ] [ qed0 ] since it contains the product of different chirality projectors . + we now calculate the cross sections for processes with helicity flip using the massive qed and then we take the massless limit after having summed the transition probability over the final phase space . an example can be found in @xcite , where the cross section for the process @xmath70 is calculated . here @xmath71 is the target ( for example another fermion ) , @xmath72 is a bremsstrahlung photon , assumed almost collinear with respect to the direction of the incident electron and @xmath73 is a set of particles produced in the reaction . + the helicity flip cross section is given by : @xmath74 where @xmath75 , @xmath76 is the energy of the incoming electron and @xmath77 is the cross section for the born process @xmath78 we see that the expression ( [ hf ] ) does not vanish in the massless limit . + the result in eq . ( [ hf ] ) coincides with the one in eq . ( [ massa0 ] ) at leading order . indeed , in @xcite the subdominant terms are not accounted for . as it will become apparent in the following section , these terms are essential for the cancellation of mass singularities . @xcite , the authors give a different interpretation of the helicity changing processes and hence come to a different conclusion about the smoothness of the zero mass limit . + as it is well know , there are two types of divergences occurring in a theory when the mass of a particle goes to zero , which will be comprehensively call in the following mass singularities . + the first type of divergences appears when we reach the phase space region , where the momentum of the massless particles vanishes : these are called infrared divergences . they occur for example in qed when the energy of the photon goes to zero . the block - nordsieck theorem @xcite assures that the infrared divergences cancel out in any inclusive cross section . + the other type of mass singularities occurs in theories with massless coupled particles , like in qed when the photon couples to a fermion , in the limit of zero fermion mass . the origin is purely kinematical : when two massless particles , say with momenta @xmath46 and @xmath79 , move parallel to each other , they have combined invariant mass equal to zero : @xmath80 even though neither @xmath46 nor @xmath79 are soft . these divergences are called collinear singularities . + if we keep the fermion mass finite and integrate over the photon emission angle , the collinear divergence does nt occur , but the possibility of a divergence in the limit @xmath81 results in the presence of the collinear logarithms , that is logarithms of the form @xmath82 , diverging for @xmath16 . + in the case of collinear singularities , the theorem by kinoshita , lee and nauenberg @xcite guarantees that these divergences cancel out if we sum the transition probability over the set of degenerate states , order by order in perturbation theory . this cancellation mechanism is analogous to that of the infrared divergences , as stated by the block - nordsieck theorem . both types of mass singularities arise because the states of a theory with massless particles are highly degenerate . the infrared divergences can be interpreted as a consequences of the fact that a state with a single charged particle is degenerate with a state made of the same particle plus a number of soft photons ; this correspond to the impossibility of distinguishing experimentally a charged particle from one accompanied by soft photons , owing to the finite resolution of the measurement apparatus . the situation of the collinear singularities is analogous : the state with a massless charged particle is degenerate with the states containing the same particle and a number of collinear photons . this corresponds to the fact that , as a consequence of the finite angular resolution , we can not establish if a massless charged particle is accompanied by collinear photons . + let us now discuss the structure of mass singularities and their cancellation in the decay rates for the radiative pion and @xmath1 decays and how the kln theorem applies to this cases . + it is useful to separate the cases of unpolarized , right - handed and left - handed outgoing lepton . + let us consider first the mass singularity cancellation mechanism for the familiar case of unpolarized radiative @xmath0 decay to the first order in @xmath83 . the differential inner bremsstrahlung contribution is given by : @xmath84\,\ln{\frac{y+a}{y - a } } \nonumber \\ & - & ( 1-y+r)^2\,\ln{\frac{y+a-2}{y - a-2 } } \bigg\}. \label{irnp}\end{aligned}\ ] ] one can easily see that the eq . ( [ irnp ] ) is divergent both in the collinear and in the infrared limits . the coefficients of the collinear logarithms do nt go to zero in the limit @xmath85 . there are also infrared divergences , because if we let @xmath50 reach its kinematical limit @xmath86 , corresponding to the photon energy going to zero , the expression ( [ irnp ] ) diverges . + the decay rate is made free from mass singularities in the ordinary way : the divergences cancellation occurs in the total inclusive decay rate , when we add all the first order contributions to the perturbative expansion , i.e. those relative to real and virtual photon emission . + the diagrams describing the real photon emission contribution were already given in fig . 4 ; the diagram for the virtual correction is drawn in fig . c [ fig5 ] the expression for the lepton energy spectrum , including the inner bremsstrahlung contribution and the virtual photon one , calculated to the leading order in @xmath87 , is given by @xcite : @xmath88 . \label{spettroy}\ ] ] @xmath89 is the lepton distribution function , given , to the first order in @xmath83 , by : @xmath90p^{(1)}(y ) , \label{defd}\ ] ] where @xmath91 is the logarithm @xmath92 diverging in the collinear limit ; if the lepton is an electron , @xmath93 . @xmath94 is the gribov - lipatov - altarelli - parisi kernel @xcite , which can be expressed in the form : @xmath95 @xmath96 is a finite term , free from infrared and collinear singularities , which has the expression : @xmath97 the differential decay rate to order @xmath83 therefore becomes : @xmath98 to calculate the inclusive decay rate , we have to integrate the expression in eq . ( [ ordalfa ] ) over @xmath50 ; to the leading order in the lepton mass , the physical region for @xmath50 is @xmath99 . the kernel @xmath100 has the property that : @xmath101 thus , when we calculate the inclusive decay rate , the coefficient of the collinear logarithm vanishes and the resulting expression is finite in the zero mass limit . + carrying out the integration over @xmath50 , we obtain the well known inclusive decay rate to order @xmath83 @xmath102 . \label{tot}\ ] ] as expected , the expression ( [ tot ] ) is finite in the collinear limit and is also free from infrared divergences , because , as usual , the infrared divergences present in the soft photon contribution and in the virtual photon contribution have cancelled each other . + let us now discuss the mass singularities in the case of the right - handed inner bremsstrahlung contribution , given in eq . ( [ ibr ] ) . it is easy to see that @xmath103 is finite in the limit @xmath104 , i.e. it is free from collinear singularities . in this limit the coefficients of both the collinear logarithms vanish . indeed , as we have observed in section [ pionpol ] , only the term related to the axial anomaly survives in the zero mass limit , that is the part of the decay rate independent of the lepton mass . + we observe that the right - handed inner bremsstrahlung contribution is free from infrared divergences as well . if we make the lepton energy @xmath50 reach its kinematical limit @xmath105 , we obtain a finite result : @xmath106 the result ( [ limir ] ) shows that the soft photon emission does not contribute to the radiative @xmath0 decay with a right - handed lepton . this is a consequence of the fact that the soft photon contribution factorizes with respect to the born decay rate , but this vanishes in the case of right - handed @xmath31 ( see eq . ( [ bornpol ] ) ) . physically , eq . ( [ limir ] ) is due to the fact that soft photons do nt carry spin , thus they can not contribute to the angular momentum balance ; therefore the process with the right - handed lepton emitting a soft photon is forbidden by angular momentum conservation . + for the same reason of angular momentum conservation , in the right - handed case also the virtual contribution identically vanishes . the virtual photon diagram ( see fig . 5 ) interferes with the born one ; the corresponding correction to the total decay rate for @xmath107 was calculated long ago by kinoshita @xcite and is given by : @xmath108 + \frac{r}{1-r } \ln{r } + 1\bigg\}\gamma_0 , \label{kin}\end{aligned}\ ] ] where @xmath109 @xmath110 is the ultraviolet cutoff and @xmath111 is the infrared one . + as usual , the virtual correction is factorized with respect to the born decay rate , but , as we have already seen , if the lepton is right - handed , this is identically zero . + in the right - handed case , the mass singularities cancellation occurs trough a mechanism different from the one working in the unpolarized decay rate . the infrared and the collinear limits give separately a finite result . in particular , the coefficient of the collinear logarithms is the lepton mass , instead of the usual correction factor coming from the soft and the virtual photon contributions , as in eq . ( [ ordalfa ] ) . in this sense , since the soft and collinear radiation factorizes with respect to the born helicity changing decay rate , the double logarithm sudakov term can be equally factorized . it could be useful to investigate the impact of the higher order terms on the radiative correction to the born amplitude . it is an open question if say hard collinear photons can be factorized and resummed . + the particular mass cancellation mechanism occurring in the right - handed radiative decay is the consequence of the combination of two constraints : the angular momentum conservation in the pion vertex and the helicity flip in the photon - lepton vertex . + the situation is completely different if we consider the radiative process with the outgoing left - handed lepton , i.e. the process without helicity flip . the differential inner bremsstrahlung contribution for left - handed outgoing lepton is given by : @xmath112 \nonumber \\ & & + \big[(a-2r)(1+r)^2 + ay(y-4r ) -2ry(1+y ) + y(5r^2+y^2 + 1)\big ] \ , \ln{\frac{y+a}{y - a } } \nonumber \\ & & + ( 1-y+r)^2(2r - a - y ) \ , \ln{\frac{y+a-2}{y - a-2 } } \bigg\ } \label{ibl}\end{aligned}\ ] ] the expression ( [ ibl ] ) contains collinear singularities , since the coefficients of the collinear logarithms do nt vanish in the limit @xmath104 , as one can see from eq . ( [ ibl ] ) . the decay rate ( [ ibl ] ) is also infrared divergent , as one can verify by taking the limit @xmath113 . in this case we do nt have the constraint constituted by the helicity flip in the photon - lepton vertex and in the pion vertex the angular momentum is conserved for soft and virtual photon emission . thus , in the left - handed case , the mass singularity cancellation occurs in the ordinary way , as in the unpolarized case , i.e. in the total inclusive decay rate , obtained by adding all the order @xmath83 contributions . + let us show how the cancellation takes place . as we have already seen ( eq . ( [ bornpol ] ) ) , in the born @xmath0 decay the outgoing lepton is left - handed , due to angular momentum conservation . thus the unpolarized and the left - handed born decay rate coincide : @xmath114 because of the factorization with respect to the born decay rate , also the unpolarized and the left - handed virtual contributions are equal : @xmath115 expressing the left - handed inner bremsstrahlung contribution in terms of the unpolarized and the right - handed ones , the total contribution to order @xmath83 to the left - handed process is given by : @xmath116 the expression ( [ totl ] ) is finite both in the infrared and in the collinear limit , because the mass singularities present in the terms between brackets cancel each other , as we have seen ( see eq . ( [ tot ] ) and @xmath117 is free from mass singularities . + let us now discuss the origin of these different cancellation mechanisms . the presence of mass singularities is a consequence of the fact that the states of a theory containing massless particles are highly degenerate . the kln theorem states that the mass singularities disappear from the transition probability when we average it over the ensemble of degenerate states . this theorem contains the block - nordsieck theorem as a special case , when we consider only the cancellation of the infrared divergences . + we can define two degeneration ensembles , one relative to the infrared divergences and one relative to the collinear singularities . we call them the infrared and the collinear ensembles , respectively . if we sum the transition probability over the states contained in the former , the infrared divergences cancel out and if we average it over the latter , we obtain a quantity free from both infrared and collinear singularities . the infrared ensemble is the one prescribed by the block - nordsieck theorem , while the second is the one prescribed by the kln theorem and contains the first as a subset . + let us now examine how the infrared and collinear ensemble are composed for the radiative pion decay in the cases of unpolarized , left - handed and right - handed outgoing lepton . this discussion concerns the issue of the degeneration of states already addressed for the unpolarized case @xcite . this issue in the case of helicity changing processes presents peculiar features . + we have seen in section [ pion ] that in the unpolarized and left - handed inner bremsstrahlung contributions there are mass singularities , indicating that we have not summed the transition probability over the entire ensemble of degenerate states . in these cases the infrared ensemble contains , to order @xmath83 , the state with a pion , a charged lepton and a neutrino and all the other states differing from this for the presence of a soft virtual or real photon , i.e. a photon with an energy @xmath118 , where @xmath119 is an infrared cut off , tipically the measurement apparatus resolution . the collinear ensemble is constituted by all the states of the infrared ensemble plus the states with a hard photon moving parallel to the pion or the lepton . clearly the degeneration arises in the limit @xmath120 . + according to the kln theorem the fact that @xmath103 is finite both in the infrared and in the collinear limits means that in the right - handed case , calculating the differential decay rate ( i.e. summing over the photon polarization and integrating over the photon energy and emission angle ) , we have already averaged over the set of degenerate states relative to this process . let us now consider how the degeneration ensemble for the right - handed radiative decay is composed . to obtain @xmath121 we have not averaged over the infrared subspace of the collinear ensemble , but this is enough to render the transition probability free from mass singularities . indeed in this case the infrared ensemble is empty , owing to the constraints imposed both by the angular momentum conservation in the pion vertex and by the helicity flip in the photon - lepton vertex . thus in this case the degeneration ensemble contains only the states with the outgoing lepton accompanied by hard collinear photons . + we conclude that imposing to the outgoing lepton a polarization opposed to the one prescribed by the vertex preceding the photon emission , implies a reduction of the degeneration subspace . this fact has two consequences : the first is that both the infrared and the collinear limits are finite and the second is that these limits are disconnected , since the collinear degeneration subspace is constituted only by the states with the pion and the outgoing lepton accompanied by hard collinear photons . thus in this case we have a particular application of the kln theorem . + in the radiative pion decay , due to the angular momentum conservation in the pion vertex , there is no room for a right - handed lepton . for such a channel , soft and virtual photon contributions are zero . this result is valid independently of the lepton mass . + let us now consider a more general case , by loosing the value of the angular momentum of the decaying state . as an example , we study the radiative @xmath1 decay in a lepton - antilepton @xmath122 pair , in which the lepton is in a definite helicity state . + the @xmath1-leptons vertex is : @xmath123 with @xmath124 where @xmath125 is the weinberg angle and @xmath126 is the @xmath1 mass . + we have chosen this process , since , by varying the constants @xmath127 and @xmath128 , we can control the structure of the @xmath1-leptons coupling ; thus it is possible to point out the role played by the conservation law occurring in this vertex in the chiral limit in the collinear singularity cancellation . if we set @xmath129 , we require that in the limit of zero lepton mass , the @xmath1 couples to a left - handed lepton . + we calculate the decay rate for the process in which the lepton is right - handed . at the born level this is given by : @xmath130 - 2g_vg_a(1 - 4r ) \bigg\}. \label{bornz}\end{aligned}\ ] ] if in eq . ( [ bornz ] ) we set @xmath129 , @xmath131 vanishes in the chiral limit , since there is nt the term related to the axial anomaly . + let us now study the decay process with the lepton emitting a real photon ( see fig . 6 ) c [ fig6 ] and evaluate the probability that the outgoing lepton is right - handed . the electromagnetic interaction does nt couples states with different chirality , hence the decay rate is expected to vanish for @xmath104 . + the decay rate for the process described by the diagram of fig . 6 , differential with respect to the lepton energy , is given by : @xmath132 \nonumber \\ & + & \frac{(1-y)}{2(1-y+r)}\ , \big[(g^2_v+g^2_a)a(2r - y ) + 2g_vg_a(y^2 - 2r)\big ] \nonumber \\ & + & \frac{2}{(y-1)}\ , \big[2(g^2_v-2g^2_a)ar + ( g^2_v+g^2_a)a + 2g_vg_a(4r - y^2+y-1)\big ] \nonumber \\ & + & \big[g^2_v+g^2_a + 2g_vg_a\,\frac{1}{a}\,\big(4r^2+r(y - y^2 + 1)-y\big)\big ] \,\big[\ln{\frac{y+a}{y - a } } - \ln{\frac{y+a-2}{y - a-2}}\big]\bigg\}. \nonumber \\ \label{rz}\end{aligned}\ ] ] here @xmath133 @xmath50 is the usual dimensionless variable : @xmath134 where @xmath135 is the lepton energy and @xmath136 the physical region for @xmath50 is @xmath137 the result obtained , as given by the emission of the photon by a single leg , is gauge dependent . to have a gauge independent amplitude , the contribution of the diagram b ) of fig . 7 must be added . for the purpose of the polarized amplitude , however , the helicity flip contribution of the diagram b ) of fig . 7 gives zero in the massless limit and is therefore negligible in our discussion . + from now on we consider the case @xmath129 , to have the condition of chirality conservation in the @xmath1 vertex for @xmath120 . taking this limit in eq . ( [ rz ] ) , we see that @xmath138 does not vanish : @xmath139 the result of this limit is the contribution related to the axial anomaly , which has the same form of the one found in the pion case . + let us now discuss the mass singularities cancellation mechanism for the @xmath1 decay case . if we keep the lepton mass different from zero , the helicity is not fixed by the interaction occurring before the photon emission , even if we set @xmath129 . thus , for @xmath140 , the soft and virtual photons contribution are different from zero and diverge in the infrared limit . indeed , if we let the lepton energy reach its kinematical limit , @xmath141 , we see that @xmath138 diverges . we expect the infrared divergences to cancel , if we add all the first order contributions , given by the diagrams of fig . 6 and 7 c [ fig7 ] and calculate the totally inclusive decay rate . ( [ collz ] ) shows that the collinear limit gives a finite result . thus , we conclude that , as in the case of the radiative pion decay , evaluating @xmath138 we have already summed over all the collinear degeneration subspace . as a consequence , the collinear and infrared limit are disconnected . + if we take the limit @xmath142 in eq . ( [ collz ] ) , we obtain : @xmath143 in general , if a quantity is finite in the collinear limit , it is finite also in the infrared limit , since the collinear subspace contains the infrared one . ( [ irz ] ) indicates that in the massless limit the soft photon contribution is zero . indeed , it is factorized with respect to the born decay rate , which , for @xmath104 and @xmath129 , vanishes . as we have discussed in section [ anomaly ] , the presence of the anomalous term is directly connected to the emission of the photon , hence it vanishes in the infrared limit . + the virtual photon contribution vanishes in the zero mass limit , as well . indeed , it is factorized respect to the born decay rate , which goes to zero as @xmath104 . the virtual correction factor can produce only a logarithmic collinear singularity , not a power - like one , needed for the cancellation of the chiral suppression . + we observe that taking the infrared limit @xmath144 in eq . ( [ hf ] ) , gives a finite result ( indeed the cross section vanishes ) . this is a consequence of the fact that the cross section ( [ hf ] ) has been calculated to the leading order in the lepton mass . from the eq . ( [ rz ] ) , we see that , for @xmath145 , the infrared divergent term is given by : @xmath146 and it is proportional to the lepton mass . performing the calculation , neglecting the mass terms , as done in ref . @xcite , means imposing the chirality conservation law in the @xmath1 vertex ; thus the soft photon contribution is zero and the infrared divergences disappear . to the leading order in the lepton mass , we have only the anomalous term , which vanishes in the infrared limit . + as done for the pion case , we now examine the composition of the degeneration ensembles for the radiative @xmath1 decay with the right - handed lepton . ( [ irz ] ) indicates that , in the chiral limit , the soft photons do nt contribute to the process with the right - handed outgoing lepton , since , not carrying spin , they can not contribute to the helicity flip . + as we have already said , the virtual photon contribution vanishes in the zero mass limit . in the limit @xmath120 , also the diagram with the photon emitted by @xmath147 does nt contribute , since , clearly , it violates the chirality conservation in the @xmath1 vertex . + the collinear degeneration arises in the massless limit . the result ( [ collz ] ) shows that , in this limit , the collinear ensemble is constituted only by the states with the lepton accompanied by hard collinear photons , just as in the case of the pion decay . thus for @xmath120 the infrared ensemble is empty and also the states with the antilepton accompanied by a hard collinear photon do nt contribute . we have shown that the dolgov and zakharov treatment of the axial anomaly can be extended to processes characterized by a lepton which changes helicity by emitting a photon , as it was already noticed in @xcite . the corresponding decay rates do nt vanish in the chiral limit , due to a term independent of the lepton mass ; we interpret the presence of this term as related to the axial anomaly . this can be seen as a signal of the anomalous symmetry breaking in processes different from the usual ones , like the @xmath3 decay . + we have computed the rates corresponding to the @xmath0 and @xmath1 radiative decays . we have analysed their infrared and collinear limits . it results essential to keep the terms of all orders in the lepton mass , since the cancellation of the infrared and collinear divergences takes place among these terms . we have examined the connection between the polarization of the outgoing leptons and the application of the kln theorem . + we have shown that for the helicity changing processes the cancellation of the collinear singularities occurs through a mechanism different from the usual one of real and virtual compensation . the coefficients in front of the collinear terms go to zero in the chiral limit , producing the finiteness of the distribution . we have found , however , a difference between the pion case and the more general case of the @xmath1 decay . the former represents a particular case due to the angular momentum conservation in the pion vertex . as a consequence , the virtual and real soft photon contribution are zero , even if the lepton mass is kept different from zero . the inner bremsstrahlung contribution is finite both in the infrared and in the collinear limits . + in the @xmath1 case , the decay rate diverges in the infrared limit , since , for @xmath140 , the soft photon contributions are not zero . however in the collinear limit , the result is finite , despite the fact that the collinear degenerate states arise only in the zero lepton mass limit . in this limit the virtual and real soft photon contributions do vanish . in order to make the collinear limit finite , it is therefore sufficient to sum over degenerate states made of the changing helicity lepton accompanied by a hard collinear photon . the transition probability becomes finite after summing over the photon final phase space . + we noticed that in the helicity changing processes the collinear limit results disconnected from the infrared one . the contributions coming from the virtual photon emission and from the emission of photons by particles different from the one changing helicity , are zero . + this situation is due to the fact that the born part of the process fixes the fermion chirality in the zero mass limit , while , after the photon emission , it is in a state of opposite chirality ; this reduces the collinear ensemble . + we conclude that the collinear singularity cancellation mechanism for helicity changing processes is controlled by the anomalous breaking of the chiral symmetry . the axial anomaly implies that the collinear limit gives a finite result , independent of the fermion mass . + the extension of the above remarks to other gauge theories like qcd , is possible . it could allow a more systematic and complete treatment of the infrared and collinear singularities . + 99 s. adler , phys . rev . * 177 * ( 1969 ) 2426 ; + j.s . bell and r. jackiw , nuovo cimento * a60 * ( 1969 ) 47 . k. fujikawa , phys . * 42 * ( 1979 ) 1195 . dolgov and v.i . zakharov , nucl . * b27 * ( 1971 ) 525 . gribov , hungarian academy of sciences report kfki-1981 - 66 ( 1981 ) . lectures given at the institute for theoretical physics , santa barbara , 1988 . mueller , phys . lett . * b234 * ( 1990 ) 517 . r. jackiw , in `` lectures on current algebra and its application '' , princeton university press , princeton , new jersey ( 1972 ) ; 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we discuss a connection between axial anomaly and polarized radiative processes . by comparison with the corresponding unpolarized cases , we consider some physical outputs for the @xmath0 and @xmath1 polarized radiative decays . we analyse in detail the pattern of mass singularity cancellation . uprf-00 - 10 + june 2000 axial anomaly effects in pion and @xmath1 radiative decays 1.0 cm luca trentadue and michela verbeni + .2 cm _ dipartimento di fisica , universit di parma , + infn gruppo collegato di parma , 43100 parma , italy _ + pacs 11.30.rd , 13.20.cz , 11.15.bt

arxiv

lease take a look at the images in the top row of fig . [ fig : fig1 ] . which object stands out the most ( i.e. , is the most salient one ) in each of these scenes ? the answer is trivial . there is only one object , thus it is the most salient one . now , look at the images in the third row . these scenes are much more complex and contain several objects , thus it is more challenging for a vision system to select the most salient object . this problem , known as _ salient object detection ( and segmentation ) _ , has recently attracted a great deal of interest in computer vision community . the goal is to simulate the astonishing capability of human attention in prioritizing objects for high - level processing . such a capability has several applications in recognition ( e.g. , @xcite ) , image and video compression ( e.g. , @xcite ) , video summarization ( e.g. , @xcite , media re - targeting and photo collage ( e.g. , @xcite ) , image quality assessment ( e.g. , @xcite ) , image segmentation ( e.g. , @xcite ) , content - based image retrieval and image collection browsing ( e.g. , @xcite ) , image editing and manipulating ( e.g. , @xcite ) , visual tracking ( e.g. , @xcite ) , object discovery ( e.g. , @xcite ) , and human - robot interaction ( e.g. , @xcite ) . a large number of saliency detection methods have been proposed in the past 7 years ( since @xcite ) . in general , a salient object detection model involves two steps : 1 ) _ selecting objects to process _ ( i.e. , determining saliency order of objects ) , and 2 ) _ segmenting the object area _ ( i.e. , isolating the object and its boundary ) . so far , models have bypassed the first challenge by focusing on scenes with single objects ( see fig . [ fig : fig1 ] ) . they do a decent job on the second step as witnessed by very high performances on existing biased datasets ( e.g. , on asd dataset @xcite ) which contain low - clutter images with often a single object at the center . however , it is unclear how current models perform on complex cluttered scenes with several objects . despite the volume of past research , this trend has not been yet fully pursued , mainly due to the lack of two ingredients : 1 ) suitable benchmark datasets for scaling up models and model development , and 2 ) a widely - agreed objective definition of the most salient object . in this paper , we strive to provide solutions for these problems . further , we aim to discover which component might be the weakest link in the possible failure of models when migrating to complex scenes . some related topics , closely or remotely , to visual saliency modeling and salient object detection include : object importance @xcite , object proposal generation @xcite , memorability @xcite , scene clutter @xcite , image interestingness @xcite , video interestingness @xcite , surprise @xcite , image quality assessment @xcite , scene typicality @xcite , aesthetic @xcite , and attributes @xcite . one of the earliest models , which generated the _ first wave _ of interest in image saliency in computer vision and neuroscience communities , was proposed by itti _ et al . _ this model was an implementation of earlier general computational frameworks and psychological theories of bottom - up attention based on center - surround mechanisms . in @xcite , itti _ et al . _ showed examples where their model was able to detect spatial discontinuities in scenes . subsequent behavioral ( e.g. , @xcite ) and computational studies ( e.g. , @xcite ) started to predict fixations with saliency models to verify models and to understand human visual attention . second wave _ of interest appeared with works of liu _ et al . _ @xcite and achanta _ et al . _ @xcite who treated saliency detection as a binary segmentation problem with 1 for a foreground pixel and 0 for a pixel of the background region . since then it has been less clear where this new definition stands as it shares many concepts with other well - established computer vision areas such as general segmentation algorithms ( e.g. , @xcite ) , category independent object proposals ( e.g. , @xcite ) , fixation prediction saliency models ( e.g. @xcite ) , and general object detection methods . this is partly because current datasets have shaped a definition for this problem , which might not totally reflect full potential of models to _ select and segment salient objects in an image with an arbitrary level of complexity_. reviewing all saliency detection models goes beyond the scope of this paper ( see @xcite ) . some breakthrough efforts are as follows . et al . _ @xcite introduced a conditional random field ( crf ) framework to combine multi - scale contrast and local contrast based on surrounding , context , and color spatial distributions for binary saliency estimation . et al . _ @xcite proposed subtracting the average color from the low - pass filtered input for saliency detection . et al . _ @xcite used a patch - based approach to incorporate global context , aiming to detect image regions that represent the scene . et al . _ @xcite proposed a region contrast - based method to measure global contrast in the lab color space . in @xcite , wang _ et al . _ estimated local saliency , leveraging a dictionary learned from other images , and global saliency using a dictionary learned from other patches of the same image . et al . _ @xcite observed that decomposing an image into perceptually uniform regions , which abstracts away unnecessary details , is important for high quality saliency detection . in @xcite , et al . _ utilized the difference between the color histogram of a region and its immediately neighboring regions for measuring saliency . et al . _ @xcite defined a measure of saliency as the cost of composing an image window using the remaining parts of the image , and tested it on pascal voc dataset @xcite . this method , in its essence , follows the same goal as in @xcite . et al . _ @xcite proposed a graphical model for fusing generic objectness @xcite and visual saliency for salient object detection . shen and wu @xcite modeled an image as a low - rank matrix ( background ) plus sparse noises ( salient regions ) in a feature space . more recently , margolin _ et al . _ @xcite integrated pattern and color distinctnesses with high - level cues to measure saliency of an image patch . some studies have considered the relationship between fixations and salincy judgments similar to @xcite . for example , xu et al . @xcite investigated the role of high - level semantic knowledge ( e.g. , object operability , watchability , gaze direction ) and object information ( e.g. , object center - bias ) for fixation prediction in free viewing of natural scenes . they constructed a large dataset called `` object and semantic images and eye - tracking ( osie ) '' . indeed they found an added value for this information for fixation prediction and proposed a regression model ( to find combination weights for different cues ) that improves fixation prediction performance . koehler et al . @xcite collected a dataset known as the ucsb dataset . this dataset contains 800 images . one hundred observers performed an explicit saliency judgment task , 22 observers performed a free viewing task , 20 observers performed a saliency search task , and 38 observers performed a cued object search task . observers completing the free viewing task were instructed to freely view the images . in the explicit saliency judgment task , observers were instructed to view a picture on a computer monitor and click on the object or area in the image that was most salient to them . _ salient _ was explained to observers as something that stood out or caught their eye ( similar to @xcite ) . observers in the saliency search task were instructed to determine whether or not the most salient object or location in an image was on the left or right half of the scene . finally , observers who performed the cued object search task were asked to determine whether or not a target object was present in the image . then , they conducted a benchmark and introduced models that perform the best on each of these tasks . a similar line of work to ours in this paper has been proposed by mishra _ et al . _ @xcite where they combined monocular cues ( color , intensity , and texture ) with stereo and motion features to segment a region given an initial user - specified seed point , practically ignoring the first stage in saliency detection ( which we address here by automatically generating a seed point ) . ultimately , our attempt in this work is to bridge the interactive segmentation algorithms ( e.g. , @xcite ) and saliency detection models and help transcend their applicability . perhaps the most similar work to ours has been published by li _ _ @xcite . in their work , they offer two contributions . _ first _ , they collect an eye movement dataset using annotated images from the pascal dataset @xcite and call their dataset pascal - s . _ second _ , they propose a model that outperforms other state - of - the - art salient object detection models on this dataset ( as well as four other benchmark datasets ) . their model decouples the salient object detection problem into two processes : 1 ) _ a segment generation process _ , followed by 2 ) _ a saliency scoring mechanism _ using fixation prediction . here , similar to li _ et al . _ , we also take advantage of eye movements to measure object saliency but instead of first fully segmenting the scene , we perform a shallow segmentation