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the transport properties of nonlinear non - equilibrium dynamical systems are far from well - understood@xcite . consider in particular so - called ratchet systems which are asymmetric periodic potentials where an ensemble of particles experience directed transport@xcite . the origins of the interest in this lie in considerations about extracting useful work from unbiased noisy fluctuations as seems to happen in biological systems@xcite . recently attention has been focused on the behavior of deterministic chaotic ratchets@xcite as well as hamiltonian ratchets@xcite . chaotic systems are defined as those which are sensitively dependent on initial conditions . whether chaotic or not , the behavior of nonlinear systems including the transition from regular to chaotic behavior is in general sensitively dependent on the parameters of the system . that is , the phase - space structure is usually relatively complicated , consisting of stability islands embedded in chaotic seas , for examples , or of simultaneously co - existing attractors . this can change significantly as parameters change . for example , stability islands can merge into each other , or break apart , and the chaotic sea itself may get pinched off or otherwise changed , or attractors can change symmetry or bifurcate . this means that the transport properties can change dramatically as well . a few years ago , mateos@xcite considered a specific ratchet model with a periodically forced underdamped particle . he looked at an ensemble of particles , specifically the velocity for the particles , averaged over time and the entire ensemble . he showed that this quantity , which is an intuitively reasonable definition of ` the current ' , could be either positive or negative depending on the amplitude @xmath0 of the periodic forcing for the system . at the same time , there exist ranges in @xmath0 where the trajectory of an individual particle displays chaotic dynamics . mateos conjectured a connection between these two phenomena , specifically that the reversal of current direction was correlated with a bifurcation from chaotic to periodic behavior in the trajectory dynamics . even though it is unlikely that such a result would be universally valid across all chaotic deterministic ratchets , it would still be extremely useful to have general heuristic rules such as this . these organizing principles would allow some handle on characterizing the many different kinds of behavior that are possible in such systems . a later investigation@xcite of the mateos conjecture by barbi and salerno , however , argued that it was not a valid rule even in the specific system considered by mateos . they presented results showing that it was possible to have current reversals in the absence of bifurcations from periodic to chaotic behavior . they proposed an alternative origin for the current reversal , suggesting it was related to the different stability properties of the rotating periodic orbits of the system . these latter results seem fundamentally sensible . however , this paper based its arguments about currents on the behavior of a _ single _ particle as opposed to an ensemble . this implicitly assumes that the dynamics of the system are ergodic . this is not true in general for chaotic systems of the type being considered . in particular , there can be extreme dependence of the result on the statistics of the ensemble being considered . this has been pointed out in earlier studies @xcite which laid out a detailed methodology for understanding transport properties in such a mixed regular and chaotic system . depending on specific parameter value , the particular system under consideration has multiple coexisting periodic or chaotic attractors or a mixture of both . it is hence appropriate to understand how a probability ensemble might behave in such a system . the details of the dependence on the ensemble are particularly relevant to the issue of the possible experimental validation of these results , since experiments are always conducted , by virtue of finite - precision , over finite time and finite ensembles . it is therefore interesting to probe the results of barbi and salerno with regard to the details of the ensemble used , and more formally , to see how ergodicity alters our considerations about the current , as we do in this paper . we report here on studies on the properties of the current in a chaotic deterministic ratchet , specifically the same system as considered by mateos@xcite and barbi and salerno@xcite . we consider the impact of different kinds of ensembles of particles on the current and show that the current depends significantly on the details of the initial ensemble . we also show that it is important to discard transients in quantifying the current . this is one of the central messages of this paper : broad heuristics are rare in chaotic systems , and hence it is critical to understand the ensemble - dependence in any study of the transport properties of chaotic ratchets . having established this , we then proceed to discuss the connection between the bifurcation diagram for individual particles and the behavior of the current . we find that while we disagree with many of the details of barbi and salerno s results , the broader conclusion still holds . that is , it is indeed possible to have current reversals in the absence of bifurcations from chaos to periodic behavior as well as bifurcations without any accompanying current reversals . the result of our investigation is therefore that the transport properties of a chaotic ratchet are not as simple as the initial conjecture . however , we do find evidence for a generalized version of mateos s conjecture . that is , in general , bifurcations for trajectory dynamics as a function of system parameter seem to be associated with abrupt changes in the current . depending on the specific value of the current , these abrupt changes may lead the net current to reverse direction , but not necessarily so . we start below with a preparatory discussion necessary to understand the details of the connection between bifurcations and current reversal , where we discuss the potential and phase - space for single trajectories for this system , where we also define a bifurcation diagram for this system . in the next section , we discuss the subtleties of establishing a connection between the behavior of individual trajectories and of ensembles . after this , we are able to compare details of specific trajectory bifurcation curves with current curves , and thus justify our broader statements above , after which we conclude . the goal of these studies is to understand the behavior of general chaotic ratchets . the approach taken here is that to discover heuristic rules we must consider specific systems in great detail before generalizing . we choose the same @xmath1-dimensional ratchet considered previously by mateos@xcite , as well as barbi and salerno@xcite . we consider an ensemble of particles moving in an asymmetric periodic potential , driven by a periodic time - dependent external force , where the force has a zero time - average . there is no noise in the system , so it is completely deterministic , although there is damping . the equations of motion for an individual trajectory for such a system are given in dimensionless variables by @xmath2 where the periodic asymmetric potential can be written in the form @xmath3 + \frac{1}{4 } \sin [ 4\pi ( x -x_0 ) ] \bigg ] .\ ] ] in this equation @xmath4 have been introduced for convenience such that one potential minimum exists at the origin with @xmath5 and the term @xmath6 . ( a ) classical phase space for the unperturbed system . for @xmath7 , @xmath8 , two chaotic attractors emerge with @xmath9 ( b ) @xmath10 ( c ) and a period four attractor consisting of the four centers of the circles with @xmath11.,title="fig:",width=302 ] the phase - space of the undamped undriven ratchet the system corresponding to the unperturbed potential @xmath12 looks like a series of asymmetric pendula . that is , individual trajectories have one of following possible time - asymptotic behaviors : ( i ) inside the potential wells , trajectories and all their properties oscillate , leading to zero net transport . outside the wells , the trajectories either ( ii ) librate to the right or ( iii ) to the left , with corresponding net transport depending upon initial conditions . there are also ( iv ) trajectories on the separatrices between the oscillating and librating orbits , moving between unstable fixed points in infinite time , as well as the unstable and stable fixed points themselves , all of which constitute a set of negligible measure . when damping is introduced via the @xmath13-dependent term in eq . [ eq : dyn ] , it makes the stable fixed points the only attractors for the system . when the driving is turned on , the phase - space becomes chaotic with the usual phenomena of intertwining separatrices and resulting homoclinic tangles . the dynamics of individual trajectories in such a system are now very complicated in general and depend sensitively on the choice of parameters and initial conditions . we show snapshots of the development of this kind of chaos in the set of poincar sections fig . ( [ figure1]b , c ) together with a period - four orbit represented by the center of the circles . a broad characterization of the dynamics of the problem as a function of a parameter ( @xmath14 or @xmath15 ) emerges in a bifurcation diagram . this can be constructed in several different and essentially equivalent ways . the relatively standard form that we use proceeds as follows : first choose the bifurcation parameter ( let us say @xmath0 ) and correspondingly choose fixed values of @xmath16 , and start with a given value for @xmath17 . now iterate an initial condition , recording the value of the particle s position @xmath18 at times @xmath19 from its integrated trajectory ( sometimes we record @xmath20 . this is done stroboscopically at discrete times @xmath21 where @xmath22 and @xmath23 is an integer @xmath24 with @xmath25 the maximum number of observations made . of these , discard observations at times less than some cut - off time @xmath26 and plot the remaining points against @xmath27 . it must be noted that discarding transient behavior is critical to get results which are independent of initial condition , and we shall emphasize this further below in the context of the net transport or current . if the system has a fixed - point attractor then all of the data lie at one particular location @xmath28 . a periodic orbit with period @xmath29 ( that is , with period commensurate with the driving ) shows up with @xmath30 points occupying only @xmath31 different locations of @xmath32 for @xmath27 . all other orbits , including periodic orbits of incommensurate period result in a simply - connected or multiply - connected dense set of points . for the next value @xmath33 , the last computed value of @xmath34 at @xmath35 are used as initial conditions , and previously , results are stored after cutoff and so on until @xmath36 . that is , the bifurcation diagram is generated by sweeping the relevant parameter , in this case @xmath0 , from @xmath27 through some maximum value @xmath37 . this procedure is intended to catch all coexisting attractors of the system with the specified parameter range . note that several initial conditions are effectively used troughout the process , and a bifurcation diagram is not the behavior of a single trajectory . we have made several plots , as a test , with different initial conditions and the diagrams obtained are identical . we show several examples of this kind of bifurcation diagram below , where they are being compared with the corresponding behavior of the current . having broadly understood the wide range of behavior for individual trajectories in this system , we now turn in the next section to a discussion of the non - equilibrium properties of a statistical ensemble of these trajectories , specifically the current for an ensemble . the current @xmath38 for an ensemble in the system is defined in an intuitive manner by mateos@xcite as the time - average of the average velocity over an ensemble of initial conditions . that is , an average over several initial conditions is performed at a given observation time @xmath39 to yield the average velocity over the particles @xmath40 this average velocity is then further time - averaged ; given the discrete time @xmath39 for observation this leads to a second sum @xmath41 where @xmath25 is the number of time - observations made . for this to be a relevant quantity to compare with bifurcation diagrams , @xmath38 should be independent of the quantities @xmath42 but still strongly dependent on @xmath43 . a further parameter dependence that is being suppressed in the definition above is the shape and location of the ensemble being used . that is , the transport properties of an ensemble in a chaotic system depend in general on the part of the phase - space being sampled . it is therefore important to consider many different initial conditions to generate a current . the first straightforward result we show in fig . ( [ figure2 ] ) is that in the case of chaotic trajectories , a single trajectory easily displays behavior very different from that of many trajectories . however , it turns out that in the regular regime , it is possible to use a single trajectory to get essentially the same result as obtained from many trajectories . further consider the bifurcation diagram in fig . ( [ figure3 ] ) where we superimpose the different curves resulting from varying the number of points in the initial ensemble . first , the curve is significantly smoother as a function of @xmath0 for larger @xmath44 . even more relevant is the fact that the single trajectory data ( @xmath45 ) may show current reversals that do not exist in the large @xmath44 data . current @xmath38 versus the number of trajectories @xmath44 for @xmath7 ; dashed lines correspond to a regular motion with @xmath46 while solid lines correspond to a chaotic motion with @xmath47 . note that a single trajectory is sufficient for a regular motion while the convergence in the chaotic case is only obtained if the @xmath44 exceeds a certain threshold , @xmath48.,title="fig:",width=302 ] current @xmath38 versus @xmath0 for different set of trajectories @xmath44 ; @xmath45 ( circles ) , @xmath49 ( square ) and @xmath50 ( dashed lines ) . note that a single trajectory suffices in the regular regime where all the curves match . in the chaotic regime , as @xmath44 increases , the curves converge towards the dashed one.,title="fig:",width=302 ] also , note that single - trajectory current values are typically significantly greater than ensemble averages . this arises from the fact that an arbitrarily chosen ensemble has particles with idiosyncratic behaviors which often average out . as our result , with these ensembles we see typical @xmath51 for example , while barbi and salerno report currents about @xmath52 times greater . however , it is not true that only a few trajectories dominate the dynamics completely , else there would not be a saturation of the current as a function of @xmath44 . all this is clear in fig . ( [ figure3 ] ) . we note that the * net * drift of an ensemble can be a lot closer to @xmath53 than the behavior of an individual trajectory . it should also be clear that there is a dependence of the current on the location of the initial ensemble , this being particularly true for small @xmath44 , of course . the location is defined by its centroid @xmath54 . for @xmath45 , it is trivially true that the initial location matters to the asymptotic value of the time - averaged velocity , given that this is a non - ergodic and chaotic system . further , considering a gaussian ensemble , say , the width of the ensemble also affects the details of the current , and can show , for instance , illusory current reversal , as seen in figs . ( [ current - bifur1],[current - bifur2 ] ) for example . notice also that in fig . ( [ current - bifur1 ] ) , at @xmath55 and @xmath56 , the deviations between the different ensembles is particularly pronounced . these points are close to bifurcation points where some sort of symmetry breaking is clearly occuring , which underlines our emphasis on the relevance of specifying ensemble characteristics in the neighborhood of unstable behavior . however , why these specific bifurcations should stand out among all the bifurcations in the parameter range shown is not entirely clear . to understand how to incorporate this knowledge into calculations of the current , therefore , consider the fact that if we look at the classical phase space for the hamiltonian or underdamped @xmath57 motion , we see the typical structure of stable islands embedded in a chaotic sea which have quite complicated behavior@xcite . in such a situation , the dynamics always depends on the location of the initial conditions . however , we are not in the hamiltonian situation when the damping is turned on in this case , the phase - space consists in general of attractors . that is , if transient behavior is discarded , the current is less likely to depend significantly on the location of the initial conditions or on the spread of the initial conditions . in particular , in the chaotic regime of a non - hamiltonian system , the initial ensemble needs to be chosen larger than a certain threshold to ensure convergence . however , in the regular regime , it is not important to take a large ensemble and a single trajectory can suffice , as long as we take care to discard the transients . that is to say , in the computation of currents , the definition of the current needs to be modified to : @xmath58 where @xmath59 is some empirically obtained cut - off such that we get a converged current ( for instance , in our calculations , we obtained converged results with @xmath60 ) . when this modified form is used , the convergence ( ensemble - independence ) is more rapid as a function of @xmath61 and the width of the intial conditions . armed with this background , we are now finally in a position to compare bifurcation diagrams with the current , as we do in the next section . our results are presented in the set of figures fig . ( [ figure5 ] ) fig . ( [ rev - nobifur ] ) , in each of which we plot both the ensemble current and the bifurcation diagram as a function of the parameter @xmath0 . the main point of these numerical results can be distilled into a series of heuristic statements which we state below ; these are labelled with roman numerals . for @xmath7 and @xmath8 , we plot current ( upper ) with @xmath62 and bifurcation diagram ( lower ) versus @xmath0 . note that there is a * single * current reversal while there are many bifurcations visible in the same parameter range.,title="fig:",width=302 ] consider fig . ( [ figure5 ] ) , which shows the parameter range @xmath63 chosen relatively arbitrarily . in this figure , we see several period - doubling bifurcations leading to order - chaos transitions , such as for example in the approximate ranges @xmath64 . however , there is only one instance of current - reversal , at @xmath65 . note , however , that the current is not without structure it changes fairly dramatically as a function of parameter . this point is made even more clearly in fig . ( [ figure6 ] ) where the current remains consistently below @xmath53 , and hence there are in fact , no current reversals at all . note again , however , that the current has considerable structure , even while remaining negative . for @xmath66 and @xmath8 , plotted are current ( upper ) and bifurcation diagram ( lower ) versus @xmath0 with @xmath62 . notice the current stays consistently below @xmath53.,title="fig:",width=302 ] current and bifurcations versus @xmath0 . in ( a ) and ( b ) we show ensemble dependence , specifically in ( a ) the black curve is for an ensemble of trajectories starting centered at the stable fixed point @xmath67 with a root - mean - square gaussian width of @xmath68 , and the brown curve for trajectories starting from the unstable fixed point @xmath69 and of width @xmath68 . in ( b ) , all ensembles are centered at the stable fixed point , the black line for an ensemble of width @xmath68 , brown a width of @xmath70 and maroon with width @xmath71 . ( c ) is the comparison of the current @xmath38 without transients ( black ) and with transients ( brown ) along with the single - trajectory results in blue ( after barbi and salerno ) . the initial conditions for the ensembles are centered at @xmath67 with a mean root square gaussian of width @xmath68 . ( d ) is the corresponding bifurcation diagram.,title="fig:",width=302 ] it is possible to find several examples of this at different parameters , leading to the negative conclusion , therefore , that * ( i ) not all bifurcations lead to current reversal*. however , we are searching for positive correlations , and at this point we have not precluded the more restricted statement that all current reversals are associated with bifurcations , which is in fact mateos conjecture . we therefore now move onto comparing our results against the specific details of barbi and salerno s treatment of this conjecture . in particular , we look at their figs . ( 2,3a,3b ) , where they scan the parameter region @xmath72 . the distinction between their results and ours is that we are using _ ensembles _ of particles , and are investigating the convergence of these results as a function of number of particles @xmath44 , the width of the ensemble in phase - space , as well as transience parameters @xmath73 . our data with larger @xmath44 yields different results in general , as we show in the recomputed versions of these figures , presented here in figs . ( [ current - bifur1],[current - bifur2 ] ) . specifically , ( a ) the single - trajectory results are , not surprisingly , cleaner and can be more easily interpreted as part of transitions in the behavior of the stability properties of the periodic orbits . the ensemble results on the other hand , even when converged , show statistical roughness . ( b ) the ensemble results are consistent with barbi and salerno in general , although disagreeing in several details . for instance , ( c ) the bifurcation at @xmath74 has a much gentler impact on the ensemble current , which has been growing for a while , while the single - trajectory result changes abruptly . note , ( d ) the very interesting fact that the single - trajectory current completely misses the bifurcation - associated spike at @xmath75 . further , ( e ) the barbi and salerno discussion of the behavior of the current in the range @xmath76 is seen to be flawed our results are consistent with theirs , however , the current changes are seen to be consistent with bifurcations despite their statements to the contrary . on the other hand ( f ) , the ensemble current shows a case [ in fig . ( [ current - bifur2 ] ) , at @xmath77 of current reversal that does not seem to be associated with bifurcations . in this spike , the current abruptly drops below @xmath53 and then rises above it again . the single trajectory current completely ignores this particular effect , as can be seen . the bifurcation diagram indicates that in this case the important transitions happen either before or after the spike . all of this adds up to two statements : the first is a reiteration of the fact that there is significant information in the ensemble current that can not be obtained from the single - trajectory current . the second is that the heuristic that arises from this is again a negative conclusion , that * ( ii ) not all current reversals are associated with bifurcations . * where does this leave us in the search for ` positive ' results , that is , useful heuristics ? one possible way of retaining the mateos conjecture is to weaken it , i.e. make it into the statement that * ( iii ) _ most _ current reversals are associated with bifurcations . * same as fig . ( [ current - bifur1 ] ) except for the range of @xmath0 considered.,title="fig:",width=302 ] for @xmath78 and @xmath8 , plotted are current ( upper ) and bifurcation diagram ( lower ) versus @xmath0 with @xmath62 . note in particular in this figure that eyeball tests can be misleading . we see reversals without bifurcations in ( a ) whereas the zoomed version ( c ) shows that there are windows of periodic and chaotic regimes . this is further evidence that jumps in the current correspond in general to bifurcation.,title="fig:",width=302 ] for @xmath7 and @xmath79 , current ( upper ) and bifurcation diagram ( lower ) versus @xmath0.,title="fig:",width=302 ] however , a * different * rule of thumb , previously not proposed , emerges from our studies . this generalizes mateos conjecture to say that * ( iv ) bifurcations correspond to sudden current changes ( spikes or jumps)*. note that this means these changes in current are not necessarily reversals of direction . if this current jump or spike goes through zero , this coincides with a current reversal , making the mateos conjecture a special case . the physical basis of this argument is the fact that ensembles of particles in chaotic systems _ can _ have net directed transport but the details of this behavior depends relatively sensitively on the system parameters . this parameter dependence is greatly exaggerated at the bifurcation point , when the dynamics of the underlying single - particle system undergoes a transition a period - doubling transition , for example , or one from chaos to regular behavior . scanning the relevant figures , we see that this is a very useful rule of thumb . for example , it completely captures the behaviour of fig . ( [ figure6 ] ) which can not be understood as either an example of the mateos conjecture , or even a failure thereof . as such , this rule significantly enhances our ability to characterize changes in the behavior of the current as a function of parameter . a further example of where this modified conjecture helps us is in looking at a seeming negation of the mateos conjecture , that is , an example where we seem to see current - reversal without bifurcation , visible in fig . ( [ hidden - bifur ] ) . the current - reversals in that scan of parameter space seem to happen inside the chaotic regime and seemingly independent of bifurcation . however , this turns out to be a ` hidden ' bifurcation when we zoom in on the chaotic regime , we see hidden periodic windows . this is therefore consistent with our statement that sudden current changes are associated with bifurcations . each of the transitions from periodic behavior to chaos and back provides opportunities for the current to spike . however , in not all such cases can these hidden bifurcations be found . we can see an example of this in fig . ( [ rev - nobifur ] ) . the current is seen to move smoothly across @xmath80 with seemingly no corresponding bifurcations , even when we do a careful zoom on the data , as in fig . ( [ hidden - bifur ] ) . however , arguably , although subjective , this change is close to the bifurcation point . this result , that there are situations where the heuristics simply do not seem to apply , are part of the open questions associated with this problem , of course . we note , however , that we have seen that these broad arguments hold when we vary other parameters as well ( figures not shown here ) . in conclusion , in this paper we have taken the approach that it is useful to find general rules of thumb ( even if not universally valid ) to understand the complicated behavior of non - equilibrium nonlinear statistical mechanical systems . in the case of chaotic deterministic ratchets , we have shown that it is important to factor out issues of size , location , spread , and transience in computing the ` current ' due to an ensemble before we search for such rules , and that the dependence on ensemble characteristics is most critical near certain bifurcation points . we have then argued that the following heuristic characteristics hold : bifurcations in single - trajectory behavior often corresponds to sudden spikes or jumps in the current for an ensemble in the same system . current reversals are a special case of this . however , not all spikes or jumps correspond to a bifurcation , nor vice versa . the open question is clearly to figure out if the reason for when these rules are violated or are valid can be made more concrete . a.k . gratefully acknowledges t. barsch and kamal p. singh for stimulating discussions , the reimar lst grant and financial support from the alexander von humboldt foundation in bonn . a.k.p . is grateful to carleton college for the ` sit , wallin , and class of 1949 ' sabbatical fellowships , and to the mpipks for hosting him for a sabbatical visit , which led to this collaboration . useful discussions with j .- m . rost on preliminary results are also acknowledged . p. hnggi and bartussek , in nonlinear physics of complex systems , lecture notes in physics vol . 476 , edited by j. parisi , s.c . mueller , and w. zimmermann ( springer verlag , berlin , 1996 ) , pp.294 - 308 ; r.d . asturmian , science * 276 * , 917 ( 1997 ) ; f. jlicher , a. ajdari , and j. prost , rev . mod . phys . * 69 * , 1269 ( 1997 ) ; c. dring , nuovo cimento d*17 * , 685 ( 1995 ) s. flach , o. yevtushenko , and y. zolotaryuk , phys . rev . lett . * 84 * , 2358 ( 2000 ) ; o. yevtushenko , s. flach , y. zolotaryuk , and a. a. ovchinnikov , europhys . lett . * 54 * , 141 ( 2001 ) ; s. denisov et al . e * 66 * , 041104 ( 2002 )

in 84 , 258 ( 2000 ) , mateos conjectured that current reversal in a classical deterministic ratchet is associated with bifurcations from chaotic to periodic regimes . this is based on the comparison of the current and the bifurcation diagram as a function of a given parameter for a periodic asymmetric potential . barbi and salerno , in 62 , 1988 ( 2000 ) , have further investigated this claim and argue that , contrary to mateos claim , current reversals can occur also in the absence of bifurcations . barbi and salerno s studies are based on the dynamics of one particle rather than the statistical mechanics of an ensemble of particles moving in the chaotic system . the behavior of ensembles can be quite different , depending upon their characteristics , which leaves their results open to question . in this paper we present results from studies showing how the current depends on the details of the ensemble used to generate it , as well as conditions for convergent behavior ( that is , independent of the details of the ensemble ) . we are then able to present the converged current as a function of parameters , in the same system as mateos as well as barbi and salerno . we show evidence for current reversal without bifurcation , as well as bifurcation without current reversal . we conjecture that it is appropriate to correlate abrupt changes in the current with bifurcation , rather than current reversals , and show numerical evidence for our claims .

arxiv

one surprising result that has come out of the more than 200 extrasolar planet discoveries to date is the wide range of eccentricities observed . unlike our own solar system , many of the extrasolar planets which are not tidally locked to their host stars have moderate eccentricities ( @xmath1 ) , and 15 planets have high eccentricities ( @xmath0 ) . these observations have spawned several theories as to the origin of highly eccentric extrasolar planets . one such method , planet - planet scattering , occurs when multiple jovian planets form several astronomical units ( au ) from the host star and then interact , leaving one in an eccentric orbit and often ejecting the other @xcite . this method has been proposed to explain the architecture of the @xmath2 and planetary system @xcite , which contains a hot jupiter as well as two jovian planets in moderately eccentric orbits . @xcite suggested a merger scenario in which inner protoplanets perturb each other and merge to form a single massive , eccentric planet with @xmath3 and @xmath4 au . interactions with stellar companions are another possible way to boost a planet s eccentricity . of the 15 stars hosting a planet with @xmath0 , six are also known to possess stellar - mass companions in wide binary orbits : hd 3651 @xcite , hd 20782 @xcite , hd 80606 , hd 89744 @xcite , 16 cyg b , and hd 222582 @xcite . if the inclination angle between the planetary orbit and a stellar companion is large , the kozai mechanism @xcite can induce large - amplitude oscillations in the eccentricity of the planet ( e.g. malmberg et al . these oscillations can be damped by general relativistic effects and by interaction with other planets , and hence are most effective in systems with a single planet in an orbit @xmath51 au from the host star @xcite . the kozai mechanism has been suggested to explain the high eccentricity of 16 cyg bb @xcite and hd 80606b @xcite . @xcite found the inclination of 16 cyg b orbiting the system barycenter to lie between 100 and 160 degrees , where 90 degrees is an edge - on orientation . however , it is the difference in inclination between the orbital planes of the planetary and stellar companion that is critical in determining the importance of the kozai mechanism , and the inclination of the planet s orbit is generally not known for non - transiting systems . of the 192 known planetary systems , 23 ( 12% ) are multi - planet systems . recent discoveries of additional objects in systems known to host at least one planet @xcite suggest that multiple - planet systems are common . of particular interest are systems which host a jovian planet and a low - mass `` hot neptune , '' e.g. 55 cnc ( = hd 75732 ) , gj 876 , @xmath6 arae ( = hd 160691 ) , gl 777a ( = hd 190360 ) . motivated by the discoveries of hot neptunes in known planetary systems , we have undertaken an intensive survey of selected single - planet systems to search for additional low - mass companions . three of the planetary systems discussed in this paper ( hd 3651 , hd 80606 , hd 89744 ) are part of this campaign . the excellent radial - velocity precision of the high resolution spectrograph on the hobby - eberly telescope ( het ) , combined with queue - scheduling , allow us to time the observations in such a way as to minimize phase gaps in the orbit of the known planet , and also to act quickly on potential new planet candidates . the use of the het in this manner is discussed further in @xcite with regard to the discovery of hd 37605b . in this work , we aim to combine observational limits on additional planets in known planetary systems with dynamical constraints obtained by n - body simulations . the observations address the question : what additional planets are ( or are not ) in these systems ? the dynamical simulations can answer the question : where are additional planets possible ? section 2 describes the observations and the test particle simulations for six highly eccentric planetary systems : hd 3651 , hd 37605 , hd 45350 , hd 80606 , hd 89744 , and 16 cyg b. we have chosen these systems based on two criteria : ( 1 ) each hosts a planet with @xmath0 , and ( 2 ) each has been observed by the planet search programs at mcdonald observatory . in 3 , we present and discuss the results of the updated orbital fits , dynamical simulations , and detection limit computations . five of the six stars considered in this work have been observed with the mcdonald observatory 9.2 m hobby - eberly telescope ( het ) using its high resolution spectrograph ( hrs ) @xcite . a full description of the het planet search program is given in @xcite . for 16 cyg b , observations from mcdonald observatory were obtained only with the 2.7 m harlan j. smith ( hjs ) telescope ; the long - term planet search program on this telescope is described in @xcite . all available published data on these systems were combined with our data from mcdonald observatory in the orbit fitting procedures . to place constraints on the architecture of planetary systems , we would like to know where additional objects can remain in stable orbits in the presence of the known planet(s ) . we performed test particle simulations using swifthal / swift.html . ] @xcite to investigate the dynamical possibility of additional low - mass planets in each of the six systems considered here . low - mass planets can be treated as test particles since the exchange of angular momentum with jovian planets is small . we chose the regularized mixed - variable symplectic integrator ( rmvs3 ) version of swift for its ability to handle close approaches between massless , non - interacting test particles and planets . particles are removed if they are ( 1 ) closer than 1 hill radius to the planet , ( 2 ) closer than 0.05 au to the star , or ( 3 ) farther than 10 au from the star . since the purpose of these simulations is to determine the regions in which additional planets could remain in stable orbits , we set this outer boundary because the current repository of radial - velocity data can not detect objects at such distances . the test particle simulations were set up following the methods used in @xcite , with the exception that only initially circular orbits are considered in this work . for each planetary system , test particles were placed in initially circular orbits spaced every 0.002 au in the region between 0.05 - 2.0 au . we have chosen to focus on this region because the duration of our high - precision het data is currently only 2 - 4 years for the objects in this study . the test particles were coplanar with the existing planet , which had the effect of confining the simulation to two dimensions . input physical parameters for the known planet in each system were obtained from our keplerian orbit fits described in 3.1 , and from recent literature for 16 cyg b @xcite and hd 45350 @xcite . the planetary masses were taken to be their minimum values ( sin @xmath7 ) . the systems were integrated for @xmath8 yr , following @xcite and allowing completion of the computations in a reasonable time . we observed that nearly all of the test - particle removals occurred within the first @xmath9 yr ; after this time , the simulations had essentially stabilized to their final configurations . we present updated keplerian orbital solutions for hd 3651b , hd 37605b , hd 80606b , and hd 89744b in table 1 . a summary of the data used in our analysis is given in table 2 , and the het radial velocities are given in tables 3 - 6 . the velocity uncertainties given for the het data represent internal errors only , and do not include any external sources of error such as stellar `` jitter . '' the parameters for the remaining two planets , hd 45350b and 16 cyg bb , are taken from @xcite and @xcite , respectively . radial velocity measurements from the het are given for hd 45350 in @xcite , and velocities for 16 cyg b from the hjs telescope are given in @xcite . as in @xcite , all available published data were combined with those from mcdonald , and the known planet in each system was fit with a keplerian orbit using gaussfit @xcite , allowing the velocity offset between each data set to be a free parameter . examination of the residuals to our keplerian orbit fits revealed no evidence for additional objects in any of the six systems in this study . the saturn - mass ( m sin @xmath10 ) planet hd 3651b was discovered by @xcite using observations from lick and keck . we fit these data , which were updated in @xcite , in combination with observations from the hjs and het at mcdonald observatory . the het data , which consist of multiple exposures per visit , were binned using the inverse - variance weighted mean value of the velocities in each visit . the standard error of the mean was added in quadrature to the weighted rms about the mean velocity to generate the error bar of each binned point ( n=29 ) . the rms about the combined fit for each dataset is : lick & keck6.6 , het9.4 , hjs12.2 . the fitted orbital parameters for hd 3651b are of comparable precision to those reported in @xcite , and agree within 2@xmath11 . the recent discovery of a t dwarf companion to hd 3651 @xcite prompts an interesting exercise : can the radial - velocity trend due to this object be detected in the residuals after removing the planet ? we detect a slope of @xmath12 yr@xmath13 , indicating that we are indeed able to discern a trend which is possibly due to the binary companion . however , the reduced @xmath14 of the orbital solution is not significantly improved by the inclusion of a linear trend ( @xmath15=0.18 ) . the parameters given in table 1 were obtained from the fit which did not include a trend . we present 23 new het observations for hd 37605 obtained since its announcement by @xcite . the data now span a total of 1065 days . the best fit is obtained by including an acceleration of @xmath16 yr@xmath13 , indicating a distant orbiting body . such a finding would lend support to the hypothesis that very eccentric single - planet systems originate by interactions within a wide binary system . the shortest period that this outer companion could have and still remain consistent with the observed acceleration and its uncertainty over the timespan of the observations is about 40 yr , assuming a circular orbit . this object would then have a minimum mass in the brown dwarf range . the planet orbiting hd 80606 , first announced by @xcite , is the most eccentric extrasolar planet known , with @xmath17 ( table 1 ) . we have fit the coralie data in combination with the keck data given in @xcite and 23 observations from het . the extreme velocity variations induced by this planet greatly increase the sensitivity of orbit fits to the weighting of individual measurements . since the uncertainties of the het velocities given in tables 3 - 6 represent internal errors only , we experimented with adding 1 - 7 of radial - velocity `` jitter '' in quadrature before fitting the data for hd 80606 . for all of these jitter values , the fitted parameters remained the same within their uncertainties . table 1 gives the parameters derived from a fit which added 3.5 of jitter @xcite to the het data . the rms about the combined fit is : coralie18.7 , het7.5 , keck5.6 . @xcite noted that the eccentricity @xmath18 and the argument of periastron @xmath19 had to be held fixed in their fit to the keck data alone . however , the large number of measurements included in this work allowed gaussfit to converge with all parameters free . for hd 89744b , we combine data from the het with 6 measurements from the hjs telescope and lick data from @xcite . the het data were binned in the same manner as for hd 3651 , resulting in n=33 independent visits . the rms about the combined fit for each dataset is : lick17.1 , het10.7 , hjs9.5 . as with hd 3651b , our derived parameters agree with those of @xcite within 2@xmath11 . the scatter about our fit remains large , most likely due to the star s early spectral type ( f7v ) , which hinders precision radial - velocity measurements due to the smaller number of spectral lines . for example , the f7v star hd 221287 was recently found to host a planet @xcite ; despite the superb instrumental precision of the harps spectrograph , that orbital solution has a residual rms of 8.5 . the results of the dynamical simulations are shown in figures 1 - 3 . the survival time of the test particles is plotted against their initial semimajor axis . as shown in figure 1 , the short - period planets hd 3651 and hd 37605 sweep clean the region inside of about 0.5 au . in both of these systems , however , a small number of test particles remained in low - eccentricity orbits near the known planet s apastron distance , near the 1:2 mean - motion resonance ( mmr ) . in the hd 3651 system , particles remained stable beyond about 0.6 au , which is not surprising given the low mass of the planet . for hd 37605 , two distinct strips of stability are seen in fig . 1 , corresponding to the 1:2 and 1:3 mmrs . the eccentricity of the test particles in the region of the 1:2 mmr oscillated between 0.00 and 0.06 . particles in 1:3 mmr oscillated in eccentricity with a larger range , up to @xmath20 , which is expected due to secular forcing . as with hd 3651 , the region beyond about 0.8 au was essentially unaffected by the planet . figure 2 shows the results for the hd 45350 and hd 80606 systems . the long period ( 963.6 days ) and relatively large mass ( m sin @xmath21=1.8 ) of hd 45350b restricted stable orbits to the innermost 0.2 au . these test particles oscillated in eccentricity up to @xmath22 . the 4 planet orbiting hd 80606 removed all test particles to a distance of about 1.5 au , and only beyond 1.75 au did test particles remain in stable orbits for the duration of the simulation ( @xmath8 yr ) . a region of instability is evident at 1.9 au due to the 8:1 mmr . figure 3 shows that hd 89744b eliminated all test particles except for a narrow region near the 8:3 resonance . for the 16 cyg b system , only particles inside of about 0.3 au remained stable , leaving open the possibility of short - period planets . the surviving particles oscillated in eccentricity up to @xmath23 , but these simulations treat the star as a point mass , and hence tidal damping of the eccentricity is not included . our results are consistent with those of @xcite , who investigated dynamical stability in extrasolar planetary systems and found that no test particles survived in the habitable zones of the hd 80606 , hd 89744 , and 16 cyg b systems . three of these systems ( hd 3651 , hd 80606 , hd 89744 ) were monitored intensely with the het as part of a larger effort to search for low - mass , short period planets . no evidence was found for any such objects in these or any of the six systems in this work . we then asked what limits can be set on additional planets using the high - precision het data we have obtained . the procedure for determining companion limits was identical to the method described in @xcite , except that in this work , the best - fit keplerian orbit for the known planet ( see 3.1 ) was removed before performing the limits computations . in this way , we determined the radial - velocity amplitude @xmath24 for which 99% of planets would have been detected in the residuals . the eccentricity of the injected test signals was chosen to be the mean eccentricity of the surviving particles from the simulations described in 3.2 . only the regions in which test particles survived were considered in these limits computations . the results of these computations were highly varied , reflecting the differing observing strategies employed for these six objects . in particular , hd 3651 , hd 80606 , and hd 89744 were monitored intensely with the het as part of a search for short - period objects , whereas hd 37605 and hd 45350 were only observed sporadically after the known planet orbits were defined and published @xcite , and 16 cyg b has only been observed with the hjs telescope at a frequency of at most once per month . the companion limits are shown in figures 4 - 6 ; planets with masses above the solid line can be ruled out by the data with 99% confidence . not surprisingly , the tightest limits were obtained for hd 3651 ( figure 4 ) , which had a total of 195 measurements , including 29 independent het visits . for periods less than about 1 year , we can exclude planets with m sin @xmath21 2 neptune masses . similar results were obtained for 16 cyg b ( n=161 ) , where the limits approach a neptune mass ( figure 6 ) . since the detection limits generally improve with the addition of more data and with higher - quality data , we can define a quantity to measure the goodness of the limits . a simple choice would be @xmath25 , where @xmath26 is the total number of observations , and @xmath27 is the mean uncertainty of the radial - velocity measurements . the values of @xmath26 and @xmath27 are given in table 2 . in the hd 45350 system , the results of the dynamical simulations complement those of the detection limit determinations . very tight limits are obtained in close orbits ( @xmath280.2 au ) . in this region , test particles were stable ( fig . 2 ) and our observations can exclude planets with m sin _ i _ between about 1 and 4 neptune masses . similar results were obtained for the 16 cyg b system , in which test particles remained stable inward of @xmath29 0.3 au . in that region , planets of 1 - 3 neptune masses can be excluded by our limits determinations ( fig . 6 ) . in most of the limits determinations , there are multiple `` blind spots '' evident where the periodogram method failed to significantly recover the injected signals . typically this occurs at certain trial periods for which the phase coverage of the observational data is poor , and often at the 1-month and 1-year windows . for none of hd 37605 ( fig . 4 ) , hd 80606 ( fig . 5 ) , or hd 89744 ( fig . 6 ) could additional companions be ruled out below about 0.7 , and for most orbital periods tested , the limits were substantially worse . one possible explanation for this result is that the sampling of the observations was poorly distributed in phase for many of the injected test signals , making significant recovery by the periodogram method difficult . this is evidenced by the `` jagged '' regions in the plots . also , the intrinsic scatter for those three systems was too large to permit tight limits determination . this is certainly reasonable for the f7 star hd 89744 . the three systems with the best limits ( hd 3651 , hd 45350 , and 16 cyg b ) also had the lowest rms scatter about their orbital solutions ( mean=@xmath30 ; table 1 ) . in contrast , the mean rms for the remaining three systems was @xmath31 . additional factors such as a paucity of data ( hd 37605 ) and short time baselines ( hd 80606 , hd 89744 ) made the determination of useful companion limits challenging for some of the planetary systems in this study . we have shown that for a sample of six highly eccentric extrasolar planetary systems , there is no evidence for additional planets . test particle simulations show that there are regions detectable by current surveys ( i.e. for @xmath32 au ) where additional objects can exist . for hd 3651 and hd 37605 , we find that protected resonances are also present . combining these simulations with detection limits computed using new high - precision het data combined with all available published data is particularly effective for the hd 3651 and hd 45350 systems . additional short - period planets can be ruled out down to a few neptune masses in the dynamically stable regions in these systems . this material is based upon work supported by the national aeronautics and space administration under grant nos . nng04g141 g and nng05g107 g issued through the terrestrial planet finder foundation science program . we are grateful to the het tac for their generous allocation of telescope time for this project . we also would like to thank barbara mcarthur for her assistance with gaussfit software . we thank the referee , greg laughlin , for his careful review of this manuscript . this research has made use of nasa s astrophysics data system ( ads ) , and the simbad database , operated at cds , strasbourg , france . the hobby - eberly telescope ( het ) is a joint project of the university of texas at austin , the pennsylvania state university , stanford university , ludwig - maximilians - universit " at mnchen , and georg - august - universit " at g " ottingen the het is named in honor of its principal benefactors , william p. hobby and robert e. eberly . lllllllll hd 3651 b & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 7.1 + hd 37605 b & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 13.0 + hd 45350 b & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 9.1 + hd 80606 b & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 13.5 + hd 89744 b & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 14.4 + 16 cyg b b & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 10.6 + lllll hd 3651 & 163 & 3.4 & & @xcite + hd 3651 & 3 & 6.1 & & hjs + hd 3651 & 29 & 2.1 & & het + hd 3651 ( total ) & 195 & 3.2 & 7083 & + hd 37605 ( total ) & 43 & 2.9 & 1065 & het + hd 45350 & 38 & 2.8 & & @xcite + hd 45350 & 28 & 4.2 & & het + hd 45350 & 47 & 8.9 & & hjs + hd 45350 ( total ) & 113 & 5.7 & 2265 & + hd 80606 & 61 & 13.7 & & @xcite + hd 80606 & 46 & 5.1 & & @xcite + hd 80606 & 23 & 2.5 & & het + hd 80606 ( total ) & 130 & 8.7 & 2893 & + hd 89744 & 50 & 11.2 & & @xcite + hd 89744 & 33 & 3.2 & & het + hd 89744 & 6 & 9.4 & & hjs + hd 89744 ( total ) & 89 & 8.1 & 2687 & + 16 cyg b & 95 & 6.3 & & @xcite + 16 cyg b & 29 & 19.7 & & hjs phase ii + 16 cyg b & 37 & 7.4 & & hjs phase iii + 16 cyg b ( total ) & 161 & 9.0 & 6950 & + lrr [ tbl-3 ] 53581.87326 & -19.1 & 2.9 + 53581.87586 & -19.4 & 2.7 + 53581.87846 & -20.7 & 2.7 + 53600.79669 & -11.5 & 2.4 + 53600.79860 & -15.5 & 3.0 + 53600.80050 & -22.8 & 2.9 + 53604.79166 & -15.8 & 1.9 + 53604.79356 & -18.8 & 2.1 + 53604.79548 & -21.3 & 2.1 + 53606.78169 & -19.3 & 1.8 + 53606.78360 & -14.8 & 2.1 + 53606.78551 & -24.0 & 1.8 + 53608.77236 & -18.8 & 1.9 + 53608.77426 & -18.0 & 1.9 + 53608.77617 & -18.8 & 1.8 + 53615.96280 & -28.0 & 2.6 + 53615.96471 & -31.9 & 2.4 + 53615.96662 & -37.8 & 2.5 + 53628.74050 & -6.8 & 2.2 + 53628.74240 & -14.5 & 2.4 + 53628.74431 & -5.5 & 2.2 + 53669.61012 & -18.2 & 2.1 + 53669.61203 & -19.2 & 2.2 + 53669.61394 & -17.7 & 2.4 + 53678.78954 & -10.6 & 2.4 + 53678.79141 & -8.6 & 2.3 + 53678.79332 & -2.3 & 2.1 + 53682.78423 & -15.4 & 2.2 + 53682.78609 & -15.0 & 2.3 + 53682.78801 & -11.9 & 2.3 + 53687.77684 & 11.3 & 2.2 + 53687.77875 & 8.7 & 2.2 + 53687.78066 & 15.9 & 2.2 + 53691.75967 & 9.6 & 2.2 + 53691.76158 & 20.3 & 2.1 + 53691.76349 & 15.9 & 2.0 + 53696.75837 & 16.1 & 1.8 + 53696.76028 & 18.6 & 1.8 + 53696.76220 & 20.0 & 2.0 + 53694.75275 & 18.0 & 1.9 + 53694.75466 & 15.1 & 2.0 + 53694.75656 & 17.8 & 2.0 + 53955.83401 & -0.5 & 1.9 + 53955.83593 & -1.2 & 2.0 + 53955.83785 & 1.3 & 1.9 + 53956.82850 & 0.4 & 2.0 + 53956.83046 & -1.0 & 2.0 + 53956.83236 & -5.4 & 2.2 + 53957.82201 & -2.1 & 2.0 + 53957.82392 & -1.3 & 2.0 + 53957.82583 & -3.6 & 2.0 + 53973.80721 & 9.8 & 7.3 + 53973.81020 & 3.5 & 2.3 + 53973.81200 & -3.5 & 2.0 + 53976.78393 & -10.4 & 2.4 + 53976.78586 & -5.4 & 2.1 + 53976.78778 & -6.7 & 2.3 + 53978.97197 & -3.8 & 2.6 + 53985.95886 & -9.0 & 2.3 + 53985.96079 & 4.3 & 3.3 + 53987.95335 & -8.3 & 2.2 + 53987.95527 & -8.0 & 2.2 + 53987.95719 & -12.0 & 2.3 + 53989.73817 & -13.2 & 2.2 + 53989.74009 & -13.2 & 2.1 + 53989.74203 & -18.6 & 2.1 + 54003.70719 & 2.0 & 2.2 + 54003.70915 & 4.7 & 2.4 + 54005.68297 & 7.0 & 2.5 + 54005.68488 & 11.1 & 2.0 + 54005.68690 & 10.2 & 2.1 + 54056.77919 & -7.5 & 2.2 + 54056.78110 & -11.5 & 2.1 + 54056.78302 & -9.6 & 2.3 + 54062.55119 & 20.1 & 1.8 + 54062.55312 & 21.9 & 2.0 + 54062.55505 & 20.9 & 2.0 + 54064.54710 & 12.8 & 2.0 + 54064.54902 & 16.7 & 2.1 + 54064.55094 & 16.6 & 2.1 + 54130.55316 & 19.1 & 2.4 + 54130.55508 & 16.9 & 2.5 + 54130.55701 & 17.6 & 2.5 + lrr [ tbl-4 ] 53002.67151 & 487.6 & 3.8 + 53003.68525 & 495.5 & 3.0 + 53006.66205 & 496.2 & 3.0 + 53008.66407 & 501.3 & 2.9 + 53010.80477 & 499.8 & 2.9 + 53013.79399 & 482.1 & 2.6 + 53042.72797 & 269.7 & 2.8 + 53061.66756 & 489.0 & 2.6 + 53065.64684 & 479.0 & 2.8 + 53071.64383 & 463.8 & 2.6 + 53073.63819 & 460.4 & 2.6 + 53082.62372 & 422.8 & 2.5 + 53083.59536 & 422.2 & 2.8 + 53088.59378 & 418.6 & 4.0 + 53089.59576 & 379.1 & 2.2 + 53092.59799 & 343.7 & 2.5 + 53094.58658 & 323.2 & 2.4 + 53095.58642 & 302.1 & 2.4 + 53096.58744 & 302.1 & 3.2 + 53098.57625 & 193.8 & 2.7 + 53264.95137 & 164.9 & 3.0 + 53265.94744 & 112.9 & 3.0 + 53266.94598 & 113.2 & 3.7 + 53266.95948 & 74.6 & 3.6 + 53266.97396 & 119.2 & 8.0 + 53283.92241 & 471.6 & 2.7 + 53318.81927 & 213.3 & 3.0 + 53335.92181 & 496.9 & 2.6 + 53338.90602 & 493.9 & 2.6 + 53377.81941 & 109.1 & 2.7 + 53378.81189 & 214.6 & 2.7 + 53379.80225 & 338.3 & 2.6 + 53381.64429 & 436.1 & 2.7 + 53384.64654 & 482.9 & 2.8 + 53724.85584 & 468.2 & 2.6 + 53731.69723 & 435.4 & 2.7 + 53738.67472 & 404.3 & 2.6 + 53743.81020 & 400.5 & 2.6 + 53748.64724 & 348.4 & 2.7 + 54039.85015 & 272.5 & 3.1 + 54054.96457 & 437.4 & 2.7 + 54055.95279 & 422.0 & 2.9 + 54067.76282 & 376.4 & 2.6 + lrr [ tbl-5 ] 53346.88103 & -20.8 & 3.0 + 53358.02089 & -49.5 & 2.7 + 53359.82400 & -60.4 & 3.0 + 53361.02985 & -64.7 & 2.5 + 53365.03079 & -77.4 & 2.4 + 53373.98282 & -88.4 & 3.0 + 53377.80112 & -105.5 & 2.4 + 53379.75230 & -109.3 & 2.7 + 53389.74170 & -115.3 & 2.5 + 53391.74400 & -129.4 & 2.4 + 53395.72763 & -146.4 & 2.3 + 53399.72518 & -158.4 & 2.5 + 53401.72497 & -174.7 & 2.7 + 53414.67819 & -219.8 & 3.0 + 53421.85529 & 261.0 & 2.2 + 53423.86650 & 322.1 & 2.0 + 53424.85231 & 245.9 & 2.1 + 53432.87120 & 87.5 & 1.9 + 53433.60628 & 70.0 & 2.1 + 53446.79322 & 4.5 & 1.9 + 54161.85400 & -109.5 & 2.8 + 54166.83797 & -119.3 & 2.4 + 54186.76189 & -184.2 & 2.3 + lrr [ tbl-6 ] 53709.89685 & -184.5 & 2.3 + 53723.85188 & -238.6 & 2.2 + 53723.85367 & -238.2 & 2.5 + 53723.85546 & -227.7 & 2.3 + 53727.84394 & -238.9 & 2.5 + 53727.84573 & -244.9 & 2.4 + 53727.84752 & -242.9 & 2.6 + 53736.81887 & -257.6 & 2.5 + 53736.82100 & -248.2 & 2.9 + 53736.82315 & -253.4 & 2.4 + 53738.03261 & -246.7 & 2.8 + 53738.03441 & -243.3 & 2.4 + 53738.03620 & -236.0 & 2.5 + 53738.80860 & -240.5 & 2.6 + 53738.81040 & -258.9 & 2.4 + 53738.81219 & -249.3 & 2.5 + 53734.81795 & -242.8 & 2.6 + 53734.81973 & -243.9 & 2.8 + 53734.82152 & -248.5 & 2.4 + 53742.79119 & -252.0 & 2.8 + 53742.79299 & -257.2 & 2.8 + 53742.79479 & -239.7 & 2.8 + 53751.78199 & -257.4 & 2.9 + 53751.78378 & -263.1 & 2.5 + 53751.78558 & -268.0 & 2.3 + 53753.78155 & -273.1 & 2.5 + 53753.78381 & -278.7 & 2.5 + 53753.78607 & -266.4 & 2.4 + 53755.76038 & -286.6 & 2.3 + 53755.76218 & -266.5 & 2.6 + 53755.76397 & -274.9 & 2.7 + 53746.81506 & -257.1 & 1.9 + 53746.81778 & -250.9 & 2.1 + 53746.82051 & -245.2 & 2.3 + 53757.77002 & -277.6 & 2.4 + 53757.77181 & -280.3 & 2.4 + 53757.77360 & -288.7 & 2.2 + 53797.64609 & -439.8 & 3.1 + 53797.64834 & -462.6 & 2.8 + 53797.65059 & -452.5 & 2.9 + 53809.62428 & -658.6 & 2.4 + 53809.62700 & -658.8 & 2.5 + 53809.62972 & -659.2 & 2.3 + 53837.76359 & -304.3 & 3.0 + 53837.76670 & -324.0 & 2.9 + 53837.78731 & -308.6 & 2.7 + 53837.79077 & -285.2 & 2.6 + 53866.69987 & -215.9 & 1.7 + 53866.70329 & -228.3 & 1.7 + 53866.70670 & -220.4 & 1.8 + 53868.68349 & -251.6 & 3.8 + 53868.68562 & -208.6 & 2.9 + 53868.68777 & -247.4 & 9.7 + 53875.66956 & -215.7 & 1.6 + 53883.65565 & -213.8 & 1.8 + 53883.65837 & -209.2 & 1.7 + 53883.66109 & -200.4 & 1.7 + 53890.63776 & -203.4 & 1.7 + 53890.63954 & -202.6 & 1.9 + 53890.64134 & -203.2 & 1.9 + 53893.62959 & -193.8 & 2.0 + 53893.63139 & -189.3 & 1.9 + 53893.63318 & -189.7 & 1.8 + 54047.94811 & -375.2 & 4.8 + 54047.94991 & -353.2 & 4.5 + 54047.95172 & -362.6 & 4.4 + 54050.96248 & -415.0 & 2.6 + 54050.96453 & -423.0 & 2.5 + 54050.96657 & -420.1 & 2.4 + 54052.96488 & -426.8 & 2.3 + 54052.96762 & -437.1 & 2.5 + 54052.97035 & -447.6 & 2.5 + 54056.94606 & -468.0 & 3.0 + 54056.94786 & -466.4 & 2.6 + 54056.94964 & -479.4 & 2.8 + 54063.92981 & -599.1 & 2.1 + 54063.93166 & -594.8 & 2.3 + 54063.93348 & -592.3 & 2.4 + 54073.91213 & -685.8 & 2.8 + 54073.91476 & -688.7 & 2.9 + 54073.91739 & -704.4 & 2.7 + 54122.01039 & -220.8 & 2.5 + 54122.01243 & -219.1 & 2.6 + 54122.01447 & -218.4 & 2.8 + 54129.74214 & -215.7 & 2.6 + 54129.74491 & -224.4 & 3.0 + 54129.74768 & -223.7 & 3.1 + 54160.65850 & -189.5 & 3.2 + 54160.66031 & -181.8 & 2.7 + 54160.66212 & -204.8 & 3.2 + 54163.66458 & -213.9 & 3.1 + 54163.66643 & -200.8 & 2.9 + 54163.66828 & -208.0 & 3.2 + 54165.88148 & -208.5 & 2.7 +

long time coverage and high radial velocity precision have allowed for the discovery of additional objects in known planetary systems . many of the extrasolar planets detected have highly eccentric orbits , which raises the question of how likely those systems are to host additional planets . we investigate six systems which contain a very eccentric ( @xmath0 ) planet : hd 3651 , hd 37605 , hd 45350 , hd 80606 , hd 89744 , and 16 cyg b. we present updated radial - velocity observations and orbital solutions , search for additional planets , and perform test particle simulations to find regions of dynamical stability . the dynamical simulations show that short - period planets could exist in the hd 45350 and 16 cyg b systems , and we use the observational data to set tight detection limits , which rule out additional planets down to a few neptune masses in the hd 3651 , hd 45350 , and 16 cyg b systems .

arxiv

the lep experiments at the resonance of @xmath1-boson have tested the standard model ( sm ) at quantum level , measuring the @xmath1-decay into fermion pairs with an accuracy of one part in ten thousands . the good agreement of the lep data with the sm predictions have severely constrained the behavior of new physics at the @xmath1-pole . taking these achievements into account one can imagine that the physics of @xmath1-boson will again play the central role in the frontier of particle physics if the next generation @xmath1 factory comes true with the generated @xmath1 events several orders of magnitude higher than that of the lep . this factory can be realized in the gigaz option of the international linear collider ( ilc)@xcite . the ilc is a proposed electron - positron collider with tunable energy ranging from @xmath12 to @xmath13 and polarized beams in its first phase , and the gigaz option corresponds to its operation on top of the resonance of @xmath1 boson by adding a bypass to its main beam line . given the high luminosity , @xmath14 , and the cross section at the resonance of @xmath1 boson , @xmath15 , about @xmath16 @xmath1 events can be generated in an operational year of @xmath17 of gigaz , which implies that the expected sensitivity to the branching ratio of @xmath1-decay can be improved from @xmath18 at the lep to @xmath19 at the gigaz@xcite . in light of this , the @xmath1-boson properties , especially its exotic or rare decays which are widely believed to be sensitive to new physics , should be investigated comprehensively to evaluate their potential in probing new physics . among the rare @xmath1-decays , the flavor changing ( fc ) processes were most extensively studied to explore the flavor texture in new physics @xcite , and it was found that , although these processes are severely suppressed in the sm , their branching ratios in new physics models can be greatly enhanced to @xmath19 for lepton flavor violation decays @xcite and @xmath20 for quark flavor violation decays @xcite . besides the fc processes , the @xmath1-decay into light higgs boson(s ) is another type of rare process that was widely studied , e.g. the decay @xmath21 ( @xmath22 ) with the particle @xmath0 denoting a light higgs boson was studied in @xcite , the decay @xmath23 was studied in the two higgs doublet model ( 2hdm)@xcite and the minimal supersymmetric standard model ( mssm)@xcite , and the decay @xmath4 was studied in a model independent way @xcite , in 2hdm@xcite and also in mssm@xcite . these studies indicate that , in contrast with the kinematic forbidden of these decays in the sm , the rates of these decays can be as large as @xmath18 in new physics models , which lie within the expected sensitivity of the gigaz . in this work , we extend the previous studies of these decays to some new models and investigate these decays altogether . we are motivated by some recent studies on the singlet extension of the mssm , such as the next - to - minimal supersymmetric standard model ( nmssm ) @xcite and the nearly minimal supersymmetric standard model ( nmssm ) @xcite , where a light cp - odd higgs boson @xmath0 with singlet - dominant component may naturally arise from the spontaneous breaking of some approximate global symmetry like @xmath24 or peccei - quuin symmetry @xcite . these non - minimal supersymmetric models can not only avoid the @xmath25-problem , but also alleviate the little hierarchy by having such a light higgs boson @xmath0 @xcite . we are also motivated by that , with the latest experiments , the properties of the light higgs boson are more stringently constrained than before . so it is worth updating the previous studies . so far there is no model - independent lower bound on the lightest higgs boson mass . in the sm , it must be heavier than @xmath26 gev , obtained from the null observation of the higgs boson at lep experiments . however , due to the more complex structure of the higgs sector in the extensions of the sm , this lower bound can be significantly relaxed according to recent studies , e.g. , for the cp - odd higgs boson @xmath0 we have @xmath27 gev in the nmssm @xcite , @xmath28 gev in the nmssm @xcite , and @xmath29 gev in the lepton - specific 2hdm ( l2hdm ) @xcite . with such a light cp - odd higgs boson , the z - decay into one or more @xmath0 is open up . noting that the decay @xmath30 is forbidden due to bose symmetry , we in this work study the rare @xmath1-decays @xmath6 ( @xmath22 ) , @xmath31 and @xmath4 in a comparative way for four models , namely the type - ii 2hdm@xcite , the l2hdm @xcite , the nmssm and the nmssm . in our study , we examine carefully the constraints on the light @xmath0 from many latest experimental results . this work is organized as follows . in sec . ii we briefly describe the four new physics models . in sec . iii we present the calculations of the rare @xmath1-decays . in sec . iv we list the constraints on the four new physics models . in sec . v we show the numerical results for the branching ratios of the rare @xmath1-decays in various models . finally , the conclusion is given in sec . as the most economical way , the sm utilizes one higgs doublet to break the electroweak symmetry . as a result , the sm predicts only one physical higgs boson with its properties totally determined by two free parameters . in new physics models , the higgs sector is usually extended by adding higgs doublets and/or singlets , and consequently , more physical higgs bosons are predicted along with more free parameters involved in . the general 2hdm contains two @xmath32 doublet higgs fields @xmath33 and @xmath34 , and with the assumption of cp - conserving , its scalar potential can be parameterized as@xcite : @xmath35,\end{aligned}\ ] ] where @xmath36 ( @xmath37 ) are free dimensionless parameters , and @xmath38 ( @xmath39 ) are the parameters with mass dimension . after the electroweak symmetry breaking , the spectrum of this higgs sector includes three massless goldstone modes , which become the longitudinal modes of @xmath40 and @xmath1 bosons , and five massive physical states : two cp - even higgs bosons @xmath41 and @xmath42 , one neutral cp - odd higgs particle @xmath0 and a pair of charged higgs bosons @xmath43 . noting the constraint @xmath44 with @xmath45 and @xmath46 denoting the vacuum expectation values ( vev ) of @xmath33 and @xmath34 respectively , we choose @xmath47 as the input parameters with @xmath48 , and @xmath49 being the mixing angle that diagonalizes the mass matrix of the cp - even higgs fields . the difference between the type - ii 2hdm and the l2hdm comes from the yukawa coupling of the higgs bosons to quark / lepton . in the type - ii 2hdm , one higgs doublet @xmath34 generates the masses of up - type quarks and the other doublet @xmath33 generates the masses of down - type quarks and charged leptons ; while in the l2hdm one higgs doublet @xmath33 couples only to leptons and the other doublet @xmath34 couples only to quarks . so the yukawa interactions of @xmath0 to fermions in these two models are given by @xcite @xmath50 with @xmath51 denoting generation index . obviously , in the type - ii 2hdm the @xmath52 coupling and the @xmath53 coupling can be simultaneously enhanced by @xmath54 , while in the l2hdm only the @xmath53 coupling is enhanced by @xmath55 . the structures of the nmssm and the nmssm are described by their superpotentials and corresponding soft - breaking terms , which are given by @xcite @xmath56 where @xmath57 is the superpotential of the mssm without the @xmath25 term , @xmath58 and @xmath59 are higgs doublet and singlet superfields with @xmath60 and @xmath61 being their scalar component respectively , @xmath62 , @xmath63 , @xmath64 , @xmath65 , @xmath66 and @xmath67 are soft breaking parameters , and @xmath68 and @xmath69 are coefficients of the higgs self interactions . with the superpotentials and the soft - breaking terms , one can get the higgs potentials of the nmssm and the nmssm respectively . like the 2hdm , the higgs bosons with same cp property will mix and the mass eigenstates are obtained by diagonalizing the corresponding mass matrices : @xmath70 where the fields on the right hands of the equations are component fields of @xmath71 , @xmath72 and @xmath61 defined by @xmath73 @xmath74 and @xmath75 are respectively the cp - even and cp - odd neutral higgs bosons , @xmath76 and @xmath77 are goldstone bosons eaten by @xmath1 and @xmath78 , and @xmath79 is the charged higgs boson . so both the nmssm and nmssm predict three cp - even higgs bosons , two cp - odd higgs bosons and one pair of charged higgs bosons . in general , the lighter cp - odd higgs @xmath0 in these model is the mixture of the singlet field @xmath80 and the doublet field combination , @xmath81 , i.e. @xmath82 and its couplings to down - type quarks are then proportional to @xmath83 . so for singlet dominated @xmath0 , @xmath84 is small and the couplings are suppressed . as a comparison , the interactions of @xmath0 with the squarks are given by@xcite @xmath85 i.e. the interaction does not vanish when @xmath86 approaches zero . just like the 2hdm where we use the vevs of the higgs fields as fundamental parameters , we choose @xmath68 , @xmath69 , @xmath87 , @xmath88 , @xmath66 and @xmath89 as input parameters for the nmssm@xcite and @xmath68 , @xmath54 , @xmath88 , @xmath65 , @xmath90 and @xmath91 as input parameters for the nmssm@xcite . about the nmssm and the nmssm , three points should be noted . the first is for the two models , there is no explicit @xmath92term , and the effective @xmath25 parameter ( @xmath93 ) is generated when the scalar component of @xmath59 develops a vev . the second is , the nmssm is actually same as the nmssm with @xmath94@xcite , because the tadpole terms @xmath95 and its soft breaking term @xmath96 in the nmssm do not induce any interactions , except for the tree - level higgs boson masses and the minimization conditions . and the last is despite of the similarities , the nmssm has its own peculiarity , which comes from its neutralino sector . in the basis @xmath97 , its neutralino mass matrix is given by @xcite @xmath98 where @xmath99 and @xmath100 are @xmath101 and @xmath102 gaugino masses respectively , @xmath103 , @xmath104 , @xmath105 and @xmath106 . after diagonalizing this matrix one can get the mass eigenstate of the lightest neutralino @xmath107 with mass taking the following form @xcite @xmath108 this expression implies that @xmath107 must be lighter than about @xmath109 gev for @xmath110 ( from lower bound on chargnio mass ) and @xmath111 ( perturbativity bound ) . like the other supersymmetric models , @xmath107 as the lightest sparticle acts as the dark matter in the universe , but due to its singlino - dominated nature , it is difficult to annihilate sufficiently to get the correct density in the current universe . so the relic density of @xmath107 plays a crucial way in selecting the model parameters . for example , as shown in @xcite , for @xmath112 , there is no way to get the correct relic density , and for the other cases , @xmath107 mainly annihilates by exchanging @xmath1 boson for @xmath113 , or by exchanging a light cp - odd higgs boson @xmath0 with mass satisfying the relation @xmath114 for @xmath115 . for the annihilation , @xmath54 and @xmath25 are required to be less than 10 and @xmath116 respectively because through eq.([mass - exp ] ) a large @xmath87 or @xmath25 will suppress @xmath117 to make the annihilation more difficult . the properties of the lightest cp - odd higgs boson @xmath0 , such as its mass and couplings , are also limited tightly since @xmath0 plays an important role in @xmath107 annihilation . the phenomenology of the nmssm is also rather special , and this was discussed in detail in @xcite . in the type - ii 2hdm , l2hdm , nmssm and nmssm , the rare @xmath1-decays @xmath118 ( @xmath22 ) , @xmath3 and @xmath4 may proceed by the feynman diagrams shown in fig.[fig1 ] , fig.[fig2 ] and fig.[fig3 ] respectively . for these diagrams , the intermediate state @xmath119 represents all possible cp - even higgs bosons in the corresponding model , i.e. @xmath41 and @xmath42 in type - ii 2hdm and l2hdm and @xmath41 , @xmath42 and @xmath120 in nmssm and nmssm . in order to take into account the possible resonance effects of @xmath119 in fig.[fig1](c ) for @xmath2 and fig.[fig3 ] ( a ) for @xmath11 , we have calculated all the decay modes of @xmath119 and properly included the width effect in its propagator . as to the decay @xmath121 , two points should be noted . one is , unlike the decays @xmath6 and @xmath11 , this process proceeds only through loops mediated by quarks / leptons in the type - ii 2hdm and l2hdm , and additionally by sparticles in the nmssm and nmssm . so in most cases its rate should be much smaller than the other two . the other is due to cp - invariance , loops mediated by squarks / sleptons give no contribution to the decay@xcite . in actual calculation , this is reflected by the fact that the coupling coefficient of @xmath122 differs from that of @xmath123 by a minus sign ( see eq.([asqsq ] ) ) , and as a result , the squark - mediated contributions to @xmath121 are completely canceled out . with regard to the rare decay @xmath11 , we have more explanations . in the lowest order , this decay proceeds by the diagram shown in fig.[fig3 ] ( a ) , and hence one may think that , as a rough estimate , it is enough to only consider the contributions from fig.[fig3](a ) . however , we note that in some cases of the type - ii 2hdm and l2hdm , due to the cancelation of the contributions from different @xmath119 in fig.[fig3 ] ( a ) and also due to the potentially largeness of @xmath124 couplings ( i.e. larger than the electroweak scale @xmath125 ) , the radiative correction from the higgs - mediated loops may dominate over the tree level contribution even when the tree level prediction of the rate , @xmath126 , exceeds @xmath20 . on the other hand , we find the contribution from quark / lepton - mediated loops can be safely neglected if @xmath127 in the type - ii 2hdm and the l2hdm . in the nmssm and the nmssm , besides the corrections from the higgs- and quark / lepton - mediated loops , loops involving sparticles such as squarks , charginos and neutralinos can also contribute to the decay . we numerically checked that the contributions from squarks and charginos can be safely neglected if @xmath127 . we also calculated part of potentially large neutralino correction ( note that there are totally about @xmath128 diagrams for such correction ! ) and found they can be neglected too . since considering all the radiative corrections will make our numerical calculation rather slow , we only include the most important correction , namely that from higgs - mediated loops , in presenting our results for the four models . one can intuitively understand the relative smallness of the sparticle contribution to @xmath11 as follows . first consider the squark contribution which is induced by the @xmath129 interaction ( @xmath130 denotes the squark in chirality state ) and the @xmath131 interaction through box diagrams . because the @xmath132 interaction conserves the chirality of the squarks while the @xmath133 interaction violates the chirality , to get non - zero contribution to @xmath11 from the squark loops , at least four chiral flippings are needed , with three of them provided by @xmath131 interaction and the rest provided by the left - right squark mixing . this means that , if one calculates the amplitude in the chirality basis with the mass insertion method , the amplitude is suppressed by the mixing factor @xmath134 with @xmath135 being the off diagonal element in squark mass matrix . next consider the chargino / neutralino contributions . since for a light @xmath0 , its doublet component , parameterized by @xmath84 in eq.([mixing ] ) , is usually small , the couplings of @xmath0 with the sparticles will never be tremendously large@xcite . so the chargino / neutralino contributions are not important too . in our calculation of the decays , we work in the mass eigenstates of sparticles instead of in the chirality basis . for the type - ii 2hdm and the l2hdm , we consider the following constraints @xcite : * theoretical constraints on @xmath136 from perturbativity , unitarity and requirements that the scalar potential is finit at large field values and contains no flat directions @xcite , which imply that @xmath137 * the constraints from the lep search for neutral higgs bosons . we compute the signals from the higgs - strahlung production @xmath138 ( @xmath139 ) with @xmath140 @xcite and from the associated production @xmath141 with @xmath142 @xcite , and compare them with the corresponding lep data which have been inputted into our code . we also consider the constraints from @xmath138 by looking for a peak of @xmath143 recoil mass distribution of @xmath1-boson @xcite and the constraint of @xmath144 mev when @xmath145 @xcite . + these constraints limit the quantities such as @xmath146 \times br ( h_i \to \bar{b } b ) $ ] on the @xmath147 plane with the the subscript @xmath148 denoting the coupling coefficient of the @xmath149 interaction . they also impose a model - dependent lower bound on @xmath150 , e.g. , @xmath151 for the type - ii 2hdm ( from our scan results ) , @xmath152 for the l2hdm@xcite , and @xmath153 for the nmssm @xcite . these bounds are significantly lower than that of the sm , i.e. @xmath154 , partially because in new physics models , unconventional decay modes of @xmath155 such as @xmath156 are open up . as to the nmssm , another specific reason for allowing a significantly lighter cp - even higgs boson is that the boson may be singlet - dominated in this model . + with regard to the lightest cp - odd higgs boson @xmath0 , we checked that there is no lower bound on its mass so long as the @xmath157 interaction is weak or @xmath155 is sufficiently heavy . * the constraints from the lep search for a light higgs boson via the yukawa process @xmath158 with @xmath22 and @xmath61 denoting a scalar @xcite . these constraints can limit the @xmath159 coupling versus @xmath160 in new physics models . * the constraints from the cleo - iii limit on @xmath161 and the latest babar limits on @xmath162 . these constraints will put very tight constraints on the @xmath163 coupling for @xmath164 . in our analysis , we use the results of fig.8 in the second paper of @xcite to excluded the unfavored points . * the constraints from @xmath165 couplings . since the higgs sector can give sizable higher order corrections to @xmath165 couplings , we calculate them to one loop level and require the corrected @xmath165 couplings to lie within the @xmath166 range of their fitted value . the sm predictions for the couplings at @xmath1-pole are given by @xmath167 and @xmath168 @xcite , and the fitted values are given by @xmath169 and @xmath170 , respectively@xcite . we adopt the formula in @xcite to the 2hdm in our calculation . * the constraints from @xmath171 leptonic decay . we require the new physics correction to the branching ratio @xmath172 to be in the range of @xmath173 @xcite . we use the formula in @xcite in our calculation . + about the constraints ( 5 ) and ( 6 ) , two points should be noted . one is all higgs bosons are involved in the constraints by entering the self energy of @xmath171 lepton , the @xmath174 vertex correction or the @xmath175 vertex correction , and also the box diagrams for @xmath176@xcite . since the yukawa couplings of the higgs bosons to @xmath171 lepton get enhanced by @xmath54 and so do the corrections , @xmath54 must be upper bounded for given spectrum of the higgs sector . generally speaking , the lighter @xmath0 is , the more tightly @xmath54 is limited@xcite . the other point is in the type - ii 2hdm , @xmath177 , b - physics observables as well as @xmath178 decays discussed above can constraint the model in a tighter way than the constraints ( 5 ) and ( 6 ) since the yukawa couplings of @xmath171 lepton and @xmath179 quark are simultaneously enhanced by @xmath54 . but for the l2hdm , because only the yukawa couplings of @xmath171 lepton get enhanced ( see eq.[yukawa ] ) , the constraints ( 5 ) and ( 6 ) are more important in limiting @xmath54 . * indirect constraints from the precision electroweak observables such as @xmath180 , @xmath181 and @xmath182 , or their combinations @xmath183 @xcite . we require @xmath184 to be compatible with the lep / sld data at @xmath185 confidence level@xcite . we also require new physics prediction of @xmath186 is within the @xmath187 range of its experimental value . the latest results for @xmath188 are @xmath189 ( measured value ) and @xmath190 ( sm prediction ) for @xmath191 gev @xcite . in our code , we adopt the formula for these observables presented in @xcite to the type - ii 2hdm and the l2hdm respectively . + in calculating @xmath180 , @xmath181 and @xmath182 , we note that these observables get dominant contributions from the self energies of the gauge bosons @xmath1 , @xmath192 and @xmath193 . since there is no @xmath194 coupling or @xmath195 coupling , @xmath0 must be associated with the other higgs bosons to contribute to the self energies . so by the uv convergence of these quantities , one can infer that , for the case of a light @xmath0 and @xmath196 , these quantities depend on the spectrum of the higgs sector in a way like @xmath197 at leading order , which implies that a light @xmath0 can still survive the constraints from the precision electroweak observables given the splitting between @xmath150 and @xmath198 is moderate@xcite . * the constraints from b physics observables such as the branching ratios for @xmath199 , @xmath200 and @xmath201 , and the mass differences @xmath202 and @xmath203 . we require their theoretical predications to agree with the corresponding experimental values at @xmath187 level . + in the type - ii 2hdm and the l2hdm , only the charged higgs boson contributes to these observables by loops , so one can expect that @xmath198 versus @xmath54 is to be limited . combined analysis of the limits in the type - ii 2hdm has been done by the ckmfitter group , and the lower bound of @xmath204 as a function of @xmath87 was given in fig.11 of @xcite . this analysis indicates that @xmath198 must be heavier than @xmath205 at @xmath185 c.l . regardless the value of @xmath54 . in this work , we use the results of fig.11 in @xcite to exclude the unfavored points . as for the l2hdm , b physics actually can not put any constraints@xcite because in this model the couplings of the charged higgs boson to quarks are proportional to @xmath206 and in the case of large @xmath54 which we are interested in , they are suppressed . in our analysis of the l2hdm , we impose the lep bound on @xmath198 , i.e. @xmath207@xcite . * the constraints from the muon anomalous magnetic moment @xmath208 . now both the theoretical prediction and the experimental measured value of @xmath208 have reached a remarkable precision , but a significant deviation still exists : @xmath209 @xcite . in the 2hdm , @xmath208 gets additional contributions from the one - loop diagrams induced by the higgs bosons and also from the two - loop barr - zee diagrams mediated by @xmath0 and @xmath155@xcite . if the higgs bosons are much heavier than @xmath25 lepton mass , the contributions from the barr - zee diagrams are more important , and to efficiently alleviate the discrepancy of @xmath208 , one needs a light @xmath0 along with its enhanced couplings to @xmath25 lepton and also to heavy fermions such as bottom quark and @xmath171 lepton to push up the effects of the barr - zee diagram@xcite . the cp - even higgs bosons are usually preferred to be heavy since their contributions to @xmath208 are negative . + in the type - ii 2hdm , because @xmath54 is tightly constrained by the process @xmath210 at the lep@xcite and the @xmath178 decay@xcite , the barr - zee diagram contribution is insufficient to enhance @xmath208 to @xmath187 range around its measured value@xcite . so in our analysis , we require the type - ii 2hdm to explain @xmath208 at @xmath211 level . while for the l2hdm , @xmath54 is less constrained compared with the type - ii 2hdm , and the barr - zee diagram involving the @xmath171-loop is capable to push up greatly the theoretical prediction of @xmath208@xcite . therefore , we require the l2hdm to explain the discrepancy at @xmath187 level . + unlike the other constraints discussed above , the @xmath208 constraint will put a two - sided bound on @xmath54 since on the one hand , it needs a large @xmath54 to enhance the barr - zee contribution , but on the other hand , too large @xmath54 will result in an unacceptable large @xmath208 . * since this paper concentrates on a light @xmath0 , the decay @xmath212 is open up with a possible large decay width . we require the width of any higgs boson to be smaller than its mass to avoid a too fat higgs boson@xcite . we checked that for the scenario characterized by @xmath213 , the coefficient of @xmath214 interaction is usually larger than the electroweak scale @xmath125 , and consequently a large decay width is resulted . for the nmssm and nmssm , the above constraints become more complicated because in these models , not only more higgs bosons are involved in , but also sparticles enter the constraints . so it is not easy to understand some of the constraints intuitively . take the process @xmath199 as an example . in the supersymmetric models , besides the charged higgs contribution , chargino loops , gluino loops as well as neutralino loops also contribute to the process@xcite , and depending on the susy parameters , any of these contributions may become dominated over or be canceled by other contributions . as a result , although the charged higgs affects the process in the same way as that in the type - ii 2hdm , charged higgs as light as @xmath215 is still allowed even for @xmath216@xcite . since among the constraints , @xmath208 is rather peculiar in that it needs new physics to explain the discrepancy between @xmath217 and @xmath218 , we discuss more about its dependence on susy parameters . in the nmssm and the nmssm , @xmath208 receives contributions from higgs loops and neutralino / chargino loops . for the higgs contribution , it is quite similar to that of the type - ii 2hdm except that more higgs bosons are involved in@xcite . for the neutralino / chargino contribution , in the light bino limit ( i.e. @xmath219 ) , it can be approximated by@xcite @xmath220 for @xmath221 with @xmath222 being smuon mass . so combining the two contributions together , one can learn that a light @xmath0 along with large @xmath54 and/or light smuon with moderate @xmath87 are favored to dilute the discrepancy . because more parameters are involved in the constraints on the supersymmetric models , we consider following additional constraints to further limit their parameters : * direct bounds on sparticle masses from the lep1 , the lep2 and the tevatron experiments @xcite . * the lep1 bound on invisible z decay @xmath223 ; the lep2 bound on neutralino production @xmath224 and @xmath225@xcite . * dark matter constraints from the wmap relic density 0.0975 @xmath226 0.1213 @xcite . note that among the above constraints , the constraint ( 2 ) on higgs sector and the constraint ( c ) on neutralino sector are very important . this is because in the supersymmetric models , the sm - like higgs is upper bounded by about @xmath227 at tree level and by about @xmath228 at loop level , and that the relic density restricts the lsp annihilation cross section in a certain narrow range . in our analysis of the nmssm , we calculate the constraints ( 3 ) and ( 5 - 7 ) by ourselves and utilize the code nmssmtools @xcite to implement the rest constraints . we also extend nmssmtools to the nmssm to implement the constraints . for the extension , the most difficult thing we faced is how to adapt the code micromegas@xcite to the nmssm case . we solve this problem by noting the following facts : * as we mentioned before , the nmssm is actually same as the nmssm with the trilinear singlet term setting to zero . so we can utilize the model file of the nmssm as the input of the micromegas and set @xmath229 . * since in the nmssm , the lsp is too light to annihilate into higgs pairs , there is no need to reconstruct the effective higgs potential to calculate precisely the annihilation channel @xmath230 with @xmath61 denoting any of higgs bosons@xcite . we thank the authors of the nmssmtools for helpful discussion on this issue when we finish such extension@xcite . with the above constraints , we perform four independent random scans over the parameter space of the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively . we vary the parameters in following ranges : @xmath231 for the type - ii 2hdm , @xmath232 for the l2hdm , @xmath233 for the nmssm , and @xmath234 for the nmssm . in performing the scans , we note that for the nmssm and the nmssm , some constraints also rely on the gaugino masses and the soft breaking parameters in the squark sector and the slepton sector . since these parameters affect little on the properties of @xmath0 , we fix them to reduce the number of free parameters in our scan . for the squark sector , we adopt the @xmath235 scenario which assumes that the soft mass parameters for the third generation squarks are degenerate : @xmath236 800 gev , and that the trilinear couplings of the third generation squarks are also degenerate , @xmath237 with @xmath238 . for the slepton sector , we assume all the soft - breaking masses and trilinear parameters to be 100 gev . this setting is necessary for the nmssm since this model is difficult to explain the muon anomalous moment at @xmath239 level for heavy sleptons@xcite . finally , we assume the grand unification relation @xmath240 for the gaugino masses with @xmath241 being fine structure constants of the different gauge group . with large number of random points in the scans , we finally get about @xmath242 , @xmath243 , @xmath244 and @xmath242 samples for the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively which survive the constraints and satisfy @xmath245 . analyzing the properties of the @xmath0 indicates that for most of the surviving points in the nmssm and the nmssm , its dominant component is the singlet field ( numerically speaking , @xmath246 ) so that its couplings to the sm fermions are suppressed@xcite . our analysis also indicates that the main decay products of @xmath0 are @xmath247 for the l2hdm@xcite , @xmath248 ( dominant ) and @xmath247 ( subdominant ) for the type - ii 2hdm , the nmssm and the nmssm , and in some rare cases , neutralino pairs in the nmssm@xcite . in fig.[fig4 ] , we project the surviving samples on the @xmath249 plane . this figure shows that the allowed range of @xmath54 is from @xmath250 to @xmath251 in the type - ii 2hdm , and from @xmath252 to @xmath253 in the l2hdm . just as we introduced before , the lower bounds of @xmath254 come from the fact that we require the models to explain the muon anomalous moment , while the upper bound is due to we have imposed the constraint from the lep process @xmath255 , which have limited the upper reach of the @xmath256 coupling for light @xmath61 @xcite(for the dependence of @xmath256 coupling on @xmath54 , see sec . this figure also indicates that for the nmssm and the nmssm , @xmath54 is upper bounded by @xmath257 . for the nmssm , this is because large @xmath87 can suppress the dark matter mass to make its annihilation difficult ( see @xcite and also sec . ii ) , but for the nmssm , this is because we choose a light slepton mass so that large @xmath54 can enhance @xmath208 too significantly to be experimentally unacceptable . we checked that for the slepton mass as heavy as @xmath258 , @xmath259 is still allowed for the nmssm . in fig.[fig5 ] and fig.[fig6 ] , we show the branching ratios of @xmath260 and @xmath261 respectively . fig.[fig5 ] indicates , among the four models , the type - ii 2hdm predicts the largest ratio for @xmath260 with its value varying from @xmath262 to @xmath263 . the underlying reason is in the type - ii 2hdm , the @xmath264 coupling is enhanced by @xmath54 ( see fig.[fig4 ] ) , while in the other three model , the coupling is suppressed either by @xmath265 or by the singlet component of the @xmath0 . fig.[fig6 ] shows that the l2hdm predicts the largest rate for @xmath266 with its value reaching @xmath5 in optimum case , and for the other three models , the ratio of @xmath261 is at least about one order smaller than that of @xmath267 . this feature can be easily understood from the @xmath268 coupling introduced in sect . we emphasize that , if the nature prefers a light @xmath0 , @xmath260 and/or @xmath269 in the type - ii 2hdm and the l2hdm will be observable at the gigaz . then by the rates of the two decays , one can determine whether the type - ii 2hdm or the l2hdm is the right theory . on the other hand , if both decays are observed with small rates or fail to be observed , the singlet extensions of the mssm are favored . in fig.[fig7 ] , we show the rate of @xmath3 as the function of @xmath270 . this figure indicates that the branching ratio of @xmath121 can reach @xmath271 , @xmath272 , @xmath273 and @xmath274 for the optimal cases of the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively , which implies that the decay @xmath121 will never be observable at the gigaz if the studied model is chosen by nature . the reason for the smallness is , as we pointed out before , that the decay @xmath121 proceeds only at loop level . comparing the optimum cases of the type - ii 2hdm , the nmssm and the nmssm shown in fig.5 - 7 , one may find that the relation @xmath275 holds for any of the decays . this is because the decays are all induced by the yukawa couplings with similar structure for the models . in the supersymmetric models , the large singlet component of the light @xmath0 is to suppress the yukawa couplings , and the @xmath0 in the nmssm has more singlet component than that in the nmssm . next we consider the decay @xmath11 , which , unlike the above decays , depends on the higgs self interactions . in fig.[fig8 ] we plot its rate as a function of @xmath270 and this figure indicates that the @xmath276 may be the largest among the ratios of the exotic @xmath1 decays , reaching @xmath277 in the optimum cases of the type - ii 2hdm , the l2hdm and the nmssm . the underlying reason is , in some cases , the intermediate state @xmath119 in fig.[fig3 ] ( a ) may be on - shell . in fact , we find this is one of the main differences between the nmssm and the nmssm , that is , in the nmssm , @xmath119 in fig.[fig3 ] ( a ) may be on - shell ( corresponds to the points with large @xmath278 ) while in the nmssm , this seems impossible . so we conclude that the decay @xmath11 may serve as an alternative channel to test new physics models , especially it may be used to distinguish the nmssm from the nmssm if the supersymmetry is found at the lhc and the @xmath11 is observed at the gigaz with large rate . before we end our discussion , we note that in the nmssm , the higgs boson @xmath0 may be lighter than @xmath279 without conflicting with low energy data from @xmath178 decays and the other observables ( see fig.[fig4]-[fig8 ] ) . in this case , @xmath0 is axion - like as pointed out in @xcite . we checked that , among the rare @xmath1 decays discussed in this paper , the largest branching ratio comes from @xmath280 which can reach @xmath281 . since in this case , the decay product of @xmath0 is highly collinear muon pair , detecting the decay @xmath280 may need some knowledge about detectors , which is beyond our discussion . in this paper , we studied the rare @xmath1-decays @xmath2 ( @xmath7 ) , @xmath282 and @xmath4 in the type - 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various new physics models predict a light cp - odd higgs boson ( labeled as @xmath0 ) and open up new decay modes for @xmath1-boson , such as @xmath2 , @xmath3 and @xmath4 , which could be explored at the gigaz option of the ilc . in this work we investigate these rare decays in several new physics models , namely the type - ii two higgs doublet model ( type - ii 2hdm ) , the lepton - specific two higgs doublet model ( l2hdm ) , the nearly minimal supersymetric standard model ( nmssm ) and the next - to - minimal supersymmetric standard model ( nmssm ) . we find that in the parameter space allowed by current experiments , the branching ratios can reach @xmath5 for @xmath6 ( @xmath7 ) , @xmath8 for @xmath3 and @xmath9 for @xmath4 , which implies that the decays @xmath10 and @xmath11 may be accessible at the gigaz option . moreover , since different models predict different patterns of the branching ratios , the measurement of these rare decays at the gigaz may be utilized to distinguish the models .