using superpixels . we then only focus on segmenting the object that is most likely to attract attention . in other words , the two steps are similar to li _ et al . _ but are performed in the reverse order . this can potentially lead to better efficiency as the first expensive segmentation part is now only an approximation . we also offer another dataset which is complimentary to li _ et al . _ s dataset and together both datasets ( and models ) could hopefully lead to a paradigm shift in the salient object detection field to avoid using simple biased datasets . further , we situate this field among other similar fields such as general object detection and segmentation , objectness proposal generation models , and saliency models for fixation prediction . several salient object detection datasets have been created as more models have been introduced in the literature to extend capabilities of models to more complex scenes . table [ tab : db ] lists properties of 19 popular salient object detection datasets . although these datasets suffer from some biases ( e.g. , low scene clutter , center - bias , uniform backgrounds , and non - ambiguous objects ) , they have been very influential for the past progress . unfortunately , recent efforts to extend existing datasets have only increased the number of images without really addressing core issues specifically background clutter and number of objects . majority of datasets ( in particular large scale ones such as those derived from the msra dataset ) have scenes with often one object which is usually located at the image center . this has made model evaluation challenging since some high - performing models that emphasize image center fail in detecting and segmenting the most salient off - center object @xcite . we believe that now is the time to move on to more versatile datasets and remedy biases in salient object datasets . in this section , we briefly explain how salient object detection models differ from fixation prediction models , what people consider the most salient object when they are explicitly asked to choose one , what are the relationships between these judgments and eye movements , and what salient object detection models actually predict . we investigate properties of salient objects from humans point of view when they are explicitly asked to choose such objects . we then study whether ( and to what extent ) saliency judgments agree with eye movements . while it has been assumed that eye movements are indicators of salient objects , so far few studies ( e.g. , @xcite ) have directly and quantitatively confirmed this assumption . moreover , the level of agreement and cases of disagreement between fixations and saliency judgments have not been fully explored . some studies ( e.g. , @xcite ) , have shown that human observers choose to annotate salient objects or regions first but they have not asked humans explicitly ( labelme data was analyzed in @xcite ) and they have ignored eye movements . knowing which objects humans consider as salient is specially crucial when outputs of a model are going to be interpreted by humans . there are two major differences between models defining saliency as `` where people look '' and models defining saliency as `` which objects stand out '' . _ first _ , the former models aim to predict points that people look in free - viewing of natural scenes usually for 3 to 5 seconds while the latter aim to detect and segment salient objects ( by drawing pixel - accurate silhouettes around them ) . in principle a model that scores well on one problem should not score very well on the other . an optimal model for fixation prediction should only highlight those points that a viewer will look at ( few points inside an object and not the whole object region ) . since salient object detection models aim to segment the whole object region they will generate a lot of false positives ( these points belong to the object but viewers may not fixate at them ) when it comes to fixation prediction . on the contrary , a fixation prediction model will miss a lot of points inside the object ( i.e. , false negatives ) when it comes to segmentation . _ second _ , due to noise in eye tracking or observers saccade landing ( typically around 1 degrees and @xmath0 30 pixels ) , highly accurate pixel - level prediction maps are less desired . in fact , due to these noises , sometimes blurring prediction maps increases the scores @xcite . on the contrary , producing salient object detection maps that can accurately distinguish object boundaries are highly desirable specially in applications . due to these , different evaluation and benchmarks have been developed for comparing models in these two categories . in practice , models , whether they address segmentation or fixation prediction , are applicable interchangeably as both entail generating similar saliency maps . for example , several researches have been thresholding saliency maps of their models , originally designed to predict fixations , to detect and segment salient proto - objects ( e.g. , @xcite ) . in our previous study @xcite , we addressed what people consider as the most outstanding ( i.e. , salient ) object in a scene . while in @xcite we studied the explicit saliency problem from a behavioral perspective , here we are mainly interested in constructing computational models for automatic salient object detection in arbitrary visual scenes . a total of 70 students ( 13 male , 57 female ) undergraduate usc students with normal or corrected - to - normal vision in the age range between 18 and 23 ( mean = 19.7 , std = 1.4 ) were asked to draw a polygon around the object that stood out the most . participants annotations were supposed not to be too loose ( general ) or too tight ( specific ) around the object . they were shown an illustrative example for this purpose . participants were able to relocate their drawn polygon from one object to another or modify its outline . we were concerned with the case of selection of the single most salient object in an image . stimuli were the images from the dataset by bruce and tsotsos ( 2005 ) @xcite 681 pixels . images in this dataset have been presented at random to 20 observers ( in a free - viewing task ) for 4 sec each , with 2 sec of delay ( a gray mask ) in between . ] . see fig . [ fig : sampleio - am ] for sample images from this dataset . we first measured the degree to which annotations of participants agree with each other using the following quantitative measure : @xmath1 where @xmath2 and @xmath3 are annotations of @xmath4-th and @xmath5-th participants , respectively ( out of @xmath6 participants ) over the @xmath7-th image . above measure has the well - defined lower - bound of 0 , when there is no overlap in segmentations of users , and the upper - bound of 1 , when segmentations have perfect overlap . [ fig : exp1].left shows histogram of @xmath8 values . participants had moderate agreement with each other ( mean @xmath9 ; std @xmath10 ; significantly above chance ) . inspection of images with lowest @xmath8 values shows that these scenes had several foreground objects while images with highest annotation agreement had often one visually distinct salient object ( e.g. , a sign , a person , or an animal ; see fig . [ fig : sampleio - am ] ) . we also investigated the relationship between explicit saliency judgments and freeviewing fixations as two indicators of visual attention . here we used shuffled auc ( sauc ) score to tackle center - bias in eye movement data @xcite . for each of 120 images , we showed that a map built from annotations of 70 participants explains fixations of free viewing observers significantly above chance ( sauc of @xmath11 , chance @xmath12 , @xmath13-test @xmath14 ; fig . [ fig : exp1].right ) . the prediction power of this map was as good as the itti98 model @xcite . hence , we concluded that explicit saliency judgments agree with fixations . [ fig : sampleio - am ] shows high- and low - agreement cases between fixations and annotations . here , we merge annotations of all 70 participants on each image , normalize the resultant map to [ 0 1 ] , and threshold it at 0.7 to build our first benchmark saliency detection dataset ( called bruce - a ) . prevalent objects in bruce and tsotsos dataset are man - made home supplies in indoor scenes ( see @xcite for more details on this dataset ) . similar results , to link fixations with salient objects , have been reported by koehler _ _ @xcite . as in @xcite , they asked observers to click on salient locations in natural scenes . they showed high correlation between clicked locations and observers eye movements ( from a different group of subjects ) in free - viewing . while the most salient @xcite , important @xcite , or interesting @xcite object may tell us a lot about a scene , eventually there is a subset of objects that can minimally describe a scene . this has been addressed in the past somewhat indirectly in the contexts of saliency @xcite , language and attention @xcite , and phrasal recognition @xcite . based on results from our saliency judgment experiment @xcite , we then decided to annotate scenes of the dataset by judd _ et al . _ the reason for choosing this dataset is because it is currently the most popular dataset for benchmarking fixation prediction models @xcite . it contains eye movements of 15 observers freely viewing 1003 scenes from variety of topics . thus , using fixations we can easily determine which object , out of several annotated objects , is the most salient one . we only used 900 images from the judd dataset and discarded images without well - defined objects ( e.g. , mosaic tiles , flames ) or images with very cluttered backgrounds ( e.g. , nature scenes ) . figure [ fig : figdiscarded ] shows examples of discarded scenes . we asked 2 observers to manually outline objects using the labelme @xcite open annotation tool ( http://new-labelme.csail.mit.edu/ ) . observers were instructed to accurately segment as many objects as possible following three rules : 1 ) discard reflection of objects in mirrors , 2 ) segment objects that are not separable as one ( e.g. , apples in a basket ) , and 3 ) interpolate the boundary of occluded objects only if doing otherwise may create several parts for an occluded object . these cases , however , did happen rarely . observers were also told that their outline should be good enough for somebody to recognize the object just by seeing the drawn polygon . observers were paid for their effort . [ fig:1 ] shows sample images and their annotated objects . to determine which object is the most salient one , we selected the object at the peak of the human fixation map . here we explore some summary statistics of our data . on average , 36.93% of an image pixels was annotated by the 1st observer with a std of 29.33% ( 44.52% , std=29.36% for the 2nd observer ) . 27.33% of images had more than 50% of their pixels segmented by the 1st observer ( 34.18% for the 2nd ) . the number of annotated objects in a scene ranged from 1 to 31 with median of 3 for the 1st observer ( 1 to 24 for the 2nd observer with median of 4 ; fig . [ fig : stat].left ) . the median object size was 10% of the total image area for the 1st observer ( 9% for the 2nd observer ) . [ fig : stat].left ( inset ) shows the average annotation map for each observer over all images . it indicates that either more objects were present at the image center and/or observers tended to annotate central objects more . overall , our data suggests that both observers agree to a good extent with each other . finally , in order to create one ground truth segmentation map per image , we asked 5 other observers to choose the best of two annotations ( criteria based on selection of annotated objects and boundary accuracy ) . the best annotation was the one with max number of votes ( 611 images with 4 to 1 votes ) . next , we quantitatively analyzed the relationship between fixations and annotations ( note that we explicitly define the most salient object as the one with the highest fraction of fixations on it ) . we first looked into the relationship between the object annotation order and the fraction of fixations on objects . [ fig : stat].middle shows fraction of fixations as a function of object annotation order . in alignment with previous findings @xcite we observe that observers chose to annotate objects that attract more fixations . but here , unlike @xcite which used saliency models to demonstrate that observers prioritize annotating interesting and salient objects , we used actual eye movement data . we also quantized the fraction of fixations that fall on scene objects over the judd - a annotations , and observed that in about 55% of images , the most salient object attracts more than 50% of fixations ( mean fixation ratio of 0.54 ; image background=0.45 ; fig . [ fig : stat].right ) . the most salient object ranged in size from 0.1% to 90.2% of the image size ( median=10.17% ) . the min and max aspect ratio ( w / h ) of bounding boxes fitted to the most salient object were 0.04 and 13.7 , respectively ( median=0.94 ) . judd dataset is known to be highly center - biased @xcite , in terms of eye movements @xcite , due to two factors : 1 ) the tendency of observers to start viewing the image from the center ( a.k.a viewing strategy ) , and 2 ) tendency of photographers to frame interesting objects at the image center ( a.k.a photographer bias ) . here we verify the second factor by showing the average annotation map of the most salient object in fig . [ fig : meps ] . our datasets seem to have relatively less center - bias compared to msra-5k and cssd datasets . note that other datasets mentioned in table i are also highly center - biased . to count the number of images with salient objects at the image center , we defined the following criterion . an image is on - centered if its most salient object overlaps with a normalized ( to [ 0 1 ] ) central gaussian filter with @xmath15 . this gaussian filter is resized to the image size and is then truncated above 0.95 . utilizing this criterion , we selected 667 and 223 on - centered and off - centered scenes , respectively . partitioning data in this manner helps scrutinize performance of models and tackle the problem of center - bias . to further explore the amount of center - bias in bruce - a and judd - a datasets , we first calculated the euclidean distance from center of bounding boxes , fitted to object masks , to the image center . we then normalized this distance to the half of the image diagonal ( i.e. , image corner to image center ) . [ fig : arearatio].left shows the distribution of normalized object distances . as opposed to msra-5k and cssd datasets that show an unusual peak around the image center , objects in our datasets are further apart from the image center . [ fig : arearatio].right shows distributions of normalized object sizes . a majority of salient objects in bruce - a and judd - a datasets occupy less than 10% of the image . on average , objects in our datasets are smaller than msra-5k and cssd making salient object detection more challenging . we also analyzed complexity of scenes on four datasets . to this end , we first used the popular graph - based superpixel segmentation algorithm by felzenszwalb and huttenlocher @xcite to segment an image into contiguous regions larger than 60 pixels each ( parameter settings : @xmath16 = 1 , segmentation coefficient @xmath17 = 300 ) . the basic idea is that the more superpixels an image contains , the more complex and cluttered it is @xcite . by analogy to scenes , an object with several superpixels is less homogeneous , and hence is more complex ( e.g. , a person vs. a ball ) . [ fig : stats ] shows distributions of number of superpixels on the most salient object , the background , and the entire scene . if a superpixel overlapped with the salient object and background , we counted it for both . in general , complexities of backgrounds and whole scenes in our datasets , represented by blue and red curves , are much higher than in the other two datasets . the most salient object in judd - a dataset on average contains more superpixels than salient objects in msra-5k and cssd datasets , even with smaller objects . the reason why number of superpixels is low on the bruce - a dataset is because of its very small salient objects ( see fig . [ fig : stats].right ) . further , we inspected types of objects in judd - a images . we found that 45% of images have at least one person in them and 27.2% have more than two people . on average each scene has 1.56 persons ( std = 3.2 ) . in about 27% of images , annotators chose a person as the most salient object . we also found that 280 out of 900 images ( 31.1% ) had one or more text in them . other frequent objects were animals , cars , faces , flowers , and signs . in general , it is agreed that for good saliency detection , a model should meet the following three criteria : 1 ) _ high detection rate_. there should be a low probability of failing to detect real salient regions , and low probability of falsely detecting background regions as salient regions , 2 ) _ high resolution_. saliency maps should have high or full resolution to accurately locate salient objects and retain original image information as much as possible , and 3 ) _ high computational efficiency_. saliency models with low processing time are preferred . here , we analyze these factors by proposing a simple baseline salient object detection model . we propose a straightforward model to serve two purposes : 1 ) _ to assess the degree to which our data can be explained by a simple model_. this way our model can be used for measuring bias and complexity of a saliency dataset , and 2 ) _ to gauge progress and performance of the state of the art models_. by comparing performance of best models relative to this baseline model over existing datasets and our datasets , we can judge how powerful and scalable these models are . note that we deliberately keep the model simple to achieve above goals . our model involves the following two steps : + _ * step 1 * _ : given an input image , we compute a saliency map and an over - segmented region map . for the former , we use a fixation prediction model ( traditional saliency models ) to find spatial outliers in scenes that attract human eye movements and visual attention . here , we use two models for this purpose : aws @xcite and hounips @xcite , which have been shown to perform very well in recent benchmarks and to be computationally efficient @xcite . as controls , we also use the generic _ objectness _ measure by alexe _ et al . _ @xcite , as well as the human fixation map to determine the upper - bound performance . the reason for using fixation saliency models is to obtain an quick initial estimation of locations where people may look in the hope of finding the most salient object . these regions are then fed to the segmentation component in the next step . it is critical to first limit the subsequent expensive processes onto the right region . for the latter , as in the previous section , we use the fast and robust algorithm by felzenszwalb and huttenlocher @xcite with same parameters as in section [ statistics ] . _ * step 2 * _ : the saliency map is first normalized to [ 0 1 ] and is then thresholded at @xmath18 ( here @xmath18 = 0.7 ) . then all unique image superpixels that spatially overlap with the truncated saliency map are included . here we discarded those superpixels that touch the image boundary because they are highly likely to be part of the background . finally , after this process , the holes inside the selected region will be considered as part of the salient object ( e.g. , filling in operation ) . [ fig : samples ] illustrates the process of segmentation and shows outputs of our model for some images from msra-5k , bruce - a , and judd - a datasets . the essential feature of our simple model is dissociating saliency detection from segmentation , such that now it is possible to pinpoint what might be the cause of mistakes or low performance of a model , i.e. , or . this is particularly important since almost all models have confused these two steps and have faded the boundary . note that currently there is no training stage in our model and it is manually constructed with fixed parameters . the second stage in our model is where more modeling contribution can be made , for example by devising more elaborate ways to include or discard superpixels in the final segmentation . one strategy is to learn model parameters from data . some features to include in a learning method are size and position of a superpixel , a measure of elongatedness , a measure of concavity or convexity , distance between feature distributions of a superpixel and its neighbors , etc . to some extent , some of these these features have already been utilized in previous models @xcite . another direction will be expanding our model to multi scale ( similar to @xcite ) . we exhaustively compared our model to 8 state of the art methods which have been shown to perform very well on previous benchmarks @xcite . these models come from 3 categories allowing us to perform cross - category comparison : 1 ) _ salient object detection models _ including cbsal @xcite , svo @xcite , pca @xcite , goferman @xcite , and fts @xcite , 2 ) _ generic objectness measure _ by alexe _ et al . _ @xcite , and 3 ) _ fixation prediction models _ including aws @xcite and hounips @xcite . we use two widely adopted metrics : * * precision - recall ( pr ) curve : * for a saliency map @xmath19 normalized to @xmath20 $ ] , we convert it to a binary mask @xmath21 with a threshold @xmath22 . @xmath23 and @xmath24 are then computed as follows given the ground truth mask @xmath25 : @xmath26 [ eqn : precision_recall ] to measure the quality of saliency maps produced by several algorithms , we vary the threshold @xmath22 from 0 to 255 . on each threshold , @xmath23 and @xmath24 values are computed . finally , we can get a precision - recall ( pr ) curve to describe the performance of different algorithms . + we also report the f - measure defined as : @xmath27 + here , as in @xcite and @xcite , we set @xmath28 to weigh precision more than recall . * * receiver operating characteristics ( roc ) curve : * we also report the false positive rate ( @xmath29 ) and true positive rate ( @xmath30 ) during the thresholding a saliency map : @xmath31 where @xmath32 and @xmath33 denote the opposite ( complement ) of the binary mask @xmath21 and ground - truth , respectively . the roc curve is the plot of @xmath30 versus @xmath29 by varying the threshold @xmath22 . results are shown in fig . [ fig : bruce ] . consistent with previous reports over the msra-5k dataset @xcite , cbsal , pca , svo , and alexe models rank on the top ( with f - measures above 0.55 and aucs above 0.90 ) . fixation prediction models perform lower at the level of the map . fts model ranked on the bottom again in alignment with previous results . our models work on par with the best models on this dataset with all f - measures above 0.70 ( max with alexe model about 0.73 ) . moving from this simple dataset ( because our simple models ranked on the top ; see also the analysis in section [ statistics ] ) to more complex datasets ( middle column in fig . [ fig : bruce ] ) we observed a dramatic drop in performance of all models . the best performance now is 0.24 belonging to the pca model . we observed about 72% drop in performance averaged over 5 models ( cbsal , fts , svo , pca , and alexe ) from msra-5k to bruce - a dataset . note in particular how map model is severely degraded here ( poorest with f measure of 0.1 ) since objects are now less at the center . our best model on this dataset is the salbase - human ( f - measure about 0.31 ) . surprisingly , auc results are still high on this dataset since objects are small thus true positive rate is high at all levels of false positive rate ( see also performance of map ) . patterns of results over judd - a dataset are similar to those over bruce - a with all of our models performing higher than others . the lowest performance here belongs to fts followed by the two fixation prediction models . our salbase - human model scores the best with the f - measure about 0.55 . among our models that used a model to pick the most salient location , salbase - aws scores higher over bruce - a and judd - a datasets possibly because aws is better able to find the most salient location . the average drop from msra-5k to judd - a dataset is @xmath0 41% ( for 5 saliency detection models ) . [ fig : f - measure2 ] shows that these findings are robust to f - measure parameterization . tables [ tab : db - fmeasure ] and [ tab : db - auc ] summarize the f - measure and auc of models . .f - measure accuracy of models . performance of the best model is highlighted in boldface font . [ cols="^,^,^,^ " , ] to study the dependency of results on saliency map thresholding ( i.e. , how many superpixels to include ) , we varied the saliency threshold @xmath18 and calculated f - measure for salbase - human and salbase - aws models ( see fig . [ fig : salthreshold ] ) . we observed that even higher scores are achievable using different parameters . for example , since objects in the judd - a dataset are larger , a lower threshold yields a better accuracy . the opposite holds over the bruce - a dataset . to investigate the dependency of results on segmentation parameters , we varied the parameters of the segmentation algorithm from too fine ( @xmath16 = 1 , k = 100 , min = 20 ; many segments ; over - segmenting ) to too coarse ( @xmath16 = 1 , k = 1000 , min = 800 ; fewer segments ; under - segmenting ) . both of these settings yielded lower performances than results in fig . [ fig : bruce ] . results with another parameter setting with @xmath16 = 1 , k = 500 , and min = 50 are shown in fig . [ fig : res_500_50 ] . scores and trends are similar to those shown in fig . [ fig : bruce ] , with salbase - human and salbase - aws being the top contenders . . stars correspond to points shown in fig . [ fig : bruce ] . note that even higher accuracies are possible with different thresholds over our datasets . corresponding f - measure values over msra-5k for salbase - aws model are : 0.62 , 0.67 , 0.70 , 0.73 , and 0.62.,width=321,height=143 ] analysis of cases where our model fails , shown in fig . [ fig : failure2 ] , reveals four reasons : _ first _ , on bruce - a dataset when humans look at an object more but annotators chose a different object . _ second _ , when a segment that touches the image border is part of the salient object . _ third _ , when the object segment falls outside the thresholded saliency map ( or a wrong one is included ) . _ fourth _ , when the first stage ( i.e. , fixation prediction model ) pick the wrong object as the most salient one ( see fig . [ fig : rrr ] , first column ) . regarding the first problem , care must be taken in assuming what people look is what they choose as the most salient object . although this assumption is correct in a majority of cases ( fig . [ fig : exp1 ] ) , it does not hold in some cases . with respect to the second and third problems , future modeling effort is needed to decide which superpixels to include / discard to determine the extent of an object . the fourth problem points toward shortcomings of fixation prediction models . indeed , in several scenes where our model failed , people and text were the most salient objects . person and text detectors were not utilized in the saliency models employed here . [ fig : rrr ] shows a visual comparison of models over 12 scenes from the judd - a dataset . cbsal and svo generate more visually pleasant maps . goferman highlights object boundaries more than object interiors . pca generates center - biased maps . some models ( e.g. , goferman , fts ) generate sparse saliency maps while some others generate smoother ones ( e.g. , svo , cbsal ) . aws and hounips models generate pointy maps to better account for fixation locations . in this work , we showed that : 1 ) explicit human saliency judgments agree with free - viewing fixations ( thus extending our previous results in @xcite ) , 2 ) our new benchmark datasets challenge existing state - of - the - art salient object detection models ( in alignment with li _ et al . _ s dataset @xcite ) , and 3 ) a conceptually simple and computationally efficient model ( @xmath00.2 s for @xmath34 saliency and segmentation maps on a pc with a 3.2 ghz intel i7 cpu and 6 gb ram using matlab ) wins over the state of the art models and can be used as a baseline in the future . we also highlighted a limitation of models which is the main reason behind their failure on complex scenes . they often segment the wrong object as the most salient one . previous modeling effort has been mainly concentrated on biased datasets with images containing objects at the center . here , we focused on this shortcoming and described how unbiased salient object detection datasets can be constructed . we also reviewed datasets that can be used for saliency model evaluation ( in addition to datasets in table [ tab : db ] ) and measured their statistics . no dataset exists so far that has all of object annotations , eye movements , and explicit saliency judgments . bruce - a has fixations , and only explicit saliency judgments but not all object labels . judd - a , osie , and pascal - s datasets have annotations and fixations but not explicit saliency judgments . here , we chose the object that falls at the peak of the fixation map as the most salient one . ucsb dataset lacks object annotations but it has fixations and saliency judgments using clicks ( as opposed to object boundaries in bruce - a ) . future research by collecting all information on a large scale dataset will benefit salient object detection research . here we suggested that the most salient object in a scene is the one that attracts the majority of fixations ( similar to @xcite ) . one can argue that the most salient object is the one that observers look at first . while in general , these two definitions may choose different objects , given the short presentation times in our datasets ( 3 sec on judd , 4 sec on bruce ) we suspect that both suggestions will yield to similar results . our model separates detection from segmentation . a benefit of this way of modeling is that it can be utilized for other purposes ( e.g. , segmenting interesting or important objects ) by replacing the first component of our model . further , augmented with a top - down fixation selection strategy , our model can be used as an active observer ( e.g. , @xcite ) . our analysis suggests two main reasons for model performance drop over the judd - a dataset : the _ first reason _ that the literature has focused so far is to avoid incorrectly segmenting the object region ( i.e. , increasing true positives and reducing false positives ) . therefore , low performance is partially due to inaccurately highlighting ( segmenting ) the salient object . the _ second reason _ that we attempted to highlight in this paper ( we believe is the main problem causing performance drop as models performed poorly on judd - a compared to msra-5k ) is segmenting the wrong object ( i.e. , not the most salient object ) . note that although here we did not consider the latest proposed salient object detection models in our model comparison ( e.g. , @xcite ) , we believe that our results are likely to generalize compared to newer models . the rationale is that even recent models have also used the asd dataset @xcite ( which is highly center - biased ) for model development and testing . nontheless , we encourage future works to use our model ( as well as li et al.s model ) as a baseline for model benchmarking . two types of cues can be utilized for segmenting an object : appearance @xcite ( i.e. , grouping contiguous areas based on surface similarities ) and boundary @xcite ( i.e. , cut regions based on observed pixel boundaries ) . here we mainly focused on the appearance features . taking advantage of both region appearance and contour information ( similar to @xcite ) for saliency detection ( e.g. , growing the foreground salient region until reaching the object boundary ) is an interesting future direction . in this regard , it will be helpful to design suitable measures for evaluating accuracy of models for detecting boundary ( e.g. , @xcite ) . our datasets allow more elaborate analysis of the interplay between saliency detection , fixation prediction , and object proposal generation . obviously , these models depend on the other . on one hand , it is critical to correctly predict where people look to know which object is the most salient one . on the other hand , labeled objects in scenes can help us study how objects guide attention and eye movements . for example , by verifying the hypotheses that some parts of objects ( e.g. , object center @xcite ) or semantically similar objects @xcite ) attract fixations more , better fixation prediction models become feasible . l. itti , c. koch , and e. niebur . a model of saliency - based visual attention for rapid scene analysis . _ ieee trans . pami _ , 1998 . n. d. b. bruce and j. k. tsotsos . saliency based on information maximization . _ nips _ , 2005 . x. hou and l. zhang . dynamic attention : searching for coding length increments . _ nips _ , 2008 . a. garcia - diaz , x. r. fdez - vidal , x. m. pardo , and r. dosil . decorrelation and distinctiveness provide with human - like saliency . _ acivs _ , 2009 . h. jiang , j. wang , z. yuan , y. wu , n. zheng , and s. li . salient object detection : a discriminative regional feature integration approach . _ ieee conference on computer vision and pattern recognition _ , 2013 . a. c. berg , t. l. berg , h. daume , j. dodge , a. goyal , x. han , a. mensch , m. mitchell , a. sood , k. stratos et al . , `` understanding and predicting importance in images , '' in ieee conference on computer vision and pattern recognition ( cvpr ) , 2012 , pp . 3562 - 3569 . x. han , a. mensch , m. mitchell , a. sood , k. stratos et al . , `` understanding and predicting importance in images , '' in ieee conference on computer vision and pattern recognition ( cvpr ) , 2012 , pp . 3562 - 3569 . s. dhar , v. ordonez , and t. l. berg , `` high level describable attributes for predicting aesthetics and interestingness , '' in ieee conference on computer vision and pattern recognition ( cvpr ) , 2011 , pp . 1657 - 1664 . z. wang , a. c. bovik , h. r. sheikh , and e. p. simoncelli , `` image quality assessment : from error visibility to structural similarity , '' ieee transactions on image processing , vol . 600 - 612 , 2004 . g. kulkarni , v. premraj , s. dhar , s. li , y. choi , a. c. berg , and t. l. berg , `` baby talk : understanding and generating simple image descriptions , '' in ieee conference on computer vision and pattern recognition ( cvpr ) , 2011 , pp . 1601 - 1608 . u. rutishauser , d. walther , c. koch , and p. perona , `` is bottom - up attention useful for object recognition ? '' in proceedings of the ieee conference on computer vision and pattern recognition , 2004 , vol . 2 , 2004 , pp . ii-37 . c. guo and l. zhang , `` a novel multiresolution spatiotemporal saliency detection model and its applications in image and video compression , '' ieee trans . on image processing , 1 , pp . 185 - 198 , 2010 . ma , x .- s . hua , l. lu , and h .- j . zhang , `` a generic framework of user attention model and its application in video summarization , '' ieee transactions on multimedia , vol . 907 - 919 , 2005 . a. ninassi , o. le meur , p. le callet , and d. barbba , `` does where you gaze on an image affect your perception of quality ? applying visual attention to image quality metric , '' in ieee conference on image processing ( icip ) , vol . 2 , 2007 , pp . ii-169 . j. li , m. levine , x. an , x. xu , and h. he , `` visual saliency based on scale - space analysis in the frequency domain , '' ieee trans . on pattern analysis and machine intelligence , 996 - 1010 , 2013 . s. frintrop , g. m. garca , and a. b. cremers , `` a cognitive approach for object discovery , '' icpr , 2014 . d. meger , p .- e . forssen , k. lai , s. helmer , s. mccann , t. southey , m. baumann , j. j. little , and d. g. lowe , `` curious george : an attentive semantic robot , '' robotics and autonomous systems , vol . 56 , no . 6 , pp . 503 - 511 , 2008 . ali borji received his bs and ms degrees in computer engineering from petroleum university of technology , tehran , iran , 2001 and shiraz university , shiraz , iran , 2004 , respectively . he did his ph.d . in cognitive neurosciences at institute for studies in fundamental sciences ( ipm ) in tehran , iran , 2009 and spent four years as a postdoctoral scholar at ilab , university of southern california from 2010 to 2014 . he is currently an assistant professor at university of wisconsin , milwaukee . his research interests include visual attention , active learning , object and scene recognition , and cognitive and computational neurosciences .