arxiv

supersymmetry ( susy ) is one of the most attractive extensions of the standard model . this symmetry solves the naturalness problem and predicts gauge coupling unification at the gut scale @xmath1 . it also predicts the existence of superpartner of the standard model ( sm ) particles . from the naturalness argument , their masses should be below tev range , hence these particles will be discovered at tevatron or large hadron collider ( lhc ) . mechanisms of susy breaking and its mediation to the minimal supersymmetric standard model ( mssm ) sector are one of the most important problems in the susy phenomenology . in many models , this dynamics is related to high energy physics far above the electroweak(ew ) scale , e.g. , gut scale or planck scale . once the mechanism is specified , mass spectrum and flavor structure of susy particle at the ew scale can be determined by a small number of parameters . hence it may be possible to confirm or exclude the mechanism by direct search or flavor - changing - neutral - current ( fcnc ) experiments in near future . if susy breaking is mediated by gravity , the structure of susy breaking masses of scalars are determined by khler potential . in the present paper , we focus on the no - scale type khler potential , in which the hidden sector and the observable sector are separated as follows : @xmath2 where @xmath3 and @xmath4 are hidden sector fields and observable sector fields , respectively . this type of khler potential is originally investigated in ref . @xcite with @xmath5 and @xmath6 . characteristic features of the khler potential eq.([eq : noscalekahler ] ) is that all scalar masses and trilinear scalar couplings ( a - terms ) vanish as the cosmological constant vanishes@xcite . the only source of susy breaking is gaugino masses . hence this scenario is highly predictive , various phenomenological consequences are obtained with a few parameters . the separation in eq.([eq : noscalekahler ] ) implies that couplings of the hidden sector and the observable sector is flavor blind , and contributions of susy particles to fcnc are suppressed . therefore this khler potential is also interesting from the viewpoint of the susy flavor problem . the no - scale structure of the khler potential is obtained in various models . it has been shown that in some classes of string theory , for example weakly coupled @xmath7 heterotic string theory , khler potential becomes the no - scale type@xcite . if the hidden sector and the observable sector are separated in the superspace density in the supergravity lagrangian , the khler potential is indeed given as in eq . ( [ eq : noscalekahler ] ) . in the two cases , the gaugino masses can be induced if the hidden sector fields couple to the gauge multiplets via the gauge kinetic function . recently it has been pointed out that the form eq.([eq : noscalekahler ] ) is realized naturally in a five - dimensional setting with two branes , namely , sequestered sector scenario@xcite . in this scenario , the hidden sector fields live on one brane and the visible sector fields live on the other . it has been shown that the form of the khler potential of the effective theory obtained by dimensional reduction is indeed eq.([eq : noscalekahler])@xcite . if the sm gauge fields dwell in the bulk , gaugino mediate the susy breaking on the hidden sector brane to the visible sector brane and the no - scale boundary condition is given at the compactification scale of the fifth dimension ( gaugino mediation @xcite ) . in the no - scale scenario , degrees of freedom of susy particle mass spectrum is limited because only non - zero soft susy breaking masses are gaugino masses and higgs mixing mass @xmath8 at the energy scale where the boundary condition is given . hence phenomenological aspects of this scenario have been investigated in the literature , mainly focusing on the mass spectrum . direct search bounds and the cosmological constraint ( i.e. , a charged particle can not be the lsp if the r - parity is conserved ) were considered and allowed region in the parameter space was identified . for the boundary condition , the following three cases were considered . first , universal gaugino masses are given at the gut scale . in this case , the cosmological constraint is severe and only the region @xmath9 and @xmath10 is allowed since stau tends to be light@xcite . the second case is that universal gaugino masses are given above the gut scale . and the third case is that non - universal gaugino masses are given at the gut scale . in this case wino , higgsino or sneutrino can be the lsp . in the latter two cases , it is shown that the cosmological constraint is not severer than the first case . in the present paper , current limits from the lightest higgs mass @xmath11 and the branching ratio for @xmath0 are also used to constrain the no - scale scenario . combining these constraints , we will show that almost all the parameter region is excluded when universal gaugino masses are given at the gut scale . however , when the boundary condition is given above the gut scale , relatively large parameter region is allowed . we also consider the case that the non - universal gaugino masses are given at the gut scale . we will show that these constraints are important when the higgsino - like neutralino is the lsp . this paper is organized as follows . in section [ sec : noscalsebc ] , we review some phenomenological aspects of the no - scale models , especially indications of the direct search bounds and the cosmological bound . in section [ sec : higgsbsgamma ] , we further constrain these models from the higgs mass bound and @xmath12 result . indications of these bounds for the tevatron are also discussed . our conclusions are given in section [ sec : conclusions ] . in this section , we briefly review phenomenological aspects of susy models with no - scale boundary condition , mainly indications of the cosmological bound and direct search limit at lep 2 . we consider the following three cases . * universal gaugino masses are given at the gut scale . hereafter we call this case the minimal scenario . * universal gaugino masses are given above the gut scale @xmath13 . throughout this paper , we take the minimal su(5 ) to be the theory above the gut scale as a typical example . * non - universal gaugino masses are given at the gut scale . once one of the above boundary conditions is given , mass spectrum of susy particles at the ew scale and their contributions to fcnc can be calculated . in this paper we solve the one - loop level rges to obtain the soft susy breaking mass parameters at the ew scale . the higgsino mass parameter @xmath14 is determined by potential minimum condition at the one - loop level . first , we discuss the minimal scenario . in this case , the following boundary condition is given at the gut scale , @xmath15 where @xmath16 is the common scalar mass and @xmath17 is universal trilinear scalar coupling . with this boundary condition , bino and right - handed sleptons are lighter than other susy particles . their masses are approximately , @xmath18 from eq.([eq : minimalmneumtau ] ) we see that the charged right - handed slepton is the lsp if the d - term @xmath19 is negligible , i.e. , @xmath20 . hence this parameter region is excluded by the cosmological consideration . on the other hand , lep 2 experiments yields the upper bound on the cross section for smuon pair production , @xmath21 for @xmath22 and @xmath23@xcite , so the parameter region @xmath24 is excluded in fig . [ fig : limitmimposmu ] and [ fig : limitmimnegmu ] , allowed region of the parameter space are shown in the @xmath25 plane . the regions above the dash - dotted line and the left side of the dash - dot - dotted line are excluded by cosmological bound and lep 2 bound on smuon pair production , respectively . therefore the minimal scenario is constrained severely . next we see the case that the universal gaugino masses are given above the gut scale . in the minimal su(5 ) case , the right - handed slepton belongs to 10-plet , so the large group factor makes slepton masses heavier . for example , when @xmath26 the bino mass and the right - handed slepton mass at the weak scale are approximately given , @xmath27 hence the cosmological constraint is not severe because the stau mass is large enough and neutralino is the lsp in the large parameter region @xcite . in the fig.[fig : limitmbc1e17posmu ] and [ fig : limitmbc1e17negmu ] , the same figures as in the fig.[fig : limitmimposmu ] and [ fig : limitmimnegmu ] are shown . unlike in the minimal case , the stau search bound at lep @xcite is also plotted because mass difference between @xmath28 and @xmath29 is larger than in the minimal case and it can be stronger than the smuon search . from these figures we see that the @xmath29 lsp is avoided unless @xmath30 is larger than about 20 . the charged stau lsp can also be avoided if gaugino masses at the gut scale are non - universal@xcite , i.e. , the following boundary condition is given , @xmath31 this boundary condition can be given naturally within the gut framework @xcite . in this case , not only bino - like neutralino , but also wino - like , higgsino - like neutralino or sneutrino can be the lsp . for @xmath32 and @xmath33 , the lsp is wino - like neutralino . for example , when @xmath34 and @xmath35 , then wino mass and charged slepton mass are ( notice that in this case the left - handed sleptons are lighter than right - handed sleptons ) ; @xmath36 the higgsino is the lsp if @xmath37 . for example , when @xmath38 and @xmath39 , then the higgsino mass and the right - handed slepton mass are @xmath40 in the two cases given above , neutral wino or higgsino is the lsp . in fact from fig.[fig : limit12r432r2posmu ] - [ fig : limit12r232r0.5negmu ] we find that neural particle is the lsp in large parameter region , thus it is cosmologically viable . in the previous section we take into account only lep 2 bound and the cosmological constraint . we find that the minimal scenario is severely constrained , but the other two scenarios are not . in this section we also include the current higgs mass bound and @xmath41 constraint . as we will see , combining the above four constraint , not only the minimal case but also the other two scenarios can be constrained more severely . we also discuss the possibility whether this scenario can be seen at the tevatron run 2 or not . before we show the numerical results , some remarks on our calculation of the higgs mass and @xmath12 are in order . it is well known that radiative correction is important when the lightest higgs mass is evaluated @xcite . in the present paper , the lightest higgs mass is evaluated by means of the one - loop level effective potential@xcite . this potential is evaluated at the renormalization point of the geometrical mean of the two stop mass eigenvalues @xmath42 . we compared our result with a two - loop result by using _ feynhiggs_@xcite , and checked that the difference between these two results is smaller than 5 gev as long as @xmath30 is bigger than 5 . when @xmath30 is close to 2 , the difference can be 7 gev . however , as we will see later , higgs mass bound plays an important rule around @xmath43 . and the two - loop effects always make the higgs mass lighter than that obtained at the one - loop level . so our conclusion is conservative and is not significantly changed by the two - loop effect . we exclude the parameter region where the lightest higgs mass is lighter than the current 95% c.l . limit from lep 2 experiments , @xmath44 @xcite . in the present paper , @xmath12 is calculated including leading order ( lo ) qcd corrections@xcite , and compare it to the current cleo measurement . in the mssm , chargino contribution can be comparable to the sm and charged higgs contributions . they interfere constructively ( destructively ) each other when @xmath45 ( @xmath46 ) . the difference between the lo and the next - to - leading order ( nlo ) result can be sizable only when cancellation among different contributions at the lo is spoiled by the nlo contributions . as we will see , however , the @xmath41 constraint is severe when the interference is constructive . in the case of destructive interference where the deviation from the nlo result may be large , this constraint is not so important . hence we expect that our conclusion is not changed significantly by the inclusion of the nlo corrections . for the experimental value , we use 95% c.l . limit from cleo , @xmath47 @xcite . first we show the numerical results for the minimal case . the case for @xmath48 is shown in fig.[fig : limitmimposmu ] . in this case , for small @xmath30 region , the stop mass is not so large that radiative correction factor @xmath49 which raises the higgs mass is small . ( for example , @xmath50 gev and @xmath51 gev for @xmath52 gev and @xmath53 ) . hence the higgs mass limit constrains this scenario severely . in fig.[fig : limitmimposmu ] , the higgs mass bound and @xmath12 constraints in the @xmath54 plane are shown . the regions below the solid line and above the dashed line are excluded by the higgs mass and @xmath12 bound , respectively . the indication of @xmath55 reported by lep 2@xcite is also shown in this figure . from the figure we find that the higgs mass bound almost excludes the region where the stau lsp is avoided . note that , as we discussed earlier , the bound we put on the higgs mass may be conservative , because the two loop correction may further reduce the higgs mass . the same figure but for @xmath56 is shown in fig.[fig : limitmimnegmu ] . now @xmath12 also constrains parameter region strongly since chargino contribution to @xmath0 interferes with sm and charged higgs ones constructively . the region above the dashed line is excluded by @xmath12 constraint . we find that only one of the two constraints is enough to exclude all the region where cosmological bound and the smuon mass bound are avoided . hence if r - parity is conserved , i.e. , the cosmological bound is relevant , this scenario with @xmath56 is excluded . next we show the numerical results in the case that the cutoff scale is larger than the gut scale . as a typical example , we choose the minimal su(5 ) as the theory above the gut scale . in fig.[fig : limitmbc1e17posmu ] and [ fig : limitmbc1e17negmu ] , results are shown for positive and negative @xmath14 , respectively . in both figures , we take @xmath58 gev . for @xmath48 case , large parameter region is allowed and susy scale @xmath59 can be as small as about 180 gev , which indicates the lsp mass @xmath60 gev . for @xmath56 , as in the minimal case , @xmath12 constraint is severer , and @xmath59 must be larger than around 280 gev . we also considered other values of the boundary scale @xmath61 from @xmath62 to @xmath63 , and checked that the behavior of the contour plot does not change so much . according to ref.@xcite , tevatron run 2 experiment can explore up to @xmath64 gev for integrated luminosity @xmath65 . hence if @xmath66 and @xmath67 , susy particles can be discovered at the experiment . in this range , trilepton from chargino - neutralino associated production @xmath68 , @xmath69 , @xmath70 is one of clean signals for susy search . notice that now two body decay @xmath71 opens . so same flavor , opposite sign dilepton from @xmath72 decay may be useful . the two body decay allows us to observe the peak edge of invariant mass of two leptons at the @xmath73 . it is expressed in terms of the neutralino masses and the slepton mass as , @xmath74 in table [ tab : mllmax ] , the dependence of @xmath73 on @xmath61 is shown . here we fix @xmath75 . notice that as @xmath61 changes , the right - handed mass changes sizably while the neutralino masses do not . hence we can obtain the mass relation among them and also cutoff scale @xmath61 , which corresponds to the compactification scale in the sequestered sector scenario , by measuring @xmath73 . on the other hand , since only @xmath76 gev is allowed for @xmath45 , the tevatron run 2 can not survey this scenario , and we have to wait lhc experiment . next , we turn to the case that gaugino masses are non - universal at the gut scale . we explore the following three cases , wino - like neutralino lsp , higgsino - like neutralino lsp and the tau sneutrino lsp . we will see that in the wino - like neutralino lsp and tau sneutrino lsp cases , constraint is not so severe even if we combine higgs mass bound and @xmath12 data , but in the higgsino - like lsp case where stops are as light as sleptons and charginos , the predicted higgs mass tends to be small , and thus the higgs mass bound becomes important . first , we discuss the wino - lsp case . the results for @xmath34 , @xmath35 are shown in fig.[fig : limit12r432r2posmu ] and fig.[fig : limit12r432r2negmu ] , for @xmath46 and @xmath45 , respectively . in this case , we obtain a relatively large higgs mass since @xmath77 is large and so are the masses of stops . hence , for @xmath46 , @xmath78 can be as small as 100 gev at @xmath43 , where the mass of the lsp @xmath79 is about 90 gev . for @xmath45 , though @xmath12 constraint is slightly severer than in the @xmath46 case , @xmath80 is allowed , which corresponds to @xmath81 . hence the wino - lsp with mass around 100 gev is allowed . examples of the mass spectrum in this case are listed as point a ( @xmath46 ) and point b ( @xmath45 ) in table [ tab : spectrum ] . at the both points , @xmath59 is chosen to be near the smallest value such that all constraints are avoided . in general , however , masses of @xmath28 and @xmath82 are highly degenerate when wino is the lsp . in fact , from table [ tab : spectrum ] , we see that the mass difference is less than 1 gev . therefore a lepton from @xmath70 is very soft and trilepton signal search is not useful because acceptance cut usually requires the smallest transverse momentum of the three leptons @xmath83 to be larger than 5 gev@xcite . recently collider phenomenology in such cases are studied in ref . it is shown that certain range of @xmath84 and @xmath85 , susy signals which are different from those in the minimal case can be detected . the high degeneracy requires to include radiative corrections to calculate @xmath85 @xcite , which is beyond of this work . it deserves detail study to estimate the mass difference in the scenario . since the constraint for the sneutrino lsp case in the @xmath86 plane is similar to those in the wino - lsp case , we show the result for @xmath46 only in fig.[fig : limit12r2.532r1.5posmu ] . in the figure , we take @xmath87 and @xmath88 . notice that the decomposition of the lsp depends on @xmath30 and the sneutrino is the lsp for @xmath89 . an example of the mass spectrum is listed as the point c in table [ tab : spectrum ] . in this case , trilepton signal comes from @xmath90 , @xmath91 , @xmath92 . since @xmath93 , @xmath94 of a lepton from @xmath82 decay is small and this signal may be hard to be detected . we may need unusual trigger to explore this scenario . next , we turn to the higgsino lsp case . higgsino lsp scenario is realized when @xmath77 is smaller than half of @xmath78 , which indicates that colored particles are lighter than in the universal gaugino mass case . hence the one - loop correction to the higgs potential which enhances the higgs mass is small and the higgs mass constraint is important . the same figures as fig.[fig : limitmimposmu ] and [ fig : limitmimnegmu ] are shown in fig.[fig : limit12r232r0.5posmu ] ( @xmath46 ) and fig.[fig : limit12r232r0.5negmu ] ( @xmath45 ) for @xmath95 and @xmath39 . in order to satisfy the higgs mass bound , @xmath78 must be larger than around 300 gev . combining the bound with @xmath12 , constraint becomes severer , especially for @xmath45 case where @xmath96 is required . example of mass spectrum in this scenario is listed as point d and e in table [ tab : spectrum ] . again we choose almost the smallest value of @xmath78 where all constraints are avoided . we see that the lsp mass must be at least @xmath97 for @xmath46 and @xmath98 for @xmath45 . hence this scenario can not be explored at the tevatron run 2 . the no - scale type boundary conditions are obtained in various types of susy models . this scenario is attractive because it is highly predictive and can be a solution to the susy flavor problem . in this paper we investigated the indication of the current higgs mass and @xmath41 constraint on susy models with the boundary condition . first we considered the minimal case where the universal gaugino mass are given at the gut scale . this scenario has been already constrained by direct search at lep and the cosmological bound severely , under the assumption of the exact r - parity . we showed that the higgs mass bound and @xmath41 constraint are also taken into account , then almost all the parameter region is excluded , leaving very narrow allowed region for @xmath48 . next we considered the case that the boundary condition is given above the gut scale . since the cosmological constraint is not severe , wide region of the parameter space is allowed . in the @xmath46 case , tevatron have a chance to observe susy signatures like trilepton events . the scale @xmath61 may be explored by measuring the peak edge of invariant mass of two leptons at the @xmath73 . however for the @xmath45 case , since @xmath99 is required , we have to wait lhc . finally we considered the case where non - universal gaugino masses are given at the gut scale . we see that the higgs mass bound is strong in the higgsino lsp case because stop masses are as light as sleptons and charginos . the mass of the higgsino - like neutralino must be larger than about 245 gev and 370 gev for @xmath46 and @xmath45 , respectively . in the wino lsp and sneutrino lsp case , the mass of the lsp can be as small as 150 gev . however , the mass difference between the lsp and parent particles produced at the collider is much smaller than in the minimal case , unusual acceptance cut may be required . the author would like to thank m. yamaguchi for suggesting the subject , fruitful discussion and careful reading of the manuscript . he also thanks t. moroi and m. m. nojiri for useful discussion . j. ellis , c. kounnas and d.v . nanopoulos , nucl . b247 ( 1984 ) 373 ; j. ellis , a.b . lahanas , d.v . nanopoulos and k. tamvakis , phys . b134 ( 1984 ) 439 ; a.b . lahanas and d.v . nanopoulos , phys . ( 1987 ) 1 . e. witten , phys . b155 ( 1985 ) 151 . t. banks and m. dine , nucl . b505 ( 1997 ) 445 ; h.p . nilles , m. olechowski and m. yamaguchi , phys . b415 ( 1997 ) 24 ; nucl . phys . b530 ( 1998 ) 43 ; z. lalak and s. thomas , nucl . phys . b515 ( 1998 ) 55 ; a. lukas , b.a . ovrut and d. waldram , nucl . phys . b532 ( 1998 ) 43 ; phys . d57 ( 1998 ) 7529 ; k. choi , h.b . kim and c. muoz , phys . d57 ( 1998 ) 7521 . y. okada , m. yamaguchi and t. yanagida , prog . 85 ( 1991 ) 1 , phys . b262 ( 1991 ) 54 ; j. ellis , g. ridolfi and f. zwirner , phys . b257 ( 1991 ) 83 ; h. e. haber and r. hempfling , phys . 66 ( 1991 ) 1815 . aleph collabolation , phys . b495 ( 2000 ) 1 ; l3 collabolation , phys . b495 ( 2000 ) 18 ; delphi collabolation , phys . b499 ( 2001 ) 23 ; opal collabolation , phys . b499 ( 2001 ) 38 ; p. igo - kemenes , talk at the lepc open session on nov . 3rd , http://lephiggs.web.cern.ch/lephiggs/talks/index.html . plane for the minimal case with @xmath48 . in the region above the dash - dotted line , the stau is the lsp . the left side of the dash - dot - dotted line is excluded by the upper bound on smuon pair production cross section at lep . the current higgs mass bound excludes the region below the solid line . in the region above the dashed line , @xmath12 is smaller than the lower limit obtained by the cleo . the shaded region is allowed . the @xmath100 curve is also shown as the dotted line . noewsb means that radiative breaking does not occur.,width=340,height=264 ] but @xmath56 . the region above the dashed line is excluded since @xmath12 is larger than the upper limit obtained by the cleo . the other lines are the same as in fig.[fig : limitmimposmu ] . notice that all region is excluded.,width=340,height=264 ] plane for universal gaugino masses , @xmath26 and @xmath48 . the left sides of the dash - dot - dotted and dash - dot - dot - dotted line are excluded by the upper bound on the smuon and stau pair production cross section , respectively . the other lines are the same as in fig.[fig : limitmimposmu ] , width=340,height=264 ] plane for the higgsino lsp case , @xmath95 , @xmath39 , @xmath101 and @xmath48 . in the region between the two dashed line , @xmath12 is smaller than the lower limit of the cleo result . , width=340,height=264 ]

no - scale structure of the khler potential is obtained in many types of supersymmetric models . in this paper , phenomenological aspects of these models are investigated with special attention to the current higgs mass bound at lep and @xmath0 result at the cleo . when the boundary condition is given at the gut scale and gaugino masses are universal at this scale , very narrow parameter region is allowed only for positive higgsino mass region if r - parity is conserved . the negative higgsino mass case is entirely excluded . on the other hand , relatively large parameter region is allowed when the boundary condition is given above the gut scale , and tevatron can discover susy signals for the positive higgsino mass case . the no - scale models with wino , higgsino or sneutrino lsp are also considered . we show that the higgs mass constraint is important for the higgsino lsp case , which requires the lsp mass to be larger than about 245 gev . 0.0 mm 0.0 mm 159.2 mm -16.0 mm 240.0 mm

arxiv

it has been known for a long time that owing to planar property and mutual focussing , colliding plane waves ( cpw ) result in spacelike singularities [ 1 ] . these singularities are somewhat weakened when the waves are endowed with a relative cross polarization prior to the collision . a solution given by chandrasekhar and xanthopoulos ( cx ) [ 2 ] , however constitutes an example contrasting this category , namely , it possesses a cauchy horizon ( ch ) instead of a spacelike singularity . naturally , this solution initiated a literature devoted entirely on the quest of stability of horizons formed hitherto . ch formed in spacetimes of cpw was shown by yurtsever to be unstable against plane - symmetric perturbations [ 3 ] . a linear perturbation analysis by cx reveals also an analogues result [ 4 ] . any such perturbation applied to a cpw spacetime will turn the ch into an essential singularity . a second factor that proved effective in weakening the strength of a singularity in cpw is the electromagnetic ( em ) field itself . in other words , the degree of divergence in the curvature scalars of colliding pure gravitational waves turn out to be stronger than the case when em field is coupled to gravity . in particular , collision of pure em waves must have a special significance as far as singularity formation is concerned . such an interesting solution was given by bell and szekeres ( bs ) which describes the collision of two linearly polarized step em waves [ 5 ] . the singularity ( in fact a ch ) formed in the interaction region of the bs solution was shown to be removable by a coordinate transformation . on the null boundaries , however it possesses esential curvature singularities which can not be removed by any means . since cross polarization and em field both play roles in the nature of resulting singularity it is worthwhile to purse these features together . this invokes a cross polarized version of the bs ( cpbs ) solution which was found long time ago [ 6,7 ] . this metric had the nice feature that the weyl scalars are all regular in the interaction region . cross polarization , however , does not remove the singularities formed on the null boundaries . in this paper we choose cpbs solution as a test ground , instead of bs , with various added test fields to justify the validity of a ch stability conjecture proposed previously by helliwell and konkowski ( hk ) [ 8,9 ] . unlike the tedious perturbation analysis of both cx and yurtsever the conjecture seems to be much economical in reaching a direct conclusion about the stability of a ch . this is our main motivation for considering the problem anew , for the case of untested solutions in cpw . in this paper we look at the spacetimes : a ) single plane wave with added colliding test fields and b ) colliding plane waves having non - singular interaction regions with test field added , fig.1 illustrates these cases . the terminology of singularities should be follwed from the classification presented by ellis and schmidt [ 10 ] . singularities in maximal four dimensional spacetimes can be divided into three basic types : quasiregular ( qr ) , scalar curvature ( sc ) and non - scalar curvature ( nsc ) . the ch stability conjecture due to hk is defined as follows . for all maximally extended spacetimes with ch , the backreaction due to a field ( whose test stress - energy tensor is @xmath0 ) will affect the horizon in one of the following manners . @xmath1 , @xmath2 and any null dust density @xmath3 are finite , and if the stress energy tensor @xmath4 in all parallel propagated orthonormal ( ppon ) frames is finite , then the ch remains non - singular . b)if @xmath1 , @xmath2 and any null dust density @xmath3 are finite , but @xmath4 diverges in some ppon frames , then an nsc singularity will be formed at the ch . c)if @xmath1 , @xmath2 and any null dust density @xmath3 diverges , then an sc singularity will be formed at the ch . expressed otherwise , the conjecture suggests to put a test field into the background geometry and study the reaction it will experience . if certain scalars diverge then in an exact back - reaction solution the field will respond with an infinite strength to the geometry ( i.e action versus reaction ) . such an infinite back - reaction will render a ch unstable and convert it into a scalar singularity . + the paper is organized as follows . in section ii , we review the cpbs solution and the correct nature of the singularity structure is presented in appendix a. section iii , deals with geodesics and test em and scalar field analyses . in section iv , we present an exact back reaction calculation for the collision of cross polarized em field coupled with scalar field . the derivation of weyl and maxwell scalars are given in appendix b. the insertion of test null dusts to the background cpbs spacetime and its exact back reaction solution is studied in section v. appendix c is devoted for the properties of this solution . the paper is concluded with a discussion in section vi . the metric that describes collision of em waves with the cross polarization was found to be [ 7 ] + @xmath5 in this representation of the metric our notations are + @xmath6 in which @xmath7 is a constant measuring the second polarization , @xmath8 are constant of energy and @xmath9 stand for the usual null coordinates . it can be seen easily that for @xmath10 the metric reduces to bs . unlike the bs metric , however , this is conformally non - flat for @xmath11 , where the conformal curvature is generated by the cross polarization . as a matter of fact this solution is a minimal extension of the bs metric . a completely different generalization of the bs solution with second polarization was given by cx [ 11 ] . their solution , however , employs an ehlers transformation and involves two essential parameters which is therefore different from ours . both solutions form ch in the interaction region . our result drown out in this paper , namely , that the horizon is unstable against added sources can also be shown to apply to the cx metric as well . as it was shown before the interaction region @xmath12 of the above metric is of type - d without scalar curvature singularities . we wish to check now the possible singularities of metric ( 1 ) . the single component of the weyl scalar suffices to serve our purpose . we find that the real part of the weyl scalar @xmath13 is given by @xmath14\end{aligned}\ ] ] where we have used the abbreviations @xmath15 as @xmath16 we obtain @xmath17 which reduces to the singularity form of the bs spacetime given by @xmath18 . we see that the same singularity remains unaffected by the introduction of the cross polarization . a similar calculation for @xmath19 gives the symmetrical singular hypersurface sitting on @xmath20 . now in order to explore the true nature of the singularity we concentrate our account on the incoming region ii @xmath21 . the metric in this region is expressed in the form @xmath22\ ] ] where @xmath23 we observe that for @xmath24 , @xmath25 is a bounded positive definite function which suggests that no additional singularities arise except the one occuring already in the bs case , namely at @xmath26 . to justify this we have calculated all riemann components in local and ppon frames ( see appendix a ) . it is found that all riemann tensor components vanish as @xmath27 . in the ppon frame , however , they are all finite and according to the classification scheme of ellis and schmidt such a singularity is called a quasiregular ( qr ) singularity . this is said to be the mildest type among all types of singularities . to check whether the qr is stable or not we consider generic test fields added to such a background geometry and study the effects . this we will do in the follwing sections . we are interested in the stability of qr singularities that are developed at @xmath28 in region ii and @xmath29 in region iii . to investigate their stability we will express geodesics and behaviour of test em and scalar fields by calculating stress - energy tensor in local and ppon frames . + our discussion on geodesics will be restricted in region ii only . we shall consider the geodesics that originate at the wave front and move toward the quasiregular singularity . solution of geodesics equations in region ii can be obtained by geodesics lagrangian method and using @xmath30 as a parameter . the results are @xmath31}{a}\tan(au ) + \frac{3p_{x_{0}}\left[5q^{2}+2 - 2\sqrt{1+q^{2}}\right]u}{4 } \nonumber \\ & & - \frac{p_{x_{0}}\left[\sqrt{1+q^{2}}-1\right]^{2}}{8a } \sin(2au ) - \frac{2p_{y_{0}}q}{a\cos(au ) } \nonumber \\ y - y_{0}&=&-\frac{2qp_{x_{0}}}{a\cos(au)}-\frac{2p_{y_{0}}\tan(au)}{a } \nonumber \\ v - v_{0}&=&\frac{\tan(au)}{a}\left[p^{2}_{x_0}(1 + 2q^2 ) + p^{2}_{y_{0}}\right ] + u\left[\frac{\epsilon}{4}(1 + 3\sqrt{1+q^{2}})\right . \nonumber \\ & & \left.-\frac{3p^2_{x_0}}{8}(5q^{2}+2 - 2\sqrt{1+q^{2}})\right ] + \frac{2p_{x_{0}}p_{y_{0}}q}{a\cos(au ) } \nonumber \\ & & + \frac { ( \sqrt{1+q^{2}}-1 ) \sin(2au)}{8a}\left[\frac{p^{2}_{x_{0}}}{2 } ( \sqrt{1+q^{2}}-1)-\epsilon\right]\end{aligned}\ ] ] where @xmath32 for null and @xmath33 for time like geodesics and @xmath34 and @xmath35 are constants of integration . it can be checked easily that for @xmath10 our geodesics agree with those of the region ii of the bs metric [ 8 ] . it is clear to see that if either @xmath36 or @xmath35 is nonzero then @xmath37 becomes positive for @xmath38 , and particles can pass from region ii to the region iv . geodesics that remain in region ii are @xmath39 where @xmath40 . the effect of cross polarization is that more geodesics remains in region ii relative to the parallel polarization case . on physical grounds this result could be anticipated because cross polarization behaves like rotation which creates a pushing out effect in the non - inertial frames . to test the stability of quasiregular singularity , let us consider a test em field whose vector potential is choosen appropriately as in [ 9 ] to be @xmath41 with arbitrary functions @xmath42 and @xmath43 . the only nonzero energy - momentum for this test em field is @xmath44\ ] ] in which a prime denotes derivative with respect to @xmath37 . both of scalars @xmath45 and @xmath2 vanish , predicting that qr singularities are not transformed into a scs . in the ppon frame . @xmath46 we find that @xmath47 are given in terms of @xmath48 by @xmath49 for@xmath50 and @xmath51 , otherwise . the divergence of this quantity predicts the occurence of nscs and therefore qr singularity must be unstable . + the stability conjecture therefore correctly finds that these qr singularities are unstable . however , the same stability conjecture does not find correctly the nature of the singularity . as we have discussed in section ii , the interior of the interaction region has no scs . the only scs is on the null boundaries . clarke and hayward have analysed these singular points for a collinear bs spacetime and found that the singularity nature of surfaces @xmath52 and @xmath53 are qr . this observation can also be used in the cross polarized version of bs spacetime , because the order of diverging terms in @xmath54 and @xmath13 are the same . the qr singularity structure formed in the incoming region of bs problem remains unchanged in the case of cross polarized version of the same problem . let us now consider the stability of these qr singularities by imposing a test scalar field in region ii which is the one of the incoming region bounded by the qr singularity . the massless scalar field equation is given by @xmath55 where we consider @xmath56 independent scalar waves so that a particular solution to this equation is obtained as in the ref ( ) @xmath57 where @xmath58 and @xmath59 are arbitrary functions . the stress energy tensor is given by @xmath60 the corresponding non - zero stress - energy tensors for the test scalar wave is obtained by taking @xmath61 as , @xmath62f(v)f'(v ) } { 8\pi f^{2 } } \nonumber \\ t_{xy}&=&t_{yx}=\frac{aq\sin(au ) \tan(au ) f(v)f'(v)}{8 \pi f^{2}}\end{aligned}\ ] ] it is observed that each component diverges as the qr singularity @xmath63 is approached . + next we consider the stress energy tensor in a ppon frame . such frame vectors are given in equation ( 11 ) . the stress - energy tensor is @xmath64 the nonzero components are ; @xmath65 \nonumber \\ t_{01}&=&t_{10}=\left(\frac{\sec^{2}(au)}{16 \pi}\right)\left[\frac{a^{2 } \tan^{2}(au)f^2}{f^{2}}-f'^{2}(v)\right ] \nonumber \\ t_{22}&=&t_{33}=\left(\frac{a\sec^{2}(au)\tan(au)f(v)f'(v)}{8\pi f^{3}}\right)\left[f^{2 } + 2q^{2}\sin^{2}(au)\right ] \nonumber \\ t_{32}&=&t_{23}=\frac{aq\sec^{3}(au)\sin^{2}(au)f'(v)f(v)\sqrt{f^{2 } + q^{2}\sin^{2}(au)}}{4\pi f^{3}}\end{aligned}\ ] ] these components also diverge as the singularity @xmath66 is approached . by the conjecture , this indicates that the qr singularity will be transformed into a curvature singularity . finally we calculate the scalar @xmath67 . @xmath68 \right\}\end{aligned}\ ] ] which also diverges as @xmath69 . from these analyses we conclude that the curvature singularity formed will be an scs . + hence , the conjecture predicts that the qr singularities of cross polarized version of bs spacetime are unstable . it is predicted that the qr singularities will be converted to scalar curvature singularities if generic waves are added . the similar results have also been obtained by hk for the bs spacetime . hk was unable to compare the validity of the conjecture by an exact back - reaction solution . in the next section we present an explicit example that represents cross - polarized em field coupled with scalar field . in the former sections , we applied hk stability conjecture to test the stability of qr singularities in the incoming region of cpbs spacetime . according to the conjecture these mild singularities are unstable . in order to see the validity of the conjecture we introduce this new solution . + let the metric be ; @xmath70\ ] ] the new solution is obtained from the electrovacuum solution . the ems solution is generated in the following manner . the lagrangian density of the system is defined by @xmath71\end{aligned}\ ] ] which correctly generate all ems field equations from a variational principle . here @xmath72 is the scalar field and we define the em potential one - form ( with coupling constant @xmath73 ) by @xmath74 where @xmath75 and @xmath76 are the components in the killing directions . variation with respect to @xmath77 and @xmath72 yields the following ems equations . @xmath78 \\ 2b_{uv}&=&-v_{v}b_{u}-v_{u}b_{v}-tanhw\left(w_{v}b_{u}+w_{u}b_{v}\right ) \nonumber \\ & & -e^{v}\left[2b_{uv}tanhw + w_{u}b_{v}+w_{v}b_{u}\right]\end{aligned}\ ] ] where @xmath79 and @xmath80 are the newmann - penrose spinors for em plane waves given as follows @xmath81 \nonumber \\ \phi_{0}&= & \frac{e^{u/2}}{\sqrt{2}}\left[e^{-v/2}\left(isinh\frac{w}{2 } + cosh\frac{w}{2}\right)a_{v}\right . \nonumber \\ & & \left.+e^{v/2}\left(icosh\frac{w}{2}+sinh\frac{w}{2}\right)b_{v}\right ] \end{aligned}\ ] ] the remaining two equations which do not follow from the variations , namely @xmath82 are automatically satisfied by virtue of integrability equations . + the metric function @xmath83 can be shifted now in accordance with + @xmath84 where @xmath85 and @xmath86 satisfy the electrovacuum em equations . integrability condition for the equation ( 31 ) imposes the constraint , @xmath87=0\ ] ] from which we can generate a large class of ems solution . as an example , for any @xmath88 satisfying the massless scalar field equation the corresponding @xmath89 is obtained from @xmath90 the only effect of coupling a scalar field to the cross polarized em wave is to alter the metric into the form , @xmath91 here @xmath92 and @xmath93 represents the solution of electrovacuum equations and @xmath89 is the function that derives from the presence of the scalar field . + it can be easily seen that for @xmath10 our solution represents pure em bs solution coupled with scalar field . it constitutes therefore an exact back reaction solution to the test scalar field solution in the bs spacetime considered by hk ( ) . it is clear to see that the weyl scalars are nonzero and scs is forming on the surface @xmath94 . this is in aggrement with the requirement of stability conjecture introduced by hk . for @xmath24 the obtained solution forms the exact back reaction solution of the test scalar field solution in the cpbs spacetime . in appendix b , we present the weyl and maxwell scalars explicitly . + from the explicit solutions we note that , the coulomb component @xmath95 remains regular but @xmath13 and @xmath54 are singular when @xmath96 or @xmath97 . this indicates that the singularity structure of the new solution is a typical scs . this result is in complete agreement with the stability conjecture . * a ) * let us assume first null test dusts moving in the cpbs background . for simplicity we consider two different cases the @xmath98 and @xmath99 projections of the spacetime . we have in the first case @xmath100 where we have used the coordinates @xmath101 according to @xmath102 the energy - momentum tensor for two oppositely moving null dusts can be chosen as @xmath103 where @xmath104 and @xmath105 are the finite energy densities of the dusts . the null propagation directions @xmath106 and @xmath107 are @xmath108 with @xmath109 @xmath110 we observe from ( 1 ) that @xmath111 which is finite for @xmath112 . the components of energy - momentum tensors in ppon frames are proportional to the expression ( 38 ) . this proportionality makes all the components of energy - momentum tensors finite . in the limit as @xmath113 which reduces our line element to the bs this expression diverges on the horizon @xmath114 . trace of the energy - momentum is obviously zero therefore we must extract information from the scalar @xmath115 . one obtains , @xmath116 the projection on @xmath99 , however is not as promising as the @xmath98 case . consider the reduced metric @xmath117 a similar analysis with the null vectors @xmath118 with @xmath119 @xmath120 yields the scalar @xmath121 from the metric ( 1 ) we see that @xmath122 which diverges on the horizon @xmath123 . the scalar @xmath124 constructed from the test dusts therefore diverges . the ppon components of the energy momentum tensors are also calculated and it is found that all of the components are proportional to the expression ( 42 ) . this indicates that the components of energy - momentum tensor diverges as the singularity is approached . when we consider the hk stability conjecture an exact back reaction solution must give a singular solution . we present now an exact back reaction solution of two colliding null shells in the interaction region of the cpbs spacetime . + * b ) * the metric given by @xmath125 where @xmath126 with @xmath127 positive constants was considered by wang [ 12 ] to represent collision of two null shells ( or impulsive dusts ) . the interaction region is transformable to the de sitter cosmological spacetime . in other words the tail of two crossing null shells is energetically equivalent to the de sitter space . it can be shown also that the choice of the conformal factor @xmath128 , with @xmath129 positive constants becomes isomorphic to the anti - de sitter space . + the combined metric of cpbs and colliding shells can be represented by @xmath130 this amounts to the substitutions @xmath131 where @xmath132 correspond to the metric functions of the cpbs solution . under this substitutions the scale invariant weyl scalars remain invariant ( or at most multiplied by a conformal factor ) because @xmath133 is the combination that arise in those scalars . the scalar curvature , however , which was zero in the case of cpbs now arises as nonzero and becomes divergent as we approach the horizon . appendix c gives the scalar quantities @xmath134 and @xmath135 . thus the exact back reaction solution is a singular one as predicted by the conjecture . it is further seen that choosing @xmath136 , which removes one of the shells leaves us with a single shell propagating in the @xmath37- direction . from the scalars we see that even a single shell gives rise to a divergent back reaction by the spacetime . the horizon , in effect , is unstable and transforms into a singularity in the presence of two colliding , or even a single propagating null shell . let us note as an alternative interpretation that the metric ( 43 ) may be considered as a colliding em waves in a de sitter background . collision of em waves creates an unstable horizon in the de sitter space which is otherwise regular for @xmath137 and @xmath138 . in this paper we have analysed the stability of qr singularities in the cpbs spacetime . three types of test fields are used to probe the stability . first we have applied test em field to the background cpbs spacetime . from the analyses we observed that the qr singularity in the incoming region becomes unstable according to the conjecture , and it is transformed into nsc singularity . this is the prediction of the conjecture . however , the exact back - reaction solution shoes that beside the true singularities on the null boundaries @xmath139 and @xmath140 . there are quasiregular singularities in the incoming regions . the interior of interaction region is singularity free and the hypersurface @xmath141 is a killing - cauchy horizon . as it was pointed out by hk in the case of colliding em step waves , conjecture fails to predict the correct nature of the singularity in the interaction region . we have also discovered the same behaviour for the cross polarized version of colliding em step waves . the addition of cross polarization does not alter the existing property . + secondly we have applied test scalar field to the background cpbs spacetime . the effect of scalar field on the qr singularity is stronger than the effect of em test field case . we have obtained that the qr singularity is unstable and transformes into a scs . to check the validity of the conjecture , we have constructed a new solution constituting an exact back reaction solution to the test scalar field in the cpbs spacetime . the solution represents the collision of cross polarized em field coupled to a scalar field . an examination of weyl and maxwell scalars shows that @xmath142 and @xmath143 diverge as the singularity is approached and unlike the test em field case the conjecture predicts the nature of singularity formed correctly . + finally , we have introduced test null dusts into the interaction region of cpbs spacetime . the conjecture predicts that the ch are unstable and transforms into scs . this result is compared with the exact back - reaction solution and observed that the conjecture works . to determine the type of singularity in the incoming region of cpbs spacetime , the riemann tensor in local and in ppon frame must be evaluated . non - zero riemann tensors in local coordinates are found as follows . @xmath144 \nonumber \\ -r_{uyuy}&= & e^{-u - v } \left [ \phi_{22 } coshw + ( i m \psi_4 ) sinhw - re \psi_4 \right ] \nonumber \\ r_{uxuy}&= & e^{-u}\left [ \phi_{22 } sinhw + ( im\psi_4)coshw\right]\end{aligned}\ ] ] where @xmath145 \nonumber \\ i m \psi_4&=&-\frac{i}{2}\left(w_{uu } -u_uw_u + m_uw_u -v^2_u coshw sinhw \right ) \nonumber \\ \phi_{22}&=&\frac{1}{4}\left(2u_{uu}-u^2_u - w^2_u - v^2_u cosh^2w \right)\end{aligned}\ ] ] note that in region ii the weyl scalar @xmath146 , therefore only @xmath143 exists . it is clear that @xmath147 in the limit @xmath27 , so that all of the components vanish @xmath148 to find the riemann tensor in a ppon frame , we define the following ppon frame vectors ; @xmath149 in this frame the non - zero components of the riemann tensors are : @xmath150 in the limit of @xmath27 , we have the following results @xmath151 which are all finite . this indicates that the apparent singularity in region ii ( one of the incoming region ) is a quasiregular singularity . in order to calculate the weyl and maxwell scalars we make use of the cx line element @xmath152 where @xmath153 @xmath154 is given in equation ( 4 ) and we have chosen @xmath155 , such that the new coordinates @xmath156 are related to @xmath157 by @xmath158 the weyl and maxwell scalars are found as @xmath159 \\ & & \nonumber \\ \psi_0&=&-e^{\gamma - i\lambda } \left [ 3r + \frac{1}{4f\sigma \sin \theta \sin \psi } \left(\sigma \sin(\psi -\theta)-\sigma_{\theta } \sin \psi \sin\theta \right . \nonumber \\ & & \nonumber \\ & & \left.\left.+ i\sin \alpha \sin^2 \theta \sin \psi \right ) \left(\gamma_{\psi}+\gamma_{\theta}\right)\right ] \\ & & \nonumber \\ 2\phi_{00}&= & e^{\gamma}\left[\frac{\cos \alpha}{\sigma^2}-\frac { \sin(\psi+\theta)(\gamma_{\theta}+\gamma_{\psi})}{2f\sin \psi \sin \theta } \right ] \\ & & \nonumber \\ 2\phi_{22}&= & e^{\gamma}\left[\frac{\cos \alpha}{\sigma^2}-\frac{\sin(\theta- \psi)(\gamma_{\theta}-\gamma_{\psi})}{2f\sin \psi \sin \theta } \right ] \\ & & \nonumber \\ -2\phi_{02}&=&e^{\gamma + i\lambda}\frac{\cos \alpha}{\sigma^2}\end{aligned}\ ] ] where @xmath160 \\ & & \\ e^{i\lambda}&=&\frac{\sin \theta + \sigma \sin \psi + i \sin \psi \sin \theta \cos \psi } { \sin \theta + \sigma \sin \psi - i \sin \psi \sin \theta \cos \psi}\end{aligned}\ ] ] the non - zero weyl and maxwell scalars for the collision of null shells in the background of cpbs spacetime are found as follows . @xmath161 \theta(u ) \theta(v ) \\ & & \nonumber \\ 4\phi e^{-m}\lambda&= & \left [ ( a\beta + \alpha b)\tan(au+bv)+(a \beta -\alpha b)\tan(au - bv)\right . \nonumber \\ & & \nonumber \\ & & \left.+\frac{4\alpha \beta}{\phi } \right ] \theta(u ) \theta(v ) \\ & & \nonumber \\ \phi_{22}&=&(\phi_{22})_{cpbs } + \left(\frac{\alpha e^m } { \phi}\right)\left [ \delta(u ) \right.\nonumber \\ & & \nonumber \\ & & \left . - \theta(u)\left(a \pi + \frac{u}{(1-u^2)(1-v^2)}\right ) \right]\\ & & \nonumber \\ \phi_{00}&=&(\phi_{00})_{cpbs } + \left(\frac{\beta e^m } { \phi}\right)\left [ \delta(v ) \right . \nonumber \\ & & \nonumber \\ & & \left.+ \theta(v)\left(b \pi - \frac{v}{(1-u^2)(1-v^2)}\right)\right ] \\ & & \nonumber \\ \phi_{02}&= & ( \phi_{02})_{cpbs}+\left ( \frac{e^m}{4fy\phi}\right ) \left[\frac{1}{f}\left(\alpha q \theta(u ) + \beta p \theta(v)\right ) \right . \nonumber \\ & & \nonumber \\ & & \left . + iq\left(\alpha l \theta(u ) + \beta k \theta(v)\right)\right]\end{aligned}\ ] ] where @xmath162 \\ & & \\ p&=&a\left[2q^2\sin(au+bv)\cos(au - bv)+f^2\left(\tan(au - bv)-\tan(au+bv)\right ) \right . \\ & & \\ & & \left . + 2f\cos(au - bv)\sin(au - bv)\left(\sqrt{1+q^2}-1\right)\right ] \\ & & \\ y&=&\left(1+\frac{q^2}{f^2}\tan(au+bv)\sin(au+bv)\cos(au - bv)\right)^{1/2}\\ & & \\ k&=&\frac{a}{\sqrt{\cos(au+bv)\cos(au - bv)}}\left[\frac{\cos(au - bv ) } { \cos(au+bv)}+\sin2au \right . \\ & & \\ & & \left . -\frac{2\left(\sqrt{1+q^2}-1\right)\sin(au+bv)\cos(au - bv ) \tan(au - bv)}{f}\right ] \\ & & \\ l&=&\frac{b}{\sqrt{\cos(au+bv)\cos(au - bv)}}\left[\frac{\cos(au - bv ) } { \cos(au+bv)}+\sin2bv \right . \\ & & \\ & & \left . + \frac{2\left(\sqrt{1+q^2}-1\right)\sin(au+bv)\cos(au - bv ) \tan(au - bv)}{f}\right ] \\ & & \\ \pi&=&\frac{\left(\sqrt{1+q^2}-1\right)\sin(2au-2bv)}{\sqrt{1+q^2}+1 + \left(\sqrt{1+q^2}-1\right)\sin^2(au - bv)}\end{aligned}\ ] ] 99 j.b . griffiths , colliding plane waves in general relativity , oxford , clarendon press ( 1991 ) . s. chandrasekhar and b.c . xanthopoulos , proc . london * a 408 * , 175 ( 1986 ) . u. yurtsever , phys . rev . * d 36 * , 1662 ( 1987 ) . s. chandrasekhar and b.c . xanthopoulos , proc . london * a 415 * , 329 ( 1988 ) . p. bell and p. szekeres , gen . * 5 * , 275 ( 1974 ) . m. halilsoy , phys . rev . * d,37 * , 2121 , ( 1988 ) . m. halilsoy , j. math . phys . * 31 * , 2694 , ( 1990 ) . d. a. konkowski . and t. m. helliwell , phys . rev . * d 43 * , 609 ( 1991 ) . d. a. konkowski . and t. m. helliwell , class . quantum grav . * 16 * , 2709 ( 1999 ) . g. f. r. ellis and b. g. schmidt , gen . * 8 * , 915 ( 1977 ) . s. chandrasekhar and b.c . xanthopoulos , proc . london * a 410 * , 311 ( 1987 ) . a. z. wang , j. math . phys . * 33 * , 1065 , ( 1992 ) . fig.1(a ) : single plane waves with added colliding test fields indicated by arrows . ch exists on the surface @xmath163 . + ( b ) : colliding plane wave spacetime with ch s in the incoming regions at @xmath164 and @xmath165 . test fields are added to test the stability of ch existing in region iv .

the quasiregular singularities ( horizons ) that form in the collision of cross polarized electromagnetic waves are , as in the linear polarized case , unstable . the validity of the helliwell - konkowski stability conjecture is tested for a number of exact backreaction cases . in the test electromagnetic case the conjecture fails to predict the correct nature of the singularity while in the scalar field and in the null dust cases the aggrement is justified .

arxiv

x - ray studies of fairly normal " galaxies , with high - energy emission not obviously dominated by a luminous active galactic nucleus ( agn ) , have recently been extended to cosmologically interesting distances in the deep field ( cdf ) surveys , which have now reached 1 ms of exposure ( cdf - n : hornschemeier et al . 2001 , hereafter paper ii ; brandt et al . 2001b , hereafter paper v ; cdf - s : tozzi et al . 2001 ; p. rosati et al . , in prep . ) . galaxies with @xmath8 are detected in appreciable numbers at 0.52 kev fluxes below @xmath9 erg @xmath6 s@xmath7 ( e.g. , paper ii ) ; the cdf - n survey goes almost two orders of magnitude fainter , detecting significant numbers of normal galaxies among the population of x - ray sources making the diffuse x - ray background ( xrb ; paper ii ; a.j . barger et al . , in prep . ) . these normal galaxies contribute as much as 510% of the xrb flux in the 0.52 kev band . the bulk of the energy density of the xrb is certainly explained by agn , but the investigation of the typical " galaxy , whether its x - ray emission is dominated by a population of x - ray binaries , hot interstellar gas , or even a low - luminosity agn , is an equally important function of deep x - ray surveys . normal galaxies are likely to be the most numerous extragalactic x - ray sources in the universe and are expected to dominate the number counts at 0.52 kev fluxes of @xmath10@xmath11 erg @xmath6 s@xmath7 ( ptak et al . 2001 ) . the cdf - n has reached the depths necessary to detect individually many normal [ @xmath12 ; @xmath13 is from 0.52 kev ] @xmath14 galaxies to @xmath15 , corresponding to a look - back time of @xmath16 gyr ( @xmath17 km s@xmath7 mpc@xmath7 , @xmath18 , and @xmath19 are adopted throughout this paper ) . reaching larger look - back times presents the exciting possibility of detecting the bulk x - ray response to the heightened star - formation rate at @xmath203 ( e.g. , madau et al . 1996 ) . one thus expects the x - ray luminosity per unit @xmath2-band luminosity to be larger at @xmath211 in the past due to the increased energy output of x - ray binary populations at @xmath203 ; this x - ray emission represents a fossil record " of past epochs of star formation ( e.g. , ghosh & white 2001 ; ptak et al . therefore , measurements of the x - ray luminosities of typical galaxies can constrain models of x - ray binary production in galaxies . while x - ray emission from individual galaxies is not easily detected at @xmath22 , it is possible to estimate the emission at their extremely faint flux levels using statistical methods such as stacking , a technique implemented successfully on the cdf - n survey data in several previous studies . these include the detection of x - ray emission from the average @xmath21 bright ( @xmath23 ) galaxy in the hubble deep field - north ( ) described in brandt et al . ( 2001a , hereafter paper iv ) and a study of x - ray emission from @xmath244 lyman break galaxies identified in the ( brandt et al . 2001c , hereafter paper vii ) . encouraged by the success of these analyses , we extend here the study of normal galaxies to the entire plus flanking fields region , now concentrating on galaxies at @xmath25 to complement the study of @xmath26 galaxies performed in paper vii . we focus on this redshift range due to the extensive spectroscopic redshift coverage ( cohen et al . 2000 and references therein ) and superb imaging which has allowed a comprehensive galaxy morphology study ( van den bergh , cohen , & crabbe 2001 ) . the cdf - n data provide extremely deep x - ray coverage over this area ( see figure 7 of paper v for the exposure map of this region ) ; the point - source detection limits in this region of the cdf - n survey in the 0.52 kev and 28 kev bands are @xmath27 erg @xmath6 s@xmath7 and @xmath28 erg @xmath6 s@xmath7 , respectively . in this study , we place observational constraints on the evolution of the ratio of x - ray luminosity to @xmath2-band luminosity of normal " spiral galaxies up to @xmath29 ; this ratio is an indicator of the current level of star formation in a galaxy ( e.g. , david , jones , & forman 1992 ; shapley et al . we also place constraints on the fraction of the diffuse xrb explained by galaxies lingering just below the cdf - n detection threshold , and thus the contribution to the xrb by normal galaxies . spectroscopic redshifts for the galaxies are drawn from the catalogs of cohen et al . ( 2000 ) , cohen ( 2001 ) , and dawson et al . ( 2001 ) in the range @xmath30 . spectroscopic redshift determination is difficult in the range @xmath31 due to the absence of strong features in the observed - frame optical band and the lack of the lyman break feature useful to identify higher redshift objects . we have therefore used the deep photometric redshift catalog of fernndez - soto , lanzetta , & yahil ( 1999 ) for the redshift interval @xmath32 , which allows some overlap in redshift space with the spectroscopic catalogs for cross - checking . the spectroscopic catalogs cover the entire hdf - n plus a substantial fraction of the flanking fields region , whereas the photometric catalog only covers the hdf - n . we shall refer to these two samples as the spectroscopic sample " and the photometric sample " throughout the rest of this letter . for the spectroscopic sample , the @xmath33-band magnitude was used to filter the sources by optical luminosity , as this is best matched to rest - frame @xmath2 over most of the redshift range under consideration here . the @xmath33 magnitudes are those given in barger et al . ( 1999 ) for the hawaii flanking fields area . for the photometric sample , the f814w ( hereafter @xmath34 ) magnitudes of fernndez - soto et al . ( 1999 ) were used . we chose galaxies which had no x - ray detection within 40 in the 0.58 kev ( full ) , 0.52 kev ( soft ) and 28 kev ( hard ) bands down to a wavdetect ( freeman et al . 2002 ) significance threshold of @xmath35 in the restricted acis grade set of paper iv . this low detection threshold ensures that our study does not include sources with x - ray emission just below the formal detection limits of paper v. we have attempted to construct a sample of galaxies similar to spiral galaxies in the local universe . to accomplish this , we have used the morphological classes of van den bergh et al . ( 2001 ) for galaxies from @xmath36 in the hdf - n and the flanking fields . to simplify the morphological filtering , we have cast objects in the van den bergh et al . ( 2001 ) catalog into the following four classes : ( 1 ) e / s0 " and e " , ( 2 ) merger " , ( 3 ) sa"sc " , including proto - spirals and spiral / irregulars , and ( 4 ) irr " , peculiar " and/or tadpole " . we then filtered the catalog to keep only classes ( 2 ) and ( 3 ) . filtering the photometric sample is more difficult due to the faintness of many of the sources and problems due to morphological evolution with redshift . we have used the spectral energy distribution ( sed ) classifications of fernndez - soto et al . ( 2001 ) to exclude all galaxies of type e " . comparison of the source lists reveals that , within the area covered by both , @xmath37 70% of galaxies identified through the two methods are in common . since the evolution of x - ray properties with redshift is of interest , we have made an effort to study objects with comparable optical luminosities at different redshifts . this is particularly important due to the non - linear relationship between x - ray luminosity and @xmath2-band luminosity for some types of spiral galaxies ( @xmath38 ; e.g. fabbiano & shapley 2001 ) . using the value of @xmath39 in the @xmath40-band as determined by blanton et al . ( 2001 ) for a large sample of galaxies in the sloan digital sky survey , we determined the value of @xmath39 in the @xmath2-band . the sloan filter @xmath40 is best matched to @xmath2 ; the resulting value of @xmath39 in the @xmath2 band is @xmath41 . to ensure that our results are not sensitively dependent upon the galaxy sed used to determine the optical properties , we have used both the sa and sc galaxy seds of poggianti et al . ( 1997 ) to calculate @xmath33 and @xmath34 vs. @xmath42 for an @xmath39 galaxy using the synthetic photometry package synphot in iraf . these calculations are shown in figure [ sample_definition ] . note the close similarity between the sa and sc tracks ; this is because the @xmath33 band corresponds to rest - frame @xmath2 in the middle of our redshift range . also plotted in figure 1 are the 151 galaxies in the spectroscopic sample with spiral or merger morphology having @xmath43 and the 651 galaxies in the photometric sample with sed class other than e " having @xmath44 . these galaxies were filtered by optical flux to lie within 1.5 mags of the @xmath39 galaxy tracks discussed above ; the galaxy samples constructed assuming sa and sc seds were identical ( or nearly so ) for all redshifts up to @xmath45 . galaxies meeting the optical magnitude filter were then divided by redshift into several bins ; these bins were constructed to ensure that there were @xmath46 galaxies per bin . the number of galaxies , median redshift , median look - back time and median optical magnitude for each bin are listed in table 1 . in figure 1 , we mark all the objects in the sc sed sample with colors indicating the different redshift bins . table 1 also includes the number of galaxies rejected from each redshift bin due to the presence of an x - ray detection within 40 ; this exclusion radius ensures that our results will not be adversely affected by the wings of the psf of very bright x - ray sources . these galaxies satisfied both the optical magnitude and morphology filtering constraints and were rejected only due to x - ray detection . this exclusion criterion is very conservative , however , considering that our astrometry is accurate to @xmath476 in the area under consideration ( see paper v ) . to allow for the off - nuclear nature of some of the x - ray sources found in normal galaxies ( e.g. , paper iv ) , we consider galaxies to be highly confident x - ray detections if the x - ray source is within 15 of the galaxy s center this matching radius is also well matched to the chandra psf . off - axis for 0.52 kev and the 83% encircled - energy radius for 0.58 kev . ] we therefore also give the number of galaxies having an x - ray detection within 15 in table 1 . the x - ray imaging data at each position were stacked in the same manner as in paper vii , keeping the 30 pixels whose centers fall within an aperture of radius 15 . the detection significance in each band was assessed by performing 100,000 monte - carlo stacking simulations using local background regions as in paper vii . a source is considered to be significantly detected if the number of counts over background exceeds that of 99.99% of the simulations . no single source in the stacking sample appeared to dominate the distribution , demonstrating the effectiveness of our selection criteria . stacking of the galaxies in the redshift bins described in table [ sample_table ] and figure [ sample_definition ] resulted in significant detections in the soft band for all of the redshift bins up to @xmath48 ( see table 2 ) . we also stacked galaxies in the redshift bin @xmath49 , but there was not a significant detection . the results for the two different spiral galaxy sed samples are nearly or exactly identical except for the detection in the highest redshift bin ( @xmath50 ) . we adopt a @xmath51 power law for the calculation of x - ray fluxes and luminosities , assuming that these galaxies are similar to spiral galaxies in the local universe and have their x - ray emission dominated by x - ray binaries ( e.g. , kim , fabbiano , & trinchieri 1992 ) . while there were several cases of significant detections in the full band , there were no highly significant detections in the hard band . given the variation of effective area and background rate with energy , the signal - to - noise ratio for sources with the assumed spectrum is highest in the soft band and lowest in the hard band , so this behavior is expected . the flux level of the soft - band detections for the spectroscopic sample is ( 56)@xmath52 erg @xmath6 s@xmath7 . the corresponding rest - frame 0.52 kev luminosities for the average galaxy are @xmath53 erg s@xmath7 for the lowest redshift bin and @xmath54 erg s@xmath7 for the highest redshift bin . for the photometric sample , the soft - band flux level of the detections is ( 35)@xmath52 erg @xmath6 s@xmath7 . we also give fluxes for the less significant detections in the full band for those redshift bins having highly significant soft - band detections . we have investigated the properties of the sources which were rejected from the stacking samples due to an x - ray detection at or near the position of the galaxy ( the numbers of such sources are given in the last column of table 1 ) . there are only 15 distinct galaxies with x - ray detections within 15 . of these 15 sources , three are broad - line agn , which are clearly not normal galaxies . one object has a photometric redshift which differs significantly from its spectroscopic redshift . since the optical properties of this object at its spectroscopic redshift place it outside our sample boundaries , we have rejected it . one object is very near another x - ray source which has been positively identified with a narrow - line agn . thus , there are a total of 10 normal " galaxies positively identified with x - ray sources within this sample , consituting a small minority of the galaxies under study here . the worst case is in the lowest redshift bin where 15% of the galaxies had x - ray detections . figure [ sbhistogram ] shows a histogram of @xmath55 values calculated for both the stacking samples and the individually x - ray detected galaxies . the individually x - ray detected galaxy set does possibly contain some lower - luminosity agn , including the agn candidate ( see papers ii and iv ) . with the exception of the objects in the redshift bin with median @xmath56 , figure [ sbhistogram ] shows that typically the x - ray luminosities of the individually detected objects are on average much higher than those of the stacked galaxies ; they are sufficiently more luminous as to appear atypical of the normal galaxy population . for the lowest redshift bin , it is plausible that our results are moderately biased by the exclusion of the x - ray detected sources . in 4 , we therefore also give results which include the individually x - ray detected objects in the sample average for this lowest redshift bin . for additional comparison , we have considered the radio properties of the individually detected galaxies and the stacked galaxies using the catalogs of richards et al . ( 1998 ) and richards ( 2000 ) . the percentage of radio detections among the individually x - ray detected galaxies is higher than that among the stacked galaxies ( @xmath57% vs. @xmath585% ) . this possible difference between the two populations is significant at the 93% level as determined using the fisher exact probability test for two independent samples ( see siegel & castellan 1988 ) . due to the x - ray luminosity difference and possible difference in radio properties , and to the fact that these galaxies constitute a small minority of those under study , we are confident we have not biased our determination of the properties of the typical galaxy by omitting these x - ray detected objects from further consideration . in figure [ lxlb]a , we plot the x - ray - to - optical luminosity ratio @xmath59 for each stacked detection , where @xmath13 is calculated for the rest - frame 0.52 kev band . we have also plotted our approximate sensitivity limit in figure [ lxlb]a , which is simply the corresponding @xmath60 x - ray luminosity detection limit achieved for a 30 ms stacking analysis divided by @xmath61 . we do not expect to detect galaxies having less x - ray emission per unit @xmath2-band luminosity than this value . for comparison with the local universe , we also plot the mean @xmath59 for spiral galaxies of comparable @xmath62 from the sample of shapley et al . this sample includes 234 spiral galaxies observed with _ einstein _ and excludes agn where the x - ray emission is clearly dominated by the nucleus . the galaxies in the shapley et al . ( 2001 ) sample all have @xmath63 ; median redshift is @xmath64 . in figure [ lxlb]b , we plot @xmath59 versus @xmath65 ; the values up to @xmath66 are consistent with what is observed in the shapley et al . sample for objects with comparable @xmath65 , although they are toward the high end of what is observed . this is consistent with figure [ lxlb]a , which shows the average @xmath59 derived from stacking being somewhat higher than that for the shapley et al . galaxies of comparable optical luminosity . there is a slight increase ( factor of 1.6 ) in the average @xmath67 from the local universe to @xmath68 . for the highest redshift bin ( @xmath50 ) , the results become somewhat sensitive to the galaxy sed one assumes for determining the optical properties . we suspect that an sc galaxy sed is more appropriate at this epoch due to the higher prevalence of star formation . adopting this sed , we find that @xmath69 has increased somewhat more substantially ( @xmath70 times ) at @xmath50 . one may also constrain star - formation models using only the x - ray properties of these galaxies . the average x - ray luminosity of the spiral galaxies in the shapley et al . ( 2001 ) sample having the same range of @xmath65 as used in this study is @xmath71 erg s@xmath7 ( converted to 0.52 kev ) . the average galaxy in our stacking sample has a luminosity @xmath72 times higher at @xmath68 . this increases to @xmath73 times higher at @xmath74 . the @xmath68 value is most likely affected by some bias due to the exclusion of legitimate normal galaxies in the lowest redshift bin ( see 3 ) . if we include these individually x - ray detected objects , then the average galaxy in our stacking sample has a luminosity @xmath75 times higher at @xmath68 , consistent with the predictions of ghosh & white ( 2001 ) that the x - ray luminosity of the typical sa - sbc spiral galaxy should be @xmath75 times higher at @xmath76 . including the x - ray detected galaxies does not significantly affect our results for the interval @xmath77 ( the difference is @xmath78% ) . however , since the x - ray luminosities of the x - ray detected objects with @xmath74 are substantially higher ( by an order of magnitude ) than the x - ray stacking averages ( see figure 2 ) , it is not appropriate to include these objects in the calculation of the average x - ray luminosity . we thus find a smaller increase in the average x - ray luminosity of galaxies at @xmath79 than the increase by a factor of @xmath80 predicted by ghosh & white ( 2001 ) . we find that the average x - ray luminosities of galaxies have not evolved upward by more than a factor of @xmath73 by @xmath74 , regardless of exclusion or inclusion of x - ray detected objects . the range of average 0.52 kev fluxes for the spiral galaxies studied here is ( @xmath46@xmath5 erg @xmath6 s@xmath7 . these x - ray fluxes are consistent with independent predictions made by ptak et al . ( 2001 ) that galaxies of this type will be detected at 0.52 kev flux levels of @xmath81 erg @xmath6 s@xmath7 ( converted from their 210 kev prediction assuming a @xmath82 power law ) . assuming a 0.52 kev xrb flux density of @xmath83 erg @xmath6 s@xmath7 deg@xmath84 ( garmire et al . 2001 ) , we have identified @xmath85% of the soft xrb as arising from spiral galaxies not yet individually detected in deep surveys . many of these objects should be sufficently bright to be detected with acis exposures of @xmath86 ms , which should be achievable in the next several years of the mission . we thank alice shapley for useful discussions and sharing data . we gratefully acknowledge the financial support of nasa grant nas 8 - 38252 ( gpg , pi ) , nasa gsrp grant ngt5 - 50247 ( aeh ) , nsf career award ast-9983783 ( wnb , dma , feb ) , and nsf grant ast99 - 00703 ( dps ) . this work would not have been possible without the enormous efforts of the entire team . ccccccc + 0.400.75 & sc & 0.635 & 6.34 & 21.73 & 29 & 12/6 + 0.750.90 & sc & 0.821 & 7.39 & 21.83 & 38 & 2/2 + 0.901.10 & sc & 0.960 & 8.05 & 22.29 & 30 & 6/4 + + + 0.501.00 & sc & 0.920 & 7.87 & 22.99 & 37 & 9/4 + 1.001.50 & sc & 1.200 & 8.97 & 24.28 & 64 & 6/2 + 1.001.50 & sa & 1.240 & 9.09 & 24.69 & 80 & 8/2 + rrrrrrrrrcrrc + 0.400.75 & sc & 49.6 & 31.7 & 99.99 & @xmath87 & 25.41 & 25.44 & 1.52 & 0.65 & 2.94 & 1.26 & 0.67 + 0.750.90 & sc & 58.0 & 36.4 & @xmath87 & @xmath87 & 33.05 & 33.09 & 1.36 & 0.57 & 4.92 & 2.07 & 1.33 + 0.901.10 & sc & 54.4 & 25.9 & @xmath87 & @xmath87 & 26.54 & 26.56 & 1.60 & 0.51 & 8.44 & 2.68 & 1.40 + + + 0.501.00 & sc & 42.0 & 33.5 & 99.77 & @xmath87 & 34.51 & 34.54 & 0.95 & 0.51 & 4.51 & 2.40 & 1.12 + 1.001.50 & sc & 44.8 & 38.5 & 98.94 & @xmath87 & 59.67 & 59.73 & 0.58 & 0.34 & 5.34 & 3.06 & 0.79 + 1.001.50 & sa & 45.6 & 34.2 & 98.29 & 99.95 & 74.61 & 74.68 & 0.48 & 0.24 & 4.70 & 2.36 & 0.81 + . the blue curves give @xmath33 vs. @xmath42 for @xmath39 sa and sc galaxies ( the lower curve at higher redshift is for the sa galaxy ) . galaxies without a colored circle were not included in the stacking sample because they either were not within the range of optical luminosity specified or because an x - ray detection was found within 40 . ( b ) the black filled circles are the photometric redshift sample of fernndez - soto et al . ( 1999 ) , excluding the e "- type galaxies . the blue curves give @xmath34 vs. @xmath42 for @xmath39 sa and sc galaxies ( the lower curve at higher redshift is for the sa galaxy ) . [ sample_definition ] ] as a function of redshift for the stacking samples . the redshift error bars indicate the full extent of the redshift bin ; the data points are at the median redshift value for that bin . the solid line indicates the 2@xmath88 x - ray sensitivity limit normalized by @xmath89 . the dashed lines above and below the solid line indicate the effect of decreasing and increasing the optical luminosity by one magnitude , respectively . objects which have less x - ray luminosity per unit @xmath2-band luminosity than this are not expected to be detected in the current data . the error bar on the shapley et al . ( 2001 ) data point indicates the dispersion of values in this sample . ( b ) @xmath59 vs. @xmath65 for both the shapley et al . ( 2001 ) sample ( open circles ) and the stacked detections presented here . the error bars on @xmath59 in both figures were calculated following the numerical method described in 1.7.3 . of lyons ( 1991 ) . the solid line in ( b ) indicates @xmath89 ; again the dashed lines correspond to one magnitude fainter and brighter than @xmath89 .

we present a statistical x - ray study of spiral galaxies in the hubble deep field - north and its flanking fields using the chandra deep field north 1 ms dataset . we find that @xmath0 galaxies with @xmath1 have ratios of x - ray to @xmath2-band luminosity similar to those in the local universe , although the data indicate a likely increase in this ratio by a factor of @xmath33 . we have also determined that typical spiral galaxies at @xmath1 should be detected in the 0.52 kev band in the flux range ( @xmath46@xmath5 erg @xmath6 s@xmath7 . 1_heao-1 _

arxiv

quantum systems can be correlated in ways that supersede classical descriptions . however , there are degrees of non - classicality for quantum correlations . for simplicity , we consider only bipartite correlations , with the two , spatially separated , parties being named alice and bob as usual . at the weaker end of the spectrum are quantum systems whose states can not be expressed as a mixture of product - states of the constituents . these are called non - separable or entangled states . the product - states appearing in such a mixture comprise a local hidden state ( lhs ) model for any measurements undertaken by alice and bob . at the strongest end of the spectrum are quantum systems whose measurement correlations can violate a bell inequality @xcite , hence demonstrating ( modulo loopholes @xcite ) the violation of local causality @xcite . this phenomenon commonly known as bell - nonlocality @xcite is the only way for two spatially separated parties to verify the existence of entanglement if either of them , or their detectors , can not be trusted @xcite . we say that a bipartite state is bell - local if and only if there is a local hidden variable ( lhv ) model for any measurements alice and bob perform . here the ` variables ' are not restricted to be quantum states , hence the distinction between non - separability and bell - nonlocality . in between these types of non - classical correlations lies epr - steering . the name is inspired by the seminal paper of einstein , podolsky , and rosen ( epr ) @xcite , and the follow - up by schrdinger @xcite , which coined the term `` steering '' for the phenomenon epr had noticed . although introduced eighty years ago , as this special issue celebrates , the notion of epr - steering was only formalized eight years ago , by one of us and co - workers @xcite . this formalization was that epr - steering is the only way to verify the existence of entanglement if one of the parties conventionally alice @xcite or her detectors , can not be trusted . we say that a bipartite state is epr - steerable if and only if it allows a demonstration of epr - steering . a state is not epr - steerable if and only if there exists a hybrid lhv lhs model explaining the alice bob correlations . since in this paper we are concerned with steering , when we refer to a lhs model we mean a lhs model for bob only ; it is implicit that alice can have a completely general lhv model . the above three notions of non - locality for quantum states coincide for pure states : any non - product pure state is non - separable , eps - steerable , and bell - nonlocal . however for mixed states , the interplay of quantum and classical correlations produces a far richer structure . for mixed states the logical hierarchy of the three concepts leads to a hierarchy for the bipartite states : the set of separable states is a strict subset of the set of non - epr - steerable states , which is a strict subset of the set of bell - local states @xcite . although the epr - steerable set has been completely determined for certain classes of highly symmetric states ( at least for the case where alice and bob perform projective measurements ) @xcite , until now very little was known about what types of states are steerable even for the simplest case of two qubits . in this simplest case , the phenomenon of steering in a more general sense i.e. within what set can alice steer bob s state by measurements on her system has been studied extensively using the so - called steering ellipsoid formalism @xcite . however , no relation between the steering ellipsoid and epr - steerability has been determined . in this manuscript , we investigate epr - steerability of the class of two - qubit states whose reduced states are maximally mixed , the so - called t - states @xcite . we use the steering ellipsoid formalism to develop a deterministic lhs model for projective measurements on these states and we conjecture that this model is optimal . furthermore we obtain two sufficient conditions for t - states to be epr - steerable , via suitable epr - steering inequalities @xcite ( including a new asymmetric steering inequality for the spin covariance matrix ) . these sufficient conditions touch the necessary condition in some regions of the space of t - states , and everywhere else the gap between them is quite small . the paper is organised as follows . in section 2 we discuss in detail the three notions of non - locality , namely bell - nonlocality , epr - steerability and non - separability . section 3 introduces the quantum steering ellipsoid formalism for a two - qubit state , and in section 4 we use the steering ellipsoid to develop a deterministic lhs model for projective measurements on t - states . in section 5 , two asymmetric steering inequalities for arbitrary two - qubit states are derived . finally in section 6 we conclude and discuss further work . two separated observers , alice and bob , can use a shared quantum state to generate statistical correlations between local measurement outcomes . each observer carries out a local measurement , labelled by @xmath1 and @xmath2 respectively , to obtain corresponding outcomes labelled by @xmath3 and @xmath4 . the measurement correlations are described by some set of joint probability distributions , @xmath5 , with @xmath1 and @xmath2 ranging over the available measurements . the type of state shared by alice and bob may be classified via the properties of these joint distributions , for all possible measurement settings @xmath1 and @xmath2 . the correlations of a _ bell - local _ state have a local hidden variable ( lhv ) model @xcite , @xmath6 for some ` hidden ' random variable @xmath7 with probability distribution @xmath8 . hence , the measured correlations may be understood as arising from ignorance of the value of @xmath7 , where the latter locally determines the statistics of the outcomes @xmath3 and @xmath4 and is independent of the choice of @xmath1 and @xmath2 . conversely , a state is defined to be bell_-nonlocal _ if it has no lhv model . such states allow , for example , the secure generation of a cryptographic key between alice and bob without trust in their devices @xcite . in this paper , we are concerned with whether the state is _ steerable _ ; that is , whether it allows for correlations that demonstrate epr - steering . as discussed in the introduction , epr - steering by alice is demonstrated if it is _ not _ the case that the correlations can be described by a hybrid lhv lhs model , wherein , @xmath9 where the local distributions @xmath10 correspond to measurements on local quantum states @xmath11 , i.e. , @xmath12.\ ] ] here @xmath13 denotes the positive operator valued measure ( povm ) corresponding to measurement @xmath2 . the state is said to be _ steerable _ by alice if there is _ no _ such model . the roles of alice and bob may also be reversed in the above , to define steerability by bob . comparing eqs . ( [ lhv ] ) and ( [ lhs ] ) , it is seen that all nonsteerable states are bell - local . hence , all bell - nonlocal states are steerable , by both alice and bob . in fact , the class of steerable states is strictly larger @xcite . moreover , while not as powerful as bell - nonlocality in general , steerability is more robust to detection inefficiencies @xcite , and also enables the use of untrusted devices in quantum key distribution , albeit only on one side @xcite . by a similar argument , a separable quantum state shared by alice and bob , @xmath14 , is both bell - local and nonsteerable . moreover , the set of separable states is strictly smaller than the set of nonsteerable states @xcite . it is important that epr - steerability of a quantum state not be confused with merely the dependence of the reduced state of one observer on the choice of measurement made by another , which can occur even for separable states . the term ` steering ' has been used with reference to this phenomenon , in particular for the concept of ` steering ellipsoid ' , which we will use in our analysis . epr - steering , as defined above , is a special case of this phenomenon , and is only possible for a subset of nonseparable states . we are interested in the epr - steerability of states for all possible _ projective _ measurements . if alice is doing the steering , then it is sufficient for bob s measurements to comprise some tomographically complete set of projectors . it is straightforward to show in this case that the condition for bob to have an lhs model , eq . ( [ lhs ] ) , reduces to the existence of a representation of the form @xmath15 = \sum_\lambda p(\lambda)\ , p(1|e,\lambda ) \,\rho_b(\lambda ) , \\ \label{reduced_p } p_e & = \operatorname{tr } [ \rho e\otimes i ] = \sum_\lambda p(\lambda ) p(1|e,\lambda).\end{aligned}\ ] ] here @xmath16 is any projector that can be measured by alice ; @xmath17 is the probability of result ` @xmath18 ' and @xmath19 is the corresponding probability given @xmath7 ; @xmath20 is the reduced state of bob s component corresponding to this result ; and @xmath21 $ ] denotes the partial trace over alice s component . note that this form , and hence epr - steerability by alice , is invariant under local unitary transformations on bob s components . determining epr - steerability in this case , where alice is permitted to measure any hermitian observable , is surprisingly difficult , with the answer only known for certain special cases such as werner states @xcite . however , in this paper we give a strong necessary condition for the epr - steerability of a large class of two - qubit states , which we conjecture is also sufficient . this condition is obtained via the construction of a suitable lhs model , which is in turn motivated by properties of the ` quantum steering ellipsoid ' @xcite . properties of this ellipsoid are therefore reviewed in the following section . an arbitrary two - qubit state may be written in the standard form @xmath22 here @xmath23 denote the pauli spin operators , and @xmath24,~ b_j=\operatorname{tr}[\rho\ , \i\otimes\sigma_j],~t_{jk}=\operatorname{tr}[\rho \,\sigma_j\otimes\sigma_k ] .\ ] ] thus , @xmath25 and @xmath26 are the bloch vectors for alice and bob s qubits , and @xmath27 is the spin correlation matrix . if alice makes a projective measurement on her qubit , and obtains an outcome corresponding to projector @xmath16 , bob s reduced state follows from eq . ( [ reduced ] ) as @xmath28 } { \operatorname{tr}[\rho \,e \otimes \i ] } .\ ] ] we will also refer to @xmath20 as bob s ` steered state ' . to determine bob s possible steered states , note that the projector @xmath16 may be expanded in the pauli basis as @xmath29 , with @xmath30 . this yields the corresponding steered state @xmath31 , with associated bloch vector @xmath32 where @xmath33 is the associated probability of result @xmath34 , @xmath35 = \frac 12 ( 1+{\boldsymbol}a\cdot{\boldsymbol}e),\ ] ] called @xmath17 previously . in what follows we will refer to the vector @xmath36 rather than its corresponding operator @xmath16 . the surface of the steering ellipsoid is defined to be the set of steered bloch vectors , @xmath37 , and in ref . @xcite it is shown that interior points can be obtained from positive operator - valued measurements ( povms ) . the ellipsoid has centre @xmath38 and the semiaxes @xmath39 are the roots of the eigenvalues of the matrix @xmath40 the eigenvectors of @xmath41 give the orientation of the ellipsoid around its centre @xcite . thus , the general equation of the steering ellipsoid surface is @xmath42 with @xmath43 being the displacement vector from the centre @xmath44 . entangled states typically have large steering ellipsoids the largest possible being the bloch ball , which is generated by every pure entangled state @xcite . in contrast , the volume of the steering ellipsoid is strictly bounded for separable states . indeed , a two - qubit state is separable if and only if its steering ellipsoid is contained within a tetrahedron contained within the bloch sphere @xcite . thus , the separability of two - qubit states has a beautiful geometric characterisation in terms of the quantum steering ellipsoid . no similar characterisation has been found for epr - steerability , to date . however , for non - separable states , knowledge of the steering ellipsoid matrix @xmath41 , its centre @xmath45 , and bob s bloch vector @xmath46 uniquely determines the shared state @xmath47 up to a local unitary transformation on alice s system @xcite , @xcite and so is sufficient , in principle , to determine the epr - steerability of @xmath47 . in this paper we find a direct connection between epr - steerability and the quantum steering ellipsoid , for the case that the bloch vectors @xmath48 and @xmath46 vanish . let @xmath49 be a singular value decomposition of the spin correlation matrix @xmath27 , for some diagonal matrix @xmath50 and orthogonal matrices @xmath51 . noting that any @xmath52 is either a rotation or the product of a rotation with the parity matrix @xmath53 , it follows that @xmath27 can always be represented in the form @xmath54 , for proper rotations @xmath55 , where the diagonal matrix @xmath56 may now have negative entries . the rotations @xmath57 and @xmath58 may be implemented by local unitary operations on the shared state @xmath47 , amounting to a local basis change . hence , all properties of a shared two - qubit state , including steerability properties in particular , can be formulated in a representation in which the spin correlation matrix has the diagonal form @xmath59 $ ] . it follows that if the shared state @xmath47 has maximally - mixed reduced states with @xmath60 , then it is completely described , up to local unitaries , by a diagonal @xmath27 , i.e. one may consider @xmath61 without loss of generality . such states are called t - states @xcite . they are equivalent to mixtures of bell states , and hence form a tetrahedron in the space parameterised by @xmath62 @xcite . entangled t - states necessarily have @xmath63 , and the set of separable t - states forms an octahedron within the tetrahedron @xcite . the t - state steering ellipsoid is centred at the origin , @xmath64 , and the ellipsoid matrix is simply @xmath65 , as follows from eqs . ( [ qse_centre ] ) and ( [ qse_mat ] ) with @xmath66 . the semiaxes are @xmath67 for @xmath68 , and are aligned with the @xmath69-axes of the bloch sphere . thus , the equation of the ellipsoid surface in spherical coordinates @xmath70 is @xmath71 , with @xmath72 we find a remarkable connection between this equation and the epr - steerability of t - states in the following subsection . without loss of generality , consider measurement by alice of hermitian observables on her qubit . such observables can be equivalently represented via projections , @xmath73 , with @xmath74 . the probability of result ` @xmath18 ' and the corresponding steered bloch vector are given by eqs . ( [ qsteerb ] ) and ( [ pe ] ) with @xmath60 , i.e. , @xmath75 hence , letting @xmath76 denote the bloch vector corresponding to @xmath77 in eq . ( [ reduced ] ) , then from eqs . ( [ reduced ] ) and , it follows there is an lhs model for bob if and only if there is a representation of the form @xmath78 for all unit vectors @xmath36 . noting further that @xmath76 can always be represented as some mixture of unit vectors , corresponding to pure @xmath11 , these conditions are equivalent to the existence of a representation of the form @xmath79 with integration over the bloch sphere . thus , the unit bloch vector @xmath80 labels both the local hidden state and the hidden variable . given lhs models for bob for any two t - states , having spin correlation matrices @xmath81 and @xmath82 , it is trivial to construct an lhs model for the t - state corresponding to @xmath83 , for any @xmath84 , via the convexity property of nonsteerable states @xcite . our strategy is to find _ deterministic _ lhs models for some set of t - states , for which the result ` @xmath18 ' is fully determined by knowledge of @xmath80 , i.e. , @xmath85 . lhs models can then be constructed for all convex combinations of t - states in this set . to find deterministic lhs models , we are guided by the fact that the steered bloch vectors @xmath86 are precisely those vectors that generate the surface of the quantum steering ellipsoid for the t - state @xcite . we make the ansatz that @xmath87 is proportional to some power of the function @xmath88 in eq . ( [ ell_eqn ] ) that defines this surface , i.e. , @xmath89^m \equiv n_t\,\left[{\boldsymbol}n^\intercal t^{-2}{\boldsymbol}n\right]^{m/2}\ ] ] for @xmath90 , where @xmath91 is a normalisation constant . further , denoting the region of the bloch sphere , for which @xmath92 by @xmath93 $ ] , the condition in eq . ( [ peint ] ) becomes @xmath94 } p({\boldsymbol}n ) { \mathop{}\!\mathrm{d}}^2{\boldsymbol}n = \frac12 $ ] . we note this is automatically satisfied if @xmath95 is a hemisphere , as a consequence of the symmetry @xmath96 for the above form of @xmath87 . hence , under the assumptions that ( i ) @xmath87 is determined by the steering ellipsoid as per eq . ( [ pform ] ) , and ( ii ) @xmath93 $ ] is a hemisphere for each unit vector @xmath36 , the only remaining constraint to be satisfied by a deterministic lhs model for a t - state is eq . ( [ beint ] ) , i.e. , @xmath97 } \left [ { \boldsymbol}n^\intercal t^{-2 } { \boldsymbol}n\right]^{m/2}\,{\boldsymbol}n { \mathop{}\!\mathrm{d}}^2{\boldsymbol}n = \frac12 t{\boldsymbol}e,\ ] ] for some suitable mapping @xmath98 $ ] . extensive numerical testing , with different values of the exponent @xmath99 , show that this constraint can be satisfied by the choices @xmath100= \{{\boldsymbol}n : { \boldsymbol}nt^{-1 } { \boldsymbol}e\geq 0\},\ ] ] for a two - parameter family of t - states . assuming the numerical results are correct , it is not difficult to show , using infinitesimal rotations of @xmath36 about the @xmath101-axis , that this family corresponds to those t - states that satisfy @xmath102 fortunately , we have been able to confirm these results analytically by explicitly evaluating the integral in eq . ( [ con ] ) for @xmath103 ( see appendix a ) . an explicit form for the corresponding normalisation constant @xmath91 is also given in appendix a , and it is further shown that the family of t - states satisfying eq . ( [ surface ] ) is equivalently defined by the condition @xmath104 this may be interpreted geometrically in terms of the harmonic mean radius of the ` inverse ' ellipsoid @xmath105 being equal to @xmath106 . equation ( [ surface ] ) defines a surface in the space of possible @xmath27 matrices , plotted in fig . 1(a ) as a function of the semiaxes @xmath107 and @xmath108 . as a consequence of the convexity of nonsteerable states ( see above ) , all t - states corresponding to the region defined by this surface and the positive octant have local hidden state models for bob . also shown is the boundary of the separable t - states ( @xmath109 @xcite ) , in red , which is clearly a strict subset of the nonsteerable t - states . the green plane corresponds to the sufficient condition @xmath110 for epr - steerable states , derived in sec . 5 below . it follows that a necessary condition for a t - state to be epr - steerable by alice is that it corresponds to a point above the sandwiched surface shown in fig . note that this condition is in fact symmetric between alice and bob , since their steering ellipsoids are the same for t - states . because of the elegant relation between our lhs model and the steering ellipsoid , and other evidence given below , we conjecture that this condition is also _ sufficient _ for epr - steerability . . * top figure ( a ) * : the red plane separates separable ( left ) and entangled ( right ) t - states . the sandwiched blue surface corresponds to the necessary condition for epr - steerability generated by our deterministic lhs model in sec . 4b : all t - states to the left of this surface are not epr - steerable . we conjecture that this condition is also sufficient , i.e. , that all states to the right of the blue surface are epr - steerable . for comparison , the green plane corresponds to the sufficient condition for epr - steerability in eq . ( [ linsuff ] ) of section 5a : all t - states to the right of this surface are epr - steerable . only a portion of the surfaces are shown , as they are symmetric under permutations of @xmath111 . * bottom figure ( b ) * : cross section through the top figure at @xmath112 , where the necessary condition can be determined analytically ( see sec . the additional black dashed curve corresponds to the non - linear sufficient condition for epr - steerability in eq . .,title="fig:",width=336 ] . * top figure ( a ) * : the red plane separates separable ( left ) and entangled ( right ) t - states . the sandwiched blue surface corresponds to the necessary condition for epr - steerability generated by our deterministic lhs model in sec . 4b : all t - states to the left of this surface are not epr - steerable . we conjecture that this condition is also sufficient , i.e. , that all states to the right of the blue surface are epr - steerable . for comparison , the green plane corresponds to the sufficient condition for epr - steerability in eq . ( [ linsuff ] ) of section 5a : all t - states to the right of this surface are epr - steerable . only a portion of the surfaces are shown , as they are symmetric under permutations of @xmath111 . * bottom figure ( b ) * : cross section through the top figure at @xmath112 , where the necessary condition can be determined analytically ( see sec . the additional black dashed curve corresponds to the non - linear sufficient condition for epr - steerability in eq . .,title="fig:",width=259 ] when @xmath113 we can solve eq . ( [ surface ] ) explicitly , because the normalisation constant @xmath91 simplifies . the corresponding equation of the @xmath108 semiaxis , in terms of @xmath114 , is given by @xmath115 ^{-1 } & u < 1 , \\ \left[1 - \frac { \sqrt{1-u^{-2}}}{2(u^2 - 1)}\ln\frac{|1-\sqrt{1-u^{-2}}|}{1+\sqrt{1-u^{-2 } } } \right ] ^{-1 } & u > 1 , \end{array } \right.\end{aligned}\ ] ] and @xmath116 for @xmath117 . 1(b ) displays this analytic epr - steerable curve through the t - state subspace @xmath118 , showing more clearly the different correlation regions . the symmetric situation @xmath119 corresponds to werner states . our deterministic lhs model is for @xmath120 in this case , which is known to represent the epr - steerable boundary for werner states @xcite . thus , our model is certainly optimal for this class of states . in the previous section a strong necessary condition for the epr - steerability of t - states was obtained , corresponding to the boundary defined in eq . ( [ surface ] ) and depicted in fig . 1 . while we have conjectured that this condition is also sufficient , it is not actually known if all t - states above this boundary are epr - steerable . here we give two sufficient general conditions for epr - steerability , and apply them to t - states . these conditions are examples of epr - steering inequalities , i.e. , statistical correlation inequalities that must be satisfied by any lhs model for bob @xcite . thus , violation of such an inequality immediately implies that alice and bob must share an epr - steerable resource . our first condition is based on a new epr - steering inequality for the spin covariance matrix , and the second on a known nonlinear epr - steering inequality @xcite . both epr - steering inequalities are further of interest in that they are asymmetric under the interchange of alice and bob s roles . suppose alice and bob share a two - qubit state with spin covariance matrix @xmath121 given by @xmath122 and that each can measure any hermitian observable on their qubit . we show in appendix [ app : cov ] that , if there is an lhs model for bob , then the singular values @xmath123 , @xmath124 , @xmath125 of the spin covariance matrix must satisfy the linear epr - steering inequality @xmath126 from @xmath127 , and using @xmath60 and @xmath128 for t - states , it follows immediately that one has the simple _ sufficient _ condition @xmath129 for the epr - steerability of t - states ( by either alice or bob ) . the boundary of t - states satisfying this condition is plotted in figs . 1 ( a ) and ( b ) , showing that the condition is relatively strong . in particular , it is a tangent plane to the necessary condition at the point corresponding to werner states ( which we already knew to be a point on the true boundary of epr - steerable states ) . however , in some parameter regions a stronger condition can be obtained , as per below . suppose alice and bob share a two - qubit state as before , where bob can measure the observables @xmath130 , @xmath131 on his qubit , with @xmath132 , for any @xmath133 $ ] , and alice can measure corresponding hermitian observables @xmath134 on her qubit , with outcomes labelled by @xmath135 . it may then be shown that any lhs model for bob must satisfy the epr - steering inequality @xcite @xmath136,\end{aligned}\ ] ] where @xmath137 denotes the probability that alice obtains result @xmath138 , and @xmath139 is bob s corresponding conditional expectation value for @xmath130 for this result . as per the first part of sec . 4a , we may always choose a representation in which the spin correlation matrix @xmath27 is diagonal , i.e. , @xmath140 $ ] , without loss of generality . making the choices @xmath141 and @xmath142 in this representation , then @xmath143 and @xmath139 are given by @xmath33 and the third component of @xmath144 in eqs . ( [ pe ] ) and ( [ qsteerb ] ) , respectively , with @xmath145 . hence , the above inequality simplifies to @xmath146,\end{aligned}\ ] ] where @xmath147 and @xmath148 are the third components of alice and bob s bloch vectors @xmath48 and @xmath46 . for t - states , recalling that @xmath149 , the above inequality simplifies further , to the nonlinear inequality @xmath150 hence , since similar inequalities can be obtained by permuting @xmath151 , we have the _ sufficient _ condition @xmath152 for the epr - steerability of t - states . the boundary of t - states satisfying this condition is plotted in fig . 1(b ) for the case @xmath112 . it is seen to be stronger than the linear condition in eq . ( [ linsuff ] ) if one semiaxis is sufficiently large . the region below both sufficient conditions is never far above the smooth curve of our necessary condition , supporting our conjecture that the latter is the true boundary . in this paper we have considered steering for the set of two - qubit states with maximally mixed marginals ( ` t - states ' ) , where alice is allowed to make arbitrary projective measurements on her qubit . we have constructed a lhv lhs model ( lhv for alice , lhs for bob ) , which describes measurable quantum correlations for all separable , and a large portion of non - separable , t - states . that is , this model reproduces the steering scenario , by which alice s measurement collapses bob s state to a corresponding point on the surface of the quantum steering ellipsoid . our model is constructed using the steering ellipsoid , and coincides with the optimal lhv lhs model for the case of werner states . furthermore , only a small ( and sometimes vanishing ) gap remains between the set of t - states that are provably non - steerable by our lhv lhs model , and the set that are provably steerable by the two steering inequalities that we derive . as such , we conjecture that this lhv lhs model is in fact optimal for t - states . proving this , however , remains an open question . a natural extension of this work is to consider lhv lhs models for arbitrary two - qubit states . how can knowledge of their steering ellipsoids be incorporated into such lhv lhs models ? investigations in this direction have already begun , but the situation is far more complex when alice and bob s bloch vectors have nonzero magnitude and the phenomenon of `` one - way steering '' may arise @xcite . finally , our lhv lhs models apply to the case where alice is restricted to measurements of hermitian observables . it would be of great interest to generalize these to arbitrary povm measurements . however , we note that this is a very difficult problem even for the case of two - qubit werner states @xcite . nevertheless , the steering ellipsoid is a depiction of all collapsed states , including those arising from povms ( they give the interior points of the ellipsoid ) and perhaps this can provide some intuition for how to proceed with this generalisation . sj would like to thank david jennings for his early contributions to this project . sj is funded by epsrc grant ep / k022512/1 . this work was supported by the australian research council centre of excellence ce110001027 and the european union seventh framework programme ( fp7/2007 - 2013 ) under grant agreement n@xmath153 [ 316244 ] . the family of t - states described by our deterministic lhs model in sec . 4b corresponds to the surface defined by either of eqs . ( [ surface ] ) and ( [ surface2 ] ) . this is a consequence of the following theorem , proved further below . for any full - rank diagonal matrix @xmath27 and nonzero vector @xmath154 one has @xmath155 note that substitution of eq . ( [ re ] ) into constraint ( [ con ] ) immediately yields eq . ( [ surface ] ) via the theorem ( with @xmath156 ) . further , taking the dot product of the integral in the theorem with @xmath157 , multiplying by @xmath91 , and integrating @xmath157 over the unit sphere , yields ( reversing the order of integration ) @xmath158 whereas carrying out the same operations on the righthand side of the theorem yields @xmath159 . equating these immediately implies the equivalence of eqs . ( [ surface ] ) and ( [ surface2 ] ) as desired . an explicit analytic formula for the normalisation constant @xmath91 is given at the end of this appendix . first , define @xmath160 ; that is , @xmath161 and @xmath162 noting @xmath157 in the theorem may be taken to be a unit vector without loss of generality , we will parameterise the unit vectors @xmath80 and @xmath157 by @xmath163 with @xmath164 $ ] and @xmath165 . thus , @xmath166 . further , without loss of generality , it will be assumed that @xmath167 points into the northern hemisphere , so that @xmath168 . then @xmath169 $ ] and @xmath170 . the surface of integration is a hemisphere bounded by the great circle @xmath171 . in the simple case where @xmath172 , the boundary curve has the parametric form @xmath173 for @xmath174 . hence , the boundary curve in the generic case can be constructed by applying the orthogonal operator @xmath175 , that rotates @xmath167 from @xmath176 to @xmath177 , to the vector @xmath178 . that is , @xmath179 and the boundary curve has the form @xmath180 for the purposes of integrating over the hemisphere , it is convenient to vary @xmath181 from @xmath182 to @xmath183 and @xmath184 from @xmath182 to its value @xmath185 on the boundary curve . from the above expression for the boundary , and using @xmath186 and @xmath187 , it follows that @xmath188 and @xmath189 . the last equation be rearranged to read @xmath190 , and after squaring both sides this equation solves to give @xmath191^{1/2}}.\]]now , @xmath192 assumes its maximum value when @xmath193 , which according to the relation @xmath188 and the fact that @xmath194 $ ] should correspond to @xmath195 . so we take the upper sign in the last equation , yielding @xmath196^{1/2}}\\ & = \frac{-\sin \alpha\cos ( \phi -\beta)}{[\cos ^{2}\alpha+\sin ^{2}\alpha\cos ^{2}(\phi -\beta)]^{1/2 } } .\end{aligned}\ ] ] it follows immediately that @xmath197^{1/2}},\end{aligned}\ ] ] with the choice of sign fixed by the fact that @xmath198 and ( by assumption ) @xmath168 . the surface integral for @xmath199 in eq . ( [ qv ] ) can now be written in the form : @xmath200 to evaluate the the third component of @xmath199 , note that the integral over @xmath184 , @xmath201can be evaluated explicitly by making the substitution @xmath202 , as @xmath203 for any @xmath204 , yielding @xmath205 after inserting the expressions for @xmath206 and @xmath207 derived earlier , we have @xmath208 we now need to integrate the last expression over @xmath181 . introducing new constants @xmath209 the full surface integral simplifies to a form that may be evaluated by mathematica ( or by contour integration over the unit circle in the complex plane ) : @xmath210the indeterminate sign here is fixed by examining the case @xmath211 and @xmath212 , for which @xmath213 and the integrand reduces to @xmath214 , which integrates to give @xmath215 . so , unsurprisingly , we choose the positive sign . this yields the third component of surface integral to be @xmath216_3 = \frac{\pi \cos \alpha}{c[ab\cos ^{2}\alpha+c(a\sin ^{2}\beta+b\cos ^{2}\beta)\sin ^{2}\alpha]^{1/2}}.\end{aligned}\ ] ] the integrals over @xmath184 in the remaining two components of @xmath199 in eq . ( [ qvnew ] ) are unfortunately not so straightforward . however , there is a simple trick that allows us to calculate both surface integrals explicitly , and that is to differentiate the integrals with respect to the parameters @xmath217 and @xmath218 . since the only dependence on @xmath217 and @xmath218 comes through the function @xmath219 , this eliminates the need to integrate over @xmath184 . in fact we only need to differentiate with respect to one of these parameters , choose @xmath217 . to see this , note that @xmath220where @xmath221 can be evaluated by making use of the equations and . in fact , @xmath222^{1/2}}\right)\end{aligned}\ ] ] @xmath223^{3/2}}.\end{aligned}\ ] ] inserting the last two equations and the expressions for @xmath224 and @xmath225 into the integrals above , and using the constants @xmath226 and @xmath227 defined earlier , then gives : @xmath228^{2}}{\mathop{}\!\mathrm{d}}\phi \nonumber \\ \label{xy1 } ~&=&\cos ^{2}\alpha\int_{0}^{2\pi } \frac{(\sin \beta\sin \phi \cos \phi + \cos \beta\cos ^{2}\phi , \sin \beta\sin ^{2}\phi + \cos \beta\sin \phi \cos \phi ) } { ( l \cos ^{2}\phi + m \sin ^{2}\phi + 2n \sin \phi \cos \phi ) ^{2}}{\mathop{}\!\mathrm{d}}\phi .\end{aligned}\ ] ] consequently , there are three separate integrals we need to evaluate and these can be done in mathematica ( or by complex contour integration ) : @xmath229 using the values we have for @xmath230 we substitute these back into equation and integrate over @xmath217 to obtain @xmath231_1 = \pi\int \frac{\cos ^{2}\alpha(m\cos \beta - n\sin \beta ) } { ( lm - n^2)^{3/2}}d\alpha\end{aligned}\ ] ] @xmath232^{1/2}},\end{aligned}\ ] ] and @xmath233_2 = \pi\int \frac{\cos ^{2}\alpha(l\sin \beta -n\cos \beta ) } { ( lm - n^2)^{3/2}}d\alpha\\ \label{qv2 } & = \frac{b^{-1}\pi \sin\alpha\sin\beta}{[ab \cos^2\alpha + c\sin^2\alpha(b\cos^2\beta + a\sin^2\beta)]^{1/2}}.\end{aligned}\ ] ] the absence of integration constants can be confirmed by noting that these expressions vanish for @xmath211 i.e. , when the vector @xmath234 is aligned with the @xmath101-axis as they should by symmetry . note the denominators of eqs . and simplify to @xmath235 . combining this with eqs . ( [ qv3 ] ) and ( [ qv1])-([qv2 ] ) , we have @xmath236 and so setting @xmath237 , the theorem follows as desired . finally , the normalisation constant @xmath91 in eq . ( [ surface ] ) may be analytically evaluated using mathematica . under the assumption that @xmath238 , denote @xmath239 . we find @xmath240+a(b+c)k[c]+ib(c - a)(e[a_1,b]-e[a_2,b])+ic(a+b)(f[a_1,b]-f[a_2,b])\right\}\big),\end{aligned}\ ] ] to demonstrate the linear epr - steering inequality in eq . ( [ lin ] ) , let @xmath253 denote some dichotomic observable that alice can measure on her qubit , with outcomes labelled by @xmath135 , where @xmath157 is any unit vector . we will make a specific choice of @xmath253 below . define the corresponding covariance function @xmath254 if there is an lhs model for bob then , noting that one may take @xmath255 in eq . ( [ lhs ] ) to be deterministic without loss of generality , there are functions @xmath256 such that @xmath257\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v$ ] , where @xmath258 , and the hidden state @xmath11 has corresponding bloch vector @xmath76 . now , the bloch sphere can be partitioned into two sets , @xmath259\cdot{\boldsymbol}v \geq 0\}$ ] and @xmath260\cdot{\boldsymbol}v < 0\}$ ] , for each value of @xmath7 . hence , noting @xmath261 , @xmath262 is equal to @xmath263\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right.\\ & ~ & \left . + \int_{s_-(\lambda ) } { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\ , [ \alpha_{{\boldsymbol}v}(\lambda ) - \bar{\alpha}_{{\boldsymbol}v}]\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right\}\end{aligned}\ ] ] @xmath264\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right.\\ & ~ & -\left . \int_{s_-(\lambda ) } { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\ , [ 1 + \bar{\alpha}_{{\boldsymbol}v}]\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right\}\\ & = & \sum_\lambda p(\lambda ) \int { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\ , |[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v|\end{aligned}\ ] ] @xmath265\cdot{\boldsymbol}v\\ & = & \sum_\lambda p(\lambda ) |{\boldsymbol}n(\lambda)-{\boldsymbol}b|\,\int { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\ , |{\boldsymbol}v\cdot { \boldsymbol}w(\lambda)|,\end{aligned}\ ] ] where @xmath266 denotes the unit vector in the @xmath267 direction , and the last line follows by interchanging the summation and integration in the second term of the previous line . the integral in the last line can be evaluated for each value of @xmath7 by rotating the coordinates such that @xmath268 is aligned with the @xmath101-axis , yielding @xmath269 . hence , the above inequality can be rewritten as @xmath270^{1/2 } \\ \label{cav } & \leq & \frac{1}{2}\sqrt{1-{\boldsymbol}b\cdot{\boldsymbol}b},\end{aligned}\ ] ] where the second and third lines follow using the schwarz inequality and @xmath271 , respectively . note , by the way , that the first inequality is tight for the case @xmath272\cdot{\boldsymbol}v\right)$ ] . now , making the choice @xmath273 with @xmath274 , one has from eqs . ( [ cjk ] ) and ( [ cv ] ) that @xmath275 combining with eq . ( [ cav ] ) immediately yields the epr - steering inequality @xmath276 finally , this inequality may similarly be derived in a representation in which local rotations put the spin covariance matrix @xmath121 in diagonal form , with coefficients given up to a sign by the singular values of @xmath121 ( similarly to the spin correlation matrix @xmath27 in sec . 4a ) . since @xmath277 is invariant under such rotations , eq . ( [ lin ] ) follows . a. j. bennet , d. a. evans , d. j. saunders , c. branciard , e. g. cavalcanti , h. m. wiseman , and g. j. pryde , `` arbitrarily loss - tolerant einstein - podolsky - rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole , '' phys . x * 2 * , 031003 ( 2012 ) . c. branciard , e. g. cavalcanti , s. p. walborn , v. scarani , and h. m. wiseman , `` one - sided device - independent quantum key distribution : security , feasibility , and the connection with steering , '' phys . a * 85 * , 010301 ( 2012 ) . for ellipsoids of separable states , there is a further ambiguity in the ` chirality ' of alice s local basis , that is , we may determine @xmath278 up to a local unitary and a partial transpose on alice s system @xcite .

the question of which two - qubit states are steerable ( i.e. permit a demonstration of epr - steering ) remains open . here , a strong necessary condition is obtained for the steerability of two - qubit states having maximally - mixed reduced states , via the construction of local hidden state models . it is conjectured that this condition is in fact sufficient . two provably sufficient conditions are also obtained , via asymmetric epr - steering inequalities . our work uses ideas from the quantum steering ellipsoid formalism , and explicitly evaluates the integral of @xmath0 over arbitrary unit hemispheres for any positive matrix @xmath1 .

arxiv

the visionary who first thought of using the spin polarization of a single electron to encode a binary bit of information has never been identified conclusively . folklore has it that feynman mentioned this notion in casual conversations ( circa 1985 ) , but to this author s knowledge there did not exist concrete schemes for implementing spintronic logic gates till the mid 1990s . encoding information in spin may have certain advantages . first , there is the possibility of lower power dissipation in switching logic gates . in charge based devices , such as metal oxide semiconductor field effect transistors , switching between logic 0 and logic 1 is accomplished by moving charges into and out of the transistor channel . motion of charges is induced by creating a potential gradient ( or electric field ) . the associated potential energy is ultimately dissipated as heat and irretrievably lost . in the case of spin , we do not have to _ move _ charges . in order to switch a bit from 0 to 1 , or vice versa , we merely have to toggle the spin . this may require much less energy . second , spin does not couple easily to stray electric fields ( unless there is strong spin - orbit interaction in the host material ) . therefore , spin is likely to be relatively immune to noise . finally , it is possible that spin devices may be faster . if we do not have to move electrons around , we will not be limited by the transit time of charges . instead , we will be limited by the spin flip time , which could be smaller . in 1994 , we proposed a concrete scheme for realizing a classical universal logic gate ( nand ) using three spins placed in a weak magnetic field @xcite . by `` three spins '' , we mean the spin orientations of three conduction band electrons , each confined in a semiconductor quantum dot . the system is shown schematically in fig . exchange interaction is allowed only between nearest neighbor spins ( second nearest neighbor interaction is considered too weak to have any effect ) . because of the magnetic field , the spin orientation in any quantum dot becomes a _ binary variable_. the spin polarization is either along the magnetic field , or opposite to the field . to understand this , consider the hamiltonian of an isolated dot : @xmath0 where @xmath1 is the unperturbed hamiltonian in the absence of the magnetic field , @xmath2 is the magnetic field , @xmath3 is the land g - factor of the quantum dot material , @xmath4 is the bohr magneton , and @xmath5 is the pauli spin matrix . if the magnetic field is directed along the z - direction , then @xmath6 diagonalizing the above hamiltonian yields the eigenspinors ( 1,0 ) and ( 0,1 ) which are + z and -z polarized spins . therefore , the spin orientation is a binary variable ; it is either parallel or anti - parallel to the applied magnetic field . in the presence of exchange interaction between two electrons confined to two _ separate _ potentials ( such as two different quantum dots ) , the anti - ferromagnetic ordering , or the singlet state , ( i.e. two neighboring spins are anti - parallel ) is preferred over the ferromagnetic ordering , or triplet state ( two spins are parallel ) @xcite . we will assume that the tendency to preserve this anti - ferromagnetic ordering is _ stronger _ than the tendency for all spins to line up along the magnetic field . this merely requires that the exchange splitting energy @xmath7 ( energy difference between triplet and singlet states ) exceed the zeeman splitting energy @xmath8 . we ensure this by reducing the potential barrier between neighboring dots to enhance the exchange , while at the same time , making the magnetic field sufficiently weak to reduce the zeeman energy . under this scenario , the ground state of the array has the spin configuration shown in fig . we will call `` upspin '' the spin orientation directed along the magnetic field and `` downspin '' the opposite orientation . we encode logic 1 in the upspin state . furthermore , we will consider the two edge dots in fig . 1(a ) as input ports to a logic gate , and the middle dot as the output port . it is obvious that when the two inputs are logic 1 , the output will be logic 0 when the system reaches ground state ( anti - ferromagnetic ordering ) . next , consider the situation when the two inputs are logic 0 ( see fig . the output must be logic 1 in order to conform to the anti - ferromagnetic ordering . however , there is a subtle issue . 1(b ) is actually _ not _ the ground state of the system , fig . this is because of the weak magnetic field . the difference between fig . 1(a ) and fig . 1(b ) is that in the former case , _ two _ spins are aligned parallel to the magnetic field , while in the latter , _ two _ spins are aligned anti - parallel to the magnetic field . therefore , if the system is left in the state of fig . 1(b ) , it must ultimately decay to the state in fig . 1(a ) , according to the laws of thermodynamics . but that may take a very long time because of three reasons . first , the system must emit some energy carrying entity to decay . this entity is most likely a phonon . however , phonon emissions in quantum dots are suppressed by the `` phonon bottleneck '' effect @xcite . second , phonons do not couple easily to spin unless we have a strongly pyroelectric material as the host . finally , if spins flip one at a time ( all three spins flipping simultaneously is very unlikely ) , then in order to access the state in fig 1(a ) , the state in fig . 1(b ) will have to go through a state where two neighboring spins will be parallel . such a state is much higher in energy than either fig . 1(a ) or fig . therefore , fig . 1(a ) and fig . 1(b ) are separated by an energy barrier , making fig . 1(b ) a long lived metastable state . as long as the input bit rate is high enough so that inputs change much more rapidly than the time it takes for the metastable state to decay to the global ground state of fig . 1(a ) , we need not worry about this issue . what happens if one of the inputs is logic 1 , and the other is logic 0 as shown in fig . 1(c ) ? here the magnetic field comes in handy to break the tie . in this case , logic 1 is preferred as the output since the all other things being equal , a spin would prefer to line up parallel to the magnetic field , rather than anti - parallel . thus , when either input is logic 0 , the ouput is logic 1 . we have realized the truth table in table 1 . .truth table of a spintronic nand gate [ cols="^,^,^",options="header " , ] the reader will recognize that this is the truth table of a nand gate , which is one of two universal boolean logic gates . since we can realize a nand , we can realize any arbitrary boolean logic circuit ( combinational or sequential ) by connecting nand gates . a number of different logic devices ( half adders , flip - flops , etc . ) were designed and illustrated in ref . @xcite . these devices have been extensively studied by others @xcite using time independent simulations . the time - independent simulations address the steady state behaviors and therefore do not directly reveal a serious problem with these devices that was already recognized in ref . @xcite . in the following section , we explain this problem . at the time these logic gates were proposed , it was also realized that they have a severe shortcoming that precludes their use in pipelined architectures @xcite . to understand the nature of the problem , consider three inverters ( not gates ) in series . a single not gate is the simplest device ; it is realized by two exchange coupled spins , one of which is the input and the other is the output . because of the anti - ferromagnetic ordering , the output is always the logic complement of the input . 2(a ) shows three conventional inverters in series and fig . 2(b ) shows the corresponding spintronic realization . the input to the first inverter is logic 1 and the output of the last inverter is logic 0 , as it should be . but now , let us suddenly change the input at the first inverter to logic 0 at time @xmath9 = 0 . the situation at time @xmath9 = 0 + is shown in fig . we expect that ultimately the output of the last inverter will become logic 1 . unfortunately , this can not happen . in fig . 2(c ) , the second spin from the left finds its left neighbor asking it to flip ( because of the exchange interaction that enforces anti - ferromagnetic ordering between two neighboring spins ) while its right neighbor is asking it to stay put because of the same exchange interaction . both influences from the left and from the right are exactly equally strong and the second cell is stuck in a logic indeterminate state that it can not get out of @xcite . rolf landauer later termed it a metastable state that prevents decay to the desired ground state @xcite . in fact , if we take the external magnetic field into account , then there is a preference for the second cell to actually _ not _ flip in response to the input since there is a slight preference for the upspin state because of the external magnetic field . in this case , the logic signal can not propagate from the input to the output and the circuit simply does not work ! similar situations were examined in ref . the real problem is that exchange interaction is _ bidirectional _ which can not ensure _ unidirectional _ flow of logic signal from the input to the output of the logic device . this unidirectionality is a required attribute of any logic device ( for the five necessary requirements of a classical logic device , see ref . @xcite ) . we can think of desperate measures to enforce the unidirectionality . for example , we can claim that if we hold the input at the first inverter ( leftmost cell in fig . 2(c ) ) to logic 0 , and do not let go , then the second cell which is equally likely to follow its left neighbor and right neighbor , will have no option but to ultimately follow its left neighbor since it is adamant and persistent ( we are not letting go of the input ) . this will happen in spite of the magnetic field since the exchange energy is larger than the zeeman splitting . in this case , we are trying to enforce unidirectionality via the input signal itself ( note that the input device does indeed break the inversion symmetry of the system in fig . this possible remedy was studied theoretically in ref . @xcite which reached the conclusion that it does not always work . in fact , the process of logic signal propagation under this scenario is inefficient thermally assisted random walk and the final logic state , if reached , can be destroyed by thermal fluctuations . the idea of using the input device to enforce unidirectionality was also implicitly used in the experiment of ref . @xcite . while this may work for a few cells ( as it did in ref . @xcite ) , it will obviously not work for a large number of cells since the influence of the input decays with increasing distance from the input . ultimately , the remote cells that are far from the input , will not feel the input s effect and remain stuck in metastable states , producing wrong answers to simple logic problems . in ref . @xcite , one solution that was offered to break this impasse was to progressively increase the distance between cells . this makes the influence of the left neighbor always stronger than that of the right neighbor since the strength of the exchange interaction has an exponential dependence on the separation between neighboring cells . this is not an elegant solution since ultimately the exchange splitting energy will become smaller than the zeeman splitting energy , at which point the paradigm will no longer work . in 1996 , we proposed a more elegant solution @xcite . this was inspired by the realization that in charge coupled device ( ccd ) arrays , there is no inherent unidirectionality , yet charge is made to propagate from one device to the next unidirectionally . this is achieved by _ clocking_. we mentioned that unidirectionality can be imposed in time or space @xcite and clocking imposes unidirectionality in time . however , a cursory examination revealed that normal clocking will not work in our case . say , we put gate pads on the barriers between neighboring cells . initially , all the the barriers are high and opaque so that there is no overlap between the wavefunctions of adjacent electrons and hence no exchange interaction between neighboring spins . now , we lower the first barrier by applying a positive potential to gate 1 as shown in fig . this allows the wavefunctions of electrons on either side of the gate pad to leak out into the barrier , overlap . and cause an exchange interaction . exchange causes the second spin to assume a polarization anti - parallel to the input spin orientation since the singlet state is lower in energy than the triplet . in other words , the second cell switches . at this point , if we let go of the input , raise the first barrier back up , and lower the second barrier by applying a positive potential to gate 2 , then either the third cell switches in response to the second ( which is shown in the upper branch ) , or the second cell switches in response to the third ( which is shown in the lower branch ) . the upper branch is the desired state , but it is equally likely that the lower branch will result since both branches obey the ant - ferromagnetic ordering between the two exchange coupled cells ( cell 2 and 3 ) . therefore , a simple sequential clock will not work . what is required is that both gate 1 and gate 2 have positive potentials while the input is applied . now the first three cells assume the correct polarizations as shown in fig . then , the input is removed , gate 1 is returned to zero potential and positive potentials are applied to gates 2 and 3 . this causes the first four cells to assume the correct polarization , and so on . this situation is shown in fig . 2(f ) and is the desired configuration . thus by lowering _ two adjacent barriers _ pairwise at the same time , we can propagate the input state through a linear array . in other words , we will need a _ three - phase clock _ , a single phase will not work . the three clock pulse trains for a three - phase clock are shown in fig . each train is phase shifted from the previous one by @xmath10 radians . such a situation is not unusual since charge coupled device arrays also need a multi - phase clock ( push clock , drop clock ) to work @xcite . while , multi - phase clocking can make these devices work , it is hardly an attractive solution since one needs to fabricate gate pads between every two cells . the separation between the cells may need to be @xmath11 5 nm in order to have sufficient exchange coupling . aligning a gate pad to within a space of 5 nm is a major lithography challenge . furthermore , the gate potentials are lowered and raised by moving charges into and out of the gate pads , leading to considerable energy dissipation that completely negates the advantage of using spins . therefore , these devices present interesting physics , but at this time , do not appear to be serious candidates for practical applications . so far , we have discussed the use of spin in classical irreversible logic gates . these logic gates dissipate a minimum of @xmath12 amount of energy per bit flip @xcite . let us assume that we can make quantum dots with a density of 10@xmath13 @xmath14 . quantum dots self assembled by electrochemical techniques in our own lab ( and in many other labs ) can achieve this density today . we show a raw atomic force micrograph of quantum dots self assembled in our lab in fig . 3 . the dark areas are the dots and the surrounding light areas are the barriers . the dot diameter in this micrograph is 50 nm and the dot density is close to 10@xmath15 @xmath14 . by using slightly different synthesis conditions , we can actually achieve densities exceeding 10@xmath13 @xmath14 . let us now assume that we can flip the spin in a quantum dot in 1 psec . then the minimum power that will be dissipated per unit area will exceed @xmath16 /(1 psec ) = 3 kw/@xmath17 ( actually most of the power will be dissipated in the clock cycles , which we have ignored ) . this dissipation is at least 30 times more than what the pentium iv chip dissipates @xcite . although removal of 1 kw/@xmath17 of heat from a chip was demonstrated almost two decades ago , removing 3 kw/@xmath17 from a chip is still a major challenge in heat sinking . the obvious way to overcome ( or , rather , circumvent ) this challenge is to develop reversible logic gates that are not constrained by the landauer @xmath12 barrier . in 1996 , we devised a logically and physically reversible quantum inverter using two exchange coupled spins @xcite . this device is very similar to the single electron parametron idea @xcite and can be viewed as a single spin parametron . since either spin could exist in a coherent superposition of two orthogonal spin states ( call them `` upspin '' and `` downspin '' states ) , this would also be a `` qubit '' . later , loss and divincenzo devised a universal quantum gate using two exchange coupled spins in two closely spaced quantum dots @xcite . recently , experimental demonstration of coherent transfer of electron spins between quantum dots coupled by conjugated molecules has been demonstrated , opening up real possibilities in this area @xcite . the idea of using a single electron or nuclear spin to encode a qubit and then utilizing this to realize a universal quantum gate , has taken hold @xcite . the motivation for this is the realization that spin coherence times in solids is much larger than charge coherence time . charge coherence times in semiconductors tend to saturate to about 1 nsec as the temperature is lowered @xcite . this is presumably due to coupling to zero point motion of phonons which can not be eliminated by lowering temperature @xcite . on the other hand , electron spin coherence times of 100 nsec in gaas at 5 k has already been reported @xcite and much higher coherence times are expected for nuclear spins in silicon @xcite . therefore , spin is obviously the preferred vehicle to encode qubits in solids . using spin to carry out all optical quantum computing has also appeared as a viable and intriguing idea @xcite . the advantage of the all - optical scheme over the electronic scheme is that we do not have to read single electron spins _ electrically _ to read a qubit . electrical read out is extremely difficult @xcite , although some schemes have been proposed for this purpose @xcite . recently , some experimental progress has been made in this direction @xcite , but reading a single qubit in the solid state still remains elusive , . the difficult part is that electrical read out requires making contacts to individual quantum dots , which is an engineering challenge . in contrast , optical read out does not require contacts . the qubit is read out using a quantum jump technique @xcite which requires monitoring the fluorescence from a quantum dot . recently , it has been verified experimentally that the spin state of an electron in a quantum dot can be read by circularly polarized light @xcite . therefore , optical read out appears to be a more practical approach . in this article we have provided a brief history of the use of single electron spin in computing . we have indicated where and why spin may have an advantage over charge in implementing the type of devices and architectures discussed here . s. bandyopadhyay and v. p. roychowdhury , proceedings of the international conference on superlattices and microstructures , liege , belgium , 1996 , also in superlat . 3 , 411416 ( 1997 ) . t. calarco , a. datta , p. fedichev , e. pazy and p. zoller , phys . rev . a , vol . 68 , no . 1 , 012310 - 1 012310 - 21 ( 2003 ) ; e. pazy , e. biolattia , t. calarco , i. d amico , p. zanardi , f. rossi and p. zoller , europhys . 62 , 175 ( 2003 ) .

this article reviews the use of single electron spins to compute . in classical computing schemes , a binary bit is represented by the spin polarization of a single electron confined in a quantum dot . if a weak magnetic field is present , the spin orientation becomes a binary variable which can encode logic 0 and logic 1 . coherent superposition of these two polarizations represent a qubit . by engineering the exchange interaction between closely spaced spins in neighboring quantum dots , it is possible to implement either classical or quantum logic gates .

arxiv

cataclysmic variables ( cvs ) are short - period binaries containing a white dwarf ( wd ) primary ( with mass @xmath2 ) and a low mass main sequence secondary ( with mass @xmath3 ) . the secondary fills its roche lobe and transfers mass to the wd through the inner lagrangian ( @xmath4 ) point . the main features of the orbital period distribution of cvs with hydrogen rich donors are the lack of systems in the 2 - 3 hr period range ( the so - called period gap ) and the sharp cut off of the distribution at around 77 minutes , as can be seen in figure [ combined ] ( upper frame ; e.g. ritter & kolb 1998 ) . so far theoretical models have been unable to reproduce the precise position of the observed short - period cut - off and observed shape of the cv orbital period distribution near this cut - off . this is summarised in figure [ combined ] . systems that evolve under the influence of gravitational radiation ( gr ; kraft et al . 1962 ) as the only sink of orbital angular momentum ( am ) reach a minimum period at @xmath5 minutes ( figure[combined ] , middle frame ; paczyski 1971 ; kolb & baraffe 1999).the probability of finding a system within a given period range is proportional to the time taken to evolve through this region . we thus have n(p ) , for the number @xmath6 of systems found within a given orbital period range around @xmath7 , and @xmath8 is the secular period derivative at this period . we thus expect an accumulation of systems ( a spike ) at @xmath9 where @xmath10 ( figure [ combined ] , lower frame ) , while no such spike is present in the observed distribution ( figure[combined ] , upper frame ) . the orbital period evolution reflects the radius evolution of the mass donor , which in turn is governed by two competing effects . mass transfer perturbs thermal equilibrium and expands the star . thermal relaxation reestablishes thermal equilibrium and contracts the star back to its equilibrium radius . the minimum period occurs where the two corresponding time scales , the mass transfer time @xmath11 and the thermal ( kelvin - helmholtz ) time @xmath12 are about equal ( e.g. paczyski 1971 ; king 1988 ) . if @xmath13 then the star is able to contract in response to mass loss , but if @xmath14 the star will not shrink rapidly enough and will become oversized for its mass . the position of the minimum period is therefore affected by the assumed mass transfer rate , and in particular by the assumed rate of orbital angular momentum ( am ) losses . in this paper we investigate ways to increase the period minimum by increasing the mass transfer rate , and investigate ways to `` hide '' the spike by introducing a spread of @xmath9 values in the cv population . in particular , we study the effect of a form of consequential am loss ( caml ) where the am is lost as a consequence of the mass transferred from the secondary , i.e. @xmath15 ( see e.g. webbink 1985 ) . in section [ theory ] we outline our general model assumptions and introduce the prescription for caml . in section [ sec22 ] we present detailed calculations of the long - term evolution of cvs , and in section [ comptest ] we compare the observed short period cv period distribution with various theoretically synthesized model distributions based on the calculations in section 2 . in this section we investigate possible solutions to the mismatch between the theoretical and observed minimum orbital period in cvs . the orbital am loss rate @xmath16 of a cv can be written as the sum of two terms , = _ sys+_caml , where @xmath17 denotes the `` systemic '' am loss rate , such as gravitational wave radiation , that is independent of mass transfer , while @xmath18 is an explicit function of the mass transfer rate . we have = 0 and _ caml0_20 we consider the general case in which the caml mechanism , along with nova mass ejections , causes a fraction of the transferred mass to leave the system . this fraction may be greater than unity as the primary may lose more mass during a nova outburst than was accreted since the last outburst . we employ a generic prescription of the effect of a caml mechanism , thus avoiding the need to specify its physical nature . possible caml mechanisms include a magnetic propeller , i.e. a system containing a rapidly spinning magnetic wd where some of the transferred material gains angular momentum from the wd spin by interaction with the wd s magnetic field ( see e.g. wynn , king & horne 1997 ) , and an accretion disc wind ( see e.g. livio & pringle 1994 ) . our caml prescription largely follows the notation of king & kolb ( 1995 ) . the am is assumed to be lost via mass loss that is axis - symmetrical with respect to an axis a fixed at the wd centre but perpendicular to the orbital plane . we define @xmath19 as the total fraction of mass lost from the secondary that leaves the system . we assume further that a fraction @xmath20 ( @xmath21 ) of the transferred mass leaves the system with some fraction @xmath22 of the angular momentum it had on leaving the @xmath4 point . we also consider mass that is lost from the system via nova mass ejections , which over the long term can be considered as an isotropic wind from the primary ( see e.g. kolb et al . this material will carry away the specific orbital angular momentum of the primary and will account for the fraction ( @xmath23 ) of the mass loss . we thus obtain _ caml = b^2_2 + , where we define @xmath24 as the caml efficiency . for comparison with king & kolb ( 1995 ) we equate this to [ eq : jdotcaml ] _ caml = j , > 0 , and obtain [ eq : nufinal ] = ( 1+q)()^2 + . for our calculations shown below we use the approximation 1-+-,^3=. this is an adaptation of the expression given in kopal ( 1959 ) and is accurate to within 1% over the range @xmath25 . in this subsection we present calculations of the long - term evolution of cvs as they approach and evolve beyond the period minimum . for the computations we used the stellar code by mazzitelli ( 1989 ) , adapted to cvs by kolb & ritter ( 1992 ) . some of these evolutionary sequences are the basis for the theoretical cv period distributions we present in section [ comptest ] below . we calculated the evolution of individual systems that are subject to caml according to equations [ eq : jdotcaml ] and [ eq : nufinal ] . we chose @xmath26 and initial donor mass @xmath27 , with a range of caml efficiencies @xmath28 as shown in figure [ fig : fullcaml ] . the systems initially evolve from longer periods towards the period bounce ( right to left ) at almost constant mass transfer rate . the minimum period increases with increasing caml efficiency to a maximum of around 70 min for @xmath29 . mass transfer stability sets an upper limit on the caml efficiency . an obvious upper limit is 1 , where all the angular momentum of the transferred material is ejected from the system . although the ejected material may carry more angular momentum than was transferred ( as in the case of a propeller system where additional angular momentum is taken from the spin of the wd ) this does not affect the net loss of orbital angular momentum . the maximum caml efficiency still compatible with mass transfer stability could be smaller than unity . the stability parameter @xmath30 which enters the expression for steady - state mass transfer , equation [ eq : stab ] ( e.g. king & kolb 1995 ) must be greater than zero ; this defines an upper limit on @xmath31 . [ eq : stab ] -_2=m_2 ( ) a plot of @xmath30 against @xmath32 for an initially marginally stable system ( @xmath33 , @xmath34 and @xmath35 ) is given in figure [ fig : dq ] . the system initially exhibits cycles of high mass transfer rate @xmath36 ( @xmath30 close to 0 ) and very low mass transfer rate @xmath37 . the high states are short lived , on the order of @xmath38 years ( see figure [ fig : highmdot ] ) . the system finally stabilizes with @xmath39 . at around @xmath40 @xmath30 starts to decrease further but always remains positive , settling at a value around @xmath41 . the tidal deformation of the secondary may have an effect on the period minimum . calculations by renvoiz , baraffe , kolb & ritter ( 2002 ) , [ see also kolb 2002 ] using 3dimensional sph models suggest that the secondary is deformed in the non - spherical roche lobe such that its volume equivalent radius is around 1.06 times that of the same star in isolation . we mimic this effect in our 1-dimensional stellar structure code by multiplying the calculated radius by a deformation factor @xmath42 before the mass transfer rate is determined from the difference between the radius and the roche lobe radius via -_2=_0(- ) . here @xmath43 is the mass transfer rate of a binary in which the secondary just fills its roche potential and @xmath44 is the photospheric pressure scale height of the secondary ( see e.g. ritter 1988 ) . figure [ fig : barraffe ] shows the effect on the minimum period and mass transfer rate for systems with various deformation factors @xmath42 , ranging from 1 ( no deformation ) to 1.24 . the mass transfer rate is seen to decrease with increasing deformation . this can be understood from the functional dependence on orbital period and donor mass in the usual quadrupole formula for the am loss rate due to gravitational radiation ( see e.g. landau & lifschitz 1958 ) . although the quadrupole formula is strictly valid only if both components are point masses , rezzolla , ury & yoshida ( 2001 ) found that the gr rate obtained using a full 3-dimensional representation of the donor star differs from the point mass approximation by less than a few percent . it can be seen from the figure that with the deformation factor 1.06 the minimum period increases from around 65 min to around 69 min , consistent with renvoiz et al ( 2002 ) for geometrical effects alone . a deformation factor of around 1.18 was required to raise the minimum period to the observed value of @xmath45 min . this is somewhat larger than the intuitive expectation ( ) ^=()^=1.12 from kepler s law and roche geometry . in our calculations we consider the simple case in which only the geometrical deformation effects are taken into account . the inclusion of the thermal effects considered by renvoiz et al ( 2002 ) have the likely effect of reducing @xmath9 , possibly by around 2% compared to the case with purely geometrical effects one possible physical mechanism that could cause a deformation factor above the value of 1.06 is magnetic pressure inside the star , as suggested by dantona ( 2000 ) . we note that patterson ( 2000 ) claims to find observational evidence for `` bloated '' secondaries in short period cvs . on the basis of donor mass estimates from the observed superhump excess period he finds that the donors have @xmath46 larger radii than predicted from 1 dimensional . , non deformed stellar models if gravitational radiation is the only am sink . even if true , this observation can not distinguish between an intrinsic deformation of the donor star or the non - equilibrium caused by orbital am losses in excess of the gr rate . to test the statistical significance of the theoretically predicted accumulation of systems near the period minimum ( `` period spike '' ) we calculated the period distributions of model populations for various assumptions about evolutionary parameters . for each parameter a series of evolutionary tracks were generated , typically around 20 . as systems evolve after the minimum period a point is reached ( typically when @xmath47 falls below @xmath48 ) where numerical fluctuations in @xmath47 become so large that the henyey scheme no longer converges . the stellar code uses tables to interpolate / extrapolate the opacities and equation of state for each iteration , and in this region the extrapolations become very uncertain . to extend the tracks we used a semi - analytical method as follows . the tracks were terminated at a value of @xmath49 , where @xmath50 is the mass transfer rate at the minimum period for the track . the radius of the star for the final part of the track is approximated by r_2=r_0m_2^ , where @xmath51 and @xmath52 are assumed to be constant . the values of @xmath51 and @xmath52 were determined from the final few data points for each track . ( @xmath52 takes a typical value of around 0.15 for systems beyond the period bounce . ) to generate the extension to the track we then calculated @xmath7 from the roche lobe condition , and @xmath53 by assuming stationarity as in section [ minpnumeric ] ( see figure [ combined ] , middle frame for an example of an extended track ) . we weight the chances of observation to the brighter systems by assuming , 1.0 . for the detection probability . we tested the calculated model parent distributions for various values of the free parameter @xmath54 against the observed cv period distribution . a k - s ( kolmogorov - smirnov ) test is insensitive to the differences between the parent distributions . the greatest difference in the cumulative distribution functions ( cdfs ) of the observed and modelled distributions occur at the boundaries of the cdfs , i.e. in the least sensitive region for the k - s test ( press et al 1992 ) . we thus decided to use the following modified @xmath55 test . for each parent distribution 10000 model samples each containing 134 systems were generated . ; ritter & kolb 1998 , internal update june 2001 , as of july 2002 the number of systems in this period range is now 152 though this does alter the values given by the @xmath55 test , the trends and hence the results remain unaltered ] each sample was tested against the model parent distribution using a @xmath55 test , with 1 , 2 and 4 minute bins . this range bridges the need for good resolution and significance of the @xmath55 test which requires a minimum number of cvs per bin . the observed period distribution was tested against the model parent distribution also , giving the reduced @xmath55 value @xmath56 . the fraction @xmath22 of generated samples with a reduced @xmath55 value less than @xmath56 was used as a measure of the significance level of rejecting the hypothesis that the observed distribution is drawn from the parent distribution . in the following we quote the rejection probability pr=@xmath22 . kolb & baraffe ( 1999 ) noted that the observed distribution of non - magnetic cvs ( figure [ fig : mnmcvs ] , middle frame ) , and the observed distribution of magnetic cvs ( figure [ fig : mnmcvs ] , lower frame ) show no significant difference below the period gap . to test and quantify this we compared these distributions for @xmath57 min , giving a reduced @xmath55 probability of 0.1213 . hence we can not rule out that the distributions are drawn from the same underlying parent distribution . this is borne out by the results of comparing both distributions with a parent distribution that is flat in @xmath7 ( see also table [ tab : tabcomb ] , entries f and g ) which give similar rejection probabilities ( pr=0.709 and pr=0.781 , respectively ) . we thus find no significant difference between the two distributions . in the following we therefore test models against the combined magnetic and non - magnetic distribution of observed systems . the lack of any distinct features in the combined observed period distribution ( figure [ fig : mnmcvs ] , upper frame ) does indeed suggest an essentially flat distribution for the underlying parent distribution . the flat distribution gives pr = 0.552 ( for the 1 minute bin width , see table [ tab : tabcomb ] ) . we use this value as a benchmark for the models discussed below . [ cols="^,^,^,^,^,^,^ " , ] king , schenker & hameury ( 2002 ) constructed a ( nearly ) flat period distribution by superimposing individual idealized pdfs with different bounce periods @xmath58 according to a suitably tailored weighting . for the double box - shaped idealised pdfs modelled on the pdf shown in our figure [ combined ] ( lower frame ) the required weighting is @xmath59 $ ] ( @xmath60 is the observed minimum period ) . this weighting function effectively mirrors the shape of the sharply peaked individual pdfs . king et al . ( 2002 ) found that the range @xmath61 is sufficient to wash out the period spike . it is clear that this procedure involves a certain degree of fine - tuning for @xmath62 if the shape of the input pdf is given . such a fine - tuning must surprise as the two functions involved presumably represent two very different physical effects . we applied the weighting @xmath62 quoted in king et al . ( 2002 ) to our non - idealized model pdfs that involve the caml efficiency and the deformation factor as a means to vary @xmath58 . the weighting produced a marginally worse fit ( @xmath63 versus @xmath64 ; 1 minute binning ) for the caml pdfs compared to the parent population based on a flat caml efficiency spectrum we discussed earlier . in part this is due to the fact that the upper limit on @xmath31 does not allow a big enough range of @xmath58 . in the case of the deformation factor pdfs the fit marginally improved ( @xmath65 versus @xmath66 ; 1 minute binning , @xmath67 ) . it is possible to optimise the fit by adding systems with deformation factors up to 1.42 , and by using the weighting @xmath68 $ ] , but this still gives the fairly large value @xmath69 ( see also figure [ fig : kings ] ) . however , such a parent population is inconsistent with the observed distribution for longer periods . as can be seen from figure [ fig : barraffe ] systems that are subject to larger deformation factors would evolve into the period gap , hence the gap would be populated in this model . for completeness we show in figure [ fig : grsum1 ] the result of the superposition suggested by king , schenker & hameury ( 2002 ) if realistic rather than idealised pdfs are used . this model assumes additional systemic am losses ( @xmath70 ; no caml , no deformation factor , @xmath71 ) as the control parameter for varying @xmath58 , and the weighting as in king et al . the pronounced feature just above 2 hrs orbital period is the result of the adiabatic reaction of the donor stars at turn - on of mass transfer ( see e.g. ritter & kolb 1992 ) . such a feature is absent in the observed distribution . if deformation effects are taken into account the additional am losses required to wash out the @xmath9 spike would cover a similar range but at a smaller magnitude . the resulting period distribution would be similar to the one shown in figure [ fig : grsum1 ] we have investigated mechanisms that could increase the bounce period for cvs from the canonical theoretical value @xmath0 min to the observed value @xmath72 min , and ways to wash out the theoretically predicted accumulation of systems near the minimum period ( the period spike ) . unlike king , schenker & hameury 2002 we focussed on effects other than increased systemic angular momentum ( am ) losses , i.e. we assume that gravitational radiation is the only systemic sink of orbital am . we find that even a maximal efficient consequential am loss ( caml ) mechanism can not increase the bounce period sufficiently . as the real cv population is likely to comprise systems with a range of caml efficiencies we would in any case expect to have a distribution of systems down to @xmath0 min , rather than the observed sharp cut - off . we considered donor stars that are `` bloated '' due to intrinsic effects , such as the tidal deformation found in 3-dim . sph simulations of roche - lobe filling stars . an implausibly large deformation factor of around 1.18 is needed to obtain a bounce period of @xmath45 min . a possible alternative identification of @xmath9 as an age limit rather than a period bounce ( king & schenker 2002 ) would limit the donor mass in any cv in a cv population dominated by hydrogen rich , unevolved systems to @xmath73 . any system with donor mass much less than this would either have an orbital period less than 78 minutes or would have already evolved beyond the period minimum . there are indeed systems with suspected @xmath74 ; good candidates are wz sge ( @xmath75 ; patterson et al 1998 ) and oy car ( @xmath76 ; pratt et al . 1999 ) . it is also possible that systems die or fade before reaching the period bounce , and hence become undetectable as cvs . the fact that the very different groups of non - magnetic and magnetic cvs show almost identical values of @xmath9 ( see figure [ fig : mnmcvs ] ) strongly suggests that the physical cause for the potential fading would have to be rooted in the donor stars or the evolution rather than the accretion physics or emission properties of the systems . even if the bounce period problem is ignored we find in all synthesized model populations ( except for the age limit model ) a pronounced remaining feature due to the accumulation of systems near the bounce . we employ a modified @xmath55 test to measure the `` goodness '' of fit against the observed sample . an f - test ( press et al 1992 ) was also applied to the majority of @xmath77 models and the same general trends observed . none of our synthesised model populations fits as well as the distribution which is simply flat in orbital period ( rejection probability @xmath78 ) . only models where brighter systems carry a far greater weight than expected in a simple magnitude limited sample ( selection factor @xmath79 with @xmath80 rather than @xmath81 ) achieve similar values for @xmath82 . however , most of our models with @xmath83 canot be rejected unambiguously on the basis of this test . models designed to `` wash out '' the period spike by introducing a large spread of the caml efficiency do generally better than population models based on donor stars that are subject to a large spread of intrinsic deformation factors . for all models the rejection probability decreases if the full wd mass spectrum is taken into account , as this introduces an additional spread in the bounce period . model populations where all cvs form at long orbital periods ( chiefly above the period gap ) give a much better fit than models that include newborn cvs with small donor mass . adding these systems to the population introduces a general increase of the orbital period distribution towards short periods , thus making the period spike more pronounced . this suggests that most cvs must have formed at long periods and evolved through the period gap to become short - period cvs . this is consistent with independent evidence that cv secondary stars are somewhat evolved ( baraffe & kolb 2000 ; schenker et al . 2002 ; thorstensen et al 2002 ) . recently , king , schenker & hameury ( 2002 ) constructed a flat orbital period distribution by superimposing idealised pdfs that describe subpopulations of cvs with a fixed initial donor mass and initial wd mass , but different bounce periods . this superposition required a strongly declining number of systems with increasing bounce periods . we repeated this experiment with a realistic pdf , but failed to obtain a markedly improved fit . in conclusion , we find that the period minimum problem and the period spike problem remain an open issue . it is possible to construct cv model populations where the period spike is washed out sufficiently so that it can not be ruled out unambiguously on the basis of an objective statisticial test against the observed cv period distribution . we thank graham wynn , andrew king and isabelle baraffe for useful discussions . andrew conway and chris jones who gave advice on the statistical analysis . we also thank andrew norton for a critical reading of the paper and the referee jean - marie hameury for useful comments .

we investigate if consequential angular momentum losses ( caml ) or an intrinsic deformation of the donor star in cvs could increase the cv bounce period from the canonical theoretical value @xmath0 min to the observed value @xmath1 min , and if a variation of these effects in a cv population could wash out the theoretically predicted accumulation of systems near the minimum period ( the period spike ) . we are able to construct suitably mixed cv model populations that a statisticial test can not rule out as the parent population of the observed cv sample . however , the goodness of fit is never convincing , and always slightly worse than for a simple , flat period distribution . generally , the goodness of fit is much improved if all cvs are assumed to form at long orbital periods . the weighting suggested by king , schenker & hameury ( 2002 ) does not constitute an improvment if a realistically shaped input period distribution is used . binaries : close stars : evolution stars : mass - loss novae , cataclysmic variables .

arxiv

stephan s quintet ( hereafter sq ) is a strongly interacting compact group which has produced a highly disturbed intragroup medium ( igm ) @xcite through a complex sequence of interactions and harrassment @xcite . this interplay has produced a large - scale intergalactic shock - wave , first observed as a narrow filament in the radio continuum @xcite , and subsequently detected in the x - ray @xcite . the high - velocity ( @xmath61000 km s@xmath13 ) collision of an intruder galaxy , ngc 7318b , with the intergalactic medium of the group @xcite is believed to be responsible for the shock - heating of the x - ray emitting gas . optical emission line ratios and observed broad linewidths provide evidence that the region is powered by strong shocks and not star formation ( xu et al . 2003 ; duc et al . 2010 , in preparation ) . the main elements of stephan s quintet are shown in figure 1 . central to the system is the primary shock , as defined by the 20 cm radio continuum emission . ngc 7318b , the intruder galaxy , lies to the west of the shock , while the large seyfert 2 galaxy @xcite , lies to the east . other members of the group are also indicated . the peculiar extranuclear star formation region , named sq - a @xcite , lies at the extreme northern end of the main shock wave . ngc 7318a ( west of ngc 7318b ) is also a strongly interacting group member , with ngc 7317 further away from the core . the unexpected discovery of extremely powerful , pure - rotational h@xmath9 line emission from the center of the shock @xcite , using the _ spitzer space telescope _ @xcite , has sparked intense interest in the sq system . warm molecular hydrogen emission was found with a line luminosity exceeding the x - ray luminosity from the shock . the mid - infrared ( mir ) h@xmath9 linewidth was resolved ( @xmath14 870 km s@xmath13 ) suggesting that the h@xmath9-emitting clouds carry a large bulk - kinetic energy , tapping a large percentage of the energy available in the shock . a recent model of sq , involving the collision between two inhomogeneous gas flows , describes h@xmath9 formation out of the multiphase , shocked gas , and an efficient cooling channel for high - speed shocks as an alternative to x - ray emission @xcite since the sq detection , several other systems exhibiting similarly strong h@xmath9 emission have been discovered . @xcite find that a large - subset of the local 3cr radio galaxies have extremely dominant mir rotational h@xmath9 lines , often seen against a very weak thermal continuum . the low agn and star formation power are insufficient to drive the mir h@xmath5 emission . mechanical heating driven by the radio jet interaction with the host galaxy ism is the favoured mechanism . in addition , total h@xmath9 luminosities in the range @xmath15 ergs s@xmath13 have been detected in some central cluster galaxies out to @xmath16 ( * ? ? ? * g. de messieres - university of virginia , private communication ) and in filaments in clusters @xcite . the study of nearby prototypes may provide valuable insight into the nature of these more distant systems . the large scale ( @xmath6 30 kpc ) of the sq shock is well - suited for such a study . in this paper i , we extend the single pointing observations of @xcite to full spectral maps of sq using the irs instrument on the _ spitzer _ space telescope , hereafter _ spitzer_. in paper ii , we will present detailed 2-dimensional excitation maps of the h@xmath5 emission across the face of the x - ray emitting shock , and compare them with models . in addition , several other papers are being prepared by our team which will discuss the relationship between the uv / x - ray emission and emission from dust . in section 2 , we present our observations and data reduction methods . in section 3 we discuss the mapping results for various detected lines . in sections 4 , 5 and 6 we present our discussion on the properties of the system and in section 7 the implication for galaxy formation . in section 8 we present our conclusions . additional material is included as appendices , with a discussion on ngc 7319 in appendix a , and a reanalysis of the high resolution mir spectrum of the shock , as well as renanalysis of the x - ray data presented in appendix b and c , respectively . throughout this paper , we assume a systemic velocity of @xmath17 for the group , corresponding to a distance of 94 mpc with @xmath18 . mid - infrared spectroscopy of the shock region in sq was obtained using the irs instrument @xcite onboard _ spitzer_. observations were done in low - resolution mapping mode , using the short - low ( @xmath19 ; @xmath20 ) and long - low ( @xmath21 ; @xmath22 ) modules and taken on january 11 2008 and december 10 2007 , respectively . figure 1 displays the outline of the areas observed superimposed on a composite image of the group . the sl spectral mapping consists of two separate , partially overlapping maps , centered north and south on the x - ray emission associated with the shock . the map was constructed with 23 steps of 2.8(0.75 @xmath2 slit width ) perpendicular to the slit and one parallel step of 7.2 . observations consisted of 60s integrations with 5 cycles per step . the ll module was used to map an area of @xmath23 using 21 steps of 8.0 ( 0.75 @xmath2 slit width ) perpendicular to the slit and a parallel step of 24.0 . an integration time of 120s was used with 3 cycles per step . primary data reductions were done by the _ spitzer _ science center ( ssc ) pipeline , version s17.0.4 and s17.2.0 for sl and ll respectively , which performs standard reductions such as ramp fitting , dark current subtraction and flat - fielding . background subtraction for ll data was performed by subtracting dedicated off - source observations , with the same observing mode , taken shortly after the mapping sequence . in the case of sl , where for scheduling reasons the dedicated `` off '' observations were too far away in time to be optimal , backgrounds were generated from observations at the periphery of the map that contain no spectral line signatures . examination of the pipeline products showed that the stray - light - corrected images ( that account for potential spillover from the peak - up arrays onto the sl1/2 spectral apertures ) contained wavelength - dependent over - corrections in some of the images , seemingly due to a high background of cosmic rays during the sl portion of the mapping . to negate this effect , the alternative flat - fielded images ( also available from the science pipeline ) that are uncorrected for crosstalk and straylight removal were used . it was determined that after background subtraction , any uncorrected stray light in the sl spectral extraction areas was unmeasurable at the @xmath24% level . thus stray - light correction was unneccesary , and the resulting basic calibrated data ( bcd ) images used were of better quality than the standard bcds . for all modules , individual frames for each pointing were median - combined and obvious `` bad '' pixels were replaced using customised software that allows for manual `` average '' replacement . the spectra were assembled into spectral cubes for each module using the software tool , cubism @xcite . further bad pixel removal was performed within cubism . spectral maps were generated by making continuum maps on either side of a feature and subtracting the averaged continuum map from the relevant emission line map . one dimensional spectra were further extracted from the data cubes using matched apertures . broad - band images at 16 and 24@xmath25 m were obtained with the _ spitzer _ irs blue peak - up imager ( pui ) and the mips instrument . the pui was obtained in a 5 x 5 map with 3 cycles of 30s duration each on 2007 december 10 . the mips instrument @xcite on _ spitzer _ obtained 24 imaging of sq on 2008 july 29 , achieving a spatial resolution of @xmath66 . primary data reduction was done by the _ spitzer _ science center ( ssc ) science pipeline ( version s18.0.2 ) run through the mopex software , and for the mips image a smooth 2d polynomial background was further removed to correct for a large - scale background gradient . _ spitzer _ irac 3.6 , 4.5 , 5.8 and 8.0 data of sq ( p.i . houck ) were obtained from the ssc archive ; these were reduced using science pipeline version s18.0.2 . the final mosaics have a pixel scale of 0.61 . the pure rotational transitions of molecular hydrogen can be excited by several mechanisms . these include fuv ( far ultraviolet ) induced pumping , and possible additional collisional heating , of the h@xmath5 in photodissociation regions associated with star formation ( e. g. black & van dishoeck 1987 , hollenbach & tielens 1997 ) , hard x - rays penetrating and heating regions within molecular clouds , which in turn excite h@xmath5 via collisions with electrons or hydrogen atoms ( lepp & mccray 1983 ; draine & woods 1992 ) and finally collisional excitation of h@xmath5 due to acceleration produced by shocks ( e.g. shull & hollenbach 1978 ) . the pure rotational mir line ratios are not especially good diagnostics for distinguishing between these mechanisms since all three mechanisms discussed can lead to well thermalized level distributions of lower - level rotational states . the rotational h@xmath5 emission lines do , however , allow us to trace gas at different temperatures and compare with model predictions ( this will be the main emphasis of paper ii ) . higher level transitions 0 - 0 s(3)-s(5 ) tend to trace warmer gas , whereas the s(0 ) and s(1 ) lines are sensitive to the `` coolest '' warm h@xmath5 . although the line ratios themselves can not be used directly as shock diagnostics , in stephan s quintet the distribution of large - scale x - ray and radio emission , plus optical emission line diagnostics , provide strong evidence that the giant filament seen in figure 1 is the result of a strong shock . in appleton et al . ( 2006 ) this fact was used to reveal the surprising association of detected h@xmath5 emission with the shock . however , in this paper we can make a more definitive association of the emission with the shock by means of spectral maps . the spectral cubes were used to extract maps of all the pure rotational emission lines of molecular hydrogen that were detected , namely the 0 - 0 s(0)28.22 , s(1)17.03 , s(2)12.28 , s(3)9.66 , s(4)8.03 and s(5)6.91 lines . specific intensity contour maps of these lines are presented in figure 2 overlaid on a @xmath26-band image of the region from @xcite . the s(0 ) and s(1 ) lines were mapped by the ll modules , while s(2 ) - s(5 ) transitions were mapped by the sl modules of the irs . as indicated in figure 1 , the sl observations were concentrated on the main shock to provide high signal to noise ( s / n ) measurements there . as a result , these maps do not fully cover sq - a or ngc 7319 . we note that the s(4 ) line at 8.03 ( fig . 2e ) is faint , and also suffers from contamination from the pah bands at 7.7 and 8.6 . the contours indicate powerful , widespread emission running north - south along the shock ridge ( see fig . 1 ) . in addition we detect strong emission from the star forming region , sq - a , as well as associated with ngc 7319 . we discuss further details of the agn - like mir emission lines from ngc 7319 in appendix a. figures 2a - d reveal a new h@xmath5 structure running eastward from the `` main '' shock ridge . in what follows , we refer to this feature as the `` bridge '' . this structure is observed faintly in the _ chandra _ @xcite and xmm @xcite x - ray images and detected as faint h@xmath27 emission by @xcite , but is not strong in radio continuum images . as is evident in figure 2 , there is distinct variation in the distribution of the warm h@xmath5 emission . the brightest 0 - 0 s(0 ) emission ( fig . 2a ) appears to be concentrated towards the north of the shock , whereas the 0 - 0 s(1 ) transition emission ( fig . 2b ) appears more concentrated towards the center . the s(0 ) and s(1 ) maps demonstrate that the h@xmath5 in the bridge terminates in a large clump a few arcsecs west of the nucleus of the seyfert 2 galaxy ngc 7319 , and in a small detour to the north ( especially in the s(1 ) map which has the highest s / n ) . the s(2 ) through s(5 ) lines clearly show that the warm h@xmath5 emission breaks into clumps in the shock . despite the limited coverage compared to the ll mapping , the base of the `` bridge '' is visible and sq - a is partially covered . sq - a is fully covered by the sl2 module ( because of fortuitous `` off observation '' coverage ) and hence the s(5 ) emission line reveals that in sq - a the h@xmath5 emission is also clumpy . the molecular - line maps provide considerable information about the excitation of the h@xmath5 along and across the x - ray shock , but these discussions will be deferred to a full 2-d modelling of the h@xmath5 excitation in paper ii ( appleton et al . , in preparation ) . instead we shall limit ourselves to global properties of the h@xmath5 here . in section [ spec ] we shall present spectra of some selected regions of the emission and discuss a global excitation diagram for the shock . to further demonstrate the close connection between the h@xmath5 emission and the main global shock - wave in sq , we now consider the distribution of warm h@xmath5 in relation to the x - ray and radio emission . figure [ fig : xray_rad]a and b show the s(0 ) and s(3 ) contours overlaid on an xmm - newton x - ray image @xcite of stephan s quintet . the warm molecular hydrogen is distributed along the length of the main north - south ( ns ) x - ray shock ridge and along the `` bridge '' , demonstrating the remarkable projected coexistence of of hot x - ray plasma ( 10@xmath28 @xmath29 t @xmath29 10@xmath30 k ) and warm h@xmath5 ( 10@xmath31 @xmath29 t@xmath29 10@xmath32 k ) . although the h@xmath5 appears to follow the x - ray , there are subtle differences . the cooler s(0 ) line has emission concentrated to the north and follows the x - ray less closely compared to the warmer s(3 ) line . the s(3 ) line exhibits a clear correspondence to the x - ray , notably at the center of the shock , where we find peaks at both wavelengths . thus the region of greatest shock heating , as traced by the stronger x - ray emission , appears to correspond to the higher - j h@xmath5 transitions , perhaps implying a causal connection . the intergalactic star formation region sq - a , is essentially absent in x - ray emission , as seen in figure [ fig : xray_rad]a , but is strongly detected in h@xmath9 . we find a similar picture in the radio continuum ( fig . [ fig : xray_rad]c & d ) with the s(0 ) line demonstrating correspondence with the main shock , but dominated by emission in the north where we observe less powerful radio emission . the s(3 ) ( and s(2)see fig . 2c ) line presents a much tighter correlation with regions of the shock that are more radio luminous than the lower - j transitions . the radio emission is quite likely sensitive to the most compressed regions of the shock where cosmic ray particles are accelerated more strongly @xcite , whereas the brightest x - ray patches are likely due to the fastest regions of the shock @xcite . as already mentioned the `` bridge '' emission is detected only faintly in the x - ray and is weak or absent at radio continuum wavelengths . this noticeable difference compared to the main shock likely implies that the conditions that give rise to strong synchrotron emission in the main shock are absent in the bridge . emission from fine - structure lines provide key diagnostics that trace the interplay between the various constituents of the shocked interstellar medium . in @xcite the spectra were limited to the very core of the shock and only weak emission was detected from all but two metal lines , namely @xmath10neii@xmath11$]12.81and @xmath10siii@xmath11$]34.82 ( this data has been reanalysed and is presented in appendix b ) . in this section we discuss the spatial distribution of emission from the @xmath10feii@xmath11$]25.99 , @xmath10oiv@xmath11$]25.89 , @xmath10siii@xmath11$]33.48 , @xmath10siii@xmath11$]34.82 , @xmath10neii@xmath11$]12.81 and @xmath10neiii@xmath11$]15.56 lines . in figure [ fig : fs ] we present the specific intensity contours of the @xmath10feii@xmath11$]25.99 and @xmath10oiv@xmath11$]25.89 blend , @xmath10siii@xmath11$]33.48 and @xmath10siii@xmath11$]34.82 emission lines . given the low spectral resolution of the sl and ll modules of _ spitzer _ , we can not distinguish between emission from @xmath10fe ii@xmath11$]25.99 and @xmath10oiv@xmath11$]25.89 . however , except in the direction of the seyfert ii galaxy ngc 7319 , the emission near 26 seen in figure [ fig : fs]a is likely to be pure @xmath10feii@xmath11 $ ] with little contamination from @xmath10oiv@xmath11 $ ] as there is no evidence from the spectra of high - excitation emission from the intragroup medium in sq . for example , @xmath10oiv@xmath11 $ ] has an excitation potential of 56 ev ( compared to 7.9 ev of @xmath10feii@xmath11 $ ] ) , and yet @xcite have shown that the @xmath10neiii@xmath11$]/@xmath10neii@xmath11 $ ] ratio is low suggesting low - excitation conditions for the ions in the shock , further supported in section [ em ] . assuming that , apart from towards ngc 7319 , @xmath10feii@xmath11 $ ] dominates the @xmath10feii@xmath11$]+@xmath10oiv@xmath11 $ ] complex , we detect faint emission from @xmath10feii@xmath11$]25.99 at the location of the center of the shock ( as defined by the x - ray `` hotspot '' in fig . [ fig : xray_rad]a ) . the energetic requirements for shocks to produce strong @xmath10feii@xmath11 $ ] emission are usually present in @xmath33 shocks while the ion abundance in @xmath34 shocks are low in comparison @xcite . we will discuss the production of @xmath10feii@xmath11 $ ] in the shock in section 6.2 . the @xmath10siii@xmath11$]33.48 distribution is presented in figure [ fig : fs]b . this fine structure line acts as a strong tracer of hii regions @xcite and we observe emission from sq - a and from other regions of star formation in the south ( see section [ sf ] ) . in the primary shock region the distributions of @xmath10feii@xmath11 $ ] and @xmath10siii@xmath11 $ ] are anti - correlated . in strong contrast to the weak @xmath10feii@xmath11 $ ] emission , there is copious @xmath10siii@xmath11$]34.82 emission ( fig . [ fig : fs]c ) , which follows the s(1 ) distribution closely with respect to the primary shock , as mapped by x - ray emission ( fig . [ fig : fs]d ) . although @xmath10siii@xmath11 $ ] is commonly found in normal hii regions , we will demonstrate below that , apart from in sq - a , the strong silicon emission does not correlate with regions of strong pah emission ( tracing star formation ) in sq , but instead closely follows the h@xmath5 and x - ray ( shock ) distributions . @xmath10siii@xmath11 $ ] acts as an efficient coolant of x - ray - irradiated gas and is predicted to be one of the top four cooling lines under these circumstances @xcite . we will discuss the excitation of @xmath10siii@xmath11 $ ] in the shock in section 6.2 . @xmath10neii@xmath11$]12.81 ( with an ionisation potential of 21.6 ev ) is also represented in the shock , as shown in figure [ fig : ne]a , although it is also emitted from some hii regions associated with star formation , such as sq - a and the star - forming region south of ngc 7318b ( see section [ sf ] ) . the @xmath10neiii@xmath11$]15.56 contours ( with an ionisation potential of 41 ev ) are shown in figure [ fig : ne]b and are associated with excitation in the shock , as well as from regions of star formation ( see section [ sf ] ) . the higher ionisation line of @xmath10neiii@xmath11 $ ] is much weaker in the shock ridge , but regions of emission correspond closely to peaks seen for the h@xmath5 s(3 ) line . the @xmath10neii@xmath11 $ ] emission , however , exhibits clearly extended emission with regions of greatest luminosity matching those seen for the h@xmath5 s(3 ) line . the @xmath10neii@xmath11 $ ] emission suggests excitation from the shock , with @xmath10neiii@xmath11 $ ] found at the location of the center of the shock , similar to what is observed for @xmath10feii@xmath11 $ ] . previous observations and spectroscopy of sq @xcite have determined that there are some regions of star formation associated with the spiral arms of the intruder galaxy ngc 7318b . we discuss in this section how these regions , which have a different spatial distribution from the shocked gas , are correlated with the pah emission we detect in the irs spectra . in figure [ fig : dust]a there is a strong correlation between the 11.3 pah distribution ( from the irs cube ) superimposed on a near - uv image from _ galex _ @xcite , which maps the uv emission from hot stars associated with weak star formation from the system . this correlation suggests that the pah molecules are excited by star formation . a similar close correlation is shown in figure [ fig : dust]b where we overlay the 11.3 contours on the irac 8 band , which is dominated by the 7.7 and 8.6 @xmath25 m feature . it is noticeable , however , that regions with strong 11.3 emission in the shock , do not appear similarly strong at 8 . this point will be addressed in section 6.4 . we compare the 11.3 pah map to the distribution of warm dust in figure [ fig : dust]c , using the mips 24@xmath25 m map of sq . again , it is clear that there is a good correlation between the pah emission and the thermal dust , most of which seems only poorly correlated with the shock ridge . the lack of conspicuous star formation in the ridge was also observed by @xcite . the main point we emphasize here is that the dust , pah and uv emission appears to be associated with previously known star formation regions and no additional star formation is observed in the shock ; this can be seen in figure [ fig : dust]d showing the h@xmath5 0 - 0 s(1 ) emission overlaid on an irac 8 image . there is little correspondence between the h@xmath5 emission in the shock and 8@xmath25 m ( hot dust plus pah ) image . this is important because it implies that there is very little triggered star formation in the molecular gas associated with the shock - excited h@xmath5 . figure [ fig : dust]c demonstrates that there is only faint dust emission at 24@xmath25 m from the shock ridge . the presence of dust in the shock is required in the model of guillard et al . ( 2009 ) to explain the formation of h@xmath5 behind the shock and we do observe evidence of depletion onto dust grains ( see section 6.2 ) . thus the faint 24 emission could be the result of destruction of very small grains ( vsgs ) , with only larger grains surviving , or indicate that the grains are cold and radiating more strongly at longer wavelengths , where _ spitzer _ has the least spatial resolution . a more detailed description of the dust and faint pah emission in the sq group ( including results from mips 70 imaging ) is discussed in separate papers ( guillard et al . 2010 . ; natale et al . 2009 , in preparation ) . guillard et al . ( 2010 ) show that the ir emission in the shock is faint due to dust being heated by a relatively low intensity uv radiation field and determine a galactic pah / vsg abundance ratio in this region . a more complete understanding of the likely existence of cool dust in the shock will require higher angular resolution and a broader wavelength coverage than that achieved by _ spitzer_. despite the faintness of emission from the main shock , the 24@xmath25 m map presents a new result , which was less obvious in previous studies , namely that the dominant regions of star formation in sq lie not in the galaxies themselves , but in two strikingly powerful , almost symmetrically disposed regions at either end of the shock . the region to the north is the well studied sq - a , but the region to the south ( which we refer to as 7318b - south ) is also very powerful and both regions lie at the ends of the shock , as defined by the h@xmath5 distribution . this may not be a coincidence , and we will discuss this further in section 6.1 . the @xmath10neii@xmath11$]12.81 , @xmath10neiii@xmath11$]15.56 and @xmath10siii@xmath11$]33.48 fine - structure lines , as mentioned above , are also tracers of star formation in sq as these lines are often associated with hii regions . the @xmath10neii@xmath11$]12.81 emission appears to follow _ both _ the h@xmath5 and the star formation regions ( see fig . [ fig : ne]a ) , appearing more extended in the south than the corresponding h@xmath5 emission and flaring out where star formation regions , especially 7318b - south , are observed optically , and through pah emission ( see fig . [ fig : dust ] ) . the warm molecular hydrogen in stephan s quintet follows the x - ray distribution in the main shock and in the `` bridge '' structure . this might suggest that the molecular hydrogen is excited directly by the x - ray heating . however , we will show that the h@xmath5 emission exceeds by at least a factor of 3 the x - ray luminosity from the various shocked filaments , thus ruling out direct x - ray excitation from the shock . to measure the strength of the h@xmath5 emission , we extract spectra from various rectangular sub - areas of sq which are defined in figure [ fig : ext ] . the spectra were extracted from cubism cubes built from each irs module , and joined to make a continous spectrum no scaling was necessary to join the spectra . figures [ fig : ext ] a , b and c indicate the spectral extraction regions of the main ns shock , a sub - region of the shock and a characteristic part of the `` bridge '' . the shock sub - region is chosen to be just north of the center of the shock , avoiding regions contaminated by star formation in the intruder . spectra for these extractions are shown in figure [ fig : spectra ] a , b and c. all three spectra , share the common property that they are dominated by molecular hydrogen emission . the shock sub - region ( fig . [ fig : spectra]b ) , unlike figure [ fig : spectra]a , is less contaminated by the star forming regions discussed in the previous section . the mir continuum of the main shock appears stronger than the shock sub - region indicating stronger emission from warm dust ; this is likely the result of contamination from star - forming regions in the main shock extraction . the bridge exhibits a similarly weak continuum emission compared to the shock sub - region . figure [ fig : spectra]b includes photometry from the irac bands , 16 peak - up image ( pui ) and mips-24 image superimposed on the irs spectrum . these are useful to probe conditions in the shock , in particular star formation ( see section 6.4 ) . stellar light in the extraction area , from an extended spiral arm of ngc 7318b , produces contamination of the shock spectrum , visible as continuum emission shortwards of 6 , in both figure 8a and b. there exists a striking similarity between the mid - ir spectrum of all three regions , showing powerful h@xmath5 lines and low excitation weak emission from fine structure lines . also the pah emission observed in the spectra of the shock regions , and in the region of the `` bridge '' in the irac 8 image ( fig . [ fig : dust]b ) , appears weak . this confirms that these properties , observed in the @xcite observations of the shock core , extend to both the full extent of the main shock and the `` bridge '' . this , and the fact that the `` bridge '' has similar x - ray properties to the main shock ( see later ) suggests that the bridge is a `` scaled - down '' version of the main shock . the weaker radio continuum emission at this location is significant . one possibility , that the `` bridge '' is older than the main shock , and the cosmic rays compressed in it have diffused away , will be discussed further in paper ii.thus the new irs observations seem to suggest that more than one large - scale group - wide shock is present in the group . this could be the result of previous tidal interactions and imply multiple shock heating events have taken place in sq , consistent with what is seen in the x - ray @xcite . figures [ fig : ext]d and [ fig : spectra]d present the extraction region and spectrum of sq - a , the extragalactic star - forming region . in this case , although h@xmath5 lines are still strong , a rising continuum and an increase in the strength of the metal lines relative to the h@xmath5 is consistent with a spectrum that is increasingly dominated by star formation a result which is already known from previous optical observations @xcite . line fluxes for all the h@xmath5 and metal lines in the spectra discussed above are presented in table [ tableh2fluxes ] and table [ tablemetalfluxes ] , respectively . we estimate the luminosity emitted from the h@xmath5 lines in the main aperture shown in figure [ fig : spectra]a . the emission from the 0 - 0 s(1 ) line alone can be calculated from table [ tableh2fluxes ] ( for d = 94 mpc ) to be 2.3 @xmath210@xmath35 ergs@xmath13 . summing the emission measured in the observed lines for the main shock ( 0 - 0 s(0 ) through s(5 ) lines ) and including an extra 28% emission from unobserved lines ( see model fit to excitation diagram below ) , yields a total h@xmath5 line luminosity from the main shock of 9.7 @xmath210@xmath35 ergs@xmath13 . this phenomenal power in the molecular hydrogen lines dwarfs by a factor of ten the next brightest mid - ir line , which is @xmath10siii@xmath11$]34.82 with a line luminosity of l@xmath36 = 0.85 @xmath210@xmath35 ergs@xmath13 . figure [ fig : excite ] presents the excitation diagram of the low - j 0 - 0 h@xmath5 transitions for the main shock extraction . the points are well fit by a model including three temperature components ( t@xmath37 = 158@xmath38k , t@xmath9 = 412@xmath39k , and t@xmath40 = 1500@xmath41k ) . it is likely that in reality , many different temperature components are present in the shock , and the three - temperature fit is only an approximation . however , it does allow us to provide an estimate of the total mass of warm h@xmath5 of 5.0@xmath4210@xmath43m@xmath44 . temperature t@xmath9 is more uncertain than formally represented by the fit because it depends on the value of the s(4 ) flux , which may be systematically too low due to pah contamination ( see fig . [ fig : excite ] ) . in paper ii we will present a more complete two - dimensional map of the excitation of the h@xmath5 in sq and explore variations in the shape of the excitation diagram along and across the shock in more detail . our observations have shown that h@xmath5 is the dominant line coolant in the mir . however , how does it compare with the most important coolant in high - speed shocks namely the x - ray emission ? @xcite suggested that the h@xmath5 emission was stronger than the x - ray emission at the shock center . this can now be evaluated over much of the inner sq group . we present a complete reanalysis of the xmm - newton observations of sq using the latest calibrations ( see appendix c for full details ) in order to determine the fluxes and luminosities of the x - ray emission to match our spectral extractions . the results indicate the striking dominance of the h@xmath5 line luminosities compared with the x - ray emission from the same regions . for the main shock , the x - ray `` bolometric '' flux of l@xmath45 = 2.8 @xmath2 10@xmath46erg s@xmath13@xmath47 corresponds to 2.95 @xmath2 10@xmath35 erg s@xmath13 , or l(h@xmath5)/l@xmath45 = 2.9 . this is a lower limit since we have not attempted to remove the contribution to the main shock aperture of an extended group - wide x - ray component upon which the emission from the shock lies . therefore it is likely that the h@xmath5 line luminosity dominates over the main shock x - ray gas by a factor @xmath48 3 . similar calculations can be done for the other regions for which h@xmath5 spectra have been extracted . for example , in the `` bridge '' region , which we have already indicated has many of the same characteristics as the main shock , we find l(h@xmath5)/l@xmath45 = 2.5 . these values demonstrate that throughout the extended regions of sq , the molecular hydrogen cooling pathway dominates over the x - ray in this shocked system . this is a very significant result , upturning the traditional view that x - ray emission always dominates cooling in the later stages of evolution in compact groups of galaxies . the fine - structure flux ratios ( see table [ tablemetalfluxes ] ) can be used to probe the conditions within the extracted regions of sq . the @xmath10siii@xmath11$]34.82/@xmath10siii@xmath11$]33.48 ratio provides an indication of the sources of excitation within the system . as mentioned previously , @xmath10siii@xmath11 $ ] is mainly a tracer of hii regions , whereas enhanced @xmath10siii@xmath11 $ ] emission can be generated via several mechanisms , including thermal excitation by x - rays ( xdrs ) , or in shocks . in the main shock , the @xmath10siii@xmath11$]34.82/@xmath10siii@xmath11$]33.48 ratio of @xmath6 4.59 is high compared to , for example , both normal galaxies ( @xmath6 1.2 ) and agn ( @xmath6 2.9 ) in the sings sample @xcite . however , this large aperture is contaminated by star formation emission from sq - a and the intruder galaxy . a better measure is given by the smaller shock sub - region , where the @xmath10siii@xmath11$]34.82/@xmath10siii@xmath11$]33.48 ratio is @xmath49 . thus it is clear that the @xmath10siii@xmath11 $ ] emission is well outside the normal range of values , even for local well - studied agn . using the upper limit found for @xmath10siii@xmath11$]33.48 in the `` bridge '' structure , we find a ratio of @xmath50 again values well outside the range of normal galaxy disk emission . indeed , these high values are typical of galactic supernova remnants where shock excitation is well determined ( e.g. * ? ? ? * ; * ? ? ? we will argue in the next section that silicon is being ionised in regions experiencing fast shocks @xmath51km@xmath0 and depleted onto dust grains . the @xmath10feii@xmath11$]26.0/@xmath10siii@xmath11$]34.82 ratio for the main shock and sub - region ( @xmath6 0.12 ) is in agreement with values found by @xcite for their sample of snr . for the extragalactic star forming region sq - a , @xmath10siii@xmath11$]34.82/@xmath10siii@xmath11$]33.48 is @xmath61.49 , only slightly higher than the average of @xmath6 1.2 found for star - forming regions in the sings sample @xcite ; another indication that sq - a is dominated by star formation . in star - forming galaxies the @xmath10neiii@xmath11$]15.56/@xmath10neii@xmath11$]12.81 ratio can be used as a measure of the hardness of the radiation field as it is sensitive to the effective temperature of the ionising sources . in the main shock , we find a value of @xmath52 which would be considered typical compared to those found in starburst systems , which range from @xmath53 @xcite , and in supernovae remnants ranging from @xmath54 @xcite . the shock subregion has a ratio of only @xmath55 indicating a lower intensity radiation field north of the shock center . however , it is clear from its spatial distribution relative to the 8 and h@xmath5 emission , that most of the neon is not originating from star formation , and so shocks are an obvious source of excitation . the @xmath10neii@xmath11$]12.81/@xmath10neiii@xmath11$]15.56 ratio can be used to estimate the shock velocity using the mappings shock model library of @xcite . in the main shock this ratio corresponds to shock speeds of between 100 and 300kms@xmath13 ( using preshock densities of @xmath56 @xmath57 and magnetic parameter @xmath58 = 1 and 3.23 the nominal equipartition value ) . we do , however , have contamination from starforming regions in the main shock and can not disentangle this emission from that produced in the shock . to address this we use the @xmath10neii@xmath11$]12.81 and @xmath10neiii@xmath11$]15.56 maps to mask areas associated with star formation ( see section 3.3 ) and determine a lower limit for the @xmath10neii@xmath11$]12.81/@xmath10neiii@xmath11$]15.56 ratio in the shock of @xmath6 4.54 . this value corresponds to a shock velocity of @xmath6 150kms@xmath13 . the average electron density is determined from the @xmath10siii@xmath11$]18.71/@xmath10siii@xmath11$]33.48 ( two lines of the same ionisation state ) ratio . for sq - a we find a ratio of @xmath60.56 , and in the main shock @xmath60.41 . this corresponds to an electron density of @xmath59 @xmath57 @xcite for both , i.e. in the low - density limit for this diagnostic @xcite . our observations have shown that the molecular hydrogen and x - ray emitting plasma appear to follow a similar distribution , and we have ruled out the possibility that this is a consequence of x - rays heating the h@xmath5 , since the h@xmath5 has the dominant luminosity . how then can we explain the similar distributions ? are these results consistent with the hypothesis that the shock is formed where the intruding galaxy ngc 7318b collides with a pre - existing tidal filament of hi drawn out of ngc 7319 in a previous interaction with another group member @xcite ? this basic mechanical picture appears plausible as can be seen in figure [ fig : hi+h2 ] which shows that the h@xmath5 distribution `` fills in '' the gap in the hi tidal tail as observed by the vla @xcite . the implication is that the hi has been converted into both a hot x - ray component and a warm h@xmath5 component by the collision of the intruder with the now missing hi . part of the puzzle of how this high - speed ( @xmath60 @xmath61 kms@xmath13 ) shock can lead to both x - ray and very strong molecular line emission is presented in a model by @xcite . the high - speed collision of ngc 7318b with the hi filament ( assumed to be composed of a multiphase medium ) leads to multiple shocks passing through and compressing denser clumps ( which become dusty nucleation sites for h@xmath5 formation ) as opposed to the lower - density gas , which is shock - heated to x - ray tempertures . the h@xmath5 therefore forms in denser clouds experiencing slower shocks . thus the coexistence of both hot x - ray gas , and cooler molecular material is a natural consequence of the multiphase medium of the pre - shocked material . modeling of the h@xmath5 excitation by @xcite demonstrates that the emission can be reproduced by low velocity ( @xmath62 ) magnetohydrodynamic shocks within the dense ( @xmath63 ) h@xmath5 gas . the denser clouds survive long enough to be heated by turbulence in the hot - gas component , tapping into the large available kinetic energy of the shock . this picture is consistent with both the broad h@xmath5 linewidth ( 870kms@xmath13 ) measured in irs high - resolution spectrometer observations of @xcite , and the velocity center of the warm h@xmath5 ( based on new irs spectral calibrations - see appendix b ) which places the gas at intermediate velocities between the intruder and the group igm . both these measurements are consistent with h@xmath5 being accelerated in a turbulent post - shocked layer . intermediate pre - shock densities and post - shock temperatures result in regions of hi and hii that have cooled , but where the dust content has been destroyed @xcite . pre - existing giant molecular clouds ( gmcs ) which may have been embedded in the hi gas , would be rapidly compressed and collapse quickly , thus forming stars.this mechanism , proposed for sq - a by @xcite , might also apply to 7318b - south . however , if this was the case , it would have to explain why two such gmcs happened to be positioned at the extreme ends of the current shock an unlikely coincidence . more probable , however , is that the geometry of the shock somehow favors the collapse of clouds at the ends of the shock perhaps in regions where the turbulent heating is less efficient . as outlined above , the combination of emission detected in the shock region of sq can be understood in terms of a spectrum of shock velocities . the fastest shock velocities ( @xmath64 km@xmath0 ) are associated with the lowest density pre - shock regions and the post - shock x - ray emitting plasma . these are fast @xmath33 shocks and represent a discontinous change of hydrodynamic variables and are often dissociative @xcite . @xmath34 shocks have a broad transition region such that the transition from pre - shock to post - shock is continuous and are usually non - dissociative @xcite . the lowest velocity shocks associated with the turbulent h@xmath5 emission are @xmath62 @xmath34 shocks @xcite . @xcite find optical emission line ratios consistent with shock models that do not include a radiative precursor @xcite . the @xmath10feii@xmath11$]25.99 emission associated with the shock region in sq is relatively weak , but coincides with the most energetic part of the shock as traced by the x - rays ( fig . [ fig : fs ] ) . we also detect abundant @xmath10siii@xmath11$]34.82 emission associated with the main shock . silicon and iron have very similar first and second ionization potentials . their first ionization potentials ( 7.9 and 8.15 ev for fe and si , respectively ) are lower than that of hydrogen but their second ionization potential is higher ( 16.19 and 16.35 ev ) . the mir @xmath10feii@xmath11 $ ] and @xmath10siii@xmath11 $ ] line emission observed from the sq shock could thus arise from predominantly neutral , as well as ionized gas . we discuss the contribution from the ionized gas using the @xmath10feii@xmath11$](@xmath65m)/@xmath10neii@xmath11$](@xmath66 m ) and @xmath10siii@xmath11$](@xmath67m)/@xmath10neii@xmath11$](@xmath66 m ) line ratios . the high @xmath10neii@xmath11$](@xmath66m)/@xmath10neiii@xmath11$](@xmath68 m ) mid - ir line ratio ( see section 5 ) implies that @xmath10neii@xmath11 $ ] is the dominant ionization state of ne in the sq shock . unlike fe and si , ne is not much depleted on dust @xcite . because the @xmath10neii@xmath11$]@xmath66 m line has a high critical density ( @xmath69@xmath57 , ho and keto 2007 ) , the neon line strength scales with the emission measure of the ionised gas . the optical line emission from the sq shock is discussed in detail in xu et al . the high values of the @xmath10oi@xmath11$](6300 ) and @xmath10nii@xmath11$](6584 ) lines to h@xmath27 line ratios are evidence of shock ionization . the optical @xmath10sii@xmath11$](6716/6731 ) line ratio as well as the mir @xmath10siii@xmath11 $ ] line ratio correspond to the low density limit ( see section 5 ) and comparison with the shock models of allen et al . ( 2008 ) constrain the pre - shock gas density to be about @xmath70@xmath57 or smaller . in figure [ fig : sil_ne ] we indicate the region ( in grey ) corresponding to the observed mir ratios ( using the upper and lower limits determined for the @[email protected]\mu$]m line emission as discussed in section 5 ) . for comparison the expected emission from the shock models of @xcite are displayed for a pre - shock gas density of @xmath70@xmath57 and two values of the magnetic parameter , @xmath58 = 0.5 and 3 . for clarity of the figure , only shock velocities from 100 to 300kms@xmath13 are used and higher velocities shocks which do not match the observed @xmath10neii@xmath11$]/@xmath10neiii@xmath11 $ ] line emission ratio ( see section 5 ) are discarded . the observed @xmath10feii@xmath11$](@xmath65m)/@xmath10neii@xmath11$](@xmath66 m ) and @xmath10siii@xmath11$](@xmath67m)/@xmath10neii@xmath11$](@xmath66 m ) ratios are both smaller than the shock values . the iron and silicon lines are not dominant cooling lines of ionizing shocks . the gas abundances of these elements do not impact the thermal structure of the shock and the line intensities roughly scale with the gas phase abundances . we thus interpret the offset between the irs observation and model values as evidence for fe and si depletion . we consider the magnitude of the depletions indicated by the arrow ( @xmath650% and @xmath660% for fe and si , respectively ) in figure [ fig : sil_ne ] as lower limits since there could be a contribution to the @xmath10feii@xmath11 $ ] and @xmath10siii@xmath11 $ ] line emission from non - ionizing @xmath33-shocks into molecular gas @xcite . such shocks could also contribute to the @xmath10oi@xmath11$](6300 ) line emission as discussed in guillard et al . forthcoming observations of the h@xmath9 ro - vibrational line emission in the near - ir should allow us to estimate whether they may be significant . a contribution from non - ionizing shocks to the @xmath10feii@xmath11 $ ] and @xmath10siii@xmath11 $ ] line emission will raise the depletion of both fe and si as well as the fe / si depletion ratio because non - ionizing shocks do not produce @xmath10neii@xmath11 $ ] line emission and the @xmath10feii@xmath11$](@xmath71m)/@xmath10siii@xmath11$](@xmath72 m ) line emission ratio in @xmath33-shocks within dense gas is larger than for shocks plotted in figure [ fig : sil_ne ] . a key aspect to understanding the emission we observe in sq , as well as a test of the proposed model of h@xmath5 excitation , is the amount and distribution of cold h@xmath5 , as measured by co. a reservoir of cold h@xmath5 associated with the warm gas observed in sq would provide key insight into conditions within the intragroup medium . in figure [ fig : co ] we show the bima co ( 1 - 0 ) integrated intensity contours from @xcite , shown with the h@xmath5 s(0 ) and s(1 ) emission contours , overlaid on an optical image of sq . these interferometric observations use the large primary beam of 110 , centered on the shock , to determine areas of high column densities in the group . the co traces areas of known star formation seen at 24 and in the nuv ( see section [ sf ] ) , notably sq - a and 7318b - south . as observed by @xcite , the co around ngc 7319 is concentrated in two regions . the dominant complex is north of the nucleus residing in a dusty tidal feature . the nuclear co is elongated perpendicular to the stellar disk suggesting a deficiency of ongoing star formation in the disk . the co distribution does not correlate with the location of warm h@xmath5 emission , particularly around ngc 7319 . the cold h@xmath5 complexes are clearly offset from the concentrations of warm h@xmath9 . even the s(0 ) emission line ( see fig . [ fig : co]a ) , which follows the coldest warm h@xmath9 , does not have peaks corresponding to the strongest co detections . in a forthcoming paper ( guillard et al . 2010 , in preparation ) , we will report the recent detection of @xmath73co(1 - 0 ) and ( 2 - 1 ) emission , associated with the warm h@xmath9 in the sq shock , using the single - dish 30 m iram telescope . these observations suggest that most of the co emission in the shock has been missed by interferometers ( because of the broad linewidth ) and show that the co emission is both present in the shock , and extends along the h@xmath9-emitting bridge and towards ngc 7319 . the kinematics of the co gas lying outside of star - forming regions , in the new observations , appears to be highly disturbed with a broad linewidth in agreement with the appleton et al . ( 2006 ) interpretation that the mir h@xmath5 lines were intrinsically very broad and resolved by the high resolution module of irs . the co data also agrees with our re - analysis of the appleton et al . ( 2006 ) data ( see appendix b ) using more recent and reliable wavelength calibration , which places the bulk of the h@xmath5 gas at velocities intermediate to that of the intruder and the group supporting the idea that the h@xmath5 gas is accelerated in the shock . section [ sf ] discussed the star forming regions observed in sq and noted that there was very little evidence for star formation in the shock associated with the warm h@xmath5 emission . the main shock contains a large quantity of warm molecular hydrogen ( 5.0 @xmath210@xmath43m@xmath44 ) providing a reservoir of fuel for star formation once it cools ( see section [ spec ] ) . we investigate star formation in the warm h@xmath9-dominated medium by considering the shock sub - region , chosen to avoid star forming regions in the intruder galaxy , but likely still subject to some contamination from these regions . pah emission is a classical tracer of star formation , but the molecules are fragile and easily destroyed in hard radiation fields . in the spectra of the main shock and subregion ( fig . [ fig : spectra]a and b ) the pah emission bands at 6.2 , 7.7 and 8.6 are far weaker compared to the 11.3 bands , which when strong are predominantly produced by neutral pah molecules @xcite . in the shock subregion we find an upper limit flux for the 6.2 pah of @xmath74 wm@xmath75 and fluxes of @xmath76 wm@xmath75 and @xmath77 wm@xmath75 for the 7.7 and 11.3 bands respectively . this corresponds to a 7.7/11.3 pah ratio of 0.35 , very low compared to the median value found for the sings sample of 3.6 @xcite . the suppression of the 7.7/11.3 pah ratio has been observed in agn environments ( e.g. low - luminosity agn in the sings sample of * ? ? ? * ) and is favored to be the result of selective destruction of pah molecules small enough to emit at 7.7 . pah processing in the shock due to larger molecules being less fragile than smaller ones is discussed further in guillard et al . an alternative explanation is that the pah molecules are chiefly large and neutral in the shock , producing enhanced 11.3 emission in comparison to the smaller pah molecules . a detailed comparison of the dust and pah emission properties can be found in guillard et al . ( 2010 ) . dust emission in the mid- and far - infrared can be used to infer the amount of star formation taking place @xcite . we measure a 24 flux in this region of 0.408 mjy , corresponding to a spectral luminosity ( @xmath78 ) of @xmath79 . this low luminosity is consistent with the weak mir continuum ( fig . [ fig : spectra]b ) , arising from emission from vsgs heated by the uv radiation field . we can combine this with a measurement of the h@xmath27 emission in this region to obtain a star formation rate ( sfr ) , given that they are complimentary ( h@xmath27 tracing the young stellar population and 24 as a measure of dust - absorbed stellar light ) . we find @xmath80 , but caution that h@xmath27 emission in sq is also the result of shock - excitation @xcite and must be considered an upper limit for measuring star formation . when we combine @xmath81 with @xmath82 , using the relation of @xcite , we find a sfr of @xmath290.05@xmath83 . a further measure of star formation can be obtained from the pah strength . using the relation of @xcite , derived from the starburst sample of @xcite , we can use the 7.7 flux density as a measure of star formation . the shock subregion has a 7.7 flux density of 0.73 mjy which corresponds to a sfr of @xmath84 , in good agreement with our previous calculation , but also an upper limit as some pah emission is contamination from known star formation regions in the group . a point of caution , however , is that this star formation indicator may be biased given that we detect suppressed 7.7 emission compared to the 11.3 pah . comparing the @xmath85(7.7)/@xmath82 ratio to the galaxies in sings , we find that the value of @xmath86 is typical of star - forming galaxies @xcite , which suggests that both measures of star formation are low , but self - consistent . the low _ upper limits _ for the sfr in the shock suggests that star formation is depressed in the shock apart from in the igm starburst , sq - a , which has a sfr of @xmath87 at the velocity of the group @xcite . this would be consistent with a picture of molecular hydrogen being reheated by mhd shocks in the turbulent medium . since the jeans mass increases with gas temperature and turbulence and shearing morions will prevent collapse , the cold molecular gas clouds may be too short - lived or undersized to facilitate collapse and produce significant star formation @xcite . the extent of a cold reservoir of molecular gas in the shock is a key consideration in this scenario , as discussed in section [ co ] . the present observations suggest that molecular line cooling in dense clumps dissipates a significant fraction of the kinetic energy available in high - speed shocks . for reference , and to provide a glimpse of what this group might look like at high redshift when filling a single beam , we present in figure [ fig : full ] the spectrum extracted for the entire group ( i.e. including the three neighboring galaxies and the shock ) . the total h@xmath5 line emission from the whole group exceeds @xmath8 and is still the most dominant mir coolant . indeed the luminosity of the rotational h@xmath5 lines is sufficient that it could be detected at high redshift with future far - infrared or sub - mm instrumentation like spica or safir ( see appleton et al . how likely is it , however , that high - speed shocks play a role in the assembly of galaxies ? there is growing evidence that galaxies at high redshift are turbulent @xcite and increasingly clumpy ( e.g. conselice et al . 2005 , elmegreen & elmegreen 2005 ) . indeed bournaud & elmegreen ( 2009 ) discuss the importance of the growth instabilities in massive gas clumps in forming disks at z @xmath88 1 , and favor at least a large fraction of the clump systems being formed in smooth flows , perhaps similar to those discussed by dekel et al . to what extent the build - up of these disks is truly `` smooth '' is not yet clear since the medium is likely to be a multiphase one . in a more standard picture , gas flowing into the more massive dark halos will experience strong shocks , most likely in an inhomogenious medium ( e. g. greif et al . 2008 ) thus it begs the question of how important h@xmath9 cooling may be in these different cases . models of the collapse of the first structures predict that strong metal lines soon dominate the cooling over molecular hydrogen when the first stars pollute the environment . it is therefore usually assumed that , except at very early stages , molecular hydrogen is a minority coolant in gas that forms the first major structures ( e.g. bromm et al . 2001 , santoro and shull 2006 ) . however , our observations show that , under the right conditions , even in high metallicity environments , molecular hydrogen can be extremely powerful in this case dominating by a factor of ten over the usually powerful @[email protected] \mu$]m line in strong shocks . if there are situations at high redshift where strong shocks propagate into a clumpy , multiphase medium , then our observations imply that molecular hydrogen cooling can not be assumed to be negligible . on the other hand , this will not be a trivial problem . our best model of stephan s quintet @xcite involves the formation of h@xmath5 in a complex multiphase turbulent medium in which shocks destroy dust in some places , but allow survival in others - thus encouraging h@xmath5 formation . in the early universe , this enhanced cooling , which has so far been neglected , will depend on the distribution and nature of the first dust grains , in concert with the formation , temperature and abundance of gas , and the feedback effects from the first stars and agn . in this paper we have presented the results of the mid - infrared spectral mapping of the stephan s quintet system using the _ spitzer space telescope_. we highlight here our five main conclusions : * the powerful h@xmath5 emission detected by @xcite surprisingly represents only a small fraction of the group - wide warm h@xmath5 ( with a lower limit luminosity of @xmath89 spread over @xmath90 kpc@xmath7 ) that dominates the mid - infrared emission of the system . there is evidence for another shock - excited feature , the so - called h@xmath5 bridge between the main shock and ngc 7319 , which is likely a remnant of past tidal interactions within the group . the spatial variation in the distribution of the h@xmath5 0 - 0 line ratios implies differences in temperature and excitation in the shocked system - this will be explored fully in paper ii . * the global l(h@xmath5)/l@xmath91 ratio in the main shock is @xmath92 , and @xmath62.5 in the new `` bridge '' feature . the results confirm that mir h@xmath9 lines are a stronger coolant than x - ray emission over the shock structures , indicating a new cooling pathway seen on a large scale in sq . this modifies the traditional view that x - rays dominate cooling at all times in the later stages of compact group evolution . since h@xmath5 forms on the surfaces of dust grains , we expect dust emission associated with these regions , but a low intensity radiation field produces only weak emission at 24 . * following earlier interpretations of nebular line ratios in the optical , we interpret infrared ionic lines within the framework of fast ( @xmath93 km@xmath0 ) ionizing shocks . comparison between the @xmath10neii@xmath11 $ ] , @xmath10neiii@xmath11 $ ] , @xmath10siii@xmath11 $ ] and @xmath10feii@xmath11 $ ] line intensities implies that both silicon and iron are depleted onto dust . this result implies that dust is not destroyed in the shock . * star formation in sq is dominated by sq - a and 7318b - south , located at the extreme ends of the shock ridge seen at radio wavelengths , suggesting they are both shock triggered starbursts . however , regions dominated by warm h@xmath5 emission exhibit very low star formation rates , consistent with a turbulent model where h@xmath5 is significantly reheated and cool clouds are too short - lived or undersized to collapse . * in sq we observe the projected coexistence of @xmath10siii@xmath11 $ ] and h@xmath5 being produced by @xmath6200km@xmath0 and @xmath620km@xmath0 velocity shocks , respectively . our observational results are consistent with a model of a multiphase postshock medium produced by a galaxy - wide collision @xcite . the cooling pathway of warm h@xmath5 emission we observe group - wide in sq is clearly a significant , albeit surprising , mechanism in shock systems . to determine the overall dominant cooling mechanism in sq , we require an inventory of lines and continuum processes at all wavelengths . early shock models @xcite predict that , apart from the rotational emission from h@xmath5 , contributions from lines such as @xmath10oi@xmath11$]63.2 , @xmath10cii@xmath11$]157.7 and the thz spectrum of h@xmath50 @xcite could be significant . we hope to explore this chemistry more fully , and the detailed distribution of cool dust , using the capabilities of the _ herschel _ space observatory . in addition , we can not rule out strong uv line - cooling . stephan s quintet provides the ideal laboratory for probing a mechanism potentially crucial in systems ranging from ulirgs to radio galaxies to supernova remnants . mec is supported by nasa through an award issued by jpl / caltech under program 40142 . we thank tom jarrett for use of his irs pixel cleaning software and irac / mips photometry software . aaaaa allen , m.g . , groves , b.a . , dopita , m.a . , sutherland , r.s . , & kewley , l.j . 2008 , , 178 , 20 allen , r.j . , & hartsuiker , j.w . 1972 , nature , 239 , 324 allen , r.j . , & sullivan , w.t.,iii . 1980 , , 84 , 181 appleton , p. , et al . 2009 , astro2010 : the astronomy and astrophysics decadal survey , science white papers , 2010 , 2 appleton , p.n . et al . 2006 , , 639 , l51 appleton , p.n . , charmandaris , v. , horrelou , c. , mirabel , i.f . , ghigo , f. , higdon , j.l . , & lord , s. 1999 , , 527 , 143 bernard - salas , j. et al . 2009 , , in press ( arxiv : astro - ph/0908.2821 ) black , j. h. , & van dishoeck , e. f. 1987 , , 322 , 412 bournaud , f. , & elmegreen , b. g. 2009 , , 694 , l158 brandl , b.r . 2006 , , 653 , 1129 bromm , v. , ferrara , a. , coppi , p. s. , & larson , r. b. 2001 , , 328 , 969 buchanan , c. l. , gallimore , j. f. , odea , c. p. , baum , s. a. , axon , d. j. , robinson , a. , elitzur , m. , & elvis , m. 2006 , , 132 , 401 calzetti , d. , et al . , 2007 , , 666 , 870 conselice , c. j. , et al . 2005 , , 633 , 29 dale , d.a . , et al . 2006 , , 646 , 161 dekel , a. , sari , r. , & ceverino , d. 2009 , , 703 , 785 deo , r.p . , et al . 2007 , , , 671 , 124 dopita , m. a. , & sutherland , r. s. 1995 , , 455 , 468 draine , b.t . , roberge , w.g . , & dalgarno , a. 1983 , , 264 , 485 draine , b. t. , & woods , d. t. 1992 , , 387 , 732 draine , b.t . , & li a. 2001 , , 551 , 807 durret , f. 1994 , s , 105 , 57 egami , e. , rieke , g.h . , fadda , d. & hines , d.c . 2006 , , 652 , l21 elmegreen , b. g. , & elmegreen , d. m. 2005 , , 627 , 632 frster schreiber , n. m. , et al . 2006 , , 645 , 1062 gao , y. & xu , c. 2000 , , 542 , l83 genzel , r. , et al . 2008 , , 687 , 59 greif , t. h. , johnson , j. l. , klessen , r. s. , & bromm , v. 2008 , , 387 , 1021 guillard , p. , boulanger , f. , cluver , m. , appleton , p.n . , pineau des forts , g. , & ogle , p. 2010 , , submitted guillard , p. , boulanger , f. , pineau des forts , g. , & appleton , p.n . 2009 , , 502 , 515 hatch , n.a . , crawford , c.s , fabian , a.c . , & johnstone , r.m . 2005 , , 358 , 765 helou , g. , & bicay , m.d . 1993 , , 415 , 93 hewitt , j.w . , rho , j. , andersen , m. , & reach , w.t . 2009 , , 694 , 1266 ho , l.c . , & keto , e. 2007 , , 658 , 314 hollenbach , d. , & mckee c.f . 1989 , , 342 , 306 hollenbach , d. j. , & tielens , a. g. g. m. 1997 , , 35 , 179 houck , j.r . , weedman , d.w . , le floch , & hao , l. 2007 , , 671 , 323 houck , j.r . 2004 , , 154 , 18 kennicutt jr . , 1998 , , 36 , 189 lepp , s. , & mccray , r. 1983 , , 269 , 560 maloney , p.r . , hollenbach , d.j . , & tielens , a.g.g.m . 1996 , , 466 , 561 martn - 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hacyan , d. , verdes - montenegro , l. , trincheri , g. , xu , c. , & pietsch , w. 2001 , , 122 , 2993 trinchieri , g. , sulentic , j. , breitschwerdt , d. , & pietsch , w. , 2003 , , 401 , 173 trinchieri , g. , sulentic , j. , pietsch , w. , & breitschwerdt , d. 2005 , , 444 , 697 verma , a. , lutz , d. , sturm , e. , sternberg , a. , genzel , r. , & vacca , w. 2003 , , 403 , 829 weedman , d. , et al . 2005 , , 633 , 706 werner , m. w. , et al . 2004 , , 154 , 1 williams , b.a . , yun , m.s . , & verdes - montenegro , l. 2002 , , 123 , 2417 xanthopoulos , e. , muxlow , t.w.b . , thomasson , p. , & garrington , s.t . , 2004 , , 353 , 1117 xu , c. , sulentic , j.w . , & tuffs , r. 1999 , , 512 , 178 xu , c.k . , lu , n. , condon , j.j . , dopita , m. , & tuffs , r.j . 2003 , , 595 , 665 xu , c.k . , et al . 2005 , , 619 , l95 yun , m.s . , verdes - montenegro , l. , del olmo , a. , & perea , j. 1997 , , 475 , l21 in this section we discuss the results pertaining to ngc 7319 , a seyfert 2 galaxy @xcite lying to the east in the sq group ( see fig . the specific intensity contours for the h@xmath5 s(0 ) and h@xmath5 s(1 ) lines ( fig . 2a and b ) show emission associated with the galaxy , seemingly connected to the rest of the group by the h@xmath5 `` bridge '' discussed in section [ molhy ] . figure [ fig : xray_rad ] shows that the nucleus of ngc 7319 produces strong x - ray emission @xcite and is prominent at radio wavelengths @xcite . however , we observe an offset between the peak of the h@xmath5 emission near ngc 7319 and the seyfert nucleus , which suggests that the `` bridge '' is a separate structure and not being excited by the agn . the agn in ngc 7319 does not have a well - collimated jet , but two extended lobes with compact hotspots , asymetrically distributed along the minor axis of the galaxy @xcite . this structure runs ne / sw and its orientation compared to the h@xmath5 filament ( which runs ew ) is not consistent with causing the excitation of the h@xmath5 bridge . this is also evident from the relatively weak power in the agn , as inferred by the emission line diagnostics discussed below , and in the x - ray where it is only a factor of @xmath62 greater than the emission associated with the main shock and `` bridge '' . in figure [ fig : fs ] we present the specific intensity contours of the @xmath10feii@xmath11$]25.99 ( blended with @xmath10oiv@xmath11$]25.89@xmath25 m ) , @xmath10siii@xmath11$]33.48 and @xmath10siii@xmath11$]34.82 emission lines . given the low spectral resolution of the sl and ll modules of _ spitzer _ , we can not distinguish between emission from @xmath10fe ii@xmath11 $ ] and @xmath10oiv@xmath11 $ ] . the agn in ngc 7319 is likely to produce both , with @xmath10feii@xmath11 $ ] emission likely originating in x - ray dissociation regions ( xdrs ) surrounding the agn @xcite . prominent emission from @xmath10siii@xmath11$]33.48 and @xmath10siii@xmath11$]34.82 are due to the high excitation conditions associated with the agn . dense pdrs and x - ray dominated regions , powered by agn , show strong @xmath10siii@xmath11 $ ] emission at 34.82 , while @xmath10siii@xmath11 $ ] 33.48 emission acts as a tracer of hii regions . we now focus on the emission line properties of ngc 7319 . figure [ fig : agn ] shows the spectrum extracted from the galaxy and the measured properties of the line ratios are listed in table [ tableh2fluxes ] and [ tablemetalfluxes ] . for the first time in the sq group , we observe a spectrum which is no longer dominated by h@xmath5 emission , but instead the brightest lines are the high - excitation @xmath10oiv@xmath11$]25.91@xmath25 m and @xmath10feii@xmath11$]25.98 blended lines , as well as the @xmath10siii@xmath11$]34.81@xmath25 m line . @xmath10nev@xmath11 $ ] is prominent at both 14.32 and 24.30@xmath25 m a line typically seen in agn . also , unlike the majority of the extended shocked regions , there is a rising thermal continuum more typical of a starforming galaxy than a classical seyfert galaxy , although as various studies have shown ( buchanan et al . 2006 ; deo et al . 2007 ) , seyfert ii galaxies exhibit a variety of mir spectral characteristics at long wavelengths . ngc 7319 s rising continuum is similar to that seen in the seyfert ii galaxy ngc 3079 ( deo et al . 2007 ) , and likely represents a dominant starburst component in the far - ir . spitzer imaging reveals not only a bright nucleus , but also extended emission regions in the galaxy . using the @xmath10siii@xmath11$]34.81@xmath25m/@xmath10siii@xmath11$]33.48 ratio as a probe of excitation sources , we find a ratio of @xmath61.85 which is low compared to the average value found for agn galaxies ( @xmath94 ) in the _ spitzer _ infrared nearby galaxy sample ( sings ) , but high compared to star - forming regions ( @xmath95 ) in the same sample @xcite . this suggests a relatively weak agn . the @xmath10neiii@xmath11$]15.56/@xmath10neii@xmath11$]12.81 ratio is a measure of radiation field strength and the value of 0.97 indicates a typical radiation field strength compared to other agn ( the sample of weedman et al . 2005 shows a range of @xmath60.17 to 1.9 ) . the average electron density , estimated from the @xmath10siii@xmath11$]18.71/@xmath10siii@xmath11$]33.48 ratio of @xmath60.56 , is @xmath59 @xmath57 @xcite , in the low - density limit for this diagnostic @xcite . the properties of the h@xmath5 emission of ngc 7319 are limited to the two long wavelength lines observed by irs - ll ( the galaxy lies outside the region mapped by sl ) . as a result , the excitation diagram contains only two points allowing only an approximate idea of the h@xmath5 mass , since without the sl wavelength coverage to provide information on possible warmer components , we are likely to overestimate the temperature of the h@xmath5 by fitting a straight line to the points . any warmer component would contribute to the flux of the 0 - 0 s(1 ) line thus leading to a reduction of the temperature ( and increase in h@xmath5 mass ) for any cooler component . however , to provide a guide , we estimate the temperature of h@xmath5 @xmath96171@xmath97 k ( assuming the gas is in thermal equilibrium and thus an ortho - para ratio of 2.65 ) and a total h@xmath5 mass of 3.0@xmath9810@xmath3m@xmath4 . we consider this a lower limit to the total warm mass if ( as is likely ) more than one component is present . gao & xu ( 2000 ) estimated the _ cold _ molecular hydrogen mass of ngc 7319 based on @xmath73co ( 1 - 0 ) observations as @xmath88 3.6 @xmath210@xmath99m@xmath4 , a factor of roughly ten greater than the warm h@xmath5 mass . such a ratio is not atypical of large spiral galaxies @xcite . the high resolution spectrum at the center of the shock in sq , obtained by @xcite , has been reanalysed using the latest calibrations available for the irs instrument ( ssc pipeline version s17 ) and is included as figure [ fig : hr ] . this has shown that the h@xmath5 gas lies at a velocity of 6360(@xmath100100 ) km s@xmath13 , between the velocity of the group ( 6600 kms@xmath13 ) and the velocity of the intruder ( 5700 kms@xmath13 ) . this is consistent with a model of gas being accelerated by the shock , as well as the turbulence demonstrated by the broad linewidth of the h@xmath5 in the shock ( 860 kms@xmath13 ) . the new sh spectrum shows that with the improved calibrations , the 11.3 pah feature is detected , although faint . this is consistent with what is found in the larger extractions that show enhanced 11.3 pah emission compared to ionised pah features emitting at 6.2 and 7.7 . in this section we present the x - ray fluxes in the extraction regions shown in figure [ fig : ext ] . the reanalysis of archival data is necessary to obtain accurate fluxes and luminosities for the x - ray emission over the various apertures matched to our spectral extractions . we use the xmm - newton epic - pn data ( see * ? ? ? * for observational details ) to obtain the most sensitive measurements . a calibrated event file was generated and filtered using standard quality flags , and subsequently cleaned of background flares . a 0.32 kev image was then extracted and corrected for instrumental response . all point - like sources were masked out to @xmath101 in the analysis of diffuse emission , and the local background level was estimated within a @xmath102 arcmin@xmath7 rectangular region away from the group core . for each region in fig . 7 , resulting background - subtracted photon count rates were converted to 0.32 kev and bolometric " ( 0.00110 kev ) x - ray fluxes assuming an absorbed thermal plasma model of metallicity 0.4 solar , with temperature as estimated from the map of osullivan et al . ( 2009 ) , and an absorbing galactic hi column density of @xmath103 @xmath47 . for ngc7319 , which harbors an agn , an absorbed power - law spectrum of photon index @xmath104 was assumed instead . results are listed in table [ x - ray ] . lcccc + region & count rate ( 0.3 - 2 kev ) & flux ( 0.3 - 2 kev ) & bolometric flux ( 0.001 - 10 kev ) & adopted spectral model + & ( photonss@xmath105 ) & ergs@xmath13@xmath47 & ergs@xmath13@xmath47 & + + main shock & 0.102 & 1.8 @xmath106 & 2.8 @xmath106 & t = 0.7 kev + shock sub - region & 0.017 & 3.3 @xmath107 & 5.1 @xmath107 & t = 0.8 kev + bridge & 0.017 & 3.0 @xmath107 & 4.5 @xmath107 & t = 0.6 kev + ngc 7319 & 0.051 & 1.1 @xmath106 & 3.7 @xmath106 & @xmath108 = 1.7 + sq - a & 0.013 & 2.4 @xmath107 & 3.7 @xmath107 & t = 0.6 kev + we note that for the main shock ( with luminosity 1.9 @xmath210@xmath35ergs@xmath13 in the 0 - 3 - 2 kev band ) we are within a factor of 2 of the 0.5 - 2 kev luminosity @xmath63.1 @xmath210@xmath35ergs@xmath13 obtained for a similar , but larger extraction of the shock by @xcite . @xcite obtain a 0.5 - 2 kev surface brightness of 0.07 l@xmath4pc@xmath75 in the main shock compared to our value of 0.1 l@xmath4pc@xmath75 for the 0.3 - 2 kev surface brightness . lccccccc + target region & aperture & h@xmath5 0 - 0 s(0 ) & h@xmath5 0 - 0 s(1 ) & h@xmath5 0 - 0 s(2 ) & h@xmath5 0 - 0 s(3 ) & h@xmath5 0 - 0 s(4 ) & h@xmath5 0 - 0 s(5 ) + & ( arcsec@xmath7 ) & @xmath10928.22@xmath25 m & @xmath10917.03@xmath25 m & @xmath10912.28@xmath25 m & @xmath1099.66@xmath25 m & @xmath1098.03@xmath25 m & @xmath1096.91@xmath25 m + + main shock & 2307 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + shock sub - region & 242 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + bridge & 413 & [email protected] & [email protected] & & & & + ngc 7319 & 1302 & [email protected] & [email protected] & & & & + sq - a & 671 & [email protected] & [email protected] & & & & [email protected] + lcccccccc + target region & @xmath10neii@xmath11 $ ] & @xmath10nev@xmath11 $ ] & @xmath10neiii@xmath11 $ ] & @xmath10siii@xmath11 $ ] & @xmath10nev@xmath11 $ ] & @xmath10feii@xmath11$]+@xmath10oiv@xmath11 $ ] & @xmath10siii@xmath11 $ ] & @xmath10siii@xmath11 $ ] + & @xmath10912.81@xmath25 m & @xmath10914.32@xmath25 m & @xmath10915.56@xmath25 m & @xmath10918.71@xmath25 m & @xmath10924.32@xmath25 m & @xmath10925.99 + 25.89@xmath25 m & @xmath10933.48@xmath25 m & @xmath10934.82@xmath25 m + + main shock & [email protected] & @xmath290.9 & [email protected] & [email protected] & @xmath290.14 & [email protected] & [email protected] & [email protected] + shock sub - region & [email protected] & @xmath290.18 & [email protected] & @xmath290.12 & @xmath290.14 & [email protected] & [email protected] & [email protected] + bridge & & & [email protected] & @xmath290.1 & @xmath290.15 & [email protected] & @xmath290.22 & [email protected] + ngc 7319 & & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 9.24 @xmath1000.49 + sq - a & & @xmath290.8 & [email protected] & [email protected] & @xmath290.18 & @xmath290.17 & [email protected] & [email protected] +

we present results from the mid - infrared spectral mapping of stephan s quintet using the _ spitzer space telescope_. a 1000km@xmath0 collision ( @xmath1yr ) has produced a group - wide shock and for the first time the large - scale distribution of warm molecular hydrogen emission is revealed , as well as its close association with known shock structures . in the main shock region alone we find 5.0 @xmath210@xmath3m@xmath4 of warm h@xmath5 spread over @xmath6480kpc@xmath7 and additionally report the discovery of a second major shock - excited h@xmath5 feature , likely a remnant of previous tidal interactions . this brings the total h@xmath5 line luminosity of the group in excess of @xmath8 . in the main shock , the h@xmath5 line luminosity exceeds , by a factor of three , the x - ray luminosity from the hot shocked gas , confirming that the h@xmath9-cooling pathway dominates over the x - ray . @xmath10siii@xmath11$]34.82emission , detected at a luminosity of 1/10th of that of the h@xmath9 , appears to trace the group - wide shock closely and in addition , we detect weak @xmath10feii@xmath11$]25.99 emission from the most x - ray luminous part of the shock . comparison with shock models reveals that this emission is consistent with regions of fast shocks ( @xmath12km@xmath0 ) experiencing depletion of iron and silicon onto dust grains . star formation in the shock ( as traced via ionic lines , pah and dust emission ) appears in the intruder galaxy , but most strikingly at either end of the radio shock . the shock ridge itself shows little star formation , consistent with a model in which the tremendous h@xmath5 power is driven by turbulent energy transfer from motions in a post - shocked layer which suppresses star formation . the significance of the molecular hydrogen lines over other measured sources of cooling in fast galaxy - scale shocks may have crucial implications for the cooling of gas in the assembly of the first galaxies .