salient object detection or salient region detection models , diverging from fixation prediction models , have traditionally been dealing with locating and segmenting the most salient object or region in a scene . while the notion of most salient object is sensible when multiple objects exist in a scene , current datasets for evaluation of saliency detection approaches often have scenes with only one single object . we introduce three main contributions in this paper : first , we take an in - depth look at the problem of salient object detection by studying the relationship between where people look in scenes and what they choose as the most salient object when they are explicitly asked . based on the agreement between fixations and saliency judgments , we then suggest that the most salient object is the one that attracts the highest fraction of fixations . second , we provide two new less biased benchmark datasets containing scenes with multiple objects that challenge existing saliency models . indeed , we observed a severe drop in performance of 8 state - of - the - art models on our datasets ( 40% to 70% ) . third , we propose a very simple yet powerful model based on superpixels to be used as a baseline for model evaluation and comparison . while on par with the best models on msra-5k dataset , our model wins over other models on our data highlighting a serious drawback of existing models , which is convoluting the processes of locating the most salient object and its segmentation . we also provide a review and statistical analysis of some labeled scene datasets that can be used for evaluating salient object detection models . we believe that our work can greatly help remedy the over - fitting of models to existing biased datasets and opens new venues for future research in this fast - evolving field . shell : xx ieee transactions on image processing salient object detection , explicit saliency , bottom - up attention , regions of interest , eye movements

arxiv

in their papers @xcite , @xcite , @xcite , ozsvth and szab constructed a decorated topological quantum field theory ( tqft ) in @xmath0 dimensions , called heegaard floer theory . ( strictly speaking , the axioms of a tqft need to be altered slightly . ) in its simplest version ( called hat ) , to a closed , connected , oriented three - manifold @xmath1 and a @xmath2 structure @xmath3 on @xmath1 one associates a vector space @xmath4 over the field @xmath5 also , to a connected , oriented four - dimensional cobordism from @xmath6 to @xmath7 decorated with a @xmath2 structure @xmath8 , one associates a map @xmath9 the maps @xmath10 can be used to detect exotic smooth structures on @xmath11-manifolds with boundary . for example , this can be seen by considering the nucleus @xmath12 of the elliptic surface @xmath13 i.e. a regular neighborhood of a cusp fiber and a section , cf . @xcite . let @xmath14 be the result of a log transform with multiplicity @xmath15 ( @xmath16 , odd ) on a regular fiber @xmath17 , cf . * section 3.3 ) . then @xmath18 and @xmath19 are homeomorphic 4-manifolds ( with @xmath20 ) , having as boundary the brieskorn sphere @xmath21 however , they are not diffeomorphic : this can be shown using the donaldson or seiberg - witten invariants ( see @xcite , @xcite , @xcite ) , but also by comparing the hat heegaard floer invariants @xmath22 and @xmath23 , where @xmath24 and @xmath25 are the cobordisms from @xmath26 to @xmath27 obtained by deleting a @xmath11-ball from @xmath18 and @xmath19 , respectively . indeed , the arguments of fintushel - stern @xcite and szab - stipsicz @xcite can be easily adapted to show that @xmath24 and @xmath25 have different hat heegaard floer invariants ; one needs to use the computation of @xmath28 due to ozsvth - szab @xcite , and the rational blow - down formula of roberts @xcite . ( it is worth noting that the maps @xmath29 give no nontrivial information for closed 4-manifolds , cf . @xcite ; exotic structures on those can be detected with the mixed heegaard floer invariants of @xcite . ) the original definitions of the vector spaces @xmath30 and the maps @xmath29 involved counting pseudoholomorphic disks and triangles in symmetric products of riemann surfaces ; the riemann surfaces are related to the three - manifolds and cobordisms involved via heegaard diagrams . in @xcite , sarkar and the third author showed that every three - manifold admits a heegaard diagram that is nice in the following sense : the curves split the diagram into elementary domains , all but one of which are bigons or rectangles . using such a diagram , holomorphic disks in the symmetric product can be counted combinatorially , and the result is a combinatorial description of @xmath31 for any @xmath32 as well as of the hat version of heegaard floer homology of null homologous knots and links in any three - manifold @xmath1 . a similar result was obtained in @xcite for all versions of the heegaard floer homology of knots and links in the three - sphere . the goal of this paper is to give a combinatorial procedure for calculating the ranks of the maps @xmath33 when @xmath24 is a cobordism between @xmath6 and @xmath7 with the property that the induced maps @xmath34 and @xmath35 are surjective . note that this case includes all cobordisms for which @xmath36 is torsion , as well as all those consisting of only 2-handle additions . roughly , the computation of the ranks of @xmath33 goes as follows . the cobordism @xmath24 is decomposed into a sequence of one - handle additions , two - handle additions , and three - handle additions . using the homological hypotheses on the cobordism and the @xmath37-action on the heegaard floer groups we reduce the problem to the case of a cobordism map corresponding to two - handle additions only . then , given a cobordism made of two - handles , we show that it can be represented by a multi - pointed triple heegaard diagram of a special form , in which all elementary domains that do not contain basepoints are bigons , triangles , or rectangles . in such diagrams all holomorphic triangles of maslov index zero can be counted algorithmically , thus giving a combinatorial description of the map on @xmath38 we remark that in order to turn @xmath30 into a fully combinatorial tqft ( at least for cobordisms satisfying our hypothesis ) , one ingredient is missing : naturality . given two different nice diagrams for a three - manifold , the results of @xcite show that the resulting groups @xmath30 are isomorphic . however , there is not yet a combinatorial description of this isomorphism . thus , while the results of this paper give an algorithmic procedure for computing the rank of a map @xmath39 the map itself is determined combinatorially only up to automorphisms of the image and the target . in fact , if one were to establish naturality , then one could automatically remove the assumption on the maps on @xmath40 , and compute @xmath33 for any @xmath24 , simply by composing the maps induced by the two - handle additions ( computed in this paper ) with the ones induced by the one- and three - handle additions , which are combinatorial by definition , cf . @xcite . the paper is organized as follows . in section [ sec : triangles ] , we define a multi - pointed triple heegaard diagram to be nice if all non - punctured elementary domains are bigons , triangles , or rectangles , and show that in a nice diagram holomorphic triangles can be counted combinatorially . below ) in @xcite , using slightly different methods . ] we then turn to the description of the map induced by two - handle additions . for the sake of clarity , in section [ sec : two ] we explain in detail the case of adding a single two - handle : we show that its addition can be represented by a nice triple heegaard diagram with a single basepoint and , therefore , the induced map on @xmath30 admits a combinatorial description . we then explain how to modify the arguments to work in the case of several two - handle additions . this modification uses triple heegaard diagrams with several basepoints . in section [ sec : last ] , we discuss the additions of one- and three - handles , and put the various steps together . finally , in section [ sec : example ] we present the example of @xmath41 surgery on the trefoil . throughout the paper all homology groups are taken with coefficients in @xmath42 unless otherwise noted . we would like to thank peter ozsvth and zoltn szab for helpful conversations and encouragement . in particular , several key ideas in the proof were suggested to us by peter ozsvth . this work was done while the third author was an exchange graduate student at columbia university . he is grateful to the columbia math department for its hospitality . he would also like to thank his advisors , robion kirby and peter ozsvth , for their continuous guidance and support . finally , we would like to thank the referees for many helpful comments , and particularly for finding a critical error in section [ sec : last ] of a previous version of this paper . the goal of this section is to show that under an appropriate condition ( `` niceness '' ) on triple heegaard diagrams , the counts of holomorphic triangles in the symmetric product are combinatorial . we start by reviewing some facts from heegaard floer theory . a triple heegaard diagram @xmath43 consists of a surface @xmath44 of genus @xmath45 together with three @xmath46-tuples of pairwise disjoint embedded curves @xmath47 @xmath48 in @xmath44 such that the span of each @xmath46-tuple of curves in @xmath49 is @xmath45-dimensional . if we forget one set of curves ( for example @xmath50 ) , the result is an ( ordinary ) heegaard diagram @xmath51 by the condition on the spans , @xmath52 , @xmath53 and @xmath54 each has @xmath55 connected components . by a multi - pointed triple heegaard diagram @xmath56 , then , we mean a triple heegaard diagram @xmath57 as above together with a set @xmath58 of @xmath55 points in @xmath44 so that exactly one @xmath59 lies in each connected component of @xmath52 , @xmath53 and @xmath54 . to a heegaard diagram @xmath60 one can associate a three - manifold @xmath61 . to a triple heegaard diagram @xmath62 , in addition to the three - manifolds @xmath61 , @xmath63 and @xmath64 , one can associate a four - manifold @xmath65 such that @xmath66 ; see @xcite . associated to a three - manifold @xmath1 is the heegaard floer homology group @xmath31 . this was defined using a heegaard diagram with a single basepoint in @xcite . in @xcite , ozsvth and szab associated to the data @xmath67 , called a multi - pointed heegaard diagram , a floer homology group @xmath68 by counting holomorphic disks in @xmath69 with boundary on the tori @xmath70 and @xmath71 it is not hard to show that @xmath72 ( here , @xmath73 is the @xmath74-torus , and @xmath75 means ordinary ( singular ) homology . ) the decomposition is not canonical : it depends on a choice of paths in @xmath44 connecting @xmath59 to @xmath76 for @xmath77 . the heegaard floer homology groups decompose as a direct sum over @xmath2-structures on @xmath61 , @xmath78 more generally , there is a decomposition @xmath79 associated to the triple heegaard diagram @xmath80 together with a @xmath2-structure @xmath8 on @xmath65 , is a map @xmath81 the definition involves counting holomorphic triangles in @xmath82 with boundary on @xmath83 , @xmath84 and @xmath85 , cf . @xcite and @xcite . two triple heegaard diagrams are called _ strongly equivalent _ if they differ by a sequence of isotopies and handleslides . it follows from ( * ? ? ? * proposition 8.14 ) , the associativity theorem ( * ? ? ? * theorem 8.16 ) , and the definition of the handleslide isomorphisms that strongly equivalent triple heegaard diagrams induce the same map on homology . call a @xmath86-pointed triple heegaard diagram _ split _ if it is obtained from a singly - pointed heegaard triple diagram @xmath87 by attaching ( by connect sum ) @xmath74 spheres with one basepoint and three isotopic curves ( one alpha , one beta and one gamma ) each , to the component of @xmath88 containing @xmath89 . we call @xmath87 the _ reduction _ of the split diagram . the following lemma is a variant of ( * ? ? ? * proposition 3.3 ) . [ lemma : splitdiagrams ] every triple heegaard diagram @xmath80 is strongly equivalent to a split one . reorder the alpha circles so that @xmath90 are linearly independent . let @xmath91 be the connected component of @xmath52 containing @xmath92 . since @xmath90 are linearly independent , one of the curves @xmath93 , @xmath94 , must appear in the boundary of @xmath91 with multiplicity exactly @xmath95 . by handlesliding this curve over the other boundary components of @xmath91 , we can arrange that the resulting @xmath93 bounds a disk containing only @xmath92 . repeat this process with the other @xmath59 , @xmath96 , being sure to use a different alpha curve in the role of @xmath93 at each step . we reorder the curves @xmath97 so that @xmath93 encircles @xmath98 . now repeat the entire process for the beta and gamma curves . finally , choose a path @xmath99 in @xmath44 from @xmath59 to @xmath76 for each @xmath100 . move the configuration @xmath101 along the path @xmath99 by handlesliding ( around @xmath93 , @xmath102 or @xmath103 ) the other alpha , beta and gamma curves that are encountered along the path @xmath99 . the result is a split triple heegaard diagram . note that its reduction is obtained from the original diagram @xmath80 by simply forgetting some of the curves and basepoints . the maps from are compatible with the isomorphism , in the following sense . given a triple heegaard diagram @xmath80 , let @xmath87 be the reduction of a split diagram strongly equivalent to @xmath80 . then the following diagram commutes : @xmath104^ { } \ar[d]^{\cong } & \hf\left(y_{\alpha,\gamma } , { \mathfrak{t}}|_{y_{\alpha,\gamma}}\right)\otimes h_*\left(t^k\right)\ar[d]^{\cong}\\ \hf(\sigma,\alphas,\betas,{\mathbf{z}},{\mathfrak{t}}|_{y_{\alpha,\beta } } ) \otimes\hf(\sigma,\betas,\gammas,{\mathbf{z } } , { \mathfrak{t}}|_{y_{\beta,\gamma } } ) \ar[r]^(.62){{\hat{f}}_{\sigma,\alphas,\betas,\gammas,{\mathbf{z}},{\mathfrak{t } } } } & \hf(\sigma,\alphas,\gammas,{\mathbf{z}},{\mathfrak{t}}|_{y_{\alpha,\gamma } } ) . } \ ] ] here , the map in the first row is @xmath105 on the @xmath30-factors and the usual intersection product @xmath106 on the @xmath75-factors . the vertical isomorphisms are induced by the strong equivalence . the proof that the diagram commutes follows from the same ideas as in @xcite : each of the @xmath74 spherical pieces in the split diagram contributes a @xmath107 to the @xmath108 factors above ; moreover , a local computation shows that the triangles induce intersection product maps @xmath109 which tensored together give the intersection product on @xmath110 it is tempting to assert that the map @xmath111 induced by a singly - pointed triple heegaard diagram depends only on @xmath65 and @xmath8 . however , this seems not to be known . fix a triple heegaard diagram @xmath112 as above . the complement of the @xmath113 curves in @xmath44 has several connected components , which we denote by @xmath114 and call _ elementary domains_. the _ euler measure _ of an elementary domain @xmath115 is @xmath116 a _ domain _ in @xmath44 is a two - chain @xmath117 with @xmath118 its euler measure is simply @xmath119 as mentioned above , the maps @xmath120 induced by the triple heegaard diagram @xmath121 are defined by counting holomorphic triangles in @xmath122 with respect to a suitable almost complex structure . according to the cylindrical formulation from @xcite , this is equivalent to counting certain holomorphic embeddings @xmath123 where @xmath124 is a riemann surface ( henceforth called the source ) with some marked points on the boundary ( which we call corners ) , and @xmath125 is a fixed disk with three marked points on the boundary . the maps @xmath126 are required to satisfy certain boundary conditions , and to be generically @xmath46-to-@xmath95 when post - composed with the projection @xmath127 more generally , we will consider such holomorphic maps @xmath128 which are generically @xmath129-to-@xmath95 when post - composed with @xmath130 these correspond to holomorphic triangles in @xmath131 where @xmath129 can be any positive integer . we will be interested in the discussion of the index from @xcite . although this discussion was carried - out in the case @xmath132 , @xmath133 , it applies equally well in the case of arbitrary @xmath74 and @xmath129 with only notational changes . in the cylindrical formulation , one works with an almost complex structure on @xmath134 so that the projection @xmath135 is holomorphic , and the fibers of @xmath136 are holomorphic . it follows that for @xmath137 holomorphic , @xmath138 is a holomorphic branched cover . the map @xmath139 need not be holomorphic , but since the fibers are holomorphic , @xmath139 is a branched map . fix a model for @xmath125 in which the three marked points are @xmath140 corners , and a conformal structure on @xmath44 with respect to which the intersections between alpha , beta and gamma curves are all right angles . since @xmath126 is holomorphic , the conformal structure on @xmath124 is induced via @xmath138 from the conformal structure on @xmath125 . it makes sense , therefore , to talk about branch points of @xmath139 on the boundary and at the corners , as well as in the interior . generically , while there may be branch points of @xmath139 on the boundary of @xmath124 , there will not be branch points at the corners . suppose @xmath128 is as above . denote by @xmath141 the projection to @xmath142 there is an associated domain @xmath143 in @xmath144 where the coefficient of @xmath145 in @xmath143 is the local multiplicity of @xmath146 at any point in @xmath147 by @xcite , the _ index _ of the linearized @xmath148 operator at the holomorphic map @xmath126 is given by @xmath149 for simplicity , we call this the index of @xmath150 note that , by the riemann - hurwitz formula : @xmath151 where @xmath152 is the ramification index ( number of branch points counted with multiplicity ) of @xmath153 ( here , branch points along the boundary count as half an interior branch point . ) from and we get an alternate formula for the index : @xmath154 note that it is not obvious how to compute @xmath155 from @xmath143 . a combinatorial formula for the index , purely in terms of @xmath143 , was found by sarkar in @xcite . however , we will not use it here . fix a multi - pointed triple heegaard diagram @xmath156 . recall that a domain is a linear combination of connected components of @xmath157 . the _ support _ of a domain is the union of those components with nonzero coefficients . if the support of a domain @xmath158 contains at least one @xmath59 then @xmath158 is called _ punctured _ ; otherwise it is called _ unpunctured_. [ def : nice ] an elementary domain is called * good * if it is a bigon , a triangle , or a rectangle , and * bad * otherwise . the multi - pointed triple heegaard diagram @xmath56 is called * nice * if every unpunctured elementary domain is good . this is parallel to the definition of nice heegaard diagrams ( with just two sets of curves ) from @xcite . a multi - pointed heegaard diagram @xmath159 is called _ nice _ if , among the connected components of @xmath160 all unpunctured ones are either bigons or squares . note that a bigon , a triangle , and a rectangle have euler measure @xmath161 and @xmath162 respectively . since @xmath163 is additive , every unpunctured positive domain ( not necessarily elementary ) in a nice diagram must have nonnegative euler measure . a quick consequence of this is the following : [ lemma : forget ] if @xmath80 is a nice triple heegaard diagram , then if we forget one set of curves ( for example , @xmath50 ) , the resulting heegaard diagram @xmath159 is also nice . in order to define the triangle maps it is necessary to assume the triple heegaard diagram is weakly admissible in the sense of ( * ? ? ? * definition 8.8 ) . in fact , nice diagrams are automatically weakly admissible , cf . corollary [ lemma : admissible ] below . our goal is to give a combinatorial description of the holomorphic triangle counts for nice triple diagrams . [ easternorthodox ] let @xmath164 be a nice multi - pointed triple heegaard diagram . fix a generic almost complex structure @xmath165 on @xmath134 as in ( * ? ? ? * section 10.2 ) . let @xmath166 be a @xmath165-holomorphic map of the kind occurring in the definition of @xmath167 . in particular , assume @xmath126 is an embedding , of index zero , and such that the image of @xmath146 is an unpunctured domain . then @xmath124 is a disjoint union of @xmath129 triangles , and the restriction of @xmath146 to each component of @xmath124 is an embedding . since the image of @xmath146 is unpunctured and positive , we have @xmath168 by , we get @xmath169 this means that at least one component of @xmath124 is topologically a disk . let @xmath170 be such a component . it is a polygon with @xmath171 vertices . we will show that @xmath172 and that @xmath173 is an embedding . let us first show that @xmath170 is a triangle . the index of the @xmath174 operator at a disconnected curve is the sum of the indices of its restrictions to each connected component . therefore , in order for an index zero holomorphic curve to exist generically , the indices at every connected component , and in particular at @xmath175 must be zero . applying to @xmath176 we get @xmath177 if @xmath178 then by we have @xmath179 hence , the map @xmath146 has no interior branch points . if @xmath180 then @xmath170 is mapped locally diffeomorphically by @xmath139 to @xmath44 . the image must have negative euler measure , which is a contradiction . so , suppose @xmath181 . the preimages of the alpha , beta , and gamma curves cut @xmath170 into several connected components . without loss of generality , assume that the boundary branch point is mapped to an alpha circle . then , along the corresponding edge of @xmath170 there is a valence three vertex @xmath182 , as shown in figure [ figure : hexagon ] . let @xmath183 denote the edge in the interior of @xmath170 meeting @xmath182 . since there is only one boundary branch point , the other intersection point of the edge @xmath183 with @xmath184 is along the preimage of a beta or gamma circle . it follows that one of the connected components @xmath185 of @xmath186 is a hexagon or heptagon . smoothing the vertex @xmath182 of @xmath185 we obtain a pentagon or hexagon which is mapped locally diffeomorphically by @xmath139 to @xmath44 . the image , then , has negative euler measure , again a contradiction . therefore , @xmath172 so @xmath170 is a triangle . furthermore , by , @xmath187 which means that there are no ( interior or boundary ) branch points at all thus , just as in the hypothetical hexagon case above , the preimages of the alpha , beta , and gamma curves must cut @xmath170 into 2-gons , 3-gons , and 4-gons , all of which have nonnegative euler measure . since the euler measure of @xmath170 is @xmath188 there can be no bigons ; in fact , @xmath170 must be cut into several rectangles and exactly one triangle . it is easy to see that the only possible tiling of @xmath170 of this type is as in figure [ fig : tiling ] , with several parallel preimages of segments on the alpha curves , several parallel beta segments , and several parallel gamma segments . we call the type of a segment ( @xmath189 , @xmath190 or @xmath191 ) its color . the tiling consists of one triangle and six different types of rectangles , according to the coloring of their edges in clockwise order ( namely , @xmath192 , @xmath193 , @xmath194 , @xmath195 , @xmath196 , and @xmath197 ) . we claim that the images of the interiors of each of these rectangles by @xmath146 are disjoint . because of the coloring scheme , only rectangles of the same type can have the same image . suppose that two different @xmath192 rectangles from @xmath170 have the same image in @xmath142 ( the cases @xmath193 , @xmath194 are exactly analogous . ) let @xmath198 and @xmath199 be the two rectangles ; suppose that @xmath198 is closer to the central triangle than @xmath200 and @xmath199 is closer to the @xmath189 boundary of @xmath201 because of the way the rectangles are colored , the upper edge of @xmath198 must have the same image as the upper edge of @xmath202 hence the @xmath192 rectangle right above @xmath198 has the same image as the one right above @xmath202 iterating this argument , at some point we get that the central triangle has the same image as some @xmath192 rectangle , which is impossible . now suppose that two different @xmath195 rectangles , @xmath198 and @xmath200 have the same image . ( the cases @xmath196 and @xmath197 are exactly analogous . ) there are two cases , according to whether the upper edge of @xmath198 has the same image as the upper edge of @xmath200 or as the lower edge of @xmath202 suppose first that the upper edge of @xmath198 has the same image as the upper edge of @xmath199 . by the _ @xmath190-height _ of @xmath203 we mean the minimal number of beta arcs that an arc in @xmath204 starting in @xmath203 , going right , and ending at a gamma arc must cross . ( the diagram is positioned in the plane as in figure [ fig : tiling ] . ) since @xmath139 is a local homeomorphism , and @xmath198 and @xmath199 have the same image , it is clear that the @xmath190-height of @xmath198 and the @xmath190-height of @xmath199 are equal . by the _ @xmath189-height _ of @xmath203 we mean the minimal number of alpha arcs that an arc in @xmath205 starting in @xmath206 going up , and ending at a gamma arc must cross . again , it is clear that the @xmath189-heights of @xmath198 and @xmath199 must be equal . but this implies that @xmath198 and @xmath199 are equal . now , suppose that the upper edge of @xmath198 has the same image as the lower edge of @xmath199 . there is a unique rectangle @xmath91 in @xmath124 with boundary contained in @xmath207 , containing @xmath198 and @xmath199 , and with one corner the same as a corner of @xmath198 and the opposite corner the same as a corner of @xmath199 . it is easy to see that @xmath139 maps antipodal points on the boundary of @xmath91 to the same point in @xmath44 . it follows that @xmath208 is a two - fold covering map . but then @xmath139 must have a branch point somewhere inside @xmath91 a contradiction . see figure [ fig : grid ] . finally , suppose some arc @xmath209 on @xmath210 has the same image as some other arc @xmath211 in @xmath124 . if @xmath211 is in the interior of @xmath124 then any rectangle ( or triangle ) adjacent to @xmath209 has the same image as some rectangle ( or triangle ) adjacent to @xmath211 . we have already ruled this out . if @xmath211 is on @xmath210 , then either any rectangle ( or triangle ) adjacent to @xmath209 has the same image as some rectangle ( or triangle ) adjacent to @xmath211 or there is a branch point somewhere on @xmath210 . we have already ruled out both of these cases . we have thus established that @xmath170 is an embedded triangle . by forgetting @xmath170 , we obtain a holomorphic map to @xmath212 still of index zero , but such that its post - composition with @xmath135 is generically @xmath213-to-@xmath95 rather than @xmath129-to-@xmath214 the result then follows by induction on @xmath215 observe that , in proposition [ easternorthodox ] above , even though each of the @xmath129 triangles is embedded , some of their domains may overlap . it turns out that they may do so only in a specific way , however : [ allow ] suppose @xmath209 is an index zero homology class represented by a union of embedded holomorphic triangles , in a nice triple diagram . suppose the union of triangles corresponds to an embedded holomorphic curve in @xmath216 . then any two triangles in @xmath209 are either disjoint in @xmath44 or overlap in @xmath44 `` head to tail '' as shown in figure [ fig : overlap ] . let @xmath217 and @xmath218 be two of the triangles in the domain @xmath209 . for a generic representative of @xmath209 to exist , the pair must also have index zero , and be embedded in @xmath134 . we already know that @xmath217 and @xmath218 are tiled as in figure [ fig : tiling ] . this strongly restricts how @xmath217 and @xmath218 can overlap . one way for @xmath217 and @xmath218 to overlap is for @xmath217 to be entirely contained inside @xmath218 . in this case , it is not hard to see that the two holomorphic triangles in @xmath134 intersect in one interior point . indeed , the intersection number of two holomorphic curves in a @xmath11-manifold is invariant in families . if we deform the heegaard diagram so that the boundary of @xmath217 in @xmath44 is a single point ( i.e. , the alpha , beta and gamma circles involved in @xmath219 intersect in an asterisk , with vertex the `` triangle '' @xmath217 ) then obviously @xmath220 is a single point . it follows that the same is true for the original triangles @xmath217 and @xmath218 . another way that @xmath217 and @xmath218 might overlap is `` head to head '' as shown on the left side of figure [ fig : headhead ] . it is then possible to decompose @xmath221 into a pair of rectangles @xmath91 and @xmath222 , and two new embedded triangles @xmath217 and @xmath218 , as shown in figure [ fig : headhead ] . an immersed rectangle in @xmath44 has index at least @xmath95 , since it admits a generic holomorphic representative . so , each of @xmath223 and @xmath224 has index at least @xmath95 . similarly , the pair of triangles @xmath225 has index at least @xmath226 . so , by additivity of the index , the whole domain has index at least @xmath227 a contradiction . using these two observations , and the rulings of @xmath217 and @xmath218 , it is then elementary to check that the only possible overlap in index zero is `` head to tail '' as in figure [ fig : overlap ] . it follows that , for a nice heegaard diagram , we can combinatorially describe the generic holomorphic curves of index @xmath226 . if @xmath158 is the domain of a generic holomorphic curve of index @xmath226 then @xmath228 has @xmath129 components , each of which bounds an embedded triangle in @xmath44 . each pair of triangles must either be disjoint or overlap as shown in figure [ fig : overlap ] . any such @xmath158 clearly has a unique holomorphic representative with respect to a split complex structure @xmath229 on @xmath134 . further , it is well known that these holomorphic curves are transversally cut out , and so persist if one takes a small perturbation of @xmath229 . in summary , to count index zero holomorphic curves in @xmath134 with respect to a generic perturbation of the split complex structure , it suffices to count domains @xmath158 which are sums of @xmath129 embedded triangles in @xmath44 , overlapping as allowed in the statement of lemma [ allow ] . in ( * ? ? ? * definition 4.2 ) , ozsvth and szab associate to a four - dimensional cobordism @xmath24 consisting of two - handle additions certain kinds of triple heegaard diagrams . the cobordism @xmath24 from @xmath6 to @xmath7 corresponds to surgery on some framed link @xmath230 denote by @xmath231 the number of components of @xmath232 . fix a basepoint in @xmath6 . let @xmath233 be the union of @xmath232 with a path from each component to the basepoint . the boundary of a regular neighborhood of @xmath233 is a genus @xmath231 surface , which has a subset identified with @xmath231 punctured tori @xmath234 , one for each link component . a singly - pointed triple heegaard diagram @xmath235 is called _ subordinate to @xmath233 _ if * @xmath236 describes the complement of @xmath233 , * @xmath237 are small isotopic translates of @xmath238 , * after surgering out the @xmath239 , the induced curves @xmath102 and @xmath103 ( for @xmath240 ) lie in the punctured torus @xmath234 . * for @xmath240 , the curves @xmath102 represent meridians for the link components , disjoint from all @xmath241 for @xmath242 , and meeting @xmath103 in a single transverse intersection point . * for @xmath240 , the homology classes of the @xmath103 correspond to the framings of the link components . a related construction is as follows . given @xmath6 and @xmath232 as above , choose a multi - pointed heegaard diagram @xmath243 for @xmath244 as in @xcite , of some genus @xmath45 ; here , @xmath245 and @xmath246 . precisely , the sets @xmath247 and @xmath248 are collections of distinct points on @xmath44 disjoint from the alpha and the beta curves , with the following two properties : first , each connected component of @xmath249 and @xmath250 contains a single @xmath59 and a corresponding @xmath251 . second , if @xmath252 is an @xmath231-tuple of embedded arcs in @xmath250 connecting @xmath59 to @xmath251 ( @xmath253 ) , and @xmath254 is an @xmath231-tuple of embedded arcs in @xmath249 connecting @xmath59 to @xmath251 ( @xmath253 ) then the link @xmath232 is the union of small push offs of @xmath252 and @xmath254 into the two handlebodies ( induced by the beta and alpha curves , respectively ) . next , we attach handles @xmath255 to @xmath256 connecting @xmath59 to @xmath251 ( for @xmath253 ) , and obtain a new surface @xmath44 . we choose a new @xmath102 ( @xmath257 ) to be the belt circle of the handle @xmath258 , and a new @xmath93 to be the union of the core of @xmath258 with @xmath259 , so @xmath93 intersects @xmath102 in one point . choose @xmath103 ( @xmath260 ) to be a small isotopic translate of @xmath102 , intersecting @xmath102 in two points . let @xmath261 ( @xmath257 ) be the union of @xmath262 with a core of the handle @xmath258 . obtain @xmath103 ( @xmath257 ) by applying dehn twists to @xmath261 around @xmath102 ; the framing of the link is determined by the number of dehn twists . in this fashion we obtain a multi - pointed triple heegaard diagram @xmath263 with @xmath264 curves of each kind . note that @xmath67 is an @xmath231-pointed heegaard diagram for @xmath6 , @xmath265 is a @xmath231-pointed heegaard diagram for @xmath7 , and @xmath266 is a @xmath231-pointed heegaard diagram for @xmath267 . the new circles @xmath93 , @xmath257 , are part of a maximal homologically linearly independent subset of @xmath268 , and similarly for @xmath102 and @xmath103 ( @xmath257 ) . consequently , by the proof of lemma [ lemma : splitdiagrams ] , there is a split diagram strongly equivalent to @xmath269 whose reduction @xmath87 is obtained from @xmath270 by forgetting @xmath271 of the @xmath93 ( respectively @xmath102 , @xmath103 ) , @xmath272 , as well as @xmath273 . it is then not hard to see that @xmath87 is a triple heegaard diagram subordinate to a bouquet for @xmath232 . in this section we will show that for any two - handle addition , one can construct a _ triple heegaard diagram strongly equivalent to a diagram @xmath274 as above . this involves finessing the diagram for the link @xmath232 inside @xmath1 and then , after adding the handles and the new curves , modifying the diagram in several steps to make it nice . for the most part we will focus on the case when we add a single two - handle . in the last subsection we will explain how the arguments generalize to several two - handles . to keep language concise , in this section we will refer to elementary domains as _ let @xmath159 be a multi - pointed heegaard diagram . recall that the diagram is called nice if all unpunctured regions are either bigons or squares . [ lemma : bad_region_adjacent ] on a nice heegaard diagram @xmath159 , for any alpha circle @xmath93 with an arbitrary orientation , there exists a punctured region @xmath158 which contains an edge @xmath163 belonging to @xmath93 , and such that @xmath158 is on the left of @xmath93 . the same conclusion holds for each beta circle . suppose a half - neighborhood on the left of the alpha circle @xmath93 is disjoint from all the punctured regions . then immediately to the left of @xmath93 we only have good regions . there are two possibilities as indicated in figure [ fig : bad_region_adjacent ] . if there is a bigon region on the left of @xmath275 then the other edge is some beta edge @xmath276 . the region on the other side of @xmath276 must be a bigon region or a square since otherwise we would have a punctured region on the left of @xmath93 . if we reach a square , we continue to consider the next region . eventually we will reach a bigon region since the number of regions are finite and we will not reach the same region twice . all regions involved form a disk bounded by @xmath93 ( as in figure [ fig : bad_region_adjacent ] ( a ) ) . in particular , this means @xmath93 is null homologous . this contradicts the fact that the @xmath277 alpha circles represent linearly independent classes in @xmath278 in the second case , there are no bigon regions . then on the left of @xmath93 , we see a chain of squares , as in figure [ fig : bad_region_adjacent ] ( b ) . the opposite edges on these squares give another alpha circle , say @xmath279 . then @xmath93 and @xmath279 are homologous to each other in @xmath278 this contradicts the same fact as in the previous case . recall that in order to define the triangle maps it is necessary for the triple heegaard diagram to be weakly admissible in the sense of ( * ? ? ? * definition 8.8 ) . [ lemma : admissible ] if @xmath80 is a nice multi - pointed triple heegaard diagram then @xmath80 is weakly admissible . by definition , the diagram is weakly admissible if there are no nontrivial domains @xmath158 supported in @xmath280 with nonnegative multiplicity in all regions , and whose boundary is a linear combination of alpha , beta , and gamma curves . suppose such a domain @xmath158 exists , and consider a curve appearing with a nonzero multiplicity in @xmath281 without loss of generality , we can assume this is an alpha curve , and all regions immediately to its left have positive multiplicity in @xmath282 by lemma [ lemma : forget ] , the diagram @xmath67 is nice . lemma [ lemma : bad_region_adjacent ] now gives a contradiction . let @xmath283 be a three - manifold together with a knot @xmath284 . we choose a singly pointed heegaard diagram @xmath285 for @xmath1 together with an additional basepoint @xmath286 such that the two basepoints determine the knot as in @xcite . after applying the algorithm from @xcite to the heegaard diagram , we can assume that the heegaard diagram is nice , with @xmath287 the ( usually bad ) region containing the basepoint @xmath288 furthermore , the algorithm in @xcite also ensures that @xmath287 is a polygon . we denote by @xmath289 the region containing @xmath290 note that either @xmath291 or @xmath289 is good . throughout this section , we will suppose that @xmath289 and @xmath287 are two different regions , and that @xmath289 is a rectangle . the case when @xmath291 corresponds to surgery on the unknot , which is already well understood . the case when @xmath289 is a bigon can be avoided by modifying the original diagram by a finger move . ( alternately , this case can be treated similarly to the case that @xmath289 is a rectangle . ) let @xmath24 be the four manifold with boundary obtained from @xmath292 $ ] by adding a two handle along @xmath293 in @xmath294 , with some framing . @xmath24 gives a cobordism between @xmath1 and @xmath295 , where @xmath295 is obtained from @xmath1 by doing the corresponding surgery along @xmath293 . now we are ready to describe our algorithm to get a nice triple heegaard diagram for the cobordism @xmath296 let @xmath252 be an embedded arc in @xmath44 connecting @xmath297 and @xmath298 in the complement of beta curves , and @xmath254 be an embedded arc connecting @xmath297 and @xmath298 in the complement of alpha curves . the union of @xmath252 and @xmath254 is a projection of the knot @xmath284 to the surface @xmath144 where @xmath44 is viewed as a heegaard surface in @xmath299 for convenience , we will always assume that @xmath252 and @xmath254 do not pass through any bigon regions , and never leave a rectangle by the same edge through which they entered ; this can easily be achieved . in this step , we modify the doubly pointed heegaard diagram @xmath300 to make @xmath301 embedded in @xmath144 while preserving the niceness of the heegaard diagram . typically , @xmath252 and @xmath254 have many intersections . we modify the diagram inductively by stabilization at the first intersection @xmath302 on @xmath254 ( going from @xmath297 to @xmath298 ) to remove that intersection , while making sure that the new diagram is still nice . a neighborhood of @xmath252 and the part on @xmath254 from @xmath297 to @xmath15 are shown in figure [ fig : knot_embedded_before ] . in the same picture , if we continue the chain of rectangles containing @xmath252 , we will end up with a region @xmath303 which is either a bigon or the punctured region @xmath287 . to get rid of the intersection point @xmath15 , we stabilize the diagram as in figure [ fig : knot_embedded_after ] . more precisely , we do a stabilization followed some handleslides of the beta curves and an isotopy of the new beta curve . after these moves , the number of intersection points decreases by one and the diagram is still nice . if we iterate this process , in the end we get a nice heegaard diagram in which @xmath252 and @xmath254 only intersect at their endpoints . furthermore , the bad region @xmath287 is still a polygon . our goal in steps 2 and 3 is to describe a particular triple heegaard diagram for the cobordism @xmath24 . starting with the alpha and the beta curves we already have , for each beta curve @xmath102 we will add a gamma curve @xmath103 ( called its twin ) which is isotopic to @xmath102 and intersects it in exactly two points . ( after this , we will add some more curves in the next step . ) for any beta curve @xmath102 , by lemma [ lemma : bad_region_adjacent ] we can choose a region @xmath145 so that @xmath145 is adjacent to the punctured region @xmath287 with the common edge on @xmath102 . if @xmath304 , then we add @xmath103 close and parallel to @xmath102 as in figure [ fig : adding_gamma_curve_simple ] ( a ) , and make a finger move as in figure [ fig : adding_gamma_curve_simple ] ( b ) . here and after , without further specification , we make the convention that the thick arcs are alpha arcs , the thin ones are beta arcs , and the interrupted ones are gamma arcs . suppose now that @xmath145 is different from @xmath287 . then @xmath145 has to be a good region . if @xmath145 is not a bigon , since the complement of the beta curves in @xmath44 is connected , we can connect @xmath145 with @xmath287 without intersecting beta curves , via an arc traversing a chain of rectangles , as indicated in the figure [ fig : adding_gamma_curve_before ] . then we do a finger move of the curve @xmath102 as indicated in figure [ fig : adding_gamma_curve_after ] . note that the knot remains embedded in @xmath44 . now we have a bigon region . we then add the gamma curve @xmath103 as shown in figure [ fig : adding_gamma_curve_after ] . note that for each pair @xmath102 and @xmath103 , we either have one sub - diagram of the form in figure [ fig : adding_gamma_curve_simple ] ( b ) , or one sub - diagram of the form in figure [ fig : beta_gamma_bigon ] . observe also that during this process , no bad region other than @xmath287 is created . after step 2 , the knot is still embedded in the heegaard diagram . in other words , we can use arcs to connect @xmath297 to @xmath298 by paths in the complement of alpha curves , and in the complement of beta curves so that the two arcs do not intersect except for the end points @xmath298 and @xmath297 , and do not pass through any bigons . we will see two chains of squares , as indicated in figure [ fig : stabilization ] . we do a stabilization of the heegaard diagram by adding a handle with one foot in each of @xmath287 and @xmath289 . we add the additional beta circle @xmath305 to be the meridian of the handle , which we push along @xmath252 until it reaches @xmath306 we also push @xmath305 through the opposite alpha edge of @xmath289 , into the adjacent region . then , we connect the two feet in the complement of alpha curves along @xmath254 and get a new alpha circle @xmath307 . finally , we add the surgery gamma circle @xmath308 as in figure [ fig : stabilization ] . the result is a triple heegaard diagram ( with @xmath309 curves ) which represents surgery along the knot @xmath284 , with a particular framing ; the framing is the sum of the number of twists of @xmath308 around the handle and a constant depending only on the original heegaard diagram . note that , depending on the framing , the local picture around the two feet of the handle may also look like figure [ fig : stabilization_anti ] , in which case instead of the octagon region @xmath310 from figure [ fig : stabilization ] we have two hexagon bad regions @xmath311 and @xmath312 . after the stabilization , @xmath307 and @xmath308 separate @xmath287 into several regions ; among these , @xmath313 and @xmath314 are ( possibly ) bad but all other regions are good . we end up with a diagram with four ( or five ) bad regions : @xmath313 , @xmath314 , @xmath310 ( or @xmath311 and @xmath312 ) , and @xmath303 . ( in some cases , @xmath313 or @xmath314 might be good , or , if there is little winding of @xmath315 some of @xmath310 , @xmath316 and @xmath314 might coincide . the argument in these cases is a simple adaptation of the one we give below . ) we will kill the badness of @xmath317 @xmath318 and @xmath310 ( or @xmath311 and @xmath312 ) , while the region @xmath313 will be the one containing the basepoint @xmath297 for our final triple heegaard diagram . we push the finger in @xmath303 across the opposite alpha edge until we reach a bigon , @xmath319 or a region of type @xmath320 as in figure [ fig : beta_gamma_bigon ] . in this case ( figure [ fig : dprime_bigon ] ( a ) ) , our finger move will kill the badness of @xmath303 , as indicated in figure [ fig : dprime_bigon ] ( b ) , and does not create any new bad regions . this is completely similar to case 1 . the finger move kills the badness of @xmath303 , and does not create any new bad regions . let us suppose the topmost region in figure [ fig : beta_gamma_bigon ] is @xmath313 . the case when the topmost region is @xmath314 is completely similar . the regions involved look like figure [ fig : dprime_ds_bad ] . if on the left @xmath313 is on top of @xmath314 , we isotope the diagram to look as in figure [ fig : dprime_ds_nice ] . the case when on the left of figure [ fig : beta_gamma_bigon ] @xmath314 is on top of @xmath313 is similar , except that we do the double finger move on the other side of @xmath305 . we have now killed the badness of @xmath303 . if there are any bigons between beta and gamma curves adjacent to @xmath314 as in figure [ fig : beta_gamma_bigon ] or figure [ fig : adding_gamma_curve_simple ] ( b ) , also shown in figures [ fig : dw2_special_handleslide ] ( a ) resp . ( c ) , we do a `` handleslide '' ( more precisely , a handleslide followed by an isotopy ) of @xmath308 over each @xmath103 ( @xmath322 ) involved as indicated in figures [ fig : dw2_special_handleslide ] ( b ) resp . the intersection of @xmath308 and @xmath307 has the pattern as figure [ fig : dw2_two_patterns ] ( a ) or ( b ) . in case ( a ) , we do nothing . in case ( b ) , we do the finger move as in figure [ fig : dw2_two_patterns ] ( c ) . now among the possibly bad regions generated from @xmath314 , we have a unique one whose boundary has an intersection of a beta curve with a gamma curve , namely the one near @xmath305 as in figure [ fig : dw2_beta_gamma_crossing ] ( a ) . ( see also figures [ fig : stabilization ] and [ fig : dprime_ds_nice ] . ) we then do a finger move as in figure [ fig : dw2_beta_gamma_crossing ] ( b ) . note that this finger move will not create any badness other than that of @xmath313 . after these special handleslides and finger moves , the region @xmath314 is divided into several possibly bad regions @xmath323 these bad regions are all adjacent to @xmath313 via arcs on @xmath308 and , furthermore , there are no intersection points of beta and gamma curves on their boundaries . we seek to kill the badness of @xmath324 using the algorithm in @xcite . the algorithm there consisted of inductively decreasing a complexity function defined using the unpunctured bad regions . in our situation , we apply a simple modification of the algorithm to the heegaard diagram made of the alpha and the gamma curves ; the modification consists of the fact that we do not deal with the bad region(s ) @xmath310 ( or @xmath311 and @xmath312 ) , but rather only seek to eliminate the badness of @xmath325 thus , in the complexity function we do not include terms that involve the badness and distance of @xmath310 ( or @xmath311 and @xmath312 ) . since all the @xmath326 s are adjacent to the preferred ( punctured ) region @xmath313 via arcs on @xmath315 the algorithm in @xcite prescribes doing finger moves of @xmath308 through alpha curves , and ( possibly ) handleslides of @xmath308 over other gamma curves . we do all these moves in such a way as not to tamper with the arrangements of twin beta - gamma curves , i.e. as not to introduce any new intersection points between @xmath308 and @xmath327 for any @xmath328 . ( in other words , we can think of fattening @xmath329 before applying the algorithm , so that they include their respective twin beta curves . ) in particular , regions of type @xmath320 are treated as bigons . the fact that the algorithm in @xcite can be applied in this fashion is based on the following two observations : * our fingers or handleslides will not pass through the regions adjacent to @xmath305 , except possibly @xmath313 itself . ( this is one benefit of the modification performed in figure [ fig : dw2_beta_gamma_crossing ] . ) * we will not reach any squares between @xmath102 and @xmath330 nor the `` narrow '' squares created in figure [ fig : dw2_special_handleslide ] . in the end , all the badness of @xmath324 is killed . we arrive at a heegaard diagram which might still have some bad regions coming from regions of type @xmath331 as in figure [ fig : dw2_special_bad_region ] ( a ) . we kill these bad regions using the finger moves indicated in figure [ fig : dw2_special_bad_region ] ( b ) . after these moves , the only remaining bad regions are @xmath310 ( or @xmath311 and @xmath312 ) , and the preferred bad region @xmath332 our remaining task is to kill the badness of @xmath310 or @xmath333 . recall that depending on the pattern of the intersection of @xmath307 and @xmath308 ( cf . figure [ fig : dw2_two_patterns ] ) , there are two cases : either we have an octagon bad region @xmath310 , or two hexagon bad regions @xmath311 and @xmath312 . in the first case , one possibility is that a neighborhood of @xmath334 looks as in figure [ fig : dstar_bad ] . we then do the finger moves indicated in figure [ fig : dstar_nice ] . it is routine to check that the new diagram is isotopic to the one in figure [ fig : dstar_bad ] . similarly , in the second case , one possibility is that a neighborhood of @xmath334 looks as in figure [ fig : dstar_bad_case2 ] . in this case , we do the finger moves indicated in figure [ fig : dstar_nice_case2 ] . however , the actual picture on the heegaard diagram may differ from figure [ fig : dstar_bad ] or [ fig : dstar_bad_case2 ] in several ( non - essential ) ways . one possible difference is that at the bottom of the figure [ fig : dstar_bad_case2 ] , the extra gamma curve on top of @xmath313 might be on the right rather than on the left ; however , we can still push the two fingers starting from @xmath313 on each side of @xmath307 . another possible difference is that at the very left of figures [ fig : dstar_bad ] and [ fig : dstar_bad_case2 ] , the curve @xmath308 may have an upward rather than a downward hook , i.e. look as in figure [ fig : dw_1and2_switched](c ) rather than ( a ) . if so , instead of the beta finger from the left in figures [ fig : dstar_bad ] and [ fig : dstar_bad_case2 ] ( cf . also figure [ fig : dw_1and2_switched](b ) ) , we push a beta - gamma finger as in figure [ fig : dw_1and2_switched](d ) . finally , instead of the situations shown in figures [ fig : dstar_bad ] and [ fig : dstar_bad_case2 ] , we might have the same pictures reflected in a horizontal axis . if so , we apply similar finger moves and arrive at the reflections of figures [ fig : dstar_nice ] and [ fig : dstar_nice_case2 ] . in all cases , the finger moves successfully kill the badness of all regions other than @xmath316 in which we keep the basepoint @xmath288 the result is a nice triple heegaard diagram for the cobordism @xmath24 . we now explain how the arguments in this section can be extended to a cobordism @xmath24 which consists of the addition of several two - handles . we view @xmath24 as surgery along a link @xmath335 of @xmath231 components . we start with a multi - pointed heegaard diagram @xmath336 together with another set of basepoints @xmath248 describing the pair @xmath337 as in @xcite . each of the two sets of curves ( @xmath268 and @xmath338 ) has @xmath339 elements . applying the algorithm in @xcite we can make this diagram nice , i.e. such that all regions not containing one of the @xmath297 s are either bigons or rectangles . for @xmath340 we denote by @xmath341 the region containing @xmath342 as in step 1 of section [ sec : boss ] , we inductively remove intersection points between the various components of the projection of @xmath232 to @xmath44 . this projection consists of arcs @xmath262 and @xmath343 with endpoints at @xmath59 and @xmath344 , such that each @xmath262 is disjoint from the beta curves , and each @xmath343 is disjoint from the alpha curves . instead of figure [ fig : knot_embedded_before ] we have the situation in figure [ fig : link_embedded_before ] . again , we stabilize and perform an isotopy to obtain a good diagram with one fewer intersection point , as in figure [ fig : knot_embedded_after ] . iterating this process ( on all link components ) , we can assume that the projection of @xmath232 is embedded in the heegaard surface . we then add twin gamma curves as in step 2 of section [ sec : boss ] . for this we need to do several isotopies of the beta curves as in figure [ fig : adding_gamma_curve_after ] . in that figure , if the region on the top left is @xmath345 the one on the right might be @xmath346 for @xmath347 ; however , the isotopy can be done as before . next , we stabilize the heegaard diagram @xmath231 times ( once for each link component ) to obtain a triple diagram for the cobordism , as in step 3 of section [ sec : boss ] . we then do the analogue of step 4 by pushing @xmath231 fingers to kill the badness of the regions of type @xmath348 . since the fingers only pass through rectangles , they do not intersect each other . the only change is that in figures [ fig : dprime_ds_bad ] and [ fig : dprime_ds_nice ] , the region on the very right may contain a different puncture @xmath59 than the one on the left . by contrast , in case 2 of step 4 , the region on the right in figure [ fig : dprime_bigon ] contains the same puncture as the region on the left , since they lie in the same connected component of the complement of the beta circles . at the end of step 4 , the beta curves split the heegaard surface into @xmath231 connected components @xmath349 . we then do the analogue of step 5 in section [ sec : boss ] . note that this step ( except for the very last bit , figure [ fig : dw2_special_bad_region ] ) only involves moving gamma curves through alpha curves . ( here , we think of the move in figure [ fig : dw2_special_handleslide ] as a single step , rather than as a handleslide followed by an isotopy . ) therefore , we can perform the moves in this step once for each connected component of @xmath232 , independently of each other , because the moves take place in the corresponding component @xmath350 in the situation considered in figure [ fig : dw2_special_bad_region ] , the gamma curves cross a beta curve ; however , the special region @xmath321 is part of a unique @xmath351 , so we can perform the isotopy of the gamma curves as before , without interference from another @xmath352 finally , for step 6 , note that in all the previous steps we have not destroyed the property that the projection of @xmath232 to the heegaard surface is embedded . more precisely , in the part of the stabilized heegaard diagram shown in figure [ fig : stabilization ] , we take a component of the link projection to be a loop starting in @xmath310 , near the upper foot of the handle , going down along @xmath307 until it reaches @xmath313 , then going inside @xmath313 until it reaches the intersection of @xmath308 and @xmath353 and then going along a sub - arc of @xmath308 to its original departure . these paths remain embedded , and disjoint from each other , throughout steps 4 and 5 . ( indeed , neither @xmath305 nor this sub - arc of @xmath308 is moved during these steps . ) it then suffices to note the finger moves in step 6 take place in a neighborhood of the projection of the corresponding link component ( the path considered above ) . therefore , these finger moves can be done without interfering with each other . the result is a nice multi - pointed triple heegaard diagram for @xmath24 . let @xmath24 be a @xmath11-dimensional cobordism from @xmath354 to @xmath355 , and @xmath8 a @xmath2-structure on @xmath24 . choose a self - indexing morse function on @xmath24 . this decomposes @xmath24 as a collection of one - handle additions which together form a cobordism @xmath356 , followed by some two - handle additions forming a cobordism @xmath357 , and three - handle additions forming a cobordism @xmath358 , in this order . let @xmath6 and @xmath7 be the intermediate three - manifolds , so that @xmath359 the map @xmath360 from @xcite is defined as the composition @xmath361 of maps associated to each of the pieces @xmath356 , @xmath357 and @xmath358 . we will review the definitions of these three maps in sections [ section : onethreehandles ] and [ sec : twohandlemaps ] . first we review a few facts about the @xmath362-action on the hat heegaard floer invariants . in ( * ? * section 4.2.5 ) , ozsvth and szab constructed an action of the group @xmath363 on @xmath31 . in @xcite ( see also ( * ? ? ? * section 2 ) ) , they also showed that the cobordism maps @xmath22 extend to maps @xmath364 moreover , if @xmath24 is a cobordism from @xmath6 to @xmath7 ( endowed with a @xmath2 structure @xmath8 ) and we denote by @xmath365 the natural inclusions , then for any @xmath366 of the form @xmath367 one has @xmath368 this equality has the following immediate corollaries : [ l1 ] if @xmath369 then @xmath370 for any @xmath371 [ l2 ] if @xmath372 then @xmath373 for any @xmath374 consider now a 3-manifold of the form @xmath375 , and let @xmath376 denote the torsion @xmath2-structure on @xmath377 . then for any @xmath2-structure @xmath3 on @xmath1 there is an isomorphism @xmath378 as @xmath379$]-modules , cf . * theorem 1.5 ) . here , the action of @xmath380 on @xmath4 is trivial , as is the action of @xmath362 on @xmath381 . further , the action of @xmath382 on @xmath383 is exactly the given by cap product @xmath384 . next , we review the definition of the heegaard floer maps induced by one- and three - handle additions , cf . * section 4.3 ) . suppose that @xmath356 is a cobordism from @xmath354 to @xmath6 built entirely from @xmath95-handles . let @xmath8 be a @xmath2-structure on @xmath356 . the map @xmath385 is constructed as follows . if @xmath386 are the @xmath95-handles in the cobordism , for each @xmath387 pick a path @xmath388 in @xmath354 , joining the two feet of the handle @xmath389 this induces a connected sum decomposition @xmath390 , where the first homology of each @xmath391 factor is generated by the union of @xmath388 with the core of the corresponding handle . further , the restriction of @xmath8 to the @xmath392-summands in @xmath6 is torsion . it follows that @xmath393 . let @xmath394 be the generator of the top - graded part of @xmath395 . then the heegaard floer map induced by @xmath356 is given by @xmath396 it is proved in ( * ? ? ? * lemma 4.13 ) that , up to composition with canonical isomorphisms , @xmath397 does not depend on the choices made in its construction , such as the choice of the paths @xmath398 dually , suppose that @xmath358 is a cobordism from @xmath7 to @xmath355 built entirely from @xmath399-handles . let @xmath8 be a @xmath2-structure on @xmath358 . the map @xmath400 is constructed as follows . one can reverse @xmath358 and view it as attaching @xmath95-handles on @xmath355 to get @xmath401 after choosing paths between the feet of these @xmath95-handles in @xmath402 we obtain a decomposition @xmath403 ( where @xmath129 is the number of @xmath399-handles of @xmath358 ) . further , the restriction of @xmath8 to the @xmath392-summands in @xmath7 is torsion . it follows that @xmath404 . let @xmath405 be the generator of the lowest - graded part of @xmath406 . then the heegaard floer map induced by @xmath358 is given by @xmath407 and @xmath408 for any homogeneous generator @xmath409 of @xmath406 not lying in the minimal degree . again , the map is independent of the choices made in its construction . let @xmath357 be a two - handle cobordism from @xmath6 to @xmath7 , corresponding to surgery on a framed link @xmath232 in @xmath6 , and let @xmath8 be a @xmath2-structure on @xmath357 . let @xmath87 be a triple heegaard diagram subordinate to a bouquet @xmath233 for @xmath232 , as in the beginning of section [ sec : two ] . then , in particular , @xmath410 is a heegaard diagram for @xmath6 , @xmath411 is a heegaard diagram for @xmath7 , and @xmath412 is a heegaard diagram for @xmath413 . ( here , @xmath45 is the genus of @xmath414 and @xmath231 the number of components of @xmath232 . ) the @xmath2-structure @xmath8 induces a @xmath2-structure ( still denoted @xmath8 ) on the four - manifold @xmath415 specified by @xmath416 ( note that @xmath415 can be viewed as a subset of @xmath357 . ) consequently , there is an induced map @xmath417 as discussed in section [ sec : prels ] . the @xmath2-structure @xmath418 is necessarily torsion , so @xmath419 let @xmath394 denote the generator for the top - dimensional part of @xmath420 . then we define ( cf . * section 4.1 ) ) the map @xmath421 by @xmath422 . now , consider instead the nice , @xmath231-pointed triple heegaard diagram @xmath80 constructed in section [ sec : two ] . as discussed in the beginning of section [ sec : two ] , @xmath80 is strongly equivalent to a split triple heegaard diagram whose reduction @xmath87 is subordinate to a bouquet @xmath233 as above . let @xmath423 be the generator for the top - dimensional part of @xmath424 . then , by diagram , with @xmath425 , we have @xmath426 in light of corollary [ lemma : admissible ] and proposition [ easternorthodox ] , the rank of the map @xmath120 can be computed combinatorially . further , since the triple heegaard diagram @xmath80 is nice , so are each of the three ( ordinary ) heegaard diagrams it specifies . consequently , by @xcite , the element @xmath427 can be explicitly identified ( as can a representative for @xmath423 in @xmath428 ) . therefore , the rank of @xmath429 can be computed combinatorially . recall that @xmath24 is a @xmath11-dimensional cobordism from @xmath354 to @xmath355 , @xmath8 a @xmath2-structure on @xmath24 , and that @xmath24 is decomposed as a collection of one - handle additions @xmath356 , followed by some two - handle additions @xmath357 , and three - handle additions @xmath358 , with @xmath6 and @xmath7 the intermediate three - manifolds , so that @xmath359 as in section [ sec : action ] , we consider the maps @xmath430 [ onto1 ] if @xmath431 is surjective , then @xmath432 the cobordism @xmath356 consists of the addition of some @xmath95-handles @xmath433 as in section [ section : onethreehandles ] , we choose paths @xmath388 in @xmath354 joining the two feet of the handle @xmath389 the union of @xmath388 with the core of @xmath255 produces a curve in @xmath356 , which in turn gives an element @xmath434 since @xmath24 is obtained from @xmath435 by adding @xmath399-handles , we have @xmath436 , so the hypothesis implies that the map @xmath437 is surjective . hence there exist disjoint , embedded curves @xmath262 in @xmath354 ( disjoint from all the @xmath438 ) such that @xmath439 ) = -e_i , i=1 , \dots , n.$ ] we can connect sum @xmath388 and @xmath262 to get new paths @xmath440 in @xmath354 between the two feet of @xmath389 using the paths @xmath440 we get a connected sum decomposition @xmath441 as in section [ section : onethreehandles ] , with the property that the inclusion of the summand @xmath442 in @xmath443 is trivial . since we can view @xmath435 as obtained from @xmath357 by adding @xmath399-handles ( which do not affect @xmath444 ) , it follows that the inclusion of @xmath380 in @xmath445 is trivial . lemma [ l1 ] then says that @xmath446 for any @xmath447 thus the kernel of @xmath448 contains all elements of the form @xmath449 , where @xmath450 and @xmath451 is any homogeneous element not lying in the top grading of @xmath452 on the other hand , from section [ section : onethreehandles ] we know that the image of @xmath453 consists exactly of the elements @xmath454 where @xmath394 is the top degree generator of @xmath455 therefore , @xmath456 this gives the desired result . [ onto2 ] if @xmath457 is surjective , then @xmath458 this is similar to the proof of lemma [ onto1 ] . a suitable choice of paths enables us to view @xmath7 as @xmath459 such that the inclusion of the summand @xmath460 in @xmath445 is trivial . lemma [ l2 ] then says that @xmath461 for any @xmath462 and @xmath463 in other words , every element in the image of @xmath448 must be of the form @xmath464 where @xmath465 and @xmath405 is the lowest degree generator of @xmath455 on the other hand , from section [ section : onethreehandles ] we know that the kernel of the map @xmath466 does not contain any nonzero elements of the form @xmath467 let @xmath24 be a cobordism from @xmath354 to @xmath355 , and @xmath8 a @xmath2-structure on @xmath24 . assume that the maps @xmath431 and @xmath457 from formula are surjective . then in each ( relative ) grading @xmath468 the rank of @xmath469 can be computed combinatorially . the map @xmath10 is , by definition , the composition @xmath470 . lemmas [ onto1 ] and [ onto2 ] imply that @xmath471 or , equivalently , @xmath472 using lemma [ onto1 ] again , the expression on the right is the same as the rank of @xmath473 thus , the maps @xmath10 and @xmath448 have the same rank . as explained in section [ sec : twohandlemaps ] , the rank of @xmath448 can be computed combinatorially . note that the relative gradings on the generators of the chain complexes are also combinatorial , using the formula for the maslov index in ( * ? ? ? * corollary 4.3 ) . this completes the proof . in fact , using sarkar s remarkable formula for the maslov index of triangles ( * ? ? ? * theorem 4.1 ) , the absolute gradings on the heegaard floer complexes can be computed combinatorially , and so the rank of @xmath10 in each absolute grading can be computed as well . we give a nice triple heegaard diagram for the cobordism from the three - sphere to the poincar homology sphere , viewed as the @xmath41 surgery on the right - handed trefoil . the right - handed trefoil knot admits the nice heegaard diagram shown in figure [ fig : poincare_trefoil ] , which is isotopic to ( * ? ? ? * figure 14 ) . applying the algorithm described in section [ sec : two ] , we obtain the nice triple heegaard diagram shown in figure [ fig : poincare_nice ] . we leave the actual computation of the cobordism map to the interested reader .

in a previous paper , sarkar and the third author gave a combinatorial description of the hat version of heegaard floer homology for three - manifolds . given a cobordism between two connected three - manifolds , there is an induced map between their heegaard floer homologies . assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism ( modulo torsion ) . under this assumption , we present a procedure for finding the rank of the induced heegaard floer map combinatorially , in the hat version .

arxiv

it is the purpose of this paper to extend the results of yetter @xcite , generalizing classical results of gerstenhaber @xcite and gerstenhaber and schack @xcite on the infinitesimal deformation theory of associative algebras and poset - indexed diagrams of associative algebras to a deformation theory for arbitrary pasting diagrams of @xmath0-linear categories , @xmath0-linear functors , and natural transformations . in particular , in @xcite the standard result that obstructions are cocycles was established only for the simplest parts of pasting diagrams : for pasting diagrams in which no compositions either 1- or 2-dimensional occur . in this paper we will establish it for deformation complexes of pasting diagrams in general , by first giving a detailed and rigorous exposition of a method developed heuristically by shrestha @xcite , then applying the method to prove that obstructions are all cocycles in a family of pasting diagrams sufficient to imply the result in general . along the way to proving that obstructions are closed in general , we will have occasion to consider deformations of pasting diagrams in which specified functors or natural transformations are required to be deformed trivially . although in the present work such conditions will be used only to reduce the problem of showing obstructions are cocycles to simple instances , the ability to handle deformations subject to such restrictions could well be useful in other settings . we will also make explicit a point overlooked in the statements and proofs of @xcite theorems 8.2 and 8.3 : the cochain maps constructed in those theorems depend on choices of association for 2-dimenensional compositions in the pasting digaram . however , as we establish here , the cochain maps are independent of those choices , up to algebraic homotopy , and thus , the isomorphism type of the deformation complex for a pasting diagram is well - defined in either the homotopy category or derived category of cochain complexes over @xmath0 . at the level of the hochschild cochain complexes @xmath1 , the interesting cochain maps are described fully in @xcite . our purpose in this section is , rather , to describe chain maps induced by pasting compositions on the full deformation complexes @xmath2 , when @xmath3 is a composable pasting diagram . proposition 4.5 of @xcite constructs two chain maps : if @xmath4 , and @xmath5 are functors , then there is a cochain map @xmath6 given by @xmath7 similarly if @xmath8 and @xmath9 are functors , there is a cochain map @xmath10 given by @xmath11 proposition 4.6 of @xcite gives two more : if @xmath12 is a natural transformation , then post- ( resp . pre- ) composition by @xmath13 induces a cochain map @xmath14 ( resp . @xmath15 for any functor @xmath16 . here @xmath17 and @xmath18 and , @xcite proposition 4.7 give a cochain map , which ties together all of the deformations when a natural transformation , its source and target , and their common source and target are deformed simultaneously : let @xmath19 be a natural transformation between @xmath0-linear functors @xmath20 . let @xmath21 denote the cone on the cochain map @xmath22 : c^\bullet({\cal a})\oplus c^\bullet({\cal b } ) \rightarrow c^\bullet(f ) \oplus c^\bullet(g).\ ] ] the cochain groups are @xmath23 with coboundary operators given by @xmath24\ ] ] proposition 4.7 was omitted in the statement of the proposition , though it is plainly present in the proof given in @xcite ] was then let @xmath19 be a natural transformation , then @xmath25 : c^\bullet({\cal a } \stackrel{f}{g } { \cal b } ) \rightarrow c^\bullet(f , g)\ ] ] is a cochain map . in @xcite the chain maps of propositions 8.1 , 8.2 and 8.3 were used only to construct the deformation complex of a general pasting diagram . in fact , they can be assembled into chain maps from the deformation complex of a composable pasting diagram to the simpler pasting diagram in which the compositions have been carried out . propositions [ chainmapfrom2comp ] , [ chainmapfrom1precomp ] and [ chainmapfrom1postcomp ] give the map explicitly in the cases of a single 2 composition , precomposition of a natural transformation by a functor , and postcompostion of a natural transformation by a functor , respectively . in each case the proof begins by collecting the maps from the propositions of @xcite with values in the direct summands corresponding to cells of the pasting diagram in which the compositions have been performed , together with identity maps for those cells which remain from the original diagram , and arranging their summands in the correct places of a matrix of maps . what is indicated in the sketches of proofs following each proposition are the main difficulties in the unedifying calculation which shows that the result is , in fact , a chain map between the deformation complexes . [ chainmapfrom2comp ] let @xmath26 be @xmath0-linear functors , and @xmath19 and @xmath27 be natural transformations . let @xmath3 be the pasting diagram consisting of both @xmath28 and @xmath13 and their ( iterated ) sources and targets , and let @xmath29 be the pasting diagram consisting of the 2-dimensional composition @xmath30 and its ( iterated ) sources and targets . then there is a chain map @xmath31 induced by the 2-dimensional composition . in particular if the summands of @xmath32 ( resp . @xmath33 ) are given in the order @xmath34 ( resp . @xmath35 @xmath36 is given by @xmath37\ ] ] ( sketch ) the only subtlety in the completely computational verification that this is a chain map involves one coordinate in which a relation of the sort in gerstenhaber and voronov @xcite or the proof of proposition 4.7 in @xcite is needed . [ chainmapfrom1precomp ] let @xmath38 and @xmath39 be @xmath0-linear functors , and @xmath40 be a natural transformation . let @xmath3 be the pasting diagram consisting of @xmath28 and its ( iterated ) sources and targets together with @xmath41 and @xmath42 , and let @xmath29 be the pasting diagram consisting of the 1-dimensional composition @xmath43 and its ( iterated ) sources and targets . then there is a chain map @xmath44 induced by the 1-dimensional composition . in particular if the summands of @xmath32 ( resp . @xmath33 ) are given in the order @xmath45 ( resp . @xmath46 @xmath47 is given by @xmath48\ ] ] ( sketch ) here the `` hardest '' verification is a coordinate which vanishes by the naturality of @xmath43 . [ chainmapfrom1postcomp ] let @xmath49 and @xmath50 be @xmath0-linear functors , and @xmath19 be a natural transformation . let @xmath3 be the pasting diagram consisting of @xmath28 and its ( iterated ) sources and targets , together with @xmath51 and @xmath52 , and let @xmath29 be the pasting diagram consisting of the 1-dimensional composition @xmath53 and its ( iterated ) sources and targets . then there is a chain map @xmath54 induced by the 2-dimensional composition . in particular if the summands of @xmath32 ( resp . @xmath33 ) are given in the order @xmath55 ( resp . @xmath56 @xmath57 is given by @xmath58\ ] ] ( sketch ) here , almost all of the coordinates in the bottom row present some minor `` difficulty '' , one depends on a gerstenhaber - voronov - style relation , while most of the others require unpacking the definitions of pullback or pushforward maps along functors and natural transformations to see some commutativity relation between them . in this section we use the cochain maps of the previous section to give explicit examples of deformation complexes associated to various shapes commutative pasting diagrams . in one example we see that cochain maps induced by two different associations of the 2-dimensional composition are not actually equal , but only chain homotopic ( or , to put it another way , are not equal in the abelian category of cochain complexes , but _ are _ in the homotopy category of cochain complexes and thus _ a fortiori _ in the derived category ) . in the examples , we use commutative pasting diagrams because only in that context is the effect of the compositions of 2-arrows evident : if no commutativities are enforced , the iterative cone construction given explicitly in @xcite proposition 4.7 suffices to describe the entire structure of the deformation complex of the pasting diagram , as indeed it does in the case of commutative pasting diagrams in which no compositions occur ( for example , the commutative `` pillow '' consisting of two categories a pair of parallel functors between them , and two copies of the same natural transformation between the functors ) . only when compositions are involved do the cochain maps of the previous section play a role . in what follows , we leave zero entries in matrices of maps giving coboundaries blank , as this seems to improve readability . we begin with the simplest example , a `` commutative pillow with triangular cross - section '' : [ triangularpillow ] consider the pasting diagram given by three functors @xmath59 and three natural transformations @xmath60 , @xmath61 and @xmath62 , with the obvious 0- , 1- and 2-cells and a single 3-cell enforcing the condition @xmath63 . the deformation complex of the pasting diagram is then given by @xmath64 @xmath65 with coboundary given by @xmath66\ ] ] note that here we have chosen to consider the composition @xmath67 to be the source of the 3-cell and the single natural transformation @xmath68 to be the target . for the opposite choice the complex would be the same , except that the non - diagonal entries of the last row would be negated . the complex @xmath69 of example [ triangularpillow ] is related to the map @xmath70 of proposition [ chainmapfrom2comp ] by [ babyqi ] @xmath69 is quasi - isomorphic to the ( dual ) mapping cylinder on @xmath70 . this follows immediately by applying the following general lemma about ( dual ) mapping cylinders of chain maps between mapping cones . let @xmath71 and @xmath72 be chain maps , and suppose @xmath73 is a chain map between their mapping cones of the form @xmath74 .\ ] ] then the ( dual ) mapping cylinder on @xmath75 , @xmath76 \oplus cone(g)[1 ] \oplus cone(g)$ ] with differential given by @xmath77 , \ ] ] is quasi - isomorphic to @xmath78 with differential given by @xmath79 .