arxiv

the short - lived radioisotope ( slri ) @xmath4al was alive during the formation of the first refractory solids in the solar nebula , the ca- , al - rich inclusions ( cais ) found in primitive chondritic meteorites . this means that at least some of the solar system s slris may have been injected into either the presolar cloud ( e.g. , boss & keiser 2012 ; boss 2012 ) or the solar nebula ( ouellette et al . 2007 , 2010 ; dauphas & chaussidon 2011 ) by a supernova or agb star shock wave . in either case , injection occurred as a single event that was spatially heterogeneous , which would potentially reduce the usefulness of @xmath4al as a spatially homogeneous chronometer ( dauphas & chaussidon 2011 ) for precise studies of the earliest phases of planet formation ( macpherson et al . 2012 ; cf . krot et al . previous models ( e.g. , boss 2011 , 2012 ) have shown how such initial spatial isotopic heterogeneity can be substantially reduced in a marginally gravitationally unstable ( mgu ) disk , as a result of the large - scale inward and outward transport and mixing of gas and particles small enough to move with the gas ( e.g. , boss et al . 2012 ) . other elements and their isotopes suggest a similarly well - mixed solar nebula ( e.g. , os : walker 2012 ; fe : wang et al . 2013 ) . the stable oxygen isotopes , on the other hand , appear to have been spatially heterogeneous in the solar nebula during the early phases of planet formation ; e.g. , small refractory particles from comet 81p / wild 2 have normalized @xmath5o/@xmath6o ratios that span the entire solar system range of @xmath0 6% variations ( nakashima et al . 2012 ) . the leading explanation for generating these oxygen anomalies is uv photodissociation of co molecules at the surface of the outer solar nebula ( e.g. , podio et al . 2013 ) , where self - shielding could lead to isotopic fractionation between gas - phase and solid - phase oxygen atoms ( e.g. , lyons & young 2005 ; krot et al . co self - shielding on the irregular , corrugated outer surface of the disk would also lead to initial spatial heterogeneity , though the process would be continuous in time , rather than a single - shot event like a supernova shock wave . furthermore , the very existence of refractory particles in comet 81p / wild 2 ( brownlee et al . 2006 ; simon et al . 2008 ; nakamura et al . 2008 ) , which are thought to have formed close to the protosun , implies that these small particles experienced large - scale outward transport from the inner solar nebula to the comet - forming regions of the outer solar nebula . mgu disks offer a means to accomplish this early large - scale transport ( e.g. , boss 2008 , 2011 ; boss et al . 2012 ) . marginally gravitationally unstable disks are likely to be involved in the fu orionis outbursts experienced by young solar - type stars ( e.g. , zhu et al . 2010b ; vorobyov & basu 2010 ; martin et al . mgu disk models ( e.g. , boss 2011 ) can easily lead to the high mass accretion rates ( @xmath7 yr@xmath8 ) needed to explain fu orionis events . fu orionis outbursts are believed to last for about a hundred years and to occur periodically for all low mass protostars ( hartmann & kenyon 1996 ; miller et al . mgu models are also capable of offering an alternative mechanism ( disk instability ) for gas giant planet formation ( e.g. , boss 2010 ; meru & bate 2012 ; basu & vorobyov 2012 ) . however , the magnetorotational instability ( mri ) is likely to be involved in fu orionis outbursts as well ( zhu et al . 2009a ) , with mri operating in the ionized innermost disk layers as well as at the disk s surfaces . zhu et al . ( 2009c , 2010a , b ) have constructed one- and two - dimensional ( axisymmetric ) models of a coupled mgu - mri mechanism , with mgu slowly leading to a build - up of mass in the innermost disk , which then triggers a rapid mri instability and an outburst . alternatively , mri may operate in the outermost disk , partially ionized by cosmic rays , leading to a build - up of mass in the dead zone at the intermediate disk midplane , thus triggering a phase of mgu transport . such a coupled mechanism may be crucial for achieving outbursts in t tauri disks , where the disk masses are expected to be smaller than at earlier phases of evolution . we present here several new sets of mgu disk models that examine the time evolution of isotopic heterogeneity introduced in either the inner or outer solar nebula , by either a single - shot event or a continuous injection process , for a variety of disk and central protostar masses , including protostars with m dwarf masses . low mass exoplanets are beginning to be discovered around an increasingly larger fraction of m dwarfs ( bonfils et al . 2013 ; dressing & charbonneau 2013 ; kopparapu 2013 ) , with a number of these being potentially habitable exoplanets , elevating the importance of understanding mixing and transport processes in m dwarf disks . the numerical models were computed with the same three dimensional , gravitational hydrodynamics code that has been employed in previous mgu disk models ( e.g. , boss 2011 ) . complete details about the code and its testing may be found in boss & myhill ( 1992 ) . briefly , the code performs second - order - accurate ( in both space and time ) hydrodynamics on a spherical coordinate grid , including radiative transfer in the diffusion approximation . a spherical harmonic ( @xmath9 ) expansion of the disk s density distribution is used to compute the self - gravity of the disk , with terms up to and including @xmath10 . the radial grid contains 50 grid points for the 10 au disk models and 100 grid points for the 40 au disk models . all models have 256 azimuthal grid points , and effectively 45 theta grid points , given the hemispherical symmetry of the grid . the theta grid is compressed around the disk s midplane to provide enhanced spatial resolution , while the azimuthal grid is uniformly spaced . the jeans length constraint is used to ensure adequate resolution . the inner boundary absorbs infalling disk gas , which is added to the central protostar , while the outer disk boundary absorbs the momentum of outward - moving disk gas , while retaining the gas on the active grid . the central protostar wobbles in such a manner as to preserve the center of mass of the entire system . the time evolution of a color field is calculated ( e.g. , boss 2011 ) in order to follow the mixing and transport of isotopes carried by the disk gas or by small particles , which should move along with the disk gas . the equation for the evolution of the color field density @xmath11 ( e.g. , boss 2011 ) is identical to the continuity equation for the disk gas density @xmath12 @xmath13 where @xmath14 is the disk gas velocity and @xmath15 is the time . the total amount of color is conserved in the same way that the disk mass is conserved , as the hydrodynamic equations are solved in conservation law form ( e.g. , boss & myhill 1992 ) . the initial disk density distributions are based on the approximation derived by boss ( 1993 ) for a self - gravitating disk orbiting a star with mass @xmath16 @xmath17 @xmath18,\ ] ] where @xmath19 and @xmath20 are cylindrical coordinates , @xmath21 is the gravitational constant , @xmath22 is the midplane density , @xmath23 is the surface density , @xmath24 ( cgs units ) and @xmath25 . the initial midplane density is @xmath26 while the initial surface density is @xmath27 the parameters @xmath28 and @xmath29 and the reference radius @xmath30 are defined in table 1 for the various disk models explored in this paper . the total amount of mass in the models does not change during the evolutions ; the initial infalling disk envelope accretes onto the disk , and no further mass is added to the system across the outer disk boundary at @xmath31 . the outer disk surfaces are thus revealed to any potential source of uv irradiation . for the 10 au outer radius disks listed in tables 2 and 3 , the initial disk temperature profiles ( figures 1 and 2 ) are based on the boss ( 1996 ) temperature profiles , with variations in the assumed outer disk temperature @xmath32 , chosen in order to study the effect of varied minimum values of the @xmath2 stability parameter . values of @xmath33 indicate marginally gravitationally unstable disks . the inner disks are all highly @xmath2 stable , with @xmath34 . for the 40 au outer radius disks listed in table 4 , the initial disk temperatures are uniform at the specified outer disk temperature @xmath32 , leading to similar initial @xmath2 values throughout the disks . for all of the models , the temperature of the infalling envelope is 50 k. the initial color field is added to the surface of the initial disk in an azimuthal sector spanning either 45 degrees ( 10 au outer radius disks ) or 90 degrees ( 40 au outer radius disks ) in a narrow ring of width 1 au , centered at the injection radii listed in the tables . these models are intended to represent one - time , single - shot injections of isotopic heterogeneity , such as supernova - induced rayleigh - taylor fingers carrying live @xmath4al ( e.g. , boss & keiser 2012 ) . table 4 lists both single - shot and continuous injection models , where in the latter case the color is added continuously to the same location on the disk surface throughout the evolution , crudely simulating ongoing photodissociation of co ( e.g. , lyons & young 2005 ) possibly leading to stable oxygen isotope fractionation between the gas and solid phases . the color field in the latter case is intended to represent isotopically distinct gas or small particles resulting from the uv photochemistry . note that in both the single - shot and continuous injection models , the total amount of color added is arbitrary ( e.g. , the color field in the injection volume is simply set equal to 1 ) , and is intended to be scaled to whatever value is appropriate for the isotope(s ) under consideration . the color field is a massless , passive tracer that has no effect on the disk s dynamics , so the total amount of color added is irrelevant for the disk s subsequent evolution . the models seek to follow the deviations from uniformity of the color field , not the absolute amounts of color added ; the evolution of the dispersion of the color field about its mean radial value , divided by the mean radial value at each instant of time , is the goal of these models . observations of the dg tau disk by podio et al . ( 2013 ) have shown that dg tau itself irradiates its disk s outer layers from 10 au to 90 au with a strong uv flux , sufficient for significant uv photolysis and the formation of observable water vapor . much higher levels of uv irradiation can occur for protoplanetary disks that form in stellar clusters containing massive stars ( e.g. , walsh et al . 2013 ) , an environment that has been suggested for our own solar system ( e.g. , dauphas & chaussidon 2011 ) in order to explain the evidence for live slris found in primitive meteorites . the fact that molecular hydrogen constitutes the great majority of a disk s mass , yet can not be directly detected , except at the star - disk boundary region , means that estimates of disk masses are uncertain at best ( e.g. , andrews & williams 2007 ) , as they are typically based on an assumed ratio between the amount of mm - sized dust grains and the total disk mass . isella et al . ( 2009 ) estimated that low- and intermediate - mass pre - main - sequence stars form with disk masses ranging from 0.05 to 0.4 @xmath1 . dg tau s disk has a mass estimated to be as high as 0.1 @xmath1 ( podio et al . recently , the mass of the tw hydra disk was revised upward to at least 0.05 @xmath1 ( bergin et al . 2013 ) . these and other observations suggest that the mgu disk masses assumed in these models may be achieved in some fraction of protoplanetary disks , and perhaps in the solar nebula as well . in fact , miller et al . ( 2011 ) detected a fu orionis outburst in the classical t tauri star lkh@xmath35 188-g4 . disk masses are typically thought to be @xmath36 for such stars . the fact that a fu orionis event occurred in lkh@xmath35 188-g4 shows that even the disks around class ii - type objects can experience instabilities leading to rapid mass accretion , e.g. , mgu disk phases . we present results for a variety of protostellar and protoplanetary disk masses , varied initial minimum @xmath2 stability parameters , and varied injection radii , for disks of two different sizes . table 2 shows the initial conditions for the models with a @xmath37 disk in orbit around a @xmath38 protostar . the disks extend from 1 au to 10 au , as in the models by boss ( 2008 , 2011 ) . the main difference from these previous models is that the disk mass ( @xmath37 ) is considerably lower than that of the previous models ( @xmath39 ) . as a result , the initial minimum @xmath2 values are considerably higher than in the previous models , ranging from 2.2 to 3.1 , compared to the previous range of 1.4 to 2.5 . the present models are thus less gravitationally unstable initially than the disks previously considered , with the goal being to learn whether or not the previous results will change for higher values of @xmath40 . figure 1 displays the initial midplane temperature profiles for these models . only the outermost regions of the disks are cool enough to be gravitationally unstable , but the models show that this is sufficient to result in qualitatively similar behavior for all of the table 2 models . figures 3 and 4 show the equatorial plane distribution of the color / gas ratio ( @xmath41 ) for model 1.0 - 2.6 - 9 . this ratio is plotted , as it is equivalent to the @xmath4al/@xmath42al and @xmath5o/@xmath6o ratios measured by cosmochemists , i.e. , the abundance of an injected or photolysis product species , divided by that of a species that was prevalent in the pre - injection disk . figure 3 shows that the initial disk surface injection at 9 au has resulted in the rapid transport of the color field downward to the disk s midplane , as well as inward to close to the inner disk boundary at 2 au . the vigorous three dimensional motions of a mgu disk are responsible for this large - scale transport in just 34 yr . at this time , the color / gas ratio is still highly heterogeneous , but figure 4 shows that only 146 yr later , the color / gas ratio has been strongly homogenized throughout the entire disk midplane . figure 5 shows the evolution of the dispersion of the ratio of the color density to the gas density for models 1.0 - 2.6 - 9 and 1.0 - 2.6 - 2 at two times . these models differ only in the injection radius , either 9 au or 2 au . the dispersion plotted in figure 5 is defined to be the square root of the sum of the squares of the color field divided by the gas density , subtracted by the azimuthal average of this ratio at a given orbital radius , divided by the square of the azimuthal average at that radius , normalized by the number of azimuthal grid points , and plotted as a function of radius in the disk midplane . figure 5 shows that the isotopic dispersion is a strong function of orbital radius and time , with the dispersion initially being relatively large ( i.e. , at 180 yr , in spite of the apparent homogeneity seen in figure 4 at the same time ) as a result of the isotopes traveling downward and radially inward and outward . however , the dispersion decreases dramatically in the outer disks for both models by 777 yr to a value of @xmath0 1% to 2% . in fact , the dispersion in both models 1.0 - 2.6 - 9 and 1.0 - 2.6 - 2 evolves toward essentially the same radial distribution by this time , showing that the exact location of the injection location has little effect on the long term evolution of the distribution : that is controlled solely by the evolution of the underlying mgu disk , which is identical for these two models ( i.e. , the color fields are passive tracers , and have no effect on the disk s evolution ) . note that any small refractory grains present in the initial disk will be carried along with the disk gas , so that some of the grains that start out at 2 au will be transported to the outermost disk , in the same manner that some of the gas is transported outward . most of the gas and dust , however , is accreted by the growing protostar . figure 6 shows the results for three models with varied @xmath40 , i.e. , models 1.0 - 2.6 - 9 , 1.0 - 2.9 - 9 , and 1.0 - 3.1 - 9 , all after 1370 yr . it can be seen that in spite of the variation in the initial degree of instability , the dispersions in the outermost disks all converge to similar values of @xmath0 1% to 2% . this suggests that mgu disk evolutions are not particularly sensitive to the exact choice of the initial @xmath2 profile , a result that was also found by boss ( 2011 ) for somewhat more massive disks . as also found by boss ( 2011 ) , the dispersions in the innermost disks are significantly higher ( @xmath0 10% to 20% ) than in the outermost disks , a direct result of the stronger mixing associated with the cooler outer disks , in spite of the longer orbital periods in the outer disks . table 3 shows the initial conditions for the models with either @xmath43 disks around @xmath44 protostars , or @xmath45 disks around @xmath46 protostars . in either case , the disks extend from 1 au to 10 au . these models are of interest for exploring how conditions might vary between disks around g dwarfs and m dwarfs , with possible ramifications for the habitability of any rocky planets that form ( e.g. , raymond et al . 2007 ) around m dwarfs . figure 2 shows the initial midplane temperature profiles for these models . figure 7 shows the time evolution of the dispersion for model 0.1 - 1.8 - 2 , appropriate for a late m dwarf protostar . as in all the models , it can be seen that the initially highly heterogeneous disk becomes rapidly homogenized , in this case by about 5000 yr . note that this time scale is considerably longer than that for g dwarf disk mixing and transport processes , as a result of the longer keplerian orbital periods for lower mass , m dwarf protostars . as in the g dwarf protostar models ( e.g. , figure 5 ) , the inner disk dispersion is higher than in the outer disk , though in these models ( with a lower initial @xmath47 ) the inner disk dispersion drops to @xmath0 5% to 10% , compared to @xmath0 1% to 2% in the outer disk . figure 8 shows the same behavior for model 0.1 - 1.8 - 9 , which differs from the previous model shown in figure 7 only in having the injection occur at 9 au instead of 2 au . as in the g dwarf disks , the dispersions for both of these models evolve toward essentially identical radial distributions : the underlying mgu disk evolution determines the outcome for the dispersions . similar results hold for the models with @xmath46 protostars , i.e. , early m dwarf disks . we now turn to a consideration of the consequence of single - shot versus continuous injection at the surface of much larger outer radius ( 40 au ) disks than have been considered to date for g dwarf stars ; boss ( 2007 ) considered disks extending from 4 au to 20 au in radius . table 4 shows the initial conditions for the models with a @xmath48 disk around a @xmath38 protostar , with the disks extending from 10 au to 40 au . because of the much larger inner and outer disk radii for this set of models , these models can be calculated for times as long as @xmath49 yr ( table 4 ) . such times are still considerably less than the typical ages ( @xmath50 yr ) of t tauri stars , implying that in order for mgu disks to occur at such late phases , a prior phase of coupled mri - mgu evolution might be required to make the present results relevant . figures 9 and 10 display the evolution of the dispersions for models 1.0 - 1.1 - 40 - 20 and 1.0 - 1.1 - 40 - 20c , differing only in that the former model has single - shot injection while the latter model has continuous injection , intended to simulate a disk with ongoing uv photolysis and fractionation at the outer disk surface . for model 1.0 - 1.1 - 40 - 20 , it can be seen that the evolution is similar to that of the previous single - shot models : a rapid drop in the dispersion , followed by homogenization to @xmath0 1% to 2% away from the inner disk boundary . the higher dispersions seen near the outer disk boundaries ( @xmath0 40 au ) are largely caused by the unphysical pile - up of considerable disk mass at 40 au and should be discounted . however , for the continuous injection model shown in figure 10 , it can be seen that the dispersions throughout the disk even after @xmath3 yr can be as high as @xmath0 20% , consistent with the much larger variation in stable oxygen isotope ratios , compared to slri ratios . in a calculation with finer spatial grid resolution , as well as perhaps sub - grid mixing processes , one might expect even stronger homogenization to occur , so the dispersion levels obtained from the present models should be considered to be upper bounds . the total amount of color added during the continuous injection models is large , compared to single - shot injection models : for model 1.0 - 1.4 - 40 - 20c , for example , after 200 yr , the total amount of color injected has increased by a factor of @xmath0 90 compared to the single - shot total , and by another factor of @xmath0 60 after 27000 yr . figures 11 and 12 compare the results for continuous injection at either 20 au or 30 au , respectively , i.e. , for models 1.0 - 1.1 - 40 - 20c and 1.0 - 1.1 - 40 - 30c . in spite of the different injection radii , figures 11 and 12 show that even at a relatively early phase ( 405 yr ) of evolution , the midplane color / gas ratios look somewhat similar ; as before , the mgu disk evolution is the same for both models , and that is the primary determinant of the long term evolution . finally , similar results as those shown in figures 9 - 12 were obtained for the other models listed in table 4 . these models show that the main factor in determining the radial dispersion profile is whether the injection occurs in a single - shot or continuously ; in the latter case , the mgu disk does its best to homogenize the color field , but the fact that spatial heterogeneity is being continuously injected limits the degree to which this heterogeneity can be reduced . while dust grains in the interstellar medium are overwhelmingly amorphous , crystalline silicate grains have been found in a late m dwarf ( sst - lup3 - 1 ) disk at distances ranging from inside 3 au to beyond 5 au , in both the midplane and surface layers ( mern et al . such crystalline silicate grains are likely to have been produced by thermal annealing in the hottest regions of the disk , well inside of 1 au ( sargent et al . 2009 ) . again , outward transport seems to be required to explain the observations , and the results for the models with a 0.1 @xmath1 protostar suggest that mgu phases in low mass m dwarf disks may be responsible for these observations . in fact , crystalline mass fractions in protoplanetary disks do not appear to correlate with stellar mass , luminosity , accretion rate , disk mass , or the disk to star ratio ( watson et al . these results also appear to be consistent with the results of the present models , which show that mgu disk phases are equally capable of relatively rapid large - scale mixing and transport , regardless of the stellar or disk mass , or the exact value of the @xmath2 stability parameter . these models have shown a rather robust result , namely that a phase of marginal gravitational instability in disks and stars with a variety of masses and disk temperatures can lead to relatively rapid inward and outward transport of disk gas and small grains , as required to drive the protostellar mass accretion associated with fu orionis events , as well as to explain the discovery of refractory grains in comet 81p / wild 2 . a mgu disk phase driving a fu orionis outburst is astronomically quite likely to have occurred for our protosun , and cosmochemically convenient for explaining the relative homogeneity of @xmath4al/@xmath42al ratios derived from a supernova injection event , and the range of @xmath5o/@xmath6o ratios derived from sustained uv self - shielding at the surface of the outer solar nebula . low - mass stars , from g dwarfs to m dwarfs , may well experience a similar phase of mgu disk mixing and transport . in this context , it is worthwhile to note that fu orionis itself , the prototype of the fu orionis outburst phenomenon , has a mass of only @xmath51 ( zhu et al . 2007 , 2009b ; beck & aspin 2012 ) , i.e. , the mass of a m dwarf , suggesting that m dwarf protoplanetary disks may experience evolutions similar to that of the solar nebula , with possible implications for the habitability of any resulting planetary system ( e.g. , raymond et al . 2007 ; bonfils et al . 2013 ; dressing & charbonneau 2013 ; kopparapu 2013 ) . i thank jeff cuzzi for his comments , the referee for a number of suggested improvements , and sandy keiser , michael acierno , and ben pandit for their support of the cluster computing environment at dtm . this work was partially supported by the nasa origins of solar systems program ( nnx09af62 g ) and is contributed in part to the nasa astrobiology institute ( nna09da81a ) . some of the calculations were performed on the carnegie alpha cluster , the purchase of which was partially supported by a nsf major research instrumentation grant ( mri-9976645 ) . 1.0 & 0.019 & @xmath52 & @xmath53 & 1.0 & 1.0 & 10.0 + 0.5 & 0.018 & @xmath54 & @xmath55 & 1.0 & 1.0 & 10.0 + 0.1 & 0.016 & @xmath56 & @xmath57 & 1.0 & 1.0 & 10.0 + 1.0 & 0.13 & @xmath58 & @xmath59 & 4.0 & 10.0 & 40.0 + 1.0 - 2.2 - 2 & 15 & 2.2 & 2 & 2520 + 1.0 - 2.2 - 9 & 15 & 2.2 & 9 & 2520 + 1.0 - 2.6 - 2 & 20 & 2.6 & 2 & 2043 + 1.0 - 2.6 - 9 & 20 & 2.6 & 9 & 2043 + 1.0 - 2.9 - 2 & 25 & 2.9 & 2 & 1200 + 1.0 - 2.9 - 9 & 25 & 2.9 & 9 & 1400 + 1.0 - 3.1 - 2 & 30 & 3.1 & 2 & 1100 + 1.0 - 3.1 - 9 & 30 & 3.1 & 9 & 1300 + 1.0 - 1.1 - 40 - 20 & 30 & 1.1 & 20 & single - shot & 25000 + 1.0 - 1.1 - 40 - 30 & 30 & 1.1 & 30 & single - shot & 24500 + 1.0 - 1.4 - 40 - 20 & 50 & 1.4 & 20 & single - shot & 24000 + 1.0 - 1.4 - 40 - 30 & 50 & 1.4 & 30 & single - shot & 15000 + 1.0 - 1.1 - 40 - 20c & 30 & 1.1 & 20 & continuous & 19500 + 1.0 - 1.1 - 40 - 30c & 30 & 1.1 & 30 & continuous & 19800 + 1.0 - 1.4 - 40 - 20c & 50 & 1.4 & 20 & continuous & 27000 + 1.0 - 1.4 - 40 - 30c & 50 & 1.4 & 30 & continuous & 27000 +

analyses of primitive meteorites and cometary samples have shown that the solar nebula must have experienced a phase of large - scale outward transport of small refractory grains as well as homogenization of initially spatially heterogeneous short - lived isotopes . the stable oxygen isotopes , however , were able to remain spatially heterogenous at the @xmath0 6% level . one promising mechanism for achieving these disparate goals is the mixing and transport associated with a marginally gravitationally unstable ( mgu ) disk , a likely cause of fu orionis events in young low - mass stars . several new sets of mgu models are presented that explore mixing and transport in disks with varied masses ( 0.016 to 0.13 @xmath1 ) around stars with varied masses ( 0.1 to 1 @xmath1 ) and varied initial @xmath2 stability minima ( 1.8 to 3.1 ) . the results show that mgu disks are able to rapidly ( within @xmath3 yr ) achieve large - scale transport and homogenization of initially spatially heterogeneous distributions of disk grains or gas . in addition , the models show that while single - shot injection heterogeneity is reduced to a relatively low level ( @xmath0 1% ) , as required for early solar system chronometry , continuous injection of the sort associated with the generation of stable oxygen isotope fractionations by uv photolysis leads to a sustained , relatively high level ( @xmath0 10% ) of heterogeneity , in agreement with the oxygen isotope data . these models support the suggestion that the protosun may have experienced at least one fu orionis - like outburst , which produced several of the signatures left behind in primitive chondrites and comets .

arxiv

as in our previous analysis of run 1a data @xcite , we conduct a general search for new particles with a narrow natural width that decay to dijets . in addition , we search for the following particles summarized in fig . [ fig_particles ] : axigluons @xcite from chiral qcd ( @xmath7 ) , excited states @xcite of composite quarks ( @xmath8 ) , color octet technirhos @xcite ( @xmath9 ) , new gauge bosons ( @xmath10,@xmath11 ) , and scalar @xmath12 diquarks @xcite ( @xmath13 and @xmath14 ) . using four triggers from run 1a and 1b , we combine dijet mass spectra above a mass of 150 gev / c@xmath1 , 241 gev / c@xmath1 , 292 gev / c@xmath1 , and 388 gev / c@xmath1 with integrated luminosities of = 7.5 in = 3.3 in = 3.3 in .089 pb@xmath0 , 1.92 pb@xmath0 , 9.52 pb@xmath0 , and 69.8 pb@xmath0 respectively . jets are defined with a fixed cone clustering algorithm ( r=0.7 ) and then corrected for detector response , energy lost outside the cone , and underlying event . we take the two highest @xmath15 jets and require that they have pseudorapidity @xmath16 and a cms scattering angle @xmath17| < 2/3 $ ] . the @xmath18 cut provides uniform acceptance as a function of mass and reduces the qcd background which peaks at @xmath19 . in fig . [ fig_dijet ] the dijet mass distribution is presented as a differential cross section in bins of the mass resolution ( @xmath20% ) . at high mass the data is systematically higher than a prediction from pythia plus a cdf detector simulation , similar to the inclusive jet @xmath21 spectrum @xcite . to search for new particles we determine the qcd background by fitting the data to a smooth function of three parameters @xcite ; fig . [ fig_dijet ] shows the fractional difference between the data and the fit ( @xmath22 ) . we note upward fluctuations near 200 gev / c@xmath1 ( @xmath23 ) , 550 gev / c@xmath1 ( @xmath24 ) and 850 gev / c@xmath1 ( @xmath25 ) . for narrow resonances it is sufficient to determine the mass resolution for only one type of new particle because the detector resolution dominates the width . in fig . [ fig_resonance ] we show the mass resolution for excited quarks ( q * ) from pythia plus a cdf detector simulation ; the long tail at low mass comes from gluon radiation . for each value of new particle mass in 50 gev / c@xmath1 steps , we perform a binned maximum likelihood fit of the data to the background parameterization and the mass resonance shape . in fig . [ fig_resonance ] we display the best fit and 95% confidence level upper limit for a 550 gev / c@xmath1 resonance . for the mass region @xmath26 gev / c@xmath1 , there are 2947 events in the data , @xmath27 events ( @xmath28 ) in the background for the fit without a resonance , @xmath29 events ( @xmath30 ) in the background for the fit that includes the resonance , and the value of the resonance cross section from the fit is @xmath31 pb ( statistical ) . in fig . [ fig_cos ] we study the angular distribution of the fluctuation in the mass region @xmath26 gev / c@xmath1 . the angular distribution is compatible with both qcd alone , and with = 3.3 in = 3.3 in qcd + 5% excited quark ( best fit ) . this amount of excited quark is coincidentally the same as found in the mass fit . although the fluctuation is interesting , we conclude it is not yet = 3.3 in = 3.3 in statistically significant , and proceed to set limits on new particle production . = 3.3 in = 3.3 in from the likelihood distribution including experimental systematic uncertainties @xcite we obtain the 95% cl upper limit on the cross section for new particles shown in fig . [ fig_dijet_limit ] . we compare this to the cross section for axigluons ( excluding @xmath32 gev / c@xmath1 ) , excited quarks ( excluding @xmath33 gev / c@xmath1 ) , technirhos ( excluding @xmath34 gev / c@xmath1 ) , w@xmath35 ( excluding @xmath36 gev / c@xmath1 ) , z@xmath35 ( excluding @xmath37 gev / c@xmath1 ) , and e6 diquarks ( excluding @xmath38 gev / c@xmath1 ) . the calculations are lowest order @xcite using cteq2l parton distributions @xcite and one - loop @xmath39 and require @xmath16 and @xmath40 . the large mass of the top quark suggests that the third generation may be special . topcolor @xcite assumes that the top mass is large mainly because of a dynamical @xmath5 condensate generated by a new strong dynamics coupling to the third generation . here the @xmath41 of qcd is a low energy symmetry arising from the breaking of an @xmath42 coupling to the third generation and an @xmath43 coupling to the first two generations only . there are then massive color octet bosons , topgluons @xmath44 , which couple largely to @xmath4 and @xmath5 . the topgluon is strongly produced and decays mainly to the third generation ( @xmath45 ) with a relatively large natural width ( @xmath46 ) . here we search for the topgluon in the @xmath4 channel . an additional @xmath47 symmetry is introduced @xcite to keep the @xmath48 quark light while the top quark is heavy ; this leads to a topcolor @xmath6 , which again couples largely to @xmath4 and @xmath5 . the topcolor @xmath6 is electroweakly produced and decays mainly to the third generation ( @xmath49 ) with a narrow natural width ( @xmath50 ) . here we search for the topcolor @xmath6 in both the the @xmath4 and @xmath5 channel ; the @xmath5 channel is the most sensitive because the coupling to @xmath5 is larger . we start with the dijet search in 19 pb@xmath0 of run 1a data @xcite and additionally require at least one of the two leading jets be tagged as a bottom quark . the b - tag requires a displaced vertex in the the secondary vertex detector @xcite . the @xmath4 event efficiency is @xmath51% independent of dijet mass . from fits to the @xmath52 distribution , we estimate that the sample is roughly 50% bottom , 30% charm , and 20% mistags of plain jets . pythia predicts that 1/5 of these bottom quarks are direct @xmath4 , and the rest are from gluon splitting and flavor excitation . consequentially , only about @xmath53% @xmath54 10% of our sample is direct @xmath4 . we expect both the purity and efficiency to increase when we use the run 1b dataset and a new tagging algorithm @xcite . with higher tagging efficiency we should be able to make better use of double b - tagged events like the one in fig . [ fig_btag_event ] . = 2.6 in = 2.6 in in fig . [ fig_btag ] we show the b - tagged dijet mass distribution corrected for the @xmath4 efficiency . also shown is the untagged dijet mass distribution from run 1a , and both are well fit with our standard parameterization @xcite . the b - tagged dijet data has an upward fluctuation near 600 gev / c@xmath1 . we model the shape of a narrow resonance using pythia z@xmath3 production and a cdf detector simulation . in fig . [ fig_btag ] we fit the b - tagged data to a 600 gev / c@xmath1 narrow resonance , and find a cross section of @xmath55 pb ( statistical ) . note that this is comparable to the dijet fluctuation in both mass and rate . however , there are only 8 events in the last two data bins of fig . [ fig_btag ] , and the fluctuation is only a @xmath56 effect , so we proceed to set limits on new particle production . we perform two kinds of fits for the limits . first , narrow resonances are modelled as described above , and the mass resolution in the cdf detector is shown in fig . [ fig_b_bbar_res ] . second , wide resonances characteristic of topgluons @xcite , including interference with normal gluons , was incorporated into pythia and a cdf detector simulation . the mass resolution in fig . [ fig_b_bbar_res ] displays destructive interference to the left of the resonance ; models with destructive interference on the right side of the resonance will be considered in the future . = 3.2 in = 3.2 in limits on new particle production are shown in fig . [ fig_btag_limit ] . the theoretical cross sections are lowest order and use cteq2l parton distributions . for narrow resonances the production cross sections are nt large enough for us to set mass limits at this time . for topgluons the production cross sections @xcite are larger , and we are able to exclude at 95% cl topgluons of width @xmath57 in the mass region @xmath58 gev / c@xmath1 , @xmath59 for @xmath60 gev / c@xmath1 , and @xmath61 for @xmath62 gev / c@xmath1 . = 3.0 in = 3.0 in = 7.7 in to search for new particles decaying to @xmath5 we start with the data sample from the top mass measurement @xcite . there we used top decays to w + four jets with at least one b - tag , and found 19 events on a background of @xmath63 , resulting in a top mass of @xmath64(stat)@xmath65(sys ) gev / c@xmath1 . that analysis fit the entire event for the top hypothesis , discarding events with @xmath66 ( poor fit ) . here we add the additional constraint that the top mass is 176 gev / c@xmath1 , which significantly enhances our resolution of the @xmath5 mass . two of the 19 events fail the @xmath66 cut when the top mass constraint is added to the fit , leaving us with 17 events . the @xmath5 mass distribution expected from a narrow resonance , normalized to the topcolor @xmath6 predicted rate @xcite , is shown in fig . [ fig_ttbar ] . here we used pythia @xmath67 . also in fig . [ fig_ttbar ] is the monte carlo distribution of the background , on the left standard model top production from herwig , and on the right qcd w + jets background from vecbos with parton showers from herwig . all monte carlos include a cdf detector simulation . on the left in fig . [ fig_ttbar ] , the comparison of the topcolor z@xmath3 to sm @xmath5 simulations illustrates that in this data sample we are sensitive to topcolor @xmath6 up to a mass of roughly 600 gev / c@xmath1 . finally , on the right in fig . [ fig_ttbar ] , we present the @xmath5 candidate mass distribution from cdf compared to the total standard model prediction . given the statistics the agreement is quite good overall . the small shoulder of 6 events on a background of @xmath68 in the region @xmath69 gev / c@xmath1 is in an interesting mass region , given the dijet and @xmath4 search results , but is not statistically significant . upper limits on the @xmath5 cross section as a function of @xmath5 mass , and on a topcolor @xmath6 , are currently in progress . = 3.1 in = 3.1 in we have searched for new particles decaying to dijets , @xmath4 , and @xmath5 . in the dijet channel we set the most significant direct mass exclusions to date on the hadronic decays of axigluons ( excluding @xmath70 gev / c@xmath1 ) , excited quarks ( excluding @xmath71 gev / c@xmath1 ) , technirhos ( excluding @xmath34 gev / c@xmath1 ) , w@xmath3 ( excluding @xmath36 gev / c@xmath1 ) , z@xmath35 ( excluding @xmath37 gev / c@xmath1 ) , and for the first time e6 diquarks ( excluding @xmath38 gev / c@xmath1 ) . in the @xmath4 channel we set the first limits on topcolor , excluding a model of topgluons for width @xmath57 in the mass region @xmath58 gev / c@xmath1 , @xmath59 for @xmath60 gev / c@xmath1 , and @xmath61 for @xmath62 gev / c@xmath1 . the search for topcolor in the @xmath5 channel has just begun and limits are in progress . limits are only a consolation prize ; the main emphasis of our search is to explore the possibility of a signal . although we do not have significant evidence for new particle production , the @xmath72 gev / c@xmath1 region shows upward fluctuations in all three channels . we can not ignore the exciting possibility that these apparently separate fluctuations may be the first signs of a new physics beyond the standard model . the remaining integrated luminosity for run 1b , currently being accumulated and analyzed , has the potential to either kill the fluctuations or reveal what may be the most interesting new physics in a generation . + f. abe et al . ( cdf ) , phys . lett . * 74 * , 3538 ( 1995 ) . p. frampton and s. glashow , phys . b190 * , 157 ( 1987 ) . u. baur et al . , int . j. mod . phys a2 , 1285(1987 ) & pr*d42 * , 815(1990 ) . k. lane et al . , pr*d44 * , 2768(1991 ) & phys . lett . * b327 * , 129(1994 ) . j. hewett and t. rizzo , phys . rep . * 183 * , 193 ( 1989 ) . talk by anwar bhatti in these proceedings . parameterization @xmath73 with parameters a , n and p. for new gauge bosons we use a k - factor to account for higher order terms . j. botts et al . ( cteq collaboration ) phys . lett . * b304 * , 159 ( 1993 ) . c. hill and s. parke , phys . rev . * d49 * , 4454 ( 1994 ) . c. hill , phys . b345 * , 483 ( 1994 ) . f. abe et al . ( cdf ) , phys . rev . * d50*,2966 ( 1994 ) . f. abe et al . ( cdf ) , phys . lett . * 74 * , 2626 ( 1995 ) . we model the interference between normal gluons and topgluons using hybrid model c in phys . rev . * d49 * , 4454 ( 1994 ) . g. burdman , c. hill , and s. parke private communication .

we present three searches for new particles at cdf . first , using 70 pb@xmath0 of data we search the dijet mass spectrum for resonances . there is an upward fluctuation near 550 gev / c@xmath1 ( 2.6@xmath2 ) with an angular distribution that is adequately described by either qcd alone or qcd plus 5% signal . there is insufficient evidence to claim a signal , but we set the most stringent mass limits on the hadronic decays of axigluons , excited quarks , technirhos , w@xmath3 , z@xmath3 , and e6 diquarks . second , using 19 pb@xmath0 of data we search the b - tagged dijet mass spectrum for @xmath4 resonances . again , an upward fluctuation near 600 gev / c@xmath1 ( 2 @xmath2 ) is not significant enough to claim a signal , so we set the first mass limits on topcolor bosons . finally , using 67 pb@xmath0 of data we search the top quark sample for @xmath5 resonances like a topcolor @xmath6 . other than an insignificant shoulder of 6 events on a background of 2.4 in the mass region 475 - 550 gev / c@xmath1 , there is no evidence for new particle production . mass limits , currently in progress , should be sensitive to a topcolor z@xmath3 near 600 gev / c@xmath1 . in all three searches there is insufficient evidence to claim new particle production , yet there is an exciting possibility that the upward fluctuations are the first signs of new physics beyond the standard model . fermilab - conf-95/152-e + cdf / pub / exotic / public/3192 + + * search for new particles decaying to dijets , + @xmath4 , and @xmath5 at cdf + * _ fermilab ms 318 + batavia , il 60510 + _

arxiv

the first observational evidence that the universe had entered a period of accelerated expansion was obtained when supernovae type ia ( snia ) were found to be fainter than expected @xcite . this fact has been confirmed by many independent observations such as temperature anisotropies of the cosmic microwave background ( cmb ) @xcite , inhomogeneities in the matter distribution @xcite , the integrated sachs wolfe ( isw ) effect @xcite , baryon acoustic oscillations ( bao ) @xcite , weak lensing ( wl ) @xcite , and gamma - ray bursts @xcite . within the framework of general relativity ( gr ) , the accelerated expansion is driven by a new energy density component with negative pressure , termed dark energy ( de ) . the nature of this unknown matter field has given rise to a great scientific effort in order to understand its properties . the observational evidence is consistent with a cosmological constant @xmath0 being the possible origin of the dark energy ( de ) driving the present epoch of the accelerated expansion of our universe and a dark matter ( dm ) component giving rise to galaxies and their large scale structures distributions @xcite . the dm is assumed to have negligible pressure and temperature and is termed cold . thanks to the agreement with observations the model is commonly known as @xmath0cdm , to indicate the nature of its main components . while favored by the observations , the model is not satisfactory from the theoretical point of view : the value of the cosmological constant is many orders of magnitude smaller than what it was estimated in particle physics @xcite . it was suggested soon that de could be dynamic , evolving with time @xcite . this new cosmological model also suffers from a severe fine - tune problem known as _ coincidence problem _ @xcite that can be expressed with the following simple terms : if the time variation of matter and de are very different why are their current values so similar ? cosmological models where dm and de do not evolve separately but interact with each other were first introduced to justify the currently small value of the cosmological constant @xcite but they were found to be very useful to alleviate the coincidence problem . in this review we will summarize the theoretical developments of this field and the observational evidence that constrains the dm / de interaction and could , eventually , lead to its detection . the emergence of galaxies and large scale structure is driven by the growth of matter density perturbations which themselves are connected to the anisotropies of the cmb @xcite . an interaction between the components of the dark sector will affect the overall evolution of the universe and its expansion history , the growth matter and baryon density perturbations , the pattern of temperature anisotropies of the cmb and the evolution of the gravitational potential at late times would be different than in the concordance model . these observables are directly linked to the underlying theory of gravity @xcite and , consequently , the interaction could be constrained with observations of the background evolution and the emergence of large scale structure . this review is organized as follows : in this introduction we describe the concordance model and we discuss some of its shortcomings that motivates considering interactions within the dark sector . since the nature of de and dm are currently unknown , in sec . [ sec : sec2 ] we introduce two possible and different approaches to describe the de and the dm : fluids and scalar fields . based on general considerations like the holographic principle , we discuss why the interaction within the dark sector is to be expected . in sec . [ sec : sec3 ] we review the influence of the interaction on the background dynamics . we find that a dm / de interaction could solve the coincidence problem and satisfy the second law of thermodynamics . in sec . [ sec : sec4 ] the evolution of matter density perturbations is described for the phenomenological fluid interacting models . in sec . [ sec : sec5 ] we discuss how the interaction modifies the non - linear evolution and the subsequent collapse of density perturbations . in sec . [ sec : sec6 ] we describe the main observables that are used in sec . [ sec : sec7 ] to constrain the interaction . finally , in sec . [ sec : sec8 ] we describe the present and future observational facilities and their prospects to measure or constrain the interaction . in table [ table : acronyms ] we list the acronyms commonly used in this review . @llll acronym & meaning & acronym & meaning + a - p & alcock - packzynki & ksz & kinematic sunyaev - zeldovich + bao & baryon accoustic oscillations & lbg & lyman break galaxies + cdm & cold dark matter & lhs & left hand side ( of an equation ) + cl & confidence level & lisw & late integrated sachs - wolfe + cmb & cosmic microwave background & lss & large scale structure + de & dark energy & mcmc & monte carlo markov chain + detf & dark energy task force & rhs & right hand side ( of an equation ) + dm & dark matter & rsd & redshift space distortions + eos & equation of state & sl & strong lensing + eisw & early integrated sachs - wolfe & snia & supernova type ia + frw & friedman - robertson - walker & tsz & thermal sunyaev - zeldovich + isw & integrated sachs - wolfe & wl & weak lensing + the current cosmological model is described by the friedmann - robertson - walker ( frw ) metric , valid for a homogeneous and isotropic universe @xcite @xmath1 , \label{eq : frw - canonical}\ ] ] where @xmath2 is the scale factor at time @xmath3 , the present time is denoted by @xmath4 and the scale factor is normalized to @xmath5 ; @xmath6 is the gaussian curvature of the space - time . we have chosen units @xmath7 but we will reintroduce the speed of light when needed . a commonly used reparametrization is the _ conformal time _ , defined implicitly as @xmath8 . in terms of this coordinate , the line element is @xmath9 . \label{eq : frw - conformal}\ ] ] if we describe the matter content of the universe as a perfect fluid with mean energy density @xmath10 and pressure @xmath11 , friedmann s equations are @xcite @xmath12 where @xmath13 is the hubble function and @xmath14 are the energy density and pressure of the different matter components , related by an equation of state ( eos ) parameter @xmath15 . in terms of the conformal time , the expression @xmath16 is used . usually densities are measured in units of the critical density : @xmath17 with @xmath18 . the curvature term can be brought to the right hand side ( rhs ) by defining @xmath19 . as a matter of convention , a sub - index `` 0 '' denotes the current value of any given quantity . due to the historically uncertain value of the hubble constant , its value is usually quoted as @xmath20kms@xmath21mpc@xmath21 so the parameter @xmath22 encloses the observational uncertainty on the value of the hubble constant . [ cols="^,^ " , ] the phenomenological description of the interaction between dark sectors was introduced in sec . [ sec : sec2 ] and the linear perturbation theory of the model was discussed in sec . [ sec : sec4 ] . for the sake of simplicity in our subsequent discussion , we will review only models with a de eos parameter @xmath23 . the results for a variable eos have been reported in @xcite . we restrict our analysis to the models that satisfy the stability condition ( sec . [ sec : sec4.stability ] ) . the interaction kernels were summarized in table [ table : models ] . the results of the mcmc analysis using different data sets are shown in fig . [ figure2_sec8 ] . when the coupling is proportional to the energy density of the de , the data constraints the value of the interaction parameter to be in the range @xmath24 . when the coupling is proportional to the energy density of the dm or the total energy density of the dark sector , the constraint is much tighter and the coupling are positive at the 68% confidence level ( cl ) . including additional data tightens the constraints on the cosmological parameters compared with the cmb data alone . results for different models can be found in @xcite . the conclusion of these studies shows that interacting dm / de models are compatible with observations . in @xcite it was found evidence of the existence of interaction but with a low cl . more recently the planck collaboration also found that coupled dm / de were in agreement with observations @xcite . in the next section we will further discuss the evidence in favor of an interactions . as a function of the interaction parameters @xmath25 for models iii and i , ii , respectively . in the left panel we used the cosmological parameters from table v , xi and xii of @xcite . in the right panel , labeled boss , the cosmological parameters are those of table [ table : boss_parameters ] . the horizontal line corresponds to the boss measured value @xmath26 with the shaded areas representing @xmath27 and @xmath28 cl . ] together with cmb data , low redshift observations like luminosity distances from snia have been used to test the dm / de interaction . at low redshifts the deviations from @xmath0cdm are not very pronounced and it is easier to establish the existence of a dynamical de component or even an interacting one using high redshift observations . recently , the analysis of boss data presented evidence against the concordance model by measuring the bao in the redshift range @xmath29 from the correlation function of the ly@xmath30 forest from high redshift quasars @xcite . their result indicates a @xmath31 deviation ( from _ _ planck__+wmap ) and @xmath32 deviation ( from act / spt ) from @xmath0cdm at @xmath33 . while the statistical evidence is still not significant , if confirmed , this result can not be explained by a dynamical de component , and it suggests a more exotic type of de . an interacting de appears as a simple and efficient solution to explain the boss result . if de and dm interact and the former transfers energy to the latter , as required to alleviate the coincidence problem ( see sec [ sec : sec2.direction ] ) and indicated by the data @xcite , it would explain the value of the hubble parameter , @xmath34 , obtained by the boss collaboration , value that is smaller than the expectation from @xmath0cdm @xcite . it would explain the discrepancy of the angular diameter distance at high redshifts . let us now briefly summarize which of the models given in table [ table : models ] can explain the boss results . let us consider a universe filled only with dm , de and baryons . we can use the hubble parameter obtained from the friedmann equation and compare it with the value obtained by the boss collaboration for different sets of adjusted cosmological parameters . we can also compare the constraints for @xmath35 and @xmath36 given by the boss experiment with constraints from cmb adjusted data using @xmath0cdm and the interaction model . to carry out this analysis we need to establish first the evolution with redshift of the energy densities of each component , specially de and dm since due to the interaction they are not independently conserved . for the models i and ii , they behave as @xcite @xmath37 \rho_d^0}{\xi_2+\omega_d}+\rho_c^0\right\ } . \label{int_de}\end{aligned}\ ] ] the baryonic density is given by the usual expression @xmath38 . for the model iii , the evolution is @xmath39 from these solutions , it is easy to establish that when the energy is transferred from the de to the dm , the energy density of the dm is always smaller than what it would have been in the standard @xmath0cdm model . since @xmath40 is the dominant component at redshifts @xmath41 and it is smaller than in the concordance model , so it would be the hubble parameter , as indicated by the boss data . ccc parameter & bestfit & @xmath42 + @xmath22 & @xmath43 & @xmath44 + @xmath45 & @xmath46 & @xmath47 + @xmath48 & @xmath49 & @xmath50 + @xmath51 & @xmath52 & @xmath53 + [ table : boss_parameters ] to make the previous statement more quantitative we took two sets of values for the cosmological parameters @xmath54 and @xmath55 : ( 1 ) the values used by boss collaboration , obtained from the planck collaboration analysis of the @xmath0cdm model and listed in table [ table : boss_parameters ] , and ( 2 ) the values derived by @xcite by fitting dm / de interacting fluid models to the _ planck _ , bao , snia and @xmath55 indicated above . using both data sets , the hubble parameter at @xmath33 has been computed using eqs . ( [ int_dm],[int_de ] ) for the cosmological models listed in table [ table : models ] . the results are shown in fig . ( [ fig : all_models ] ) . the right panel corresponds to the cosmological parameters of the boss collaboration and the left panel to those of @xcite . the figure shows the measured value @xmath35 and its @xmath27 and @xmath28 contours . in both panels , the @xmath0cdm model that corresponds to the case of no interaction is always outside the @xmath27cl . while still not significant , it does show that the data prefers an dm / de interacting model with positive interaction . further improvements on the data could help to establish the existence of an interaction . is the critical energy density today . the attractor solutions of @xmath56 does not depend on the initial conditions at the early time of the universe . the purple lines represent the density evolution of cosmological model with different initial conditions . solid circles represent the density contrast @xmath56 today . values change with the initial conditions but they are bounded in two attractor solutions @xmath57 in @xmath58 plane.,width=480,height=432 ] one motivation to study interacting dm / de models is to alleviate the coincidence problem . only at the present time the dm / de ratio in the @xmath59 model is of order unity , demanding a fine - tuning on the initial conditions at the planck scale of 90 orders of magnitude ( see sec . [ sec : sec1.coincidence ] ) . let us examine if once interacting de models are confronted with observations , the goal can be satisfied . we will concentrate on the analysis of the phenomenological fluid model . the results shown in fig . [ figure2_sec8 ] suggest that positive coupling parameters are compatible with the data and positive values work in the direction of solving the cosmological coincidence problem @xcite and similar conclusions have also been reached for the field description of the de @xcite . let us particularize our analysis for fluid models , and in particular for model iv . in sec . [ sec : sec2.solution ] we have demonstrated how this model solves the coincidence problem . as illustrated in fig . [ fig : ratios](a ) the ratio has two attractor solutions @xmath60 ; the past solution and future ratios are given in eq . ( [ r+- ] ) . when the coupling is @xmath61 , the ratios behave asymptotically as @xmath62 i.e. , the behavior of the attractor solutions of the ratio @xmath56 only depends on the coupling constant @xmath63 and does not depend on the initial conditions at the planck scale . this solution of the coincidence problem is illustrated in fig . [ figure5_sec7 ] . purple solid lines represent the evolution of the energy densities in units of the critical energy density today , @xmath64 , with different initial conditions . the density contrast @xmath56 at present is different for different initial conditions but all the curves are bounded by the two attractors solutions @xmath65 and @xmath66 . in this particular case we fixed @xmath67 and @xmath68 . during the whole thermal history of the universe , the dm to de ratio takes values within the range @xmath69 ; it changes much less than that in the @xmath59 model , thus the cosmological coincidence problem is greatly alleviated . the discovery that the expansion of the universe is accelerating has led to large observational programs being carried out to understand its origin . new facilities are being designed and built aiming to measure the expansion history and the growth of structure in the universe with increasing precision out to greater redshifts . since the interaction in the dark sectors changes the expansion history of the universe and the evolution of matter and radiation density perturbations , peculiar velocities and gravitational fields , these new facilities will not only test the current period of accelerated expansion but also explain the nature on the interaction between dark sectors . thus , current and future observations could and will be used to set up constraints on interactions between dark sectors and clarify the nature of dm / de interactions . observations of type ia supernova , baryon acoustic oscillations ( bao ) , gravitational lensing , redshift - space distortions and the growth of cosmic structure probe the evolution of the universe at @xmath70 . in parallel , the physics of dm / de interactions at recombination can be probed by the cmb radiation power spectrum while the isw effect and lensing pattern of the cmb sky are sensitive to the growth of matter at lower redshifts . the de task force ( detf ) was established to advise the different u.s . funding agencies on future de research . their report categorized different experimental approaches by introducing a quantitative `` figure of merit '' that is sensitive to the properties of de , including its evolution with time @xcite . using this figure of merit , they evaluated ongoing and future de studies based on observations of baryon acoustic oscillations , galaxy clusters , supernova and weak lensing . the detf categorized the different experiments by their different degree of development . stage i referred to the discovery experiments , stage ii to the on - going experiments at that time when the report was elaborated ( circa 2006 ) , stage iii was defined as the next generation that are currently in full operation . they also looked forward to a stage iv generation of more capable experiments . examples of stage ii surveys are the canada - france - hawaii telescope ( cfht ) legacy survey , with observations of snia @xcite and weak lensing @xcite and that ended in 2009 , the essence @xcite and sdss - ii @xcite supernova surveys and bao measurements from the sdss @xcite . while new observations continue to be expanded and improved with more recent instruments , the chft lensing survey remains the largest weak lensing survey to date . in this section we will briefly review projects that are currently operating or under construction ( stage iii and iv ) . all of these facilities share the common feature of surveying wide areas to collect large samples of galaxies , clusters , and/or supernovae and they will help clarify the nature of the interaction between dark sectors . more details can be found in @xcite . the existing and planned ground based de experiments collect data on snia , galaxy clustering and gravitational lensing . wide - field imaging is used to measure weak gravitational lensing and clustering of galaxies in bins of photometrically estimated redshifts and wide - field spectroscopy , to map the clustering of galaxies , quasars and the ly-@xmath30 forest and measure distances and expansion rates with bao and the history of structure growth with redshift - space distortions ( rsd ) . type ia supernovae are searched to determine the distance - redshift relation . the 6-degree field galaxy survey ( 6dfgs ) has mapped the nearby universe over @xmath71 deg@xmath72 of the southern sky with galactic latitude @xmath73 . the median redshift of the survey is @xmath74 . it is the largest redshift survey of the nearby universe , reaching out to @xmath75 . the survey data includes images , spectra , photometry , redshifts and a peculiar velocity survey of a subsample of 15,000 galaxies . the final release of redshift data is given in @xcite . the baryon oscillation spectroscopic survey ( boss ) is currently the largest spectroscopic redshift survey worldwide , mapping @xmath76 deg@xmath72 up to @xmath77 . boss is the largest of the four surveys that comprise sdss - iii and has been in operation for 5 years since 2009 . its goals are to measure angular diameter distance and expansion rate using bao , using 1.5 million galaxies @xcite . using ly-@xmath30 lines towards a dense grid of high - redshift quasars , it has pioneered a method to measure bao at redshifts @xmath78 $ ] . the analysis of sdss data release 9 has provided a measurement of the bao scale at @xmath79 with a precision of 2 - 3% @xcite . this survey will be followed by the extended boss ( eboss ) that will be operating for six years and will extend the boss survey to higher redshifts . similar to boss , the hobby - eberly telescope de experiment ( hetdex ) at the austin mcdonald observatory has the goal of providing percent - level constraints on the hubble parameter and angular diameter distance on the redshift range @xmath80 $ ] by using a combination of bao and power spectrum shape information . it will be achieved by surveying 0.8 million @xmath81 emitting galaxies on a field of view of 420deg@xmath72 @xcite . the javalambre physics of the accelerating universe astronomical survey ( j - pas ) is a new astronomical facility dedicated to mapping the observable universe in 56 colors @xcite . the starting date for this multi - purpose astrophysical survey is 2015 . in five years , j - pas will cover @xmath82deg@xmath72 using a system of 54 narrow band and 2 broad - band filters in the range @xmath83 nm . the filter system was optimized to accurately measure photometric redshifts for galaxies up to @xmath84 . the main instruments are a 2.5 m telescope located at el pico del buitre ( teruel , spain ) and a 1.2 giga - pixel camera . the main goals of the survey are to measure angular and radial components of bao from the galaxy clustering , determine the evolution of the cosmic volume from cluster counts and luminosity distances from snia . the filter system will permit to determine the redshifts of the observed supernovae . the camera is not optimized to measure galaxy elipticities so weak lensing studies would require ellipticity measurements obtained from other surveys . the jpas telescope will measure bao from high redshift quasars to achieve a better precision than boss @xcite , open the possibility of using the test described in sec . [ sec : sec7.evidence ] to disproof the concordance model . the panoramic survey telescope and rapid response system ( pan - starrs ) describes a facility with a cosmological survey among its major goals . the final goal is to use four coordinated telescopes to carry out survey of the full sky above dec=@xmath85 @xcite that will go a factor @xmath86 deeper than the sdss imaging survey . the survey will provide data on high redshift sn , galaxy clustering and gravitational lensing . for that purpose , in addition to the wide survey , an ultra - deep field of @xmath87deg@xmath72 will be observed down to magnitude 27 in the @xmath88 band with photometric redshifts to measure the growth galaxy clustering . data from this facility has already been used to constrain the equation of state parameter @xcite . the wigglez de survey is a large - scale redshift survey carried out at the anglo - australian telescope and is now complete . it has measured redshifts for @xmath89 galaxies over 1000deg@xmath72 in the sky . it combines measurements of cosmic distance using bao with measurements of the growth of structure from redshift - space distortions out to redshift @xmath90 @xcite . the atacama cosmology telescope ( act ) operates at 148 , 218 and 277 ghz with full - width at half maximum angular resolutions of @xmath91 @xcite . act observes the sky by scanning the telescope in azimuth at a constant elevation of @xmath92 as the sky moves across the field of view in time , resulting in a stripe - shaped observation area . the collaboration has released two observed areas of @xmath93deg@xmath72 and @xmath94deg@xmath72 @xcite . sky maps , analysis software , data products and model templates are available through nasa legacy archive for microwave background data analysis ( lambda ) . the south pole telescope ( spt ) is a 10 m telescope designed to map primary and secondary anisotropies in the cmb , currently operating at 95 , 150 , 220 ghz with a resolution with resolution ( 1.7 , 1.2 , 1.0)@xmath95 . the noise levels are 18@xmath96k at 150ghz and @xmath97 larger for the other two channels @xcite . it has observed a region of @xmath98deg@xmath72 . data in the three frequencies were used to produce a radiation power spectra covering the multipole range @xmath99 . at present is the most precise measurement of the radiation power spectrum at @xmath100 at those frequencies ; at those angular scales the signal is dominated by the sz effect and is not so relevant to constrain models of dm / de interaction . a polarization - sensitive receiver have been installed on the spt ; data at 95 and 150 ghz has provided a measurement of the @xmath101-mode polarization power spectrum from an area of @xmath102deg@xmath72 , spanning the range @xmath103 . the resulting power spectra was consistent with predictions for the spectrum arising from the gravitational lensing of @xmath104-mode polarization @xcite . the de survey ( des ) is a wide - field imaging and supernova survey on the blanco 4 m telecope at cerro tololo ( chile ) using the de camera . it has started operations and it will continue for five years . the de spectroscopic survey instrument ( desi ) is a wide field spectroscopic instrument intended to start in 2018 and operate also for five years in the nearly twin mayall telescope at kitt peak ( arizona ) . desi will obtain spectra and redshifts for at least 18 million emission - line galaxies , 4 million luminous red galaxies and 3 million quasi - stellar objects , to probe the effects of de on the expansion history bao and measure the gravitational growth history through rsd . the resulting 3-d galaxy maps at redshift @xmath105 and ly-@xmath30 forest at @xmath106 are expected to provide the distance scale in 35 redshift bins with a one - percent precision @xcite . the imaging survey will detect 300 million galaxies , with approximately 200 million wl shape measurements , almost a two - order of magnitude improvement over the cfhtlens weak lensing survey . approved as a major cosmology survey in sdss - iv ( 2014 - 2020 ) , eboss will capitalize on this premier facility with spectroscopy on a massive sample of galaxies and quasars in the relatively uncharted redshift range that lies between the boss galaxy sample and the boss ly-@xmath30 sample . compared with boss , this new survey will focus on a smaller patch of 7500 deg@xmath72 but it will reach higher magnitudes . it will measure both the distance - redshift relation and the evolution of the hubble parameter using different density tracers ; the clustering of luminous red galaxies ( lgrs ) and emission line galaxies ( elgs ) , quasars and ly-@xmath30 systems to probe the bao scale in the redshift ranges [ 0.6,0.8 ] , [ 1,2.2 ] and [ 2.2,3.5 ] respectively and it will achieve 1 - 2% accuracy in distance measurements from baos between @xmath107 . the javalambre physics of the accelerating universe astronomical survey ( jpas ) is a new astronomical facility dedicated to mapping the observable universe in 56 colors @xcite . the starting date for this multi - purpose astrophysical survey is 2015 . in five years , jpas will cover @xmath82deg@xmath72 using a system of 54 narrow band and 2 broad - band filters in the range @xmath83 nm . the filter system was optimized to accurately measure photometric redshifts for galaxies up to @xmath84 . the main instruments are a 2.5 m telescope located at el pico del buitre ( teruel , spain ) and a 1.2 giga - pixel camera . the main goals of the survey are to measure angular and radial components of bao from the galaxy clustering , determine the evolution of the cosmic volume from cluster counts and luminosity distances from snia . the filter system will permit to determine the redshifts of the observed supernovae . the camera is not optimized to measure galaxy elipticities so weak lensing studies would require ellipticity measurements obtained from other surveys . the jpas telescope will measure bao from high redshift quasars to achieve a better precision than boss @xcite , open the possibility of using the test described in sec . [ sec : sec7.evidence ] to disproof the concordance model . the large synoptic survey telescope ( lsst ) is a wide - field , ground - based telescope , designed to image @xmath108 deg@xmath72 in six optical bands from 320 nm to 1050 nm . the telescope will be located on cerro pachn ( chile ) and it will operate for a decade allowing to detect galaxies to redshifts well beyond unity . its science goals are to measure weak and strong gravitational lensing , bao , snia and the spatial density , distribution , and masses of galaxy clusters as a function of redshift . its first light is expected on 2019 . the square kilometre array ( ska ) is a radio - facility which is scheduled to begin construction in 2018 . the hi galaxy redshift survey can provide us with accurate redshifts ( using the 21 cm line ) of millions of sources over a wide range of redshifts , making it an ideal redshift survey for cosmological studies @xcite . although technically challenging , the ska could measure the expansion rate of the universe in real time by observing the neutral hydrogen ( hi ) signal of galaxies at two different epochs @xcite . wide - field multi - object spectrograph ( wfmos ) is a camera specially devoted to galaxy surveys . it will be mounted atop the 8.2 m subaru telescope on mauna kea ( hawaii ) . one of the science goals of the wfmos camera is high precision measurements of bao . the wfmos de survey comprises two parts : a 2,000 deg@xmath72 survey of two million galaxies at redshifts @xmath109 and a high redshift survey of about half a million lyman break galaxies ( lbgs ) at redshifts @xmath110 that would probe distances and the hubble rate beyond @xmath111 ( see @xcite for more details ) . bingo is a radio telescope designed to detect bao at radio frequencies by measuring the distribution of neutral hydrogen at cosmological distances using a technique called intensity mapping . the telescope will be located in a disused , open cast , gold mine in northern uruguay . it will operate in the range [ 0.96,1.26 ] ghz to observe the redshifted 21 cm hydrogen line . it will consist of a two - mirror compact range design with a 40 m diameter primary and it will have no moving parts to provide an excellent polarization performance and very low side - lobe levels required for intensity mapping . currently , the interest on cmb ground experiments is centered on polarization . for a cosmic variance limited experiment polarization alone places stronger constraints on cosmological parameters than cmb temperature @xcite . experiments like sptpol @xcite and quixote @xcite are currently taken data aiming to characterize the polarization of the cmb and of the galactic and extragalactic sources . cmb experiments devoted to measuring polarization from the ground are also being proposed ; the scientific capabilities of a cmb polarization experiment like cmb - s4 have been considered that in combination with low redshift data would be able to constrain , among other parameters , the de equation of state and dark matter annihilation @xcite . satellite surveys usually require a dedicated facility and , consequently , are more expensive than those carried out from the ground . their significant advantage is that , by observing outside the atmosphere , the data usually contains a lower level of systematic errors . the wilkinson microwave anisotropy probe ( wmap ) was a satellite mission devoted to measure cmb temperature fluctuations at frequencies operating between 23 and 94ghz . launched on june 30 , 2001 and operated for 9 years up to the end of september 2010 . the main results and data products of the nine years of operation are described in @xcite . the final 9yr data released was soon followed by those of the planck collaboration . the planck satellite observed the microwave and sub - millimeter sky from august 12th , 2009 to oct 23rd , 2013 in nine frequencies between 30 and 857 ghz , with angular resolution between 33 and 5. its goal was to produce cmb maps both in temperature and polarization . the planck collaboration has released data on cmb temperature anisotropies , thermal sunyaev - zledovich ( tsz ) effect . the measured temperature and polarization , a catalog of sunyaev - zeldovich ( sz ) clusters and likelihood codes to assess cosmological models against the planck data @xcite and other data products can be downloaded from the planck legacy archive @xcite . the temperature - temperature , temperature - e mode and e mode - e mode power spectra are measured up to @xmath112 @xcite , allowing the cmb lensing potential @xcite and the constraint on cosmological models beyond the @xmath0cdm model @xcite . erosita will be a x - ray satellite that will be launched in 2016 . it will perform the first imaging all - sky survey in energy range 0.3 - 10 kev @xcite . the goal of erosita is the detection of @xmath113 galaxy clusters out to redshifts @xmath114 , in order to study the large scale structure in the universe and test and characterize cosmological models including de . in the soft x - ray band ( 0.5 - 2 kev ) , it will be about 20 times more sensitive than the rosat all sky survey , while in the hard band ( 2 - 10 kev ) it will provide the first ever true imaging survey of the sky at those energies . euclid is a european space agency de satellite mission scheduled for launch in 2020 . this mission is designed to perform two surveys : a wide 15,000 deg@xmath72 survey in the optical and near - infrared and a deep survey on 40 deg@xmath72 two magnitudes deeper . these facilities are not independent between each other . euclid will map the extra - galactic sky with the resolution of the hubble space telescope , with optical and near - infrared ( nir ) imaging and nir spectroscopy . photometric redshifts for the galaxies in the wide survey will be provided from ground photometry and from the nir survey . in addition , 50 million spectroscopic redshifts will be obtained . euclid data will allow to measure the expansion history and the growth of structure with great precision . a detailed quantitative forecast of euclid performance has been discussed in @xcite . the data will allow to constrain many different cosmological models ; when the growth factor is parametrized as @xmath115 the value @xmath116 corresponds to the @xmath0cdm model and euclid will measure this parameter with a precision of @xmath117 @xcite . forecasts for other parametrizations of the growth factor and for other magnitudes such as the bias , de sound speed , redshift space distortions are given in @xcite . the wide field infrared survey telescope ( wfirst ) is an american satellite mission that is currently being reviewed and expected to be launch in 2023 . this mission updates and expands earlier proposed missions like the super nova acceleration probe ( snap ) and the joint de mission ( jdem ) . like euclid , one of its primary science goals is to determine the properties of de and in many respects complements euclid . wfirst strategy is to construct a narrow and deep galaxy redshift survey of 2000 deg@xmath72 . both satellites will measure the redshift for a similar number of galaxies and will obtain a comparable precision for the baryon acoustic oscillations derived angular diameter distances and hubble constant redshift evolution @xcite . nevertheless , due to their different observing strategy will allow cross - checks that will help to identify and eliminate systematics . the combination of both data sets will significantly improve the constraints on the dark energy parameters . many synergies will come from cross - correlating data from different observations for instance , euclid , wfirst and ska have similar scientific aims but will carry observations at different wavelengths . euclid and wfirst probe the low redshift universe , through weak lensing and galaxy clustering measurements . the ska has the potential to probe a higher redshift regime and a different range in scales of the matter power spectrum , which are linear scales rather than the quasi - non - linear scales to which euclid and wfirst will be sensitive . the combination of different observations will particularly sensitive to signatures of modified gravity . cross - correlation of different data sets will help to control systematics for the primary science . the ska , wfirst and euclid will be commissioned on similar timescales offering an exciting opportunity to exploit synergies between these facilities . @xcite the cosmic origins explorer ( core ) is a stage iv full - sky , microwave - band satellite proposed to esa within cosmic vision 2015 - 2025 . core will provide maps of the microwave sky in polarization and temperature in 15 frequency bands , ranging from 45 ghz to 795 ghz , with angular resolutions from @xmath118 at 45 ghz and @xmath119 at 795 ghz , with sensitivities roughly 10 to 30 times better than planck @xcite . the polarized radiation imaging and spectroscopy mission ( prism ) is a large - class mission proposed to esa in may 2013 within the framework of the esa cosmic vision program . its main goal is to survey the cmb sky both intensity and polarization in order to precisely measure the absolute sky brightness and polarization . the mission will detect approximately @xmath120 clusters using the thermal sz effect and a peculiar velocity survey using the kinetic sz effect that comprises our entire hubble volume @xcite . nasa is carrying similar efforts through the primordial polarization program definition team ( pppdt ) that converge towards a satellite dedicated to the study of cmb polarization ( cmbpol ) @xcite . combing these complementary ground based and space based observations , we would hopefully achieve a better understanding of the nature of dm , de and the interaction within the dark sectors . we thank s. tsujikawa for comments and suggestions . e. a. wishes to thank fapesp and cnpq ( brazil ) for support and a. a. costa , e. ferreira and r. landim for discussions and suggestions . f. a. b. acknowledges financial support from the ministerio de ciencia e innovacin , grant fis2012 - 30926 and the `` programa de profesores visitantes severo ochoa '' of the instituto de astrofsica de canarias . b. w. would like to acknowledge the support by national basic research program of china ( 973 program 2013cb834900 ) and national natural science foundation of china and he wishes to thank j. h. he and x. d. xu for helpful discussions .

models where dark matter and dark energy interact with each other have been proposed to solve the coincidence problem . we review the motivations underlying the need to introduce such interaction , its influence on the background dynamics and how it modifies the evolution of linear perturbations . we test models using the most recent observational data and we find that the interaction is compatible with the current astronomical and cosmological data . finally , we describe the forthcoming data sets from current and future facilities that are being constructed or designed that will allow a clearer understanding of the physics of the dark sector .

arxiv

spectra of type 1 agn show a diversity of broad and narrow emission lines that provide direct insights into the structure and kinematics of photoionized , and otherwise excited , gas in the vicinity of the putative central massive object . broad emission lines , like much studied h@xmath0 ( e.g. , * ? ? ? * hereafter z10 ) , are thought to arise in or near an accretion disk acting as the fuel reservoir for the central supermassive black hole ( log m@xmath1 m@xmath2 ) . h@xmath0 shows a diversity of line widths as well as profile shifts and asymmetries @xcite . despite this diversity some systematics have emerged and are best highlighted via the concept of two type 1 agn populations @xcite . population a show the smallest broad - line widths fwhm h@xmath0=1000 - 4000 and includes the narrow line seyfert 1 ( nlsy1 ) sources ( fwhm @xmath3 2000 ) . a h@xmath0 profiles are currently best fit by a single lorentz function . population b sources show fwhm h@xmath0=4000 - 12000 and require two gaussians ( one unshifted and one redshifted ) for a reasonable profile description . `` broad - line '' h@xmath0 profiles as narrow as fwhm = 500 @xcite and as broad as fwhm = 40000 @xcite have been found . a is predominantly radio - quiet while pop . b involves a mix of radio - quiet and the majority of radio - loud quasars . broad- and narrow - line profile shifts are known and the phenomenology can be confusing . narrow emission lines like [ oiii]5007 are regarded as a reliable measure of the local quasar rest frame except in the case of `` blue outliers '' , usually found in sources with fwhm h@xmath0= 1500 - 3500 and weak [ oiii ] @xcite . blue outliers show [ oiii ] blueshifts as large as @xmath41000 . no pop . b sources with blueshifted [ oiii ] are known at low z ( or luminosity ) . careful use of [ oiii ] and h@xmath0 narrow line as rest frame measures suggests that broad h@xmath0 in pop . a sources rarely shows a systematic red or blue shift above the fwhm profile level . a blueshifted component or asymmetry is observed in some extreme feii strong pop . a sources @xcite . b sources show more complex line shift properties . the h@xmath0 profile usually shows two components : 1 ) a `` classical '' broad component ( bc ; fwhm = 4000 5000 ) with zero or small ( red or blue ) shift , and 2 ) a very broad ( vbc ; 10000 ) and redshifted ( @xmath51000 ) component . composites involving the 469 brightest sdss - dr5 quasars suggest that these two components represent the underlying stable structure of h@xmath0 in pop . b sources . broad feii emission has been found in type 1 quasars since the era of photographic spectroscopy in the 60s . feii emission blends are almost ubiquitous in a sample of the brightest ( usually highest s / n ) sdss quasars ( z10 ) . circumstantial evidence has accumulated supporting the assumption that feii emission arises in or near the emitting clouds that produce other low ionization lines like h@xmath0 ( see e.g. , @xcite ) . fwhm feii appears to correlate with fwhm h@xmath0 over the full range where feii can be detected ( fwhm=1000 - 12000 ) . this can be clearly seen at low @xmath6 by observing the shape ( e.g. , smoothness ) of the feii 4450 - 4700 blue blend ( and the feii multiplet 42 line at 5018 ) near [ oiii]5007 . in pop . a sources the blend resolves into individual lines while it becomes much smoother in pop . b sources . sources with the strongest feii emission also show a weakening of h@xmath0 emission as expected if the latter is collisionally quenched in the same dense medium where strong feii emission can be produced @xcite . obviously systematic line shifts place important constraints on models for the geometry and kinematics of the broad line region . the most famous example involves a systematic blueshift of high ionization lines ( e.g. , civ 1549 ) relative to low ionization lines ( e.g. , balmer ) especially in pop . a sources ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? evidence was recently advanced ( * ? ? ? * hereafter h08 ) for the existence of a _ systematic _ redshift of feii relative to [ oiii]5007 ( and hence the balmer lines ) in a majority of type 1 quasars . this result , along with a narrower estimated feii line width , has been ascribed to feii emission arising in a region with dynamics dominated by infall and located at larger radius than the region producing the bulk of h@xmath0 . h08 argue that the amplitude of the shifts correlates inversely with source eddington ratio ( l / l@xmath7@xmath8@xmath9 ) . interpretations for such an feii redshift have already appeared @xcite reflecting the potential importance of such a first - order kinematic signature . having worked on line spectra and profile shifts for many years we were surprised by the h08 claims and decided to test the hypothesis of a systematic feii redshift . could we have missed it ? first let us consider what we know . a quasars show relatively symmetric unshifted lorentz - like h@xmath0 profiles with fwhm@xmath34000 . in our work using the brightest ( @xmath10 17.5 or @xmath11 17.5 ; @xcite ) sdss dr5 quasars we processed spectra for @xmath4260 pop . a sources ( from a sample of 469 quasars ; z10 ) and we found no evidence for a systematic shift of feii lines relative to h@xmath0 or . such an feii shift should be easiest to detect in the brightest pop . a sdss spectra with narrowest broad - line profiles and strongest feii emission . it is immediately suspicious that more and larger feii redshifts are claimed for pop . b sources . in only one pop . a source in our sample sdss j0946 + 0139 do we find a large h@xmath0 peak ( 90@xmath12 intensity level ) redshift of 1830 . this source is similar to oq208 ( @xcite and discussed in h08 ) which shows @xmath13 @xmath142000 . sdss j0946 is the only pop . a source with a large feii redshift in our z10 sample ( 1/260 ) . z10 found 19 quasars with an h@xmath0 peak ( 9/10 fractional intensity ) blueshifted more than -320 and 4 sources with the peak redshift more than + 320 . the remaining 241 pop . a sources showed no significant h@xmath0 peak shift ( figure 8 of z10 ) . best feii template fits to these sources show no significant difference in centroid redshift between feii and h@xmath0 . there are two possible causes of small and spurious h@xmath0 ( or feii ) shifts : 1 ) host galaxy contamination and 2 ) blue outliers.except in rare cases host galaxy contamination is unlikely to induce systematic redshifts with the amplitudes reported by h08 . extreme blue outliers with [ oiii ] blueshifts in the range 400 - 1000 are rare and therefore can not be the cause of the large and systematic shifts reported in h08 . in fact h08 selection criteria rejected sources likely to be seriously affected by 1 ) or 2 ) . h08 chose 4000 + sources from sdss dr5 with computed s / n @xmath15 10 . z10 also used dr5 where @xmath494% of sources show s / n @xmath15 10 . our sample was magnitude - limited with a slightly shallower redshift upper limit ( z=0.7 instead of 0.8 ) . why do we reach different conclusions about feii shifts ? a big part of the answer could involve how s / n was computed . compute s / n over the range 44305500@xmath16 . this procedure overestimates the quality of the data because it includes major emission lines in the computation . we compute s / n in the range 5600 - 5800@xmath16 , which is free of strong lines and represents as close as one can approach to an estimate of continuum s / n near h@xmath0 . using our range the h08 sample shows mean and median s / n values of 10.6 and 7.4 , respectively ; approx . 65% show s / n @xmath3 10 . we find that only 182 spectra of our bright sample are included in the h08 s . the majority of the h08 s sources are lower s / n than those in our sample . one can not estimate reliable feii line shifts using individual sdss spectra for sources fainter than about g@xmath417 - 17.5 . in rough order of importance our studies indicate that the accuracy of feii shift measures depends on : 1 ) feii strength and feii / h@xmath0 profile widths , 2 ) spectral s / n and 3 ) if estimates depend heavily on fits to the 4430 - 4680 blend , strength of heii 4686 emission . typical individual spectra used by h08 _ show _ too low s / n to allow convincing conclusions about feii shift and width typical parameter uncertainties for individual sources are much larger than the ones connected with our high s / n composites ( for a typical a2 source with s / n@xmath1720 uncertainties of shift estimates are larger than @xmath181500 ) . individual source spectra with large quoted feii redshift and s / n near the sample median were extracted from the h08 sample and specfit modelled . using an feii template with fixed shifts ranging from zero up to the largest values quoted by h08 , @xmath19 can not distinguish between zero and e.g. , 1000 redshift in the majority of the sources . the best recourse is to generate high s / n composite spectra . h08 argue that one can not confirm or refute the existence of a systematic feii redshift using composite spectra because of the large dispersion of fwhm , shifts and flux values for both h@xmath0 and feii . this is likely true for composites generated from random subsamples of sources but not true if one generates composites over more limited ranges of parameter values . one can generate binned composites over limited ranges of fwhm h@xmath0 and feii strength using the 4de1 formalism ( @xcite ; z10 ) . 4de1 bins a2 ( fwhm h@xmath0= 1000 - 4000 , 0.5 @xmath20 @xmath20 1.0 ) and b1 ( fwhm h@xmath0=4000 8000 , @xmath20 0.5 ) are of particular interest because they include the largest numbers of sources in random samples . specfit analysis ( @xcite ; details in @xcite ) of an a2 median composite involving n = 130 z10 sources ( s / n 90 ) gives a best - fit consistent with zero feii redshift . the situation for the b1 composite ( n=131 sources from z10 ; s / n 110 ) is less constraining because lines are broader and feii weaker . table 1 reports feii template shifts and 2@xmath21 uncertainties for specfit tests discussed in this paper . we also report peak shifts of bc extracted from the best specfit solutions along with `` core '' shifts measured at the centroid of the line peak after nc subtraction . in no case do we find a significant shift between feii and the rest frame or between feii and . we also do not find any significant feii shifts if we restrict to sources with l / l@xmath7 ratio @xmath22 ( h08 suggested the shifts might be largest for low l / l@xmath7 sources ) . since we find no evidence for systematic feii redshifts in our z10 bright quasar sample composites it is useful to generate feii shift composites using the h08 sample . we generate them within the 4de1 context thereby restricting the ranges of fwhm h@xmath0 , feii relative strength ( and likely also fwhm feii ) for each composite . since the distribution of feii shifts shown in h08 is continuous we focus on the sources with largest quoted shift values . if these shifts are not confirmed then smaller shifts are even less likely to be real . we therefore focus on constructing median composites for all h08 sources falling in 4de1 bins a2 and b1 with h08 feii redshift estimates @xmath23 ( figure 1 ) . two composites were constructed for each spectra bin : 1 ) one with no restriction on feii width ( h08 do not constrain fwhm feii in their template fits so it is sometimes very different from fwhm h@xmath0 ) and 2 ) one with feii width constrained to the fwhm range of h@xmath0 in a particular bin ( i.e. , @xmath24 for a2 and @xmath25 for b1 ) . upper and lower panels of figure 1 show bins a2 ( fwhm h@xmath0 @xmath3 4000 ) and b1 ( fwhm h@xmath0=4000 - 8000 ) , respectively ( n=156 for bin a2 and n=240 for bin b1 ) . the s / n @xmath4 55 - 60 for both composites . spectra show best - fit specfit models superimposed . the left and center panels involved feii templates fixed to the best fit and 1500 shifts , respectively . our template prescription is described in @xcite graphical results for the best - fits are shown in the right panels of figure 1 . fits were performed over the range @xmath17 4470 5450 , where feii and continuum emission account for the total flux making it the safest region for normalized @xmath26 computations . @xmath26 values are shown for the range of adopted feii shifts . in order to estimate confidence intervals we considered a set of fits with displacements @xmath27= + @xmath28 , for integer @xmath29 , along with the best fit and a few additional @xmath27 cases in proximity to the minimum @xmath26 . one can see a clear preference for zero or near - zero fits . the significance of @xmath26 variations is described by @xmath30 statistics appropriate for ratios of @xmath26 values @xcite . given the large number of degrees of freedoms in the sampling range ( 4500 4630 , 5040 5090 , 5310 5360 ) any @xmath26 differences between two fits become significant at a 95% confidence level if @xmath31 . the @xmath26 values indicate that zero shift and `` best shift '' values in table 1 are not significantly different . all fits involving shifts @xmath5 500 are statistically significant . the middle panel of figure 1 upper row demonstrates visually that the fit with @xmath13 = + @xmath32 ( and even more the fits with larger displacement ) do not reproduce the observed feii emission . both the residuals and @xmath19 results rule out any systematic redshift for at least half of the h08 sample ( pop . note especially the fits to the two relatively isolated multiplet 42 feii lines between h@xmath0 and [ oiii]4959 and on the red wing of [ oiii]5007 . the redshifted fit fails to include the blue side of the 4450 4700 blend and the red side is confused by the frequent presence of heii 4686 . the latter line is not mentioned in the h08 study leaving us to conclude that it was not included in their fits . it can certainly give the impression of a redshift of the feii blue blend , which is the most useful feii diagnostic in the optical spectra of low redshift quasars ( the red feii blend is frequently affected by coronal lines as well as mgii host galaxy absorption in lower redshift sources ) . pioneering principal component analysis of the bqs survey @xcite found that heii4686 equivalent width anticorrelates with sources luminosity ( it is eigenvector 2 ) . there is a tendency for the h08 sources with largest feii redshifts to favor a smaller and lower ( @xmath33 ) range of source luminosity than those with near zero shifts ( @xmath34 ) . thus the effect of heii will tend to play a larger role in the sources where the largest feii redshifts have been found . show composite spectra for five bins of feii redshift in their figure 12 . the three bins involving largest feii redshift sources show a prominent heii signature that , if not subtracted , will increase the apparent significance of any assumed feii redshift . only the bin involving sources with no feii redshift ( within the uncertainties ) shows no evidence of heii emission . figure 9a of h08 suggests that a larger fraction of quasars with fwhm h@xmath35 4000 ( population b ) show large feii redshifts . the lower panels of our figure 1 show specfit models superimposed on a b1 median composite constructed from all h08 sources with quoted feii redshift greater than 1000 . the situation is certainly more ambiguous than for the a2 composite . it is hard to identify individual feii features . lines are broad , feii is weak and under these conditions there are serious limitations on the reliability of fwhm and shift estimates for feii ( cf . fig . 3 of @xcite ) . the same analysis as done for a2 composite shows much poorer constraints on the feii shift . the best fit yields @xmath13 @xmath36 but is not distinguishable from a zero shift solution . if one actually computes @xmath26 values over the ranges 4474 4640 , 5040 5105 , 5320 5400 , the @xmath26 monotonically increases from 0 shift ( figure 1 , lower rightmost panel ) , although the increase remains insignificant until @xmath17 1100 , where @xmath37 . b1 feii is too faint and the lines are too broad to make inferences about line shifts and widths . the claim of large feii shifts are not , and can not be , confirmed . @xcite recently report an feii study of sdss quasars and any feii redshifts they measure ( their figure 16 ) are much smaller than those reported by h08 ( the average feii shift relative to the narrow lines is 100 @xmath18 240 ) . returning to our previous list of major sources of uncertainty for feii shift and fwhm estimates leads us to suggest that low spectral s / n and above average heii strength are the culprits . the fit to the 4430 4680 blue blend drives the best fit @xmath19 results . the exclusion of heii4686 from the h08 fits likely results in a tendency for heii to `` redshift '' the blue feii blend . this effect in a typically low luminosity sample , where heii is stronger than average , likely drove the conclusion that feii was systematically redshifted . we tested this conclusion omitting the heii line from our fits to the bin a2 and b1 composites generated from the h08 sample . feii shifts in lines 2 and 5 of table 1 increase from -60 to + 770km / s and from 730 to 1570km / s , respectively . the more constraining a2 results suggest that heii can produce the entire systematic redshift claimed by h08 . we _ do not _ confirm large feii redshifts relative to narrow [ oiii ] and broad h@xmath0 emission in type 1 agn but can not rule out the existence of small red ( or blue ) shifts in particular subsamples . fitting median composites built from spectra with large claimed feii shifts ( @xmath23 ) indicates small shifts with an upper limit @xmath17 300 for bin a2 . in the case of b1 the best fit suggests @xmath17700 but the shift is very poorly constrained . in both cases the shifts are not significantly different from 0 . these results do not support the origin of feii emission from a dynamical disjoint region from the one emitting the broad core of . our result also challenges the usefulness of feii shift as orientation parameter . small systematic shifts of feii with respect to the rest frame seem plausible but a reliable analysis is possible only on spectra of high s / n ratio . lcccccc a2 - z10 & -40 @xmath18 90 & 10 & 0 & @xmath38 & 130 + a2 - h08 , @xmath39 & -60 @xmath18 400 & 80 & 90 & 70 & 156 + a2 - h08 , @xmath40 & -160 @xmath18 375 & 100 & 10 & 45 & 194 + b1 - z10 & -340 @xmath18 400 & -150 & -80 & @xmath38 & 131 + b1 - h08 , @xmath39 & + 730@xmath41 & 0 & 100 & @xmath38 & 240 + b1 - h08 , @xmath40 & + 180 @xmath18 450 & -80 & 40 & @xmath38 & 410 + b1 - h08 @xmath42 z10 , @xmath40 & -150 @xmath18 470 & -170 & -140 & 45 & 22 +

we test the recent claim by hu et al . ( 2008 ) that feii emission in type 1 agn shows a systematic redshift relative to the local source rest frame and broad - line h@xmath0 . we compile high s / n median composites using sdss spectra from both the hu et al . sample and our own sample of the 469 brightest dr5 spectra . our composites are generated in bins of fwhm h@xmath0 and feii strength as defined in our 4d eigenvector 1 ( 4de1 ) formalism . we find no evidence for a systematic feii redshift and consistency with previous assumptions that feii shift and width ( fwhm ) follow h@xmath0 shift and fwhm in virtually all sources . this result is consistent with the hypothesis that feii emission ( quasi - ubiquitous in type 1 sources ) arises from a broad - line region with geometry and kinematics the same as that producing the balmer lines .

arxiv

nuclear multifragmentation is presently intensely studied both theoretically and experimentally . due to the similitude existent between the nucleon - nucleon interaction and the van der waals forces , signs of a liquid - gas phase transition in nuclear matter are searched . while the theoretical calculations concerning this problem started at the beginning of 1980 @xcite , the first experimental evaluation of the nuclear caloric curve was reported in 1995 by the aladin group @xcite . a wide plateau situated at around 5 mev temperature lasting from 3 to 10 mev / nucleon excitation energy was identified . the fact was obviously associated with the possible existence of a liquid - gas phase transition in nuclear matter and generated new motivations for further theoretical and experimental work . similar experiments of eos @xcite and indra @xcite followed shortly . using different reactions they obtained slightly different caloric curves , the plateau - like region being absent in the majority of cases . factors contributing to these discrepancies are both the precision of the experimental measurements and the finite - size effects of the caloric curve manifested through the dependency of the equilibrated sources [ @xmath5 sequence on the reaction type . concerning the first point of view , recent reevaluations of the aladin group concerning the kinetic energies of the emitted neutrons brought corrections of about 10 @xmath6 ( in the case of the reaction @xmath1au+@xmath1au , 600 mev / nucleon ) . more importantly however it was proven that the energies of the spectator parts are growing with approximately 30 @xmath6 in the bombarding energy interval 600 to 1000 mev / nucleon . on the other side , the universality of the quantity @xmath7 subject to the bombarding energy variation ( which was theoretically proven @xcite to be a signature of statistical equilibrium ) suggests that for the above - mentioned reactions the equilibrated sources sequence [ @xmath5 should be the same . consequently , we deal with an important nonequilibrium part included in the measured source excitation energies which may belong to both pre - equilibrium or pre - break - up stages @xcite . the smm calculations suggest a significant quantity of nonequilibrium energy even in the case of the 600 mev / nucleon bombarding energy reaction @xcite . thus , the necessity of accurate theoretical descriptions of the break - up stage and of the sequential secondary particle emission appears to be imperative in order to distinguish between the equilibrium and nonequilibrium parts of the measured excitation energies . these approaches should strictly obey the constrains of the physical system which , in the case of nuclear multifragmentation , are purely microcanonic . as we previously underlined @xcite , in spite of their success in reproducing some experimental data , the two widely used statistical multifragmentation models ( smm @xcite and mmmc @xcite ) are not strictly satisfying the microcanonical rules . the present paper describes some refinements and improvements brought to the sharp microcanonical multifragmentation model proposed in @xcite and also the employment of the model in its new version in the interpretation of the recent experimental data of the aladin group @xcite . the improvements brought to the model @xcite are presented in section ii . section iii presents the new evaluations of temperature curves and the first evaluations ( performed with this model ) of heat capacities at constant volume ( @xmath8 ) represented as a function of system excitation energy and temperature and also the comparison between the model predictions and the recent experimental heli isotopic temperature curve [ @xmath9 @xcite . conclusions are drawn in section iv . the improvements brought to the microcanonical multifragmentation model concerns both the _ break - up _ stage and the _ secondary particle emission _ stage . + ( i ) _ primary break - up refinements _ + comparing to the version of ref.@xcite the present model has the following new features : + ( a ) the experimental discrete energy levels are replacing the level density for fragments with @xmath10 ( in the previous version of the model a thomas fermi type level density formula was used for all particle excited states ) . in this respect , in the statistical weight of a configuration and the correction factor formulas @xcite the level density functions are replaced by the degeneracies of the discrete levels , @xmath11 ( here @xmath12 denotes the spin of the @xmath13th excited level ) . as a criterion for level selection ( i.e. the level life - time must be greater than the typical time of a fragmentation event ) we used @xmath14 1 mev , where @xmath15 is the width of the energy level . + ( b ) in the case of the fragments with @xmath16 the level density formula is modified as to take into account the strong decrease of the fragments excited states life - time ( reported to the standard duration of a fragmentation event ) with the increase of their excitation energy . to this aim the thomas fermi type formula @xcite is completed with the factor @xmath17 ( see ref.@xcite ) : @xmath18 where @xmath19 , @xmath20 and @xmath21 . + ( ii ) _ inclusion of the secondary decay stage _ + for the @xmath22 nuclei it was observed that the fragments excitation energies are sufficiently small such as the sequential evaporation scheme is perfectly applicable . according to weisskopf theory @xcite ( extended as to account for particles larger than @xmath23 ) , the probability of emitting a particle @xmath24 from an excited nucleus is proportional to the quantity : @xmath25 where @xmath26 are the stable excited states of the fragment @xmath24 subject to particle emission ( their upper limit is generally around 7 - 8 mev ) , @xmath27 is the kinetic energy of the formed pair in the center of mass ( c.m . ) frame , @xmath28 is the degeneracy of the level @xmath13 , @xmath29 and @xmath30 are respectively the reduced mass of the pair and the separation energy of the particle @xmath24 and finally @xmath31 is the inverse reaction cross - section . due to the specificity of the multifragmentation calculations we considered the range of the emitted fragments @xmath24 up to the @xmath32 limit . for the inverse reaction cross - section we have used the optical model based parametrization from ref . the sequential evaporation process is simulated by means of standard monte carlo ( see for example @xcite ) . for nuclei with @xmath33 ( the only excited states of @xmath34 nuclei taken into consideration are few states higher than 20 mev belonging to the @xmath23 particle ) depending on their amount of excitation we consider _ secondary break - up _ for @xmath35 and weisskopf evaporation otherwise ( here @xmath36 is the excitation energy of the fragment @xmath37 and @xmath38 is its binding energy ) . the microcanonical weight formulas have the usual form @xcite excepting the level density functions which are here replaced by the discrete levels degeneracies . due to the reduced dimensions of the @xmath39 systems , the break - up channels are countable ( and a classical monte carlo simulation is appropriate ) when a mean field approach is used for the coulomb interaction energy . in this respect , the wigner - seitz approach @xcite is employed for the coulomb interaction : @xmath40 where @xmath41 and @xmath42 denotes the mass and the charge of the source nucleus , the resulting fragments have the index @xmath13 , @xmath43^{1/3}$ ] and @xmath44 . here @xmath45 denotes the break - up volume and @xmath46 the volume of the nucleus at normal density . it should be added that @xmath47 is the radius of the source nucleus at break - up and @xmath48 is the radius of fragment @xmath13 at normal density . for each event of the primary break - up simulation , the entire chain of evaporation and secondary break - up events is monte carlo simulated . using the improved version of the microcanonical multifragmentation model , the caloric curves corresponding to two freeze - out radii ( r=2.25 a@xmath49 and r=2.50 a@xmath49 fm ) are reevaluated for the case of the source nucleus ( 70 , 32 ) ( the microcanonical caloric curves evaluated with the initial version of the model are given in ref . these are presented in fig . 1 ( a ) . one can observe that the main features of the caloric curve from refs . @xcite are reobtained . thus , one can recognize the liquid - like region at the beginning of the caloric curve , then a large plateau - like region and finally the linearly increasing gas - like region . one may also notice that the caloric curve behavior at the freeze - out radius variation is maintained : the decrease of the freeze - out radius leads to a global lifting of the caloric curve . as it is well known , the curves of the constant volume heat capacity ( @xmath8 ) as a function of system excitation energy ( @xmath50 ) and as a function of temperature ( @xmath51 ) may provide important information concerning the transition region and the transition order . for this reason the curves @xmath52 and @xmath53 have been evaluated ( see fig . 1 ( a ) and fig . 1 ( b ) ) . we remind that the constant volume heat capacity ( @xmath8 ) is calculable in the present model using the formula @xcite : @xmath54 ^ 2\right>+t^2\left<\left(\frac32 n_c- \frac52\right ) \frac1{k^2}\right>.\ ] ] it can be observed that the @xmath52 curve has a sharp maximum around 4.5 mev / nucleon excitation energy for both considered freeze - out radii . this suggests that a phase transition exists in that region . the transition temperatures can be very well distinguished by analyzing the @xmath53 . one can observe two sharp - peaked maxima pointing the transition temperatures corresponding to the two considered freeze - out radii . in order to make a direct comparison between the calculated heli isotopic temperature and the recent experimental results @xcite one has to deduce the sequence of excitation energy as a function of the system dimension [ @xmath5 . this is done as in refs . @xcite using as matching criterion the simultaneously reproduction of the @xmath55 and @xmath56 curves . this couple of curves can fairly well identify the dimension and the excitation of the equilibrated nuclear source @xcite . here @xmath2 stands for the multiplicity of intermediate mass fragments and is defined as the number of fragments with @xmath57 from a fragmentation event while @xmath3 denotes the charge asymmetry of the two largest fragments and , for one fragmentation event is defined as @xmath58 with @xmath59 where @xmath60 is the maximum charge of a fragment and @xmath61 is the second largest charge of a fragment in the respective event . @xmath4 represents the _ bound charge _ in one fragmentation event and is defined as the sum of the charges of all fragments with @xmath62 . the simultaneous fit of the calculated curves @xmath55 and @xmath56 on the corresponding experimental data ( @xmath1au+@xmath1au at 1000 mev / nucleon ) is given in fig . the agreement is very good . the equilibrated source sequence [ @xmath5 we used for this purpose is given in fig . 3 together with the experimental evaluations of the excitation energies as a function of source dimension for the reaction @xmath1au+@xmath1au at 600 , 800 and 1000 mev / nucleon . the theoretically obtained sequence is relatively close to the experimental line corresponding to 600 mev / nucleon bombarding energy . the deviations between the calculated equilibrated source sequence and the three experimental lines suggest that the experimental evaluations contain a quantity of non - equilibrium energy which grows with increasing the bombarding energy . as suggested in ref . @xcite , its origin may be situated in both the pre - equilibrium and pre - break - up stage . these deviations are exclusively due to the neutron kinetic energies which , reevaluated @xcite from the 1995 data @xcite , are much larger . it should also be pointed that apart from the smm predictions @xcite , the quantity of non - equilibrium energy predicted by the present model is smaller and thus the model predicted equilibrated source sequence is closer to the experimental line of the 600 mev / nucleon bombarding energy reaction . after evaluating the sequence of the equilibrated sources a direct comparison the heli calculated isotopic temperature curve with the ones recently evaluated by the aladin group @xcite is performed . to this purpose the uncorrected albergo temperature is used : @xmath63 $ ] , the experimental predictions being divided by @xmath64 ( which is the factor used in the aladin evaluation of the heli caloric curve chosen as to average the qsm , gemini and mmmc models predictions ) . the result is represented in fig . 4 as a function of @xmath4 . it can be observed that the agreement between the calculated @xmath65 and the experimental data corresponding to the @xmath1au+@xmath1au reaction at 600 and 1000 mev / nucleon bombarding energy is excellent on the entire range of @xmath4 . in comparison , the smm model predicts in the region @xmath66 a curve steeper than the experimental data . sumarizing , the microcanonical multifragmentation model from ref . @xcite is improved by refining the primary break - up part and by including the secondary particle emission . the caloric curve rededuced with the new version of the model preserves its general aspect @xcite manifesting an important plateau - like region . the transition regions are clearly indicated by the sharp maxima of the @xmath52 and @xmath53 curves . the model proves the ability of simultaneously fitting the `` definitory '' characteristics of the nuclear multifragmentation phenomenon @xmath55 and @xmath56 . evaluating the equilibrated source sequence @xmath67 [ by using the criterion of reproducing both @xmath68 and @xmath69 versus @xmath70 , a nonequilibrium part of the experimentally evaluated excitation energy growing with the increase of the bombarding energy is identified . the direct comparison of the calculated heli caloric curve shows an excellent agreement with the experimental heli curves recently evaluated by the aladin group . g. fai and j. randrup , nucl . a * 404 * , 551 ( 1983 ) ; comp . phys . comm . * 42 * , 385 ( 1986 ) ; l. p. csernai , g. fai and j. randrup , phys . b * 140 * , 149 ( 1984 ) ; g. fai , l. p. csernai , j. randrup and h. stocker , phys . b * 164 * , 265 ( 1985 ) . j. pochodzalla _ et al . _ , lett . * 75 * , 1040 ( 1995 ) . j. a. hauger _ et al . _ , lett . * 77 * , 235 ( 1996 ) . y. g. ma _ et al . b * 390 * , 41 ( 1997 ) . a. schuttauf _ _ , nucl . phys . * a607 * , 457 ( 1996 ) . j. pochodzalla , progr . . phys . * 39 * , 443 ( 1997 ) . w. trautmann _ et al . _ , gsi - preprint 98 - 20 , march 1998 . hongfei xi _ et al . _ , a * 359 * , 397 ( 1997 ) . w. f. j. mueller , proceedings of the workshop on multifragmentation , hirschegg ( austria ) ( 1999 ) , gsi . al . h. raduta and ad . r. raduta , phys . c * 59 * , 323 ( 1999 ) . al . h. raduta and ad . r. raduta , nucl . phys . * a647 * , 12 ( 1999 ) . j. p. bondorf , a. s. botvina , a. s. iljinov , i. n. mishustin and k. sneppen , phys . rep . * 257 * , 133 ( 1995 ) . d. h. e. gross , rep . . phys . * 53 * , 605 ( 1990 ) . al . h. raduta and ad . r. raduta , phys . c * 55 * , 1344 ( 1997 ) . h. raduta and ad . r. raduta , phys . c * 56 * , 2059 ( 1997 ) . j. randrup and s. koonin , nucl . a471 * , 355c ( 1987 ) . v. weisskopf , phys . rev . * 52 * , 295 ( 1937 ) . a. s. botvina , a. s. iljinov , i. n. mishustin , j. p. bondorf , r. donangelo and k. sneppen , nucl . phys . a475 * , 663 ( 1987 ) . i. dostrovsky , z. fraenkel and g. friedlander , phys . rev . * 116 * , 683 ( 1959 ) . al . h. raduta and ad . r. raduta , phys . c * 59 * , r1855 ( 1999 ) .

the microcanonical multifragmentation model from [ al . h. raduta and ad . r. raduta , phys . rev . c * 55 * , 1344 ( 1997 ) ; * 56 * , 2059 ( 1997 ) ; * 59 * , 323 ( 1999 ) ] is refined and improved by taking into account the experimental discrete levels for fragments with @xmath0 and by including the stage of sequential decay of the primary excited fragments . the caloric curve is reevaluated and the heat capacity at constant volume curve is represented as a function of excitation energy and temperature . the sequence of equilibrated sources formed in the reactions studied by the aladin group ( @xmath1au+@xmath1au at 600 , 800 and 1000 mev / nucleon bombarding energy ) is deduced by fitting simultaneously the model predicted mean multiplicity of intermediate mass fragments ( @xmath2 ) and charge asymmetry of the two largest fragments ( @xmath3 ) versus bound charge ( @xmath4 ) on the corresponding experimental data . calculated heli isotopic temperature curves as a function of the bound charge are compared with the experimentally deduced ones .

arxiv

for more than 60 years it has been well known that the quiet solar corona is heated to a temperature of about 12 million kelvins while the visible surface of the sun is roughly 250 times cooler ( grotrian 1939 ; edlen 1942 ; phillips , 1995 ) . it has been also recognized that magnetic fields or waves play a key rle in the heating of the solar corona so that somehow convective energy in the photosphere is converted to thermal energy in the corona via magnetic fields or wave energy . the primary energy source for this heating must lie in the convection zone below the solar photosphere ( e.g. bray et al . , 1991 ; golub & pasachoff 1998 ; aschwanden 2004 ) where there is 100 times as much energy available than that required to heat the corona ( @xmath0300w / m@xmath1 : withbroe & noyes 1977 , aschwanden 2001 ) . currently , the debate centres on whether the energy to heat the corona derives from dissipation of magneto - hydrodynamic ( mhd ) waves ( e.g. hollweg , 1981 ) or from numerous small - scale magnetic reconnection events giving rise to nanoflares ( parker , 1988 , aschwanden 2004 ) . it has been found theoretically that the interaction of the magnetic field with convective flows in or below the photosphere can produce two types of magnetic disturbances in coronal structures . firstly , the buffeting of magnetic flux concentrations in the photosphere by granulation generates mhd waves which can propagate into magnetic flux tubes and dissipate their energy in the chromosphere or corona ( e.g. ofman et al . 1998 ) . secondly , in coronal loops the random motions of magnetic loop foot - points can produce twisting and braiding of coronal field lines , which generates field - aligned electric currents that can be dissipated resistively ( e.g. parker 1972 , 1983 ; heyvaerts & priest 1983 ; van ballegooijen 1990 ) . the main difference between these processes is that plasma inertia plays a key rle in wave propagation , but is unimportant for the dynamics of field - aligned currents along coronal loops . thus these types of magnetic heating mechanisms can be crudely classified as either wave - heating or current - heating mechanisms . there are theoretical arguments for both mechanisms , but the observational evidence for nano - flare heating is perhaps looking less convincing than before . extrapolation of the number spectra of small flares down to microflares has been made to nano - flares but the total energy , while tantalisingly close , is most probably less than the required amount ( parnell et al . several theoretical studies showed that only high - frequency mhd waves ( @xmath2hz ) are capable of significant heating ( e.g. porter et al . 1994 , aschwanden 2004 ) . these waves have been sought using the fe xiv green " line at 530.3 nm ( emitted at @xmath3 k ) and the fe x red " line at 637.5 nm ( @xmath4 k ) . observations of high - frequency intensity oscillations of the coronal structures have been made by pasachoff and colleagues ( green line : pasachoff et al . , 1995 , 2000 , 2002 ) , by ruin and minarovjech ( green and red lines : ruin & minarovjech , 1991 , 1994 ) , by rudawy , phillips and colleagues ( green line , total eclipses in 1999 and 2001 ; phillips et al . , 2000 , williams et al . , 2001 ; rudawy et al . , 2001 ; williams et al . , 2002 ; rudawy et al . , 2004 ) , and by singh et al . ( green and red lines : singh et al . , 2009 ) . phillips and rudawy and their colleagues with their secis ( solar eclipse coronal imaging system ) ccd camera instrument have obtained the highest time resolution up to now . the results of these investigations are somewhat contradictory , with both positive and negative observations of oscillations ( e.g. pasachoff & landman 1984 ; koutchmy et al . 1994 ; cowsik et al . 1999 ; williams et al . 2001 , 2002 ; rudawy et al . space missions capable of comparable time resolution measurements have not been available up to the present time , so using ground - based equipment is still the only way of making such observations ( aschwanden , 2004 , klimchuk , 2006 ) . in this paper , we describe a set - up that will be used to search for high - frequency coronal oscillations . we shall be making observations using the mid - sized coronagraph ( belonging to the astronomical institute of slovak academy of sciences in tatranska lomnica : lexa , 1963 ) at lomnicky peak observatory . observations will be made in the fe xiv green line using the secis instrument ( phillips et al . 2000 ) , now owned by the astronomical institute at the university of wrocaw , poland . the observational system on the lomnicky peak observatory consists of 3 main instrumental parts : the 20-cm lyot - type coronagraph , secis instrument ( two fast - frame - rate ccd cameras , auxiliary electronics systems and dedicated computer ) and a special opto - mechanical interface between the coronagraph and secis cameras . the lomnicky peak coronagraph ( lexa , 1963 ) , made by carl zeiss jena , is located at the summit of lomnicky peak ( 2634 m altitude ) , allowing observations in the light of prominent coronal visible - light emission lines out to a significant distance beyond the solar limb . the front part of its optical system consists of a single objective lens ( bk7 glass , @xmath5=1.71 m , @xmath6=17.0 m , @xmath7200 mm , @xmath83 m ) and a primary diaphragm which obscures the lens to a final clear aperture of 195 mm . the focal lengths of the objective for the wavelength of the green line are 2975 and 2980 mm for the axial and outer light beams respectively . the central part of the optical system has an artificial moon ( the occulting disk ) which is a fat mirror inclined with respect to the optical axis and reflects the solar disk light out of the coronagraph tube . the artificial moon is fixed in front of a field lens in a hole in the center of the lens , and can be changed to similar ones with various diameters . behind the field lens there is a re - imaging triplet lens in order to correct , at least partially , geometric aberrations of the primary lens and to focus a difraction image of the primary diaphragm on lyot s stop . lyot s stop lies between the second and the third lenses of the triplet and blocks the scattered light coming from the primary diaphragm . a particular feature of the optical system is a four - lens imaging objective of 9 cm aperture . the combined action of both objectives creates the final image of the corona with diameter @xmath9 mm . the coronagraph is equipped with a fast optical automatic guider . it detects offsets of the actual pointing using two photodiode pairs in an anti - parallel connection providing closed - loop correction signals to the drives . it precisely stabilizes the relative position of the occulting disk against the solar image , ensuring stable position of the field of view . the two secis cameras are connected to the lomnicky peak coronagraph using a new , special opto - mechanical interface . the optical set - up is shown schematically in fig.1 , with fig.2 showing the components in the rigid light - tight box . the coronal image formed by the coronagraph is to the left in fig.1 , at the focus of the entrance lens ( marked l300/82 in fig.1 : diameter and focal length are d=82 mm , f=300 mm ) . this lens forms a parallel light - beam which then passes to a beam splitter ( bspl , shorter dimension = 50 mm ) . the reflected beam from the beam splitter then passes through a broad - band and neutral density filter combination ( marked nd / wb filter " in the figure ) , is reflected again from a flat mirror ( m3 ) , and finally is brought to a focus on to the ccd camera ( marked ccd wl " ) by an achromatic lens ( l120/50 ) . the parallel beam transmitted by the beam splitter is reflected by a flat mirror ( m1 ) , passes a narrow - band fexiv 530.3 nm interference filter ( fe5303 " ) , is reflected by a second flat mirror ( m2 ) , then brought to a focus on to the ccd camera ( marked ccd fe5303 " ) by the acromatic lens ( l120/50 ) . fig.1 shows the ray diagram for the configuration . in this figure , an on - axis pair of rays is shown by thick lines . the thin lines represent rays from one extreme of the coronagraph image . the angle between them ( greatly exaggerated in the figure ) is no more than about 0.5 degrees . with such a small angle , a negligible wavelength shift is produced by the interference filter ( fe5303 ) . the optical system was designed taking into account the actual optical parameters of the lomnicky peak coronagraph and the desired spatial scale of the images on the ccd chips . it was optimized to avoid any vignetting ( to keep entire fields of view as bright as possible ) and to limit geometric or chromatic aberrations of the system . in order to minimize the total cost of the interface , all optical and mechanical elements as well as adjustable optical mounts were general - purpose stock elements , selected from the melles griot company catalogue . the beam splitter ( bspl ) reflects only 10% of the light to the broad - band channel , transmitting the remaining light to the narrow - band ( green - line ) channel . the broad - band filter was selected to be centred on the green line wavelength but having a larger range of transmission ( central wavelength 530.0 nm , bandpass fwhm 10 nm ) . a neutral - density filter ( nd ) in the white - light optical channel was selected to ensure equal exposure times in the narrow - band and broad - band channels . final images in both channels are formed by lenses with the same focal lengths , so giving the same spatial scale . the entire optical system is mounted in a rigid box attached to a rear plate of the coronagraph . a light - tight black aluminium box acting as a rigid optical frame for the optical components was manufactured by the workshop of the astronomical institute at the university of wrocaw ( fig.2 ) . the optical system was assembled and pre - aligned in the box ; apart from focusing of the system , no other special alignments of the optical components are needed at the telescope ( fig.3 ) . three narrow - band filters with passbands around the fexiv 530.3 nm green line are available ( two made by barr associates , inc . , and one by andover corporation ) . all filters have fwhm passbands of @xmath10 nm and diameters of 50 mm . the filter chosen for use in the optical system is fitted with a thermostatic device to maintain the required temperature under variable ambient temperatures . the secis instrument was built , tested , and calibrated between 1997 and 1999 in a british polish collaboration to search for short - period coronal light oscillations . it was used with great success during the total eclipse seen from bulgaria in 1999 , from zambia in 2001 ( see phillips et al . , 2000 ; williams et al . , 2001 ; williams et al . , 2002 ; rudawy et al . , 2004 ) , and most recently from libya in 2006 . the ccd cameras ( manufactured by eev , chelmsford , u.k . ) are high - performance cameras designed specifically for scientific and machine vision applications . the image sensor is a monochrome 512@xmath11512 pixel frame transfer ccd . this device has square ( 15@xmath12 m @xmath13 m ) pixels , and can be driven at a non - interlaced frame rate up to 70 frames per second . the cameras digitise the signal from the ccd to nominally 12 bits and provide a real dynamic range of over 1000:1 . the camera electronics operate the cameras in an asynchronous " mode , where a trigger pulse from the control electronics commands one of the cameras to capture an image at a precise moment and with a precise exposure period . this feature allows the two cameras to capture accurately synchronised images . the data are captured and stored on a personal computer . the computer system captures the synchronised digital video streams from the two ccd cameras and reconstitutes the video images , storing them for more detailed analysis . the computer has dual pentium processors , 128 megabytes of memory , and four 9 gb disk drives . it is able to run a set of observations consisting of up to about 10000 images for each camera . the image processing software allows the replay of the video , and the cropping to sub - sequences and regions of interest . these selections can then be exported to files in fits format . more detailed information about secis and its first scientific application are described by phillips et al . ( 2000 ) . since 2003 secis has also been used for making high time resolution spectral observations of solar flares over the profile of the h-@xmath14 ( 656.3 nm ) line using the multi - channel subtractive double pass ( msdp ) imaging - spectrograph ( mein , 1977 and 1991 ) and large coronagraph ( with 530 mm main objective ) or horizontal telescope ( with 150 mm main objective ) installed at biakow observatory ( astronomical institute at the university of wrocaw ) ( radziszewski et al . , 2006 , 2007a , 2007b , 2008 ) . preliminary tests of the entire system were performed in april 2009 , though the lack of solar activity at that time meant that no prominent coronal structures were detectable . the filters manufactured by barr associates , inc . , have 50 mm clear diameter and have quarter - wave flatness specification . they are coated with ion - assist refractory oxide coatings that greatly reduce the wavelength shift with ambient temperature and filter longevity and produce higher transmittance . to investigate possible degradation since their last use during the 2006 eclipse , the filter passband widths and central wavelengths were tested for thermal stability using disk centre solar light and a small spectrograph with a dispersion of 0.9nm / mm connected directly to the coronagraph ( minarovjech , 2009 ) . examples of the mean reference disk centre spectra taken with and without the narrow - band filter are shown in fig.4 . the tests showed that the filters have to be heated to a fairly high temperature ( 4550 degrees celsius ) in order to tune the filter passband to the green - line wavelength . the dependence of the main passband parameters on the filter temperature is displayed in fig.5 . first , the filter passband transmission , the width ( fwhm ) of its passband , and its central wavelength averaged over the length ( 7.2 mm ) of the spectrograph slit were examined as a function of filter temperature ; these are indicated by the thick lines in fig.5 . these measurements show that a typical maximum transmission of the filters is @xmath010 % , and that the fwhm of the passband is about 0.31 nm for an ambient temperature resulting in the filter central wavelength to equal the green line wavelength . by taking short lengths of 0.8 mm at nine positions along the spectrograph slit ( which is aligned along the radial direction of the filter ) , we also examined the variation of the same quantities as a function of radial distance over the filter ; these are the thin lines plotted in fig.5 . there is a similar dependence on temperature in these individual measurements to the averaged results . the filter passband transmission varies by up to about 10% from the mean value , the passband wavelength position by up to 0.02 nm , and the passband width by up to only 4% . the background is very stable apart from one outlying measurement . test observations were taken in both the broad - band and narrow - band channels . with solar activity at an extremely low level , no coronal structures were visible in the green line at that time ( 2009 april 7 at 06:40 ut ) but the tests were nevertheless useful in that the optical and photometric quality of the data could be examined . examples of snapshots selected from the data in both channels are shown in figs.6 and 7 . inspection of the images shows that the image quality was very good . moreover , the different character of the radial gradients illustrates the very good spectral blocking of the narrow - band filter . the internal instrumental scattered light in the coronagraph and all the optical parts of secis was found to be sufficiently low to allow the required data acquisition . this confirms that the instrument itself is ready to measure prominent active region coronal loops above the solar limb when they appear under coronal " skies , i.e. with low degree of light scatter by the earth atmosphere . more detailed inspection of the data ( figs.8 and 9 ) shows that the noise level was low . the photon count level in the narrow - band channel within the portion of the image occupied by the artificial moon was measured to be at a very low level , averaging 8 dn s@xmath15 , only slightly more than the dark current level of 23 dn s@xmath15 . a bright coronal active region is expected to have a high signal - to - noise ratio , though experience from the 1999 and 2001 eclipses with the secis cameras suggests that the cameras are very unlikely to reach saturation levels . the secis instrument installed at the lomnicky peak observatory lyot coronagraph will allow data to be acquired that may result in an improved knowledge of where in the corona mhd waves are generated and/or dissipated . in particular , the signatures of high - frequency mhd waves involved in coronal heating may be observed . a considerable improvement in our knowledge of a long - standing problem of solar physics could be made by such observations , with implications for the physics of active regions , flares , the solar wind , and solar activity , as well as mechanisms of solar - terrestrial relationships . we acknowledge the anonymous referee for comments which helped to improve the paper . the work of j. a. and j. r. was supported partly by the slovak research and development agency under the contract no . apvv-0066 - 06 which also covered all expenses related to the secis instrument at the lomnicky peak observatory ( slovakia ) . authors are obliged for the support of the astronomical institute , slovak academy of sciences staff , namely assistants k. mank , r. maura , p. havrila , p. bendk , and the workshop assistant j. klein . p.r . was supported by the polish ministry of science and higher education , grant number n203 022 31/2991 . this research has made use of nasa s astrophysics data system .

heating mechanisms of the solar corona will be investigated at the high - altitude solar observatory lomnicky peak of the astronomical institute of sas ( slovakia ) using its mid - size lyot coronagraph and post - focal instrument secis provided by astronomical institute of the university of wrocaw ( poland ) . the data will be studied with respect to the energy transport and release responsible for heating the solar corona to temperatures of mega - kelvins . in particular investigations will be focused on detection of possible high - frequency mhd waves in the solar corona . the scientific background of the project , technical details of the secis system modified specially for the lomnicky peak coronagraph , and inspection of the test data are described in the paper .

arxiv

water vapor is an important molecule for the chemistry of interstellar and circumstellar clouds . the 6@xmath8 - 5@xmath9 masing transition of h@xmath3o at 22 ghz , which arises from levels around 700k , has been used since its detection by cheung et al . ( 1969 ) to trace high excitation gas around star forming regions and evolved stars . the size of the emitting regions at that frequency is typically of the order of a few milliarcseconds ( a few 10@xmath10 cm ) . hence , no information has been obtained from this line on the role of h@xmath3o at large spatial scales . other h@xmath3o lines have been detected from ground or airborne based telescopes like the 3@xmath11 - 2@xmath12 transition at 183 ghz ( waters et al . , 1980 ; cernicharo et al . 1990 , 1994 , 1996 ; gonzlez - alfonso et al . 1994 , 1998 ) , the 4@xmath13 - 3@xmath14 transition at 380 ghz ( phillips , kwan and huggins 1980 ) , the 10@xmath15 - 9@xmath16 transition at 321 ghz ( menten , melnick and phillips 1990a ) and the 5@xmath17 - 4@xmath18 transition at 325 ghz ( menten et al . also , the 1@xmath19 - 2@xmath20 transition of h@xmath21o at 547 ghz has been observed by zmuidzinas et al . ( 1994 ) . among these lines only the 3@xmath11 - 2@xmath12 transition at 183 ghz has been used to map the emission of h@xmath3o at very large spatial scale ( cernicharo et al . 1994 , hereafter referred to as cer94 ) . the map of the orion molecular cloud shown in cer94 is 6 orders of magnitude larger than the size of the spots detected at 22 ghz and for the first time an h@xmath3o abundance estimate was derived for the different large scale components of the orion molecular cloud . the iso satellite has provided the opportunity to observe thermal lines of water in the middle and far - infrared ( see the reviews by van dishoeck 1997 ; cernicharo 1997 ; and cernicharo et al 1997a,1998 ) . mapping of the sgrb2 molecular cloud by cernicharo et al . ( 1997b ) has definitely shown that water vapor is an ubiquitous molecule in molecular clouds with an abundance of 10@xmath7 . maps of the emission of several h@xmath3o lines in orion irc2 have been obtained by cernicharo et al . ( 1997a , 1998 , 1999 ) . observations of the central position have been also obtained by van dishoeck et al . ( 1998 ) , gonzlez - alfonso et al . ( 1998 ) and harwit et al . however , the iso observations of h@xmath3o have drawbacks . in addition to the limited spectral resolution and the high opacity of the thermal lines of h@xmath3o , the poor angular resolution provided by iso in the far - infrared prevents any detailed study of the spatial structure and physical conditions of the h@xmath3o emitting regions . an important aid in deriving h@xmath3o abundances could come from the observation of another masing transition of h@xmath3o with similar properties to those of the 183 ghz line . the 325 ghz line of h@xmath3o was observed by menten et al ( 1990b ) in the direction of orion irc2 and other molecular clouds . however , no maps were obtained . here we report the detection of extended water emission at 325 ghz and show that the h@xmath3o abundance is @xmath4 10@xmath5 in the * plateau*. the present data show the importance of ground - based observations of h@xmath3o in deriving the abundance of h@xmath3o in molecular clouds and in providing useful contraints on the physical conditions of the emitting regions . our ground - based observations provide much finer spatial resolution than iso or swas , and an estimate of x(h@xmath3o ) as accurate as that obtained from the extremely optically thick h@xmath3o lines observed in the submillimeter and far infrared domains . the observations were performed with the 10.4 m telescope of the caltech submillimeter observatory at the summit of mauna kea ( hawaii ) on april 1@xmath22 1998 . the receiver , a helium - cooled sis mixer operating in double - sideband mode ( dsb ) , was tuned at the frequency of the 5@xmath04@xmath1 line of h@xmath23o ( 325.152919 ghz ) . the h@xmath3o line was placed in the upper sideband ( usb ) to minimize atmospheric noise from the image sideband ( which was at 322.35 ghz ) . lines in the signal sideband are severely attenuated relative to those in the image sideband , due to the atmospheric h@xmath24 line . therefore it was necessary to check the sideband origin of a given line . the tuning frequency was shifted by 100 mhz to confirm that the central feature in the spectrum of orion - irc2 was the water vapor line . the backend consisted of a 1024 channel acousto - optic spectrometer covering a bandwidth of 500 mhz ( @xmath25v=1.1 kms@xmath26 ) . figure 1 shows the observed spectrum which matches very well the previous observation by menten et al . ( 1990b ) except for the line intensity ( see below ) . the pointing was determined by observing the same line towards the o - rich evolved star vy cma and was found to be accurate to 5 . the weather conditions were very stable during the observations with an atmospheric pressure and temperature of 620 mb and -1.4@xmath27c respectively . the relative humidity was measured to be 4 - 5% . the measured opacity from tipping scans at 225 ghz was @xmath40.025 . during the same night we performed broadband fourier transform spectroscopy ( fts ) measurements of the atmospheric absorption with the fts described in serabyn and weisstein ( 1995 ) . model calculations using the multi - layer atmospheric radiative transfer model atm ( cernicharo 1985,1988 ; pardo 1996 ) yielded an estimated precipitable water vapor column above the telescope of @xmath28200 @xmath29 m , which corresponds to a zenith transmission at 325.15 ghz of @xmath28 60% ( the corresponding value for the image side bande was @xmath28 87% , hence the line intensities for the image sideband features are overstimated by a factor @xmath28 2 ) . the heterodyne h@xmath3o data were calibrated using an absorber at ambient temperature . the calculated system noise temperature , for the signal sideband , was @xmath42100 k. the orion spectrum shown in figure 1 shows that the lines from the image sideband are weaker than the 325 ghz h@xmath3o line , i.e. , just the opposite of that occurring in the spectrum of menten et al . ( 1990b ) . rather than a variation of the maser emission ( the h@xmath3o line profile in figure 1 is identical to that shown in menten et al . 1990b ) we think that this difference is due to much better atmospheric transmission during our observations . we estimate that our intensity scale is correct to within 20 - 30% . the orion - irc2 map was carried out in position switching mode by using the on - the - fly procedure with an off position 5 away in azimuth . the spatial distribution of the 325 ghz emission is shown in figure 2 together with that of ch@xmath2oh ( from the image side band ) and the 183 ghz emission from cer94 . integrated intensity maps for selected velocity intervals are also shown in figure 2 . in order to compare the line profiles of the h@xmath3o lines at 183 and 325 ghz we reobserved a few positions at 183 ghz with the 30-m iram telescope in january 1999 . the weather conditions were also excellent with a zenith opacity at this frequency of @xmath4 1 . the spectra , together with those obtained in 1994 , are shown in figure 3 . the observed 5@xmath17 - 4@xmath18 line profile towards the center position looks similar to that of the 3@xmath11 - 2@xmath12 line observed by cer94 ( see spectra in figure 3 ) . however , the antena temperature of the line is 20 times weaker and the line profile , although covering the same velocity range , is shifted towards the red . taking into account the different beam sizes of the iram-30 m telescope at 183 ghz and the cso at 325 ghz , and the extension of the emission in the latter line , we estimate that the main beam brightness temperature ratio , @xmath30=t@xmath31(183)/t@xmath31(325 ) is 10 - 20 if both lines were observed with a telescope of @xmath32 beam size . on the other hand , t@xmath31(183)@xmath33 k. both @xmath30 and t@xmath31(183 ) are well determined and can not be related to a calibration problem as the atmospheric conditions were extremely good during our observations at both frequencies . similar values for * r * , i.e. , @xmath341020 , are also found at other positions in the cloud ( see figure 2 ) and represent a real difference in the brightness temperatures of the two lines . only at position @xmath35=-12 , @xmath36=48 the peak temperature of the 5@xmath17 - 4@xmath18 transition approaches that of the 3@xmath11 - 2@xmath12 ( the 5@xmath17 - 4@xmath18 line is , however , narrower ) . the 325 ghz emission at this position presents a local maximum clearly visible in the velocity maps ( see figure 2 ) . like the 3@xmath11 - 2@xmath12 line , the 5@xmath17 - 4@xmath18 transition is masing in nature . there are some narrow features at 325 ghz but with intensities of only a few k , i.e. , much weaker than those reported at 183 ghz by cer94 . these features agree in velocity with those found at 183 ghz . however , @xmath30 changes drastically from feature to feature , a fact that reveals the maser nature of the emission . outside the central region the lines are very narrow ( @xmath373 - 5 kms@xmath26 ) . some of the 3@xmath11 - 2@xmath12 narrow velocity components have antenna temperatures above 2000 k and are probably a few arcseconds in size ( cer94 , gonzlez - alfonso 1995 ) . the observations at this frequency performed in january 1999 clearly indicate a variation in the intensity of some of these features with respect to those of cer94 . however , at positions where the line is dominated by the plateau emission ( @xmath35=28 , @xmath36=-16 and @xmath35=12 , @xmath36=-16 for example ; see figure 3 ) and outside the central region ( @xmath35=-12 , @xmath36=48 ; figure 3 ) the line shape and intensity do not show any significant change between both epochs . a possible explanation for the 183 ghz and 325 ghz emission being spatially extended could be that it arises from many masing point like sources strongly diluted in the beam . this is ruled out by the results of cer94 where even the strong features at 183 ghz ( t@xmath382000 - 4000 k ) show indication of some spatial extent ( see above ) . the densities needed to reproduce the observed brightness temperatures of the maser spots at 22 ghz would result on a thermal or suprathermal 183 ghz line . consequently , if the 183 ghz emission was arising from the same region than that of the 22 ghz line very large column densities will be needed to reproduce the observed 183 ghz and 325 ghz intensities . in addition , the weak and extended emission observed at 183 ghz by cer94 clearly indicates the presence of water vapour coexistent with the molecular gas in the orion molecular ridge . in order to understand the behavior of the two masing lines we have modeled the radiative transfer of the rotational levels of p - h@xmath3o for the physical conditions of the orion molecular cloud . the radiative transfer method is described in gonzlez - alfonso & cernicharo ( 1997 ) , and the model consists of a molecular shell with diameter of @xmath39 cm ( size of 15@xmath40 at 450 pc ) which expands at a constant velocity of 25 km s@xmath26 . collisional rates between water vapor and helium were taken from _ green _ , maluendes & mclean ( 1993 ) . the helium abundance was assumed to be 0.1 , and the rates were corrected to take into account the collisions between h@xmath23o and h@xmath23 . we calculated the statistical equilibrium populations of the lowest 45 rotational levels of para - h@xmath3o for different temperatures ( t@xmath41=100 , 150 , 200 and 300 k ) , column densities n(p - h@xmath3o ) , and volume densities ( n(h@xmath3)=@xmath42 , @xmath43 and @xmath44 @xmath45 ) . figure 4 shows the main beam brightness temperatures ( t@xmath31 ) ( as observed by a telescope of 15@xmath40 beam size ) for the 183 and 325 ghz para - water lines ( thin and broad lines , respectively ) , together with @xmath30 ( dashed lines ) . these t@xmath31 were computed from the integrated intensity by assuming that the spectral emission is gaussian - shaped . inspection of fig . 4 indicates that the line intensity ratio @xmath30 increases with n(p - h@xmath3o ) for low column densities ( which depend on t@xmath41 and n(h@xmath3 ) ) . both masers are unsaturated in these conditions , but the higher opacity of the 183 ghz transition makes this line more sensitive to variations of n(p - h@xmath3o ) . for higher values of n(p - h@xmath3o ) , the 183 ghz line becomes saturated and the exponential amplification of the 325 maser line yields a decrease of @xmath30 . finally , when both the 183 and 325 ghz lines are saturated , @xmath30 approaches a nearly constant value or even decreases below 1 for high n(h@xmath3 ) and low t@xmath41 . the maser at 183 ghz is quenched for these later conditions , although t@xmath31 can still remain above t@xmath41 due to the suprathermal excitation of the line ( see cer94 ) . even for relatively low column densities ( n(p - h@xmath3o)@xmath46 @xmath47 ) , low temperatures ( 100 k ) and volume density ( @xmath42 @xmath45 ) , the 183 ghz line has an intensity larger than 10 k ( see cer94 ) . however , the possibility of appreciable amplification for the 5@xmath17 - 4@xmath18 line is much more restricted than for the 183 ghz line , due to the high energy of the levels involved in the 325 ghz line ( @xmath48 k ) , and to the higher frequency and einstein coefficient of this transition . this fact explains the difference in spatial extent between both transitions , so that the the 325 ghz line is spatially restricted to the plateau while the 183 ghz line is in addition detected in the ridge ( cer94 ) . the water vapor column density that fits the observed log * r*@xmath49 depends strongly on the assumed values of n(h@xmath3 ) and t@xmath41 . the higher n(h@xmath3 ) and t@xmath41 , the lower n(p - h@xmath3o ) that is needed to obtain an appreciable amplification of the 325 ghz line . 4 shows that * r*@xmath50 is obtained in different panels for n(p - h@xmath3o ) ranging from @xmath51 @xmath47 ( n(h@xmath3)=@xmath44 @xmath45 , t@xmath41=300 k ) to @xmath52 @xmath47 ( n(h@xmath3)=@xmath42 @xmath45 , t@xmath41=100 k ) . however , some of these models are not compatible with the observed intensities . for n(h@xmath3)@xmath53@xmath43 @xmath45 and t@xmath41@xmath54150 k , we find that * r*@xmath55 yields t@xmath31(325 ) in excess of @xmath56 k and t@xmath31(183)@xmath54@xmath57 k. both the observed intensities and * r * are only compatible with more moderate values of n(h@xmath3 ) and/or t@xmath41 . the physical reason is that , for high values of n(h@xmath3 ) and/or t@xmath41 , the collisional pumping of the 183 ghz line becomes so efficient that the emission in this line reaches high intensities for column densities that provide t@xmath31(325 ) of 50100 k. of course , the plateau may have regions with very high densities and temperatures ( which will give rise , for example , to the emission at 22 ghz and to the narrow spectral features observed at 183 and 325 ghz ) , but these will be much smaller than the observed size of the cloud . the widespread emission from the plateau we observe at 183 ghz and 325 ghz is only compatible with moderate values of n(h@xmath3 ) and t@xmath41 . in our models , the 22 ghz line will have intensities similar to those already observed in orion ( genzel et al , 1981 ) only for the highest kinetic temperatures and volume densities in figure 4 . the brightest spots at 22 ghz could be correlated with the narrow features at 183 ghz , and with the relatively weak lines at 325 ghz . brightness temperatures above 10@xmath58 k can be obtained for large column densities , t@xmath59@xmath54150 k and n(h@xmath3)@xmath5410@xmath58 @xmath45 . lower limits for n(h@xmath3 ) and t@xmath41 can be obtained from the observations of other molecular lines ( e.g. , blake et al . 1987 ) , so that we adopt n(h@xmath3)@xmath53@xmath43 @xmath45 and t@xmath41=100150 k. for these conditions we obtain n(p - h@xmath3o ) in the range @xmath60@xmath61 @xmath47 , and hence n(h@xmath3o ) in the range @xmath62@xmath52 @xmath47 . the water vapor abundance can be derived from the co data taken with similar angular resolution ( see cer94 ) . for the intermediate velocity wind cer94 derived a co column density of 10@xmath63 @xmath47 . hence , the x(h@xmath3o)/x(co ) abundance ratio in the plateau is around 1 , i.e. , x(h@xmath3o)@xmath28@xmath64 . in the ridge molecular cloud the 325 ghz is very weak . our models and the 183 ghz data provide an estimate for x(h@xmath3o ) of a few 10@xmath6 - 10@xmath7 ( see cer94 ) which is in good agreement with our iso results ( see cernicharo et al . 1998 , 1999 ) . the comparison of several masing transitions arising in relatively low energy levels of h@xmath3o allows us to constrain the physical conditions of the different emitting regions . so far , ground - based observations of these transitions with large radio telescopes are the only means to obtain the spatial distribution of h@xmath3o in interstellar clouds . j. cernicharo and e. gonzlez - alfonso acknowledge spanish dges for this research under grants pb96 - 0883 and esp98 - 1351e . pardo gratefully acknowledges the financial support of the _ observatoire de paris - meudon _ , _ cnes _ and _ mto - france_. the cso is supported by nsf contract # ast-9615025 .

we present observations of the 5@xmath04@xmath1 transition of water vapor _ at 325.15 ghz _ taken with the cso telescope towards orion irc2 . the emission is more extended than that of other molecular species such as ch@xmath2oh . however , it is much less extended than the emission of water vapor at 183.31 ghz reported by cernicharo et al ( 1994 ) . a comparison of the line intensities at 325.15 ghz and 183.31 ghz puts useful constraints on the density and temperature of the emitting regions and allows an estimate of h@xmath3o abundance , x(h@xmath3o ) , of @xmath410@xmath5 in the plateau and @xmath410@xmath6 - 10@xmath7 in the ridge . 21.75 cm -0.5 cm = 20pt ism : molecules ism : individual ( orion irc2 ) line : profiles masers radio lines : ism submillimeter

arxiv

threshold nets are obtained by assigning a weight @xmath1 , from a distribution @xmath2 , to each of @xmath3 nodes and connecting any two nodes @xmath4 and @xmath5 whose combined weights exceed a certain threshold , @xmath6 : @xmath7 @xcite . threshold nets can be produced of ( almost ) arbitrary degree distributions , including scale - free , by judiciously choosing the weight distribution @xmath2 and the threshold @xmath6 , and they encompass an astonishingly wide variety of important architectures : from the star graph ( a simple cartoon " model of scale - free graphs consisting of a single hub ) with its low density of links , @xmath8 , to the complete graph . studied extensively in the graph - theoretical literature @xcite , they have recently come to the attention of statistical and non - linear physicists due to the beautiful work of hagberg , swart , and schult @xcite . , and ( b ) its box representation , highlighting modularity . nodes are added one at a time from bottom to top , @xmath9 s on the left and @xmath10 s on the right.,scaledwidth=35.0% ] hagberg _ et al_. , exploit the fact that threshold graphs may be more elegantly encoded by a two - letter sequence , corresponding to two types of nodes , @xmath9 and @xmath10 @xcite . as new nodes are introduced , according to a prescribed sequence , nodes of type @xmath9 connect to none of the existing nodes , while nodes of type @xmath10 connect to all of the nodes , of either type : @xmath11 and @xmath12 . in fig . [ graph_box](a ) we show an example of the threshold graph obtained from the sequence @xmath13 . note the _ modular _ structure of threshold graphs : a subsequence of @xmath14 consecutive @xmath10 s gives rise to a @xmath15-clique , while nodes in a subsequence of @xmath9 s connect to @xmath10 nodes thereafter , but not among one another . we highlight this modularity with a diagram of boxes ( similar to @xcite ) : oval boxes enclose nodes of type @xmath9 , that are not connected among themselves , while rectangular boxes enclose @xmath16-cliques of @xmath10-nodes @xcite . a link between two boxes means that all of the nodes in one box are connected to all of the nodes in the other , fig . [ graph_box](b ) . given the sequence of a threshold net , there exist fast algorithms to compute important structural benchmarks , besides its modularity , such as degree distribution , triangles , betweenness centrality , and the spectrum and eigenvectors of the graph laplacian @xcite . the latter are a crucial determinant of dynamics and synchronization and have applications to graph partitioning and mesh processing @xcite . perhaps more importantly , it becomes thus possible to _ design _ threshold nets with a particular degree distribution , spectrum of eigenvalues , etc . , @xcite . despite their malleability , threshold nets are limited in some obvious ways , for example their diameter is 1 or 2 , regardless of the number of nodes @xmath3 . our idea consists of studying the broader class of nets that can be constructed from a sequence ( formed from two or more letters ) by deterministic rules of connectivity on their own right . it is truly this property that gives the nets all their desired attributes : modularity ( as in everyday life complex nets ) , easily computable structural measures including the possibility of design and a high degree of compressibility . roughly speaking , each additional letter to the alphabet allows for an increase of one link in the nets diameter , so that the three - letter nets possess diameter 3 or 4 ( some of the new types of two - letter nets have diameter 3 ) . this modest increase is very significant , however , in view of the fact that the diameter of many everyday life complex nets is not much larger than that @xcite . sequence nets gain us much latitude in the types of nets that can be described in this elegant fashion , while retaining much of the analytical appeal of threshold nets . another unusual property of sequence nets is that any ensemble of sequence nets admits a natural ordering ; simply list them alphabetically according to their sequences . one may use this ordering for exploring eigenvalues and other structural properties of sequence nets . in this paper , we make a first stab at the general class of _ sequence nets_. in section [ two - letter ] we explore systematically all of the possible rules for creating connected sequence nets from a two - letter alphabet . applying symmetry arguments , we find that threshold nets are only one of three equivalence classes , characterized by the highest level of symmetry . we then discuss the remaining two classes , showing that also then there is a high degree of modularity and that various structural properties can be computed easily . curiously , the new classes of two - letter sequence nets can be related to a generalized form of threshold nets , where the difference @xmath17 , rather than the sum of the weights , is the one compared to the threshold @xmath6 . in section [ three - letter ] we derive all possible forms of connected three - sequence nets . symmetry arguments lead us to the discovery of 30 distinct equivalence classes . among these classes , we identify a natural extension of threshold nets to three - letter sequence nets . despite the enlarged alphabet , 3-letter sequence nets do retain many of the desirable properties of threshold and 2-letter sequence nets . we also show that at least some of the 3-letter sequence nets can be mapped into threshold nets with _ two _ thresholds , instead of one . we conclude with a summary and discussion of open problems in section [ conclude ] . consider graphs that can be constructed from sequences @xmath18 of the two letters @xmath9 and @xmath10 . we can represent any possible rule by a @xmath19 matrix * r * whose elements indicate whether nodes of type @xmath4 connect to nodes of type @xmath5 : @xmath20 if the nodes connect , and 0 otherwise ( @xmath21 stands for @xmath22 , respectively ) . [ graph_box ] gives an example of the graph obtained from the sequence @xmath13 , applying the _ threshold _ rule @xmath23 . since each element can be @xmath24 or @xmath25 independently of the others , there are @xmath26 possible rules . we shall disregard , however , the four rules that fail to connect between @xmath9 and @xmath10 , @xmath27 for they yield simple _ disjoint _ graphs of the two types of nodes : @xmath28 yields isolated nodes only , @xmath29 yields one complete graph of type @xmath9 and one of type @xmath10 , @xmath30 yields a complete graph of type @xmath9 and isolated nodes of type @xmath10 , etc . applied to the sequence @xmath13 ( a ) , and from @xmath31 applied to the reverse - inverted sequence @xmath32 ( b ) , are identical.,scaledwidth=35.0% ] the list of remaining rules can be shortened further by considering two kinds of symmetries : ( a ) permutation , and ( b ) time reversal . _ permutation _ is the symmetry obtained by permuting between the two types of nodes , @xmath33 . thus , a permuted rule ( @xmath34 and @xmath35 ) acting on a permuted sequence ( @xmath36 ) yields back the original graph @xcite . _ time reversal _ is the symmetry obtained by reversing the arrows ( time " ) in the connectivity rules , or taking the transpose of @xmath37 . the transposed rule acting on the reversed sequence @xmath38 yields back the original graph . the two symmetry operations are their own inverse and they form a symmetry group . in particular , one may combine the two symmetries : a rule with @xmath34 applied on a reversed sequence with inverted types @xmath39 yields back the original graph , see fig . [ time_reversal ] . all of the four rules @xmath40 are equivalent and generate threshold graphs . @xmath41 is the rule for threshold graphs exploited by hagberg et al . , @xcite , and @xmath42 is equivalent to it by permutation . @xmath31 is obtained from @xmath41 by time reversal and permutation ( fig . [ time_reversal ] ) , and @xmath43 is obtained from @xmath41 by time reversal . the two rules @xmath44 are equivalent , by either permutation or time reversal , and generate non - trivial bipartite graphs that are different from threshold nets ( fig . [ abgraphs ] ) . the rule @xmath45 generates complete bipartite graphs . however , the complete bipartite graph @xmath46 can also be produced by applying @xmath47 to the sequence @xmath48 of @xmath49 @xmath9 s followed by @xmath50 @xmath10 s , so the rule @xmath51 is a `` degenerate '' form of @xmath47 . one could see that this is the case at the outset , because of the symmetrical relations @xmath52 , @xmath11 : these render the ordering of the @xmath9 s and @xmath10 s in the graph s sequence irrelevant . by the same principle , @xmath53 and @xmath54 are degenerate forms of @xmath41 and @xmath42 , respectively . they yield threshold graphs with segregated sequences of @xmath9 s and @xmath10 s . the two rules @xmath55 are equivalent , by either permutation or time reversal , and generate non - trivial graphs different from threshold graphs and graphs produced by @xmath47 ( fig . [ abgraphs ] ) . finally , the rule @xmath56 is a degenerate form of @xmath57 ( or @xmath58 ) and yields only complete graphs ( which are threshold graphs , so @xmath59 is subsumed also in @xmath60 ) . , applying rules @xmath47 ( a ) , @xmath41 ( b ) , and @xmath57 ( c ) . note the figure - background symmetry of ( a ) and ( c ) : the graphs are the inverse , or complement of one another ( see text ) . the inverse of the threshold graph ( b ) is also a ( two - component ) threshold graph , obtained from the same sequence and applying the rule @xmath42 ( @xmath41 s complement).,scaledwidth=47.0% ] to summarize , @xmath41 , @xmath47 , and @xmath57 are the only two - letter rules that generate different classes of non - trivial connected graphs . there is yet another amusing type of symmetry : applying @xmath47 and @xmath57 to the same sequence yields _ complement _ , or _ inverse _ graphs nodes are adjacent in the inverse graph if and only if they are _ not _ connected in the original graph . the figure - background symmetry manifest in the rules @xmath47 and @xmath57 ( @xmath61 ) is also manifest in the graphs they produce ( fig . [ abgraphs]a , c ) . on the other hand , the inverse of threshold graphs are also threshold graphs . also , the complement of a threshold rule applied to the complement ( inverted ) sequence yields back the original graph . in this sense , threshold graphs have maximal symmetry . @xmath47-graphs are typically less dense , and @xmath57-graphs are typically denser than threshold graphs . possible connections between nodes of type @xmath9 and @xmath10 . ( b ) three equivalent representations of the threshold rule @xmath41 . the second and third diagram are obtained by label permutation and time - reversal , respectively . ( c ) diagrams for @xmath47 and @xmath57 . note how they complement one another to the full set of connections in part ( a).,scaledwidth=25.0% ] the connectivity rules have an additional useful interpretation as directed graphs , where the nodes represent the letters of the sequence alphabet , a directed link , e , g . , from @xmath9 to @xmath10 indicates the rule @xmath52 , and a connection of a type to itself is denoted by a self - loop ( fig . [ graph_notation ] ) . because the rules are the same under permutation of types , there is no need to actually label the nodes : all graph isomorphs represent the same rule . likewise , time - reversal symmetry means that graphs with inverted arrows are equivalent as well . note that the direction of self - loops is irrelevant in this respect , so we simply take them as undirected . we shall make use of this notation , extensively , for the analysis of 3-letter sequence nets in section [ three - letter ] . a very special property of sequence nets is the fact that any arbitrary ensemble of such nets possesses a natural ordering , simply listing the nets alphabetically according to their sequences . in contrast , think for example of the ensemble of erds - rnyi random graphs of @xmath3 nodes , where links are present with probability @xmath49 : there is no natural way to order the @xmath62 graphs in the ensemble @xcite . plotting a structural property against the alphabetical ordering of the ensemble reveals some inner structure of the ensemble itself , yielding new insights into the nature of the nets . as an example , in fig . [ eigs_2threshold ] we show @xmath63 , the second smallest eigenvalue , for the ensemble of connected threshold nets containing @xmath64 nodes ( there are @xmath65 graphs in the ensemble , since their sequences must all start with the letter @xmath9 ) . notice the beautiful pattern followed by the eigenvalues plotted in this way , which resembles a fractal , or a cayley tree : the values within the first half of the graphs in the @xmath1-axis repeat in the second half , and the pattern iterates as we zoom further into the picture . nodes , plotted against their alphabetical ordering.,scaledwidth=45.0% ] structural properties of the new classes of two - letter sequence nets , @xmath47 and @xmath57 , are as easily derived as for threshold nets . here we focus on @xmath47 alone , which forms a subset of bipartite graphs . the analysis for @xmath57 is very similar and often can be trivially obtained from the complementary symmetry of the two classes . all connected sequence nets in the @xmath47 class must begin with the letter @xmath9 and end with the letter @xmath10 . a sequence of this sort may be represented more compactly @xcite by the numbers of @xmath9 s and @xmath10 s in the alternating layers , @xmath66 . we assume that there are @xmath3 nodes and @xmath14 layers ( @xmath14 is even ) . we also use the notation @xmath67 and @xmath68 for the total number of @xmath9 s and @xmath10 s , as well as @xmath69 and likewise for @xmath70 . finally , since all the nodes in a layer have identical properties we denote any @xmath9 in the @xmath4-th layer by @xmath71 and any @xmath10 in the @xmath5-th layer by @xmath72 . with this notation in mind we proceed to discuss several structural properties . : since @xmath9 s connect only to subsequent @xmath10 s ( and @xmath10 s only to preceding @xmath9 s ) the degree @xmath73 of the nodes is given by @xmath74 : there are no triangles in @xmath47 nets so the clustering of all nodes is zero . : every @xmath9 is connected to the last @xmath10 , so the distance between any two @xmath9 s is 2 . every @xmath10 is connected to the first @xmath9 in the sequence , so the distance between any two @xmath10 s is also 2 . the distance between @xmath75 and @xmath76 is 1 if @xmath77 ( they connect directly ) , and 3 if @xmath78 ( @xmath75 links to @xmath79 , that links to @xmath80 , that links to @xmath76 ) . : because of the time - reversal symmetry between @xmath9 and @xmath10 , it suffices to analyze @xmath10 nodes only . the result for @xmath9 can then be obtained by simply reversing the creation sequence and permuting the letters . the vertex betweenness @xmath81 of a node @xmath82 is defined as : @xmath83 where @xmath84 is the number of shortest paths from node @xmath85 to @xmath86 ( @xmath87 ) , excluding the cases that @xmath88 or @xmath89 . @xmath90 is the number of shortest paths from @xmath85 to @xmath86 that goes through @xmath82 . the factor @xmath91 appears for undirected graphs since each pair is counted twice in the summation . the betweenness of @xmath10 s can be calculated from lower layers to higher layers recursively . in the first b - layer @xmath92 and @xmath93 for @xmath94 . the second term on the rhs accounts for the shortest paths from layer @xmath95 to itself and all previous layers of @xmath9 , and the third term corresponds to paths from @xmath95 to @xmath72 to @xmath71 ( @xmath96 ) to @xmath97 . although this recursion can be solved explicitly it is best left in this form , as it thus highlights the fact that the betweenness centrality increases from one layer to the next . in other words , the networks are _ modular _ , where each additional @xmath10-layer dominates all the layers below . : unlike threshold nets , for @xmath47 nets the eigenvalues are _ not _ integer , and there seems to be no easy way to compute them . instead , we focus on the second smallest and largest eigenvalues , @xmath63 and @xmath98 , alone , for their important dynamical role : the smaller the ratio @xmath99 the more susceptible the network is to synchronization @xcite . consider first @xmath63 . for @xmath47 it is easy to show that both the _ vertex _ and _ edge connectivity _ are equal to @xmath100 . then , following an inequality in @xcite , @xmath101 the upper bound seems stricter and is a reasonable approximation to @xmath63 ( see fig . [ l2bounds ] ) . nets with @xmath64 against their alphabetical ordering ( solid curve ) , and their upper and lower bounds ( broken lines).,scaledwidth=40.0% ] for @xmath98 , using theorem 2.2 of @xcite one can derive the bounds @xmath102 but they do not seems very useful , numerically . playing with various structural properties of the nets , plotted against their alphabetical ordering , we have stumbled upon the approximation @xmath103 where @xmath104 is the average degree of the graph , see fig . [ l2approx ] . the approximation is exact for bipartite _ complete _ graphs ( @xmath105 ) and the relative error increases slowly with @xmath3 ; it is roughly at 10% for @xmath106 . nets with @xmath64 against their alphabetical ordering ( solid curve ) , and its approximated value ( broken line).,scaledwidth=40.0% ] in @xcite it was shown that threshold graphs have a mapping to a sequence net , with a unique sequence ( under the threshold rule " @xmath41 ) ; and conversely , for any @xmath41-sequence net there exists a set of weights @xmath107 of the nodes ( not necessarily unique ) , such that connecting any two nodes that satisfy @xmath7 reproduces the sequence net . here we establish a similar relation between @xmath47- ( or @xmath57- ) sequence nets and a different kind of threshold net , where connectivity is decided by the difference @xmath17 rather than the sum of the weights . we begin with the mapping of a weighted set of nodes to a @xmath47-sequence net . let a set of @xmath3 nodes have weights @xmath107 ( @xmath108 ) , taken from some probability density , and we assume @xmath109 , without loss of generality . denote nodes with @xmath110 as type @xmath9 and nodes with @xmath111 as type @xmath10 . finally , connect any two nodes @xmath4 and @xmath5 that satisfy @xmath112 . the resulting graph can be constructed by a unique sequence under the rule @xmath47 , obtained as follows . for convenience , rewrite the set of weights as @xmath113 where the first @xmath114 weights correspond to @xmath9-nodes and the rest to @xmath10-nodes . denote the creation sequence by @xmath18 and determine the @xmath115 by the algorithm ( in pseudo - code ) : set @xmath116 , @xmath117 for @xmath118 , do : 0.4 cm if @xmath119 0.8 cm set @xmath120 and @xmath121 0.4 cm else 0.8 cm set @xmath122 and @xmath123 end . it is understood that if the @xmath124 are exhausted before the end of the loop , the remainder @xmath10-nodes are automatically affixed to the end of the sequence ( and similarly for the @xmath125 ) . for example , using this algorithm we find that the difference - threshold " graph resulting from the set of weights @xmath1261,2,3,5,7,16,17,20@xmath127 and @xmath128 , can be reproduced from the sequence @xmath13 , with the rule @xmath47 . consider now the converse problem : given a graph created from the sequence @xmath18 with the rule @xmath47 , we derive a ( non - unique ) set of weights @xmath129 such that connecting any two nodes with @xmath112 results in the same graph . rewrite first the creation sequence into its compact form @xmath130 , and assign weights @xmath131 for nodes @xmath9 in layer @xmath131 , weights @xmath132 for nodes @xmath10 in layer @xmath0 , and set the threshold at @xmath133 . for example , the sequence @xmath13 has a compact representation @xmath134 , with @xmath135 layers , so the three @xmath9 s in layer @xmath25 have weights @xmath25 , the two @xmath10 s in layer @xmath136 have weights @xmath137 , the two @xmath9 s in layer @xmath138 have weights @xmath138 , and the single @xmath10 in layer @xmath139 has weight @xmath140 . the weights @xmath141 , with connection threshold @xmath142 , reproduce the original graph . sequence graphs obtained from the rule @xmath57 can be also mapped to difference threshold graphs in exactly the same way , only that the criterion for connecting two nodes is then @xmath143 , instead of @xmath112 , as for @xmath47 . the mapping of sequence nets to generalized threshold graphs may be helpful in the analysis of some of their properties , for example , for finding the _ isoperimetric number _ of a sequence graph @xcite . with a three - letter alphabet , @xmath144 , there are at the outset @xmath145 possible rules . again , these can be reduced considerably , due to symmetry . because the rule matrix has 9 entries ( an odd number ) no rule can be identical to its complement . thus , we can limit ourselves to rules with no more than 4 non - zero entries and apply symmetry arguments to reduce their space at the very end we can then add the complements of the remaining rules . in fig . [ 3nets ] we list all possible three - letter rules with two , three , and four interactions . rules that lead to disconnected graphs , and symmetric rules ( by label permutation or time - reversal ) have been omitted from the figure . and @xmath146 ) , and rules 3 , 12 , 13 , and 14 are degenerate cases of rules 2 , 6 , 7 , and 6 , respectively . this leaves us with fifteen distinct three - letter rules ( underlined ) , and their fifteen complements , for a total of 30 different classes of three - letter sequence nets.,scaledwidth=40.0% ] rule 2 @xcite is in fact not new : identifying nodes of type @xmath9 and @xmath146 ( as marked in rule 1 of the figure ) we can easily see that the rule is identical to the two - letter rule 8 . in the same fashion , rule 7 is the same as the two - letter threshold rule 4 . rule 3 is a degenerate form of 2 : because of the double connection @xmath147 and @xmath148 , the order at which @xmath10 and @xmath146 appear in the sequence relative to one another is inconsequential . ( on the other hand , the order of the @xmath10 s relative to @xmath9 s _ is _ important , since @xmath9 s connect only to those @xmath10 s that appear earlier in the sequence . ) then , given a sequence one can rearrange it by moving all the @xmath146 s to the end of the list . if we now apply 2 , @xmath52 and @xmath148 , then we get the same graph as from the original sequence under the rule 3 . the same consideration applies to rules , and , that are degenerate forms of 6 , 7 and 8 ( or 6 ) , respectively . we are thus left with only 15 distinct rules with fewer than 5 connections . to these one should add their complements , for a total of 30 distinct three - letter rules . note the resemblance of , , and to two - letter threshold nets . seems like a particularly symmetrical generalization and we will focus on it in much of our discussion below . while one can easily establish wether a graph is connected or not , _ a posteriori _ , with a burning algorithm that requires @xmath149 steps , it is useful to have shortcut rules that tell us how to avoid bad sequences at the outset : knowing that two - letter threshold graphs are connected if and only if their sequence ends with @xmath10 , deals with the question most effectively . analogous criteria exist for three - letter sequence graphs but they are a bit more complicated . for example , three - letter sequences interpreted with lead to connected graphs if and only if they satisfy : _ ( 1 ) the first a and the first c in the sequence appear before the last b. ( 2 ) the sequence does not start with b_. ( we assume that the sequence contains all three letters . ) for 1the requirements are : _ ( 1 ) the first a in the sequence must appear after the first b. ( 2 ) the last c in the sequence must appear before the last b. ( 3 ) the last a in the sequence must appear after the first c , and there ought to be at least one b between the two . _ similar criteria exist for all other three - letter rules and can be found by inspection . structural properties of three - letter sequence nets are analyzed as easily as those of two - letter nets , here we list , as an example , a few basic attributes of sequence nets . we use a notation similar to that of section [ new_classes ] . : @xmath9 and @xmath146 nodes form complete subgraphs , while @xmath10 nodes connect to all preceding @xmath9 s and @xmath146 s . thus the degree of the nodes are : @xmath150 : since the @xmath9 nodes make a subset complete graph @xmath151 , and likewise for @xmath146 , @xmath152 . the @xmath10 s do not connect among themselves , but they all connect to the nodes in the first layer ( which does not consist of @xmath10 s ) , so @xmath153 . for the distance of @xmath9 nodes from @xmath10 , we have @xmath154 where @xmath155 is the index of the first @xmath9-layer and @xmath156 is the index of the last @xmath10-layer . the first line follows since @xmath10 s are directly connected to preceding @xmath9 s and @xmath146 s . the second , and third and fourth lines are illustrated in fig . [ distance]a and b , respectively . the distance @xmath157 follows the very same pattern . finally , inspecting all different cases one finds @xmath158 in nets . ( a ) if @xmath159 and the first @xmath9 is below @xmath72 the distance is 2 . ( b ) if the first @xmath9 is above @xmath72 , then the first @xmath146 must be below ( @xmath10 ca nt start the sequence ) ; in that case if @xmath71 is below the last @xmath10 the distance is 3 , and otherwise the distance is 4 . only the relevant parts of the complete net are shown.,scaledwidth=40.0% ] : we have found no obvious way to compute the eigenvalues , despite the similarities between nets and two - letter threshold nets . however , plots of the eigenvalues against the alphabetical ordering of the nets once again reveals intriguing fractal patterns , and one can hope that these might be exploited at the very least to produce good bounds and approximations . in fig . [ r_r18 ] we plot the ratio @xmath160 for nets with @xmath161 against their alphabetical ordering . the @xmath1-axis includes sequences of nets that are not connected : in this case @xmath162 and synchronization is not possible . these cases show as gaps in the plot , for example , the big gap in the center corresponds to disconnected sequences that start with the letter @xmath10 ( see section [ connect ] ) . for nets consisting of @xmath161 nodes , against their alphabetical ordering . note the gap near the center , which corresponds to sequences of disconnected graphs . note also the mirror symmetry this is due to the mirror symmetry of the rule itself.,scaledwidth=40.0% ] some of the three - letter sequence nets can be mapped to generalized forms of threshold nets . for example , the following scheme yields a _ two_-threshold net , equivalent to three - letter sequence nets generated by the rule . let the nodes be assigned weights @xmath163 , from a random distribution , and connect any two nodes @xmath4 and @xmath5 that satisfy @xmath164 or @xmath165 . identifying nodes with weight @xmath166 with @xmath9 , nodes with @xmath167 with @xmath10 , and nodes with @xmath168 with @xmath146 , we see that all @xmath9 s connect to one another and all @xmath146 s connect to one another but the @xmath10 s do not , and @xmath9 s and @xmath146 s do not connect ; nodes of type @xmath9 and @xmath10 may or may not connect , and likewise for nodes of type @xmath146 and @xmath10 . to reflect the actual connections , the nodes of type @xmath9 and @xmath10 may be arranged in a sequence according to the algorithm in @xcite , for the threshold rule . also the nodes of type @xmath146 and @xmath10 may be arranged in a sequence , to reflect the actual connections , with the very same algorithm . because there are no connections between @xmath9 and @xmath146 the two results may be trivially merged . note , however , that once the @xmath9-@xmath10 sequence is established the order of the @xmath10 s is set , so the direction of connections between @xmath146 and @xmath10 ( @xmath169 or @xmath170 ) is _ not _ arbitrary . in our example , the mapping is possible to but not to . we have introduced a new class of nets , sequence nets , obtained from a sequence of letters and fixed rules of connectivity . two - letter sequence nets contain threshold nets , and in addition two newly discovered classes . the class can be mapped to a difference - threshold " net , where nodes @xmath4 and @xmath5 are connected if their weights difference satisfies @xmath143 . this type of net may be a particularly good model for social nets , where the weights might measure political leaning , economical status , number of offspring , etc . , and agents tend to associate when they are closer in these measures . we have shown that the structural properties of the new classes of two - letter sequence nets can be analyzed with ease , and we have introduced an ordering in ensembles of sequence nets that is useful in visualizing and studying their various attributes . we have fully classified 3-letter sequence nets , and looked at a few examples , showing that they too can be analyzed simply . the diameter of sequence nets grows linearly with the number of letters in the alphabet and for a 3-letter alphabet it is already 3 or 4 , comparable to many everyday life complex nets . realistic diameters might be achieved with a modest expansion of the alphabet . there remain numerous open questions : applying symmetry arguments we have managed to reduce the class of 3-leter nets to just 30 types , but we have not ruled out the possibility that some overlooked symmetry might reduce the list further ; the question of which sequences lead to connected nets can be studied by inspection for small alphabets , but we have no comprehensive approach to solve the problem in general ; we have shown how to map sequence nets to generalized types of threshold nets , in some cases is such a mapping always possible ? is there a systematic way to find such mappings for any sequence rule ? ; what kinds of nets would result if the connectivity rules applied only to the @xmath50 preceding letters , instead of to _ all _ preceding letters ? etc . we hope to tackle some of these questions in future work .

we study a new class of networks , generated by sequences of letters taken from a finite alphabet consisting of @xmath0 letters ( corresponding to @xmath0 types of nodes ) and a fixed set of connectivity rules . recently , it was shown how a binary alphabet might generate threshold nets in a similar fashion [ hagberg et al . , phys . rev . e 74 , 056116 ( 2006 ) ] . just like threshold nets , sequence nets in general possess a modular structure reminiscent of everyday life nets , and are easy to handle analytically ( i.e. , calculate degree distribution , shortest paths , betweenness centrality , etc . ) . exploiting symmetry , we make a full classification of two- and three - letter sequence nets , discovering two new classes of two - letter sequence nets . the new sequence nets retain many of the desirable analytical properties of threshold nets while yielding richer possibilities for the modeling of everyday life complex networks more faithfully .

arxiv

variable selection methods based on penalty theory have received great attention in high - dimensional data analysis . a principled approach is due to the lasso of @xcite , which uses the @xmath0-norm penalty . @xcite also pointed out that the lasso estimate can be viewed as the mode of the posterior distribution . indeed , the @xmath1 penalty can be transformed into the laplace prior . moreover , this prior can be expressed as a gaussian scale mixture . this has thus led to bayesian developments of the lasso and its variants @xcite . there has also been work on nonconvex penalization under a parametric bayesian framework . @xcite derived their local linear approximation ( lla ) algorithm by combining the expectation maximization ( em ) algorithm with an inverse laplace transform . in particular , they showed that the @xmath2 penalty with @xmath3 can be obtained by mixing the laplace distribution with a stable density . other authors have shown that the prior induced from a penalty , called the nonconvex log penalty and defined in equation ( [ eqn : logp ] ) below , has an interpretation as a scale mixture of laplace distributions with an inverse gamma mixing distribution @xcite . recently , @xcite extended this class of laplace variance mixtures by using a generalized inverse gaussian mixing distribution . related methods include the bayesian hyper - lasso @xcite , the horseshoe model @xcite and the dirichlet laplace prior @xcite . in parallel , nonparametric bayesian approaches have been applied to variable selection @xcite . for example , in the infinite gamma poisson model @xcite negative binomial processes are used to describe non - negative integer valued matrices , yielding a nonparametric bayesian feature selection approach under an unsupervised learning setting . the beta - bernoulli process provides a nonparametric bayesian tool in sparsity modeling @xcite . additionally , @xcite proposed a nonparametric approach for normal variance mixtures and showed that the approach is closely related to lvy processes . later on , @xcite constructed sparse priors using increments of subordinators , which embeds finite dimensional normal variance mixtures in infinite ones . thus , this provides a new framework for the construction of sparsity - inducing priors . specifically , @xcite discussed the use of @xmath4-stable subordinators and inverted - beta subordinators for modeling joint priors of regression coefficients . @xcite established the connection of two nonconvex penalty functions , which are referred to as log and exp and defined in equations ( [ eqn : logp ] ) and ( [ eqn : exp ] ) below , with the laplace transforms of the gamma and poisson subordinators . a subordinator is a one - dimensional lvy process that is almost surely non - decreasing @xcite . in this paper we further study the application of subordinators in bayesian nonconvex penalization problems under supervised learning scenarios . differing from the previous treatments , we model latent shrinkage parameters using subordinators which are defined as stochastic processes of regularization parameters . in particular , we consider two families of compound poisson subordinators : continuous compound poisson subordinators based on a gamma random variable @xcite and discrete compound poisson subordinators based on a logarithmic random variable @xcite . the corresponding lvy measures are generalized gamma @xcite and poisson measures , respectively . we show that both the gamma and poisson subordinators are limiting cases of these two families of the compound poisson subordinators . since the laplace exponent of a subordinator is a bernstein function , we have two families of nonconvex penalty functions , whose limiting cases are the nonconvex log and exp . additionally , these two families of nonconvex penalty functions can be defined via composition of log and exp , while the continuous and discrete compound poisson subordinators are mixtures of gamma and poisson processes . recall that the latent shrinkage parameter is a stochastic process of the regularization parameter . we formulate a hierarchical model with multiple regularization parameters , giving rise to a bayesian approach for nonconvex penalization . to reduce computational expenses , we devise an ecme ( for expectation / conditional maximization either " ) algorithm @xcite which can adaptively adjust the local regularization parameters in finding the sparse solution simultaneously . the remainder of the paper is organized as follows . section [ sec : levy ] reviews the use of lvy processes in bayesian sparse learning problems . in section [ sec : gps ] we study two families of compound poisson processes . in section [ sec : blrm ] we apply the lvy processes to bayesian linear regression and devise an ecme algorithm for finding the sparse solution . we conduct empirical evaluations using simulated data in section [ sec : experiment ] , and conclude our work in section [ sec : conclusion ] . our work is based on the notion of bernstein and completely monotone functions as well as subordinators . let @xmath5 with @xmath6 . the function @xmath7 is said to be completely monotone if @xmath8 for all @xmath9 and bernstein if @xmath10 for all @xmath9 . roughly speaking , a _ subordinator _ is a one - dimensional lvy process that is non - decreasing almost surely . our work is mainly motivated by the property of subordinators given in lemma [ lem : subord ] @xcite . [ lem : subord ] if @xmath11 is a subordinator , then the laplace transform of its density takes the form @xmath12 where @xmath13 is the density of @xmath14 and @xmath15 , defined on @xmath16 , is referred to as the _ laplace exponent _ of the subordinator and has the following representation @xmath17 \nu ( d u).\ ] ] here @xmath18 and @xmath19 is the lvy measure such that @xmath20 . conversely , if @xmath15 is an arbitrary mapping from @xmath21 given by expression ( [ eqn : psi ] ) , then @xmath22 is the laplace transform of the density of a subordinator . it is well known that the laplace exponent @xmath15 is bernstein and the corresponding laplace transform @xmath23 is completely monotone for any @xmath24 @xcite . moreover , any function @xmath25 , with @xmath26 , is a bernstein function if and only if it has the representation as in expression ( [ eqn : psi ] ) . clearly , @xmath15 as defined in expression ( [ eqn : psi ] ) satisfies @xmath27 . as a result , @xmath15 is nonnegative , nondecreasing and concave on @xmath16 . we are given a set of training data @xmath28 , where the @xmath29 are the input vectors and the @xmath30 are the corresponding outputs . we now discuss the following linear regression model : @xmath31 where @xmath32 , @xmath33^t$ ] , and @xmath34 is a gaussian error vector @xmath35 . we aim at finding a sparse estimate of the vector of regression coefficients @xmath36 by using a bayesian nonconvex approach . in particular , we consider the following hierarchical model for the regression coefficients @xmath37 s : @xmath38 & \stackrel{iid}{\sim } p(\eta_j ) , \\ \sigma & \sim\iga(\alpha_{\sigma}/2 , \beta_{\sigma}/2),\end{aligned}\ ] ] where the @xmath39 s are referred to as latent shrinkage parameters , and the inverse gamma prior has the following parametrization : @xmath40 furthermore , we regard @xmath39 as @xmath41 , that is , @xmath42 . here @xmath43 is defined as a subordinator . let @xmath44 , defined on @xmath16 , be the laplace exponent of the subordinator . taking @xmath45 , it can be shown that @xmath46 defines a nonconvex penalty function of @xmath47 on @xmath48 . moreover , @xmath46 is nondifferentiable at the origin because @xmath49 and @xmath50 . thus , it is able to induce sparsity . in this regard , @xmath51 forms a prior for @xmath47 . from lemma [ lem : subord ] it follows that the prior can be defined via the laplace transform . in summary , we have the following theorem . [ thm : lapexp00 ] let @xmath15 be a nonzero bernstein function on @xmath16 . if @xmath52 , then @xmath46 is a nondifferentiable and nonconvex function of @xmath47 on @xmath53 . furthermore , @xmath54 where @xmath43 is some subordinator . recall that @xmath14 is defined as the latent shrinkage parameter @xmath55 and in section [ sec : blrm ] we will see that @xmath56 plays the same role as the regularization parameter ( or tuning hyperparameter ) . thus , there is an important connection between the latent shrinkage parameter and the corresponding regularization parameter ; that is , @xmath57 . because @xmath58 , each latent shrinkage parameter @xmath39 corresponds to a local regularization parameter @xmath59 . therefore we have a nonparametric bayesian formulation for the latent shrinkage parameters @xmath39 s . it is also worth pointing out that @xmath60 where @xmath61 denotes a laplace distribution with density given by @xmath62 , then @xmath63 defines the proper density of some random variable ( denoted @xmath64 ) . subsequently , we obtain a proper prior @xmath65 for @xmath47 . moreover , this prior can be regarded as a laplace scale mixture , i.e. , the mixture of @xmath66 with mixing distribution @xmath67 . if @xmath68 , then @xmath69 is not a proper density . thus , @xmath70 is also improper as a prior of @xmath47 . however , we still treat @xmath70 as the mixture of @xmath66 with mixing distribution @xmath67 . in this case , we employ the terminology of pseudo - priors for the density , which is also used by @xcite . obviously , @xmath71 is bernstein . it is an extreme case , because we have that @xmath72 , @xmath73 and that @xmath74 , where @xmath75 denotes the dirac delta measure at @xmath56 , which corresponds to the deterministic process @xmath76 . we can exclude this case by assuming @xmath77 in expression ( [ eqn : psi ] ) to obtain a strictly concave bernstein function . in fact , we can impose the condition @xmath78 . this in turn leads to @xmath77 due to @xmath79 . in this paper we exploit laplace exponents in nonconvex penalization problems . for this purpose , we will only consider a subordinator without drift , i.e. , @xmath77 . equivalently , we always assume that @xmath80 . we here take the nonconvex log and exp penalties as two concrete examples ( also see * ? ? ? the log penalty is defined by @xmath81 while the exp penalty is given by @xmath82 clearly , these two functions are bernstein on @xmath16 . moreover , they satisfy @xmath27 and @xmath83 . it is also directly verified that @xmath84 \nu(du ) } , \ ] ] where the lvy measure @xmath19 is given by @xmath85 the corresponding subordinator @xmath86 is a gamma subordinator , because each @xmath14 follows a gamma distribution with parameters @xmath87 , with density given by @xmath88 we also note that the corresponding pseudo - prior is given by @xmath89 furthermore , if @xmath90 , the pseudo - prior is a proper distribution , which is the mixture of @xmath91 with mixing distribution @xmath92 . as for the exp penalty , the lvy measure is @xmath93 . since @xmath94 d b } = \infty,\ ] ] then @xmath95 $ ] is an improper prior of @xmath47 . additionally , @xmath96 is a poisson subordinator . specifically , @xmath14 is a poisson distribution with intensity @xmath97 taking values on the set @xmath98 . that is , @xmath99 which we denote by @xmath100 . in this section we explore the application of compound poisson subordinators in constructing nonconvex penalty functions . let @xmath101 be a sequence of independent and identically distributed ( i.i.d . ) real valued random variables with common law @xmath102 , and let @xmath103 be a poisson process with intensity @xmath104 that is independent of all the @xmath105 . then @xmath106 , for @xmath24 , follows a compound poisson distribution with density @xmath107 ( denoted @xmath108 ) , and hence @xmath43 is called a compound poisson process . a compound poisson process is a subordinator if and only if the @xmath105 are nonnegative random variables @xcite . it is worth pointing out that if @xmath109 is the poisson subordinator given in expression ( [ eqn : possion ] ) , it is equivalent to saying that @xmath14 follows @xmath110 . we particularly study two families of nonnegative random variables @xmath111 : nonnegative continuous random variables and nonnegative discrete random variables . accordingly , we have continuous and discrete compound poisson subordinators @xmath109 . we will show that both the gamma and poisson subordinators are limiting cases of the compound poisson subordinators . in the first family @xmath111 is a gamma random variable . in particular , let @xmath112 and the @xmath111 be i.i.d . from the @xmath113 distribution , where @xmath114 , @xmath115 and @xmath116 . the compound poisson subordinator can be written as follows @xmath117 the density of the subordinator is then given by @xmath118 we denote it by @xmath119 . the mean and variance are @xmath120 respectively . the laplace transform is given by @xmath121 where @xmath122 is a bernstein function of the form @xmath123.\ ] ] the corresponding lvy measure is given by @xmath124 notice that @xmath125 is a gamma measure for the random variable @xmath126 . thus , the lvy measure @xmath127 is referred to as a generalized gamma measure @xcite . the bernstein function @xmath128 was studied by @xcite for survival analysis . however , we consider its application in sparsity modeling . it is clear that @xmath128 for @xmath114 and @xmath116 satisfies the conditions @xmath129 and @xmath130 . also , @xmath131 is a nonnegative and nonconvex function of @xmath47 on @xmath48 , and it is an increasing function of @xmath132 on @xmath133 . moreover , @xmath131 is continuous w.r.t . @xmath47 but nondifferentiable at the origin . this implies that @xmath131 can be treated as a sparsity - inducing penalty . we are interested in the limiting cases that @xmath134 and @xmath135 . [ pro : first ] let @xmath136 , @xmath128 and @xmath137 be defined by expressions ( [ eqn : first_tt ] ) , ( [ eqn : first ] ) and ( [ eqn : first_nu ] ) , respectively . then 1 . @xmath138 and @xmath139 ; 2 . @xmath140 and @xmath141 ; 3 . @xmath142 and @xmath143 . this proposition can be obtained by using direct algebraic computations . proposition [ pro : first ] tells us that the limiting cases yield the nonconvex log and exp functions . moreover , we see that @xmath14 converges in distribution to a gamma random variable with shape @xmath144 and scale @xmath145 , as @xmath146 , and to a poisson random variable with mean @xmath144 , as @xmath147 . it is well known that @xmath148 degenerates to the log function @xcite . here we have shown that @xmath122 approaches to exp as @xmath147 . we list another special example in table [ tab : exam ] when @xmath149 . we refer to the corresponding penalty as a _ linear - fractional _ ( lfr ) function . for notational simplicity , we respectively replace @xmath150 and @xmath151 by @xmath145 and @xmath152 in the lfr function . the density of the subordinator for the lfr function is given by @xmath153 we thus say each @xmath14 follows a squared bessel process without drift @xcite , which is a mixture of a dirac delta measure and a randomized gamma distribution @xcite . we denote the density of @xmath14 by @xmath154 . lllll & bernstein functions & lvy measures @xmath137 & subordinators @xmath14 & priors + log & @xmath155 & @xmath156 & @xmath157 & proper@xmath158 + exp & @xmath159 $ ] & @xmath160 & @xmath161 & improper + lfr & @xmath162 & @xmath163 & @xmath164 & improper + cel & @xmath165 $ ] & @xmath166 & @xmath167 & improper + + + in the second case , we consider a family of discrete compound poisson subordinators . particularly , @xmath111 is discrete and takes values on @xmath168 . and it is defined as logarithmic distribution @xmath169 , where @xmath170 and @xmath171 , with probability mass function given by @xmath172 moreover , we let @xmath173 have a poisson distribution with intensity @xmath174 , where @xmath114 . then @xmath14 is distributed according to a negative binomial ( nb ) distribution @xcite . the probability mass function of @xmath14 is given by @xmath175 which is denoted as @xmath176 . we thus say that @xmath14 follows an nb subordinator . let @xmath177 and @xmath178 . it can be verified that @xmath179 has the same mean and variance as the @xmath119 distribution . the corresponding laplace transform then gives rise to a new family of bernstein functions , which is given by @xmath180.\ ] ] we refer to this family of bernstein functions as _ compound exp - log _ ( cel ) functions . the first - order derivative of @xmath181 w.r.t . @xmath182 is given by @xmath183 the lvy measure for @xmath181 is given by @xmath184 the proof is given in appendix 1 . we call this lvy measure a _ generalized poisson measure _ relative to the generalized gamma measure . like @xmath128 , @xmath181 can define a family of sparsity - inducing nonconvex penalties . also , @xmath181 for @xmath114 , @xmath185 and @xmath116 satisfies the conditions @xmath186 , @xmath187 and @xmath188 . we present a special cel function @xmath189 as well as the corresponding @xmath14 and @xmath137 in table [ tab : exam ] , where we replace @xmath190 and @xmath150 by @xmath152 and @xmath145 for notational simplicity . we now consider the limiting cases . [ pro:8 ] assume @xmath137 is defined by expression ( [ eqn : second_nu ] ) for fixed @xmath185 and @xmath116 . then we have that 1 . @xmath191 and @xmath192 . @xmath193 and @xmath194 . 3 . @xmath195 and @xmath142 . 4 . @xmath196 and @xmath197 notice that @xmath198 this shows that @xmath127 converges to @xmath199 , as @xmath200 . analogously , we obtain the second part of proposition [ pro:8]-(d ) , which implies that as @xmath200 , @xmath14 converges in distribution to a gamma random variable with shape parameter @xmath144 and scale parameter @xmath145 . an alternative proof is given in appendix 2 . proposition [ pro:8 ] shows that @xmath181 degenerates to exp as @xmath147 , while to log as @xmath200 . this shows an interesting connection between @xmath128 in expression ( [ eqn : first ] ) and @xmath181 in expression ( [ eqn : second ] ) ; that is , they have the same limiting behaviors . we note that for @xmath201 , @xmath202\ ] ] which is a composition of the log and exp functions , and that @xmath203\ ] ] which is a composition of the exp and log functions . in fact , the composition of any two bernstein functions is still bernstein . thus , the composition is also the laplace exponent of some subordinator , which is then a mixture of the subordinators corresponding to the original two bernstein functions @xcite . this leads us to an alternative derivation for the subordinators corresponding to @xmath122 and @xmath204 . that is , we have the following theorem whose proof is given in appendix 3 . [ thm : poigam ] the subordinator @xmath14 associated with @xmath128 is distributed according to the mixture of @xmath205 distributions with @xmath206 mixing , while @xmath14 associated with @xmath181 is distributed according to the mixture of @xmath207 distributions with @xmath208 mixing . additionally , the following theorem illustrates a limiting property of the subordinators as @xmath145 approaches 0 . [ thm : limit ] let @xmath209 be a fixed constant on @xmath210 $ ] . 1 . if @xmath211 where @xmath212 $ ] or @xmath213 , then @xmath14 converges in probability to @xmath56 , as @xmath214 . 2 . if @xmath215 where @xmath216\ ] ] or @xmath213 , then @xmath14 converges in probability to @xmath56 , as @xmath214 . the proof is given in appendix 4 . since @xmath14 converges in probability to @xmath56 " implies @xmath14 converges in distribution to @xmath56 , " we have that @xmath217 finally , consider the four nonconvex penalty function given in table [ tab : exam ] . we present the following property . that is , when @xmath213 and for any fixed @xmath116 , we have @xmath218 \leq\frac{s}{\gamma s { + } 1 } \leq\frac{1}{\gamma } [ 1 { - } \exp ( { - } \gamma s ) ] \leq\frac { 1}{\gamma } \log\big({\gamma } s { + } 1 \big ) \leq s,\ ] ] with equality only when @xmath219 . the proof is given in appendix 5 . this property is also illustrated in figure [ fig : penalty ] . in table [ tab : exam ] with @xmath220 and @xmath71 . ] we apply the compound poisson subordinators to the bayesian sparse learning problem given in section [ sec : levy ] . defining @xmath221 , we rewrite the hierarchical representation for the joint prior of the @xmath37 under the regression framework . that is , we assume that @xmath222 & \stackrel{ind}{\sim } & l(b_j|0 , \sigma ( 2\eta_j)^{-1 } ) , \\ f_{t^{*}(t_j)}(\eta_j ) & { \propto } & \eta_j^{-1 } f_{t(t_j)}(\eta_j),\end{aligned}\ ] ] which implies that @xmath223 the joint marginal pseudo - prior of the @xmath37 s is given by @xmath224 we will see in theorem [ thm : poster ] that the full conditional distribution @xmath225 is proper . thus , the maximum _ a posteriori _ ( map ) estimate of @xmath226 is based on the following optimization problem : @xmath227 clearly , the @xmath59 s are local regularization parameters and the @xmath228 s are latent shrinkage parameters . moreover , it is interesting that @xmath43 ( or @xmath55 ) is defined as a subordinator w.r.t . @xmath56 . the full conditional distribution @xmath229 is conjugate w.r.t . the prior , which is @xmath230 . specifically , it is an inverse gamma distribution of the form @xmath231.\ ] ] in the following experiment , we use an improper prior of the form @xmath232 ( i.e. , @xmath233 ) . clearly , @xmath229 is still an inverse gamma distribution in this setting . additionally , based on @xmath234 \prod_{j=1}^p \exp(-\frac { \eta _ j}{\sigma } |b_j|)\vadjust{\eject}\ ] ] and the proof of theorem [ thm : poster ] ( see appendix 6 ) , we have that the conditional distribution @xmath235 is proper . however , the absolute terms @xmath236 make the form of @xmath237 unfamiliar . thus , a gibbs sampling algorithm is not readily available and we resort to an em algorithm to estimate the model . notice that if @xmath238 is proper , the corresponding normalizing constant is given by @xmath239 d |b_j|= 2 \int_{0}^{\infty } \exp\big [ -t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big ] d ( |b_j|/\sigma),\ ] ] which is independent of @xmath240 . also , the conditional distribution @xmath241 is independent of the normalizing term . specifically , we always have that @xmath242 which is proper . as shown in table [ tab : exam ] , except for log with @xmath243 which can be transformed into a proper prior , the remaining bernstein functions can not be transformed into proper priors . in any case , our posterior computation is directly based on the marginal pseudo - prior @xmath244 . we ignore the involved normalizing term , because it is infinite if @xmath244 is improper and it is independent of @xmath240 if @xmath244 is proper . given the @xmath245th estimates @xmath246 of @xmath247 in the e - step of the em algorithm , we compute @xmath248 p(\eta_j|b_j^{(k ) } , \sigma ^{(k ) } , t_j ) } d \eta_j + \log p(\sigma ) \\ & \propto-\frac{n+\alpha_{\sigma}}{2 } \log\sigma{- } \frac{\|{\bf y } { -}{\bf x}{\bf b}\|_2 ^ 2 + \beta_{\sigma}}{2 \sigma } - ( p+1 ) \log \sigma \\ & \quad- \frac{1 } { \sigma } \sum_{j=1}^p here we omit some terms that are independent of parameters @xmath240 and @xmath226 . in fact , we only need to compute @xmath249 in the e - step . considering that @xmath250 and taking the derivative w.r.t . @xmath236 on both sides of the above equation , we have that @xmath251 the m - step maximizes @xmath252 w.r.t.@xmath253 . in particular , it is obtained that : @xmath254 the above em algorithm is related to the linear local approximation ( lla ) procedure @xcite . moreover , it shares the same convergence property given in @xcite and @xcite . subordinators help us to establish a direct connection between the local regularization parameters @xmath59 s and the latent shrinkage parameters @xmath39 s ( or @xmath41 ) . however , when we implement the map estimation , it is challenging how to select these local regularization parameters . we employ an ecme ( for expectation / conditional maximization either " ) algorithm @xcite for learning about the @xmath37 s and @xmath59 s simultaneously . for this purpose , we suggest assigning @xmath59 gamma prior @xmath255 , namely , @xmath256 because the full conditional distribution is also gamma and given by @xmath257 \sim\ga\big(\alpha_{t } , 1/[\psi(|b_j|/\sigma ) + \beta_{t}]\big).\ ] ] recall that we here compute the full conditional distribution directly using the marginal pseudo - prior @xmath238 , because our used bernstein functions in table [ tab : exam ] can not induce proper priors . however , if @xmath238 is proper , the corresponding normalizing term would rely on @xmath59 . as a result , the full conditional distribution of @xmath59 is possibly no longer gamma or even not analytically available . figure [ fig : graphal0]-(a ) depicts the hierarchical model for the bayesian penalized linear regression , and table [ tab : alg ] gives the ecme procedure where the e - step and cm - step are respectively identical to the e - step and the m - step of the em algorithm , with @xmath258 . the cme - step updates the @xmath59 s with @xmath259 in order to make sure that @xmath260 , it is necessary to assume that @xmath261 . in the following experiments , we set @xmath262 . we conduct experiments with the prior @xmath263 for comparison . this prior is induced from the @xmath264-norm penalty , so it is a proper specification . moreover , the full conditional distribution of @xmath59 w.r.t . its gamma prior @xmath265 is still gamma ; that is , @xmath257 \sim\ga\big({\alpha_t}{+}2 , \ ; 1/({\beta_t } { + } \sqrt{|b_j|/\sigma})\big).\ ] ] thus , the cme - step for updating the @xmath59 s is given by @xmath266 the convergence analysis of the ecme algorithm was presented by @xcite , who proved that the ecme algorithm retains the monotonicity property from the standard em . moreover , the ecme algorithm based on pseudo - priors was also used by @xcite . . the basic procedure of the ecme algorithm [ cols= " < , < " , ] our analysis is based on a set of simulated data , which are generated according to @xcite . in particular , we consider the following three data models small , " medium " and large . " data s : : : @xmath267 , @xmath268 , @xmath269 , and @xmath270 is a @xmath271 matrix with @xmath272 on the diagonal and @xmath273 on the off - diagonal . data m : : : @xmath274 , @xmath275 , @xmath276 has @xmath277 non - zeros such that @xmath278 and @xmath279 , and @xmath280 . data l : : : @xmath281 , @xmath282 , @xmath283 , and @xmath284 ( five blocks ) . for each data model , we generate @xmath285 data matrices @xmath286 such that each row of @xmath286 is generated from a multivariate gaussian distribution with mean @xmath287 and covariance matrix @xmath270 , @xmath288 , or @xmath289 . we assume a linear model @xmath290 with multivariate gaussian predictors @xmath286 and gaussian errors . we choose @xmath240 such that the signal - to - noise ratio ( snr ) is a specified value . following the setting in @xcite , we use @xmath291 in all the experiments . we employ a standardized prediction error ( spe ) to evaluate the model prediction ability . the minimal achievable value for spe is @xmath272 . variable selection accuracy is measured by the correctly predicted zeros and incorrectly predicted zeros in @xmath292 . the snr and spe are defined as @xmath293 for each data model , we generate training data of size @xmath294 , very large validation data and test data , each of size @xmath295 . for each algorithm , the optimal global tuning parameters are chosen by cross validation based on minimizing the average prediction errors . with the model @xmath292 computed on the training data , we compute spe on the test data . this procedure is repeated @xmath296 times , and we report the average and standard deviation of spe and the average of zero - nonzero error . we use `` '' to denote the proportion of correctly predicted zero entries in @xmath226 , that is , @xmath297 ; if all the nonzero entries are correctly predicted , this score should be @xmath298 . we report the results in table [ tab : toy2 ] . it is seen that our setting in figure [ fig : graphal0]-(a ) is better than the other two settings in figures [ fig : graphal0]-(b ) and ( c ) in both model prediction accuracy and variable selection ability . especially , when the size of the dataset takes large values , the prediction performance of the second setting becomes worse . the several nonconvex penalties are competitive , but they outperform the lasso . moreover , we see that log , exp , lfr and cel slightly outperform @xmath264 . the @xmath264 penalty indeed suffers from the problem of numerical instability during the em computations . as we know , the priors induced from lfr , cel and exp as well as log with @xmath299 are improper , but the prior induced from @xmath264 is proper . the experimental results show that these improper priors work well , even better than the proper case . vs. @xmath300 on data s " and data m " where @xmath301 is the permutation of @xmath302 such that @xmath303 . ] recall that in our approach each regression variable @xmath37 corresponds to a distinct local tuning parameter @xmath59 . thus , it is interesting to empirically investigate the inherent relationship between @xmath37 and @xmath59 . let @xmath304 be the estimate of @xmath59 obtained from our ecme algorithm ( alg 1 " ) , and @xmath305 be the permutation of @xmath306 such that @xmath307 . figure [ fig : tb1 ] depicts the change of @xmath308 vs.@xmath300 with log , exp , lfr and cel on data s " and data m. " we see that @xmath308 is decreasing w.r.t . moreover , @xmath308 becomes 0 when @xmath300 takes some large value . a similar phenomenon is also observed for data l. " this thus shows that the subordinator is a powerful bayesian approach for variable selection . in this paper we have introduced subordinators into the definition of nonconvex penalty functions . this leads us to a bayesian approach for constructing sparsity - inducing pseudo - priors . in particular , we have illustrated the use of two compound poisson subordinators : the compound poisson gamma subordinator and the negative binomial subordinator . in addition , we have established the relationship between the two families of compound poisson subordinators . that is , we have proved that the two families of compound poisson subordinators share the same limiting behaviors . moreover , their densities at each time have the same mean and variance . we have developed the ecme algorithms for solving sparse learning problems based on the nonconvex log , exp , lfr and cel penalties . we have conducted the experimental comparison with the state - of - the - art approach . the results have shown that our nonconvex penalization approach is potentially useful in high - dimensional bayesian modeling . our approach can be cast into a point estimation framework . it is also interesting to fit a fully bayesian framework based on the mcmc estimation . we would like to address this issue in future work . consider that @xmath310 & = \log\big[1-\frac{1}{1{+}\rho } \exp(-\frac{\rho}{1{+}\rho } \gamma s)\big ] - \log\big[1-\frac{1}{1{+}\rho}\big ] \\ & = \sum_{k=1}^{\infty } \frac{1}{k ( 1{+}\rho)^k } \big[1- \exp\big ( { -}\frac{\rho}{1{+}\rho } k \gamma s\big)\big ] \\ & = \sum_{k=1}^{\infty } \frac{1}{k ( 1{+}\rho)^k } \int_{0}^{\infty } ( 1- \exp(- u s ) ) \delta_{\frac{\rho k \gamma}{1{+}\rho}}(u ) d u.\end{aligned}\ ] ] we thus have that @xmath311 . we here give an alternative proof of proposition [ pro:8]-(d ) , which is immediately obtained from the following lemma . let @xmath312 take discrete value on @xmath313 and follow negative binomial distribution @xmath314 . if @xmath315 converges to a positive constant as @xmath316 , @xmath317 converges in distribution to a gamma random variable with shape @xmath315 and scale @xmath272 . since @xmath318 we have that @xmath319 notice that @xmath320 and @xmath321 this leads us to @xmath322 similarly , we have that @xmath323 consider a mixture of @xmath324 with @xmath325 mixing . that is , @xmath326 letting @xmath327 , @xmath328 and @xmath329 , we have that @xmath330 we now consider a mixture of @xmath331 with @xmath332 which is @xmath333 . let @xmath334 , @xmath335 , @xmath336 and @xmath337 . thus , @xmath338 since @xmath339=1 $ ] , we only need to consider the case that @xmath213 . recall that @xmath119 , whose mean and variance are @xmath340 whenever @xmath213 . by chebyshev s inequality , we have that @xmath341 hence , we have that @xmath342 similarly , we have part ( b ) . we first note that @xmath343 which implies that @xmath344 for @xmath345 . subsequently , we have that @xmath346 \leq0 $ ] . as a result , @xmath347 for @xmath345 . as for @xmath348 , it is directly obtained from that @xmath349 since @xmath350 = \frac{\gamma}{\exp(\gamma s ) } - \frac{\gamma}{1+\gamma s}<0 $ ] for @xmath345 , we have that @xmath351 for @xmath345 . first consider that @xmath352 \prod_{j=1}^p \sigma^{-1 } \exp\big(-t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big).\ ] ] to prove that @xmath353 is proper , it suffices to obtain that @xmath354 \prod_{j=1}^p \sigma^{-1 } \exp \big(-t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big ) d { \bf b } < \infty}.\ ] ] it is directly computed that @xmath355 \nonumber \\ & = \exp\big [ { - } \frac{1}{2 \sigma } ( { \bf b}{- } { \bf z})^t { \bf x}^t { \bf x } ( { \bf b}- { \bf z } ) \big ] \times\exp\big[- \frac{1}{2 \sigma } { \bf y}^t ( { \bf i}_n - { \bf x } ( { \bf x}^t { \bf x})^{+ } { \bf x}^t ) { \bf y}\big],\end{aligned}\ ] ] where @xmath356 and @xmath357 is the moore - penrose pseudo inverse of matrix @xmath358 @xcite . here we use the well - established properties that @xmath359 and @xmath360 . notice that if @xmath358 is nonsingular , then @xmath361 . in this case , we consider a conventional multivariate normal distribution @xmath362 . otherwise , we consider a singular multivariate normal distribution @xmath363 @xcite , the density of which is given by @xmath364.\ ] ] here @xmath365 , and @xmath366 , @xmath367 , are the positive eigenvalues of @xmath358 . in any case , we always write @xmath368 . thus , @xmath369 d{\bf b } < \infty}$ ] . it then follows the propriety of @xmath370 because @xmath371 \prod _ { j=1}^p \exp\big ( { - } t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big)\leq\exp\big [ { - } \frac{1}{2 \sigma } \|{\bf y}- { \bf x}{\bf b}\|_2 ^ 2 \big].\ ] ] we now consider that @xmath372 \prod_{j=1}^p \exp\big(-t_j \psi \big(\frac{|b_j| } { \sigma } \big ) \big).\ ] ] let @xmath373 { \bf y}$ ] . since the matrix @xmath374 is positive semidefinite , we obtain @xmath375 . based on expression ( [ eqn : pf01 ] ) , we can write @xmath376 \varpropto n({\bf b}|{\bf z } , \sigma({\bf x}^t { \bf x})^{+ } ) { \iga}(\sigma|\frac{\alpha_{\sigma } { + } n{+}2p{-}q}{2 } , \nu{+ } \beta_{\sigma}).\ ] ] subsequently , we have that @xmath377 d { \bf b } d \sigma } < \infty,\ ] ] and hence , @xmath377 \prod_{j=1}^p \exp\big(-t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big ) d { \bf b}d \sigma } < \infty.\ ] ] therefore @xmath378 is proper . thirdly , we take @xmath379 } { \sigma ^{\frac{n+\alpha_{\sigma}+2p}{2 } + 1 } } \prod_{j=1}^p \big\{\exp \big({-}t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big ) \frac { t_j^{{\alpha_t}{- } 1 } \exp({- } { \beta_t } t_j)}{\gamma({\alpha_t } ) } \big\ } \\ & \triangleq f({\bf b } , \sigma , { \bf t}).\end{aligned}\ ] ] in this case , we compute @xmath380 } { \sigma^{\frac{n+\alpha_{\sigma}+2p}{2 } + 1 } } \prod _ { j=1}^p \frac{1 } { \big({\beta_t } { + } \psi\big(\frac{|b_j| } { \sigma } \big ) \big)^{{\alpha_t } } } d { \bf b}d \sigma}.\ ] ] similar to the previous proof , we also have that @xmath381 because @xmath382 . as a result , @xmath383 is proper . finally , consider the setting that @xmath384 . that is , @xmath385 and @xmath386 . in this case , if @xmath387 , we obtain @xmath388 and @xmath389 . as a result , we use the inverse gamma distribution @xmath390 . thus , the results still hold . polson , n. g. and scott , j. g. ( 2010 ) . `` shrink globally , act locally : sparse bayesian regularization and prediction . '' in bernardo , j. m. , bayarri , m. j. , berger , j. o. , dawid , a. p. , heckerman , d. , smith , a. f. m. , and west , m. ( eds . ) , _ bayesian statistics 9_. oxford university press . the authors would like to thank the editors and two anonymous referees for their constructive comments and suggestions on the original version of this paper . the authors would especially like to thank the associate editor for giving extremely detailed comments on earlier drafts . this work has been supported in part by the natural science foundation of china ( no . 61070239 ) .