\ ] ] include the last complex into the ( dual ) mapping cylinder by @xmath80 , the cokernel is plainly isomorphic to @xmath81})$ ] , which is acylic , so the subcomplex is quasi - isomorphic to the ( dual ) mapping cylinder as claimed . continuing with pasting diagrams which only involve 2-dimensional compositions , consider next the two `` pillows with square cross - sections '' : consider the pasting diagram given by four functors @xmath82 and four natural transformations @xmath60 , @xmath61 , @xmath83 and @xmath84 , with the obvious 0- , 1- and 2-cells and a single 3-cell enforcing the condition @xmath85 . the deformation complex of the pasting diagram is then given by @xmath86 @xmath87 with coboundary given by @xmath88\ ] ] where @xmath89 consider the pasting diagram given by four functors @xmath90 and four natural transformations @xmath60 , @xmath61 , @xmath91 and @xmath92 , with the obvious 0- , 1- and 2-cells and a single 3-cell enforcing the condition @xmath93 . the deformation complex of the pasting diagram is then given by @xmath94 @xmath95 with coboundary given by @xmath96\ ] ] where @xmath97 now , observe that in this example , we made a choice : we computed the bottom row , and in paricular the map @xmath98 by left - associating @xmath28 , @xmath13 and @xmath68 . had we right - associated them , the bottom row , other than @xmath98 would have remained the same ( though it would have been more natural to write @xmath99 , @xmath100 and @xmath101 as the names for the other maps ) . the second entry , rather than being @xmath98 would have been @xmath102 . now , it is easy to see that the chain maps from @xmath103 @xmath104 to @xmath105 given by the entries of the bottom row other than @xmath106 are not equal , and their difference has a single non - zero entry @xmath107 . there is however a contracting homotopy for this difference , given by the map with all by the second entry zero , and that entry given by @xmath108 , due to the relationship between the brace and coboundary discovered in the classical case by gerstenhaber and voronov @xcite and seen to apply in the categorical setting in @xcite . thus in the homotopy category ( or the derived category , if one prefers ) our choice of left - association was a matter of indifference . it is easy to compute more examples in which no 1-dimensional compositions occur , for instance a `` pillow '' with a pentagonal cross - section with a triple composition equal to a double composition , by combining the techniques of the last three examples . rather than write this out explicitly , we now turn to examples involving 1-dimensional composition : [ equalspost1comp ] consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , three functors @xmath20 and @xmath50 , two natural transformations @xmath60 and @xmath110 and a single 3-cell enforcing the condition that @xmath111 . the deformation complex of the pasting diagram is given by @xmath112 @xmath113 with coboundary given by @xmath114\ ] ] [ equalspre1comp ] consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , three functors @xmath115 and @xmath39 , two natural transformations @xmath116 and @xmath117 and a single 3-cell enforcing the condition that @xmath118 . the deformation complex of the pasting diagram is given by @xmath112 @xmath119 with coboundary given by @xmath120\ ] ] consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , four functors @xmath20 , @xmath50 and @xmath121 , and three natural transformations @xmath60 , @xmath122 and @xmath123 , with the obvious 0- , 1- and 2- cells and a single 3-cell enforcing the condition that @xmath124 . the deformation complex of the pasting diagram is given by @xmath125 @xmath126 with coboundary given by @xmath127^ * & -id & d_{h(f),k } \end{array } \right]\ ] ] where @xmath128 consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , four functors @xmath115 , @xmath39 and @xmath121 , and three natural transformations @xmath129 , @xmath130 and @xmath131 , with the obvious 0- , 1- and 2- cells and a single 3-cell enforcing the condition that @xmath132 . the deformation complex of the pasting diagram is given by @xmath125 @xmath133 with coboundary given by @xmath134^ * & \sigma_*(f^ * ) & -id & d_{h(f),k } \end{array } \right]\ ] ] where @xmath135 consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , four functors @xmath136 , @xmath137 and @xmath138 , and three natural transformations @xmath139 , @xmath130 and @xmath140 , with the obvious 0- , 1- and 2- cells and a single 3-cell enforcing the condition that @xmath141 . the deformation complex of the pasting diagram is given by @xmath125 @xmath142 with coboundary given by @xmath143 _ * & \sigma^*(k _ * ) & -id & d_{h(f),k } \end{array } \right]\ ] ] where @xmath144 consider the pasting diagram consisting of three categories @xmath42 , @xmath109 and @xmath52 , four functors @xmath136 , @xmath145 and @xmath146 , and three natural transformations @xmath147 , @xmath148 and @xmath149 , with the obvious 0- , 1- and 2- cells and a single 3-cell enforcing the condition that @xmath150 . the deformation complex of the pasting diagram is given by @xmath125 @xmath151 with coboundary given by @xmath152 _ * & \sigma^*(g^ * ) & -id & d_{f , k(g ) } \end{array } \right]\ ] ] where @xmath153 first , observing that the map @xmath154 is simply the identity in the case where the composable pasting diagram @xmath3 is simply the 2-cell @xmath28 ( with its sources and targets ) , and that the ( dual ) mapping cylinder on a map is the cone on the direct sum of the map and the identity on its target we have if @xmath3 is a pasting diagram with a single 3-cell the boundary of which has domain ( resp . codomain ) which is a composable pasting diagram @xmath29 and codomain ( resp . domain ) which is a single 2-cell , then @xmath155 is quasi - isomorphic to the ( dual ) mapping cylinder of the map of proposition 8.3 of @xcite induced by the composition . observe this is a generalization of proposition [ babyqi ] and in special instances relates the ( dual ) mapping cylinder of the map in proposition [ chainmapfrom1postcomp ] ( resp . [ chainmapfrom1precomp ] ) to the complex of example [ equalspost1comp ] ( resp . [ equalspre1comp ] ) . the following is a trivial observation about the deformation complex constructed in proposition 8.3 of @xcite and the remarks following : [ uniontopushout ] if @xmath156 is a pasting diagram which is the union of pasting diagrams @xmath157 and @xmath158 , then the deformation complex @xmath155 is the pushout of the induced inclusions of deformation complexes @xmath159 for @xmath160 @xmath161 . in @xcite shrestha developed a method of using polygons with edges labeled by arrow - valued operations to simultaneously encode cocycle conditions , the formulas for obstructions , and the condition that the next term of a deformation cobound the obstruction . suitable cell - decompositions of the surface of a 2-sphere into such polygons and `` trivial '' polygons which encode tautologous equalities in place of cocycle conditions and zero ( as a difference of identical sums ) in place of an obstruction , then provide a convenient method of proof for standard obstructions - are - cocycle results . as an example , consider the cocycle conditions , formulas for obstructions , and cobounding conditions in the case of the deformation of composition , writing the undeformed composition as @xmath162 to match the notation for the degree @xmath163 deformation term as @xmath164 : the cocycle condition is , of course , @xmath165 the formula for the degree @xmath163 obstruction is @xmath166 . and the condition that @xmath167 cobounds @xmath168 is @xmath169 now consider the square , with oriented edges labeled by the compositions occuring in the expression of the associativity of composition : @xmath170^{a(bc ) } & \circ\\ .\ar[r]_{ab}\ar[u]^{bc } & \ar[u]_{(ab)c.}}.\ ] ] + each of the above formulas can be obtained from ( or represented by ) the square by adding all possible terms obtained by assigning degrees to the edges in such a way that the degrees are chosen from a particular set and the sum of degrees along each of the oriented parts of the boundary has a specified value , and taking the difference of the expressions thus obtained for each of the oriented parts of the boundary . gerstenhaber s proof @xcite that obstructions to the deformation of an associative algebra are hochschild cocycles can then be described in terms of the cube below by first noting that the formula for the coboundary of the degree @xmath163 obstruction can be written as a signed sum of the obstruction - type expressions ( with the edge - degrees ranging from @xmath171 to @xmath172 ) , five faces representing the usual terms of the coboundary , and one representing @xmath171 , `` prolonged '' by pre- or post - composing with the degree @xmath171 label on another edge so that all of the expressions represent compositions of terms from the deformation of an iterated composition of all four maps , and thus , all representing parallel maps , can all be added . @xmath173_{(\overline{ab}c)d } & & & \odot \\ .\ar[ru]_{\overline{ab}c}\ar[rrr]^{cd } & & & .\ar[ru]^{\overline{ab}(cd ) } & \\ & & & & \\ & .\ar@{-->}[uuu]_{a(bc)}\ar@{-->}[rrr]^{(bc)d } & & & .\ar[uuu]^{a(b\overline{cd})}\\ \circ\ar[uuu]_{ab}\ar@{-->}[ru]_{bc}\ar[rrr]^{cd } & & & .\ar[ru]^{b\overline{cd}}\ar[uuu]^{ab } & } \ ] ] + the proof then proceeds by iteratively `` clearing '' edges shared by two squares with the same sign by rewriting terms involving that edge s label contributed by one square using the cobounding and cocycle condition from the other square sharing the edge until the expression is reduced to two copies of the obstruction - type expression on an equatorial hexagon , with opposite signs . this observation , that gerstenhaber s proof can be encoded by such a figure , then motivates a sequence of definitions and lemmas that allow for similar encoding of more general and complex proofs of the same sort . shrestha s technique @xcite is most quickly and rigorously described by labeling 1-cells of certain pasting schemes and computads ( cf . @xcite , @xcite , @xcite ) with well - formed expressions in a particular ( essentially ) algebraic theory : a _ directed polygon _ @xmath174 is a 2-computad with a single 2-cell , all of whose 0- and 1-cells lie in the boundary of the 2-cell . a _ whiskered polygon _ @xmath175 is a 2-computad with a single 2-cell @xmath174 , whose underlying 1-computad is the union of two 1-dimensional pasting schemes ( directed paths ) , with the same 0-domain and 0-codomain , whose underlying cell complex is contractible . we refer to the 1-dimensional pasting scheme consisting of the path from the 0-domain of @xmath175 to the 0-domain of @xmath174 as the _ domain whisker _ ( note : it may be simply a 0-cell ) , and the 1-dimensional pasting scheme consisting of the path from the 0-codomain of @xmath174 to the 0-codomain of @xmath175 as the _ codomain whisker _ ( again it may simply be a 0-cell ) . a _ tiled sphere _ is a 3-computad with a single 3-cell , all of whose 0- , 1- and 2-cells lie in the boundary of the 3-cell . in applying the method in a given circumstance , one needs to label the edges of tiled spheres with arrow - valued operations from a theory associated to the pasting diagram ( or more general structure ) . for the present application the following suffice : [ theory_of_diagram ] to any pasting diagram @xmath3 , _ theory of _ @xmath3 , @xmath176 , is the essentially algebraic theory with types @xmath177 and @xmath178 for each category @xmath52 in the diagram ( the objects of @xmath52 and the arrows of @xmath52 respectively ) and operations * @xmath179 of arity @xmath177 and type @xmath178 , * @xmath180 and @xmath181 of arity @xmath178 and type @xmath177 , and * @xmath182 of arity @xmath183 and type @xmath178 for each category @xmath52 in @xmath3 ; * @xmath184 of arity @xmath177 and type @xmath185 and * @xmath186 of arity @xmath178 and type @xmath187 for each functor @xmath188 in @xmath3 ; and * @xmath28 of arity @xmath177 and type @xmath187 for each natural transformation @xmath60 for @xmath189 in @xmath3 . and , with axioms the equations expressing the axioms of categories for each 6-tuple @xmath190 , the functoriality of each @xmath191 and the naturality of each @xmath28 . we term operations of type @xmath178 for any @xmath52 `` arrow - valued '' and those of type @xmath177 for any @xmath52 `` object - valued '' , and refer to the model of @xmath176 given by @xmath3 as the `` tautologous model '' . in other applications of the method , as in @xcite @xmath176 may be replaced with an extension of the theory ( for instance including operations of arity @xmath192 and type @xmath193 encoding monoidal product of two arrows ) . we conjecture that broader generalizations , for instance to deformations of @xmath163-tuple categories and ( weak ) @xmath163- categories with a @xmath0-linear structure on their @xmath163-arrows , and to suitable pasting diagrams of these , or even to models of other sorts of essentially algebraic theories ( cf . @xcite ) with appropriate linearizations of parts of the structure , can be described , but do not pursue this possibility here . notice a subtlety in the description of the axioms of @xmath176 in definition [ theory_of_diagram ] : the only axioms are those inherited from the axioms of categories , functors and natural transformations . in the definition [ parallels ] , on the contrary , all of the equations which hold in the diagram @xmath3 are enforced . [ parallels ] the _ theory of parallels _ @xmath194 for a pasting diagram @xmath3 is the essentially algebraic theory with the same types and object valued operations as @xmath176 and with arrow - valued operations given by all set functions @xmath195 with the same domain and codomain as the instantiation of an arrow - valued operation @xmath196 of @xmath176 in the tautologous model and satisfying @xmath197 and @xmath198 ( we call @xmath195 _ a parallel _ for @xmath196 ) , and all equations that hold among ( iterated generalized ) compositions of these functions as axioms . notice in neither definition did we include any addition or scalar multiplication operations induced by the @xmath0-linear structure on the categories , even though we are only applying these constructions to pasting diagrams of @xmath0-linear categories , @xmath0-linear functors and natural transformations . however , when applied to such a pasting diagram , the vector space structure on the hom - sets induces a vector space structure on the set of parallels for any arrow - valued operation @xmath196 of @xmath176 . given a ( countably infinite ) stock of variables of each type in the theory it is evident what is meant by a well - formed formula ( wff ) of the theory . we now make some technical definitions : an _ equivalent _ of a wff is any wff of the same type in the same variables such that for every instantiation of the variables in the two formulas , the values are equal . in what follows , for brevity we will refer to well - formed subformulas of a wff as wfss . when considered as wfss of a fixed wff @xmath175 , repetitions of equal wffs are considered distinct wfss . with this convention , the following ( a generality about wfss of a wff in any formal system ) is immediate : the wfss of a wff @xmath175 form a partially ordered set @xmath199 under @xmath200 when @xmath201 is a wfs of @xmath202 , and the incidence diagram of @xmath199 is a tree . thus @xmath199 admits a natural number valued depth function @xmath203 for which @xmath204 and whenever @xmath202 covers @xmath201 , @xmath205 . the minimal elements ( in the order - theoretic sense , not just those of maximum depth ) for @xmath199 are instances of variables . we denote the subposet of non - minimal wfss of @xmath175 by @xmath206 a _ well - formed labeling _ of a whiskered polygon ( resp . tiled sphere ) is an assignment @xmath207 to each directed 1-cell of arrow - valued wff in @xmath176 for some pasting diagram ( or an appropriate extension of this theory ) with the properties * edges are not labeled with variables ( i.e. all labels involve an operation in the theory ) . * for each 1-pasting scheme ( directed path ) @xmath208 from the 0-domain of the whiskered polygon ( resp . tiled sphere ) the labels @xmath209 can be iteratively replaced , beginning at the 0-codomain , with labels @xmath210 such that * * if the only proper wfss of @xmath211 are variables then @xmath212 . * * for all @xmath213 @xmath214 is equivalent to @xmath211 . * * in the list of replacement labels @xmath210 , for each @xmath214 , if @xmath215 is a wff occuring @xmath216 times as a non - minimal wfs of @xmath214 , then there exist @xmath216 distinct @xmath217 s with @xmath218 such that @xmath219 . * if @xmath208 is a maximal directed path in a whiskered polygon ( resp . tiled sphere ) , the replacement labels @xmath210 are precisely the non - minimal wfss of @xmath220 . * for every 2-cell , and every maximal extension of the 2-cell to a whiskered polygon , the labels of the final edges of the two paths from the 0-domain to the 0-codomain of the whiskered polygon are equivalents . ( note this condition is vacuously true unless the 0-codomain of the 2-cell is the 0-codomain of the whiskered polygon . ) the following is then immediate : for any maximal path @xmath208 in a whiskered polygon ( resp . tiled sphere ) the totally ordered set of wff s @xmath221 is a totalization of the partially ordered set @xmath222 . # 1#2#3#4#5 @font ( 3885,7414)(1816,-7874 ) ( 2206,-5356 ) ( 0,-1)1275 ( 2206,-6646 ) ( 4,-3)1112 ( 3316,-7456 ) ( 1 , 0)1530 ( 4426,-4186 ) ( 4,-3)1080 ( 5536,-5041 ) ( 0,-1)1605 ( 5558,-6669)(-1,-1)735 ( 2926,-4111)(-2 , 3)750 ( 2206,-2956 ) ( 0 , 1)1275 ( 2206,-1666 ) ( 4 , 3)1112 ( 3316,-856 ) ( 1 , 0)1530 ( 4426,-4126 ) ( 4 , 3)1080 ( 5536,-3271 ) ( 0 , 1)1605 ( 5558,-1643)(-1 , 1)735 ( 3016,-4156 ) ( 1 , 0)1350 ( 3001,-4261 ) ( 0,-1)1350 ( 2963,-5657)(-3,-4)642 ( 3031,-4231 ) ( 1,-1)765 ( 3841,-5011 ) ( 1 , 0)1560 ( 3061,-5641 ) ( 3,-2)675 ( 3751,-5056 ) ( 0,-1)975 ( 3796,-6091 ) ( 3,-4)945 ( 2251,-2986 ) ( 1 , 0)1320 ( 3946,-1696 ) ( 1 , 0)1485 ( 3871,-1726)(-2 , 3)550 ( 3608,-2913 ) ( 1 , 4)285 ( 4321,-4111)(-2 , 3)750 ( 1831,-2326)(0,0)[lb ] ( 2176,-1126)(0,0)[lb ] ( 2641,-2806)(0,0)[lb ] ( 3871,-2371)(0,0)[lb ] ( 3706,-1246)(0,0)[lb ] ( 3661,-631)(0,0)[lb ] ( 2161,-3706)(0,0)[lb ] ( 5296,-1096)(0,0)[lb ] ( 3991,-3361)(0,0)[lb ] ( 3241,-3976)(0,0)[lb ] ( 5041,-3886)(0,0)[lb ] ( 5686,-2566)(0,0)[lb ] ( 2071,-4621)(0,0)[lb ] ( 5071,-4501)(0,0)[lb ] ( 5671,-6061)(0,0)[lb ] ( 1831,-6046)(0,0)[lb ] ( 2161,-7411)(0,0)[lb ] ( 3541,-7801)(0,0)[lb ] ( 5356,-7261)(0,0)[lb ] ( 2746,-5131)(0,0)[lb ] ( 4426,-5281)(0,0)[lb ] ( 2581,-6391)(0,0)[lb ] ( 3496,-4516)(0,0)[lb ] ( 3841,-5581)(0,0)[lb ] ( 2971,-6121)(0,0)[lb ] ( 4201,-6406)(0,0)[lb ] ( 2416,-5491)(0,0)[lb ] ( 2956,-3421)(0,0)[lb ] ( 4111,-4606)(0,0)[lb ] ( 4351,-1981)(0,0)[lb ] an example of a well - formed labeling of a tiled sphere is given in figure [ functor_obstructions_are_cocycles ] . ( the boundaries of the two octogons bordering the unbounded region should be identified along the edges with the same labels , and the 3-cell inserted to fill the resulting topological sphere . ) this particular example arises in applying the method to show that the obstructions to deforming ( the arrow part of ) a functor , when the compositions in its source and target categories are also being deformed , are cocycles . note that in the labeling , at one point notation has been abused and @xmath223 has been used in place of either of the equivalent wffs @xmath224 or @xmath225 . it is also easy to see along which paths the label - replacement axiom wf1 results in non - trival replacements of labels ( as , for instance along any path ending in the top right - most edge in which @xmath226 occurs , rather than @xmath227 : the label of the top right - most edge must be replaced with @xmath228 ) . now each arrow - valued operation is a part of the algebraic structure which is subject to infinitesimal deformation , by being replaced by a formal power - series ( or polynomial in @xmath229 with @xmath230 for some @xmath163 ) whose coefficients are arrow - valued operations of the same type and arity ( e.g. the arrow part of a functor @xmath188 is replaced with @xmath231 where for each @xmath232 , @xmath233 is a map from @xmath234 to @xmath235 ) . from this , we abstract consider any set of arrow - valued operations @xmath236 in @xmath176 for some @xmath3 . fix @xmath237 . to each arrow - valued operation @xmath238 associate a sequence of parallels @xmath239 in the sense of definition [ parallels ] ( truncated at @xmath240 if @xmath241 ) with @xmath242 . we call @xmath239 _ the degree @xmath0 parallel of @xmath243 _ , and a choice @xmath244 of such a sequence for every operation in @xmath236 , _ a degree @xmath163 family of parallels for @xmath236 _ now , for any well - formed labeling of a whiskered polygon or tiled sphere with , let @xmath236 be the set of all arrow - valued operations occuring in the labels of the edges , closed under equivalence . every family of parallels for @xmath236 , then gives rise to many labelings of the maximal paths of the whiskered polygon or tiled sphere by wffs from @xmath194 by replacing the labels along the path using wf1 , then for each edge , chosing a degree @xmath0 and replacing the last - applied operation @xmath243 in the label on that edge with @xmath239 in the label for that edge and all edges later in the path for which the label on the given edge as a wfs . in particular , any such choice of degrees for each edge along a path from the 0-domain to the 0-codomain creates a new wff in @xmath194 . suppose the sequence of labels on a maximal path was @xmath245 , where each @xmath246 denotes a well - formed expression all of whose iterated inputs have an equivalent among the labels earlier in the sequence , and by abuse of notion also that well - formed expression s last - applied operation . a choice of degrees @xmath247 then produces a new sequence of well - formed expression @xmath248 in which each instance of an operation has been replaced with its parallel of the chosen degree in that and all later edge - labels . note : in general these new labelings are not well - formed labelings . at this point , recall that all of this is taking place in a linear setting , so that parallel arrows ( and thus parallels ) can be added . we will now define several different expressions which a system of parallels associates to a whiskered polygon . [ conditions_and_obstructions ] for any well - formed labeling of a whiskered polygon @xmath175 , @xmath236 the set of operations occuring in the replacement labelings of both paths , and @xmath244 a family of parallels for @xmath236 , let @xmath249 ( resp . @xmath250 ) be the replacement labeling on the maximal path which traverses domain ( resp . codomain ) of the 2-cell . the _ cocycle - type condition _ associated to @xmath175 is the equation @xmath251 the _ @xmath252 order obstruction - type expression _ associated to @xmath175 is the expression @xmath253 and , the _ @xmath252 order cobounding - type condition _ associated to @xmath175 is the expression @xmath254 in each case , the string of parallels should be understood as naming the well - formed experssion obtained by the iterated substitution of the named parallel for the corresponding operation in the well - formed expression labeling the last edge of the path . the following then provides the basis for shrestha s method : [ elementary ] for any well - formed labeling of a tiled sphere @xmath255 with 3-cell @xmath256 , @xmath236 the set of operations occuring in the replacement labelings of all paths , and @xmath244 a family of parallels for @xmath236 , the domain ( resp . codomain ) of @xmath256 can be expressed as the union of whiskered polygons with the same 0-domain and 0-codomain as @xmath255 , one with each 2-cell in @xmath257 ( resp . @xmath258 ) as its 2-cell in such a way that the sum of the @xmath252 order obstruction - type expressions for the whiskered polygons is the @xmath252 order obstruction - type expression for the ( whiskered ) polygon consisting of a single 2-cell and the union of the 1-pasting schemes @xmath259 and @xmath260 . once the combinatorial structure of power s proof of the the uniqueness of pasting compositions @xcite is recalled , the result is immediate sums cancel in pairs leaving only the difference giving the desired @xmath252 order obstruction - type expression . unfortunately , as it stands , the result is not immediately applicable . recall the cube encoding gerstenhaber s proof . any face is part of a whiskered polygon ( with one whisker having one edge , and the other having none ) . the @xmath252 order obstruction is given by @xmath261 the terms in its coboundary are _ not _ @xmath252 order obstruction - type expressions associated to the faces of their cubes with their whiskers . the @xmath252 order obstruction type expressions are , instead , instances of @xmath262 with the obstruction and a variable as inputs , or instances of the obstruction with one of its variables replaced with @xmath262 applied to two variables . what then is the relationship between actual @xmath252 order obstructions and @xmath252 order obstruction - type expressions ? if one considers one of the whiskered polygons in the example , it is easy to see that the terms in the @xmath252 order obstruction - type expression which do not correspond to terms from the coboundary of the obstruction have a label of positive degree on the edge of the whisker . if we fix the label on the whisker edge to be of degree @xmath263 , the terms with this label are then the condition that @xmath264 cobound the degree @xmath265 obstruction ( or the cocycle condition if @xmath266 ) with @xmath267 of two variables as argument ( resp . used as input to an instance of @xmath267 ) when the non - trivial whisker is the domain ( resp . codomain ) whisker . in this case , provided the system of parallels is describing the terms of an associative deformation , the @xmath252 order obstruction - type expression is thus equal to the corresponding term in the coboundary of the @xmath252 order obstruction , since its extra terms all vanish by the cocycle and cobounding conditions . to imitate this in general , we need conditions depending only on the labels on the domain and codomain of the 2-cell in a whiskered polygon which ensures that the cocycle- and cobounding - type conditions in all lower degrees hold . as in definition [ conditions_and_obstructions ] consider a whiskered polygon @xmath175 with 2-cell @xmath174 and let @xmath249 ( resp . @xmath250 ) be the replacement labeling on the maximal path which traverses domain ( resp . codomain ) of @xmath174 . suppose the first @xmath268 edges lie in the domain whisker . in this case , for @xmath269 , @xmath246 and @xmath270 are equivalents , and the orderings on the edge labels of the domain whisker induced by restricting the partial orderings on @xmath271 and @xmath272 coincide . let @xmath273 be the resulting partially ordered set of equivalence classes of wffs . [ strong_vanishing ] given a well - formed labeling of a whiskered polygon @xmath175 with 2-cell @xmath174 , let @xmath236 be the set of operations occuring in the replacement labelings of both paths , and @xmath244 a family of parallels for @xmath236 . recall that @xmath274 is the theory of parallels for @xmath175 ( and thus includes the elements of @xmath244 . let @xmath268 and @xmath273 be as in the discussion above . and suppose the last @xmath275 edges of each path lie in the codomain whisker . the _ @xmath252 order strong vanishing condition _ associated to @xmath175 ( for @xmath276 ) is the condition that @xmath277 where the hatting of wffs indicates the result of the following process : for each maximal element of @xmath273 select a variable of the same type which does not occur in among the variables used in the labeling ; the hatted wff is the result of replacing all wfss in that equivalence class with the corresponding variable ; and the universal quantification ranges over all parallels in @xmath274 to the labels on the corresponding edge of the codomain whisker . we then have [ strong_works ] for any well - formed labeling of a whiskered polygon @xmath175 with operations @xmath278 and a family of parallels @xmath279 , the @xmath280 order strong vanishing condition implies the cocycle - type condition , the @xmath281 order strong vanishing condition implies the @xmath281 order cobounding - type condition , and , moreover , if the @xmath281 order vanishing conditions hold for all @xmath282 , then the @xmath252 order coboundary - type expression is equal to @xmath283 where @xmath268 and @xmath275 are as in the previous definition and the meanings of the sequences of wffs are as in definition [ conditions_and_obstructions ] . the cocycle - type condition simply an instantiation of the @xmath280 order strong vanishing condition , while the @xmath281 order cobounding - type condition is the sum of all instantiations of the @xmath281 order strong vanishing condition ranging over all choices of parallels for the labels of edges in the whiskers . for the last statement , notice that the terms in the @xmath252 order coboundary - type expression which are not represented in the expression of the proposition all have at least one label on an edge of one of the whiskers which is of positive degree . these terms can be partitioned into subsets which according to the degrees of the labels in the whiskers . for each choice of degrees for the labels in the whiskers , the terms are an instatiation of the strong cocycle - type condition , and thus add to zero . two sorts of whiskered polygons satisfying the strong vanishing conditions arise in practice : `` non - trivial '' whiskered polygons in which the strong vanishing conditions follow from the well - formedness of the labeling and deformation theoretic cocycle and cobounding conditions satisfied by the parallels of the labels in the boundary of the 2-cell , and `` trivial '' whiskered polygons in