in this paper we discuss bayesian nonconvex penalization for sparse learning problems . we explore a nonparametric formulation for latent shrinkage parameters using subordinators which are one - dimensional lvy processes . we particularly study a family of continuous compound poisson subordinators and a family of discrete compound poisson subordinators . we exemplify four specific subordinators : gamma , poisson , negative binomial and squared bessel subordinators . the laplace exponents of the subordinators are bernstein functions , so they can be used as sparsity - inducing nonconvex penalty functions . we exploit these subordinators in regression problems , yielding a hierarchical model with multiple regularization parameters . we devise ecme ( expectation / conditional maximization either ) algorithms to simultaneously estimate regression coefficients and regularization parameters . the empirical evaluation of simulated data shows that our approach is feasible and effective in high - dimensional data analysis .

arxiv

with the advent of high temperature superconductivity in the cuprates and the possibility of exotic gap symmetry including nodal behavior , a renewed effort to find novel experimental probes of order parameter symmetry has ensued . one result of this effort was the proposal by sauls and co - workers@xcite to examine the nonlinear current response of d - wave superconductors . they showed that a nonanalyticity in the current - velocity relation at temperature @xmath1 is introduced by the presence of nodes in the order parameter . one prediction was that an anisotropy should exist in the nonlinear current as a function of the direction of the superfluid velocity relative to the position of the node . this would be reflected in an anisotropy of a term in the inverse penetration depth which is linear in the magnetic field @xmath2 . early experimental work did not verify these predictions@xcite and it was suggested that impurity scattering@xcite or nonlocal effects@xcite may be responsible . however , a more recent reanalysis of experiment has claimed to confirm the predictions@xcite . an alternative proposal was given by dahm and scalapino@xcite who examined the quadratic term in the magnetic response of the penetration depth , which shows a @xmath3 dependence at low @xmath4 as first discussed by xu et al.@xcite . dahm and scalapino demonstrated that this upturn would provide a clear and unique signature of the nodes in the d - wave gap and that this feature could be measured directly via microwave intermodulation effects . indeed , experimental verification of this has been obtained@xcite confirming that nonlinear microwave current response can be used as a sensitive probe of issues associated with the order parameter symmetry . thus , we are led to consider further cases of gap anisotropy and turn our attention to the two - band superconductor mgb@xmath0 which is already under scrutiny for possible applications , including passive microwave filter technology@xcite . mgb@xmath0 was discovered in 2001@xcite and since this time an enormous scientific effort has focused on this material . on the basis of the evidence that is available , it is now thought that this material may be our best candidate for a classic two - band electron - phonon superconductor , with s - wave pairing in each channel@xcite . a heightened interest in two - band superconductivity has led to claims of possible two - band effects in many other materials , both old@xcite and new@xcite . our goal is to compare in detail the differences between one - band and two - band s - wave superconductors in terms of their nonlinear response , that would be measured in the coefficients defined by xu et al.@xcite and dahm and scalapino@xcite . this leads us to reconsider the one - band s - wave case , where we study issues of dimensionality , impurities , and strong electron - phonon coupling . we find new effects due to strong - coupling at both high and low @xmath4 . we then examine the situation for two - band superconductors , starting from a case of highly decoupled bands . here , we are looking for signatures of the low energy scale due to the smaller gap , the effect of integration of the bands , and the response to inter- and intraband impurities . unusual behavior exists distinctly different from the one - band case and not necessarily understood as a superposition of two separate superconductors . finally , we return to the case of mgb@xmath0 which was studied previously via a more approximate approach@xcite . in the current work , we are able to use the complete microscopic theory with the parameters and the electron - phonon spectral functions taken from band structure@xcite . in this way , we provide more detailed predictions for the nonlinear coefficient of mgb@xmath0 . in section ii , we briefly summarize the necessary theory for calculating the gap and renormalization function in two - band superconductors , from which the current as a function of the superfluid velocity @xmath5 is then derived . in section iii , we explain our procedure for extracting the temperature - dependent nonlinear term from the current and we examine the characteristic features for one - band superconductors in light of issues of dimensionality , impurity scattering and strong coupling . section iv presents the results of two - band superconductors and simple formulas are given for limiting cases which aid in illuminating the effects of anisotropy . the case of mgb@xmath0 is also discussed . we form our conclusions in section v. the superfluid current has been considered theoretically in the past by many authors for s - wave@xcite and for other order parameters , such as d - wave and f - wave@xcite . most recently , the case of two - band superconductivity has been examined@xcite with good agreement obtained between theory and experiment for the temperature dependence of the critical current@xcite . in this work , we wish to calculate the superfluid current as a function of superfluid velocity @xmath5 or momentum @xmath6 and extract from this the nonlinear term . to do this , we choose to evaluate the expression for the superfluid current density @xmath7 that is written on the imaginary axis in terms of matsubara quantities.@xcite this naturally allows for the inclusion of impurity scattering and strong electron - phonon coupling in a numerically efficient manner . written in general for two - bands having a current @xmath8 and @xmath9 , for the first and second band , respectively , we have : @xmath10 where @xmath11 is the electric charge , @xmath12 is the electron mass , @xmath4 is the temperature , @xmath13 , @xmath14 is the electron density and @xmath15 is the fermi velocity of the @xmath16th band ( @xmath17,2 ) . the @xmath18 represents an integration for the @xmath16th band which is given as @xmath19 for a 3d band and @xmath20 for a 2d band , with @xmath21 in the 2d case . also , in the expression for the current , the 3 should be changed to a 2 for 2d . this is done within a mean - field treatment and ignoring critical fluctuations near @xmath22 . here , we have taken the approximation of a spherical fermi surface in 3d and a cylindrical one in 2d as we will see further on that the differences between 2d and 3d are not significant to more than a overall numerical factor and so providing more precise fermi surface averages will not changes the results in a meaningful way . to evaluate this expression , we require the solution of the standard s - wave eliashberg equations for the renormalized gaps and frequencies @xmath23 and @xmath24 , respectively . these have been generalized to two bands and must also include the effect of the current through @xmath6 . with further details given in refs . @xcite , we merely state them here : @xmath25\nonumber\\ & \times&\biggl\langle \frac{\tilde\delta_j(m)}{\sqrt{(\tilde\omega_j(m)-is_jz)^2+\tilde\delta_j^2(m)}}\biggr\rangle_j\nonumber\\ & + & \pi\sum_jt^+_{lj}\biggl\langle \frac{\tilde\delta_j(n ) } { \sqrt{(\tilde\omega_j(n)-is_jz)^2+\tilde\delta_j^2(n)}}\biggr\rangle_j\label{eq : del}\end{aligned}\ ] ] and @xmath26 where @xmath27 sums over the number of bands and the sum over @xmath12 is from @xmath28 to @xmath29 . here , @xmath30 is the ordinary impurity scattering rate and @xmath31 indexes the @xmath31th matsubara frequency @xmath32 , with @xmath33 , where @xmath34 . the @xmath35 are coulomb repulsions , which require a high energy cutoff @xmath36 , taken to be about six to ten times the maximum phonon frequency , and the electron - phonon interaction enters through @xmath37 with @xmath38 the electron - phonon spectral functions and @xmath39 the phonon energy . note that the dimensionality does not change the gap equations when there is no current . for finite @xmath6 , it does and we will see later the result of this effect . likewise , an essential ingredient is that the current enters the eliashberg equations and provides the bulk of the nonlinear effect for temperatures above @xmath40 . indeed , at @xmath22 all of the nonlinearity arises from the gap . we now proceed to the case of one - band superconductors , to illustrate the generic features of the superfluid current and demonstrate how we extract the nonlinear term . in the section following , we will return to the two - band case . ( 250,200 ) in fig . [ fig1 ] , we illustrate that these equations reproduce the standard results for @xmath7 versus @xmath6 for a one - band superconductor in the weak coupling bcs limit . equations ( [ eq : js])-([eq : z ] ) were solved for both the 2d and 3d cases at @xmath41 and 0.95 . the @xmath1 result of past literature@xcite is recovered in the case of 3d . one sees for @xmath42 at low @xmath6 , the curve is essentially linear , reflecting the relationship of @xmath43 , with @xmath44 the superfluid density . for strong coupling , the slope would be reduced by approximately @xmath45 as the superfluid condensate is also reduced by this factor . likewise the reduction in the slope with temperature would reflect the temperature dependence of the superfluid density . indeed , to provide these curves using the eliashberg equations , we used the @xmath46 spectrum of al and made the corrections for the @xmath45 factor . al is a classic bcs weak coupling superconductor , that agrees with bcs in every way and is generally used for bcs tests of the eliashberg equations . the @xmath47 for al is 0.43 . while at low @xmath4 the curves show little deviation from linearity at low @xmath6 , and thus the nonlinear correction will be essentially zero ( exponentially so with temperature in bcs theory ) , at @xmath4 near @xmath22 , one sees that there is more curvature for @xmath48 and hence a larger nonlinear term is expected . however , while the 2d and 3d curves differ in behavior near the maximum in @xmath7 , one finds that the behavior at low @xmath6 is very similar . indeed , the nonlinearity is a very small effect on these plots and hard to discern , however , it will be borne out in our paper that the nonlinear current does not show significant differences in the @xmath4-dependence between 2d and 3d . nevertheless , we will still include both the 2d and 3d calculation in our two - band calculations as mgb@xmath0 has a 2d @xmath49-band and a 3d @xmath50-band , and there is a overall factor of 2/3 between the two in the nonlinear term due to dimensionality . to obtain the nonlinear current as @xmath48 , the general expression for the current can be expanded to second lowest order in powers of @xmath6 leading to the general formula @xmath51 , \label{eq : jexpand}\ ] ] where only first and third order terms arise . here by choice @xmath52 and the variable for the expansion was taken as @xmath53 , where @xmath54 . @xmath55 and @xmath56 are temperature - dependent coefficients which follow when solutions of the eliashberg equations ( [ eq : del ] ) and ( [ eq : z ] ) are substituted in the expression ( [ eq : js ] ) for the current . in practice , it is complicated to expand eqs . ( [ eq : js])-([eq : z ] ) to obtain an explicit form @xmath56 and so we chose to extract @xmath55 and @xmath56 numerically by solving our full set of equations with no approximations for @xmath7 versus @xmath6 . from this numerical data , we find the intercept and slope of @xmath57 versus @xmath58 for @xmath48 from which we obtain the @xmath55 and @xmath56 , respectively . ( 250,200 ) the results for @xmath59 and @xmath60 as a function of temperature are shown in fig . one sees , in the inset , @xmath59 which , in the one - band case , is just the superfluid density @xmath44 normalized to the clean bcs value at @xmath1 . there is no difference between 2d and 3d bcs . also , shown is the @xmath59 extracted for the strong electron - phonon coupling superconductor pb with no impurities and with impurity scattering of @xmath61 . one sees that strong coupling pushes the temperature dependence of the curve higher , even slightly so at @xmath1 , and this is a well - documented effect@xcite . with impurities , the superfluid density is reduced in accordance with standard theory . these curves were obtained from our @xmath7 calculations and agree exactly with bcs and eliashberg calculations done with the standard penetration depth formulas@xcite , confirming that our numerical procedure is accurate . the second term in eq . ( [ eq : jexpand ] ) gives the nonlinear current and the coefficient @xmath60 , which is a measure of this , is also shown for the four cases . [ note that @xmath60 is the same as the @xmath62 of ref . @xcite to within a constant of proportionality . ] here one does find a difference between the 2d and 3d bcs curves showing that dimensionality can affect the nonlinear current . in the case of strong coupling one finds an increase in the nonlinear piece near @xmath22 and also a finite contribution at low @xmath4 which is unexpected in the usual bcs scenario . impurities have the effect of further increasing the low @xmath4 contribution and reducing the curve near @xmath22 . near @xmath4 equal to @xmath22 ( @xmath63 ) in bcs , it can be shown analytically that @xmath64 and for 3d @xmath65 as obtained in our previous paper , ref . the value of @xmath56 for 2d is increased by a factor of @xmath66 . these numbers agree with the numerical calculations in fig . [ fig2 ] , where the 3d bcs curve goes to 0.21 for 2d and 0.14 in 3d . there are two definitions in the literature for the nonlinear coefficient : one is denoted as @xmath67 due to xu et al.@xcite and the other , @xmath68 , used by dahm and scalapino@xcite , is the one that is related to the intermodulation power in microstrip resonators . rewriting eq . ( [ eq : jexpand ] ) in the form @xmath69 , \label{eq : jexpand2}\ ] ] dahm and scalapino define@xcite @xmath70 xu , yip and sauls@xcite keep the form of eq . ( [ eq : jexpand ] ) but define a variable @xmath71 , where @xmath72 is the temperature dependent gap equal to @xmath73 . with this they identify the coefficient @xmath74 in this work , we always take @xmath75 to be the usual bcs temperature dependence of the gap function . ( 250,200 ) in fig . [ fig3 ] , we show the calculations for the @xmath67 coefficient of xu et al .. here , we have made a number of points . first , the 2d bcs curve derived from our procedure agrees with that shown by xu et al.@xcite , once again validating our numerical work for extracting the very tiny nonlinear coefficient . second , for bcs one sees a difference between 2d and 3d in the nonlinear coefficient . the 2d curve goes to 1 at @xmath22 and to 2/3 for the 3d case . the question arises as to whether the difference between 2d and 3d is simply a numerical factor and so with the dotted curve , we show the 3d case scaled up by 3/2 . we do note that there is a small difference in the temperature variations at an intermediate range of @xmath4 , but the major difference between 2d and 3d is the overall numerical factor of 2/3 . third , one might question the necessity of including the effect of the current on the gap itself and to answer this , we show the long - dashed curve where the @xmath6 dependence was omitted in the eliashberg equations ( [ eq : del ] ) and ( [ eq : z ] ) . one finds that the nonlinear coefficient is reduced substantially at temperatures above @xmath76 and disappears at @xmath22 . thus , without the @xmath6 dependence in the gap , the true nonlinear effects will not be obtained for high temperatures as the gap provides the major contribution to the nonlinearity . in the lower frame of fig . [ fig3 ] , we examine the case of pb to illustrate strong electron - phonon coupling and impurity effects . it is seen that the strong coupling increases the value at @xmath22 and also gives a finite value at low @xmath4 . the behavior at low @xmath4 is surprising in light of the bcs result@xcite , but is related to the inelastic electron - phonon scattering which appears to increase the nonlinear coefficient at small @xmath4 in a similar way to what is already known about the effect of impurities in bcs@xcite . the strong coupling behavior near @xmath22 is similar to that seen for other quantities such as the specific heat@xcite , where the downward bcs curvature is now turned concave upward to higher values at @xmath22 . impurities have the effect of reducing the nonlinearity near @xmath22 and increasing it at low @xmath4 . once again , in bcs we can provide some analytic results near and at @xmath22 for @xmath77 which provide a useful check on our numerical work . for three dimensions near @xmath22 : @xmath78 and , upon substituting for @xmath75 , @xmath79 doing the same algebra for the two - dimensional case corrects these expressions by a factor of 3/2 and gives 1 instead of 2/3 for @xmath80 . to characterize the strong - coupling effects seen in the figure for pb , we can develop a strong - coupling correction formula for @xmath81 and @xmath80 . these formulas have been provided in the past for many quantities and form a useful tool for experimentalists and others to estimate the strong coupling corrections.@xcite this was done by evaluating this quantity for ten superconductors using their known @xmath46 spectra and their @xmath22 values . we used al , v , sn , in , nb , v@xmath82ga , nb@xmath82ge , pb , pb@xmath83bi@xmath84 , and pb@xmath85bi@xmath86 . these materials were chosen to span the range of typical s - wave superconductors with strong coupling parameter @xmath87 ranging from 0.004 to 0.2 . the details of these materials and references for the spectra may be found in the review by carbotte@xcite . the parameter @xmath88 is defined as : @xmath89.\ ] ] by fitting to these materials , we arrived at the following strong coupling correction formulas for three dimensions : @xmath90\ ] ] and @xmath91 note that , even though @xmath92 is the usual form of the strong coupling correction , in this last equation , we have found no advantage in fitting with the additional parameter offered by the log factor . these formula should be seen as approximate tools to give the trend for @xmath87 for values restricted to the range of 0 to 0.2 . pb has @xmath87 value of 0.128 and is intermediate to this range , and al is a weak coupling superconductor with a value of 0.004 . ( 250,200 ) in fig . [ fig4 ] , we show the coefficient used by dahm and scalapino@xcite for the same cases as previously considered . with this coefficient one finds qualitatively similar curves . the 2d and 3d bcs curves go to zero rapidly at low temperature , but once again the strong coupling effects in pb give a finite value for @xmath68 at low @xmath4 . with impurities the tail at low temperature is raised significantly . due to the divergence in @xmath68 near @xmath22 because of the division by three powers of the superfluid density which is going to zero at @xmath22 , we prefer to work with a new quantity @xmath93 , which removes this divergence . thus , we define @xmath94 and this is shown in the inset in fig . it has the advantage of illustrating the detailed differences between the curves more clearly and providing finite values at @xmath22 which can be evaluated analytically in bcs theory . in this instance , we obtain @xmath95 for three dimensions in agreement with what we obtain from our numerical work , shown in the fig . [ fig4 ] . once again we can develop strong coupling formulas for this quantity and they are given as : @xmath96\ ] ] and @xmath97 for three dimensions . once again , there was no extra advantage to fitting @xmath98 with the usual form that includes the log factor . this last quantity @xmath99 is related to the intermodulation power in microstrip resonators and hence can be measured directly . having identified the features of one - band superconductors , we now turn to the two - band case where signatures of the two - band nature may occur in these nonlinear coefficients . the generalization of eq . ( [ eq : jexpand2 ] ) to the two - band case proceeds as follows . the total current @xmath7 is the sum of the two partial currents @xmath100 , @xmath101 with @xmath102 for the two - dimensional @xmath49- and three - dimensional @xmath50-band , respectively . for our numerical work , we do take into account the different dimensionality of the bands but , for simplicity in our analytic work below , we take them both to be three dimensional . a decision needs to be taken about the normalization of the current @xmath7 in the second term . dahm and scalapino have used @xmath103 . here instead , we prefer to use the more symmetric form @xmath104 which reduces properly to the one - band case when our two bands are taken to be identical , with @xmath105 , where @xmath31 is the total electron density per unit volume . for the combined system , eqs . ( [ eq : dougb ] ) and ( [ eq : saulsa ] ) still hold with @xmath55 and @xmath56 modified as follows : @xmath106 and @xmath107 with these definitions eq . ( [ eq : jexpand ] ) also holds with @xmath108 replacing @xmath109 and the xu , yip , and sauls variable , @xmath71 , of the one - band case is replaced by @xmath110 , with @xmath75 the usual temperature profile of the bcs gap . other choices could be made . the superfluid density @xmath44 is proportional to @xmath55 for the combined system , specifically @xmath111 is given by eq . ( [ eq : atb ] ) with the first two factors omitted . ( 250,200 ) in fig . [ fig5 ] , we show both the @xmath60 and the @xmath68 for a model which uses truncated lorentzians for the @xmath112 spectra . this same model was used in our previous work@xcite and so we refer the reader to that paper for details . also , in ref . @xcite may be found the curves for the @xmath113 , the penetration depth , and other quantities for the same parameters used here . the essential parameters of this model are @xmath114 , @xmath115 and the interband electron - phonon coupling is varied from @xmath116 ( nearly decoupled case ) to 0.1 ( more integrated case ) . in addition , the @xmath117 , @xmath118 and @xmath119 . in the nearly decoupled case of @xmath116 , it can be seen that the solid curve looks like a superposition of two separate superconductors , one with a @xmath22 which is about 0.33 of the bulk @xmath22 . the lower temperature part of this curve is primarily due to the @xmath50-band ( or band 2 ) which is three dimensional , and indeed , when examined in detail , it has the characteristic behavior of the 3d example studied in the one - band case . the part of the curve at higher temperatures above about @xmath120 is due to the @xmath49-band ( or band 1 ) which is taken to be 2d and indeed , in the case of @xmath60 it shows a dependence approaching @xmath22 that expected for 2d strong - coupling with some interband anisotropy effects . the relative scale of the two sections of the curve is set by the value of the gap anisotropy @xmath121 and the ratio @xmath122 . the overall scale on the y - axis for @xmath68 differs from that of fig . [ fig4 ] due to our choice of @xmath108 for the normalization in the nonlinear term . indeed , for nearly decoupled bands ( solid curve ) , the value of the nonlinear coefficient @xmath68 is small at reduced temperature @xmath123 just above the sharp peak due to band 2 . specifically , it is of order 0.5 . if it had been referred to @xmath124 instead of @xmath108 , it would be smaller still by a factor of 1.7 and comparable to the single band 2d bcs result at the same reduced temperature ( fig . [ fig4 ] bottom frame , solid curve ) . however , as the non - diagonal electron - phonon couplings @xmath125 and @xmath126 are increased and a better integration of two bands proceeds , @xmath68 at @xmath123 can increase by an order of magnitude as , for example , in the dashed curve . the actual scale in this region is set by the details of the electron - phonon coupling ( see later the specific case of mgb@xmath0 ) . with more integration between the bands , one finds that the sharp peak at lower @xmath4 is reduced and rounded with a tail reaching to @xmath22 . when @xmath127 , the feature characteristic of the @xmath50-band @xmath22 is almost gone in @xmath60 and absent entirely in @xmath68 , even for modest interband coupling . the same conclusion holds for the effects of interband scattering ( shown in fig . [ fig5 ] for a value of @xmath128 for the nearly decoupled case ) which also integrates the bands and eliminates the lower energy scale . however , while the structure at the lower @xmath22 is now reduced to the point of giving a monotonic curve for @xmath68 , there still remains a large nonlinear contribution well above that for the one - band s - wave case , which marks the presence of the second band . we can have further insight into these results and check our work by developing some simple analytic results in renormalized bcs theory ( rbcs ) . for a summary of the approximations of rbcs and a comparison with full numerical solution for various properties including @xmath7 , we refer the reader to our previous work@xcite . for simplicity , we take both bands to be three dimensional in the following . near @xmath129 , eqs . ( [ eq : rbcsa ] ) and ( [ eq : rbcsb ] ) are modified for each band to : @xmath130 and @xmath131 where the functions @xmath132 and @xmath133 have been derived in ref . @xcite and @xmath134 is independent of @xmath135 , where @xmath136 . the @xmath137 s depend on the microscopic parameters of the theory . in rbcs , they are @xmath138 , @xmath139 , @xmath140 , and @xmath141 , from which @xmath22 and @xmath142 follow . while the expressions obtained for the @xmath132 and @xmath133 are lengthy , and hence we do not repeat them here , they are explicit algebraic forms . it is useful in this work to consider several simplifying limits . for decoupled bands @xmath143 and @xmath144 . as the band 2 does not contribute near @xmath22 , @xmath145 and @xmath146 take on the form of the single band case ( eqs . ( [ eq : rbcsa ] ) and ( [ eq : rbcsb ] ) ) . another limiting case is the separable anisotropy model.@xcite in this model , there are only two gap values with a ratio of @xmath147 , with @xmath148 an anisotropy parameter often assumed small . in this case , @xmath149 , @xmath150 and @xmath151 , where @xmath152 and @xmath153 . as a result @xmath154 and @xmath155 in this model , taking in addition that @xmath118 and @xmath156 leads to the one - band case and this can be used as a check of our algebra . to see the consequences of this algebra for our nonlinear coefficient @xmath68 , we begin with the decoupled band case near @xmath129 for which @xmath144 and @xmath145 and @xmath146 reduce to their single band value . in this limit of @xmath157 , @xmath158 ^ 2 , \label{eq : dougdc}\ ] ] where @xmath159 is the gap anisotropy parameter @xmath121 , and @xmath142 is the gap at @xmath1 . this expression shows explicitly the corrections introduced by the two - band nature of the system over the pure one - band case . note that @xmath68 is always increased by the presence of the correction term . in ( [ eq : dougdc ] ) , @xmath159 can never be taken to be one since we have assumed band 2 is weaker than band 1 . before leaving the decoupled case , it is worth noting that @xmath68 will show a change at the band 2 critical temperature @xmath160 . for @xmath4 below @xmath160 , @xmath161 and @xmath162 will be finite while above this temperature they are both zero . when the coupling @xmath125 and @xmath126 is switched on but still small , we expect that these quantities will acquire small tails and that they vanish only at @xmath22 . this is the hallmark of nearly decoupled bands . for the anisotropic @xmath163 model near @xmath22 @xmath164 ^ 3},\ ] ] where the average gap @xmath165 is related to @xmath22 by @xmath166 $ ] . for @xmath167 this expression reduces properly to the one - band limit . therefore , it is seen that anisotropy increases @xmath68 for @xmath4 near @xmath22 . another interesting limiting case is to assume both bands are the same , i.e. isotropic gap case , but that the fermi velocities differ in the two bands . near @xmath22 , we obtain @xmath168 ^ 3 } \frac{1}{8}\frac{(v_{f1}+v_{f2})^2}{(v_{f1}v_{f2})^2}(v_{f1}^2+v_{f2}^2).\ ] ] in this case , the fermi velocity anisotropy changes the nonlinear coefficient , but when @xmath118 the expression reduces properly to the one - band result . we find that the fermi velocity anisotropy increases @xmath68 near @xmath22 , a result that is seen in one of our calculations for mgb@xmath0 shown in fig . [ fig6 ] . ( 250,200 ) with fig . [ fig6 ] , we now turn to the specific case of mgb@xmath0 , where we have used the parameters and @xmath112 given by band structure calculations , and as a result , there are , in principle , no free parameters other than varying the impurity scattering rate . the basic parameters are @xmath169 , @xmath170 , @xmath171 , @xmath172 , @xmath173 , @xmath174 , @xmath175 , @xmath176 , with a @xmath177 k and a gap anisotropy of @xmath178 . the ratio of the two density of states is @xmath179 and of the fermi velocities is @xmath180 . we have found excellent agreement between theory and experiment for these parameters , as have other authors@xcite . as we have found in our previous work , mgb@xmath0 is quite integrated between the bands . it is also an intermediate strong coupler with @xmath181 and thus there is competition between the strong coupling effects and the anisotropy@xcite . in fig . 6 , the solid curve gives the prediction for mgb@xmath0 for @xmath60 and @xmath68 . a strong nonmonotonic feature around the lower band energy scale is observed in @xmath60 , but the @xmath68 is monotonically increasing with temperature . to see the second band effects in @xmath68 , it is better to plot @xmath182 ( the inset ) which accentuates the subtle variations found at the lower energy scale associated with the @xmath50 band . also shown in the inset for the upper frame is @xmath59 , which gives the temperature dependence of the superfluid density . the solid curve agrees with our previous calculation by other means@xcite . the variation in @xmath59 appears to be sufficient to remove the bump in @xmath60 when divided by three factors of @xmath59 to obtain the definition of @xmath68 . also , shown are the effects of intraband scattering with @xmath183 for the dashed curve and @xmath184 for the dot - dashed . scattering in the @xmath50-band reduces its contribution and provides an impurity tail at low @xmath4 , as found for the one - band case . however , scattering in the @xmath49-band , while lowering @xmath60 near @xmath22 as expected , does not appear to add weight at low @xmath4 . this is because the parameters for mgb@xmath0 heavily weight the @xmath50-band and the @xmath49-band is a small component . thus , upon comparison between @xmath49- and @xmath50-band scattering , @xmath68 could be lowered at @xmath185 , for example , by putting impurities in the @xmath50-band , but it would be raised if the impurity scattering is in the @xmath49-band . the dotted curve in the figure is for pure mgb@xmath0 but where we have taken @xmath186 to mimic a case where transport may happen along the c - axis . in this instance , the bump in @xmath60 remains , but is gone in @xmath68 . we see that @xmath68 is large due to the higher power of the fermi velocity ratio that enters the calculation , and , as a result , the nonlinearity is greatly increased . as @xmath68 is a relevant quantity for microwave filter design , this study provides some insight into which factors may be used to optimize the material and reduce the nonlinear effects . study of nonlinear current response is important for device applications and for providing fundamental signatures of order parameter symmetry , such as have been examined in the case of d - wave superconductivity . in this paper , we have considered the case of two - band superconductors . in so doing , we also reexamined the one - band case and discovered that there can exist extra nonlinearity at both low and high temperatures due to strong electron - phonon coupling , for which we have provided strong - coupling correction formulas , whereas the excess nonlinearity induced by impurity scattering occurs primarily at low temperatures . at @xmath22 , impurities will give an enhanced or decreased contribution depending on the particular nonlinear coefficient discussed . in this paper , we have examined two nonlinear coefficients defined in the literature , one due to xu et al.@xcite and one defined by dahm and scalapino , with an emphasis on the latter as it is related to the intermodulation power in microstrip resonators@xcite . we have also studied issues associated with dimensionality motivated by the two - band superconductor mgb@xmath0 , which has a two - dimensional @xmath49-band and a three - dimensional @xmath50-band . within our one - band calculation , aside from an overall factor of 2/3 , we find little difference in the temperature variation of the nonlinear coefficient in mean - field between 2d and 3d . this is further reduced by strong coupling effects . for two - band superconductors , we show that for nearly decoupled bands a strong signature of the small gap @xmath50-band will appear in the nonlinear coefficients , but with increased interband coupling or interband scattering , such a signature will rapidly disappear . likewise , intraband impurities in the @xmath50-band will wash out the temperature variation of the @xmath50-band , whereas the intraband impurities in the @xmath49-band largely effect the nonlinearity at higher temperatures above the energy scale of the @xmath50-band , for the parameters typical to mgb@xmath0 . we provide a prediction for the nonlinear coefficient in mgb@xmath0 using the parameters set by band structure calculations . as the bands in mgb@xmath0 are quite integrated , we find that the nonlinear coefficient @xmath68 is monotonically increasing in contrast to a previous prediction , which was based on a number of approximations,@xcite and we find that the increased nonlinearity due to the @xmath50-band is best reduced at @xmath185 by adding impurities to the @xmath50-band . should the supercurrent sample the c - axis direction , a larger anisotropy in the fermi velocity ratio between the bands would result and this effect is found to increase the nonlinearity . finally , several simple formulas have been provided for near @xmath22 which aid in the understanding of the range of behavior observed in the numerical calculations . we await experimental verification of our predictions . we thank dr . ove jepsen for supplying us with the mgb@xmath0 electron - phonon spectral functions . djs would also like to acknowledge useful discussions with thomas dahm . ejn acknowledges funding from nserc , the government of ontario ( prea ) , and the university of guelph . jpc acknowledges support from nserc and the ciar . djs acknowledges nsf support under grant no . dmr02 - 11166 . this research was supported in part by the national science foundation under grant no . phy99 - 07949 and we thank the hospitality of the kitp , where this work was initiated . a. maeda , y. iino , t. hanaguri , n. motohira , k. kishio , and t. fukase , phys . lett . * 74 * , 1202 ( 1995 ) ; a. maeda , t. hanaguri , y. iino , s. matsuoka , y. kokata , j. shimoyama , k. kishio , h. asaoka , y. matsushita , m. hasegawa , and h. takei , j. phys . jpn . * 65 * , 3638 ( 1996 ) ; a. bhattacharya , i. zuti , o.t . valls , a.m. goldman , u. welp , and b. veal , phys . lett . * 82 * , 3132 ( 1999 ) ; a. bhattacharya , i. zuti , o.t . valls , a.m. goldman , phys . * 83 * , 887 ( 1999 ) ; c.p . bidinosti , w.n . hardy , d.a . bonn , and r. liang , phys . rev . lett . * 83 * , 3277 ( 1999 ) ; a. carrington , r.w . giannetta , j.t . kim , and j. giapintzakis , phys . b * 59 * , r14173 ( 1999 ) . james c. booth , k.t . leung , sang young lee , j.h . lee , b. oh , h.n . lee , and s.h . moon , supercond . * 16 * , 1518 ( 2003 ) ; g. lamura , a.j . purnell , l.f . cohen , a. andreone , f. chiarella , e. di gennaro , and r. vaglio , appl . phys . lett . * 82 * , 4525 ( 2003 ) . for example , yu.a . nefyodev , a.m. shuvaev , and m.r . trunin , cond - mat/0509244 ; v. guritanu , w. goldacker , f. bouquet , y. wang , r. lortz , g. goll , and a. junod , phys . b * 70 * , 184526 ( 2004 ) ; etienne boaknin , m.a . tanatar , johnpierre paglione , d. hawthorn , f. ronning , r.w . hill , m. sutherland , louis taillefer , jeff sonier , s.m . hayden , and j.w . brill , phys . rev . lett . * 90 * , 117003 ( 2003 ) . for example , m.a . tanatar , johnpierre paglione , s. nakatsuji , d.g . hawthorn , e. boaknin , r.w . hill , f. ronning , m. sutherland , louis taillefer , c. petrovic , p.c . canfield , and z. fisk , phys . lett . * 95 * , 067002 ( 2005 ) ; p.m.c . rourke , m.a . tanatar , c.s . turel , j. berdeklis , c. petrovic , and j.y.t . wei , phys . lett . * 94 * , 107005 ( 2005 ) .

we have calculated the nonlinear current of a number of single band s - wave electron - phonon superconductors . among issues considered were those of dimensionality , strong electron - phonon coupling , impurities , and comparison with bcs . for the case of two bands , particular attention is paid to the role of anisotropy , the integration effects of the off - diagonal electron - phonon interaction , as well as inter- and intraband impurities . for the specific case of mgb@xmath0 , we present results based on the known microscopic parameters of band theory .

arxiv

classification problem is one of the most important tasks in time series data mining . a well - known 1-nearest neighbor ( 1-nn ) with dynamic time warping ( dtw ) distance is one of the best classifier to classify time series data , among other approaches , such as support vector machine ( svm ) @xcite , artificial neural network ( ann ) @xcite , and decision tree @xcite . for the 1-nn classification , selecting an appropriate distance measure is very crucial ; however , the selection criteria still depends largely on the nature of data itself , especially in time series data . though the euclidean distance is commonly used to measure the dissimilarity between two time series , it has been shown that dtw distance is more appropriate and produces more accurate results . sakoe - chiba band ( s - c band ) @xcite originally speeds up the dtw calculation and later has been introduced to be used as a dtw global constraint . in addition , the s - c band was first implemented for the speech community , and the width of the global constraint was fixed to be 10% of time series length . however , recent work @xcite reveals that the classification accuracy depends solely on this global constraint ; the size of the constraint depends on the properties of the data at hands . to determine a suitable size , all possible widths of the global constraint are tested , and the band with the maximum training accuracy is selected . ratanamahatana - keogh band ( r - k band ) @xcite has been introduced to generalize the global constraint model represented by a one - dimensional array . the size of the array and the maximum constraint value is limited to the length of the time series . and the main feature of the r - k band is the multi bands , where each band is representing each class of data . unlike the single s - c band , this multi r - k bands can be adjusted as needed according to its own class warping path . although the r - k band allows great flexibility to adjust the global constraint , a learning algorithm is needed to discover the best multi r - k bands . in the original work of r - k band , a hill climbing search algorithm with two heuristic functions ( accuracy and distance metrics ) is proposed . the search algorithm climbs though a space by trying to increase / decrease specific parts of the bands until terminal conditions are met . however , this learning algorithm still suffers from an overfitting phenomenon since an accuracy metric is used as a heuristic function to guide the search . to solve this problem , we propose two new learning algorithms , i.e. , band boundary extraction and iterative learning . the band boundary extraction method first obtains a maximum , mean , and mode of the paths positions on the dtw distance matrix , and the iterative learning , band s structures are adjusted in each round of the iteration to a silhouette index @xcite . we run both algorithms and the band that gives better results . in prediction step , the 1-nn using dynamic time warping distance with this discovered band is used to classify unlabeled data . note that a lower bound , lb_keogh @xcite , is also used to speed up our 1-nn classification . the rest of this paper is organized as follows . section 2 gives some important background for our proposed work . in section 3 , we introduce our approach , the two novel learning algorithms . section 4 contains an experimental evaluation including some examples of each dataset . finally , we conclude this paper in section 5 . our novel learning algorithms are based on four major fundamental concepts , i.e. , dynamic time warping ( dtw ) distance , sakoe - chiba band ( s - c band ) , ratanamahatana - keogh band ( r - k band ) , and silhouette index , which are briefly described in the following sections . dynamic time warping ( dtw ) @xcite distance is a well - known similarity measure based on shape . it uses a dynamic programming technique to find all possible warping paths , and selects the one with the minimum distance between two time series . to calculate the distance , it first creates a distance matrix , where each element in the matrix is a cumulative distance of the minimum of three surrounding neighbors . suppose we have two time series , a sequence @xmath0 of length @xmath1 ( @xmath2 ) and a sequence @xmath3 of length @xmath4 ( @xmath5 ) . first , we create an @xmath1-by-@xmath4 matrix , where every ( @xmath6 ) element of the matrix is the cumulative distance of the distance at ( @xmath6 ) and the minimum of three neighboring elements , where @xmath7 and @xmath8 . we can define the ( @xmath6 ) element , @xmath9 , of the matrix as : @xmath10 where @xmath11 is the squared distance of @xmath12 and @xmath13 , and @xmath9 is the summation of @xmath14 and the the minimum cumulative distance of three elements surrounding the ( @xmath6 ) element . then , to find an optimal path , we choose the path that yields a minimum cumulative distance at ( @xmath15 ) , which is defined as : @xmath16 where @xmath17 is a set of all possible warping paths , @xmath18 is ( @xmath6 ) at @xmath19 element of a warping path , and @xmath20 is the length of the warping path . in reality , dtw may not give the best mapping according to our need because it will try its best to find the minimum distance . it may generate the unwanted path . for example , in figure [ flo : dtw1 ] @xcite , without global constraint , dtw will find its optimal mapping between the two time series . however , in many cases , this is probably not what we intend , when the two time series are expected to be of different classes . we can resolve this problem by limiting the permissible warping paths using a global constraint . two well - known global constraints , sakoe - chiba band and itakura parallelogram @xcite , and a recent representation , ratanamahatana - keogh band ( r - k band ) , have been proposed , figure [ flo : dtw2 ] @xcite shows an example for each type of the constraints . [ cols="^,^ " , ] [ flo : result ] in this work , we propose a new efficient time series classification algorithm based on 1-nearest neighbor classification using the dynamic time warping distance with multi r - k bands as a global constraint . to select the best r - k band , we use our two proposed learning algorithms , i.e. , band boundary extraction algorithm and iterative learning . silhouette index is used as a heuristic function for selecting the band that yields the best prediction accuracy . the lb_keogh lower bound is also used in data prediction step to speed up the computation . we would like to thank the scientific parallel computer engineering ( space ) laboratory , chulalongkorn university for providing a cluster we have used in this contest . 1 fumitada itakura . minimum prediction residual principle applied to speech recognition . , 23(1):6772 , 1975 . eamonn j. keogh and chotirat ann ratanamahatana . exact indexing of dynamic time warping . , 7(3):358386 , 2005 . alex nanopoulos , rob alcock , and yannis manolopoulos . feature - based classification of time - series data . , pages 4961 , 2001 . chotirat ann ratanamahatana and eamonn j. keogh . making time - series classification more accurate using learned constraints . in _ proceedings of the fourth siam international conference on data mining ( sdm 2004 ) _ , pages 1122 , lake buena vista , fl , usa , april 22 - 24 2004 . chotirat ann ratanamahatana and eamonn j. keogh . three myths about dynamic time warping data mining . in _ proceedings of 2005 siam international data mining conference ( sdm 2005 ) _ , pages 506510 , newport beach , cl , usa , april 21 - 23 2005 . juan jos rodrguez and carlos j. alonso . interval and dynamic time warping - based decision trees . in _ proceedings of the 2004 acm symposium on applied computing ( sac 2004 ) _ , pages 548552 , nicosia , cyprus , march 14 - 17 2004 . peter rousseeuw . silhouettes : a graphical aid to the interpretation and validation of cluster analysis . , 20(1):5365 , 1987 . hiroaki sakoe and seibi chiba . dynamic programming algorithm optimization for spoken word recognition . , 26(1):4349 , 1978 . yi wu and edward y. chang . distance - function design and fusion for sequence data . in _ proceedings of the 2004 acm cikm international conference on information and knowledge management ( cikm 2004 ) _ , pages 324333 , washington , dc , usa , november 8 - 13 2004 .

1-nearest neighbor with the dynamic time warping ( dtw ) distance is one of the most effective classifiers on time series domain . since the global constraint has been introduced in speech community , many global constraint models have been proposed including sakoe - chiba ( s - c ) band , itakura parallelogram , and ratanamahatana - keogh ( r - k ) band . the r - k band is a general global constraint model that can represent any global constraints with arbitrary shape and size effectively . however , we need a good learning algorithm to discover the most suitable set of r - k bands , and the current r - k band learning algorithm still suffers from an overfitting phenomenon . in this paper , we propose two new learning algorithms , i.e. , band boundary extraction algorithm and iterative learning algorithm . the band boundary extraction is calculated from the bound of all possible warping paths in each class , and the iterative learning is adjusted from the original r - k band learning . we also use a silhouette index , a well - known clustering validation technique , as a heuristic function , and the lower bound function , lb_keogh , to enhance the prediction speed . twenty datasets , from the workshop and challenge on time series classification , held in conjunction of the sigkdd 2007 , are used to evaluate our approach .

arxiv

the standard model of elementary particles ( sm ) has been very successful in describing the nature at the electroweak ( ew ) scale . recently , the atlas and cms collaborations at the large hadron collider ( lhc ) have discovered a new particle @xcite , which is consistent with the sm higgs boson . this discovery also strengthens the correctness of the sm . so far , no explicit evidence of physics beyond the sm has been reported from the lhc . several groups , however , have reported an anomaly of the muon anomalous magnetic moment @xmath9 ( muon g-2 ) , which has been precisely measured experimentally @xcite and compared with state - of - the - art theoretical predictions ( for example , see @xcite and references therein ) . the estimated discrepancies between the sm predictions and the measured value are consistently more than @xmath10 , as listed in table [ tab : g-2 ] . although it is too early to conclude that this anomaly is evidence of new physics beyond the sm , we expect new particles and interactions related with the muon sector once we regard it as a hint of new physics . gauge interactions have been playing a central role to construct fundamental models in particle physics history . following this line , in this paper , we purse the possibility that the muon has a new gauge interaction beyond the sm . .measured muon g-2 ( @xmath11 ) and the estimated differences ( @xmath12 ) from the recent sm predictions in several references . [ cols="^,^",options="header " , ] from the charge assignments , renormalizable terms in a lagrangian which contribute to the lepton masses are given by @xmath13 here , @xmath14 are the yukawa couplings of the charged leptons and not related to the neutrino mass . the neutrino mass is determined from yukawa couplings @xmath15 , majorana masses @xmath16 and @xmath17 , and yukawa couplings @xmath18 and @xmath19 . note that the mass terms between the left and right - handed neutrinos are diagonal . therefore , the neutrino mixing is obtained by mixing among the right - handed neutrinos . if the majorana masses @xmath20 and @xmath21 are of the same order , the seesaw mechanism @xcite provides the observed order one neutrino mixing . from the seesaw formula , a relation between the parameters is given by @xmath22 where @xmath23 gev is the vacuum expectation value of the sm higgs , and @xmath24 is the difference between the mass squared of the left - handed neutrinos . @xmath25 and @xmath26 denote @xmath27 and @xmath20 and @xmath21 collectively . interactions given by eq . ( [ eq : mass terms ] ) break the lepton symmetry . on the other hand , @xmath28 symmetry is broken by the anomaly against the @xmath29 gauge interaction , whose effect is efficient at the early universe by the sphaleron process in the finite temperature @xcite . therefore , the baryon asymmetry is washed out if both effects are important simultaneously . let us calculate a condition such that the washout does not occur . first of all , the sphaleron process is efficient only at the temperature above the ew scale . therefore , if the baryon asymmetry is generated below the ew scale , the washout does not occur . in the following , we assume that the baryon asymmetry is produced above the ew scale and calculate the constraint on the parameters in eq . ( [ eq : mass terms ] ) . let us consider two possibilities in which the washout does not occur . 1 . @xmath30 is small , 2 . @xmath16 and @xmath17 are small . if any of the two conditions are satisfied , the lepton number is effectively conserved . therefore , one should adopt the weakest condition among them . let us discuss the two cases in detail . in the limit @xmath31 , the lepton symmetry is restored for each flavors . therefore , if the interaction by @xmath30 is inefficient , the washout of the baryon asymmetry does not occur . the most efficient interaction is shown in figure [ fig : lambda ] and its rate is given by @xmath32 where @xmath33 , @xmath34 , @xmath35 , @xmath36 are the cross section of the process , the number density of related particles , the velocity of related particles , and the yukawa coupling of the top quark , respectively . @xmath37 denotes the thermal average . by requiring that the rate is smaller than the hubble scale for @xmath38 gev , we obtain the bound @xmath39 .,width=264 ] if both majorana masses vanish , @xmath40 symmetry is restored . the most efficient interaction which induces the symmetry violation by the majorana masses is shown in figure [ fig : m ] . its rate is given by @xmath41 this rate is smaller than the hubble scale for @xmath38 gev if @xmath42 from the relation ( [ eq : seesaw ] ) , one can see that the condition ( [ eq : constraint - m ] ) is severer than the condition ( [ eq : constraint - l ] ) . therefore , it is enough to satisfy the condition ( [ eq : constraint - l ] ) in order for the washout not to occur . with the relation ( [ eq : seesaw ] ) , the condition is interpreted as @xmath43 since the right - handed neutrinos are light and weakly coupled , it is necessary to consider whether they are long - lived . if they are long - lived , they might over - close the universe , or destroy the success of the big - bang nucleosynthesis ( bbn ) . the most important decay channel is given by the diagram shown in figure . [ fig : decay of n ] . here , we have assumed that @xmath33 is heavier than the right - handed neutrinos and hence the decay mode @xmath44 is closed . the decay rate is given by @xmath45 the decay of the right - handed neutrinos is efficient around the temperature @xmath46 therefore , the right - handed neutrinos decay before the bbn begins and does not affect it . passarino - veltman functions @xcite are defined by @xmath47 \left[(k+p)^2-m_b^2+i \epsilon\right]},\nonumber \\ p^\mu b_1(a , b;p ) & = & 16\pi^2 \mu^{2\epsilon } \int \frac{d^n k}{i(2\pi)^n}\frac{k^\mu } { \left[k^2-m_a^2+i\epsilon\right]\left[(k+p)^2-m_b^2+i\epsilon\right ] } , \nonumber \\ p^\mu p^\nu b_{21}(a , b;p)&+&g^{\mu \nu}b_{22}(a , b;p ) \nonumber \\ & = & 16\pi^2 \mu^{2\epsilon } \int \frac{d^n k}{i(2\pi)^n } \frac{k^\mu k^\nu } { \left[k^2-m_a^2+i\epsilon\right]\left[(k+p)^2-m_b^2+i\epsilon\right ] } , \ ] ] @xmath48[(k+p_1)^2-m_b^2+i\epsilon][(k+p_1+p_2)^2-m_c^2+i\epsilon ] } , \nonumber \\ & & \left ( p_1^\mu c_{11}+p_2^\mu c_{12 } \right)(a , b , c;p_1,p_2 ) \nonumber \\ & = & 16\pi^2 \mu^{2\epsilon } \int \frac{d^n k}{i(2\pi)^n}\frac{k^\mu } { [ k^2-m_a^2+i\epsilon][(k+p_1)^2-m_b^2+i\epsilon][(k+p_1+p_2)^2-m_c^2+i\epsilon ] } , \nonumber \\ & & \left\{(p_1^\mu p_1^\nu c_{21 } + p_2^\mu p_2^\nu c_{22}+(p_1^\mu p_2^\nu+p_1^\nu p_2^\mu)c_{23 } + g^{\mu \nu}c_{24 } \right\ } ( a , b , c;p_1,p_2 ) \nonumber \\ & = & 16\pi^2 \mu^{2\epsilon } \int \frac{d^n k}{i(2\pi)^n}\frac{k^\mu k^\nu } { [ k^2-m_a^2+i\epsilon][(k+p_1)^2-m_b^2+i\epsilon][(k+p_1+p_2)^2-m_c^2+i\epsilon ] } , \nonumber \\\end{aligned}\ ] ] where we use dimensional regularization in @xmath49 dimensions , and @xmath50 is a renormalization scale . 99 g. aad _ et al . _ [ atlas collaboration ] , phys . b * 716 * , 1 ( 2012 ) . s. chatrchyan _ et al . _ [ cms collaboration ] , phys . b * 716 * , 30 ( 2012 ) . j. beringer _ [ particle data group collaboration ] , phys . d * 86 * , 010001 ( 2012 ) and 2013 partial update for the 2014 edition . t. aoyama , m. hayakawa , t. kinoshita and m. nio , phys . rev . lett . * 109 * , 111808 ( 2012 ) [ arxiv:1205.5370 [ hep - ph ] ] . a. czarnecki , w. j. marciano and a. vainshtein , phys . d * 67 * , 073006 ( 2003 ) [ erratum - ibid . d * 73 * , 119901 ( 2006 ) ] . k. hagiwara , r. liao , a. d. martin , d. nomura and t. teubner , j. phys . g * 38 * , 085003 ( 2011 ) [ arxiv:1105.3149 [ hep - ph ] ] . t. teubner , k. hagiwara , r. liao , a. d. martin and d. nomura , chin . c * 34 * , 728 ( 2010 ) [ arxiv:1001.5401 [ hep - ph ] ] . m. benayoun , p. david , l. delbuono and f. jegerlehner , eur . j. c * 72 * , 1848 ( 2012 ) [ arxiv:1106.1315 [ hep - ph ] ] . f. jegerlehner and r. szafron , eur . j. c * 71 * , 1632 ( 2011 ) [ arxiv:1101.2872 [ hep - ph ] ] . f. jegerlehner and a. nyffeler , phys . * 477 * , 1 ( 2009 ) [ arxiv:0902.3360 [ hep - ph ] ] . m. davier , a. hoecker , b. malaescu and z. zhang , eur . j. c * 71 * , 1515 ( 2011 ) [ erratum - ibid . c * 72 * , 1874 ( 2012 ) ] . t. moroi , phys . rev . d * 53 * , 6565 ( 1996 ) [ erratum - ibid . d * 56 * , 4424 ( 1997 ) ] . for example , see t. hambye , k. kannike , e. ma and m. raidal , phys . d * 75 * , 095003 ( 2007 ) [ hep - ph/0609228 ] ; 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in this paper , we consider phenomenology of a model with an @xmath0 gauge symmetry . since the muon couples to the @xmath0 gauge boson ( called @xmath1 boson ) , its contribution to the muon anomalous magnetic moment ( muon g-2 ) can account for the discrepancy between the standard model prediction and the experimental measurements . on the other hand , the @xmath1 boson does not interact with the electron and quarks , and hence there are no strong constraints from collider experiments even if the @xmath1 boson mass is of the order of the electroweak scale . we show an allowed region of a parameter space in the @xmath0 symmetric model , taking into account consistency with the electroweak precision measurements as well as the muon g-2 . we study the large hadron collider ( lhc ) phenomenology , and show that the current and future data would probe the interesting parameter space for this model . # 1 .75 in * muon g-2 and lhc phenomenology in the @xmath0 gauge symmetric model * .75 in keisuke harigaya@xmath2 , takafumi igari@xmath3 , mihoko m. nojiri@xmath4 , + michihisa takeuchi@xmath5 , and kazuhiro tobe@xmath6 0.25 in _ @xmath2kavli ipmu ( wpi ) , todias , university of tokyo , chiba , kashiwa , 277 - 8583 , japan _ _ @xmath3department of physics , nagoya university , aichi , nagoya 464 - 8602 , japan _ _ @xmath7theory center , kek , tsukuba , ibaraki 305 - 0801 , japan _ _ @xmath5theoretical particle physics and cosmology group , department of physics , king s college london , london wc2r 2ls , uk _ _ @xmath8kobayashi - maskawa institute for the origin of particles and the universe , + nagoya university , aichi , nagoya 464 - 8602 , japan _ .5 in kcl - ph - th/2013-*37 * kek - th-1684 ipmu 13 - 0213

arxiv

entanglement in quantum multipartite systems is a unique property in quantum world . it plays an important role in quantum information processing @xcite . therefore , the study of its essential features and dynamical behavior under the ubiquitous decoherence of relevant quantum system has attracted much attention in recent years @xcite . for example , it was found that the entanglement of qubits under the markovian decoherence can be terminated in a finite time despite the coherence of single qubit losing in an asymptotical manner @xcite . the phenomenon called as entanglement sudden death ( esd ) @xcite has been observed experimentally @xcite . this is detrimental to the practical realization of quantum information processing using entanglement . surprisingly , some further studies indicated that esd is not always the eventual fate of the qubit entanglement . it was found that the entanglement can revive again after some time of esd @xcite , which has been observed in optical system @xcite . it has been proven that this revived entanglement plays a constructive role in quantum information protocols @xcite . even in some occasions , esd does not happen at all , instead finite residual entanglement can be preserved in the long time limit @xcite . this can be due to the structured environment and physically it results from the formation of a bound state between the qubit and its amplitude damping reservoir @xcite . these results show rich dynamical behaviors of the entanglement and its characters actually have not been clearly identified . recently , lpez _ et al . _ asked a question about where the lost entanglement of the qubits goes @xcite . interestingly , they found that the lost entanglement of the qubits is exclusively transferred to the reservoirs under the markovian amplitude - damping decoherence dynamics and esd of the qubits is always accompanied with the entanglement sudden birth ( esb ) of the reservoirs . a similar situation happens for the spin entanglement when the spin degree of freedom for one of the two particles interacts with its momentum degree of freedom @xcite . all these results mean that the entanglement does not go away , it is still there but just changes the location . this is reminiscent of the work of yonac _ et al . _ @xcite , in which the entanglement dynamics has been studied in a double jaynes - cummings ( j - c ) model . they found that the entanglement is transferred periodically among all the bipartite partitions of the whole system but an identity ( see below ) has been satisfied at any time . this may be not surprising since the double j - c model has no decoherence and any initial information can be preserved in the time evolution . however , it would be surprising if the identity is still valid in the presence of the decoherence , in which a non - equilibrium relaxation process is involved . in this paper , we show that it is indeed true for such a system consisted of two qubits locally interacting with two amplitude - damping reservoirs . it is noted that although the infinite degrees of freedom of the reserviors introduce the irreversibility to the subsystems , this result is still reasonable based on the fact that the global system evolves in a unitary way . furthermore , we find that the distribution of the entanglement among the bipartite subsystems is dependent of the explicit property of the reservoir and its coupling to the qubit . the rich dynamical behaviors obtained previously in the literature can be regarded as the special cases of our present result or markovian approximation . particularly , we find that , instead of entirely transferred to the reservoirs , the entanglement can be stably distributed among all the bipartite subsystems if the qubit and its reservoir can form a bound state and the non - markovian effect is important , and the esd of the qubits is not always accompanied with the occurrence of esb of reservoirs . irrespective of how the entanglement distributes , it is found that the identity about the entanglement in the whole system can be satisfied at any time , which reveals the profound physics of the entanglement dynamics under decoherence . this paper is organized as follows . in sec . [ model ] , the model of two independent qubits in two local reservoirs is given . and the dynamical entanglement invariance is obtained based on the exact solution of the non - markovian decoherence dynamics of the qubit system . in sec . [ edd ] , the entanglement distribution over the subsystems when the reservoirs are pbg mediums is studied explicitly . a stable entanglement - distribution configuration is found in the non - markovian dynamics . finally , a brief discussion and summary are given in sec . we consider two qubits interacting with two uncorrelated vacuum reservoirs . due to the dynamical independence between the two local subsystems , we can firstly solve the single subsystem , then apply the result obtained to the double - qubit case . the hamiltonian of each local subsystem is @xcite @xmath0 where @xmath1 and @xmath2 are the inversion operators and transition frequency of the qubit , @xmath3 and @xmath4 are the creation and annihilation operators of the @xmath5-th mode with frequency @xmath6 of the radiation field . the coupling strength between the qubit and the reservoir is denoted by @xmath7 , where @xmath8 and @xmath9 are the unit polarization vector and the normalization volume of the radiation field , @xmath10 is the dipole moment of the qubit , and @xmath11 is the free space permittivity . for such a system , if the qubit is in its ground state @xmath12 and the reservoir is in vacuum state at the initial time , then the system does not evolve to other states . when the qubit is in its excited state @xmath13 , the system evolves as @xmath14 here @xmath15 denotes that the qubit jumps to its ground state and one photon is excited in the @xmath5-th mode of the reservoir . @xmath16 satisfies an integro - differential equation @xmath17 where the kernel function @xmath18 is dependent of the spectral density @xmath19 . introducing the normalized collective state of the reservoir with one excitation as @xmath20 and with zero excitation as @xmath21 @xcite , eq . ( [ t9 ] ) can be written as @xmath22 , where @xmath23 . it should be emphasized that the introducing of normalized collective state is not a reduction of present model to the j - c model @xcite , as noted in @xcite . the dynamics is given by eq . ( [ t7 ] ) , which is difficult to obtain analytically since its non - markovian nature . in general the numerical integration should be used . it is emphasized that our treatment to the dynamics of the system is exact without resorting to the widely used born - markovian approximation . to compare with the conventional approximate result , we may derive straightforwardly the master equation from eq . ( [ t9 ] ) after tracing over the degree of freedom of the reservoir @xcite , @xmath24+\gamma ( t)[2\sigma _ { -}\rho ( t)\sigma _ { + } \nonumber \\ & & -\sigma _ { + } \sigma _ { -}\rho ( t)-\rho ( t)\sigma _ { + } \sigma _ { - } ] , \label{mstt}\end{aligned}\]]where the time - dependent parameters are given by @xmath25,~\gamma ( t)=-\text{re}[\frac{% \dot{b}(t)}{b(t)}]$ ] . the time - dependent parameters @xmath26 and @xmath27 play the roles of lamb shifted frequency and decay rate of the qubit , respectively . the integro - differential equation ( [ t7 ] ) contains the memory effect of the reservoir registered in the time - nonlocal kernel function and thus the dynamics of qubit displays non - markovian effect . if the time - nonlocal kernel function is replaced by a time - local one , then eq . ( [ mstt ] ) recovers the conventional master equation under born - markovian approximation @xcite . according to the above results , the time evolution of a system consisted of two such subsystems with the initial state @xmath28 is given by @xmath29 where @xmath30 and @xmath31 are the coefficients to determine the initial entanglement in the system . from @xmath32 , one can obtain the time - dependent reduced density matrix of the bipartite subsystem qubit1-qubit2 ( @xmath33 ) by tracing over the reservoir variables . it reads @xmath34 where @xmath35 and @xmath36 . similarly , one can obtain the corresponding reduced density matrices for other subsystems like reservoir1-reservoir2 ( @xmath37 ) and qubit - reservoir ( @xmath38 , @xmath39 , @xmath40 , @xmath41 ) . using the concurrence @xcite to quantify entanglement , we can calculate the entanglement of each subsystem as @xmath42 with @xmath43 for different bipartite partitions labeled by @xmath44 as @xmath45 one can verify that @xmath46 in eqs . ( [ qq])-([qr2 ] ) satisfy an identity @xmath47 where @xmath48 is just the initial entanglement present in @xmath33 . ( [ t15 ] ) recovers the explicit form derived in a double j - c model @xcite when each of the reservoirs contains only one mode , i.e. @xmath49 , where the decoherence is absent and the dynamics is reversible . it is interesting that this identity is still valid in the present model because the reservoirs containing infinite degrees of freedom here lead to a completely out - of - phase interaction with qubit and an irreversibility . furthermore , one notes that the identity is not dependent of any detail about @xmath16 , which only determines the detailed dynamical behavior of each components in eq . ( [ t15 ] ) . this result manifests certain kind of invariant nature of the entanglement . ( [ t15 ] ) can be intuitively understood by the global multipartite entanglement of the whole system . the global entanglement carried by the subsystem @xmath50 can be straightforwardly calculated from eq . ( [ phit ] ) by generalized concurrence @xcite as @xmath48 , which , coinciding with the bipartite entanglement initially present in @xmath33 , just is the right hand side of eq . ( [ t15 ] ) . since there is no direct interaction between @xmath51 and @xmath52 , this global entanglement is conserved during the time evolution . from this point , our result is consistent with the one in refs . @xcite and @xcite . another observation of eq . ( [ t15 ] ) is that the different coefficients in the left hand side are essentially determined by the energy / information transfer among the local subsystems . explicitly , in our model the total excitation number is conserved , so the energy degradation in @xmath53 with factor @xmath16 is compensated by the energy enhancement in @xmath54 with factor @xmath55 . this causes that @xmath56 , @xmath57 , and @xmath58 , in all of which the double excitation is involved , have similar form except for the different combinations of @xmath16 and @xmath55 in eqs . ( [ qq ] ) , ( [ rr ] ) , and ( [ qr2 ] ) . the dynamical consequence of the competition of the two terms in these equations causes the sudden death / birth of entanglement characterized by the presence of negative @xmath59 . a different case happens for @xmath60 , where only single excitation is involved and no sudden death is present . with these observation , one can roughly understand why such combination in left hand side of eq . ( [ t15 ] ) gives the global entanglement . the significance of eq . ( [ t15 ] ) is that it gives us a guideline to judge how the entanglement spreads out over all the bipartite partitions . it implies that entanglement is not destroyed but re - distributed among all the bipartite subsystems and this re - distribution behavior is not irregular but in certain kind of invariant manner . the similar invariant property of entanglement evolution has also been studied in ref . @xcite . in the following we explicitly discuss the entanglement distribution , especially in the steady state , by taking the reservoir as a photonic band gap ( pbg ) medium @xcite and compare it with the previous results . we will pay our attention mainly on the consequence of the non - markovian effect on the entanglement distribution and its differences to the results in refs . @xcite and @xcite . for the pbg medium , the dispersion relation near the upper band - edge is given by @xcite @xmath61 where @xmath62 , @xmath63 is the upper band - edge frequency and @xmath64 is the corresponding characteristic wave vector . in this case , the kernel function has the form @xmath65 where @xmath66 is a dimensionless constant . in solving eq . ( [ t7 ] ) for @xmath16 , eq . ( [ kn ] ) is evaluated numerically . here we do not assume that @xmath5 is replaced by @xmath64 outside of the exponential @xcite . so our result is numerically exact . in the following we take @xmath67 as the unit of frequency . [ cols="^,^ " , ] in figs . [ bst ] and [ nbst ] , we show the entanglement evolutions of each subsystem for two typical cases of @xmath68 and @xmath69 , which correspond to the atomic frequency being located at the band gap and at the upper band of the pbg medium , respectively . in the both cases the initial entanglement in @xmath33 begins to transfer to other bipartite partitions with time but their explicit evolutions , in particular the long time behaviors , are quite different . in the former case , the entanglement could be distributed stably among all possible bipartite partitions . [ bst](a ) shows that after some oscillations , a sizeable entanglement of @xmath33 is preserved for the parameter regime of @xmath70 . remarkably , the entanglement in @xmath71 forms quickly in the full range of @xmath30 [ fig . [ bst](c ) ] and dominates the distribution . on the contrary , only slight entanglement of @xmath37 is formed in a very narrow parameter regime @xmath72 , as shown in fig . [ bst](b ) . however , when @xmath73 is located at the upper band of the pbg medium , the initial entanglement in @xmath33 is transferred completely to the @xmath37 in the long - time limit , as shown in fig . [ nbst ] . at the initial stage , @xmath74 and @xmath75 are entangled transiently , but there is no stable entanglement distribution . this result is consistent with that in refs . it is noted that the entanglement in @xmath76 comes from two parts : one is transferred from @xmath33 , the other is created by the direct interaction between @xmath53 and @xmath54 . this can be seen clearly from fig . [ bst](c ) when @xmath30 is very small , the initial entanglement of @xmath33 is very small , while that of @xmath76 is rather large , which just results from the interaction between @xmath53 and @xmath54 . another interesting point is a stable entanglement can even be formed for the non - interacting bipartite system @xmath39 [ see [ bst](d ) when @xmath77 . this entanglement transfer also results from the local interaction between @xmath53 and @xmath54 @xcite . it is not difficult to understand these rich behaviors of entanglement distribution according to eqs . ( [ qq])-([qr2 ] ) and its invariance ( [ t15 ] ) . from these equations , one can clearly see that the entanglement dynamics and its distributions in the bipartite partitions are completely determined by the time - dependent factor @xmath78 of single - qubit excited - state population . [ tong0 ] shows its time evolutions for the corresponding parameter regimes presented above . we notice that @xmath79 when @xmath73 is located at the band gap , which means that there is some excited - state population in the long - time limit . this phenomenon known as population trapping @xcite is responsible for the suppression of the spontaneous emission of two - level system in pbg reservoir and has been experimentally observed @xcite . such population trapping just manifests the formation of bound states between @xmath53 and @xmath54 @xcite , which has been experimentally verified in @xcite . consequently , @xmath53 and @xmath54 are so correlated in the bound states that the initial entanglement in @xmath33 can not be fully transferred to @xmath37 . the oscillation during the evolution is just the manifestation of the strong non - markovian effect induced by the reservoirs . on the contrary , if @xmath73 is located in the upper band , then @xmath80 and the qubits decay completely to their ground states . in this case the bound states between @xmath53 and @xmath54 are absent and , according to eq . ( [ t15 ] ) , the initial entanglement in @xmath33 is completely transferred to @xmath37 , as clearly shown in eq . ( [ rr ] ) . in addition , in refs . @xcite it was emphasized that esd of @xmath33 is always accompanied with esb of @xmath37 . however , this is not always true . to clarify this , we examine the condition to obtain esd of the qubits and the companying esb of the reservoirs . from eqs . ( [ qq ] ) and ( [ rr ] ) it is obvious that the condition is @xmath81 and @xmath82 at any @xmath83 and @xmath84 , which means @xmath85 when the bound states is absent , @xmath86 , the condition ( [ dd ] ) can be satisfied when @xmath87 . so one can always expect esd of the qubits and the companying esb of the reservoirs in the region @xmath88 , as shown in fig . [ nbst ] and refs . however , when the bound states are available , the situation changes . in particular , when @xmath89 in the full range of time evolution , no region of @xmath90 can make the condition ( [ dd ] ) to be satisfied anymore . for clarification , we present three typical behaviors of the entanglement distribution in fig . [ td ] . in all these cases the bound states are available . [ td](a ) shows the situation where the entanglement is stably distributed among all of the bipartite subsystems . [ td](b ) indicates that the entanglement of @xmath91 shows esb and revival , while the entanglement of @xmath33 does not exhibit esd . [ td](c ) shows another example that while the entanglement of @xmath33 has esd and revival @xcite , the entanglement of @xmath37 does not show esb but remains to be zero . [ td](b , c ) reveal that esd in @xmath33 has no direct relationship with esb in @xmath37 . the above discussion is not dependent of the explicit spectral density of the individual reservoir . to confirm this , we consider the reservoir in free space . the spectral density has the ohmic form @xmath92 , which can be obtained from the free - space dispersion relation @xmath93 . one can verify that the condition for the formation of bound states is : @xmath94 @xcite . in fig . [ osd ] , we plot the results in this situation . the previous results can be recovered when the bound states are absent @xcite . on the contrary , when the bound states are available , a stable entanglement is established among all the bipartite partitions . therefore , we argue that the stable entanglement distribution resulted from the bound states is a general phenomenon in open quantum system when the non - markovian effect is taken into account . in summary , we have studied the entanglement distribution among all the bipartite subsystems of two qubits embedded into two independent amplitude damping reservoirs . it is found that the entanglement can be stably distributed in all the bipartite subsystems , which is much different no matter to the markovian approximate result @xcite or to the decoherenceless double j - c model result @xcite , and an identity about the entanglement in all subsystems is always satisfied . this identity is shown to be independent of any detail of the reservoirs and their coupling to the qubit , which affect only the explicit time evolution behavior and the final distribution . the result is significant to the study of the physical nature of entanglement under decoherence . it implies an active way to protect entanglement from decoherence by modifying the properties of the reservoir via the potential usage of the newly emerged technique , i.e. quantum reservoir engineering @xcite . this work is supported by the fundamental research funds for the central universities under grant no . lzujbky-2010 - 72 , gansu provincial nsf under grant no . 0803rjza095 , the national nsf of china , the program for ncet , and the cqt wbs grant no . r-710 - 000 - 008 - 271 .

we study the entanglement dynamics of two qubits , each of which is embedded into its local amplitude - damping reservoir , and the entanglement distribution among all the bipartite subsystems including qubit - qubit , qubit - reservoir , and reservoir - reservoir . it is found that the entanglement can be stably distributed among all components , which is much different to the result obtained under the born - markovian approximation by c. e. lpez _ et al . _ [ phys . rev . lett . * 101 * , 080503 ( 2008 ) ] , and particularly it also satisfies an identity . our unified treatment includes the previous results as special cases . the result may give help to understand the physical nature of entanglement under decoherence .

arxiv

one of the most important astrophysical phenomena still lacking an explanation is the origin of the celestial gamma - ray bursts ( grb ) . these are powerful flashes of gamma - rays lasting from less than one second to tens of seconds , with isotropic distribution in the sky . they are observed above the terrestrial atmosphere with x gamma ray detectors aboard satellites @xcite . thanks to the bepposax satellite @xcite , afterglow emission at lower wavelengths has been discovered @xcite and we now know that at least long ( @xmath17s ) grb s are at cosmological distances , with measured red shifts up to 4.5 ( see , e.g. , review by djorgovski @xcite and references therein ) . among the possible explanations of these events , which involve huge energy releases ( up to @xmath18 erg , assuming isotropic emission ) , the most likely candidates are the collapse of a very massive star ( hypernova ) and the coalescence of one compact binary system ( see , e.g. , reviews by piran @xcite and mszros @xcite and references therein ) . in both cases emission of gravitational waves ( gw ) is expected to be associated with them ( e.g. ref . @xcite ) . according to several models , the duration of a gw burst is predicted to be of the order of a few milliseconds for a variety of sources , including the coalescing and merging black holes and/or neutron star binaries . therefore gw bursts can be detected by the present resonant detectors , designed to detect gw through the excitation of the quadrupole modes of massive cylinders , resonating at frequencies near 1 khz . at the distances of the grb sources ( @xmath19 gpc ) , the gw burst associated with a total conversion of 1 - 2 solar masses should have amplitude of the order of @xmath20 . the present sensitivity for 1 ms gw pulses of the best gw antennas with signal to noise ratio ( snr ) equal to unity is @xmath21 ( see e.g. ref . @xcite ) , which requires a total conversion of one million solar masses at 1 gpc . however , although detection of a gravitational signal associated with a single grb appears hopeless , detection of a signal associated with the sum of many events could be more realistic . thus we launched a program devoted to studying the presence of correlations between grb events detected with bepposax and the output signals from gravitational antennas nautilus and explorer . searching for correlation between grb and gw signals means dealing with the difference between the emission times for the two types of phenomena . furthermore , there is also the fact to consider that the time difference can vary from burst to burst . in the present analysis we use an algorithm based on cross - correlating the outputs of two gw detectors ( see @xcite ) , thus coping with the problem of the unknown possible time difference between grb and gw bursts , and also of the unmodelled noise . the rome group operates two resonant bar detectors : explorer @xcite , since 1990 , at the cern laboratories , and nautilus @xcite , since 1995 , at the infn laboratories in frascati . 0.1 in . main characteristics of the two detectors . @xmath22 indicates , for each detector , the two resonant frequencies and @xmath23 indicates the bandwidth . the relatively larger bandwidth of explorer is due to an improved readout system . [ cols="^,^,^,^,^,^,^,^ " , ] [ finale ] the agreement between the values of the simulated input signals and the values calculated using eq . ( [ uppere ] ) shows that our model is correct . having presented the experimental method and the model for the averaged correlation at zero delay time @xmath24 , we can infer the values of gw amplitude @xmath25 consistent with the observation . we note that , using eqs . [ uppere ] and [ snrc1 ] , energy @xmath26 is related to the measured cross - correlation @xmath24 by e_0=t_eff ( ) ^1/4 ( ) ^1/2 [ eq : relazione ] hence , the data are summarized by an observed average squared energy @xmath27 , at @xmath28 standard deviation from the expected value in the case of noise alone , as calculated with the aid of eq . ( [ eq : sigmar_e ] ) where we put @xmath29 . the standard deviation , expressed in terms of squared energy , is obtained from eq . ( [ eq : sigmar_e ] ) , in the case @xmath30 , which gives @xmath31 . according to the model discussed above , in the case of gw signals of energy @xmath32 , we expect @xmath33 to be a random number , modeled with a gaussian probability density function around @xmath34 with a standard deviation @xmath35 : f(e_0 ^ 2|e^2 ) , where @xmath32 is the unknown quantity we wish to infer from the observed value of @xmath26 , given in eq . [ eq : relazione ] . this probability inversion is obtained using bayes theorem ( see , e.g. , @xcite for a physics oriented introduction ) : f(e^2|e_0 ^ 2 ) f(e_0 ^ 2|e^2 ) f_(e^2 ) [ eq : bayes ] where @xmath36 is the prior probability density function of observing gw signals of squared energy @xmath34 . in fact , we are eventually interested in inferring the gw s amplitude @xmath25 , related to the energy @xmath32 by eq . ( [ upperh ] ) . therefore we have a similar equation : f(h|e_0 ^ 2 ) f(e_0 ^ 2| h ) f_(h ) [ eq : bayesh ] where @xmath37 is obtained by a transformation of @xmath38 . as prior for @xmath25 we considered a uniform distribution , bounded to non negative values of @xmath25 , obtained from eq . [ eq : bayesh ] , i.e. @xmath39 is a step function @xmath40 . this seems to us a reasonable choice and it is stable , as long as other priors can be conceived which model the positive attitude of reasonable scientists ( see ref . @xcite ) . , but rather the probability per decade of @xmath25 , i.e. researchers may feel equally uncertain about the orders of magnitudes of @xmath25 . this prior is known as jeffreys prior , but , in our case , it produces a divergence for @xmath41 in eq . [ eq : bayesh ] , a direct consequence of the infinite orders of magnitudes which are equally believed . to get a finite result we need to put a cut - off at a given value of @xmath25 . this problem is described in depth , for example , in @xcite and in @xcite . [ foot:1 ] ] the probability density function of @xmath25 is plotted in fig . [ finaleh ] . the highest beliefs are for very small values , while values above @xmath42 are practically ruled out . from fig . [ finaleh ] we obtain an expected value and standard deviation for @xmath25 of @xmath43 and @xmath44 , respectively , which fully account for what is perceived as a null result . in these circumstances , we can provide an upper limit , defined as value @xmath45 , such that there is a given probability for the amplitude of gw s to be below it , i.e. _ 0^h(ul)f(h |e_0 ^ 2)h = p_l , with @xmath46 the chosen probability level . results are plotted in fig.[fighh ] . for example , we can exclude the presence of signals of amplitudes @xmath47 with 95% probability . probabilistic results depend necessarily on the choice of prior probability density function of @xmath25 . for example , those firmly convinced that gw burst intensities should be in the @xmath48 region would never allow a 5% chance to @xmath25 above @xmath49 . therefore , in frontier research particular care has to be used , before stating probabilistic results . the bayesian approach , thanks to the factorization between likelihood and prior , offers natural ways to a present prior - independent result . the simple idea would be just to provide the likelihood for each hypothesis under investigation , in our case @xmath50 . more conveniently , it has been proposed ( @xcite-@xcite ) to publish the likelihood rescaled to the asymptotic limit , where experimental sensitivity is lost completely ; @xmath51 , in our case . indicating with @xmath52 this rescaled likelihood , we have ( h ) = . [ eq : rbur_def ] in statistics jargon , this function gives the bayes factor of all @xmath25 hypotheses with respect to @xmath53 . in intuitive terms , it can be interpreted as a `` relative belief updating ratio '' or a `` probability density function shape distortion function '' , since from eq . ( [ eq : bayesh ] ) we have = . [ eq : bayesf ] in the present case we get , numerically ( h ) = a h 0 , where @xmath54 is the rescaling factor , @xmath55 and @xmath56 the result is given in fig . [ loglinrh ] , where the choice of the log scale for @xmath25 is to remember that there are infinite orders of magnitudes where the value could be located ( and hence the problem discussed in footnote [ foot:1 ] ) . interpretation of fig . [ loglinrh ] , in the light of eqs . ( [ eq : rbur_def])-([eq : bayesf ] ) , is straightforward : up to a fraction of @xmath57 the experimental evidence does not produce any change in our belief , while values much larger than @xmath57 are completely ruled out . the region of transition from @xmath52 from 1 to zero identifies a _ sensitivity bound _ for the experiment . the exact value of this bound is a matter of convention , and could be , for example , at @xmath58 , or @xmath59 . we have @xmath60 and @xmath61 , respectively . note that these bounds have no probabilistic meaning . in any case , the full result should be considered to be the @xmath52 function , which , being proportional to the likelihood , can easily be used to combine results ( for independent datasets the global likelihood is the product of the likelihoods , and proportional constants can be included in the normalization factor ) . note that the result given in terms of scaled likelihood and sensitivity bound can not be misleading . in fact , these results are not probabilistic statements about @xmath25 and no one would imagine they were . on the other hand , confidence limits , which are not probabilistic statements on the quantity of interest , tend to be perceived as such ( see e.g. @xcite and references therein ) . using for the first time a cross - correlation method applied to the data of two gw detectors , explorer and nautilus , new experimental upper limits have been determined for the burst intensity causing correlations of gw s with grb s . analyzing the data over 47 grbs , we exclude the presence of signals of amplitude @xmath62 , with 95% probability , with a time window of @xmath63 . with the time window of @xmath64 s , we improve the previous gw upper limit to about @xmath65 . the result is also given in terms of scaled likelihood and sensitivity bound , which we consider the most complete and unbiased way of providing the experimental information . in a previous paper @xcite we had given more stringent upper limits , but this was under the hypothesis that the gw signals always occur at the same time with respect to the grb arrival time . here , instead , we only require that the time gap between the grb and the gw burst be within a given time window . similar comparison can be made with the auriga / batse result @xcite , where an upper limit `` @xmath66 with c.l . 95% '' is estimated under the assumption that gw s arrive at the grb time within a time window of @xmath3 s. finally , we remark that this method can be applied for any expected delay between grb and gw , with appropriate time shifting of the integration window with respect to the gbr arrival time , according to the prediction of the chosen model . we thank f. campolungo , r. lenci , g. martinelli , e. serrani , r. simonetti and f. tabacchioni for precious technical assistance . we also thank dr . r. elia for her contribution . given two independent detectors , let @xmath67 and @xmath68 be the measured quantities , which are the ( filtered ) data in our case . we introduce the variables @xmath69 and @xmath70 where @xmath71=t_{eff}$ ] and @xmath72=t_{eff}$ ] . we recall that the cross - correlation function is r()= [ crossa ] with summation extended up to the number of independent samples @xmath73 . we calculate @xmath74 . thus @xmath75 . we easily verify that @xmath76=0 $ ] . in the absence of any correlated signal we also have @xmath77=0 $ ] . let us calculate the variance of @xmath78 . we notice that when squaring the numerator of eq.[crossa ] and taking the average , the cross - terms vanish if the @xmath73 data are independent from each other . then , since also @xmath79 and @xmath80 are independent variables , we obtain _ r^2== = [ siga ] the previous considerations still apply to the cumulative cross - correlation @xmath81 , obtained by averaging @xmath82 independent @xmath78 . the final variance for the cross - correlation @xmath81 is _ r^2= [ sigma ] let us now consider an energy signal @xmath83 on both detectors at the same time . the expected value of the cross - correlation @xmath78 at @xmath84 will be positive . in the case of @xmath73 independent data the signal @xmath83 will appear in one datum only and we have : e[r(=0)-r(0)]== [ segnalea ] where we put @xmath85 . we note that , in eq.[segnalea ] , @xmath86=0 $ ] also when a signal is present . p. astone , c. buttiglione , s. frasca , g.v . pallottino and g. pizzella , il nuovo cimento 20,9 ( 1997 ) p.astone et al . `` gravitational astronomy '' pag 189 + ed . d.e.mcclelland and h.a.bachor , world scientific 1991 g. d agostini _ bayesian reasoning in high - energy physics : principles and applications _ cern 99 - 03 ( 1999 ) this report and related references can be found at the url : http://www-zeus.roma1.infn.it/ agostini / prob+stat.html g. dagostini , _ overcoming priors anxiety _ , invited contribution to the monographic issue of the revista de la real academia de ciencias on bayesian methods in the sciences , ed . j.m . bernardo ( 1999 ) physics/9906048 . g. dagostini , proc . xviii international workshop on maximum entropy and bayesian methods , garching ( germany ) , july 1998 , v. dose , w. von der linden , r. fischer , and r. preuss , eds , kluwer academic publishers , dordrecht , 1999 , physics/9811046 .

data obtained during five months of 2001 with the gravitational wave ( gw ) detectors explorer and nautilus were studied in correlation with the gamma ray burst data ( grb ) obtained with the bepposax satellite . during this period bepposax was the only grb satellite in operation , while explorer and nautilus were the only gw detectors in operation . no correlation between the gw data and the grb bursts was found . the analysis , performed over 47 grb s , excludes the presence of signals of amplitude @xmath0 , with 95% probability , if we allow a time delay between gw bursts and grb within @xmath1 s , and @xmath2 , if the time delay is within @xmath3 s. the result is also provided in form of scaled likelihood for unbiased interpretation and easier use for further analysis . _ @xmath4 istituto nazionale di fisica nucleare infn , rome _ + _ @xmath5 university of rome `` tor vergata '' and infn , rome 2 _ + _ @xmath6 ifsi - cnr and infn , rome _ + _ @xmath7 university of laquila and infn , rome 2 _ + _ @xmath8 iess - cnr and infn , rome _ + _ @xmath9 university of rome `` la sapienza '' and infn , rome _ + _ @xmath10 istituto nazionale di fisica nucleare infn , frascati _ + _ @xmath11 university of ferrara and iasf - cnr , bologna _ + _ @xmath12 university of ferrara , ferrara _ + _ @xmath13 university of ferrara , ferrara and ita `` i. calvi '' , finale emilia , modena _ + _ @xmath14 university of rome `` tor vergata '' and infn , frascati _ + _ @xmath15 ifsi - cnr and infn , rome 2 _ + _ @xmath16 ifsi - cnr and infn , frascati _ + +

arxiv

today , with a vast amount of publications being produced in every discipline of scientific research , it can be rather overwhelming to select a good quality work ; that is enriched with original ideas and relevant to scientific community . more often this type of publications are discovered through the citation mechanism . it is believed that an estimate measure for scientific credibility of a paper is the number of citations that it receives , though this should not be taken too literally since some publications may have gone unnoticed or have been forgotten about over time . knowledge of how many times their publications are cited can be seen as good feedback for the authors , which brings about an unspoken demand for the statistical analysis of citation data . one of the impressive empirical studies on citation distribution of scientific publications @xcite showed that the distribution is a power - law form with exponent @xmath0 . the power - law behaviour in this complex system is a consequence of highly cited papers being more likely to acquire further citations . this was identified as a _ preferential attachment _ process in @xcite . the citation distribution of scientific publications is well studied and there exist a number of network models @xcite to mimic its complex structure and empirical results @xcite to confirm predictions . however , they seem to concentrate on the total number of citations without giving information about the issuing publications . the scientific publications belonging to a particular research area do not restrict their references to that discipline only , they form bridges by comparing or confirming findings in other research fields . for instance most _ small world network models _ @xcite presented in statistical mechanics , reference a sociometry article @xcite which presents the studies of milgram on the small world problem . this is the type of process which we will investigate with a simple model that only considers two research areas and referencing within and across each other . the consideration of cross linking also makes the model applicable to _ the web of human sexual contacts _ @xcite , where the interactions between males and females can be thought of as two coupled growing networks . this paper is organized as follows : in the proceeding section the model is defined and analyzed with a rate equation approach @xcite . in the final section discussions and comparisons of findings with the existing data are presented . one can visualize the proposed model with the aid of fig . ( [ coupled ] ) that attempts to illustrate the growth mechanism . we build the model by the following considerations . initially , both networks @xmath1 and @xmath2 contains @xmath3 nodes with no cross - links between the nodes in the networks . at each time step two new nodes with no incoming links , one belonging to network @xmath1 and the other to @xmath2 , are introduced simultaneously . the new node joining to @xmath1 with @xmath4 outgoing links , attaches @xmath5 fraction of its links to pre - existing nodes in @xmath1 and @xmath6 fraction of them to pre - existing nodes in @xmath2 . the similar process takes place when a new node joins to @xmath2 , where the new node has @xmath7 outgoing links from which @xmath8 of them goes to nodes in @xmath2 and the complementary @xmath9 goes to @xmath1 . the attachments to nodes in either networks are preferential and the rate of acquiring a link depends on the number of connections and the initial attractiveness of the pre - existing nodes . we define @xmath10 as the average number of nodes with total @xmath11 number of connections that includes the incoming intra - links @xmath12 and the incoming cross - links @xmath13 in network @xmath1 at time @xmath14 . similarly , @xmath15 is the average number of nodes with @xmath16 connections at time @xmath14 in network @xmath2 . notice that the indices are discriminative and the order in which they are used is important , as they indicate the direction that the links are made . further more we also define @xmath17 and @xmath18 the average number of nodes with @xmath12 and @xmath19 incoming intra - links to @xmath1 and @xmath2 respectively . finally , we also have @xmath20 and @xmath21 to denote the average number of nodes in @xmath1 and @xmath2 with @xmath13 and @xmath22 incoming cross - links . to keep this paper less cumbersome we will only analyse the time evolution of network @xmath1 and apply our results to network @xmath2 . in addition to this , we only need to give the time evolution of @xmath23 , defined as the joint distribution of intra - links and cross - links . using this distribution we can find all other distributions that are mentioned earlier . the time evolution of @xmath23 can be described by a rate equation @xmath24\nonumber\\ & & + p_{ba}m_{b}[(k_{aa}+k_{ba}-1+a)n_{a}(k_{aa},k_{ba}-1,t)\nonumber\\ & & -(k_{aa}+k_{ba}+a)n_{a}(k_{aa},k_{ba},t)]\}+ \delta_{k_{aa}0}\delta_{k_{ba}0}.\end{aligned}\ ] ] the form of the eq . ( [ na ] ) seems very similar to the one used in @xcite . in that model the rate of creating links depends on the out - degree of the issuing nodes and the in - degree of the target nodes . here we are concerned with two different types of in - degrees namely intra- and cross - links of the nodes . on the right hand side of eq . ( [ na ] ) the terms in first square brackets represent the increase in the number of nodes with @xmath11 links when a node with @xmath25 intra - links acquires a new intra - link and if the node already has @xmath11 links this leads to reduction in the number . similarly , for the second square brackets where the number of nodes with @xmath11 links changes due to the incoming cross - links . the final term accounts for the continuous addition of new nodes with no incoming links , each new node could be thought of as the new publication in a particular research discipline . the normalization factor @xmath26 sum of all degrees is defined as @xmath27 we limit ourself to the case of preferential linear attachment rate@xcite @xmath28 shifted by @xmath29 , the initial attractiveness @xcite of nodes in @xmath1 , which ensures that there is a nonzero probability of any node acquiring a link . the nature of @xmath30 lets one to obtain , as @xmath31 @xmath32 where @xmath33 is the average total in - degree in network @xmath1 . ( [ mat ] ) implying that @xmath26 is linear in time . similarly , it is easy to show that @xmath34 is also linear function of time . we use these relations in eq . ( [ na ] ) to obtain the time independent recurrence relation @xmath35n_{a}(k_{aa},k_{ba})\nonumber\\ = p_{aa}m_{a}(k_{aa}+k_{ba}+a-1)n_{a}(k_{aa}-1,k_{ba})\nonumber\\ + p_{ba}m_{b}(k_{aa}+k_{ba}+a-1)n_{a}(k_{aa},k_{ba}-1)\nonumber\\ + ( a+<m_{a}>)\delta_{k_{aa}0}\delta_{k_{ba}0}.\end{aligned}\ ] ] the expression in eq . ( [ arec ] ) does not simplify however , it lets us to obtain the total in - degree distribution @xmath36 writing @xmath37 and since @xmath38 then @xmath39 satisfies @xmath40n_{a}(k_{a})=<m_{a}>(k_{a}+a-1)n_{a}(k_{a}-1 ) \nonumber\\ + ( a+<m_{a}>)\delta_{k_{a}0}.\end{aligned}\ ] ] solving eq . ( [ narec ] ) for @xmath41 yields , @xmath42 with @xmath43 as @xmath44 eq . ( [ nagamma ] ) gives the asymptotic behaviour of the total in - degree distribution in @xmath1 @xmath45 which is a power - law form with an exponent @xmath46 that only depends on the average total in - degree and the initial attractiveness of the nodes . similarly , we can write the total in - degree distribution in network @xmath2 for the asymptotic limit of @xmath47 as @xmath48 again , the exponent depends upon the initial attractiveness @xmath49 of nodes and the average total incoming links @xmath50 . we now move on to analyse @xmath17 , the distribution of the average number of nodes with @xmath12 intra - links in network @xmath1 . in citation network one can think of these links being issued from the same subject class as the receiving nodes and in the case of human sexual contact network , they represent the homosexual interactions . since @xmath51 which can also be written as @xmath52 , a linear function of time . then summing eq . ( [ arec ] ) over all possible values of @xmath13 @xmath53 we get @xmath54n_{aa}(k_{aa})+<m_{a } > \sum_{k_{ba}=0}^{\infty}k_{ba}n_{a}(k_{aa},k_{ba})\nonumber\\ = p_{aa}m_{a}(k_{aa}+a-1)n_{aa}(k_{aa}-1)+p_{aa}m_{a } \sum_{k_{ba}=0}^{\infty}k_{ba}n_{a}(k_{aa}-1,k_{ba})\nonumber\\ + p_{ba}m_{b}(k_{aa}+a)n_{aa}(k_{aa})+p_{ba}m_{b } \sum_{k_{ba}=0}^{\infty}(k_{ba}-1)n_{a}(k_{aa},k_{ba}-1)\nonumber\\ + ( a+<m_{a}>)\delta_{k_{aa}0}.\end{aligned}\ ] ] for large @xmath12 eq . ( [ aarec ] ) reduces to @xmath55n_{aa}(k_{aa})=\nonumber\\ p_{aa}m_{a}(k_{aa}+a-1)n_{aa}(k_{aa}-1)+(a+<m_{a}>)\delta_{k_{aa}0}.\end{aligned}\ ] ] iterating former relation for @xmath56 yields @xmath57 where @xmath58 in the asymptotic limit as @xmath59 eq . ( [ aagamma ] ) has a power - law form @xmath60 that depends upon both @xmath5 and the coupling parameter @xmath61 . similarly , the time independent recurrence relation for @xmath18 has the same form as eq . ( [ aarec ] ) with the only difference being the parameters . therefore we will simply give the power - law distribution @xmath62 where the other coupling parameter @xmath63 is revealed in the exponent . finally , the distribution of average number of nodes with incoming cross - links @xmath20 in @xmath1 can be found by summing over @xmath23 for all its intra - links @xmath64 as before @xmath65 is also linear in time . when the cross links @xmath13 are large enough , then from eq . ( [ arec ] ) we obtain @xmath66 where @xmath67 in the asymptotic limit as @xmath68 the distribution @xmath69 has a power - law form and similarly for the network @xmath2 as @xmath68 @xmath70 unlike the case in intra - links , here the exponents are inversely proportional to the coupled parameters @xmath61 and @xmath63 respectively . for the sake of simplicity , we set the number of outgoing links of the new nodes in either networks to be the same , i.e. @xmath71 . furthermore taking the rate of cross linking to be @xmath72 and the rate of intra linking @xmath73 , consequently we have @xmath74 , and @xmath75 as the coupling parameter . in the weak coupling case , the cross linking is negligibly small i.e. @xmath76 then the power - law exponent of the intra - link distribution is @xmath77 equal to total link distribution @xmath78 . this gives a solution obtained in@xcite and when @xmath79 we recover the exponent @xmath80 , the empirical findings in @xcite . thus , varying @xmath75 in @xmath83 yields any values of @xmath84 between @xmath85 and @xmath86 . on the contrary , the exponent of cross - link distribution @xmath87 decreases from @xmath86 to @xmath88 , as @xmath75 increases from @xmath89 to @xmath90 . taking @xmath79 gives @xmath91 supposing @xmath92 , which seems reasonable for consideration of citation networks , we find that @xmath93 and @xmath94 . the former result coincides with the distribution of connectivities for the electric power grid of southern california @xcite . where the system is small and the local interactions is of importance hence there seems to be some analogy to the intra - linking process . for the latter , as far as we are aware there is none empirical studies present in the published literature . now , consider the web of human sexual contacts@xcite . if we let @xmath1 to represent males and @xmath2 females that is @xmath95 and @xmath96 then @xmath97 are the power - law exponents of the degree distributions of the sexes . where @xmath98 and @xmath49 denote the male and female attractiveness respectively and usually @xmath99 is considered @xcite . by setting @xmath100 , @xmath101 and @xmath81 that is , cross links are predominant then as in @xcite we obtain @xmath102 for males and @xmath103 for females . the exponents @xmath104 and @xmath105 have been observed for the cumulative distributions in empirical study @xcite . the model we studied here seems to have the flexibility to represent variety of complex systems . we would like to thank the china scholarship council , epsrc for their financial support and geoff rodgers for useful discussions .

we introduce and solve a model which considers two coupled networks growing simultaneously . the dynamics of the networks is governed by the new arrival of network elements ( nodes ) making preferential attachments to pre - existing nodes in both networks . the model segregates the links in the networks as intra - links , cross - links and mix - links . the corresponding degree distributions of these links are found to be power - laws with exponents having coupled parameters for intra- and cross - links . in the weak coupling case the model reduces to a simple citation network . as for the strong coupling , it mimics the mechanism of _ the web of human sexual contacts_.

arxiv

the theoretical description of hadron production at large transverse momentum ( @xmath2 ) in either hadronic or nuclear collisions at high energies is traditionally framed in a two - step process that involves first a hard scattering of partons , followed by the fragmentation of the scattered parton to the detected hadron @xcite . the first part is calculable in perturbation qcd , while the second part makes use of fragmentation functions that are determined phenomenologically . such a production mechanism has recently been found to be inadequate for the production of particles at intermediate @xmath2 in heavy - ion collisions @xcite . instead of fragmentation it is the recombination of partons that is shown to be the more appropriate hadronization process , especially when the soft partons are involved . although at extremely high @xmath2 fragmentation is still dominant , it is desirable to have a universal description that can be applied to any @xmath2 , based on the same hadronization scheme . to achieve that goal it is necessary that the fragmentation process can be treated as the result of recombination of shower partons in a jet . the purpose of this paper is to take that first step , namely : to introduce the notion of shower partons and to determine their distributions in order to represent the phenomenological fragmentation functions in terms of recombination . the subject matter of this work is primarily of interest only to high - energy nuclear collisions because hadronization in such processes is always in the environment of soft partons . semi - hard shower partons initiated by a hard parton can either recombine among themselves or recombine with soft partons in the environment . in the former case the fragmentation function is reproduced , and nothing new is achieved . it is in the latter case that a very new component emerges in heavy - ion collisions , one that has escaped theoretical attention thus far . it should be an important hadronization process in the intermediate @xmath2 region . our main objective here is to quantify the properties of shower partons and to illustrate the importance of their recombination with thermal partons . the actual application of the shower parton distributions ( spd ) developed here to heavy - ion collisions will be considered elsewhere @xcite . the concept of shower partrons is not new , since attempts have been made to generate such partons in pqcd processes as far as is permitted by the validity of the procedure . two notable examples of such attempts are the work of marchesini and webber @xcite and geiger @xcite . however , since pqcd can not be used down to the hadronization scale , the branching or cascading processes terminate at the formation of color - singlet pre - hadronic clusters , which can not easily be related to our shower partons and their hadronization . we shall discuss in more detail at the end of secs . iii and iv the similarities and differences in the various approaches . the fragmentation of a parton to a hadron is not a process that can be calculated in pqcd , although the @xmath1 evolution of the fragmentation function ( ff ) is calculable . the ff s are usually parameterized by fitting the data from @xmath3 annihilations @xcite as well as from @xmath4 and @xmath5 collisions @xcite . although the qcd processes of generating a parton shower by gluon radiation and pair creation can not be tracked by perturbative methods down to low virtuality , we can determine the spd s phenomenologically in much the same way that the ff s themselves are , except that we fit the ff s , whereas the ff s are determined by fitting the data . an important difference is that both the shower partons and their distributions are defined in the context of the recombination model , which is the key link between the shower partons ( inside the black box called ff ) and the observed hadron ( outside the black box ) . in the recombination model the generic formula for a hadronization process is @xcite @xmath6 where @xmath7 is the joint distribution of a quark @xmath8 at momentum fraction @xmath9 and an antiquark @xmath10 at @xmath11 , and @xmath12 is the recombination function ( rf ) for the formation of a meson at @xmath13 . we have written the lhs of eq . ( [ 1 ] ) as @xmath14 , the invariant ff , but the rhs would have the same form if the equation were written for the inclusive distribution , @xmath15 , of a meson produced in a collisional process . in the former case of fragmentation , @xmath16 refers to the shower partons initiated by a hard parton . in the latter case of inclusive production , @xmath16 refers to the @xmath8 and @xmath10 that are produced by the collision and are to recombine in forming the detected meson . the equations for the two cases are similar because the physics of recombination is the same . in either case the major task is in the determination of the distribution @xmath16 . we now focus on the fragmentation problem and regard eq . ( [ 1 ] ) as the basis of the recombination model for fragmentation . the lhs is the ff , known from the parameterization that fits the data . the rhs has the rf that is known from previous studies of the recombination model @xcite and will be specified in the next section . thus it is possible to determine the properties of @xmath16 from eq.([1 ] ) . to facilitate that determination we shall assume that @xmath16 is factorizable except for kinematic constraints , i.e. , in schematic form we write it as @xmath17 where @xmath18 denotes the distribution of shower parton @xmath8 with momentum fraction @xmath9 in a shower initiated by a hard parton @xmath19 . the exact form with proper kinematic constraints will be described in detail in the next section . here we remark on the general implications of eqs . ( [ 1 ] ) and ( [ 2 ] ) . the important point to emphasize is that we are introducing the notion of shower partons and their momentum distributions @xmath20 . the significance of the spd is not to be found in problems that involve only the collisions of leptons and hadrons , for which the fragmentation of partons is known to be an adequate approach , and the recombination of shower partons merely reproduces what is already known . the knowledge about the spd becomes crucial when the shower partons recombine with other partons that are not in the jet but are in the ambient environment . we shall illustrate this important point later . it should be recognized that the spd that we shall determine through the use of eqs . ( [ 1 ] ) and ( [ 2 ] ) depends on the specific form of @xmath12 , which in turn depends on the wave function of the meson produced . it would be inconsistent to use our @xmath21 given below in conjunction with some approximation of the rf that differs significantly from our @xmath22 . the recombination of two shower partons must recover the ff from which the spd s are obtained . finally , we remark that @xmath21 should in principle depend on @xmath1 at which the @xmath23 is used for its determination , since @xmath1 evolution affects both . it is outside the scope of this paper to treat the @xmath1 dependence of @xmath21 . our aim here is to show how @xmath21 can be determined phenomenologically , and how it can be applied , when @xmath1 is fixed . the same method can be used to determine @xmath21 at other values of @xmath1 . in practice , the @xmath1 dependence of @xmath21 is not as important as the inclusion of the role of the shower partons in the first place at any reasonably approximate @xmath1 in heavy - ion collisions where hard partons are produced in a range of transverse momentum . in order to solve eqs . ( [ 1 ] ) and ( [ 2 ] ) for @xmath21 , we first point out that there are various @xmath24 functions corresponding to various fragmentation processes . we shall select five of them , from which we can determine five spd s . three of them form a closed set that involves no strange quarks or mesons . let us start with those three . consider the light quarks @xmath25 , @xmath26 , @xmath27 , @xmath28 , and gluon @xmath29 . they can all fragment into pions . to reduce them to three essential ff s , we consider the three basic types @xmath30 , @xmath31 and @xmath32 , that correspond to valence , sea and gluon fragmentation , respectively . if the fragmenting quark has the same flavor as that of a valence quark in @xmath33 , then the valence part of the fragmentation is described by @xmath30 , e.g. , @xmath34 , @xmath35 , @xmath36 , the subscript @xmath37 referring to the valence component . all other cases of quark fragmentation are described by @xmath31 , e.g. , @xmath38 , @xmath39 , @xmath40 . if the initiating parton is a gluon , then we have @xmath32 for any state of @xmath33 . those ff s are given by ref.@xcite in parametric form . we shall use them even though they are older than the more recent ones @xcite , which do not give the @xmath30 and @xmath31 explicitly . our emphasis here is not on accuracy , but on the feasibility of extracting the spd s from the ff s of the type discussed above . we shall determine @xmath21 from the bkk parameterization @xcite with @xmath1 fixed at 100 gev@xmath41 and demonstrate that the use of shower partons is important in heavy - ion collisions . for the spd s we use the notation @xmath42 and @xmath43 for valence and sea - quark distributions , respectively , in a shower initiated by a quark or antiquark , and @xmath44 for any light quark distribution in a gluon - initiated shower . that is , for example , @xmath42 @xmath45 , @xmath46 , @xmath47 . it should be recognized that @xmath43 also describes the sea quarks of the same flavor , such as @xmath48 , so that the overall distribution of shower quark that has the same flavor as the initiating quark ( e.g. @xmath49 ) is given by @xmath50 it is evident from the above discussion that there is a closed relationship that is independent of other unknowns . it follows from eq . ( [ 1 ] ) when restricted to sea - quark fragmentation : @xmath51 the sea - spd @xmath52 can be determined from this equation alone . in eq.([4 ] ) we have exhibited the argument of the second @xmath43 function that reflects the momentum constraint , i.e. , if one shower parton has momentum fraction @xmath9 , then the momentum fraction of the other recombining shower parton can not exceed @xmath53 , and can only be a fraction of the balance @xmath54 . symmetrization of @xmath9 and @xmath11 is automatic by virtue of the invariance of @xmath55 under the exchange of @xmath9 and @xmath11 . after @xmath52 is determined from eq . ( [ 4 ] ) , we next can obtain @xmath42 from @xmath56 where the curly brackets define the symmetrization of the leading parton momentum @xmath57 \quad . \label{6}\end{aligned}\ ] ] finally , we have the closed equation for the gluon - initiated shower @xmath58 in this non - strange sector we have 3 spd s ( @xmath43 , @xmath42 and @xmath44 ) to be determined from the 3 phenomenological ff s ( @xmath31 , @xmath30 and @xmath59 ) . in extending the consideration to the strange sector , we must make use of @xmath43 and @xmath44 determined in the above set and two new ff s @xmath60 and @xmath61 that describe the fragmentation of a non - strange quark and gluon , respectively , to a kaon . that is , we have @xmath62 @xmath63 where @xmath64 and @xmath65 are two additional spd s specifying the strange quark distributions in showers initiated by non - strange and gluon partons , respectively . @xmath66 is the rf for kaon . to complete the description of the integral equations , we now specify the rf s . they depend on the square of the wave functions of the mesons , @xmath33 and @xmath67 , whose structures in momentum space have been quantified in the valon model @xcite . unlike the case of the proton , whose structure is well studied by deep inelastic scattering so that the valon distribution can be obtained from the parton distribution functions @xcite , the rf for the pion relies on the parton distribution of the pion probed by drell - yan process @xcite . the derivation of the rf s for both @xmath33 and @xmath67 is given in @xcite ; they are @xmath68 @xmath69 the @xmath70 functions guarantee the momentum conservation of the recombining quarks and antiquarks , which are dressed and become the valons of the produced hadrons . since the recombination process involves the quarks and antiquarks , one may question the fate of the gluons . this problem has been treated in the formulation of the recombination model @xcite , where gluons are converted to quark - antiquark pairs in the sea before hadronization . that is , the sea is saturated by the conversion to carry all the momentum , save the valence parton . such a procedure has been shown to give the correct normalization of the inclusive cross section of hadronic collisions @xcite . in the present problem of parton fragmentation we implement the recombination process in the same framework , although gluon conversion is done only implicitly . what is explicit is that the gluon degree of freedom is not included in the list of shower partons . it means that in the equations for @xmath30 , @xmath31 , and @xmath32 ( and likewise in the strange sector ) only @xmath42 , @xmath43 and @xmath44 appear ; they are the spd s of quarks and antiquarks that are to recombine . those quarks and antiquarks must include the converted sea , since they are responsible for reproducing the ff s through eqs . ( [ 4 ] ) , ( [ 5 ] ) and ( [ 7 ] ) without gluons . thus the shower partons whose momentum distributions we calculate are defined by those equations that have no gluon component for recombination , and would not be the same as what one would conceptually get ( if it were possible ) in a pqcd calculation that inevitably has both quarks and gluons . it should be noted that our procedure of converting gluons to @xmath71 pairs is essentially the same as what is done in @xcite , whose branching processes terminate at the threshold of the non - perturbative regime . in that approach nearby quarks and antiquarks that are the products of the conversion from different gluons form color - singlet clusters of various invariant masses that subsequently decay ( or fragment as in strings ) sequentially through resonances to the lowest lying hadron states @xcite . similar but not identical approach is taken in @xcite , where gluons are not directly converted to @xmath71 pairs , but are either absorbed or annihilated by @xmath72 born - diagram processes . we now proceed to solve the integral equations for the five ff s , which are known from ref . those equations relate them to the five unknown spd s : @xmath42 , @xmath43 , @xmath44 , @xmath64 and @xmath65 . if those equations were algebraic , we obviously could solve them for the unknowns . being integral equations , they can nevertheless be `` solved '' by a fitting procedure that should not be regarded as being unsatisfactory for lack of mathematical rigor , since the ff s themselves are obtained by fitting the experimental data in some similar manner . indeed , the ff s in the next - to - leading order are given in parameterized forms @xcite @xmath73 where the parameters for @xmath74 gev@xmath41 are given in table i for @xmath75 and @xmath76 . .parameters in eq . ( [ 12 ] ) for @xmath77gev@xmath41 . [ cols="<,^,^,^,^",options="header " , ] it is evident from fig . 1 that all the fits are very good , except in the low @xmath13 region of @xmath78 . in the latter case we are constrained by the condition @xmath79 that is imposed by the requirement that there can be only one valence quark in the shower partons . however , the fit for @xmath80 is excellent , and that is the important region for the determination of @xmath81 . in application to @xmath82 , say , the @xmath25 quark in the shower must have both valence and sea quarks so the shower distribution for the @xmath25 quark is always the sum : @xmath83 . since @xmath52 is large at small @xmath84 , and is accurately determined , the net result for @xmath85 should be quite satisfactory . it is remarkable how well the ff s in fig . 1 are reproduced in the recombination model . the corresponding spd s that make possible the good fit are shown in fig . they have very reasonable properties , namely : ( a ) valence quark is harder ( b ) sea quarks are softer , ( c ) gluon jet has higher density of shower partons , and ( d ) the density of produced @xmath86 quarks is lower than that of the light quarks . for valence+sea quark ( solid line ) , sea quark ( dash - dot line ) and thermal partons ( dashed line).,scaledwidth=45.0% ] it is appropriate at this point to relate our approach to those of marchesini - webber @xcite and geiger @xcite , which are serious attempts to incorporate the qcd dynamics in their description of the branching and collision processes . the former is done in the momentum space only , whereas the latter is formulated in space - time as well as in momentum space . the parton cascade model of geiger is a very ambitious program that treats a large variety of processes ranging from @xmath3 annihilation @xcite to deep inelastic scattering @xcite to hadronic and nuclear collisions @xcite . the evolution of partons is tracked by use of relativistic transport equations with gain and loss terms . cluster formation takes into account the invariant distance between near - neighbor partons . cluster decay makes use of the hagedorn spectrum and the particle data table . because of the complexity of the problems both qcd models are implemented by monte carlo codes the predictive power of the models is exhibited as numerical outputs that can not easily be adapted for comparison with our results on the spd s . our approach makes no attempt to treat the qcd dynamics ; however , the spd s obtained are guaranteed to reproduce the ff s on the one hand , and are conveniently parameterized for use in other context that goes beyond fragmentation , as we shall show in the next section . from the way the color - singlet clusters are treated in the qcd models , it is clear that our shower partons do not correspond to the partons of those models at the end of their evolution processes , except in the special case when the cluster consists of only one particle . in our approach the non - perturbative part of how the shower partons dress themselves and recombine to form hadrons with the proper momentum - fraction distributions is contained in the rf s . such shower partons that are ready to hadronize are sufficiently far from other shower partons as to be independent from them . in general , they can not be identified with the @xmath8 and @xmath87 that form the color - singlet clusters in the qcd models , but are more closely related to the constituents of the final hadrons , as in the case of quarkonium formation @xcite . the distribution of those constituents in a hard - parton shower can not be displayed in the qcd models , but are determined by us by solving eqs . ( [ 4 ] ) and ( [ 9 ] ) as we have stated in the introduction , the purpose of determining the spd s is for their application to problems where the ff s are insufficient to describe the physics involved . we consider in this section two such problems as illustrations of the usefulness of the spd s . the first is when a hard parton is produced in the environment of thermal partons , as in heavy - ion collisions . the second is the determination of two - pion distribution in a jet . let us suppose that a @xmath25 quark is produced at @xmath88 gev / c in a background of thermal partons whose invariant @xmath89 distribution is @xmath90 let the parameters @xmath91 and @xmath92 be chosen to correspond to a typical situation in au+au collisions at @xmath93 gev @xcite @xmath94 the high-@xmath89 @xmath25 quark generates a shower of partons with various flavors . consider specifically @xmath25 and @xmath28 in that shower . the valence quark distribution is given by @xmath95 , while the @xmath28 sea - quark distribution ( including the ones converted from the gluons ) is given by @xmath96 . in fig . 3 we plot @xmath97 for ( a ) @xmath25 quark ( valence and sea ) in solid line , ( b ) @xmath28 sea antiquark in dash - dot line , and ( c ) @xmath28 thermal antiquark in dashed line . they correspond to @xmath98 ( invariant distributions @xmath99 , @xmath43 , and @xmath100 , respectively ) , in which @xmath101 and @xmath96 are evaluated at @xmath102 , with @xmath103 gev / c . note that the thermal distribution is higher than the shower parton distributions for @xmath104 gev / c . that makes a crucial difference in the recombination of those partons . such a thermal distribution is absent in @xmath105 collisions , whose soft partons are at least two orders of magnitude lower . in @xmath3 annihilation there are , of course , no soft partons at all . in @xmath2 arising from thermal - shower recombination ( solid line ) and shower - shower recombination , i.e. fragmentation ( dash - dot line).,scaledwidth=45.0% ] we now calculate the production of @xmath106 from the assemblage of @xmath25 and @xmath28 partons . the thermal - shower ( @xmath107 ) recombination gives rise to @xmath108 where eq . ( [ 10 ] ) has been used in an equation such as eq.([1 ] ) for @xmath109 , but expressed for @xmath110 . using eqs . ( [ 3 ] ) , ( [ 15 ] ) and the parametrizations given in table ii , the integral in eq . ( [ 17 ] ) can readily be evaluated . the result is shown by the solid line in fig . 4 . it is to be compared with the @xmath2 distribution from the fragmentation of the @xmath25 quark to @xmath106 , which is @xmath111 since this is just retracing the path in which we obtained @xmath67 and @xmath43 from the @xmath112 function in the first place , eq . ( [ 18 ] ) can more directly be identified with @xmath113 . \label{19}\end{aligned}\ ] ] the result is shown by the dash - dot line in fig . evidently , the contribution from the thermal - shower recombination is much more important than that from fragmentation in the range of @xmath2 shown . despite the fact that @xmath114 is lower than @xmath115 for @xmath116 gev / c , its dominance at @xmath117 gev / c is enough to result in the @xmath118 recombination to dominate over the @xmath119 recombination for all @xmath120 gev / c . this example demonstrates the necessity of knowing the spd s in a jet , since @xmath101 is used in eq . ( [ 17 ] ) . if @xmath121 recombination is the only important contribution as in @xmath105 collisions , then fragmentation as in eq . ( [ 19 ] ) is all that is needed , and the search for spd s plays no crucial role . in realistic problems the hard - parton momentum @xmath122 has to be integrated over the weight of the jet cross section . however , for our illustrative purpose here , that is beside the point . our next example is the study of the dihadron distribution in a jet . we need only carry out the investigation here for a jet in vacuum , since the replacement of a shower parton by a thermal parton for a jet in a medium is trivial , having seen how that is done in the replacement of eq . ( [ 18 ] ) by ( [ 17 ] ) in the case of the single - particle distribution . consider the joint distribution of two @xmath106 in a jet initiated by a hard @xmath25 quark . as we shall work in the momentum fraction variables , the value of the momentum of the initiating @xmath25 quark is irrelevant , except that it should be high . let @xmath123 and @xmath124 denote the momentum fractions of the two @xmath106 , and @xmath125 denotes that of the @xmath19th parton , @xmath126 . then , since only one @xmath25 quark can be valence , the other three quarks being in the sea , we have one @xmath67 , three @xmath43 , and two @xmath22 functions . combinatorial complications arise when we impose the condition that @xmath127 for @xmath128 . there are two methods to keep the accounting of the different orderings of the four @xmath125 . _ method 1 . _ let one ordering be @xmath129 there are 4 ! ways to rearrange the four @xmath125 in all orders . however , they are to be convoluted with @xmath130 , which is symmetric in @xmath131 , and similarly with @xmath132 . thus there are @xmath133 independent terms . since @xmath67 can appear at any one of the four positions in eq . ( [ 20 ] ) , we have altogether 24 terms . thus we have @xmath134 r_{\pi}(x_1,x_2,x_1 ) r_{\pi}(x_3,x_4,x_2 ) , \label{21}\end{aligned}\ ] ] where @xmath135 symbolizes the permutation of all @xmath125 and summing over all four positions of @xmath67 , but eliminating redundant terms that are symmetric under the interchanges of @xmath131 and @xmath136 . _ method 2 . _ let us fix the ordering in eq . ( [ 20 ] ) but permute the contributing @xmath125 to @xmath123 and @xmath124 . there are six arrangements of @xmath125 and @xmath137 in @xmath138 , while counting in @xmath139 and @xmath140 is unnecessary . let us denote the summation over them by @xmath141 . thus we have @xmath142 \left [ { 1 \over 6}\sum_q r_{\pi}(x_i , x_j , x_1 ) r_{\pi}(x_{i'},x_{j'},x_2)\right ] \label{22}\end{aligned}\ ] ] where @xmath143 denotes summing over the four positions of @xmath67 . equation ( [ 22 ] ) is equivalent to ( [ 21 ] ) . it should be noted that not all terms in these equations can be expressed in the form factorizable ff s . one example that can is @xmath144 r_{\pi}(x_1,x_2,x_1)\nonumber\\ & & \times l\left ( { x_3 \over 1-x_1-x_2 } \right)l\left ( { x_4 \over 1-x_1-x_2-x_3 } \right)r_{\pi}(x_3,x_4,x_2)\nonumber\\ & & = d^{\pi^+}_u(x_1 ) d^{\pi^+}_s\left(x_2/(1-x_1)\right ) .\label{23}\end{aligned}\ ] ] because of the presence of terms that can not be written in factorizable form , the two - particle distribution can not be adequately represented by the ff s only . correlated distribution in a @xmath25-quark initiated jet.,scaledwidth=45.0% ] using the spd s obtained in the previous section , we get the results shown in fig . 5 , which exhibits the @xmath124 distribution for four fixed values of @xmath123 . this type of correlation in parton fragmentation has never been calculated before . although the shapes of the @xmath124 distributions look similar in the log scale in fig . 5 , there is significant attenuation as @xmath145 for each value of @xmath123 . thus the effective slope becomes steeper for larger @xmath123 . recent experiments at rhic have begun to measure the distribution of particles associated with triggers restricted to a small interval . the extension of our calculation here to such problems will need the input of jet cross sections for all hard partons in heavy - ion collisions and the participation of thermal partons in the recombination . here we only demonstrate the utility of the spd s in the study of dihadron correlation . we have described the fragmentation process in the framework of recombination . the shower parton distributions obtained are shown to be useful in problems where the knowledge of the fragmentation functions alone is not sufficient to provide answers to questions concerning the interaction between a jet and its surrounding medium or between particles within a jet . such questions arise mainly in nuclear collisions at high energies . in our view the basic hadronization process is recombination , even for fragmentation in vacuum . since the recombination process can only be formulated in the framework of a model , the shower parton distributions obtained are indeed model dependent . that is a price that must be paid for the study of hadrons produced at intermediate @xmath2 where the interaction between soft and semi - hard partons can not be ignored , and where perturbative qcd is not reliable . once recombination is adopted for treating hadronization in that @xmath2 range , the extension to higher @xmath2 can remain in the recombination framework , since the fragmentation process is recovered by the recombination of two shower partons . for hadron production in heavy - ion collisions at super high energies , such as at lhc , then the high density of hard partons produced will require the consideration of recombination of hard partons from overlapping jets . thus it is sensible to remain in the recombination mode for all @xmath2 . we have shown in this paper how the spd s can be determined from the ff s . although we have determined the spd s at only one value of @xmath1 for the ff s , it is clear that the same procedure can be followed for other value of @xmath1 . the formal description of how the @xmath1 dependences of the ff s can be transferred to the @xmath1 dependences of the spd s is a problem that is worth dedicated attention . while the numerical accuracy of the spd s obtained here can still be improved , especially at lower @xmath1 , for the purpose of phenomenological applications the availability of the parametrizations given in table ii is far more important than not taking into account at all the shower partons and their interactions with the medium in the environment . the @xmath1 evolution of the spd s may have to undergo a long process of investigatory evolution of its own just as what has happened to the ff s . that can proceed in parallel to the rich phenomenology that can now be pursued in the application of the role of shower partons to heavy - ion collisions . we are grateful to s. kretzer for a helpful communication . this work was supported , in part , by the u. s. department of energy under grant no . de - fg03 - 96er40972 and by the ministry of education of china under grant no .

we develop the notion of shower partons and determine their distributions in jets in the framework of the recombination model . the shower parton distributions obtained render a good fit of the fragmentation functions . we then illustrate the usefulness of the distributions in a problem where a jet is produced in the environment of thermal partons as in heavy - ion collisions . the recombination of shower and thermal partons is shown to be more important than fragmentation . application to the study of two - particle correlation in a jet is also carried out . @xmath0 2@xmath1

arxiv

we are interested in the following nonconvex semidefinite programming problem : @xmath1 where @xmath2 is convex , @xmath3 is a nonempty , closed convex set in @xmath4 and @xmath5 ( @xmath6 ) are nonconvex matrix - valued mappings and smooth . the notation @xmath7 means that @xmath8 is a symmetric negative semidefinite matrix . optimization problems involving matrix - valued mapping inequality constraints have large number of applications in static output feedback controller design and topology optimization , see , e.g. @xcite . especially , optimization problems with bilinear matrix inequality ( bmi ) constraints have been known to be nonconvex and np - hard @xcite . many attempts have been done to solve these problems by employing convex semidefinite programming ( in particular , optimization with linear matrix inequality ( lmi ) constraints ) techniques @xcite . the methods developed in those papers are based on augmented lagrangian functions , generalized sequential semidefinite programming and alternating directions . recently , we proposed a new method based on convex - concave decomposition of the bmi constraints and linearization technique @xcite . the method exploits the convex substructure of the problems . it was shown that this method can be applied to solve many problems arising in static output feedback control including spectral abscissa , @xmath9 , @xmath10 and mixed @xmath11 synthesis problems . in this paper , we follow the same line of the work in @xcite to develop a new local optimization method for solving the nonconvex semidefinite programming problem . the main idea is to approximate the feasible set of the nonconvex problem by a sequence of inner positive semidefinite convex approximation sets . this method can be considered as a generalization of the ones in @xcite . 0.1 cm _ contribution . _ the contribution of this paper can be summarized as follows : * we generalize the inner convex approximation method in @xcite from scalar optimization to nonlinear semidefinite programming . moreover , the algorithm is modified by using a _ regularization technique _ to ensure strict descent . the advantages of this algorithm are that it is _ very simple to implement _ by employing available standard semidefinite programming software tools and _ no globalization strategy _ such as a line - search procedure is needed . * we prove the convergence of the algorithm to a stationary point under mild conditions . * we provide two particular ways to form an overestimate for bilinear matrix - valued mappings and then show many applications in static output feedback . 0.1 cm _ outline . _ the next section recalls some definitions , notation and properties of matrix operators and defines an inner convex approximation of a bmi constraint . section [ sec : alg_and_conv ] proposes the main algorithm and investigates its convergence properties . section [ sec : app ] shows the applications in static output feedback control and numerical tests . some concluding remarks are given in the last section . in this section , after given an overview on concepts and definitions related to matrix operators , we provide a definition of inner positive semidefinite convex approximation of a nonconvex set . let @xmath12 be the set of symmetric matrices of size @xmath13 , @xmath14 , and resp . , @xmath15 be the set of symmetric positive semidefinite , resp . , positive definite matrices . for given matrices @xmath16 and @xmath17 in @xmath12 , the relation @xmath18 ( resp . , @xmath19 ) means that @xmath20 ( resp . , @xmath21 ) and @xmath22 ( resp . , @xmath23 ) is @xmath24 ( resp . , @xmath25 ) . the quantity @xmath26 is an inner product of two matrices @xmath16 and @xmath17 defined on @xmath12 , where @xmath27 is the trace of matrix @xmath28 . for a given symmetric matrix @xmath16 , @xmath29 denotes the smallest eigenvalue of @xmath16 . [ de : psd_convex]@xcite a matrix - valued mapping @xmath30 is said to be positive semidefinite convex ( _ psd - convex _ ) on a convex subset @xmath31 if for all @xmath32 $ ] and @xmath33 , one has @xmath34 if holds for @xmath35 instead of @xmath36 for @xmath37 then @xmath38 is said to be _ strictly psd - convex _ on @xmath39 . in the opposite case , @xmath38 is said to be _ psd - nonconvex_. alternatively , if we replace @xmath36 in by @xmath40 then @xmath38 is said to be psd - concave on @xmath39 . it is obvious that any convex function @xmath2 is psd - convex with @xmath41 . a function @xmath42 is said to be _ strongly convex _ with parameter @xmath43 if @xmath44 is convex . the notation @xmath45 denotes the subdifferential of a convex function @xmath46 . for a given convex set @xmath39 , @xmath47 if @xmath48 and @xmath49 if @xmath50 denotes the normal cone of @xmath39 at @xmath51 . the derivative of a matrix - valued mapping @xmath38 at @xmath51 is a linear mapping @xmath52 from @xmath4 to @xmath53 which is defined by @xmath54 for a given convex set @xmath55 , the matrix - valued mapping @xmath56 is said to be differentiable on a subset @xmath16 if its derivative @xmath57 exists at every @xmath58 . the definitions of the second order derivatives of matrix - valued mappings can be found , e.g. , in @xcite . let @xmath59 be a linear mapping defined as @xmath60 , where @xmath61 for @xmath62 . the adjoint operator of @xmath8 , @xmath63 , is defined as @xmath64 for any @xmath65 . finally , for simplicity of discussion , throughout this paper , we assume that all the functions and matrix - valued mappings are _ twice differentiable _ on their domain . let us first describe the idea of the inner convex approximation for the scalar case . let @xmath42 be a continuous nonconvex function . a convex function @xmath66 depending on a parameter @xmath67 is called a convex overestimate of @xmath68 w.r.t . the parameterization @xmath69 if @xmath70 and @xmath71 for all @xmath72 . let us consider two examples . 0.1 cm _ example 1 . _ let @xmath46 be a continuously differentiable function and its gradient @xmath73 is lipschitz continuous with a lipschitz constant @xmath74 , i.e. @xmath75 for all @xmath76 . then , it is well - known that @xmath77 . therefore , for any @xmath78 we have @xmath79 with @xmath80 . moreover , @xmath81 for any @xmath51 . we conclude that @xmath82 is a convex overestimate of @xmath46 w.r.t the parameterization @xmath83 . now , if we fix @xmath84 and find a point @xmath85 such that @xmath86 then @xmath87 . consequently if the set @xmath88 is nonempty , we can find a point @xmath85 such that @xmath86 . the convex set @xmath89 is called an inner convex approximation of @xmath90 . 0.1 cm _ example 2 . _ @xcite we consider the function @xmath91 in @xmath92 . the function @xmath93 is a convex overestimate of @xmath46 w.r.t . the parameterization @xmath94 provided that @xmath95 . this example shows that the mapping @xmath96 is not always identity . let us generalize the convex overestimate concept to matrix - valued mappings . [ def : over_relaxation ] let us consider a psd - nonconvex matrix mapping @xmath97 . a psd - convex matrix mapping @xmath98 is said to be a psd - convex overestimate of @xmath38 w.r.t . the parameterization @xmath69 if @xmath99 and @xmath100 for all @xmath76 and @xmath101 in @xmath102 . let us provide two important examples that satisfy definition [ def : over_relaxation ] . _ example 3 . _ let @xmath103 be a bilinear form with @xmath104 , @xmath105 and @xmath106 arbitrarily , where @xmath16 and @xmath17 are two @xmath107 matrices . we consider the parametric quadratic form : @xmath108 one can show that @xmath109 is a psd - convex overestimate of @xmath110 w.r.t . the parameterization @xmath111 . indeed , it is obvious that @xmath112 . we only prove the second condition in definition [ def : over_relaxation ] . we consider the expression @xmath113 . by rearranging this expression , we can easily show that @xmath114 . now , since @xmath104 , by @xcite , we can write : @xmath115 note that @xmath116 . therefore , we have @xmath117 for all @xmath118 and @xmath119 . _ example 4 . _ let us consider a psd - noncovex matrix - valued mapping @xmath120 , where @xmath121 and @xmath122 are two psd - convex matrix - valued mappings @xcite . now , let @xmath122 be differentiable and @xmath123 be the linearization of @xmath122 at @xmath124 . we define @xmath125 . it is not difficult to show that @xmath126 is a psd - convex overestimate of @xmath127 w.r.t . the parametrization @xmath128 . [ re : nonunique_of_bmi_app ] _ example 3 _ shows that the `` lipschitz coefficient '' of the approximating function is @xmath129 . moreover , as indicated by _ examples _ 3 and 4 , the psd - convex overestimate of a bilinear form is not unique . in practice , it is important to find appropriate psd - convex overestimates for bilinear forms to make the algorithm perform efficiently . note that the psd - convex overestimate @xmath130 of @xmath131 in _ example 3 _ may be less conservative than the convex - concave decomposition in @xcite since all the terms in @xmath130 are related to @xmath132 and @xmath133 rather than @xmath16 and @xmath17 . let us recall the nonconvex semidefinite programming problem . we denote by @xmath134 the feasible set of and @xmath135 the relative interior of @xmath136 , where @xmath137 is the relative interior of @xmath3 . first , we need the following fundamental assumption . [ as : a1 ] the set of interior points @xmath138 of @xmath136 is nonempty . then , we can write the generalized kkt system of as follows : @xmath139 any point @xmath140 with @xmath141 is called a _ kkt point _ of , where @xmath142 is called a _ stationary point _ and @xmath143 is called the corresponding lagrange multiplier . the main step of the algorithm is to solve a convex semidefinite programming problem formed at the iteration @xmath144 by using inner psd - convex approximations . this problem is defined as follows : @xmath145 here , @xmath146 is given and the second term in the objective function is referred to as a regularization term ; @xmath147 is the parameterization of the convex overestimate @xmath148 of @xmath149 . let us define by @xmath150 the solution mapping of [ eq : convx_subprob ] depending on the parameters @xmath151 . note that the problem [ eq : convx_subprob ] is convex , @xmath152 is multivalued and convex . the feasible set of [ eq : convx_subprob ] is written as : @xmath153 the algorithm for solving starts from an initial point @xmath154 and generates a sequence @xmath155 by solving a sequence of convex semidefinite programming subproblems [ eq : convx_subprob ] approximated at @xmath156 . more precisely , it is presented in detail as follows . [ alg : a1 ] * initialization . * determine an initial point @xmath157 . compute @xmath158 for @xmath6 . choose a regularization matrix @xmath159 . set @xmath160 . * iteration @xmath161 ( @xmath162 ) * perform the following steps : * _ step 1 . _ for given @xmath156 , if a given criterion is satisfied then terminate . * _ solve the convex semidefinite program [ eq : convx_subprob ] to obtain a solution @xmath163 and the corresponding lagrange multiplier @xmath164 . * _ update @xmath165 , the regularization matrix @xmath166 ( if necessary ) . increase @xmath161 by @xmath167 and go back to step 1 . * * the core step of algorithm [ alg : a1 ] is step 2 where a general convex semidefinite program needs to be solved . in practice , this can be done by either implementing a particular method that exploits problem structures or relying on standard semidefinite programming software tools . note that the regularization matrix @xmath168 can be fixed at @xmath169 , where @xmath43 is sufficiently small and @xmath170 is the identity matrix . since algorithm [ alg : a1 ] generates a feasible sequence @xmath155 to the original problem and this sequence is strictly descent w.r.t . the objective function @xmath46 , _ no globalization strategy _ such as line - search or trust - region is needed . we first show some properties of the feasible set @xmath171 defined by . for notational simplicity , we use the notation @xmath172 . [ le : feasible_set ] let @xmath173 be a sequence generated by algorithm [ alg : a1 ] . then : * @xmath174 the feasible set @xmath175 for all @xmath176 . * @xmath177 it is a feasible sequence , i.e. @xmath178 . * @xmath179 @xmath180 . * @xmath181 for any @xmath182 , it holds that : @xmath183 where @xmath184 is the strong convexity parameter of @xmath46 . for a given @xmath156 , we have @xmath185 and @xmath186 for @xmath6 . thus if @xmath187 then @xmath188 , the statement a ) holds . consequently , the sequence @xmath189 is feasible to which is indeed the statement b ) . since @xmath163 is a solution of [ eq : convx_subprob ] , it shows that @xmath190 . now , we have to show it belongs to @xmath191 . indeed , since @xmath192 by definition [ def : over_relaxation ] for all @xmath6 , we conclude @xmath193 . the statement c ) is proved . finally , we prove d ) . since @xmath163 is the optimal solution of [ eq : convx_subprob ] , we have @xmath194 for all @xmath187 . however , we have @xmath195 due to c ) . by substituting @xmath196 in the previous inequality we obtain the estimate d ) . now , we denote by @xmath197 the lower level set of the objective function . let us assume that @xmath198 is continuously differentiable in @xmath199 for any @xmath67 . we say that the _ robinson qualification _ condition for [ eq : convx_subprob ] holds at @xmath124 if @xmath200 for @xmath6 . in order to prove the convergence of algorithm [ alg : a1 ] , we require the following assumption . [ as : a2 ] the set of kkt points of is nonempty . for a given @xmath67 , the matrix - valued mappings @xmath198 are continuously differentiable on @xmath199 . the convex problem [ eq : convx_subprob ] is solvable and the robinson qualification condition holds at its solutions . we note that if algorithm 1 is terminated at the iteration @xmath161 such that @xmath201 then @xmath156 is a stationary point of . [ th : convergence ] suppose that assumptions a.[as : a1 ] and a.[as : a2 ] are satisfied . suppose further that the lower level set @xmath199 is bounded . let @xmath202 be an infinite sequence generated by algorithm [ alg : a1 ] starting from @xmath157 . assume that @xmath203 . then if either @xmath46 is strongly convex or @xmath204 for @xmath182 then every accumulation point @xmath205 of @xmath206 is a kkt point of . moreover , if the set of the kkt points of is finite then the whole sequence @xmath207 converges to a kkt point of . first , we show that the solution mapping @xmath150 is _ closed_. indeed , by assumption a.[as : a2 ] , [ eq : convx_subprob ] is feasible . moreover , it is strongly convex . hence , @xmath208 , which is obviously closed . the remaining conclusions of the theorem can be proved similarly as ( * ? ? ? * theorem 3.2 . ) by using zangwill s convergence theorem @xcite of which we omit the details here . [ rm : conclusions ] note that the assumptions used in the proof of the closedness of the solution mapping @xmath209 in theorem [ th : convergence ] are weaker than the ones used in ( * ? ? ? * theorem 3.2 . ) . in this section , we present some applications of algorithm [ alg : a1 ] for solving several classes of optimization problems arising in static output feedback controller design . typically , these problems are related to the following linear , time - invariant ( lti ) system of the form : @xmath210 where @xmath211 is the state vector , @xmath212 is the performance input , @xmath213 is the input vector , @xmath214 is the performance output , @xmath215 is the physical output vector , @xmath216 is state matrix , @xmath217 is input matrix and @xmath218 is the output matrix . by using a static feedback controller of the form @xmath219 with @xmath220 , we can write the closed - loop system as follows : @xmath221 the stabilization , @xmath9 , @xmath222 optimization and other control problems of the lti system can be formulated as an optimization problem with bmi constraints . we only use the psd - convex overestimate of a bilinear form in _ example 3 _ to show that algorithm [ alg : a1 ] can be applied to solving many problems ins static state / output feedback controller design such as : * sparse linear static output feedback controller design ; * spectral abscissa and pseudospectral abscissa optimization ; * @xmath223 optimization ; * @xmath224 optimization ; * and mixed @xmath225 synthesis . these problems possess at least one bmi constraint of the from @xmath226 , where @xmath227 , where @xmath118 and @xmath28 are matrix variables and @xmath228 is a affine operator of matrix variable @xmath28 . by means of _ example 3 _ , we can approximate the bilinear term @xmath229 by its psd - convex overestimate . then using schur s complement to transform the constraint @xmath230 of the subproblem [ eq : convx_subprob ] into an lmi constraint @xcite . note that algorithm [ alg : a1 ] requires an interior starting point @xmath231 . in this work , we apply the procedures proposed in @xcite to find such a point . now , we summary the whole procedure applying to solve the optimization problems with bmi constraints as follows : [ scheme : a1 ] + _ step 1 . _ find a psd - convex overestimate @xmath232 of @xmath233 w.r.t . the parameterization @xmath234 for @xmath235 ( see _ example 1 _ ) . + _ step 2 . _ find a starting point @xmath157 ( see @xcite ) . + _ step 3 . _ for a given @xmath156 , form the convex semidefinite programming problem [ eq : convx_subprob ] and reformulate it as an optimization with lmi constraints . + _ step 4 . _ apply algorithm [ alg : a1 ] with an sdp solver to solve the given problem . now , we test algorithm [ alg : a1 ] for three problems via numerical examples by using the data from the comp@xmath236ib library @xcite . all the implementations are done in matlab 7.8.0 ( r2009a ) running on a laptop intel(r ) core(tm)i7 q740 1.73ghz and 4 gb ram . we use the yalmip package @xcite as a modeling language and sedumi 1.1 as a sdp solver @xcite to solve the lmi optimization problems arising in algorithm [ alg : a1 ] at the initial phase ( phase 1 ) and the subproblem [ eq : convx_subprob ] . the code is available at http://www.kuleuven.be/optec/software/bmisolver . we also compare the performance of algorithm [ alg : a1 ] and the convex - concave decomposition method ( ccdm ) proposed in @xcite in the first example , i.e. the spectral abscissa optimization problem . in the second example , we compare the @xmath10-norm computed by algorithm [ alg : a1 ] and the one provided by hifoo @xcite and penbmi @xcite . the last example is the mixed @xmath237 synthesis optimization problem which we compare between two values of the @xmath238-norm level . we consider an optimization problem with bmi constraint by optimizing the spectral abscissa of the closed - loop system @xmath239 as @xcite : @xmath240 here , matrices @xmath216 , @xmath217 and @xmath218 are given . matrices @xmath241 and @xmath220 and the scalar @xmath242 are considered as variables . if the optimal value of is strictly positive then the closed - loop feedback controller @xmath219 stabilizes the linear system @xmath243 . by introducing an intermediate variable @xmath244 , the bmi constraint in the second line of can be written @xmath245 . now , by applying scheme [ scheme : a1 ] one can solve the problem by exploiting the sedumi sdp solver @xcite . in order to obtain a strictly descent direction , we regularize the subproblem [ eq : convx_subprob ] by adding quadratic terms : @xmath246 , where @xmath247 . algorithm [ alg : a1 ] is terminated if one of the following conditions is satisfied : * the subproblem [ eq : convx_subprob ] encounters a numerical problem ; * @xmath248 ; * the maximum number of iterations , @xmath249 , is reached ; * or the objective function of is not significantly improved after two successive iterations , i.e. @xmath250 for some @xmath251 and @xmath252 , where @xmath253 . we test algorithm [ alg : a1 ] for several problems in comp@xmath236ib and compare our results with the ones reported by the _ convex - concave decomposition method _ ( ccdm ) in @xcite . -0.45 cm .computational results for in comp@xmath254ib [ cols= " < , > , > , > , > , > , > , > , > , > " , ] here , @xmath225 are the @xmath223 and @xmath224 norms of the closed - loop systems for the static output feedback controller , respectively . with @xmath255 , the computational results show that algorithm [ alg : a1 ] satisfies the condition @xmath256 for all the test problems . the problems ac11 and ac12 encounter a numerical problems that algorithm [ alg : a1 ] can not solve . while , with @xmath257 , there are @xmath258 problems reported infeasible , which are denoted by `` - '' . the @xmath224-constraint of three problems ac11 and nn8 is active with respect to @xmath257 . we have proposed a new iterative procedure to solve a class of nonconvex semidefinite programming problems . the key idea is to locally approximate the nonconvex feasible set of the problem by an inner convex set . the convergence of the algorithm to a stationary point is investigated under standard assumptions . we limit our applications to optimization problems with bmi constraints and provide a particular way to compute the inner psd - convex approximation of a bmi constraint . many applications in static output feedback controller design have been shown and two numerical examples have been presented . note that this method can be extended to solve more general nonconvex sdp problems where we can manage to find an inner psd - convex approximation of the feasible set . this is also our future research direction .

in this work , we propose a new local optimization method to solve a class of nonconvex semidefinite programming ( sdp ) problems . the basic idea is to approximate the feasible set of the nonconvex sdp problem by inner positive semidefinite convex approximations via a parameterization technique . this leads to an iterative procedure to search a local optimum of the nonconvex problem . the convergence of the algorithm is analyzed under mild assumptions . applications in static output feedback control are benchmarked and numerical tests are implemented based on the data from the compl@xmath0ib library .

arxiv

differential geometry has proven to be highly valuable in extracting the geometric meaning of continuum vector theories . of particular interest has been the dirac - khler formulation of fermionic field theory @xcite , which uses the antisymmetry inherent in the product between differential forms to describe the clifford algebra . in order to regularize calculations , we are required to introduce a discrete differential geometry scheme and it would be ideal if this had the same properties as the continuum and the correct continuum limit . however , defining such a scheme has proven to be very challenging . the difficulties are usually exhibited by the hodge star , which maps a form to its complement in the space , and the wedge product between forms . in a discretization , we would like the latter to allow the product rule to be satisfied and we would like both to be local . several discrete schemes have been proposed that address these difficulties with varying success . becher and joos @xcite used a lattice to define operators with many desirable properties , but that were non - local . to resolve the non - locality , they introduced translation operators . kanamori and kawamoto @xcite also used a lattice and introduced a specific non - commutativity between the fields and discrete forms . this allowed the product rule to be fulfilled , but they found that it became necessary to introduce a second orientation of form in order for their action to remain hermitian . instead of a lattice , balachandran _ et al _ @xcite used a quantized phase space to regularize their system , leading to a fuzzy coordinate space @xcite . in this paper , we shall build upon a proposal by adams @xcite in which he introduces two parallel lattices to maintain the locality of the hodge star and uses a finite element scheme to capture the properties of the wedge product . this proposal describes a local , discrete differential geometry for an arbitrary topological space and its formal aspects have been thoroughly studied by de beauc , samik sen , siddartha sen and czech @xcite . however , here we want to focus on its application to the dirac - khler formulation . in lattice quantum chromodynamics ( lattice qcd ) calculations , it is common to see the staggered fermion formulation used to describe fermions @xcite . this formulation addresses the problem of fermion doubling @xcite by reducing the number of degenerate fermions to @xmath0 in @xmath1 dimensional space - time . it is frequently used with the quarter - root trick @xcite to provide a description of a single fermion on the lattice , although this approach has attracted some controversy @xcite . the continuous dirac - khler formulation is regarded as providing the continuum limit for the staggered fermion formulation and so a discrete dirac - khler formulation with continuum properties can potentially offer great insight into how to develop non - degenerate , doubler - free fermionic field theories for the lattice . in this paper , we show how the two lattices of adams proposal can be used to describe chiral symmetry in the associated dirac - khler formulation . we also see how the idea of using more than one lattice can be extended to describe an exact flavour projection . we find that this necessitates the introduction of two new structures of lattice and a new operator . finally , we evaluate the path integral for this formulation , considering the effects of chiral and flavour projection . this builds on our previous work @xcite . our starting point is the _ complex _ , which is the space on which we define the discrete differential geometry . it comprises the points of the lattice , together with the links , faces , volumes and hyper - volumes between the points . each point , link , face , volume or hyper - volume is an example of a _ simplex _ and each simplex has an accompanying cochain . we denote a cochain by the vertices of its corresponding simplex . for example , we write the cochain for the simplex between vertices @xmath2 , @xmath3 , @xmath4 and @xmath5 from fig . [ twod ] as @xmath6 $ ] . each cochain is essentially a functional that acts upon a differential form of the same dimension as its simplex to give unity . for example , @xmath6 $ ] is defined such that @xmath7 the cochains act as the discrete differential forms of the theory and a general field is a linear combination of cochains . on the square @xmath8 , we write a general field as @xmath9)[a]+\tilde{\phi}([b])[b]+\tilde{\phi}([c])[c]+\tilde{\phi}([d])[d ] \\ & & + \tilde{\phi}([ab])[ab]+\tilde{\phi}([dc])[dc]+\tilde{\phi}([da])[da ] + \tilde{\phi}([cb])[cb ] \\ & & + \tilde{\phi}([abcd])[abcd ] \ . \end{array}\ ] ] to define the wedge product between cochains , we must first introduce the whitney map , which maps from the complex to the continuum , and the de rham map , which maps the other way . the whitney map , @xmath10 , replaces a cochain with a differential form of the same dimension as its accompanying simplex and introduces functions to interpolate in the regions between simplexes . for example , taking @xmath8 to be a unit square with origin @xmath2 , we introduce the interpolation functions @xmath11 where @xmath12 is the coordinate vector and this allows us to write @xmath13)[a]+\tilde{\phi}([b])[b]+\tilde{\phi}([c])[c]+\tilde{\phi}([d])[d]\right ) = \tilde{\phi}([a])\mu_1(x)\mu_2(x ) \\ \hspace{0.4 cm } + \tilde{\phi}([b])(1-\mu_1(x))\mu_2(x ) + \tilde{\phi}([c])(1-\mu_1(x))(1-\mu_2(x ) ) \\ \hspace{0.4 cm } + \tilde{\phi}([d])\mu_1(x)(1-\mu_2(x ) ) \\ w\left(\tilde{\phi}([da])[da ] + \tilde{\phi}([cb])[cb]+\tilde{\phi}([dc])[dc]+\tilde{\phi}([ab])[ab]\right ) = \\ \hspace{0.4 cm } \tilde{\phi}([da])\mu_1(x ) dx^2 + \tilde{\phi}([cb])(1-\mu_1(x))dx^2 + \tilde{\phi}([dc])(1-\mu_2(x))dx^1 \\ \hspace{0.4 cm } + \tilde{\phi}([ab])\mu_2(x ) dx^1\\ w\left(\tilde{\phi}([abcd])[abcd]\right ) = \tilde{\phi}([abcd])dx^1\wedge dx^2 . \end{array}\ ] ] the de rham map , @xmath14 , discretizes a field by integrating over the simplexes whose dimension match that of the accompanying differential form . @xmath14 also introduces a cochain of the appropriate dimension . thus , @xmath15 ) & = & \phi(x)|_{x = a } & \tilde{\phi}([b ] ) & = & \phi(x)|_{x = b}\\ \tilde{\phi}([c ] ) & = & \phi(x)|_{x = c } & \tilde{\phi}([d ] ) & = & \phi(x)|_{x = d}\\ \tilde{\phi}([dc ] ) & = & \int_{dc } \phi(x ) dx^1 & \tilde{\phi}([ab ] ) & = & \int_{ab } \phi(x ) dx^1\\ \tilde{\phi}([da ] ) & = & \int_{da } \phi(x ) dx^2 & \tilde{\phi}([cb ] ) & = & \int_{cb } \phi(x ) dx^2\\ \tilde{\phi}([abcd ] ) & = & \int_{abcd } \phi(x ) dx^1\wedge dx^2 \end{array}\ ] ] and @xmath16 & = & \tilde{\phi}([a])[a ] + \tilde{\phi}([b])[b ] + \tilde{\phi}([c])[c ] + \tilde{\phi}([d])[d ] \\ r\left[\phi(x,1)dx^1\right ] & = & \tilde{\phi}([dc])[dc]+\tilde{\phi}([ab])[ab ] \\ r\left[\phi(x,2)dx^2\right ] & = & \tilde{\phi}([da])[da]+\tilde{\phi}([cb])[cb ] \\ r\left[\phi(x,12)dx^1\wedge dx^2\right ] & = & \tilde{\phi}([abcd])[abcd ] \ . \end{array}\ ] ] the wedge product between two discrete fields , @xmath17 , now takes the form @xmath18 where @xmath19 is the wedge product of the continuum . we can take advantage of @xmath10 and @xmath14 to define the discrete exterior derivative as @xmath20 where @xmath21 is the exterior derivative from the continuum , @xmath22 . in the continuum , the hodge star is defined to be @xmath23 where we have written @xmath24 as shorthand for the product of forms @xmath25 and @xmath26 is the complement of @xmath27 in the space . @xmath28 is the levi - civita tensor . the square of the hodge star has the property @xmath29 where @xmath30 is the dimension of the form and @xmath1 the dimension of the space . to define the hodge star discretely requires the introduction of a second complex , known as the _ dual _ , in the same space as the first . the dual ( shown in fig . [ origcomp ] for two dimensions ) is aligned with the original complex so that the mid - points of complementary simplexes coincide . the hodge star is defined so that it maps a cochain from one complex to a cochain from the other complex with an aligned simplex . this gives the square of the operator , acting on a general cochain @xmath31 $ ] , the following local form : @xmath32 = ( -1)^{g(n - g)}[g]$ ] , where @xmath33 is dimension of the simplex and @xmath1 the dimension of the space . with the hodge star in place , the adjoint derivative can be defined as @xmath34 = ( -1)^{ng+n+1}*d * [ g]\ ] ] and the laplacian can be written @xmath35 , where @xmath36 is the dirac - khler operator . to define the inner product , we must introduce the barycentric subdivided complex @xcite . the vertices of this complex are formed from the midpoints of the simplexes on either the original or dual complex ( the result is the same , whichever we choose ) . these vertices are used in the construction of a new set of simplexes and a new whitney map , denoted @xmath37 , which interprets a cochain from either complex as a cochain on the barycentric subdivided complex and maps it to a product of differential forms and interpolating functions defined from the simplexes of the barycentric subdivided complex . the barycentric subdivided complex belonging to fig . [ origcomp ] is shown in fig . [ bary ] . this allows the inner product of two discrete fields , @xmath38 and @xmath39 , to be defined as @xmath40 in the dirac - khler basis , the clifford algebra is implemented with the clifford product , @xmath41 , acting on differential forms , @xmath42 formally , @xmath41 is defined to be @xmath43 where @xmath44 . for a one - form , acting upon a general field @xmath45 where @xmath46 is the contraction operator @xmath47 and @xmath48 is the linear combination of forms @xmath49 the correspondence between the dirac spinor , @xmath50 , and the dirac - khler field , @xmath48 , is established with @xmath51 where @xmath52 is defined to be @xmath53 in euclidean space - time . @xmath54 is shorthand for the product of the matrices @xmath55 , where @xmath56 take the form @xmath57 and @xmath58 are the pauli matrices . the matrix @xmath52 is pivotal to this correspondence because it has the properties @xmath59 which mean that @xmath60 and @xmath61 the components @xmath62 and @xmath63 are explicitly related by @xmath64 on the complex , we find that the fields do not exhibit the properties of eqs . ( [ corr1 ] ) and ( [ corr2 ] ) exactly . if this were the case , we would find that , referring to fig . [ twod ] , @xmath65+[ab]\right ) & \tilde{\wedge } & \left(\tilde{\phi}([ad])[ad]+\tilde{\phi}([cb])[cb]\right ) \\ & = & \int_{abcd}dx^1 dx^2 tr\left(\gamma_2\psi(x)\right)[abcd ] \ . \end{array}\ ] ] however , in the right hand side of this expression , the integration is over a domain of different dimension to that of the combination of @xmath66-matrices . using the definition of the de rham map and eq . ( [ equiv ] ) , we can see that no such field exists in the discretization , so , instead , the left hand side evaluates to @xmath67+[ab]\right ) & \tilde{\wedge } & \left(\tilde{\phi}([ad])[ad]+\tilde{\phi}([cb])[cb]\right ) \\ & = & \frac{1}{2}\left(\tilde{\phi}([da])+\tilde{\phi}([cb])\right)[abcd ] \ , \end{array}\ ] ] which is a first order approximation to the right hand side of eq . ( [ wedge ] ) . in the continuum , the columns of the four by four matrix , @xmath50 , each correspond to a separate flavour of field which can be isolated using the flavour projection @xmath68 , where @xmath69 and @xmath70 however , because the properties of eqs . ( [ corr1 ] ) and ( [ corr2 ] ) are only approximately captured on the complex , we can not use the discrete counterparts to @xmath71 to facilitate flavour projection and @xmath72 to generate exact chiral symmetry . however , we can take advantage of the relationship between the dual and the original complexes to implement an exact chiral symmetry , as we shall demonstrate in the next section . in subsequent sections , we introduce additional complexes to implement exact flavour projection . it was shown by rabin that , in the continuum , the chiral symmetry of dirac - khler fields is related to the hodge star @xcite . using equation ( [ equiv ] ) , we can show that the substitution @xmath73 is equivalent to the transformation @xmath74 where @xmath75 and @xmath76 are operators defined to be @xmath77 on the complex , the discrete fields are obtained from the continuous fields . as such , there is the potential to use @xmath78 to describe chiral symmetry . however , we can not use the formulation as it stands , because the fields associated with the simplexes from each complex are initially discretized by integrating over different domains . for example , referring to fig . [ origcomp ] , @xmath79)$ ] is obtained by sampling @xmath80 at @xmath2 and @xmath81)$ ] is obtained by integrating @xmath82 over @xmath83 . in this case , @xmath84 is not equivalent to @xmath85 because the domains do not agree . to attain this equivalence , we must modify the domain of integration used to initially discretize the fields on one of the complexes . whilst the choice is arbitrary , we will chose to modify the fields on the dual . we introduce a new de rham map , @xmath86 , that is identical to @xmath14 on the original complex , but that uses domains of integration on the dual that match simplices from the original complex . it is defined so that fields on the dual are discretized using domains of integration defined by the simplexes from the original complex related to the simplexes of the dual by their accompanying cochains and @xmath78 . formally , @xmath86 is defined to be @xmath87 & = & \sum_{h}\int_{h}\phi(x , h ) dx^h & \hspace{0.5cm}\mbox { on the original complex } \\ r_0[\phi(x , h)dx^h ] & = & \sum_{h}\int_{\bar{*}h}\phi(x , h ) dx^h & \hspace{0.5cm}\mbox { on the dual complex } . \end{array}\ ] ] here , @xmath27 is a simplex of the same space - time dimension as @xmath24 and @xmath88 maps a simplex from the dual to its counterpart on the original complex obtained as the simplex associated with the cochain @xmath89 $ ] . we continue to define the wedge product , clifford product , exterior derivative and adjoint derivative using @xmath14 . only for the initial discretization of the fields do we propose to use @xmath86 . with the fields discretized in this manner , we can generate an exact chiral symmetry with the operator @xmath90 acting on @xmath38 which has the property @xmath91 to implement chiral projection , we deconstruct @xmath78 into @xmath92 , which is the hodge star mapping cochains from the original complex to the dual , and @xmath93 which is the hodge star mapping cochains from the dual complex to the original . this allows us to write chiral projection as @xmath94 if we write @xmath95 , where @xmath96 and @xmath97 are the discrete fields on the original and dual complexes , respectively , then @xmath98 will project the right handed degrees of freedom of @xmath99 onto the original complex and the left handed degrees of freedom onto the dual . the dirac - khler field enjoys global @xmath100 flavour symmetry in the continuum . however , as can be seen from eq . ( [ dxproj ] ) , flavour projection only requires the subgroups generated by @xmath101 and @xmath102 . as we have seen , on the complex , the nive implementation of these symmetries is only approximate . however , just as we were able to use the dual and original complexes to describe an exact chiral symmetry , we can use the dual and original complexes to describe the @xmath103 symmetry needed for flavour projection . we write @xmath104 as the product of two operators @xmath105 and @xmath106 and @xmath107 as the product of the two projections @xmath108 and @xmath109 , such that @xmath110 where @xmath111 in the continuum , we can show that @xmath112 and we can use this to write @xmath106 as @xmath113 unfortunately , describing the symmetry generated by @xmath102 is more involved . when we apply @xmath114 to a general form , the form we obtain is complementary in all four dimensions to the original and this allows us to describe this process in terms of @xmath78 . if we consider applying @xmath115 to a general form , the form we obtain is complementary in the @xmath116 subspace , but equal to the original form in the @xmath117 subspace . by analogy , we need an operator that maps a form to its complement in the @xmath116 subspace , but not in the @xmath117 subspace . to this end we introduce @xmath118 , which , in the continuum , we define to be @xmath119 here , we have introduced @xmath120 as the components of @xmath27 belonging to the @xmath116 subspace and @xmath121 as the complementary operator in the @xmath116 subspace . @xmath122 is equal to @xmath27 in the @xmath117 subspace and complementary to @xmath27 in the @xmath116 subspace and @xmath123 is the complement of @xmath120 in the @xmath116 subspace . the square of @xmath118 has the property @xmath124 which is comparable to eq . ( [ starsq ] ) . with @xmath118 , we can show that @xmath125 where @xmath126 and @xmath127 is the number of components of @xmath27 in the @xmath116 subspace . to describe @xmath118 in the discretization , we are required to introduce a new complex , just as adams was required to introduce the dual to describe @xmath78 . the new complex should align with the original complex so that simplexes that are complementary in the @xmath116 subspace , but not the @xmath117 subspace , share midpoints . we christen the new complex the @xmath128 ( @xmath129-complement ) complex . in order to help visualize the alignment of the @xmath128 complex , in fig . [ 12c ] we show its analogue in two dimensions . here , the simplexes with coincident midpoints are complementary to each other in the @xmath130 direction , but not the @xmath131 direction and we would define the analogue of @xmath118 so that it maps between the following pairs of cochains from @xmath8 and @xmath83:- @xmath132 & \leftrightarrow & [ ab ] \\ { [ d ] } & \leftrightarrow & [ dc ] \\ { [ c ] } & \leftrightarrow & [ dc ] \\ { [ b ] } & \leftrightarrow & [ ab ] \\ { [ da ] } & \leftrightarrow & [ abcd ] \\ { [ cb ] } & \leftrightarrow & [ abcd ] \ . \end{array}\ ] ] it is worth mentioning that , in a two dimensional theory , we would not need to introduce this complex to isolate the flavours because the origianl and dual complexes would be sufficient . it is simply an analogue to the @xmath128 complex . in order to complete the description of flavour projection , we must go further and define a fourth complex . the term proportional to @xmath133 in eq . ( [ dxproj ] ) maps a form to its complement in the @xmath116 subspace , before mapping the resulting form to its complement in all four dimensions . the end result is a form complementary to the original in the @xmath117 subspace , but not in the @xmath116 subspace . in the continuum , this can be described as a combination of @xmath118 and @xmath78 , but to capture this map discretely requires us to introduce a fourth complex . the fourth complex is the dual of the @xmath128 complex ( or the @xmath129-complement of the dual , it can be viewed either way ) and we christen it the @xmath134 ( @xmath129 complement s dual ) complex . the two dimensional analogue of the @xmath134 complex , together with the original , dual and the analogue of the @xmath128 complex , is shown in fig . [ 12cd ] . to enable flavour projection between these four complexes , we must ensure that the fields are initially discretized using compatible domains of integration . to this end , we extend the definition of @xmath86 so that the fields on all four complexes are initially discretized using domains of integration taken from the original complex . @xmath86 becomes @xmath87 & = & \sum_h \int_{h } \phi(x , h ) dx^h & \hspace{0.5cm}\mbox { on the original complex } \\ r_0[\phi(x , h)dx^h ] & = & \sum_h \int_{\bar{*}h } \phi(x , h ) dx^h & \hspace{0.5cm}\mbox { on the dual complex } \\ r_0[\phi(x , h)dx^h ] & = & \sum_h \int_{\bar{\spadesuit}h } \phi(x , h)dx^h & \hspace{0.5cm}\mbox { on the $ 12c$ complex and } \\ r_0[\phi(x , h)dx^h ] & = & \sum_h \int_{\bar{*}\bar{\spadesuit}h } \phi(x , h)dx^h & \hspace{0.5cm}\mbox { on the $ 12cd$ complex } , \end{array}\ ] ] where @xmath135 is defined as before and @xmath136 maps a simplex to its complement in the @xmath116 subspace , but not the @xmath117 subspace . formally , @xmath137 generates the simplex associated with the cochain generated by @xmath138 $ ] . we must also extend the definition of @xmath106 from eq . ( [ projbeta ] ) to include the maps between the @xmath128 and @xmath134 complexes . if we rewrite @xmath99 as @xmath139 where @xmath140 and @xmath141 are the contributions from the @xmath128 and @xmath134 complexes , respectively , @xmath142 now takes the form @xmath143 where @xmath144 maps from the @xmath134 complex to the @xmath128 complex and @xmath145 maps the other way . to describe @xmath105 , we similarly deconstruct @xmath118 into @xmath146 which maps from the @xmath128 complex to the original , @xmath147 , which maps the other way , @xmath148 , which maps from the @xmath134 complex to the dual and @xmath149 which maps the other way . this allows us to write @xmath105 as @xmath150 and flavour projection can now be written as @xmath151 . one of the properties of the geometric discretization is that the dirac - khler operator maps the degrees of freedom from each complex in the same way . for example , referring to @xmath8 and @xmath83 from fig . [ origcomp ] , @xmath36 maps @xmath152)[abcd]$ ] to @xmath153)[da]$ ] and this is matched by the behaviour of @xmath36 on the dual which maps @xmath154)[c]$ ] to @xmath155)[dc]$ ] . consequently , the cancellation properties of @xmath152)$ ] and @xmath154)$ ] will be equally valid before and after the application of @xmath36 , so we have @xmath156\hat{\tilde{\phi}}=0 \ .\ ] ] to illustrate @xmath104 , we shall consider the effect of @xmath157 on @xmath99 . in particular , we will consider @xmath157 in stages , so first we apply @xmath158 to @xmath99 . this leaves the degrees of freedom belonging to the first and third columns of @xmath159 on the ordinary and dual complexes and the degrees of freedom belonging to the second and fourth columns of @xmath159 on the @xmath128 and @xmath134 complexes . if we now apply @xmath160 to this system , it will project between the original and dual complexes to leave the degrees of freedom belonging to the first column of @xmath159 on the original complex and the degrees of freedom belonging to the third column of @xmath159 on the dual complex . between the @xmath128 and @xmath134 complexes , @xmath160 will leave the degrees of freedom belonging to the second column of @xmath159 on the @xmath128 complex and the degrees of freedom belonging to the fourth column of @xmath159 on the @xmath134 complex . in the previous two sections , we showed that it was possible to implement exact chiral symmetry using the original and dual complexes and flavour projection using the original , dual , @xmath128 and @xmath134 complexes . however , because both projections use the original and the dual complexes in different ways , we can not implement chiral and flavour projection simultaneously using the formulation as it stands . to illustrate this point , we consider @xmath161 , which we write as @xmath162 . for the four complexes , @xmath163 becomes @xmath164 @xmath98 leaves the degrees of freedom belonging to the upper components of @xmath159 on the original and @xmath128 complexes and the degrees of freedom belonging to the lower components of @xmath159 on the dual and @xmath134 complexes . to this , we apply @xmath158 , which projects between the original and @xmath128 complexes to leave the degrees of freedom belonging to the upper components of the first and third columns of @xmath159 on the original complex and the degrees of freedom belonging to the upper components of the second and fourth columns of @xmath159 on the @xmath128 complex . @xmath158 also projects between the dual and @xmath134 complexes to leave the degrees of freedom belonging to the lower components of the first and third columns on the dual complex and the degrees of freedom belonging to the lower components of the second and fourth columns of @xmath159 on the @xmath134 complex . if we now consider applying @xmath160 to this system , we see that @xmath160 projects between the original and dual complexes to leave the degrees of freedom belonging to both the upper and lower components of the first and third columns on both complexes . it also projects between the @xmath128 and @xmath134 complexes to leave the degrees of freedom belonging to both the upper and lower components of the second and fourth columns on both complexes . because @xmath106 and @xmath163 both map between the original and dual complexes and the @xmath128 and @xmath134 complexes in different ways , we can not use these definitions for @xmath104 and @xmath163 to isolate non - degenerate , chiral dirac - khler fermions however , we can overcome this difficulty , by introducing a second set of complexes that are duplicates of the existing four . by introducing a second set , we can redefine the chiral projection , so that it maps between complexes from different sets , whilst continuing to define the flavour projection so that it maps between complexes from the same set . this arrangement allows us to use @xmath163 to place only the degrees of freedom belonging to the right handed components of @xmath159 on one set of complexes and the degrees of freedom belonging to the left handed components of @xmath159 on the other . the flavour projection within each set will now no longer mix different chiralities of field . to formally define this system , we label the first set of complexes @xmath2 and the duplicate set @xmath3 . we write the hodge star operator in the form @xmath165 , where @xmath166 labels the sets from which and to which the hodge star maps and @xmath167 label the complexes from which and to which it maps , respectively . because @xmath118 only maps between complexes of one set , it is unnecessary to modify its notation . the chiral projection operator now takes the form @xmath168 and we write @xmath169 . @xmath170 places the degrees of freedom belonging to the right handed components of @xmath159 on the complexes of set @xmath2 and the degrees of freedom belonging to the left handed components of @xmath159 on the complexes of set @xmath3 . the flavour projection operator is now @xmath171 , where @xmath172 as before , we describe the application of @xmath173 to @xmath98 in stages . @xmath174 leaves the degrees of freedom belonging to the upper components of the first and third columns of @xmath159 on the original and dual complexes of set @xmath2 and the lower components of the first and third columns on the original and dual complexes of set @xmath3 . it also leaves the degrees of freedom belonging to the upper components of the second and fourth columns of @xmath159 on the @xmath128 and @xmath134 complexes of set @xmath2 and the lower components of the second and fourth columns of @xmath159 on the @xmath128 and @xmath134 complexes of set @xmath3 . applying @xmath160 to this system will leave the degrees of freedom belonging to the upper components of the first column of @xmath159 on the original complex of set @xmath2 and the lower components of the first column of @xmath159 on the original complex of set @xmath3 . it will also leave the degrees of freedom belonging to the upper components of the third column of @xmath159 on the dual complex of set @xmath2 and the lower components of the third column of @xmath159 on the dual complex of set @xmath3 . similarly , it will leave the degrees of freedom belonging to the upper components of the second column of @xmath159 on the @xmath128 complex of set @xmath2 and the lower components of the second column of @xmath159 on the @xmath128 complex of set @xmath3 . lastly , it will also leave the degrees of freedom belonging to the upper components of the fourth column of @xmath159 on the @xmath134 complex of set @xmath2 and the lower components of the fourth column of @xmath159 on the @xmath134 complex of set @xmath3 . using these definitions , @xmath104 and @xmath163 can now be used to isolate a non - degenerate , chiral dirac - khler fermion on each complex . the flavour and chiral projections have important consequences for the path integral . if we initially consider only flavour projection , the action must include contributions from all four complexes . @xmath175 where , on each complex , we have @xmath176 @xmath159 contributes different degrees of freedom to the fields on each complex whose cochains are related by @xmath118 and @xmath78 , so we can write the path integral as the product of four path integrals @xmath177[d\tilde{\phi}_i]e^{-s_i(\tilde{\bar{\phi}}_i,\tilde{\phi}_i)}\right)^{l_0 } , \ ] ] where @xmath178 is an as yet undefined constant . because each complex lies in a separate space , we can evaluate each contribution separately and , on each complex , we have the standard result @xmath179[d\tilde{\phi}_i]\left(e^{-<\tilde{\bar{\phi}}_i , ( d-\delta)\tilde{\phi}_i>}\right)^{l_0 } = det\left[(d-\delta)\right]^{l_0 } , \ ] ] where @xmath36 is defined as a matrix operator . consequently , the full path integral takes the form @xmath180^{4l_0}\ ] ] and for this to have the correct continuum limit , we require that @xmath181 . interestingly , this means that the form of the path integral on each complex , prior to flavour projection , is @xmath179[d\tilde{\phi}_i]\left(e^{-<\tilde{\bar{\phi}}_i , ( d-\delta)\tilde{\phi}_i>}\right)^{\frac{1}{4 } } , \ ] ] which is exactly equivalent to that used in the quarter - root trick of lattice qcd @xcite . applying flavour projection to this system changes the relationship between @xmath38 and @xmath159 so that @xmath182 and @xmath183 each describe the degrees of freedom belonging to one column of @xmath184 and @xmath159 , respectively . this applies to the measures of integration as well as the fields in the action . one consequence of this projection is that several fields on each complex come to share the same structure in terms of @xmath185 , @xmath56 and @xmath159 . for example , after flavour projection , referring to fig . [ twod ] , @xmath186)=\psi(x)_1^{(1)}|_{x = a } , \hspace{1 cm } \tilde{\phi}_o([abcd])=-\int_{abcd}dx^1\wedge dx^2 \psi(x)_1^{(1 ) } \ .\ ] ] however , because these fields are integrated over different domains , they are independent and the path integral is evaluated in the same way to give @xmath187 . if we now extend the path integral so that the action permits chiral projection , we must include the contribution from both sets of complexes . the action becomes @xmath188 and , prior to flavour projection , we can evaluate the path integral separately on each complex to obtain @xmath189^{8l_0 } , \ ] ] which requires us to set @xmath190 , in order to obtain the correct continuum limit . applying both chiral and flavour projection to this system , as in the previous section , changes the relationship between @xmath38 and @xmath159 so that the path integral comes to be a product of four , single flavour , right - handed path integrals and four , single flavour , left - handed path integrals . as in the previous case , fields sharing a similar structure of @xmath185 , @xmath56 and @xmath159 are independent because their domains of integration differ . however , because the components of @xmath99 on each complex can be unambiguously described as left or right handed , the chiral projection sends half the components to zero . consequently , the effective dimension of @xmath36 is halved on each complex . here we have shown that it is possible to describe exact chiral symmetry for dirac - 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it is shown that an exact chiral symmetry can be described for dirac - khler fermions using the two complexes of the geometric discretization . this principle is extended to describe exact flavour projection and it is shown that this necessitates the introduction of a new operator and two new structures of complex . to describe simultaneous chiral and flavour projection , eight complexes are needed in all and it is shown that projection leaves a single flavour of chiral field on each .

arxiv

the need for the efficient use of the scarce spectrum in wireless applications has led to significant interest in the analysis of cognitive radio systems . one possible scheme for the operation of the cognitive radio network is to allow the secondary users to transmit concurrently on the same frequency band with the primary users as long as the resulting interference power at the primary receivers is kept below the interference temperature limit @xcite . note that interference to the primary users is caused due to the broadcast nature of wireless transmissions , which allows the signals to be received by all users within the communication range . note further that this broadcast nature also makes wireless communications vulnerable to eavesdropping . the problem of secure transmission in the presence of an eavesdropper was first studied from an information - theoretic perspective in @xcite where wyner considered a wiretap channel model . in @xcite , the secrecy capacity is defined as the maximum achievable rate from the transmitter to the legitimate receiver , which can be attained while keeping the eavesdropper completely ignorant of the transmitted messages . later , wyner s result was extended to the gaussian channel in @xcite . recently , motivated by the importance of security in wireless applications , information - theoretic security has been investigated in fading multi - antenna and multiuser channels . for instance , cooperative relaying under secrecy constraints was studied in @xcite@xcite . in @xcite , for amplify and forwad relaying scheme , not having analytical solutions for the optimal beamforming design under both total and individual power constraints , an iterative algorithm is proposed to numerically obtain the optimal beamforming structure and maximize the secrecy rates . although cognitive radio networks are also susceptible to eavesdropping , the combination of cognitive radio channels and information - theoretic security has received little attention . very recently , pei _ et al . _ in @xcite studied secure communication over multiple input , single output ( miso ) cognitive radio channels . in this work , finding the secrecy - capacity - achieving transmit covariance matrix under joint transmit and interference power constraints is formulated as a quasiconvex optimization problem . in this paper , we investigate the collaborative relay beamforming under secrecy constraints in the cognitive radio network . we first characterize the secrecy rate of the amplify - and - forward ( af ) cognitive relay channel . then , we formulate the beamforming optimization as a quasiconvex optimization problem which can be solved through convex semidefinite programming ( sdp ) . furthermore , we propose two sub - optimal null space beamforming schemes to reduce the computational complexity . we consider a cognitive relay channel with a secondary user source @xmath0 , a primary user @xmath1 , a secondary user destination @xmath2 , an eavesdropper @xmath3 , and @xmath4 relays @xmath5 , as depicted in figure [ fig : channel ] . we assume that there is no direct link between @xmath0 and @xmath2 , @xmath0 and @xmath1 , and @xmath0 and @xmath3 . we also assume that relays work synchronously to perform beamforming by multiplying the signals to be transmitted with complex weights @xmath6 . we denote the channel fading coefficient between @xmath0 and @xmath7 by @xmath8 , the fading coefficient between @xmath7 and @xmath2 by @xmath9 , @xmath7 and @xmath1 by @xmath10 and the fading coefficient between @xmath7 and @xmath3 by @xmath11 . in this model , the source @xmath0 tries to transmit confidential messages to @xmath2 with the help of the relays on the same band as the primary user s while keeping the interference on the primary user below some predefined interference temperature limit and keeping the eavesdropper @xmath3 ignorant of the information . it s obvious that our channel is a two - hop relay network . in the first hop , the source @xmath0 transmits @xmath12 to relays with power @xmath13=p_s$ ] . the received signal at the @xmath14 relay @xmath7 is given by @xmath15 where @xmath16 is the background noise that has a gaussian distribution with zero mean and variance of @xmath17 . in the af scenario , the received signal at @xmath7 is directly multiplied by @xmath18 without decoding , and forwarded to @xmath2 . the relay output can be written as @xmath19 the scaling factor , @xmath20 is used to ensure @xmath21=|w_m|^2 $ ] . there are two kinds of power constraints for relays . first one is a total relay power constraint in the following form : @xmath22 where @xmath23^t$ ] and @xmath24 is the maximum total power . @xmath25 and @xmath26 denote the transpose and conjugate transpose , respectively , of a matrix or vector . in a multiuser network such as the relay system we study in this paper , it is practically more relevant to consider individual power constraints as wireless nodes generally operate under such limitations . motivated by this , we can impose @xmath27 or equivalently @xmath28 where @xmath29 denotes the element - wise norm - square operation and @xmath30 is a column vector that contains the components @xmath31 . @xmath32 is the maximum power for the @xmath14 relay node . the received signals at the destination @xmath2 and eavesdropper @xmath3 are the superposition of the messages sent by the relays . these received signals are expressed , respectively , as @xmath33 where @xmath34 and @xmath35 are the gaussian background noise components with zero mean and variance @xmath36 , at @xmath2 and @xmath3 , respectively . it is easy to compute the received snr at @xmath2 and @xmath3 as @xmath37 where @xmath38 denotes the mutual information . the interference at the primary user is latexmath:[\ ] ] where superscript @xmath43 denotes conjugate operation . then , the received snr at the destination and eavesdropper , and the interference on primary user can be written , respectively , as @xmath44 with these notations , we can write the objective function of the optimization problem ( i.e. , the term inside the logarithm in ( [ srate ] ) ) as @xmath45 if we denote @xmath46 , @xmath47 , define @xmath48 , and employ the semidefinite relaxation approach , we can express the beamforming optimization problem as @xmath49 the optimization problem here is similar to that in @xcite . the only difference is that we have an additional constraint due to the interference limitation . thus , we can use the same optimization framework . the optimal beamforming solution that maximizes the secrecy rate in the cognitive relay channel can be obtained by using semidefinite programming with a two dimensional search for both total and individual power constraints . for simulation , one can use the well - developed interior point method based package sedumi @xcite , which produces a feasibility certificate if the problem is feasible , and its popular interface yalmip @xcite . it is important to note that we should have the optimal @xmath50 to be of rank - one to determine the beamforming vector . while proving analytically the existence of a rank - one solution for the above optimization problem seems to be a difficult task , we would like to emphasize that the solutions are rank - one in our simulations . thus , our numerical result are tight . also , even in the case we encounter a solution with rank higher than one , the gaussian randomization technique is practically proven to be effective in finding a feasible , rank - one approximate solution of the original problem . details can be found in @xcite . obtaining the optimal solution requires significant computation . to simplify the analysis , we propose suboptimal null space beamforming techniques in this section . we choose @xmath51 to lie in the null space of @xmath52 . with this assumption , we eliminate @xmath3 s capability of eavesdropping on @xmath2 . mathematically , this is equivalent to @xmath53 , which means @xmath51 is in the null space of @xmath54 . we can write @xmath55 , where @xmath56 denotes the projection matrix onto the null space of @xmath54 . specifically , the columns of @xmath56 are orthonormal vectors which form the basis of the null space of @xmath54 . in our case , @xmath56 is an @xmath57 matrix . the total power constraint becomes @xmath58 . the individual power constraint becomes @xmath59 under the above null space beamforming assumption , @xmath60 is zero . hence , we only need to maximize @xmath61 to get the highest achievable secrecy rate . @xmath61 is now expressed as @xmath62 the interference on the primary user can be written as @xmath63 defining @xmath64 , we can express the optimization problem as @xmath65 this problem can be easily solved by semidefinite programming with bisection search @xcite . in this section , we choose @xmath51 to lie in the null space of @xmath52 and @xmath66 . mathematically , this is equivalent to requiring @xmath67 , and @xmath68 . we can write @xmath69 , where @xmath70 denotes the projection matrix onto the null space of @xmath54 and @xmath71 . specifically , the columns of @xmath70 are orthonormal vectors which form the basis of the null space . in our case , @xmath70 is an @xmath72 matrix . the total power constraint becomes @xmath73 . the individual power constraint becomes @xmath74 . with this beamforming strategy , we again have @xmath75 . moreover , the interference on the primary user is now reduced to @xmath76 which is the sum of the forwarded additive noise components present at the relays . now , the optimization problem becomes @xmath77 again , this problem can be solved through semidefinite programming . with the following assumptions , we can also obtain a closed - form characterization of the beamforming structure . since the interference experienced by the primary user consists of the forwarded noise components , we can assume that the interference constraint @xmath78 is inactive unless @xmath41 is very small . with this assumption , we can drop this constraint . if we further assume that the relays operate under the total power constraint expressed as @xmath79 , we can get the following closed - form solution : @xmath80 where @xmath81 is the largest generalized eigenvalue of the matrix pair @xmath82 . and positive definite matrix @xmath83 , @xmath84 is referred to as a generalized eigenvalue eigenvector pair of @xmath82 if @xmath84 satisfy @xmath85 @xcite . ] hence , the maximum secrecy rate is achieved by the beamforming vector @xmath86 where @xmath87 is the eigenvector that corresponds to @xmath88 and @xmath89 is chosen to ensure @xmath90 . the discussion in section [ sec : op ] can be easily extended to the case of more than one primary user in the network . each primary user will introduce an interference constraint @xmath91 which can be straightforwardly included into ( [ optimal ] ) . the beamforming optimization is still a semidefinite programming problem . on the other hand , the results in section [ sec : op ] can not be easily extended to the multiple - eavesdropper scenario . in this case , the secrecy rate for af relaying is @xmath92 , where the maximization is over the rates achieved over the links between the relays and different eavesdroppers . hence , we have to consider the eavesdropper with the strongest channel . in this scenario , the objective function can not be expressed in the form given in ( [ srate ] ) and the optimization framework provided in section [ sec : op ] does not directly apply to the multi - eavesdropper model . however , the null space beamforming schemes discussed in section [ sec : null ] can be extended to the case of multiple primary users and eavesdroppers under the condition that the number of relay nodes is greater than the number of eavesdroppers or the total number of eavesdroppers and primary users depending on which null space beamforming is used . the reason for this condition is to make sure the projection matrix @xmath93 exists . note that the null space of @xmath94 channels in general has the dimension @xmath95 where @xmath4 is the number of relays . we assume that @xmath96 , @xmath97 are complex , circularly symmetric gaussian random variables with zero mean and variances @xmath98 , @xmath99 , @xmath100 and @xmath101 respectively . in this section , each figure is plotted for fixed realizations of the gaussian channel coefficients . hence , the secrecy rates in the plots are instantaneous secrecy rates . in fig . [ fig:1 ] , we plot the optimal secrecy rates for the amplify - and - forward collaborative relay beamforming system under both individual and total power constraints . we also provide , for comparison , the secrecy rates attained by using the suboptimal beamforming schemes . the fixed parameters are @xmath102 , @xmath103 , and @xmath104 . since af secrecy rates depend on both the source and relay powers , the rate curves are plotted as a function of @xmath105 . we assume that the relays have equal powers in the case in which individual power constraints are imposed , i.e. , @xmath106 . it is immediately seen from the figure that the suboptimal null space beamforming achievable rates under both total and individual power constraints are very close to the corresponding optimal ones . especially , they are nearly identical in the high snr regime , which suggests that null space beamforming is optimal at high snrs . thus , null space beamforming schemes are good alternatives as they are obtained with much less computational burden . moreover , we interestingly observe that imposing individual relay power constraints leads to small losses in the secrecy rates . in fig . [ fig:11 ] , we change the parameters to @xmath107 , @xmath108 and @xmath104 . in this case , channels between the relays and the eavesdropper and between the relays and the primary - user are on average stronger than the channels between the relays and the destination . we note that beamforming schemes can still attain good performance and we observe similar trends as before . in fig . [ fig:2 ] , we plot the optimal secrecy rate and the secrecy rates of the two suboptimal null space beamforming schemes ( under both total and individual power constraints ) as a function of the interference temperature limit @xmath41 . we assume that @xmath109 . it is observed that the secrecy rate achieved by beamforming in the null space of both the eavesdropper s and primary user s channels ( bnep ) is almost insensitive to different interference temperature limits when @xmath110 since it always forces the signal interference to be zero regardless of the value of @xmath41 . it is further observed that beamforming in the null space of the eavesdropper s channel ( bne ) always achieves near optimal performance regardless the value of @xmath41 under both total and individual power constraints . in this paper , collaborative relay beamforming in cognitive radio networks is studied under secrecy constraints . optimal beamforming designs that maximize secrecy rates are investigated under both total and individual relay power constraints . we have formulated the problem as a semidefinite programming problem and provided an optimization framework . in addition , we have proposed two sub - optimal null space beamforming schemes to simplify the computation . finally , we have provided numerical results to illustrate the performances of different beamforming schemes . a. wyner `` the wire - tap channel , '' _ bell . syst tech . j _ , vol.54 , no.8 , pp.1355 - 1387 , jan 1975 . i. csiszar and j. korner `` broadcast channels with confidential messages , '' _ ieee trans . inform . theory _ , vol.it-24 , no.3 , pp.339 - 348 , may 1978 . v. nassab , s. shahbazpanahi , a. grami , and z .- q . luo , `` distributed beamforming for relay networks based on second order statistics of the channel state information , '' _ ieee trans . on signal proc . 56 , no 9 , pp . 4306 - 4316 , g. zheng , k. k. wong , a. paulraj , and b. ottersten , `` robust collaborative - relay beamforming , '' _ ieee trans . on signal proc . 57 , no . 8 , aug . 2009 z - q luo , wing - kin ma , a.m .- c . so , yinyu ye , shuzhong zhang `` semidefinite relaxation of quadratic optimization problems '' _ ieee signal proc . magn . 3 , may 2010 j. lofberg , `` yalmip : a toolbox for modeling and optimization in matlab , '' _ proc . the cacsd conf . _ , taipei , taiwan , 2004 . s. boyd and l. vandenberghe , convex optimization . cambridge , u.k . : cambridge univ . press , 2004 .

in this paper , a cognitive relay channel is considered , and amplify - and - forward ( af ) relay beamforming designs in the presence of an eavesdropper and a primary user are studied . our objective is to optimize the performance of the cognitive relay beamforming system while limiting the interference in the direction of the primary receiver and keeping the transmitted signal secret from the eavesdropper . we show that under both total and individual power constraints , the problem becomes a quasiconvex optimization problem which can be solved by interior point methods . we also propose two sub - optimal null space beamforming schemes which are obtained in a more computationally efficient way . _ index terms : _ amplify - and - forward relaying , cognitive radio , physical - layer security , relay beamforming .