which the sets of edge labels on the domain and codomain of the 2-cell differ only by changing the choice of totalization of the partial order on wfss of the final label in the path and substitution of equivalent wffs . we formalize this in the following propositions : [ nontrivial ] given a well - formed labeling of a whiskered polygon @xmath175 with 2-cell @xmath174 , let @xmath236 be the set of operations occuring in the replacement labelings of both paths , and @xmath244 a family of parallels for @xmath236 . let @xmath268 , @xmath275 , @xmath0 , @xmath284 and @xmath273 and the hatting of wffs be as definition [ strong_vanishing ] . if @xmath285 and @xmath286 are a well - formed labeling of the directed polygon @xmath174 , then @xmath175 satisfies the @xmath287 order strong vanishing condition . if , moreover , this labeling of @xmath174 and the system of parallels satisfy the cocycle - type condition ( resp . the @xmath252 order cobounding - type condition ) then the well - formed labeling of @xmath175 and system of parallels satisfies the @xmath280 order ( resp . @xmath252 order strong vanishing condition ) . notice first that the condition that the labeling of @xmath174 be well - formed means that the wff labeling the two edges incident with the 0-codomain of @xmath174 are equivalents , and thus their difference ( as an arrow - valued operation ) vanishes . the well - formedness may thus be viewed as a zeroth order analogue of the cocycle - type and cobounding - type relations . the proposition is immediate once it is observed that at each order the strong vanishing condition on @xmath175 is simply the result of applying the parallels to labels on the codomain whisker to the terms of the expression of that order which vanishes on @xmath174 . in practice the labels on @xmath174 come from an equational condition ( associativity , functoriality , or naturality in the present work ) and the cocycle - type and cobounding - type conditions are actual cocycle and cobounding conditions derived from the requirement that deformations preserve the equational condition , while the whiskers arise in taking coboundaries of the corresponding obstruction . [ trivial ] given a well - formed labeling of a whiskered polygon @xmath175 with 2-cell @xmath174 , let @xmath268 , @xmath275 , @xmath0 , @xmath284 , and @xmath273 and the hatting of wffs be as definition [ strong_vanishing ] . if @xmath288 and @xmath285 is obtained from @xmath286 by changing the totalization of the partial ordering on the wfss of the label on the edge(s ) incident with the 0-codomain of @xmath175 and replacing wffs with equivalents , then the well - formed label of @xmath175 satisfies the strong vanishing conditions of all orders for any system of parallels . in this case for any choice of parallels labels on edges of the codomain whisker , and of degrees in a system of parallels for the corresponding @xmath289 and @xmath290 , the resulting expressions for the two paths are equivalent and thus their difference is zero , and summing over all such choices of total order @xmath291 gives the @xmath281 order strong vanishing condition for @xmath175 . we refer to whiskered polygons equipped with a labeling satifying the hypotheses of proposition [ trivial ] as trivial whiskered polygons . finally , we need to describe in general the relationship between the cohomological description of infinitesimal deformations and the evident expression of the same data in terms of systems of parallels and the various expressions given by labeling of ( whiskered ) polygons and tiled spheres . if @xmath3 is a @xmath0-linear pasting diagram , @xmath176 its theory , or an extension thereof with the same types , its deformation theory is _ polygonizable _ if the cochain group in which cocycles specify first order deformations admits a direct sum decomposition indexed a family of equational axioms of the theory each of which can be expressed as the vanishing of the difference of the values of the paths in a well - formed labeling of a directed polygon ( we call such a polygon equipped with its well - formed labeling an _ axiomatic polygon _ ) , and which , moreover satisfy * the vanishing of each direct summand of the cocycle is precisely the cocycle - type condition associated to the axiomatic polygon indexing the direct summand . * each direct summand of the @xmath252 order obstruction is the obsturuction - type expression associated to the axiomatic polygon indexing the direct summand . * the cobounding condition for the extension of a deformation to the next degree is the direct sum of the cobounding - type conditions associated to the axiomatic polygons . * and the coboundary of an obstruction admits a direct sum decomposition in which each direct summand is a signed sum of expressions specified by labeling whiskered polygons with the polygon labeled by the obstruction , and the whiskers labeled by operations of @xmath176 ( degree 0 labels in the system of parallels naming the deformation terms ) . we are now in a position to state a theorem which encapsulates shrestha s polygonal method for our purposes : if @xmath176 is the theory of a pasting diagram , or an extension thereof , and admits a polygonizable deformation theory , then all obstructions are cocycles , provided for each direct summand of the coboundary of the obstruction , there exists a tiled sphere , and a well - formed labeling of the tiled sphere with the properties * each whiskered polygon naming a summand in the signed sum of p4 occurs exactly once in the tiled sphere either in the domain ( resp . codomain ) if its sign is positive ( resp . negative ) , or with its domain and codomain swapped and in the codomain ( resp . domain ) if its sign is positive ( resp . negative ) . * every 2-cell which is not part of the whiskered polygons of s1 is trivial in the sense of proposition [ trivial ] this follows immediately from propositions [ elementary ] , [ strong_works ] , [ nontrivial ] and [ trivial ] . figures 1 through 6 then establish the following : [ singlecomposition ] if @xmath3 is a pasting diagram with a single instance of composing two natural transformations or of composing a natural transformation with a functor , then all obstructions to its deformation are cocycles . in principal , it appears , one could apply the polygonal method to directly show that obstructions to the deformation of any given ( arbitrarily complicated ) pasting diagram are cocycles however a metatheorem to this effect has proved to be beyond the authors capabilities . instead , we will approach the general problem indirectly by reducing the deformations of any pasting diagram to deformations of a related pasting diagram all of whose cells are one of a small finite set of forms , provided one can specify that certain cells are deformed trivially : the composition - free pasting diagrams for which the result was established in @xcite , those of [ singlecomposition ] and a short list of diagrams in which specified cells are deformed trivially : triangles and solid tetrahedra all of whose faces are identity natural transformations and are deformed 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4666,-3436)(0,0)[lb ] in @xcite deformations of a functor @xmath41 ( resp . a natural transformation @xmath19 ) in which the source and target are left undeformed ( or to say the same thing differently deformed trivially in the strong sense ) were shown to be governed by the hochschild complex @xmath292 ( resp . @xmath1 ) . for the desired reduction we will need to consider deformations of pasting diagrams in which a natural transformation ( in particular the identity natural transformation ) is deformed trivially in the strong sense while its domains and codomains are deformed , possibly nontrivially . as a warm - up and for potential use in other applications , let us consider first the problem of deforming the ( pasting ) diagram @xmath3 consisting of two categories @xmath42 and @xmath109 and a functor @xmath38 , subject to the requirement that @xmath41 be deformed trivially . without the restriction that @xmath41 be deformed trivially , a(n @xmath293 order ) deformation would be determined by a family of parallels for the set of operations @xmath294 , the compositions in @xmath42 and @xmath109 and the arrow - part of @xmath41 satisfying the usual cocycle and cobounding conditions . the restriction that @xmath41 be deformed trivially requires that all of the positive order parallels to @xmath41 be zero . cohomologically , the triples of degree @xmath0 parallels @xmath295 ( in the absence of the triviality requirement ) lie in @xmath155 , the mapping cone on @xmath296 and satisfy cocycle ( @xmath297 ) and cobounding ( @xmath298 ) conditions . the requirement that @xmath299 together with the coboundary condition thus implies that @xmath300 lies in @xmath301 . in fact we have first order deformations of the ( pasting ) diagram @xmath3 consisting of two categories @xmath42 and @xmath109 and a functor @xmath38 , subject to the requirement that @xmath41 be deformed trivially are classified by the second cohomology of @xmath301 . moreover all obstructions to deforming @xmath3 with @xmath41 deformed trivially are cocycles in the third cochain group of @xmath301 and an @xmath293 order deformation with @xmath41 deformed trivially can be extended to an @xmath302 order deformation with @xmath41 deformed trivially if and only if there is a degree 2 cochain in @xmath301 cobounding the degree @xmath163 obstruction . the classification of first order deformations follows from the remarks above . the rest of the proposition follows from the corresponding result in @xcite without the triviality restriction once it is shown that all obstructions , which _ a priori _ lie in the cone on @xmath303 lie in the kernel ( considered as a subcomplex under the obvious inclusion ) . as observed above , @xmath300 lies in @xmath301 . so suppose as an induction hypothesis that for all @xmath304 @xmath305 lies in @xmath301 ( or equivalently and more usefully @xmath306 ) . the degree @xmath163 obstruction @xmath168 in the cone then has coordinates @xmath307,\ ] ] @xmath308,\ ] ] and @xmath309 the last vanishes since each term involves @xmath310 for some @xmath311 ( and in the latter sum all the @xmath312 are bilinear ) . it thus remains only to show that @xmath313 of the first coordinate equals @xmath314 of the second . computing @xmath315)\ ] ] @xmath316\ ] ] @xmath317\ ] ] in each case the equality holds by the induction hypothesis , and in the second case the bilinearity @xmath312 for each @xmath0 . thus the proposition holds . a similar result holds for deformations of a diagram consisting of two parallel functors @xmath49 and a natural transformation @xmath19 between them which is to be deformed trivially : without the triviality restriction , the deformation complex is the cone on @xmath318 . with the restriction , @xmath319 becomes the deformation complex : first order deformations of the pasting diagram @xmath3 consisting of two categories a pair of parallel functors @xmath49 , their sources and target and a natural transformation @xmath19 subject to the requirement that @xmath28 be deformed trivially are classified by the second cohomology of @xmath319 . moreover all obstructions to deforming @xmath3 with @xmath28 deformed trivially are cocycles in the third cochain group of @xmath319 and an @xmath293 order deformation with @xmath28 deformed trivially can be extended to an @xmath302 order deformation with @xmath28 deformed trivially if and only if there is a degree 2 cochain in @xmath319 cobounding the degree @xmath163 obstruction . the argument is essentially identical to that for the previous proposition , except for the calculation showing that obstructions , which _ a priori _ lie in the full deformation complex of @xmath3 lie in @xmath319 . to see that all obstructions lie in @xmath319 , consider figure [ naturality_obstructions_are_cocycles ] . the condition that degree @xmath163 obstruction lies in @xmath319 is precisely the condition that the signed sum of the degree @xmath163 obstruction - type expressions associated to the non - trivial faces other than the three hexagons vanish , when all higher order parallels of @xmath28 are instantiated as 0 . but observe that the other faces are all trivial or evaluate to 0 , in all degrees greater than 0 , trivially under the hypothesis that all higher order parallels of @xmath28 are 0 , so the result follows from the general principles of the polygonal method . applying the same argument _ mutatis mutandis _ to figure [ nat_from_composition_obstructions_are_cocycles ] shows [ trivialtriangle ] first order deformations of the pasting diagram @xmath3 consisting of three categories @xmath320 and @xmath321 three functors @xmath38 , @xmath322 and @xmath323 and a natural transformation @xmath324 subject to the requirement that @xmath28 be deformed trivially are classified by the second cohomology of @xmath325 , where @xmath326:{\frak c}^\bullet(\partial d ) \rightarrow c^\bullet(g(f),h ) .\ ] ] where @xmath327 is the diagram obtained by omitting @xmath28 . moreover all obstructions to deforming @xmath3 with @xmath28 deformed trivially are cocycles in the third cochain group of @xmath325 and an @xmath293 order deformation with @xmath28 deformed trivially can be extended to an @xmath302 order deformation with @xmath28 deformed trivially if and only if there is a degree 2 cochain in @xmath325 cobounding the degree @xmath163 obstruction . in the special case where @xmath328 and @xmath329 , it is intuitively clear that the deformations of @xmath3 in which @xmath28 is deformed trivially are completely determined by the deformation complex of the simpler diagram @xmath330 . in fact we have [ reducestotwoedges ] in the deformation complex of the pasting diagram @xmath3 consisting of three categories @xmath320 and @xmath321 three functors @xmath38 , @xmath322 and @xmath331 and the natural transformation @xmath332 the complex @xmath325 where @xmath75 is as in proposition [ trivialtriangle ] is isomorphic to the deformation complex of the subdiagram @xmath330 . the obvious quotient map from the deformation complex of @xmath327 to @xmath333 is split by the obvious inclusion . the kernel of the quotient map is easily seen to be isomorphic to @xmath325 , while the quotient of the inclusion is easily seen to be the deformation complex of @xmath334 . combining propositions [ uniontopushout ] , [ trivialtriangle ] and [ reducestotwoedges ] then gives [ trivialthreecell ] let @xmath3 be a pasting diagram which could arise by the following construction : begin with a computad consisting of a composable 1-pasting diagram with any number of edges , a single edge with the same domain and codomain as the 1-pasting diagram , a pair of 2-cells both with the resulting circle as boundary ( and a 3-cell , or not ) . now , triangulate each 2-cell in a way corresponding to any parenthesization of the edges of the original 1-pasting diagram . apply a map of pasting schemes to the underlying pasting scheme of @xmath335 such that the value of every 2-cell is an identity natural transformation with domain given by two composable functors and codomain given by their composition . if @xmath3 is such a pasting diagram , then the single edge in the initial part of the construction is labeled with the composition of the initial 1-pasting diagram , and if , moreover , @xmath70 is the chain map from the deformations complex @xmath155 to the direct sum indexed by the triangular 2-cells of @xmath3 whose coordinate for a 2-cell @xmath336 with edges labeled by a composable pair of functors @xmath38 and @xmath322 and their composition @xmath337 is given by projection onto @xmath338 followed by the map @xmath75 of proposition [ trivialtriangle ] onto @xmath339 , then @xmath340 classifies the deformations of @xmath3 in which all of the identity natural transformations are deformed trivially , and is isomorphic to the deformation complex of the original composable 1-pasting diagram . the key thing to note here is that because all triangulations of disks are shellable , the pasting diagram can be constructed out of the trivial triangles of proposition [ trivialtriangle ] by iterated pushouts with diagrams previously so constructed until a pushout of the two triangulated disks along their boundary is made , and that once the triviality condition are imposed the presence or absence of the 3-cell makes no difference to the kernel . note also that a 2-dual form of proposition [ trivialthreecell ] in which all of the identity natural transformations have the composition as their domain and two composable functors as their codomain also holds with the same proof . the proofs of propositions [ trivialtriangle ] and [ reducestotwoedges ] can be extended to show [ fatpost1comp ] let @xmath3 be the pasting diagram consisting of three categories @xmath341 , functors @xmath342 , the composite functors @xmath343 and @xmath344 , natural transformations @xmath19 , @xmath345 labeling bigons , and @xmath346 and @xmath347 labeling triangles with two edges labeled by the composands as domain and one edge labeled by the composition as codomain , and a single 3-cell asserting the commutativity of the diagram . letting @xmath70 be the map with one coordinate for each triangle given by @xmath75 of proposition [ trivialtriangle ] for that 2-cell , @xmath340 classifies the deformations of @xmath3 in which the two identity natural transformations are deformed trivially , and is isomorphic to the complex of example [ equalspost1comp ] . and [ fatpre1comp ] let @xmath3 be the pasting diagram consisting of three categories @xmath341 , functors @xmath348 , the composite functors @xmath349 and @xmath350 , natural transformations @xmath19 , @xmath351 labeling bigons , and @xmath352 and @xmath353 labeling triangles with two edges labeled by the composands as domain and one edge labeled by the composition as codomain , and a single 3-cell asserting the commutativity of the diagram . letting @xmath70 be the map with one coordinate for each triangle given by @xmath75 of proposition [ trivialtriangle ] for that 2-cell , @xmath340 classifies the deformations of @xmath3 in which the two identity natural transformations are deformed trivially , and is isomorphic to the complex of example [ equalspre1comp ] . observe , that the conclusion of both proposition [ fatpost1comp ] and [ fatpre1comp ] hold regardless of which part of the boundary ( two composable functors or their composition ) is the domain and which is the codomain of each of the triangular faces labeled with identity natural transformations . we are now in a position to prove the main result : that the deformations of any pasting diagram are classified by deformation complex in which all obstructions to extending a deformation of order @xmath163 to a deformation of order @xmath354 are cocycles . to do this we will replace the diagram with a more complex diagram with simpler parts , whose deformation theory , when some of the 2-cells are required to be deformed trivially , coincides with that of the original pasting diagram . a pasting diagram is _ finely divided _ if every 2-cell is either a bigon or a triangle labeled with the identity natural transformation , and every 3-cell is of one of the following forms : * a composition - free pillow with bigonal cross - section ( imposing the equality of two natural transformations between the same pair of functors ) * a diagram of the form in proposition [ trivialthreecell ] or its 2-dual * a `` triangular pillow '' as in example [ triangularpillow ] * a diagram of the form in example [ equalspost1comp ] * a diagram of the form in proposition [ fatpost1comp ] * a diagram of the form in example [ equalspre1comp ] * a diagram of the form in proposition [ fatpre1comp ] every pasting diagram @xmath3 can be replaced with a finely divided pasting diagram @xmath355 whose underlying cell complex is a subdivision of the underlying cell complex of @xmath3 and whose deformations when all identity natural transformations labeling triangles are deformed trivially are equivalent to deformations of @xmath3 . begin by subdividing each 2-cell as follows : insert a bigon with 1-cells labeled by the 1-composition of the domain and codomain of the 2-cell and labeled by the same natural transformation as the original 2-cell , making the domain ( resp . codomain ) of the bigon coincide with that of the original 2-cell if the domain ( resp . codomain ) consists of a single 1-cell . label the complementary cell(s ) with identity natural transformations to be deformed trivially . now each of the complementary cells has a domain ( resp . codomain ) which is a sequence of 1-cells labeled by functors and a codomain ( resp . domain ) which is a single 1-cell labeled with their composition . choose a parenthesization of the composition and subdivide the 2-cell in the usual way corresponding to a parenthesization , labeling the new 1-cells with the appropriate pairwise compositions of labels already present and all 2-cells with identity natural transformation ( to be deformed trivially ) . now , for each 3-cell which is not already of one of the forms specified in the proposition , choose an order of pasting composition for the domain ( resp . codomain ) , and use this to create a subdivision of the 3-cell as follows : iteratively , for each 1-composition of a natural transformation with a functor , insert a diagram of the form in proposition [ fatpost1comp ] or [ fatpre1comp ] as appropriate and for each binary 2-composition , insert a triangular pillow . when this has been done for both the domain and codomain , there will be bigons labeled with the pasting compostion of the domain and codomain of the original 3-cell . identify their domains ( resp . codomains ) . the 3-cells of the resulting subdivision are now those explicitly inserted in the construction , a composition - free pillow with bigonal cross - section , and , we claim , 3-cells all of the form in proposition [ trivialthreecell ] . verifying the claim is a matter of keeping track of the triangular faces in such a way as to organize them into the pairs of triangulated disks with appropriate boundaries : note that immediately after the subdivision of 2-cells into a bigon and triangles , the triangulated cell(s ) ( if any the original 2-cell could have been a bigon ) are of the desired form : they have a domain ( resp . codomain ) consisting of a composable 1-pasting diagram and codomain ( resp . domain ) consisting of a single edge labeled with the composition . call such a triangulated 2-cell a `` nicely triangulated 2-cell '' . thus the remainder of the construction begins with the original 2-cells decomposed into a family of nicely triangulated 2-cells and family of bigons . throughout the rest of the construction the family of nicely triangulated 2-cells will be iteratively replaced with families with fewer nicely triangulated 2-cells ( but with more triangles in the newer ones ) . for addition of a 3-cell of the forms in propositions [ fatpost1comp ] or [ fatpre1comp ] , the triangles will have as two of their edges the single edge labeled with the composition from previously constructed nicely triangulated 2-cells . for each of the triangles , replace these two nicely triangulated 2-cells with the union of the triangle and the two nicely triangulated 2-cells , noticing that it is nicely triangulated . when the triangular pillows and the bigonal pillow are adjoined , the family of nicely triangulated 2-cells is unchanged . thus , when the construction ends , the remaining 3-cells in the decomposition must each be bounded by the union of two nicely triangulated 2-cells . we can now prove the main result ( recalling the dimension conventions of @xcite which leaves the deformation cohomology of a pasting digram in dimension @xmath356 ) : [ main ] for any pasting diagram @xmath3 of @xmath0-linear categories , @xmath0-linear functors and natural transformations , the deformations of @xmath3 are classified by the @xmath356-cohomology of the @xmath340 , where @xmath70 is the map from the deformation complex @xmath357 given by the same description as in proposition [ trivialthreecell ] , and all obstructions to extension of a degree @xmath163 deformation to a degree @xmath354 deformation are 0-cocycles . for the classification statement , first observe that the proof of proposition [ uniontopushout ] applies equally well to taking unions of pasting diagrams in which some 2-cells have been specified as deforming trivially ( provided that cells in the intersection are specified as deforming trivially in both pasting diagrams of which the union is being taken ) . the isomorphisms of propositions [ trivialthreecell ] , [ fatpost1comp ] and [ fatpre1comp ] then combine by universal property of pushouts to give an isomorphism between @xmath155 and @xmath340 in the category of chain complexes . that all obstructions are cocycles is a condition local to each cell together with its boundary , and all cells of @xmath355 are either deformed trivially , so that the obstruction vanishes by triviality , are of the forms which were shown to have vanishing obstructions in theorem [ singlecomposition ] , or are composition - free and thus have vanishing obstructions by the results of @xcite . the primary purpose of this paper , to complete the deformation theory for pasting diagrams of @xmath0-linear categories described in @xcite , was accomplished by theorem [ main ] . its primary importance , however , may lie less in the result than in the techniques used . it is the authors intent to apply the deformation theory of pasting diagrams to the still - open problem of providing a complete deformation theory for monoidal categories in which all arrow - valued elements are deformed simultaneously , rather than just the structure maps as in @xcite . this will require using extensions of @xmath358 hinted at above , in which monoidal prolongations are included as additional operations , as the basis of the polygonal method . it will also avoid the difficulties in @xcite arising from the need to intuit the correct formulas for higher differentials in a multicomplex from the scant data provided by the instances arising in the deformation theory . likewise , although for the present purpose there is something unsatisfying about our detour though finely divided pasting diagrams with specified trivially - deformed identity cells we believe the direct result , that all obstructions in the deformation complex of any pasting diagram as defined in @xcite are cocycles , though it is beyond our present ability to prove like the polygonal technique , this detour may have other applications . the difficulty with the direct result was that although 2-categories , and the pieces of them corresponding to direct summands in deformation complexes are inherently `` globular '' , pasting diagrams are `` opetopic '' . the reduction of the opetopic to trivial ( mostly simplicial ) elements and globular elements might well find application in connecting the zoo of opetopic definitions of weak n - categories with the globular approach of batanin .

we continue the development of the infinitesimal deformation theory of pasting diagrams of @xmath0-linear categories begun in @xcite . in @xcite the standard result that all obstructions are cocycles was established only for the elementary , composition - free parts of pasting diagrams . in the present work we give a proof for pasting diagrams in general . as tools we use the method developed by shrestha @xcite of representing formulas for obstructions , along with the corresponding cocycle and cobounding conditions by suitably labeled polygons , giving a rigorous exposition of the previously heuristic method , and deformations of pasting diagrams in which some cells are required to be deformed trivially . [ multiblock footnote omitted ]

arxiv

our aim in physics is not only to calculate some observable and get a correct number but mainly to understand a physical picture responsible for the given phenomenon . it very often happens that a theory formulated in terms of fundamental degrees of freedom can not answer such a question since it becomes overcomplicated at the related scale . thus a main task in this case is to select those degrees of freedom which are indeed essential . for instance , the fundamental degrees of freedom in crystals are ions in the lattice , electrons and the electromagnetic field . nevertheless , in order to understand electric conductivity , heat capacity , etc . we instead work with `` heavy electrons '' with dynamical mass , phonons and their interaction . in this case a complicated electromagnetic interaction of the electrons with the ions in the lattice is `` hidden '' in the dynamical mass of the electron and the interactions among ions in the lattice are eventually responsible for the collective excitations of the lattice - phonons , which are goldstone bosons of the spontaneously broken translational invariance in the lattice of ions . as a result , the theory becomes rather simple - only the electron and phonon degrees of freedom and their interactions are essential for all the properties of crystals mentioned above . quite a similar situation takes place in qcd . one hopes that sooner or later one can solve the full nonquenched qcd on the lattice and get the correct nucleon and pion mass in terms of underlying degrees of freedom : current quarks and gluon fields . however , qcd at the scale of 1 gev becomes too complicated , and hence it is rather difficult to say in this case what kind of physics , inherent in qcd , is relevant to the nucleon mass and its low - energy properties . in this lecture i will try to answer this question . i will show that it is the spontaneous breaking of chiral symmetry which is the most important qcd phenomenon in this case , and that beyond the scale of spontaneous breaking of chiral symmetry light and strange baryons can be viewed as systems of three constituent quarks which interact by the exchange of goldstone bosons ( pseudoscaler mesons ) , vector and scalar mesons ( which could be considered as a representation of a correlated goldstone boson exchange ) and are subject to confinement . at low temperatures and densities the @xmath6 chiral symmetry of qcd lagrangian is spontaneously broken down to @xmath7 by the qcd vacuum ( in the large @xmath8 limit it would be @xmath9 ) . a direct evidence for the spontaneously broken chiral symmetry is a nonzero value of the quark condensates for the light flavors @xmath10 which represent the order parameter . that this is indeed so , we know from three sources : current algebra , qcd sum rules , and lattice gauge calculations . there are two important generic consequences of the spontaneous breaking of chiral symmetry ( sbcs ) . the first one is an appearance of the octet of pseudoscalar mesons of low mass , @xmath11 , which represent the associated approximate goldstone bosons ( in the large @xmath8 limit the flavor singlet state @xmath12 should be added ) . the second one is that valence ( practically massless ) quarks acquire a dynamical mass , which have been called historically constituent quarks . indeed , the nonzero value of the quark condensate itself implies at the formal level that there should be rather big dynamical mass , which could be in general a moment - dependent quantity . thus the constituent quarks should be considered as quasiparticles whose dynamical mass comes from the nonperturbative gluon and quark - antiquark dressing . the flavor - octet axial current conservation in the chiral limit tells that the constituent quarks and goldstone bosons should be coupled with the strength @xmath13 @xcite , which is a quark analog of the famous goldberger - treiman relation . we can not say at the moment for sure what is the microscopical mechanism for sbcs in qcd . any sufficiently strong scalar interaction between quarks will induce the sbcs ( e.g. the instanton - induced interaction contains the scalar part , or it can be generated by monopole condensation , etc . ) . all these general aspects of sbcs are well illustrated by the nambu and jona - lasinio model @xcite , where the constituent mass is generated by the scalar part of some nonperturbative local gluonic interaction between current quarks while its pseudoscalar part gives rise to a relativistic deeply - bound pseudoscalar @xmath14 systems as goldstone bosons . accordingly one arrives at the following interpretation of light and strange baryons in the low - energy regime . the highly nonperturbative gluodynamics gives rise to correlated quark - antiquark structures in the baryon sea ( virtual mesons ) . at the same time the current valence quarks get dressed by the quark condensates and by the meson loops . the strongly - correlated quark - antiquark pairs in the pseudoscalar channel manifest themselves by virtual pseudoscalar mesons , while the weakly - correlated pairs in other channels - by vector , etc mesons . when one integrates over the meson fields in the baryon wave function one arrives at the simple qqq fock component with confined constituent quarks and with residual interaction between them mediated by the corresponding meson fields @xcite . the complimentary description of the vector - meson fields as well as of the scalar ones as arising from the correlated goldstone bosons is also possible , which does not contradict , however , to their interpretation as weakly bound @xmath14 systems . the coupling of the constituent quarks and the pseudoscalar goldstone bosons will ( in the @xmath15 symmetric approximation ) have the form @xmath16 within the nonlinear realization of chiral symmetry ( it would be @xmath17 within the linear @xmath18-model chiral symmetry representation ) . a coupling of this form , in a nonrelativistic reduction for the constituent quark spinors , will to lowest order give rise the @xmath19 structure of the meson - quark vertex , where @xmath20 is meson momentum . thus , the structure of the potential between quarks `` @xmath21 '' and `` @xmath22 '' in momentum representation is @xmath23 where @xmath24 is dressed green function for chiral field which includes both nonlinear terms of chiral lagrangian and fermion loops , @xmath25 is meson - quark formfactor which takes into account the internal structure of quasiparticles . at big distances ( @xmath26 ) , one has @xmath27 and @xmath28 . it then follows from ( [ 2 ] ) that @xmath29 , which means that the volume integral of the goldstone boson exchange ( gbe ) interaction should vanish , @xmath30 this sum rule is not valid in the chiral limit , however , where @xmath31 and , hence , @xmath32 . the sum rule ( [ 1 ] ) is trivial for the tensor component of the pseudoscalar - exchange interaction since the tensor force component automatically vanishes on averaging over the directions of @xmath33 . but for the spin - spin component the sum rule ( [ 1 ] ) indicates that there must be a strong short - range term . indeed , at big interquark separations the spin - spin component of the pseudoscalar - exchange interaction is @xmath34 , it then follows from the sum rule above that at short interquark separations the spin - spin interaction should be opposite in sign as compared to the yukawa tail and very strong : @xmath35 _ it is this short - range part of the goldstone boson exchange ( gbe ) interaction between the constituent quarks that is of crucial importance for baryons : it has a sign appropriate to reproduce the level splittings and dominates over the yukawa tail towards short distances . _ in a oversimplified consideration with a free klein - gordon green function instead of the dressed one in ( [ 2 ] ) and with @xmath36 , one obtains the following spin - spin component of @xmath37 interaction : @xmath38 in the chiral limit only the negative short - range part of the gbe interaction survives . as a representation of a multiple correlated goldstone - boson - exchange one can include the exchange by a scalar flavor - singlet `` @xmath18''-meson as well as by vector mesons . the `` @xmath18''-meson exchange does not play a principal role in baryons since it does not contain the spin - spin and tensor - force component . its attractive central potential as well as a weak spin - orbit force can be effectively included into confining interaction in baryons . however , these forces are of significant importance in nn system at medium range . the coupling of constituent quarks with the octet vector meson field @xmath39 in the @xmath40 - symmetric approximation is @xmath41 where @xmath42 and @xmath43 are the vector- and tensor coupling constants . a coupling of this form - to lowest order - will give rise the spin - spin , tensor , spin - orbit , and central interactions between the constituent quarks . it is very instructive to compare signs of the spin - spin and tensor components of pseudoscalar- and vector - exchange interactions . the spin - spin and tensor components of the pseudoscalar - exchange interaction arise from the @xmath44 structure of the pseudoscalar - meson constituent quark vertex as : @xmath45 . \label{3.2}\ ] ] the coupling of the vector - meson to constituent quark gives @xmath46 structure of the vertex . therefore the spin - spin and tensor components of the vector - meson exchange interaction arise as @xmath47 . \label{3.3}\ ] ] comparing ( [ 3.2 ] ) with ( [ 3.3 ] ) one observes that the spin - spin component of the vector - exchange has the same sign as the spin - spin component of the pseudoscalar - exchange interaction and is `` two times stronger '' while their tensor components have opposite signs . the @xmath46 structure of the vector - meson constituent quark vertex also suggests that the spin - spin component of the vector - meson - exchange interaction should satisfy the same sum rule ( [ 1 ] ) as the pseudoscalar - exchange interaction . then similar to pseudoscalar - exchange interaction there should be a short - range term in the vector - meson exchange interaction of the form ( [ 1.1 ] ) . again , if one neglects the spatial structure of the meson - quark vertex and uses a free green function for the vector - meson field , this short range term is described by a @xmath48 - function piece similar to equation ( [ 3 ] ) . summarizing , both vector- and pseudoscalar - exchange interactions produce the short - range spin - flavor force ( [ 1.1 ] ) while their tensor forces largely cancel each other in baryons . this observation is crucial for the present model . + the spin - orbit component associated with the vector - meson exchange interaction is big and empirically very important in nn system . so the question is why this spin - orbit force is big in nn system and becomes inessential in baryons , where the spin - orbit splittings are generally very small ( see e.g. at @xmath49 , @xmath50 , ... ls multiplets ) . the reason for this remarkable phenomenon is an explicit flavor dependence of the vector - meson - exchange ls force . consider for simplicity the @xmath51 case , i.e. an exchange by @xmath2- and @xmath3-mesons between u and d quarks . the @xmath2-meson is isovector and the @xmath2-meson exchange potential contains the factor @xmath52 . the @xmath3-meson is isoscalar and the @xmath3-exchange interaction does not contain the isospin - dependent factor . in @xmath53 nn partial wave the isospin of the two - nucleon system is @xmath54 and the isospin matrix element for the @xmath2-meson exchange potential is @xmath55 . thus both @xmath3- and @xmath2-meson exchange spin - orbit forces contribute with the same sign in nn system . numerically the contribution from @xmath3-exchange is 2.5 - 3 times bigger than from the @xmath2-meson exchange spin - orbit force . now let us consider @xmath53 state of two light quarks in baryon . in the present case the isospin of the quark pair is @xmath56 , due to the presence of the color part of wave function . then for the @xmath2-meson exchange one obtains @xmath57 . thus the @xmath2-meson exchange spin - orbit force obtains opposite sign and becomes strongly enhanced in baryons . as a result one observes a strong cancellation of the @xmath2- and @xmath3-meson exchange spin - orbit forces in baryons and the net weak spin - orbit interaction does not induce appreciable splittings in baryons . numerical details about both the tensor- and spin - orbit force cancellation in baryons can be found in ref . summarizing previous sections one concludes that the pseudoscalar- and vector - meson exchange interactions produce strong flavor - spin interaction ( [ 1.1 ] ) at short range while the net tensor and spin - orbit forces are rather weak . that the net spin - orbit and tensor interactions between constituent quarks _ in baryons _ should be weak also follows from the typically small splittings in ls - multiplets , which are of the order 10 - 30 mev . these small splittings should be compared with the hyperfine splittings produced by spin - spin force , which are of the order of @xmath58 splitting . thus , indeed , in baryons it is the spin - spin interaction ( [ 1.1 ] ) between constituent quarks is of crucial importance . consider first , for the purposes of illustration , a schematic model which neglects the radial dependence of the potential function @xmath59 in ( [ 1.1 ] ) , and assume a harmonic confinement among quarks as well as @xmath60 . in this model @xmath61 note , that contrary to the color - magnetic interaction from perturbative one - gluon exchange , the chiral interaction is explicitly flavor - dependent . it is this circumstance which allows to solve the long - standing problem of ordering of the lowest positive - negative parity states . if the only interaction between the quarks were the flavor- and spin - independent harmonic confining interaction , the baryon spectrum would be organized in multiplets of the symmetry group @xmath62 . in this case the baryon masses would be determined solely by the orbital structure , and the spectrum would be organized in an _ alternative sequence of positive and negative parity states , _ i.e. in this case the spectrum would be : ground state of positive parity ( @xmath63 shell , @xmath64 is the number of harmonic oscillator excitations in a 3-quark state ) , first excited band of negative parity ( @xmath65 ) , second excited band of positive parity ( @xmath66 ) , etc . the hamiltonian ( [ 4 ] ) , within a first order perturbation theory , reduces the @xmath62 symmetry down to @xmath67 , which automatically implies a splitting between the octet and decuplet baryons ( e.g. the @xmath68 resonance becomes heavier than nucleon ) . let us now see how the pure confinement spectrum above becomes modified when the gbe hamiltonian ( [ 4 ] ) is switched on . for the octet states @xmath69 , @xmath70 , @xmath71 , @xmath72 ( @xmath63 shell ) as well as for their first radial excitations of positive parity @xmath73 , @xmath74 , @xmath75 , @xmath76 ( @xmath66 shell ) the expectation value of the hamiltonian ( [ 4 ] ) is @xmath77 . for the decuplet states @xmath68 , @xmath78 , @xmath79 , @xmath80 ( @xmath63 shell ) the corresponding matrix element is @xmath81 . in the negative parity excitations ( @xmath65 shell ) in the @xmath69 , @xmath70 and @xmath71 spectra ( @xmath82 - @xmath83 , @xmath84 - @xmath85 and @xmath86 - @xmath87 ) the contribution of the interaction ( [ 4 ] ) is @xmath88 . the first negative parity excitation in the @xmath70 spectrum ( @xmath65 shell ) @xmath89 - @xmath90 is flavor singlet and , in this case , the corresponding matrix element is @xmath91 . the latter state is unique and is absent in other spectra due to its flavor - singlet nature . these matrix elements alone suffice to prove that the ordering of the lowest positive and negative parity states in the baryon spectrum will be correctly predicted by the chiral boson exchange interaction ( [ 4 ] ) . the constant @xmath92 may be determined from the n@xmath93 splitting to be 29.3 mev . the oscillator parameter @xmath94 , which characterizes the effective confining interaction , may be determined as one half of the mass differences between the first excited @xmath95 states and the ground states of the baryons , which have the same flavor - spin , flavor and spin symmetries ( e.g. @xmath73 - @xmath69 , @xmath74 - @xmath70 , @xmath75 - @xmath71 ) , to be @xmath96 mev . thus the two free parameters of this simple model are fixed and we can make now predictions . in the @xmath69 , @xmath70 and @xmath71 sectors the mass difference between the lowest excited @xmath97 states ( @xmath73 , @xmath74 , and @xmath75 ) and @xmath98 negative parity pairs ( @xmath82 - @xmath83 , @xmath84 - @xmath85 , and @xmath86 - @xmath87 , respectively ) will then be @xmath99 whereas for the lowest states in the @xmath70 system ( @xmath74 , @xmath89 - @xmath90 ) it should be @xmath100 this simple example shows how the chiral interaction provides different ordering of the lowest positive and negative parity excited states in the spectra of the nucleon and the @xmath70-hyperon . this is a direct consequence of the symmetry properties of the boson - exchange interaction @xcite . namely , completely symmetric fs state in the @xmath73 , @xmath74 and @xmath75 positive parity resonances from the @xmath66 band feels a much stronger attractive interaction than the mixed symmetry state in the @xmath82 - @xmath83 , @xmath84 - @xmath85 and @xmath86 -@xmath87 resonances of negative parity ( @xmath65 shell ) . consequently the masses of the positive parity states @xmath73 , @xmath74 and @xmath75 are shifted down relative to the other ones , which explains the reversal of the otherwise expected `` normal ordering '' . the situation is different for @xmath89 - @xmath90 and @xmath74 , as the flavor state of @xmath89 - @xmath90 is totally antisymmetric . because of this the @xmath89 - @xmath90 gains an attractive energy , which is comparable to that of the @xmath74 , and thus the ordering suggested by the confining oscillator interaction is maintained . note that the problem of the relative position of positive - negative parity states can not be solved with other types of hyperfine interactions between constituent quarks ( the colour - magnetic and instanton - induced ones ) . in the semirelativistic chiral constituent quark model @xcite the dynamical part of the hamiltonian consists of linear pairwise confining interaction with the string tension fixed to the known value 1 gev / fm from regge slopes ( which also follows from the heavy quarkonium spectroscopy and lattice calculations ) , and the chiral interaction , mediated by pseudoscalar , scalar and vector - meson exchanges . both the flavor - octet and flavor - singlet pseudoscalar- and vector - meson exchanges are taken into account . the coupling constants of constituent quarks with mesons are fixed from the empirically known @xmath101 , @xmath102 , and @xmath103 coupling constants . the `` sigma - meson '' constituent quark coupling constant is taken to be equal the pion constituent quark coupling constant , as constrained by chiral symmetry . for the constituent quark masses one takes typical values @xmath104 mev , @xmath105 mev and do not fit them . the short - range behaviour of the interaction is determined by the cut - off parameters @xmath70 in the constituent quark meson form - factors , which are taken in monopole form . in order to avoid a proliferation of free parameters , by assuming independent values of @xmath70 for each meson , we adopt the linear scaling prescription @xmath106 , where @xmath107 is meson mass , for pseudoscalar- and vector mesons . the kinetic - energy operator is taken in relativistic form , @xmath108 . the semirelativistic three - quark hamiltonian was solved along the stochastical variational method @xcite in momentum space . for the whole q - q potential the model involves a total of 4 free parameters whose numerical values are determined from the fit to all 35 confirmed low - lying states . in fig . 1 we present the ground states as well as low - lying excited states in @xmath64 , @xmath68 , @xmath70 , @xmath71 , @xmath72 , and @xmath80 spectra . from the results of fig . 1 it becomes evident that within the chiral constituent quark model a unified description of both nonstrange and strange baryon spectra is achieved in good agreement with phenomenology . it is instructive to learn how the spin - spin interaction ( [ 1.1 ] ) affects the energy levels when it is switched on and its strength is gradually increased ( fig . 2 ) . starting out from the case with confinement only , one observes that the degeneracy of states is removed and the inversion of ordering of positive and negative parity states is achieved in the @xmath64 spectrum , as well as for some states in the @xmath70 spectrum , while the ordering of the lowest positive - negative parity states is opposite in @xmath64 and @xmath70 spectra . it is clear that the fock components @xmath109 ( including meson continuum ) can not be completely integrated out in favour of the meson - exchange @xmath37 potentials for some states above or near the corresponding meson thresholds . such components in addition to the main one @xmath110 could explain e.g. an exceptionally big splitting of the flavor singlet states @xmath111 , since the @xmath89 lies below the @xmath112 threshold and can be presented as @xmath112 bound system @xcite . note , that in the case of the present approach this old idea is completely natural and does not contradict a flavor - singlet @xmath110 nature of @xmath89 , while it would be in conflict with naive constituent quark model where no room for mesons in baryons . an admixture of such components will be important in order to understand strong decays of some excited states . while technically inclusion of such components in addition to the main one @xmath110 in a coupled - channel approach is rather difficult task , it should be considered as one of the most important future directions . what is an intuitive picture of the nucleon in the low - energy regime ? the goldstone bosons as well as vector meson fields couple to valence quarks . thus the nucleon consists mostly of 3 constituent quarks which are very big objects due to their meson clouds ( see fig . 3 ) . these constituent quarks are all the time in strong overlap inside the nucleon . that is why the short - range part of chiral interaction ( which is represented by the contact term in the oversimplified representation ) is so crucially important inside baryons . when constituent quarks are well separated and there is a phase space for meson propagation , the long - range yukawa tails of meson - exchange interactions become very important . it is these parts of meson exchange which produce the necessary long- and intermediate - range attraction in two - nucleon system . if one ascribes a short - range central repulsion in the nn system to the central part of @xmath3 exchange between nucleons , then one should increase the @xmath103 coupling constant by factor 3 as compared to its empirical value , as it is usually done in phenomenological one - boson - exchange nn potentials . evidently the short - range repulsion in the nn system should be connected with the nucleon structure in the low - energy regime and within the quark picture should be related to fermi - nature of constituent quarks and to the specific interactions between them . so far , all studies of the short - range @xmath4 interaction within the constituent quark model were based on the one - gluon exchange interaction between quarks . they explained the short - range repulsion in the @xmath4 system as due to the colour - magnetic part of oge combined with quark interchanges between 3q clusters @xcite . it has been shown , however , that there is practically no room for colour - magnetic interaction in light baryon spectroscopy and any appreciable amount of colour - magnetic interaction , in addition to chiral interaction , destroys the spectrum @xcite . this conclusion is confirmed by recent lattice qcd calculations @xcite . if so , the question arises which interquark interaction is responsible for the short - range @xmath4 repulsion . below i show that the same short - range part of chiral interaction ( [ 1.1 ] ) which causes e.g. @xmath113 splitting and produces good baryon spectra , also induces a short - range repulsion in @xmath4 system when the latter is treated as 6q system @xcite . at present one can use only a simple nonrelativistic @xmath114 ansatz for the nucleon wave function when one applies the quark model to @xmath4 interaction . thus one needs first an effective nonrelativistic parametrization of chiral interaction interaction which would provide correct nucleon mass , @xmath113 splitting and the nucleon stability with this ansatz . for that one can use the nonrelativistic parametrization @xcite which satisfies approximately the conditions above . where @xmath116 is a collective ( generator ) coordinate which is the separation distance between the two wells ( it should not be mixed with the relative motion jacobi coordinate ) , @xmath117 is the lowest expectation value of the 6q hamiltonian at fixed @xmath116 , and @xmath118 is a mass of two well - separated nucleons ( @xmath119 ) calculated with the same hamiltonian . at the moment we are interested in what is the @xmath4 interaction at zero separation between nucleons . it has been proved by harvey that when @xmath120 , then in both @xmath121 and @xmath122 partial waves in the @xmath4 system only two types of orbital 6q configurations survive @xcite : @xmath123_o>$ ] and @xmath124_o>$ ] , where @xmath125_o$ ] is young diagram , describing spatial permutational symmetry in 6q system . there are a few different flavor - spin symmetries , compatible with the spatial symmetries above : @xmath126_o[33]_{fs}$ ] , @xmath127_o[33]_{fs}$ ] , @xmath127_o[51]_{fs}$ ] , @xmath127_o[411]_{fs}$ ] , @xmath127_o[321]_{fs}$ ] , and @xmath127_o[2211]_{fs}$ ] . thus , in order to evaluate the @xmath4 interaction at zero separation between nucleons it is necessary to diagonalize a @xmath5 hamiltonian in the basis above and use the procedure ( [ 6 ] ) . from the adiabatic born - oppenheimer approximation ( [ 6 ] ) we find that @xmath128 is highly repulsive in both @xmath121 and @xmath122 partial waves , with the core being of order 1 gev . this repulsion implies a strong suppression of the @xmath4 wave function in the nucleon overlap region . due to the specific flavor - spin symmetry of chiral interaction between quarks the configuration @xmath129_o[51]_{fs}$ ] becomes highly dominant among other possible 6q configurations at zero separation between nucleons ( however , the `` energy '' of this configuration is much higher than the energy of two well - separated nucleons , that is why there is a strong short - range repulsion in @xmath4 system ) . the symmetry structure of this dominant configuration induces `` an additional '' effective repulsion , related to the `` pauli forbidden state '' in this case , and the s - wave @xmath4 relative motion wave function has a node at short range@xcite . the existence of a strong repulsion , related to the energy ballance , discussed above , suggests , however , that the amplitude of the oscillating @xmath4 wave function at short range will be strongly suppressed . thus , within the chiral constituent quark model one has all the necessary ingredients to understand microscopically the @xmath4 interaction . there appears strong short - range repulsion from the same short - range part of chiral interaction which also produces hyperfine splittings in baryon spectroscopy . the long- and intermediate - range attraction in the @xmath4 system is automatically implied by the yukawa part of pion - exchange and correlated two - pion exchanges ( @xmath18-exchange ) between quarks belonging to different nucleons . the necessary tensor and spin - orbit force comes from the pseudoscalar- and vector - exchange yukawa tails . what will be a short - range interaction in other @xmath130 and @xmath131 systems ? in the chiral limit there is no difference between all octet baryons : @xmath132 . thus if one explains a strong short - range repulsion in @xmath4 system as related mostly to the spontaneous breaking of chiral symmetry , as above , then the same short - range repulsion should persist in other @xmath130 and @xmath131 systems @xcite . of course , due to the explicit chiral symmetry breaking the strength of this repulsion should be essentially different as compared to that one in @xmath4 system . one can naively expect that it will be weaker . it is my pleasure to thank d.o.riska , z.papp , w.plessas , k.varga , r. wagenbrunn , fl . stancu , and s. pepin , in collaboration with whom different results discussed in this talk have been obtained . i also thank organizers of the school for the invitation to give this lecture . a. manohar and h. georgi , nucl b234 * ( 1984 ) 189 ; d. i. diakonov and v. yu . petrov , nucl . phys . * b272 * ( 1986 ) 457 . y. nambu and g. jona - lasinio , phys . * 122 * ( 1961 ) 345 ; u. vogl and w. weise , progr . * 27 * ( 1991 ) 195 ; s. p. klevansky , rev . mod . * 64 * ( 1992 ) 649 ; t. hatsuda and t. kunihiro , phys . rep . * 247 * ( 1994 ) 221 . glozman and d. o. riska , physics reports * 268 * ( 1996 ) 263 . glozman , w. plessas , k. varga , r. wagenbrunn , in preparation . glozman , w. plessas , k. varga , r. wagenbrunn , hep - ph/9706507 . k. varga and y. suzuki , phys . rev . * c52 * ( 1995 ) 2885 ; k. varga , y. ohbayasi , and y. suzuki , phys . lett . * b396 * ( 1997 ) 1 . r. h. dalitz and a. deloff , j. phys . * g 17 * ( 1991 ) 289 ( and references therein ) . m. oka and k. yazaki , int . , vol . 1 ( quarks and nuclei , ed . w. weise ) , world scientific , singapore , p. 490 glozman , z. papp , w. plessas , k. varga , and r. wagenbrunn , phys . * c57 * ( 1998 ) [ nucl - th/9705011 ] . k. f. liu and s. j. dong - paper in preparation ( for a preliminary result see hep - lat/9411067 ) . glozman , z. papp , and w. plessas , phys . * b381 * ( 1996 ) 311 . stancu , s. pepin , and l. ya . glozman , phys . rev . * c56 * ( 1997 ) 2779 . m. harvey , nucl . phys . * a352 * ( 1981 ) 301 . v. g. neudatchin , yu . f. smirnov , and r. tamagaki , progr . . phys . * 58 * ( 1977 ) 1072 . stancu , s. pepin , and l. ya . glozman , phys . rev . * d57 * ( 1998 ) 4393 .

in the low - energy regime light and strange baryons should be considered as systems of constituent quarks with confining interaction and a chiral interaction that is mediated by goldstone bosons as well as by vector and scalar mesons . the flavor - spin structure and sign of the short - range part of the spin - spin force reduces the @xmath0 symmetry down to @xmath1 , induces hyperfine splittings and provides correct ordering of the lowest states with positive and negative parity . there is a cancellation of the tensor force from pseudoscalar- and vector - exchanges in baryons . the spin - orbit interactions from @xmath2-like and @xmath3-like exchanges also cancel each other in baryons while they produce a big spin - orbit force in nn system . a unified description of light and strange baryon spectra calculated in a semirelativistic framework is presented . it is demonstrated that the same short - range part of spin - spin interaction between the constituent quarks induces a strong short - range repulsion in @xmath4 system when the latter is treated as @xmath5 system . thus one can achieve a simultaneous understanding of a baryon structure and baryon - baryon interaction in the low - energy regime . -1 cm -0.5 cm -1 cm 23 cm * baryons in a chiral constituent quark model * l. ya . glozman _ institute for theoretical physics , university of graz , austria _

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