arxiv

the @xmath0-invariance of the maxwell equation was discovered by cunningham and bateman a century ago . however in order to quantize the maxwell field and due to gauge freedom , a gauge fixing condition is necessary . the lorenz gauge is usually used , which breaks the @xmath0 invariance . nonetheless since such a symmetry mights apear to lack physical meaning , its breaking does not bother many people @xcite . the purpose of the present paper is to demonstrate the benefits of keeping this fundamental symmetry when quantizing the maxwell field in conformally flat spaces ( cfs ) . the starting point is the following . a classical @xmath0-invariant field can not , at least locally , distinguish between two cfss @xcite . so why not maintain the @xmath0-invariance during the quantization process in a cfs ? doing so , a free field living in a cfs might behave like in a flat space and the corresponding wightman two - point functions can be related to their minkowskian counterparts . the work @xcite confirms this assertion in the special case of maxwell field in de sitter space . indeed , a new and simple two - point wightman function @xmath1 was found and which has the same physical ( gauge independent ) content as the two - point function of allen and jacobson @xcite . this is because the faraday propagator @xmath2 is the same . the present work extends to general cfss and clarify the quantum structure of the formalism developed in @xcite . we use dirac s six - cone formalism and realize all cfss as intersections of the null cone and a given surface in a six - dimensional lorentzian space . the introduction of auxiliary fields and the use of the gubta - bleuler quantization scheme are necessary to deal with gauge freedom of the maxwell field . another important ingredient is the use of a well - suited coordinate system . this allows to @xmath0-invariant cfs formulas to get a minkowskian form . the main result is a set of wightman two - point functions for maxwell and auxiliary fields . this paper is organized as follows . [ geom ] sets the coordinates systems and the geometrical construction of cfss . [ fields ] defines the fields and gives their dynamical equations . in sec . [ quantum - field ] , the dynamical system is solved , the quantum field is explicitly constructed and the two - point functions are written down . some technical details are given in appdx . [ details ] . the infinitesimal @xmath0 action on the fields @xmath3 is expanded in appdx . [ action ] and their @xmath0-invariant scalar product is given in appdx . the six - dimensional lorentzian space @xmath4 is provided with the natural orthogonal coordinates @xmath5 and equipped with the metric @xmath6 . quantities related to @xmath4 and its null cone @xmath7 are labeled with a tilde . we define a second coordinate system @xmath8 , @xmath9 where the four components @xmath10 is the so - called polyspherical coordinate system @xcite and @xmath11 . a straightforward calculation yields @xmath12 which means that the component @xmath13 carries alone the homogeneity of the @xmath14 s . using the system @xmath15 , the null cone reads @xmath16 a five - dimensional surface in @xmath4 is defined through @xmath17 where the real and smooth function @xmath18 depends only on @xmath10 and @xmath19 and is then homogeneous of degree @xmath20 . the intersection of @xmath21 and @xmath22 is a four - dimensional space @xmath23 where the index @xmath24 in @xmath25 refers to @xmath26 ) . regarding to its metric inherited from @xmath27 , precisely @xmath28 @xmath25 turns out to be a cfs . a smooth move of the surface @xmath29 , which corresponds to changing the function @xmath30 , amounts to perform a weyl rescaling . this locally relates all cfss and permits to go from one to another . note that for @xmath31 , @xmath25 reduces to minkowski space @xmath32 and accordingly the four components system @xmath10 yields the usual cartesian system . the gradients @xmath33 are extensivelly used in this article . the function @xmath30 does not depend on @xmath13 and thus @xmath34 . the choice of the function @xmath30 , including its @xmath19 dependence has to be done in such a way to ensure the invariance ( in @xmath4 ) of the surface @xmath22 under the action of the isometry group associated to the desired @xmath25 four - dimensional space . since the @xmath4 null - cone is @xmath0-invariant , the resulting @xmath25 will be invariant under its isometry group . let us consider an example : @xmath35^{-1}$ ] , where @xmath36 is a constant . the associated surface @xmath22 and thus the corresponding @xmath25 are left invariant under the action of de sitter group @xcite . also , @xmath25 is a de sitter space . in this section , we explain how to obtain the @xmath0-invariant maxwell field in @xmath25 from a six - dimensional one - form . following dirac @xcite , we consider a one - form @xmath37 defined in @xmath4 homogeneous of degree @xmath38 and which decomposes on the @xmath39 basis as @xmath40 the components @xmath41 are homogeneous of degree @xmath42 and obey to the equation @xmath43 this equation is naturally invariant under the @xmath0 action since this group has a linear action when acting in @xmath4 . we then decompose the one - form @xmath37 on the basis @xmath44 corresponding to the system @xmath45 ( [ coord+muc ] ) , with a slight but capital modification on the @xmath46 component . there are two ways , the first decomposition reads @xmath47 the second is given by @xmath48 now , identifying ( [ eq-1 ] ) with ( [ eq-2 ] ) , one obtains the relation between the fields @xmath49 and @xmath50 through @xmath51 all the fields @xmath52 and @xmath53 are by construction homogeneous of degree @xmath54 . as a consequence , @xmath55 and @xmath56 this amounts to project the fields @xmath52 on @xmath57 and @xmath53 on @xmath58 . then projecting the fields on the null cone @xmath21 yields @xmath59 thus @xmath60 and @xmath61 are respectively @xmath25 and minkowski fields . though in a slightly different maner , this relation was obtained in @xcite in the particular case of de sitter space and was called the `` extended weyl transformation '' . the fields @xmath62 and @xmath63 are auxiliary fields and the field @xmath64 is , up to the condition @xmath65 the maxwell field . this will become clear here after . let us now turn to the dynamical equations . our strategy is to transport minkowskian @xmath0-invariant equations to get @xmath0-invariant equations in the @xmath25 space . the first step is thus to write down the minkowskian equations which are obtained using the equation ( [ equation - a ] ) and the relation ( [ a(a)-m ] ) . this system reads @xmath66 the corresponding system in @xmath25 is obtained using ( [ extendedweyl - bis ] ) , @xmath67 where all contractions are performed using @xmath68 even though we are in the curved space @xmath25 . the field @xmath69 obeying to the system above is not yet the maxwell one . nevertheless , the constraint @xmath70 simplifies the system ( [ syst1m - h ] ) and leads to @xmath71 despite their minkowskian form , these equations are the maxwell equation and a conformal gauge condition on any conformally flat space . this is due to the use of the polyspherical coordinate system ( [ coord+muc ] ) , which makes apparent the flatness feature of the @xmath25 spaces . the constraint @xmath72 reduces the extended weyl transformation ( [ extendedweyl - bis ] ) into the identity @xmath73 recovering the ordinary vanishing conformal weight of the maxwell field @xmath69 . after some algebra , the covariant form of ( [ maxwellh1 ] ) takes the form @xmath74 where @xmath75 . the first line ( resp . the second one ) is the covariant maxwell ( resp . the eastwood - singer gauge @xcite ) equation in an arbitrary @xmath25 space . this conformal gauge was first derived by bayen and flato in minkowski space @xcite . its extension to curved spaces ( even cfss ) is not trivial and can be performed using adapted tools like the weyl - gauging technique @xcite or the weyl - to - riemann method @xcite . note that the system ( [ system - covariant ] ) is valid only if @xmath76 ( an @xmath0-invariant constraint ) . but the latter has to be fixed at the end of the quantization process , not at the begining . indeed , the auxiliary field @xmath77 acts as a faddeev popov ghost field and its retention during the quantization process is necessary . the constraint @xmath76 will be applied on the quantum space to select an invariant subspace of physical states and the wightman functions thus include the whole big space . this is related to the undecomposable group representation ( see appendix [ action ] ) . we now apply the gupta - bleuler quantization scheme @xcite . this can be summarized as follows . we have seen that @xmath78 is interpreted as the maxwell field in the eastwood - singer gauge ( [ system - covariant ] ) on the space @xmath79 when the constraint @xmath80 is applied . the problem is that pure gauge solutions ( @xmath81 , with @xmath82 and @xmath80 ) are orthogonal to all the solutions including themselves . as a consequence , the space of solutions is degenerate and no wightman functions can be constructed . to fix this problem , we consider the system ( [ syst1m - h ] ) , instead of ( [ system - covariant ] ) , for which @xmath83 and thus a causal reproducing kernel can be found . this means that for quantum fields @xmath84 acting on some hilbert ( or krein ) space @xmath85 , we can not impose the operator equation @xmath86 . instead , we define the subspace of physical states @xmath87 which cancels the action of @xmath88 . then the maxwell equation and the eastwood - singer gauge hold in the mean on the space @xmath89 . the task seems complicated at first sight , but thanks to the correspondence ( [ extendedweyl - bis ] ) we only need to solve the minkowskian system ( [ syst1 m ] ) , which is already done in @xcite . indeed , using the weyl equivalence between cfss , the whole structure of an @xmath90-covariant free field theory can be transported from minkowski to another cfs . in the following , we solve the dynamical equations , obtain the modes , determine the quantum field , the subspace of physical states and finally compute the two - point functions . the solutions of the minkowskian system ( [ syst1 m ] ) can be obtained from @xcite and read @xmath91 where @xmath92 are polarization vectors whose components are given by @xmath93 and verifying @xmath94 with respect to the scalar product ( [ scalarps - a ] ) . the matrix @xmath95 relates the fields @xmath96 and @xmath97 ( [ matrix - s ] ) . the scalar modes @xmath98 are solutions of the minkowskian @xmath0-invariant ( or massless ) sclalar field equation @xmath99 , @xmath100 where @xmath101 denotes the usual hyperspherical harmonics . the normalization constant @xmath102 is chosen in order to get @xmath103 with respect to the klein - gordon scalar product . as a consequence , the solutions ( [ modes - a ] ) are normalized with respect to ( [ scalarps - a ] ) , @xmath104 thus the general solution of the system ( [ syst1 m ] ) reads @xmath105 where @xmath106 are real constants . let us now turn to the modes of the system ( [ syst1m - h ] ) . they are obtained thanks to the extended weyl transformation ( [ extendedweyl - bis ] ) applied on the minkowskian modes ( [ modes - a ] ) @xmath107 these modes are normalized like ( [ norm ] ) but according to the scalar product ( [ scalarps - a - k ] ) . the general solution on @xmath79 reads @xmath108 where the @xmath109 are some real constants . note that when @xmath110 the solutions ( [ generalsolution - am ] ) and ( [ generalsolution - ak ] ) solve the maxwell equation in the eastwood - singer gauge . we can now define the quantum fields and construct the fock spaces as usual . the quantum fields corresponding to ( [ generalsolution - am ] ) and ( [ generalsolution - ak ] ) are respectively defined through @xmath111 @xmath112 where the operators @xmath113 and @xmath114 are respectively the annihilators and creators of the modes ( [ modes - a ] ) in @xmath32 and the modes ( [ modes - a - k ] ) in @xmath25 . the use of the same annihilators and creators for all cfss is highly important for our purpose . indeed , this allows to define the the same vaccuum state @xmath115 through @xmath116 for any annihilator . the one - particle states are built by applying the creators on the vacuum state @xmath117 and the multiple particle states of the fock spaces are constructed as usual . moreover , the annihilation and creation operators obey to the following algebra @xmath118 = [ \hat a_{{\scriptscriptstyle l}m ( \alpha)}^{\dag } , \hat a_{{\scriptscriptstyle l}'m ' ( \beta)}^{\dag } ] = 0 \\ & [ \hat a_{{\scriptscriptstyle l}m ( \alpha ) } , \hat a_{{\scriptscriptstyle l}'m ' ( \beta)}^{\dag } ] = -\tilde \eta_{{\scriptscriptstyle \alpha}\beta } \delta_{{\scriptscriptstyle l}l ' } \delta_{{\scriptscriptstyle m}m'}. \end{split}\ ] ] the subset of physical states in both spaces is determined thanks to the classical physical solutions ( [ generalsolution - am ] ) and ( [ generalsolution - ak ] ) verifying ( [ a+=0 ] ) . in quantum language , @xmath119 is a physical state iff @xmath120 where @xmath121 is the annihilator part of @xmath122 . this implies the equality @xmath123 for any physical states @xmath119 and @xmath124 . also , the subspace of physical states is the same in all cfss , which allows to transport physical quantities from minkowski space into the @xmath25 space . as a consequence , one obtains @xmath125 in @xmath25 and the corresponding minkowskian system in @xmath32 . the quantum fields fulfill the maxwell equation together with the eastwood - singer gauge in the mean on the physical states . we show in this part how to get the wightman two - point functions on @xmath25 from their minkowskian counterparts . the wightman functions related to the minkowskian fields @xmath126 are defined through @xmath127 their expressions are given in @xcite and read @xmath128 where @xmath129 with @xmath130 stands for the wightman two - point function related to the minkowskian massless scalar field . the wightman two - point functions related to the field @xmath131 are given by @xmath132 now , using ( [ modes - a - k ] ) , ( [ maxwelds2ptdef ] ) and ( [ wightman - k ] ) , allows to write the following capital formula @xmath133 where the @xmath134 terms read @xmath135 the wightman two - point functions ( [ dk - dm ] ) read @xmath136 where @xmath137 , @xmath138 , @xmath139 and @xmath140 . + to end this paper , let us consider an important particular case , that corresponding to de sitter space . this case is obtained by specifying @xmath141 where @xmath142 is related to the de sitterian ricci scalar through @xmath143 . the gradients ( [ upsilon ] ) read @xmath144 in this case we obtain simple expressions for the two - point functions related to the fields @xmath145 on de sitter space . the three more relevant yield @xmath146 where we have used ( de sitter is a maximally symmetric space ) the standard unit tangent vectors @xmath147 and @xmath148 , the parallel propagator along the geodesic @xmath149 and the usual function @xmath150 of the geodesic distance @xmath151 relating @xmath152 and @xmath153 , @xmath154 . see @xcite for a more precise statement . note that the two - point function @xmath155 has the same physical content with the allen and jacobson two - point function @xcite . an @xmath0-covariant quantization of the maxwell field in an arbitrary conformally flat space was presented . following dirac s six - cone formalism , all conformally flat spaces @xmath25 are realized as intersections of the null cone and a given surface @xmath29 . the quantum field was explicitly constructed using the gupta - bleuler canonical quantization scheme and the wightman two - point functions were given . the price to pay for this simplicity and the maintaining of the @xmath0 invariance during the whole quantization process was the introduction of two auxiliary fields @xmath156 and @xmath77 . as a consequence , the maxwell field @xmath69 does not propagate `` alone '' but together with its two auxiliary fields . the propagation must use all the wightman functions ( [ d - x - k - relevent ] ) and not only the `` purely '' maxwell one @xmath157 . nonetheless , in a recent work @xcite , we have used the functions ( [ d - ds ] ) to propagate the maxwell field generated by two charges of opposite sign placed at the two poles of a de sitter space . the calculations showed that only @xmath157 is involved , which trivialize the problem . one can consider to use the two - point functions ( [ d - x - k - relevent ] ) to propagate the electromagnetic field for some charge distribution given in other cfss , like flrw spaces for instance . one concludes that is much worth to maintain the @xmath0 symmetry during the whole quantization process when dealing with maxwell field in a conformally flat space . the problem then goes back to minkowski and the calculations become much easier . in fact , the classical and quantum structures of the free maxwell field are locally the same in all conformally flat spaces . the remained question is to know if this is true for other free fields and how to deal with @xmath0-invariant interactions ? i would like to thank m. novello , j. renaud and e. huguet for illuminating discussions and the cnpq for financial support . considering ( [ field - a ] ) and ( [ eq-1 ] ) , expressing the basis @xmath159 in terms of @xmath44 and then identifying both sides , one obtains the expression of @xmath160 in terms of @xmath161 . we find , after using the homogeneity properties , @xmath162 which reads @xmath163 this system can be inverted in @xmath164 following the same steps as above , one obtain the matrix linking the @xmath165 to the @xmath166 @xmath167 + a^{{\scriptscriptstyle k}}_4 [ \upsilon_c(1+x^2)-1 ] \\ & & \qquad \qquad \qquad + \upsilon_c a^{{\scriptscriptstyle k}}.x \biggr\ } \\ { \displaystyle a_\mu^{{\scriptscriptstyle k } } } & = & k \biggl\ { a^{{\scriptscriptstyle k}}_5 \left ( ( 1 -x^2 ) \upsilon_{\mu } - \frac{1}{2}x_{\mu } \right ) \\ & & + a^{{\scriptscriptstyle k}}_4 \left ( ( 1 + x^2 ) \upsilon_{\mu } + \frac{1}{2 } x_{\mu } \right ) + a^{{\scriptscriptstyle k}}_\nu \left ( \upsilon_{\mu } x^\nu + \delta_{\mu}^{\nu } \right ) \biggr\ } \\ { \displaystyle a_+^{{\scriptscriptstyle k } } } & = & { \displaystyle k \biggl\{a^{{\scriptscriptstyle k}}_5 ( 1- x^2 ) } + a^{{\scriptscriptstyle k}}_4 ( 1 + x^2 ) + a^{{\scriptscriptstyle k}}.x \biggr\}. \end{array } \right .\ ] ] this system can be obtained using the minkowskian system ( [ a(a)-m ] ) , the relation @xmath168 ( which comes out from the homogeneity properties of the fields ) and the extended weyl transformations ( [ extendedweyl - bis ] ) . this is inverted in @xmath169 the @xmath0 infinitesimal action on the field @xmath3 is given by commutators of the group generators and the field . first , we write down the infinitesimal transformations of the minkowskian fields @xmath96 which can be found in @xcite then we transport the resulting representation into @xmath25 . for any element @xmath170 , the related generator is denoted by @xmath171 and whose action on the field @xmath96 reads @xmath172 \\ & = x_{e } \ a_{{\scriptscriptstyle i}}^{{\scriptscriptstyle m } } + \left ( \sigma_{e } \right)_{{\scriptscriptstyle i}}^{{\scriptscriptstyle j } } \ a_{{\scriptscriptstyle j}}^{{\scriptscriptstyle m } } \end{split}\ ] ] where the first part represents the scalar action and the second the spinorial action . setting @xmath173 the minkowskian infinitessimal action reads @xmath174 } \ a^{{\scriptscriptstyle m}}_\tau \\ & \left ( x_{\mu\nu}^{{\scriptscriptstyle m } } \ a^{{\scriptscriptstyle m}}\right)_+ = x_{\mu\nu } a^{{\scriptscriptstyle m}}_+ , \end{aligned } \right.\ ] ] for the rotations , @xmath175 for the translations ; @xmath176}^\lambda + x^\lambda \eta_{\mu\nu } ) a^{{\scriptscriptstyle m}}_\lambda - 2\eta_{\mu\nu } a^{{\scriptscriptstyle m}}_+ \\ & \left(k_\mu^{{\scriptscriptstyle m } } \ a^{{\scriptscriptstyle m}}\right)_+ = k_\mu a^{{\scriptscriptstyle m}}_+ , \end{aligned } \right.\ ] ] for the special conformal transformations ( sct ) . finally , we have @xmath177 for the dilations . the undecomposable structure of the fields @xmath96 is made clear . under the @xmath0 action , the component @xmath178 overlaps @xmath179 which in turn overlaps @xmath180 . so we have the scheme @xmath181 the second step is to trasport the group action from minkowski to the @xmath79 space using the extended weyl transformation ( [ extendedweyl - bis ] ) @xmath182 a_{+}^{{\scriptscriptstyle k } } \end{split}\ ] ] where we have used @xmath183 for all @xmath184 . also only the second part of the last line has to be computed . note that the constraint @xmath189 ( [ a+=0 ] ) is @xmath0-invariant . this is important since this constraint defines the subset of physical states . the @xmath0-invariant scalar product for the minkowskian field @xmath96 reads @xmath190 where @xmath191 is some cauchy surface in @xmath32 and @xmath192 is a surface element . an important point is that this cauchy surface is common to all the spaces @xmath25 since they are all conformally equivalent @xcite . using ( [ lien - weyl ] ) and ( [ extendedweyl - bis ] ) , the scalar product for the field @xmath166 is obtained from ( [ scalarps - a ] ) and reads @xmath193 where the @xmath25 surface element is related to its minkoskian counterpart by @xmath194 .

we present an @xmath0-covariant quantization of the free electromagnetic field in conformally flat spaces ( cfs ) . a cfs is realized in a six - dimensional space as an intersection of the null cone with a given surface . the smooth move of the latter is equivalent to perform a weyl rescaling . this allows to transport the @xmath0-invariant quantum structure of the maxwell field from minkowski space to any cfs . calculations are simplified and the cfs wightman two - point functions are given in terms of their minkowskian counterparts . the difficulty due to gauge freedom is surpassed by introducing two auxiliary fields and using the gupta - bleuler quantization scheme . the quantum structure is given by a vacuum state and creators / annihilators acting on some hilbert space . in practice , only the hilbert space changes under weyl rescalings . also the quantum @xmath0-invariant free maxwell field does not distinguish between two cfss .

arxiv

the parallel chip - firing game or candy - passing game is a periodic automaton on graphs in which vertices , each of which contains some nonnegative number of chips , `` fire '' exactly one chip to each of their neighbors if possible . formally , let @xmath4 be an undirected graph with vertex set @xmath5 and edge set @xmath6 . define the _ parallel chip - firing game _ on @xmath4 to be an automaton governed by the following rules : * at the beginning of the game , @xmath7 chips are placed on each vertex @xmath8 in @xmath4 , where @xmath7 is a nonnegative integer . position _ of the parallel chip - firing game , denoted by @xmath9 , be the ordered pair @xmath10 containing the graph and the number of chips on each vertex of the graph . * at each _ move _ or _ step _ of the game , if a vertex @xmath8 has at least as many chips as it has neighbors , it will give ( _ fire _ ) exactly one chip to each neighbor . such a vertex is referred to as _ firing _ ; otherwise , it is _ non - firing_. all vertices fire simultaneously ( in parallel ) . we employ the notation of levine @xcite . let @xmath11 denote the step operator ; that is , @xmath12 is the position resulting after one step is performed on @xmath9 . let @xmath13 , and @xmath14 . we refer to @xmath15 as the position occurring _ after @xmath16 steps_. for simplicity , we limit our discussion to connected graphs . as the number of chips and number of vertices are both finite , there are a finite number of positions in this game . additionally , since each position completely determines the next position , it follows that for each initial position @xmath9 , there exist some positive integers @xmath17 such that for large enough @xmath18 , @xmath19 . we refer to the minimal such @xmath17 as the _ period _ @xmath20 of @xmath9 , and we refer to the set @xmath21 as one period of @xmath9 . also , we call the minimal such @xmath18 the _ transient length _ @xmath22 of @xmath9 . the parallel chip - firing game was introduced by bitar and goles @xcite in 1992 as a special case of the general chip - firing game posited by bjrner , lovsz , and shor @xcite in 1991 . they @xcite showed that the period of any position on a tree graph is 1 or 2 . in 2008 , kominers and kominers @xcite further showed that all connected graphs satisfying @xmath23 have period 1 ; they further established a polynomial bound for the transient length of positions on such graphs . their result @xcite that the set of all `` abundant '' vertices @xmath24 with @xmath25 stabilizes is particularly useful in simplifying the game . it was conjectured by bitar @xcite that @xmath26 for all games on all graphs @xmath4 . however , kiwi et . @xcite constructed a graph on which there existed a position whose period was at least @xmath27 , disproving the conjecture . still , it is thought that excluding particular graphs constructed to force long periods , most graphs still have periods that are at most @xmath28 . in 2008 , levine @xcite proved this for the complete graph @xmath1 . the parallel chip - firing game is a special case of the more general chip - firing game , in which at each step , a vertex is chosen to fire . the general chip - firing game , in turn , is an example of an _ abelian sandpile _ @xcite , and has been shown to have deep connections in number theory , algebra , and combinatorics , ranging from elliptic curves @xcite to the critical group of a graph @xcite to the tutte polynomial @xcite . bitar and goles @xcite observed that the parallel chip - firing game has `` nontrivial computing capabilities , '' being able to simulate the and , not , and or gates of a classical computer ; later , goles and margenstern @xcite showed that it can simulate any two - register machine , and therefore solve any theoretically solvable computational problem . finally , the parallel chip - firing game can be used to simulate a pile of particles that falls whenever there are too many particles stacked at any point ; this important problem in statistical physics is often referred to as the _ deterministic fixed - energy sandpile _ @xcite . the fixed - energy sandpile , in turn , is a subset of the more general study of the so - called _ spatially extended dynamical systems _ , which occur frequently in the physical sciences and even economics @xcite . such systems demonstrate the phenomenon of _ self - organized criticality _ , tending towards a `` critical state '' in which slight perturbations in initial position cause large , avalanche - like disturbances . self - organized critical models such as the abelian sandpile tend to display properties of real - life systems , such as @xmath29 noise , fractal patterns , and power law distribution @xcite . finally , the parallel chip - firing game is an example of a cellular automaton , the study of which have implications from biology to social science . in section 2 , we establish some lemmas about parallel chip - firing games on general simple connected graphs . we bound the number of chips on any single vertex in games with nontrivial period , define the notion of a complement position @xmath30 of @xmath9 and show that it has the same behavior as @xmath9 , and find a necessary and sufficient condition for a period to occur . then , in section 3 , we find , with proof , every possible period for the complete bipartite graph @xmath31 . we do so by first showing the only possible periods are of length @xmath32 or @xmath33 for @xmath34 , and then constructing games with such periods , proving our main result . finally , in section 4 , we construct positions on the complete @xmath2-partite graph @xmath35 with period @xmath17 for all @xmath36 . consider a simple connected graph @xmath4 . for each vertex @xmath8 in @xmath4 , let @xmath37 denote the number of firing neighbors @xmath38 of @xmath8 ; that is , the number of vertices @xmath38 neighboring @xmath8 satisfying @xmath39 . a step of the parallel chip - firing game on @xmath4 is then defined as follows : @xmath40 define a _ terminating _ position to be a position in which no vertices fire after finitely many moves . we begin our investigation by proving some lemmas limiting the number of chips on each vertex in a game with nontrivial period ( period greater than 1 ) . [ lem:2n-1 ] for sufficiently large @xmath18 , @xmath41 for all @xmath42 in all games with nontrivial period on a connected graph @xmath4 . kominers and kominers @xcite showed that if a vertex @xmath42 satisfies @xmath43 , then @xmath44 . they then showed that if , after sufficiently many steps @xmath18 , there still exists a vertex @xmath8 with @xmath45 , then all vertices must be firing from that step onward . since the period of a position is 1 if and only if either all or no vertices in @xmath4 are firing @xcite , @xmath46 is true for any game on @xmath4 with nontrivial period and sufficiently large @xmath18 . we further bound the number of chips on each vertex by generalizing a result of levine @xcite : [ lem : confined ] consider a vertex @xmath8 in position @xmath9 such that @xmath47 . then @xmath48 either @xmath49 or not . we consider the cases individually . if @xmath50 , then @xmath51 . so @xmath52 if instead @xmath53 , then @xmath54 . hence @xmath55 @xmath56 if a vertex @xmath8 satisfies @xmath57 , we call it _ confined_. furthermore , call a position confined if all vertices in the position are confined . note that for confined @xmath8 , @xmath58 lemmas [ lem:2n-1 ] and [ lem : confined ] imply that if @xmath59 , then @xmath60 is confined if @xmath61 , where @xmath22 is the transient length of @xmath9 ; that is , once the game reaches a position which repeats periodically , all subsequent positions are confined . we generally limit our discussion to confined positions to exclude positions with trivial periods . next , we define @xmath62 to be the indicator function of whether a vertex @xmath8 fires at step @xmath18 . we prove a lemma about positions that are equivalent , or have the same behavior , when acted upon by the step operator @xmath11 . [ lem : complement ] let the _ complement _ @xmath30 of a confined position @xmath9 be the position that results after replacing the @xmath7 chips on each vertex @xmath63 with @xmath64 chips . then @xmath65 . we begin by noticing that since @xmath9 is confined , each vertex @xmath8 has at most @xmath66 chips , so each vertex in @xmath30 has a nonnegative number of chips . observe that a vertex @xmath8 fires in @xmath30 exactly when it did not fire in @xmath9 . hence , @xmath67 , and all but @xmath37 neighbors will fire in @xmath68 . so @xmath69 this lemma means we may treat @xmath9 and @xmath30 as equivalent positions , as at any point during their firing , we may transform one into the other . this implies the following corollary : for all positions @xmath9 on @xmath4 , @xmath70 . next , we prove a proposition that characterizes a period of the game on any connected graph @xmath4 . for each position @xmath9 and vertex @xmath63 , let @xmath71 be the number of times @xmath8 fires in the first @xmath18 steps . [ prop : alleq ] the position @xmath9 on @xmath4 satisfies @xmath72 if and only if each vertex has fired the same number of times within those @xmath18 steps ; that is , iff for all vertices @xmath73 , @xmath74 if equation holds , then by equation , @xmath75 for all @xmath8 , so @xmath72 . conversely , if @xmath72 , consider the vertex @xmath76 such that @xmath77 is maximal . then , since @xmath78 for all vertices @xmath38 neighboring @xmath8 , @xmath79 but as @xmath80 , we see that @xmath81 must hold for all @xmath38 neighboring @xmath8 . since the graph is connected , we continue inductively through the entire graph to obtain equation . recall that a complete bipartite graph @xmath82 may be partitioned into two subsets of vertices , @xmath83 and @xmath84 , such that no edges exist among vertices in the same set , but every vertex in @xmath83 is connected to every vertex in @xmath84 . we refer to the sets @xmath85 as the _ sides _ of @xmath4 . define @xmath86 and @xmath87 . as stated above , bitar and goles @xcite showed that if no vertices or all vertices are firing , the period is 1 . we consider only games whose period is greater than 1 ; that is , at least one vertex is firing every turn , and not all vertices fire every turn . let @xmath88 and @xmath89 denote the number of vertices in @xmath83 and @xmath84 , respectively , that fire in @xmath9 . then , @xmath90 if @xmath91 , and @xmath92 if @xmath93 . notice that @xmath89 is the number of vertices in @xmath84 with at least @xmath94 chips , and @xmath88 is the number of vertices in @xmath83 with at least @xmath95 chips . let @xmath96 be the number of times any of the vertices in @xmath83 have fired in the first @xmath18 steps starting from , and including , @xmath15 , and define @xmath97 similarly . define @xmath98 and @xmath99 . without loss of generality , we prove facts about the vertices in @xmath83 , which also hold for vertices in @xmath84 . in the first @xmath18 steps , a vertex @xmath8 in @xmath83 fires a total of @xmath100 chips and receives @xmath101 chips . hence , @xmath102 next , we prove a lemma that bounds the number of times a vertex has fired once the position is confined . [ lem : diff1 ] let @xmath103 . if @xmath9 is confined , and @xmath104 , then for all @xmath105 , @xmath106 we prove this by induction on @xmath18 . the base case , @xmath107 , is straightforward : vertices @xmath8 and @xmath38 have each fired either 0 or 1 times . if @xmath8 fires after step @xmath108 , then @xmath109 chips , and @xmath38 also fires . now , assume @xmath110 if @xmath111 , then by equation , @xmath112 @xmath113 thus , if @xmath8 is ready to fire after step @xmath18 , then @xmath38 must be ready to fire also . it follows that @xmath114 otherwise , @xmath115 . then , since @xmath60 is confined from lemma [ lem : confined ] , by equation , @xmath116 @xmath117 by lemma [ lem : confined ] , since the degrees of both @xmath8 and @xmath38 are @xmath95 . so , if @xmath38 is ready to fire after step @xmath18 , so is @xmath8 , and equation again holds . from the above lemma , we can deduce the following : [ lem : ndiv ] if @xmath9 is confined and @xmath118 , then @xmath119 for all @xmath120 . let @xmath76 be the vertex in @xmath83 with @xmath121 minimal . by lemma [ lem : diff1 ] , for all @xmath91 , @xmath122 where @xmath123 . if @xmath124 is the number of vertices @xmath125 with @xmath126 , then @xmath127 since @xmath128 , we have @xmath129 . then @xmath130 because @xmath131 ; so @xmath132 for all @xmath91 . since @xmath133 , this implies @xmath119 for all @xmath91 . clearly , if all @xmath94 vertices in @xmath83 have fired the same number of times , then @xmath118 ; so we have found a necessary and sufficient condition for all vertices on the same side to fire the same number of times . but by proposition [ prop : alleq ] , a period is completed when all @xmath63 have fired the same number of times ; thus , we desire a relation between the sides that forces every vertex on both sides to fire the same number of times . our first step is the following lemma . [ lem : t1 ] if @xmath9 is confined , and @xmath134 for some positive integer @xmath32 , then @xmath135 for all @xmath136 . if @xmath93 is firing , then @xmath137 since @xmath9 is confined and @xmath8 is firing , @xmath138 ; and since @xmath139 is confined by lemma [ lem : confined ] , we have @xmath140 these two inequalities together imply that , for firing vertices @xmath8 , @xmath141 if @xmath136 is instead non - firing , then @xmath142 chips . @xmath143 is confined by lemma [ lem : confined ] , so @xmath144 ; since @xmath145 because @xmath8 is non - firing , we then deduce , similarly as above , that @xmath146 for non - firing vertices @xmath8 as well . therefore , for all @xmath136 , we have that @xmath147 so @xmath148 for all @xmath136 . if @xmath149 , then we can compute @xmath150 ; hence @xmath8 does not fire after step @xmath18 , and @xmath151 . if instead @xmath152 , then @xmath153 , so @xmath8 fires after step @xmath18 , and @xmath154 , and we are done . note that applying this lemma to @xmath15 also means @xmath155 . next , recalling the definition of @xmath156 in equation , we define @xmath157 for nonnegative integers @xmath16 and positive integers @xmath18 . note that by definition , @xmath158 applying lemmas [ lem : ndiv ] and [ lem : t1 ] to the position @xmath15 , we find that if @xmath159 , then @xmath160 for all vertices @xmath91 and all vertices @xmath161 . now , we give a sufficient condition for a period of a position on @xmath31 to occur . [ lem : periodt ] if @xmath9 is confined , and for some @xmath162 and @xmath163 , @xmath159 and @xmath164 for all @xmath91 , then @xmath165 if @xmath18 is chosen to be as small as possible . by lemma [ lem : ndiv ] applied to @xmath166 , since @xmath159 , @xmath167 for all @xmath91 . if for some @xmath162 , @xmath164 for all @xmath91 , then @xmath168 for all @xmath91 . but by equation , @xmath169 for all @xmath91 , @xmath161 . hence , @xmath170 for all vertices @xmath171 , which by proposition [ prop : alleq ] applied to @xmath172 implies @xmath173 , or @xmath174 . but @xmath18 is taken to be as small as possible , so @xmath165 . using this fact , we limit which periods are possible for games on @xmath31 . [ prop : t2 t ] if @xmath9 is confined and @xmath134 , then @xmath165 or @xmath175 . by iteratively applying lemma [ lem : t1 ] @xmath176 times , along with lemma [ lem : ndiv ] , we find that @xmath177 for all vertices @xmath91 and nonnegative even integers @xmath16 . expanding the sums @xmath178 and @xmath179 in terms of @xmath180 , we obtain @xmath181 for all nonnegative even integers @xmath16 and vertices @xmath91 . if @xmath164 for all @xmath91 and some @xmath182 , then the period is @xmath18 by lemma [ lem : periodt ] . otherwise , let @xmath183 be the set of vertices @xmath184 satisfying @xmath185 for all @xmath186 . the range of @xmath187 is @xmath188 , so this condition is equivalent to @xmath189 for all @xmath186 . we show that the period of @xmath9 is then @xmath190 . * case 1 . * @xmath18 is odd . consider some nonnegative integer @xmath191 , and let @xmath192 . if @xmath191 is even , let @xmath193 ; otherwise , let @xmath194 . if @xmath195 then @xmath196 and @xmath197 . but by equation , @xmath198 contradicting the assumption that @xmath185 for all @xmath186 . hence , @xmath199 for all nonnegative @xmath191 . this implies @xmath200 for all @xmath201 ; hence @xmath202 for all @xmath192 . since @xmath203 for other vertices @xmath204 , by lemma [ lem : periodt ] , @xmath175 . * case 2 . * @xmath18 is even . let @xmath192 . by equation , @xmath205 but by equation , @xmath206 for all @xmath192 . then as above , @xmath175 by lemma [ lem : periodt ] . more specifically , the following corollary holds : [ cor : lea ] let @xmath9 be a position on @xmath31 . if @xmath20 is odd , @xmath207 ; and if @xmath20 is even , @xmath208 . without loss of generality , let @xmath209 . if @xmath210 , @xmath211 . otherwise , @xmath59 . since @xmath212 , we may replace @xmath9 by @xmath213 and assume @xmath9 is confined by lemma [ lem : confined ] . by the pigeonhole principle , there must exist steps @xmath214 with @xmath215 . but then @xmath216 , so by proposition [ prop : t2 t ] applied to @xmath217 , @xmath218 or @xmath33 , where @xmath219 hence , if @xmath20 is odd , @xmath211 , and if @xmath20 is even , @xmath220 . finally , we characterize all possible periods for @xmath9 . [ prop : existpos ] there exist positions @xmath9 on @xmath82 with period @xmath32 and @xmath33 for all @xmath221 . without loss of generality , let @xmath209 . let @xmath222 be the vertices in @xmath83 , and @xmath223 be the vertices in @xmath84 . let @xmath32 be a positive integer such that @xmath224 . we represent each position @xmath9 on @xmath4 by two vectors @xmath225 @xmath226 consider the following position @xmath227 , which we claim has period @xmath32 : @xmath228 @xmath229 @xmath230 and @xmath231 fire , so @xmath232 is represented by @xmath233 @xmath234 we can see that the vertices @xmath235 with @xmath236 satisfy @xmath237 for @xmath238 . so , @xmath239 follows upon applying proposition [ prop : alleq ] , noting that after @xmath32 steps , each vertex has fired exactly once . hence , @xmath227 has period @xmath32 . next , consider the following position @xmath240 , which we claim has period @xmath33 : @xmath241 @xmath242 note that , if at any point @xmath243 , then @xmath244 , because @xmath245 and @xmath246 have the same neighbors . so , @xmath247 is represented by @xmath248 @xmath249 and @xmath250 is represented by @xmath251 @xmath234 we can see that for @xmath252 , the vertex in @xmath4 that fires ( has @xmath95 chips if it is in @xmath83 , or @xmath94 chips if it is in @xmath84 ) in position @xmath253 is @xmath254 so , after @xmath33 steps , every vertex will have fired once , and by proposition [ prop : alleq ] , @xmath255 . it remains to construct initial positions with period 1 or 2 . the trivial game with no chips on any vertex has period 1 , while the initial position where each vertex in @xmath83 has @xmath95 chips , and each vertex in @xmath84 has 0 chips , can be easily checked to have period 2 . thus , all periods @xmath256 , and @xmath257 , are achievable . combining our results , we obtain our main theorem . a nonnegative integer @xmath17 is a possible period of a position @xmath9 of the parallel chip - firing game on @xmath31 if and only if @xmath258 by corollary [ cor : lea ] , no period lengths may lie outside the sets in ; and in proposition [ prop : existpos ] , we have constructed positions with all such periods . we again use the vector notation from above to represent the positions of a parallel chip - firing game on the complete @xmath2-partite graph @xmath259 formed by joining the anticliques @xmath260 ; let the vertices in @xmath261 be @xmath262 for each @xmath263 . without loss of generality , we will assume @xmath264 . as above , we represent a position on @xmath4 by the set of vectors @xmath265 below is a representation of a position which has period @xmath266 for all @xmath267 and @xmath268 . note that a vertex in @xmath269 fires when it has at least @xmath270 chips ; here @xmath271 is the degree of any vertex in @xmath269 . for our construction , we let @xmath272 we now show that this position indeed has period @xmath266 . let @xmath273 be the set of all firing vertices in @xmath274 . it can be checked that @xmath275 @xmath276 for @xmath277 ; and @xmath278 for @xmath279 , encompassing steps @xmath280 through @xmath281 ; @xmath282 for @xmath283 ; and @xmath284 for @xmath285 . ( in fact , each vertex @xmath63 fires exactly when it contains @xmath286 chips . ) the latter two categories describe which vertices fire during steps @xmath287 through @xmath288 . but after this @xmath289 step , every vertex in @xmath4 has fired exactly once ; the last to fire is @xmath290 . hence , @xmath291 by proposition [ prop : alleq ] , and the period is @xmath292 as desired . this means all periods from @xmath293 to @xmath294 are achievable , as @xmath295 ranges from @xmath108 to @xmath296 and @xmath32 ranges from @xmath293 to @xmath297 . as an example , consider the following position on the graph @xmath298 with period @xmath299 : @xmath300 for this position , @xmath301 , and its predicted period length is @xmath302 as desired . for several graphs , a proof of bitar s conjecture that @xmath26 for all parallel chip - firing games on those graphs would be interesting ; we proved the conjecture for the complete bipartite graph . though we have constructed many periods of games on complete @xmath2-partite graphs in section 4 , there exist periods longer than those detailed . for example , take the following position on @xmath303 , which has period 5 : @xmath304 though positions with these larger periods are more difficult to characterize generally , bitar s conjecture still appears to be true for complete @xmath2-partite graphs . moreover , bounding the periods of positions on vertex - regular graphs and more general bipartite graphs are directions for further research . by doubling the length of each cycle in the graph used in the counterexample by kiwi et . @xcite , we find a counterexample on a graph containing only even cycles , that is , for the general bipartite graph . we would also like to determine which periods less than the bound are possible . levine @xcite related period lengths of games on the complete graph to the _ activity _ , defined as @xmath305 . on the other hand , we believe that period lengths are related to lengths of subcycles ( closed paths ) of the graph @xmath4 ; in particular , we conjecture that any period length of a game on @xmath4 is either a divisor of the order of some subcycle of @xmath4 , or perhaps the least common multiple of the orders of some disjoint subcycles of @xmath4 . this agrees with known results for the tree graph @xcite , complete graph @xcite , and now the complete bipartite graph . our numerical experiments have also verified this conjecture for cycle graphs and complete @xmath32-partite graphs ; in fact , my correspondence with zhai @xcite has produced a proof of this conjecture for the cycle graph . another interesting direction to pursue is observing the implications of `` reducing '' the parallel chip - firing game by removing as many chips as possible from each vertex without affecting their firing pattern ( without changing @xmath156 for all @xmath42 and @xmath306 ) . this reduction may simplify some games into being more approachable by induction . besides studying period lengths of parallel chip - firing games , an examination of the transient length of games on certain graphs would be useful in modeling real - world phenomena . studying transient positions would also help uncover what attributes determine whether a position is within a period or not , and bounding the transient length would make for more efficient computation of the period length of games on complex graphs . chip - firing games on lattices and tori have been used as cellular automaton models of the deterministic fixed - energy sandpile ( see @xcite ) . since most studies of sandpiles have been concerned with asymptotic measures such as the `` activity , '' bounding the period length of such models could serve as a measure of the fidelity of the model to the real world . the author would like to thank my mentor , yan zhang of the massachusetts institute of technology , for his teaching , guidance , and support , the center for excellence in education and the research science institute for sponsoring my research , and dr . john rickert for tips on writing and lecturing . the author would also like to thank dr . lionel levine , scott kominers , paul kominers , daniel vitek , brian hamrick , dr . dan teague , patrick tenorio , alex zhai , and dr . ming ya jiang for valuable pointers and advice on this problem and this paper .

the parallel chip - firing game is a periodic automaton on graphs in which vertices `` fire '' chips to their neighbors . in 1989 , bitar conjectured that the period of a parallel chip - firing game with @xmath0 vertices is at most @xmath0 . though this conjecture was disproven in 1994 by kiwi et . al . , it has been proven for particular classes of graphs , specifically trees ( bitar and goles , 1992 ) and the complete graph @xmath1 ( levine , 2008 ) . we prove bitar s conjecture for complete bipartite graphs and characterize completely all possible periods for positions of the parallel chip - firing game on such graphs . furthermore , we extend our construction of all possible periods for games on the bipartite graph to games on complete @xmath2-partite graphs , @xmath3 , and prove some pertinent lemmas about games on general simple connected graphs . tian - yi jiang

arxiv

thanks to a fortunate coincidence of observations by agile , _ fermi _ , and _ swift _ satellites , together with the optical observations by the vlt / fors2 and the nordic optical telescope , it has been possible to obtain an unprecedented set of data , extending from the optical - uv , through the x - rays , all the way up to the high energy ( gev ) emission , which allowed detailed temporal / spectral analyses on grb 090510 @xcite . in contrast with this outstanding campaign of observations , a theoretical analysis of the broadband emission of grb 090510 has been advanced within the synchrotron / self - synchrotron compton ( ssc ) and traditional afterglow models ( see , e.g. , sections 5.2.1 and 5.2.2 in * ? ? ? * ) . paradoxically , this same methodology has been applied in the description of markedly different type of sources : e.g. , @xcite for the low energetic long grb 060218 , @xcite for the high energetic long grb 130427a , and @xcite for the s - grf 051221a . in the meantime , it has become evident that grbs can be subdivided into a variety of classes and sub - classes @xcite , each of them characterized by specific different progenitors which deserve specific theoretical treatments and understanding . in addition every sub - class shows different episodes corresponding to specifically different astrophysical processes , which can be identified thanks to specific theoretical treatments and data analysis . in this article , we take grb 090510 as a prototype for s - grbs and perform a new time - resoved spectral analysis , in excellent agreement with the above temporal and spectral analysis performed by , e.g. , the _ fermi _ team . now this analysis , guided by a theoretical approach successfully tested in this new family of s - grbs @xcite , is directed to identify a precise sequence of different events made possible by the exceptional quality of the data of grb 090510 . this include a new structure in the thermal emission of the p - grb emission , followed by the onset of the gev emission linked to the bh formation , allowing , as well , to derive the structure of the circumburst medium from the spiky structure of the prompt emission . this sequence , for the first time , illustrates the formation process of a bh . already in february 1974 , soon after the public announcement of the grb discovery @xcite , @xcite presented the possible relation of grbs with the vacuum polarization process around a kerr - newman bh . there , evidence was given for : a ) the formation of a vast amount @xmath2-baryon plasma ; b ) the energetics of grbs to be of the order of @xmath11 erg , where @xmath12 is the bh mass ; c ) additional ultra - high energy cosmic rays with energy up to @xmath13 ev originating from such extreme process . a few years later , the role of an @xmath2 plasma of comparable energetics for the origin of grbs was considered by @xcite and it took almost thirty years to clarify some of the analogies and differences between these two processes leading , respectively , to the alternative concepts of fireball " and fireshell " @xcite . in this article we give the first evidence for the formation of a kerr newman bh , in grb 090510 , from the merger of two massive nss in a binary system . grbs are usually separated in two categories , based on their duration properties . short grbs have a duration @xmath14 s while the remaining ones with @xmath15 s are traditionally classified as long grbs . short grbs are often associated to ns - ns mergers ( see e.g. @xcite ; see also @xcite for a recent review ) : their host galaxies are of both early- and late - type , their localization with respect to the host galaxy often indicates a large offset @xcite or a location of minimal star - forming activity with typical circumburst medium ( cbm ) densities of @xmath16@xmath17 @xmath18 , and no supernovae ( sne ) have ever been associated to them . the progenitors of long grbs , on the other hand , have been related to massive stars @xcite . however , in spite of the fact that most massive stars are found in binary systems @xcite , that most type ib / c sne occur in binary systems @xcite and that sne associated to long grbs are indeed of type ib / c @xcite , the effects of binarity on long grbs have been for a long time largely ignored in the literature . indeed , until recently , long grbs have been interpreted as single events in the jetted _ collapsar _ fireball model ( see e.g. @xcite and references therein ) . multiple components evidencing the presence of a precise sequence of different astrophysical processes have been found in several long grbs ( e.g. @xcite , @xcite ) . following this discovery , further results led to the introduction of a new paradigm expliciting the role of binary sources as progenitors of the long grb - sn connection . new developments have led to the formulation of the induced gravitational collapse ( igc ) paradigm @xcite . the igc paradigm explains the grb - sn connection in terms of the interactions between an evolved carbon - oxygen core ( co@xmath19 ) undergoing a sn explosion and its hypercritical accretion on a binary ns companion @xcite . the large majority of long bursts is related to sne and are spatially correlated with bright star - forming regions in their host galaxies @xcite with a typical cbm density of @xmath20 @xmath18 @xcite . a new situation has occurred with the observation of the high energy gev emission by the _ fermi_-lat instrument and its correlation with both long and short bursts with isotropic energy @xmath21 erg , which has been evidenced in @xcite and @xcite , respectively . on the basis of this correlation the occurrence of such prolonged gev emission has been identified with the onset of the formation of a bh @xcite . as recalled above , the long grbs associated to sne have been linked to the hypercritical accretion process occurring in a tight binary system when the ejecta of an exploding co@xmath19 accretes onto a ns binary companion ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? when the hypercritical accretion occurs in a widely separated system with an orbital separation @xmath22 cm @xcite , the accretion is not sufficient to form a bh . for these softer systems with rest - frame spectral peak energy @xmath23 kev the upper limit of their observed energy is @xmath24 erg , which corresponds to the maximum energy attainable in the accretion onto a ns @xcite . such long a burst corresponds to an x - ray flash ( xrf ) . the associated x - ray afterglow is also explainable in terms of the interaction of the prompt emission with the sn ejecta ( fryer et al . , in preparation ) . in these systems no gev emission is expected in our theory and , indeed , is not observed . interestingly , a pioneering evidence for such an x - ray flash had already been given in a different context by @xcite , @xcite , and @xcite . for tighter binaries ( @xmath25 cm , @xcite ) , the hypercritical accretion onto the companion ns leads to the formation of a bh . for these harder systems with @xmath26 kev the lower limit of their observed energy is @xmath24 erg , which necessarily needs the accretion process into a bh . an associated prolonged gev emission occurs after the p - grb emission and at the beginning of the prompt emission , and originates at the onset of the bh formation @xcite . these more energetic events are referred to as binary - driven hypernovae ( bdhne ) . specific constant power - law behaviors are observed in their high energy gev , x - rays , and optical luminosity light curves @xcite . in total analogy , the formation of a bh can occur in short bursts , depending on the mass of the merged core of the binary system . when the two ns masses are large enough , the merged core can exceed the ns critical mass and the bh formation is possible . in the opposite case , a massive ns ( mns ) is created , possibly , with some additional orbiting material to guarantee the angular momentum conservation . we then naturally expect the existence of two short bursts sub - classes : authentic short grbs ( s - grbs ) , characterized by the formation of a bh @xcite , with @xmath21 erg , a harder spectrum ( see section [ epeakeiso ] ) and associated with a prolonged gev emission ( see section [ gevemission ] ) ; short gamma - ray flashes ( s - grfs ) , producing a mns @xcite , with @xmath27 erg . in this second sub - class , of course , the gev emission should not occur and , indeed , is never observed . , where @xmath28 is the critical value for vacuum polarization and @xmath29 is the electric field strength . the plot assumes a black hole mass energy @xmath30 . figure reproduced from @xcite with their kind permission . ] following the discovery of the first prototype of this s - grb class , namely grb 090227b @xcite , the first detailed analysis of such a genuine short grb originating from a binary ns merger leading to a bh was done for grb 140619b by @xcite , determining as well the estimated emission of gravitational waves . the latter has been estimated following the method applied by @xcite for grb 090227b . from the spectral analysis of the early @xmath31 s , they inferred an observed temperature @xmath32 kev of the @xmath2 plasma at transparency ( p - grb ) , a theoretically derived redshift @xmath33 , a total burst energy @xmath34 erg , a rest - frame peak energy @xmath35 mev , a baryon load @xmath36 , and an average cbm density @xmath37 @xmath18 . we turn in this article to the most interesting case of grb 090510 which has , in addition to very similar properties of the members of this new class of s - grb sources , a spectroscopically determined value of the redshift and represents one of the most energetic sources of this family both in the @xmath38-ray and in the gev ranges . actually , a first attempt to analyze grb 090510 was made by interpreting this source as a long grb @xcite . an unusually large value of the cbm density was needed in order to fit the data : this interpretation was soon abandoned when it was noticed that grb 090510 did not fulfill the nesting conditions of the late x - ray emission typical of long grbs @xcite , see also section [ xray ] and figure [ episode3 ] . in light of the recent progress in the understanding of the fireshell theory , we address the interpretation of grb 090510 as the merging of a binary ns . we give clear evidence for the validity of this interpretation . in view of the good quality of the data both in @xmath38- rays and in the gev range , we have performed a more accurate description of the p - grb , best fitted by a convolution of thermal spectra . this novel feature gives the first indication for the existence of an axially symmetric configuration of the dyadotorus emitting the @xmath2 plasma which had been previously theoretically considered and attentively searched for . this gives the first indication that indeed the angular momentum plays a role and a dyadotorus is formed , as theoretically predicted in a series of papers ( see * ? ? ? * and figure [ dyadotorus ] ) . this naturally leads to the evidence for the formation of a rotating bh as the outcome of the gravitational collapse . we turn then to the main new feature of grb 090510 which is the high energy @xmath6@xmath7 gev emission ( see figure [ 090510gev ] ) . the direct comparison of the gev emission in this source and in the bdhne 130427a shows the remarkable similarities of these two gev components ( see figure [ 090510gev ] ) . the fact that the s - grb 090510 originates from a binary ns merger and the bdhn 130427a from the igc of a sn hypercritical accretion process onto a companion ns clearly points to the bh as originating this gev emission , the reason being that these two astrophysical systems are different in their progenitors and physical process and have in the formation of a bh their unique commonality . this paper is structured as follows : in section 2 we summarize the relevant aspects of the fireshell theory and compare and contrast it with alternative approaches . in section 3 we discuss the recent progress on the ns equilibrium configuration relevant for s - grbs and bdhne . in section 4 we move on to describe the observations of grb 090510 and their analysis . the s - grb nature of grb 090510 is justified in section 5 , and we offer an interpretation of our results in section 6 . section 7 concludes this work . a standard flat @xmath39cdm cosmological model with @xmath40 and @xmath41 km s@xmath42 mpc@xmath42 is adopted throughout the paper . the fireshell scenario @xcite , has been initially introduced to describe a grb originating in a gravitational collapse leading to the formation of a kerr - newman bh . a distinct sequence of physical and astrophysical events are taken into account : * an optically thick pair plasma the fireshell of total energy @xmath43 is considered . as a result , it starts to expand and accelerate under its own internal pressure @xcite . the baryonic remnant of the collapsed object is engulfed by the fireshell the baryonic contamination is quantified by the baryon load @xmath44 where @xmath45 is the mass of the baryonic remnant @xcite . * after the engulfment , the fireshell is still optically thick and continues to self - accelerate until it becomes transparent . when the fireshell reaches transparency , a flash of thermal radiation termed proper - grb ( p - grb ) is emitted @xcite . * in grbs , the @xmath2-baryon plasma evolves from the ultra - relativistic region near the bh all the way reaching ultra - relativistic velocities at large distances . to describe such a dynamics which deals with unprecedentedly large lorentz factors and also regimes sharply varying with time , in @xcite it has been introduced the appropriate relative spacetime transformation paradigm . this paradigm gives particular attention to the constitutive equations relating four time variables : the comoving time , the laboratory time , the arrival time , and the arrival time at the detector corrected by the cosmological effects . this paradigm is essential for the interpretation of the grb data : the absence of adopting such a relativistic paradigm in some current works has led to a serious misinterpretation of the grb phenomenon . * in compliance with the previous paradigm , the interactions between the ultra - relativistic shell of accelerated baryons left over after transparency and the cbm have been considered . they lead to a modified blackbody spectrum in the co - moving frame @xcite . the observed spectrum is however non - thermal in general ; this is due to the fact that , once the constant arrival time effect is taken into account in the equitemporal surfaces ( eqts , see * ? ? ? * ; * ? ? ? * ) , the observed spectral shape results from the convolution of a large number of modified thermal spectra with different lorentz factors and temperatures . * all the above relativistic effects , after the p - grb emission , are necessary for the description of the prompt emission of grbs , as outlined in @xcite . the prompt emission originates in the collisions of the accelerated baryons , moving at lorentz factor @xmath46@xmath47 , with interstellar clouds of cbm with masses of @xmath48@xmath49 g , densities of @xmath50@xmath3 @xmath18 and size of @xmath51@xmath52 cm , at typical distances from the bh of @xmath53@xmath54 cm ( see , e.g. , @xcite for long bursts ) . our approach differs from alternative tratments purporting late activities from the central engine ( see , e.g. , the _ collapsar _ model in @xcite , @xcite , @xcite and references therein , and the _ magnetar _ model in @xcite , @xcite , @xcite , @xcite , @xcite , and references therein ) . * @xmath43 and @xmath55 are the only two parameters that are needed in a spherically symmetric fireshell model to determine the physics of the fireshell evolution until the transparency condition is fulfilled . three additional parameters , all related to the properties of the cbm , are needed to reproduce a grb light curve and its spectrum : the cbm density profile @xmath56 , the filling factor @xmath57 that accounts for the size of the effective emitting area , and an index @xmath58 that accounts for the modification of the low - energy part of the thermal spectrum @xcite . they are obtained by running a trial - and - error simulation of the observed light curves and spectra that starts at the fireshell transparency . * a more detailed analysis of pair cration process around a kerr - newman bh has led to the concept of dyadotorus @xcite . there , the axially symmetric configuration with a specific distribution of the @xmath2 , as well as its electromagnetic field , have been presented as function of the polar angle . the total spectrum at the transparency of the @xmath2plasma is a convolution of thermal spectra at different angles . this formalism describing the evolution of a baryon - loaded pair plasma is describable in terms of only three intrinsic parameters : the @xmath2 plasma energy @xmath59 , the baryon load @xmath55 , and the specific angular momentum @xmath60 of the incipient newly - formed bh . it is , therefore , independent of the way the pair plasma is created . in addition to the specific case , developed for the sake of example , of the dyadotorus created by a vacuum polarization process in an already formed kerr - newman bh , more possibilities have been envisaged in the meantime : * the concept of dyadotorus can be applied as well in the case of a pair plasma created via the @xmath61 mechanism in a ns merger as described in @xcite , @xcite , @xcite , @xcite , assuming that the created pair plasma is optically thick . the relative role of neutrino and weak interactions vs. the electromagnetic interactions in building the dyadotorus is currently topic of intense research . * equally important are the relativistic magneto - hydrodynamical process leading to a dyadotorus , indicated in the general treatment of @xcite , and leading to the birth of a kerr - newman bh , surrounded by an opposite charged magnetosphere in a system endowed with global charge neutrality . active research is ongoing . * progress in understanding the ns equilibrium configuration imposing the global charge neutrality condition , as opposed to the local charge neutrality usually assumed @xcite . a critical mass for a non - rotating ns @xmath62 has been found for the nl3 nuclear equation of state @xcite . the effects of rotation and of the nuclear equation of state on the critical mass is presented in @xcite and in @xcite . the existence of electromagnetic fields close to the critical value has been evidenced in the interface between the core and the crust in the above global neutrality model , as well as very different density distributions in the crust and in the core , which could play an important role during the ns ns mergers ( see figure [ nanda ] and * ? ? ? ( _ middle panel _ ) in the core - crust transition layer normalized to the @xmath63-meson compton wavelength @xmath64 fm . _ lower panel _ : density profile inside a ns star with central density @xmath65 , where @xmath66 is the nuclear density , from the solution of the tov equations ( locally neutral case ) and the globally neutral solution presented in @xcite . the density at the edge of the crust is the neutron drip density @xmath67 g @xmath68 . reproduced from @xcite with their kind permission . ] the above three possibilities have been developed in recent years , but they do not have to be considered exaustive for the formation of a dyadotorus endowed by the above three parameters . in conclusion the evolution in the understanding of the grb phenomenon , occurring under very different initial conditions , has evidenced the possibility of using the dyadotorus concept for describing sources of an optically thick baryon - loaded @xmath2 plasma within the fireshell treatment in total generality . the key role neutrino emission in the hypercritical accretion process onto a ns has been already examined in the literature ( see , e.g. , * ? ? ? * ; * ? ? ? the problem of hypercritical accretion in a binary system composed of a co@xmath19 and a companion ns has been studied in @xcite ( see also references therein ) . the energy released during the process , in form of neutrinos and photons , is given by the gain of gravitational potential energy of the matter being accreted by the ns and depends also on the change of binding energy of the ns while accreting both matter and on the angular momentum carried by the accreting material ( see , e.g. , @xcite and @xcite ) . for a typical ns mass of @xmath70 m@xmath71 , a value observed in galactic ns binaries , and a ns critical mass @xmath72 in the range from @xmath73 m@xmath71 up to @xmath74 depending on the equations of state and angular momentum ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * for details ) , the accretion luminosity can be as high as @xmath75@xmath76 erg s@xmath42 for accretion rates @xmath77@xmath78 s@xmath42 ( see * ? ? ? * ; * ? ? ? * for details ) . for binary systems with a separation @xmath79 cm ( @xmath80 min ) , our numerical simulations indicate that : a ) the accretion process duration lasts @xmath81 s ( see , e.g. , * ? ? ? * ; * ? ? ? * ) , b ) the ns collapses to a bh , and c ) a total energy larger than @xmath69 erg is released during the hypercritical accretion process . these systems correspond to the bdhne @xcite . for systems with larger separations the hypercritical accretion is not sufficient to induce the collapse of the ns into a bh and the value of @xmath69 erg represents a theoretical estimate of the upper limit to the energy emitted by norm in the hypercritical accretion process . this sub - class of sources corresponds to the xrfs @xcite . the same energetic considerations do apply in the analysis of the hypercritical accretion occurring in a close binary ns system undergoing merging @xcite . therefore , in total generality , we can conclude that the energy emitted during a ns ns merger leading to the formation of a bh should be larger than @xmath69 erg ( see figure [ rt_gw ] ) . the limit of @xmath69 erg clearly depends on the initial ns mass undergoing accretion , by norm assumed to be @xmath70 m@xmath71 , and on the yet unknown value of @xmath72 , for which only an absolute upper limit of @xmath82 m@xmath71 has been established for the non - rotating case @xcite . as already pointed out in @xcite , for ns ns mergers , the direct determination of the energy threshold of @xmath69 erg dividing s - grfs and s - grbs , as well as xrfs and bdhne , provides fundamental informations for the determination of the actual value of @xmath83 , for the minimum mass of the newly - born bh , and for the mass of the accreting ns . ] ) . the binary orbit gradually shrinks due to energy loss through gravitational waves emission ( yellow - brown ) . at point a , the merger occurs : the fireshell ( in red ) is created and starts its expansion . it reaches transparency at point b , emitting the p - grb ( light purple ) . the prompt emission ( deep purple ) then follows at point c. the dashed lines represent the gev emission ( delayed relative to the start of the grb ) originating in the newly - born bh . this space - time diagram well illustrates how the gev emission originates in the newly - born bh and follows a different space - time path from the prompt emission , contrary to what stated in @xcite . the prompt emission originates from the interactions of the baryons , accelerated to ultrarelativistic lorentz factors during the pair - baryon electromagnetic pulse , with the clumpy circumburst medium ( see section 2 ) . the analysis of the spiky structure of the prompt emission allows to infer the structure of the circumburst medium ( see figure [ 090510simlc ] ) . there is the distinct possibility that the gev emission prior to @xmath84 s in the arrival time may interact with the prompt emission . in this sense the work by @xcite may become of interest . ] in this section , we summarize the observations of grb 090510 as well as the data analysis . we used _ ( gbm and lat ) and _ swift_/xrt data for the purposes of this work . the _ fermi_/gbm instrument @xcite was triggered at @xmath8500:22:59.97 ut on may 10 , 2009 by the short and bright burst grb 090510 ( @xcite , trigger 263607781 / 090510016 ) . the trigger was set off by a precursor emission of duration 30 ms , followed @xmath86 0.4 s later by a hard episode lasting @xmath87 s. this grb was also detected by swift @xcite , _ fermi_/lat @xcite , agile @xcite , konus - wind @xcite , and suzaku - wam @xcite . the position given by the gbm is consistent with that deduced from _ swift _ and lat observations . during the first second after lat trigger at 00:23:01.22 ut , _ fermi_/lat detected over 50 events ( respectively over 10 ) with an energy above 100 mev ( respectively above 1 gev ) up to the gev range , and more than 150 ( respectively more than 20 ) within the first minute @xcite . this makes grb 090510 the first bright short grb with an emission detected from the kev to the gev range . observations of the host galaxy of grb 090510 , located by vlt / fors2 , provided a measurement of spectral emission lines . this led to the determination of a redshift @xmath88 @xcite . the refined position of grb 090510 obtained from the nordic optical telescope @xcite is offset by 0.7 " relative to the center of the host galaxy in the vlt / fors2 image . at @xmath89 , this corresponds to a projected distance of 5.5 kpc . the identified host galaxy is a late - type galaxy of stellar mass @xmath90 , with a rather low star - forming rate sfr @xmath91 ( @xcite and references therein ) . . lower panel : comptonized + power law best fit of the corresponding spectrum ( from @xmath92 to @xmath93 s).,title="fig : " ] . lower panel : comptonized + power law best fit of the corresponding spectrum ( from @xmath92 to @xmath93 s).,title="fig : " ] [ cols="<,^,^,^,^,^,^,^",options="header " , ] in order to determine the profile of the cbm , a simulation of the prompt emission following the p - grb has been performed . the simulation starts at the transparency of the fireshell with the parameters that we determined above . a trial - and - error procedure is undertaken , guided by the necessity to fit the light curve of grb 090510 . the results of this simulation ( reproduction of the light curve and spectrum , in the time interval from @xmath94 to @xmath95 s , and cbm profile ) are shown in figure [ 090510simlc ] . the average cbm density is found to be @xmath96 . this low value , typical of galactic halo environments , is consistent with the large offset from the center of the host and further justifies the interpretation of grb 090510 as a short grb originating in a binary ns merger . our theoretical fit of the prompt emission ( see red line in the middle plot of figure [ 090510simlc ] ) predicts a cut - off at @xmath97 mev . the spectrum at energy @xmath98 mev could be affected by the onset of the high energy power - law component manifested both in the data of the mini - calorimeter on board agile ( see top panel of figure 4 in @xcite ) and in the data points from the _ fermi_-gbm bgo - b1 detector . there is a weak precursor emission about 0.4 s before the p - grb ( or @xmath99 s in the cosmological rest frame ) . two gev photons have been detected during the precursor emission . precursors are commonly seen in long bursts : @xcite found that @xmath100 20% of them show evidence of an emission preceding the main emission by tens of seconds . short bursts are less frequently associated with precursors . no significant emission from the grb itself is expected prior to the p - grb since it marks the transparency of the fireshell but the precursor may be explainable in the context of a binary ns merger by invoking the effects of the interaction between the two nss just prior to merger . indeed , it has been suggested that precursor emission in short bursts may be caused by resonant fragmentation of the crusts @xcite or by the interaction of the ns magnetospheres @xcite . the timescale ( @xmath100 0.21 s between the precursor and the p - grb ) is consistent with a pre - merger origin of the precursor emission . from its formation to its transparency , the fireshell undergoes a swift evolution . the thermalization of the pair plasma is achieved almost instantaneously ( @xmath101 s , @xcite ) ; and the @xmath102 plasma of grb 090510 reaches the ultra - relativistic regime ( i.e. a lorentz factor @xmath103 ) in a matter of @xmath104 s , according to the numerical simulation . the radius of the fireshell at transparency , @xmath105 cm , corresponds to more than a hundred light - seconds ; however relativistic motion in the direction of the observer squeezes the light curve by a factor @xmath106 , which makes the fireshell capable of traveling that distance under the observed timescale . the spectral analysis of this precursor is limited by the low number of counts . @xcite interpreted the spectrum with a blackbody plus power law model . this leads to a blackbody temperature of @xmath107 kev . the isotropic energy contained in the precursor amounts to @xmath108 erg . an interesting feature of the fireshell model is the possibility to infer a theoretical redshift from the observations of the p - grb and the prompt emission . in the case of grb 090510 , a comparison is therefore possible between the measured redshift @xmath88 and its theoretical derivation . an agreement between the two values would in particular strengthen the validity of our p - grb choice , which would in turn strengthen our results obtained with this p - grb . the feature of redshift estimate stems from the relations , engraved in the fireshell theory , between different quantities computed at the transparency point : the radius in the laboratory frame , the co - moving frame and blue - shifted temperatures of the plasma , the lorentz factor , and the fraction of energy radiated in the p - grb and in the prompt emission as functions of @xmath55 ( see figure 4 in @xcite ) . thus , the ratio @xmath109 implies a finite range for the coupled parameters @xmath43 and @xmath55 ( last panel of figure 4 in @xcite ) . assuming @xmath110 , this ratio is known since it is equal to the ratio between the observed fluences of the respective quantities : @xmath111 with the measured values @xmath112 erg @xmath113 and @xmath114 erg @xmath113 , we find @xmath115 . in addition , knowing the couple [ @xmath116 , @xmath55 ] gives the ( blue - shifted towards the observer ) temperature of the fireshell at transparency @xmath118 ( figure 4 in @xcite , second panel ) . but we also have the following relation between @xmath118 and the observed temperature at transparency @xmath119 , linking their ratio to the redshift : @xmath120 finally , since we assume that @xmath110 , we also have an expression of @xmath43 as a function of @xmath121 using the formula of the k - corrected isotropic energy : @xmath122 where @xmath123 is the photon spectrum of the grb and the fluence @xmath124 is obtained in the full gbm energy range 8 40000 kev . the use of all these relations allows a redshift to be determined by an iterative procedure , testing at every step the value of the parameters @xmath125 and @xmath118 . the procedure successfully ends when both values are consistent according to the relations described above . in the case of grb 090510 , we find @xmath126 , which provides a satisfactory agreement with the measured value @xmath88 . grb 090510 is associated with a high - energy emission , consistently with all other observed s - grbs , i.e. energetic events with @xmath21 erg . the only case of a s - grb without gev emission , namely grb 090227b , has been explained by the absence of alignment between the lat and the source at the time of the grb emission . nevertheless evidence of some gev emission in this source has been recently obtained ( ruffini et al . , in preparation ) . the gev light curve of grb 090510 is plotted in figure [ 090510gev ] together with other s - grb light curves and showing a common power - law behavior , which goes as @xmath127 , similar to the clustering of the gev light curves found by @xcite . these s - grbs are compared with that of the bdhne 130427a which shares a similar behavior . @xcite suggest and argue that the gev emission is related to the presence of a bh and its activity . this view is supported by the fact that the gev emission is delayed with respect to the @xmath38-ray emission : it starts only after the p - grb is over . the gev emission of grb 090510 is particularly intense , reaching @xmath128 erg . such a large value , one of the largest observed among s - grbs , is consistent with the large angular momentum of the newborn bh . this energetic can not be explained in terms of nss in view of the lower value of the gravitational binding energy . the absence of gev emission in s - grfs is also confirmed from the strong upper limit to the gev emission for s - grbs imposed by the fermi - lat sensitivity . we assume for a moment that the gev emission of a s - grf is similar to that of s - grb . we then compute the observed gev flux light curve of s - grb 090510 at different redshifts , e.g. , @xmath129 and @xmath130 , which correspond to the redshifts of the s - grb 081024b and of the s - grb 140402a , respectively ( aimuratov et al . , in preparation ) . the result is that if we compare these computed flux light curves with the _ fermi_-lat sensitivity of the pass 8 release 2 version 6 instrument response functions , which is approximately @xmath131 erg @xmath113s@xmath42 , all of them are always well above the lat broadband sensitivity by a factor @xmath132 ( see figure [ gevz ] ) . this result does not depend on the choice of the source . in their rest - frame all the s - grb gev light curves follow a similar behavior . therefore , the gev emission of s - grb 090510 is always above @xmath132 times to the lat sensitivity , even at higher redshifts . if we now assume that s - grfs do conform to the same behavior of s - grbs , the absence of detection of gev emission implies that the s - grfs have necessarily fluxes at least @xmath133@xmath134 times smaller than those of s - grbs . @xmath7 gev flux light curve of the s - grbs 090510 ( red squares ) , and the corresponding ones obtained by translating this s - grb at @xmath129 ( blue circles ) and at @xmath135 ( green diamonds ) . ] in order to estimate the energy requirement of the @xmath6@xmath7 gev emission of figure [ 090510gev ] we consider the accretion of mass @xmath136 onto a kerr - newman bh , dominated by its angular momentum and endowed with electromagnetic fields not influencing the geometry , which remains approximately that of a kerr bh . we recall that if the infalling accreted material is in an orbit co - rotating with the bh spin , up to @xmath137 of the initial mass is converted into radiation , for a maximally rotating kerr bh , while this efficiency drops to @xmath138 , when the infalling material is on a counter - rotating orbit ( see ruffini & wheeler 1969 , in problem 2 of @xmath139 104 in @xcite ) . therefore , the gev emission can be expressed as @xmath140 and depends not only on the efficiency @xmath141 in the accretion process of matter @xmath136 , but also on the geometry of the emission described by the beaming factor @xmath142 ( here @xmath143 is the half opening angle of jet - like emission ) . depending on the assumptions we introduce in equation [ accretion ] , we can give constraints on the amount of accreted matter or on the geometry of the system . for an isotropic emission , @xmath144 , the accretion of @xmath145 m@xmath71 , for the co - rotating case , and of @xmath146 m@xmath71 , for the counter - rotating case , is required . alternatively , we can assume that the accreted matter comes from the crustal material from an @xmath147 m@xmath71 ns ns binary progenitor . the crustal mass from the nl3 nuclear model for each of these nss is @xmath148 m@xmath71 ( see , e.g. , * ? ? ? * and figure [ nanda ] ) . assuming that crustal material accounts also for the baryon load mass , e.g. , @xmath149 m@xmath71 , the total available mass for accretion is @xmath150 m@xmath71 . then , the presence of a beaming is necessary : from equation [ accretion ] , a half opening beaming angle @xmath151 , for co - rotating case , and @xmath152 , for the counter - rotating case , would be required . the above considerations are clearly independent from the relativistic beaming angle @xmath153 , where the lower limit on the lorentz factor @xmath154 has been derived , in a different context , by @xcite to the gev luminosity light curve ( see figure [ 090510gev ] ) . further consequences on these results for the estimate of the rate of these s - grbs will be presented elsewhere ( ruffini et al . in preparation ) . it is interesting to recall some of the main novelties introduced in this paper with respect to previous works on grb 090510 . particularly noteworthy are the differences from the previous review of short bursts by @xcite , made possible by the discovery of the high energy emission by the fermi team in this specific source @xcite . a new family of short bursts characterized by the presence of a bh and associated high energy emission when lat data are now available , comprehends grbs 081024b , 090227b , 090510 , 140402a , and 140619b ( see , e.g. , figure [ 090510gev ] ) . the excellent data obtained by the fermi team and interpreted within the fireshell model has allowed to relate in this paper the starting point of the high energy emission with the birth of a bh . our fireshell analysis assumes that the @xmath38-ray and the gev components originate from different physical processes . first , the interpretation of the prompt emission differs from the standard synchrotron model : we model the collisions of the baryon accelerated by the grb outflow with the ambient medium following a fully relativistic approach ( see section 2 ) . second , we assume that the gev emission originates from the matter accretion onto the newly - born bh and we show that indeed the energy requirement is fulfilled . this approach explains also the delayed onset of the gev emission , i.e. , it is observable only after the transparency condition , namely after the p - grb emission . the joint utilization of the excellent data from the _ fermi_-gbm nai - n6 and n7 and the bgo - b1 detectors and from the mini - calorimeter on board agile @xcite has given strong observational support to our theoretical work . grb 090510 has been analyzed in light of the recent progress achieved in the fireshell theory and the resulting new classification of grbs . we show that grb 090510 is a s - grb , originating in a binary ns merger ( see figure [ rt_gw ] ) . such systems , by the absence of the associated sn events , are by far the simplest grbs to be analyzed . our analysis indicates the presence of three distinct episodes in s - grbs : the p - grb , the prompt emission , and the gev emission . by following the precise identification of successive events predicted by the fireshell theory , we evidence for the first indication of a kerr bh or , possibly , a kerr - newman bh formation : * the p - grb spectrum of grb 090510 , in the time interval from @xmath92 to @xmath94 s , is best - fitted by a comptonized component ( see figures [ pgrb ] and [ comp ] and table [ tab : fit ] ) , which is interpreted as a convolution of thermal spectra originating in a dyadotorus ( see @xcite and @xcite , figure [ dyadotorus ] , and section 2 ) . * the prompt emission follows at the end of the p - grb ( see figure [ spectotal ] ) . the analysis of the prompt emission within the fireshell model allows to determine the inhomogeneities in the cbm giving rise to the spiky structure of the prompt emission and to estimate as well an averaged cbm density of @xmath155 @xmath18 obtained from a few cbm clouds of mass @xmath48 g and typical dimensions of @xmath53 cm ( see figure [ 090510simlc ] ) . such a density is typical of galactic halos where binary ns are expected to migrate due to large natal kicks . * the late x - ray emission of grb 090510 does not follow the characteristic patterns expected in bdhn events ( see figure [ episode3 ] and @xcite ) . * the gev emission occurs at the end of the p - grb emission and is initially concurrent with the prompt emission . this sequence occurs in both s - grbs @xcite and bdhne @xcite . this delayed long lasting ( @xmath156 s ) gev emission in grb 090510 is one of the most intense ever observed in any grb ( see figure [ 090510gev ] and * ? ? ? * ; * ? ? ? * we then consider accretion on co - rotating and counter - rotating orbits ( see ruffini & wheeler 1969 , in problem 2 of @xmath139 104 in @xcite ) around an extreme kerr bh . assuming the accretion of the crustal mass @xmath157 m@xmath71 from a @xmath147 m@xmath71 ns ns binary , fulfilling global charge neutrality ( see figure [ nanda ] ) , geometrical beaming angles of @xmath152 , for co - rotating case , and @xmath151 , for the counter - rotating case , are inferred . in order to fulfill the transparency condition , the initial lorentz factor of the jetted material has to be @xmath158 ( see section 6.6 ) . * while there is evidence that the gev emission must be jetted , no beaming appears to be present in the p - grb and in the prompt emission , with important consequence for the estimate of the rate of such events @xcite . * the energetic and the possible beaming of the gev emission requires the presence of a kerr bh , or a kerr - newman bh dominated by its angular momentum and with electromagnetic fields not influencing the geometry ( see also section 6.5 ) . * the self - consistency of the entire procedure has been verified by estimating , on the ground of the fireshell theory , the cosmological redshift of the source . the theoretical redshift is @xmath159 ( see section 6.4 ) , close to and consistent with the spectroscopically measured value @xmath10 @xcite . * the values of @xmath160 and @xmath161 of grb 090510 fulfill with excellent agreement the muruwazha relation ( see section 5.2 , figure [ calderone ] and * ? ? ? the main result of this article is that the dyadotorus manifests itself by the p - grb emission and clearly preceeds the prompt emission phase , as well as the gev emission originating from the newly - formed bh . this contrasts with the usual assumption made in almost the totality of works relating bhs and grbs in which the bh preceeds the grb emission . in conclusion , in this article , we take grb 090510 as the prototype of s - grbs and perform a new time - resoved spectral analysis , in excellent agreement with that performed by the agile and the _ fermi _ teams . now this analysis , guided by a theoretical approach successfully tested in this new family of s - grbs , is directed to identify a precise sequence of different events made possible by the exceptional quality of the data of grb 090510 . this include a new structure in the thermal emission of the p - grb emission , followed by the onset of the gev emission linked to the bh formation , allowing , as well , to derive the strucutre of the circumburst medium from the spiky structure of the prompt emission . this sequence , for the first time , illustrates the formation process of a bh . it is expected that this very unique condition of generating a jetted gev emission in such a well defined scenario of a newly - born bh will possibly lead to a deeper understanding of the equally jetted gev emission observed , but not yet explained , in a variety of systems harboring a kerr bh . among these systems we recall binary x - ray sources ( see , e.g. , * ? ? ? * and references therein ) , microquasars ( see , e.g. , * ? ? ? * and references therein ) , as well as , at larger scale , active galactic nuclei . we thank the editor and the referee for their comments which helped to improve the presentation and the contextualization of our results . we are indebted to marco tavani for very interesting comments , as well as for giving us observational supporting evidences . this work made use of data supplied by the uk _ swift _ data center at the university of leicester . m. e. , m. k. , and y. a. are supported by the erasmus mundus joint doctorate program by grant numbers 2012 - 1710 , 2013 - 1471 , and 2014 - 0707 respectively , from the eacea of the european commission . c.c . acknowledges indam - gnfm for support . m.m . acknowledges the partial support of the project n 3101/gf4 ipc-11 , and the target program f.0679 of the ministry of education and science of the republic of kazakhstan . , j. 2003 , in american institute of physics conference series , vol . 662 , gamma - ray burst and afterglow astronomy 2001 : a workshop celebrating the first year of the hete mission , ed . g. r. ricker & r. k. vanderspek , 229236 , i. b. , klebesadel , r. w. , & evans , w. d. 1975 , in annals of the new york academy of sciences , vol . 262 , seventh texas symposium on relativistic astrophysics , ed . p. g. bergman , e. j. fenyves , & l. motz , 145158

in a new classification of merging binary neutron stars ( nss ) we separate short gamma - ray bursts ( grbs ) in two sub - classes . the ones with @xmath0 erg coalesce to form a massive ns and are indicated as short gamma - ray flashes ( s - grfs ) . the hardest , with @xmath1 erg , coalesce to form a black hole ( bh ) and are indicated as genuine short - grbs ( s - grbs ) . within the fireshell model , s - grbs exhibit three different components : the p - grb emission , observed at the transparency of a self - accelerating baryon-@xmath2 plasma ; the prompt emission , originating from the interaction of the accelerated baryons with the circumburst medium ; the high - energy ( gev ) emission , observed after the p - grb and indicating the formation of a bh . grb 090510 gives the first evidence for the formation of a kerr bh or , possibly , a kerr - newman bh . its p - grb spectrum can be fitted by a convolution of thermal spectra whose origin can be traced back to an axially symmetric dyadotorus . a large value of the angular momentum of the newborn bh is consistent with the large energetics of this s - grb , which reach in the @xmath3@xmath4 kev range @xmath5 erg and in the @xmath6@xmath7 gev range @xmath8 erg , the most energetic gev emission ever observed in s - grbs . the theoretical redshift @xmath9 that we derive from the fireshell theory is consistent with the spectroscopic measurement @xmath10 , showing the self - consistency of the theoretical approach . all s - grbs exhibit gev emission , when inside the _ fermi_-lat field of view , unlike s - grfs , which never evidence it . the gev emission appears to be the discriminant for the formation of a bh in grbs , confirmed by their observed overall energetics .

arxiv

measurements at lep , sld , and the tevatron have been used extensively to limit models with physics beyond that of the standard model ( sm)@xcite . by performing global fits to a series of precision measurements , information about the parameters of new models can be inferred@xcite . the simplest example of this approach is the prediction of the @xmath3 boson mass . in the standard model , the @xmath3- boson mass , @xmath1 , can be predicted in terms of other parameters of the theory . the predicted @xmath3 boson mass is strongly correlated with the experimentally measured value of the top quark mass , @xmath4 , and increases quadratically as the top quark mass is increased . this strong correlation between @xmath1 and @xmath4 in the standard model can be used to limit the allowed region for the higgs boson mass@xcite . in a model with higgs particles in representations other than @xmath5 doublets and singlets , there are more parameters in the gauge / higgs sector than in the standard model . the sm tree level relation , @xmath6 no longer holds and when the theory is renormalized at one loop , models of this type will require extra input parameters@xcite . models with new physics are often written in terms of the sm lagrangian , @xmath7 plus an extra contribution , @xmath8 where @xmath9 represents contributions from new physics beyond the sm . phenomenological studies have then considered the contributions of @xmath7 at one - loop , plus the tree level contributions of @xmath9 . in this note , we give two specific examples with @xmath0 at tree level , where we demonstrate that this procedure is incorrect . we discuss in detail what happens in these models when the scale of the new physics becomes much larger than the electroweak scale and demonstrate explicitly that the sm is not recovered . the possibility of a heavy higgs boson which is consistent with precision electroweak data has been considered by chivukula , hoelbling and evans@xcite and by peskin and wells@xcite in the context of oblique corrections . in terms of the @xmath10 , @xmath11 and @xmath12 parameters@xcite , a large contribution to isospin violation , @xmath13 , can offset the contribution of a heavy higgs boson to electroweak observables such as the @xmath3 boson mass . the triplet model considered in this paper provides an explicit realization of this mechanism . the oblique parameter formulation neglects contributions to observables from vertex and box diagrams , which are numerically important in the example discussed here . in section [ renorm ] , we review the important features of the sm for our analysis . we discuss two examples in sections [ higgstrip ] and appendix [ lrmodel ] where the new physics does not decouple from the sm at one - loop . for simplicity , we consider only the dependence of the @xmath3 boson mass on the top quark mass and demonstrate that a correct renormalization scheme gives very different results from the sm result in these models . section [ higgstrip ] contains a discussion of the sm augmented by a real scalar triplet , and appendix [ lrmodel ] contains a discussion of a left - right @xmath14 symmetric model . in section [ nondecoupling ] , we show that the dependence on scalar masses in the w - boson mass is quadratic and demonstrate that the triplet is non - decoupling . our major results are summarized in eq . [ cc1]-[cc3 ] . these results are novel and have not been discussed in the literature before . section [ results ] contains our numerical results and section [ conc ] concludes this paper . similar results in the context of the littlest higgs model have previously been found in ref . . the one - loop renormalization of the sm has been extensively studied@xcite and we present only a brief summary here , in order to set the stage for sections [ higgstrip ] and appendix [ lrmodel ] . in the electroweak sector of the sm , the gauge sector has three fundamental parameters , the @xmath15 gauge coupling constants , @xmath16 and @xmath17 , as well as the vacuum expectation ( vev ) of the higgs boson , @xmath18 . once these three parameters are fixed , all other physical quantities in the gauge sector can be derived in terms of these three parameters and their counter terms . we can equivalently choose the muon decay constant , @xmath19 , the z - boson mass , @xmath20 , and the fine structure constant evaluated at zero momentum , @xmath21 , as our input parameters . experimentally , the measured values for these input parameters are@xcite , @xmath22 the w - boson mass then can be defined through muon decay@xcite , @xmath23\ ] ] where @xmath24 summarizes the radiative corrections , @xmath25 where @xmath26 , @xmath27 and @xmath28 is the weak mixing angle . the sm satisfies @xmath29 at tree level , @xmath30 in eq . ( [ rhodef ] ) , @xmath1 and @xmath31 are the physical gauge boson masses , and so our definition of the weak mixing angle , @xmath32 , corresponds to the on - shell scheme@xcite . it is important to note that in the sm , @xmath32 is not a free parameter , but is derived from @xmath33 the counterterms of eq . ( [ drdef ] ) are given by@xcite , @xmath34 where @xmath35 , for @xmath36 , are the gauge boson 2-point functions ; @xmath37 is defined as @xmath38 . the term @xmath39 contains the box and vertex contributions to the renormalization of @xmath40@xcite . the counterterm for @xmath41 can be derived from eq . ( [ rhodef ] ) , @xmath42 = \frac{\overline{c}_{\theta}^{2}}{\overline{s}_{\theta}^{2 } } \biggl [ \frac{\pi_{zz}(m_{z}^{2})}{m_{z}^{2 } } - \frac{\pi_{ww}(m_{w}^{2})}{m_{w}^{2 } } \biggr]\quad . \label{stdef}\ ] ] putting these contributions together we obtain , @xmath43\quad .\nonumber\end{aligned}\ ] ] these gauge boson self - energies can be found in ref . and and we note that the fermion and scalar contributions to the two - point function @xmath44 vanish . the dominant contributions to @xmath24 is from the top quark , and the contributions of the top and bottom quarks to the gauge boson self - energies are given in [ loop ] . the @xmath45 dependence in @xmath46 and @xmath47 exactly cancel , thus the difference , @xmath48 , depends on @xmath49 only logarithmically . the second term , @xmath50 , also depends on @xmath49 logarithmically . however , the quadratic @xmath45 dependence in @xmath51 and @xmath52 do not cancel . thus @xmath24 depends on @xmath49 quadratically , and is given by the well known result , keeping only the two - point functions that contain a quadratic dependence on @xmath49@xcite , @xmath53 where @xmath54 is the number of colors and the superscript @xmath55 denotes that we have included only the top quark contributions , in which the dominant contribution is quadratic . the complete contribution to @xmath24 can be approximated , @xmath56 the first term in eq . ( [ drsm ] ) results from the scaling of @xmath57 from zero momentum to @xmath20@xcite . in the numerical results , the complete contributions to @xmath24 from top and bottom quarks , the higgs boson as well as the gauge bosons are included , as given in eq . ( [ drsm1 ] ) . in this section , we consider the sm with an additional higgs boson transforming as a real triplet ( y=0 ) under the @xmath59 gauge symmetry . hereafter we will call this the triplet model ( tm)@xcite . this model has been considered at one - loop by blank and hollik@xcite and we have checked that our numerical codes are correct by reproducing their results . in addition , we derive the scalar mass dependence in this model and show that the triplet is non - decoupling by investigating various scalar mass limits . we also find the conditions under which the lightest neutral higgs can be as heavy as a tev , which has new important implications on higgs searches . these results concerning the scalar fields are presented in the next section . the @xmath58 higgs doublet in terms of its component fields is given by , @xmath60 with @xmath61 being the goldstone boson corresponding to the longitudinal component of the @xmath62 gauge boson . a real @xmath59 triplet , @xmath63 , can be written as @xmath64 , @xmath65 there are thus four physical higgs fields in the spectrum : there are two neutral higgs bosons , @xmath66 and @xmath67 , @xmath68 and the mixing between the two neutral higgses is described by the angle @xmath69 . the charged higgses @xmath70 are linear combinations of the charged components in the doublet and the triplet , with a mixing angle @xmath71 , @xmath72 where @xmath73 are the goldstone bosons corresponding to the longitudinal components of @xmath74 . the masses of these four physical scalar fields , @xmath75 , @xmath76 and @xmath77 , respectively , are free parameters in the model . the @xmath3 boson mass is given by , @xmath78 where @xmath79 is the vev of the neutral component of the @xmath59 higgs boson and @xmath80 is the vacuum expectation value of the additional scalar , leading to the relationship @xmath81 . a real triplet does not contribute to @xmath31 , leading to @xmath82 the main result of this section is to show that the renormalization of a theory with @xmath83 at tree level is fundamentally different from that of the sm . due to the presence of the @xmath58 triplet higgs , the gauge sector now has four fundamental parameters , the additional parameter being the vev of the @xmath58 triplet higgs , @xmath84 . a consistent renormalization scheme thus requires a fourth input parameter@xcite . we choose the fourth input parameter to be the effective leptonic mixing angle , @xmath85 , which is defined as the ratio of the vector to axial vector parts of the @xmath86 coupling , @xmath87 with @xmath88 and @xmath89 . this leads to the definition of @xmath85 , @xmath90 the measured value from lep is given by @xmath91@xcite . as usual the @xmath3 boson mass is defined through muon decay@xcite , @xmath92 where we have chosen @xmath93 instead of @xmath94 as an input parameter . the contribution to @xmath95 is similar to that of the sm , @xmath96 where the counter term @xmath97 is defined through @xmath98 , @xmath20 and @xmath85 as , @xmath99 unlike in the sm case where @xmath100 is defined through @xmath98 and @xmath20 as given in eq . ( [ stdef ] ) , now @xmath85 is an independent parameter , and its counter term is given by@xcite , @xmath101\ ; \biggr ] \ ; , \nonumber \\ & \equiv & \biggl(\frac{\hat{c}_{\theta}}{\hat{s}_{\theta } } \biggr ) \frac{re\biggl(\pi^{\gamma z}(m_{z}^{2})\biggr)}{m_{z}^{2 } } + \delta_{v - b}^{\prime } \ ; , \end{aligned}\ ] ] where @xmath102 is the axial part of the electron self - energy , @xmath103 and @xmath104 are the vector and axial - vector form factors of the vertex corrections to the @xmath105 coupling . these effects have been included in our numerical results@xcite . the total correction to @xmath95 in this case is then given by , @xmath106 where @xmath39 summarizes the vertex and box corrections in the tm model , and it is given by@xcite , @xmath107 , \qquad r \equiv m_{w}^{2 } / m_{z}^{2 } \ ; , \ ] ] where we show only the finite contributions in the above equation . keeping only the top quark contribution , @xmath108 where @xmath109 is the momentum cutoff in dimensional regularization and the definition of the function @xmath110 can be found in appendix a. as @xmath111 is logarithmic , the @xmath49 dependence of @xmath98 is now logarithmic . we note that this much softer relation between @xmath98 and @xmath49 is independent of the choice of the fourth input parameter . this will become clear in our second example , the left - right symmetric model , given in [ lrmodel ] . in our numerical results , we have included in @xmath95 the complete contributions , which are summarized in [ scalar ] , from the top and bottom quarks and the four scalar fields , as well as the gauge bosons , and the complete set of vertex and box corrections . as shown in [ scalar ] , @xmath95 depends on scalar masses quadratically . this has important implications for models with triplets , such as the littlest higgs model@xcite . the two point function @xmath44 does not have any scalar dependence , while @xmath112 and @xmath113 depend on scalar masses logarithmically . the quadratic dependence thus comes solely from the function @xmath114 . when there is a large hierarchy among the three scalar masses ( case ( c ) in [ scalar ] and its generalization ) , all contributions are of the same sign , and are proportional to the scalar mass squared , @xmath115 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2 } m_{h^{\pm}}^{2}}{2m_{w}^{4 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) - c_{\delta}^{2 } \frac{2m_{k^{0}}^{2}m_{h^{\pm}}^{2}}{m_{w}^{4 } } % \biggl(1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{k^{0}}^{2}}\biggr)%\biggr ) \biggr\ } \ ; , \nonumber\end{aligned}\ ] ] for @xmath116 . thus the scalar contribution to @xmath95 in this case is very large , and it grows with the scalar masses . on the other hand , when the mass splitting between either pair of the three scalar masses is small ( case ( a ) and ( b ) and their generalization ) , the scalar contributions grow with the mass splitting@xcite , @xmath117 \nonumber\\ & & \qquad + \frac{5}{72}\biggl [ s_{\delta}^{2}\frac { \bigl(m_{h^{\pm}}^{2}-m_{h^{0}}^{2}\bigr)}{m_{h^{0}}^{2 } } + 4c_{\delta}^{2 } \frac { \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr ] \biggr\ } \nonumber \ ; , \end{aligned}\ ] ] for @xmath118 , and , @xmath119 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}m_{h^{\pm}}^{2}}{2m_{w}^{4 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) + \frac{5}{18}c_{\delta}^{2 } \frac{\bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr\ } \nonumber \ ; , \end{aligned}\ ] ] for @xmath120 . cancellations can occur in this case among contributions from different scalar fields , leading to the viability of a heavier neutral higgs boson than is allowed in the sm@xcite . the non - decoupling property of the triplet is seen in eq . ( [ cc1 ] ) , ( [ cc2 ] ) and ( [ cc3 ] ) . because @xmath95 depends quadratically on the scalar masses@xcite , the scalars must be included in any effective field theory analysis of low energy physics . the scalar potential of the model with a @xmath58 triplet and an @xmath58 doublet is given by the following@xcite : @xmath121 where @xmath122 denotes the pauli matrices , and @xmath123 from the minimization conditions ( see [ minimize ] ) , @xmath124 we obtain the following conditions , @xmath125 the two mixing angles , @xmath69 and @xmath71 , in the neutral and charged higgs sectors defined in eqs . ( [ neutralmix ] ) and ( [ chargedmix ] ) , are solutions to the following two equations@xcite , @xmath126 \nonumber \label{cond1}\\ 0 & = & -\lambda_{4}v + \lambda_{3}vv^{\prime}+\tan\gamma \biggl [ \mu_{1}^{2 } - \mu_{2}^{2 } + 3 \lambda_{1 } v^{2 } - \frac{1}{2 } \lambda_{3 } v^{2 } -\lambda_{4}v^{\prime } -3\lambda_{2 } v^{\prime 2 } \\ & & \qquad + \frac{1}{2 } \lambda_{3 } v^{\prime 2 } + \lambda_{4 } v \tan\gamma - \lambda_{3 } vv^{\prime}\tan\gamma \biggr ] \ ; , \label{cond2 } \nonumber\end{aligned}\ ] ] which are obtained by minimizing the scalar potential . in terms of the parameters in the scalar potential , the masses of the four scalar fields are given by@xcite , @xmath127 this model has six parameters in the scalar sector , @xmath128 . equivalently , we can choose @xmath129 as the independent parameters . two of these six parameters , @xmath18 and @xmath130 , contribute to the gauge boson masses . when turning off the couplings between the doublet and the triplet in the scalar potential , @xmath131 , the triplet could still acquire a vev , @xmath132 , provided that @xmath133 is negative . since we have not observed any light scalar experimentally up to the ew scale , the triplet mass which is roughly of order @xmath134 has to be at least of the ew scale , @xmath135 . this is problematic because the vev of a real triplet only contributes to @xmath98 but not to @xmath31 , which then results in a contribution of order @xmath136 to the @xmath137 parameter , due to the relation , @xmath138 . for @xmath134 greater than @xmath18 , the ew symmetry is broken at a high scale . in order to avoid these problems , the parameter @xmath133 thus has to be positive so that the triplet does not acquire a vev via this mass term when @xmath139 is turned off . once the coupling @xmath140 is turned on while keeping @xmath141 , the term @xmath142 effectively plays the role of the mass term for @xmath63 and for @xmath143 . thus , similar to the reasoning given above , for @xmath144 , the coupling @xmath140 has to be positive so that it does not induce a large triplet vev . for simplicity , consider the case when there is no mixing in the neutral higgs sector , @xmath145 . in this case , when the mixing in the charged sector approaches zero , @xmath146 , the masses @xmath76 and @xmath77 approach infinity , and their difference @xmath147 approaches zero . the contribution due to the new scalars thus vanishes , and only the lightest neutral higgs contributes to @xmath95 . even though the contribution due to the new scalars vanishes , @xmath95 does not approach @xmath24 . this is because in the tm case , four input parameters are needed , while in the sm case three inputs are needed . there is no continuous limit that takes one case to the other@xcite . one way to achieve the @xmath146 limit is to take the mass parameter @xmath148 while keeping the parameter @xmath139 finite . ( [ cond1 ] ) then dictates that @xmath149 . however , satisfying eq . ( [ cond2 ] ) requires that @xmath150 . as @xmath151 , this condition implies that the dimensionless coupling constant @xmath140 has to scale as @xmath152 , which approaches infinity as @xmath146 . this can also be seen from eq . ( [ neutralmix1 ] ) . as there is no mixing in the neutral sector , @xmath153 the condition @xmath154 then follows . so , in the absence of the neutral mixing , @xmath145 , in order to take the charged mixing angle @xmath71 to zero while holding @xmath139 fixed , one has to take @xmath140 to infinity . in other words , for the triplet to decouple requires a dimensionless coupling constant @xmath140 to become strong , leading to the breakdown of the perturbation theory . alternatively , the neutral mixing angle @xmath69 can approach zero by taking @xmath155 while keeping @xmath140 and @xmath139 fixed . in this case , the minimization condition , @xmath156 where @xmath157 , implies that the charged mixing angle @xmath71 has to approach zero . this again corresponds to the case where the custodial symmetry is restored , by which we mean that the triplet vev vanishes , @xmath158 . in this case , severe fine - tuning is needed in order to satisfied the condition given in eq . ( [ cond2 ] ) . another way to get @xmath146 is to have @xmath159 , which trivially satisfies eq . ( [ cond1 ] ) . this can also be seen from eq . [ chargemix2 ] , @xmath160 eq . ( [ cond2 ] ) then gives @xmath161 . so for small @xmath140 , the masses of these additional scalar fields are of the weak scale , @xmath162 . this corresponds to a case when the custodial symmetry is restored . so unless one imposes by hand such symmetry to forbid @xmath139 , four input parameters are always needed in the renormalization . if there is a symmetry which makes @xmath163 ( to all orders ) , say , @xmath164 , then there are only three input parameters needed . so the existence of such a symmetry is crucial when one - loop radiative corrections are concerned . mass as a function of the top quark mass in the sm , tm and the lr model . the data point represents the experimental values with @xmath165 error bars@xcite . for the sm , we include the complete contributions from top and bottom quarks , the sm higgs boson with @xmath166 gev , and the gauge bosons . for the tm and the lr model , we include only the top quark contribution and the absolute normalization is fixed so that the curves intersect the data point . the @xmath167 boson mass is chosen to be @xmath168 tev in the lr model . ] mass in the tm as a function of the top quark mass for scalar masses @xmath169 gev , with @xmath93 and @xmath85 varying within their @xmath165 limits@xcite @xmath170 and @xmath171 . the solid curve indicates the prediction with @xmath93 and @xmath85 taking the experimental central values , @xmath172 and @xmath173 . the area bounded by the short dashed ( dotted ) curves indicates the prediction with @xmath174 ( @xmath175 ) and @xmath176 ( @xmath177 ) varying with its @xmath165 limits . the data point represents the experimental values with @xmath165 error bars@xcite . ] mass in the tm as a function of the top quark mass for scalar masses , @xmath75 , @xmath76 and @xmath77 , varying independently between ( a ) @xmath178 tev , ( b ) @xmath179 gev , and ( c ) @xmath180 gev . the data point represents the experimental values with @xmath165 error bars@xcite.,title="fig : " ] mass in the tm as a function of the top quark mass for scalar masses , @xmath75 , @xmath76 and @xmath77 , varying independently between ( a ) @xmath178 tev , ( b ) @xmath179 gev , and ( c ) @xmath180 gev . the data point represents the experimental values with @xmath165 error bars@xcite.,title="fig : " ] mass in the tm as a function of the top quark mass for scalar masses , @xmath75 , @xmath76 and @xmath77 , varying independently between ( a ) @xmath178 tev , ( b ) @xmath179 gev , and ( c ) @xmath180 gev . the data point represents the experimental values with @xmath165 error bars@xcite.,title="fig : " ] mass in the tm as a function of the lightest neutral higgs boson mass , @xmath75 , for various values of @xmath76 and @xmath77 . the area bounded by the two horizontal lines is the @xmath165 allowed region for @xmath98@xcite.,title="fig : " ] mass in the tm as a function of the lightest neutral higgs boson mass , @xmath75 , for various values of @xmath76 and @xmath77 . the area bounded by the two horizontal lines is the @xmath165 allowed region for @xmath98@xcite.,title="fig : " ] mass in the tm as a function of the lightest neutral higgs boson mass , @xmath75 , for various values of @xmath76 and @xmath77 . the area bounded by the two horizontal lines is the @xmath165 allowed region for @xmath98@xcite.,title="fig : " ] the previous section has presented analytic results for the triplet model , demonstrating that the dependence of the @xmath3 mass on the top quark mass is logarithmic , while the dependence on the scalar masses is quadratic . a dramatic change in the behavior of the @xmath3 mass is also observed in the @xmath181 model@xcite . for comparison with the triplet model , we summarize the results of the left - right model in appendix [ lrmodel ] . in this case , the dependence of the @xmath3 mass on the top quark mass is weakened from that of the sm since it depends on @xmath182 , where @xmath183 is the heavy charged gauge boson mass of the left - right model . the dependence of the @xmath3 mass on the top quark mass , @xmath49 , in the case of the sm , the model with a triplet higgs , and the minimal left - right model , are shown in fig . [ fg : mwmt ] . for the sm , we include the complete contributions from top and bottom quarks , the sm higgs boson with @xmath166 gev , and the gauge bosons . in this case , the @xmath49 dependence in the prediction for @xmath1 is quadratic . the range of values for the input parameter @xmath49 that give a prediction for @xmath98 consistent with the experimental @xmath165 limits@xcite , @xmath184 gev , is very narrow . it coincides with the current experimental limits@xcite , @xmath185 gev . for the triplet model and the lr model , we include only the top quark contribution . as we have shown in sec . [ higgstrip ] , the prediction for @xmath1 in the triplet model depends on @xmath49 only logarithmically . in our numerical result for the left - right model , we have used @xmath186 in the gauge sector , in addition to @xmath49 in the fermion sector , to predict @xmath98 . here we have identified @xmath187 and @xmath188 as the @xmath3 and @xmath62 bosons in the sm and consequently @xmath189 and @xmath190 . in this case , the @xmath49 dependence in the prediction for @xmath98 is similarly softer because the top quark contributions are suppressed by a heavy scale , @xmath183 . in the triplet model and the left - right model , the range of @xmath49 that gives a prediction for @xmath98 consistent with the experimental value is thus much larger , ranging from @xmath191 to @xmath192 gev . the presence of the triplet higgs thus dramatically changes the @xmath49 dependence in @xmath98 . this is clearly demonstrated in fig . [ fg : mwmt ] by the almost flat curves of the triplet and left - right symmetric models , contrary to that of the sm , which is very sensitive to @xmath49 . in fig . [ fg : mwmterr ] , we show the prediction for @xmath98 as a function of @xmath49 in the triplet model , with @xmath93 and @xmath85 varying within the @xmath165 limits@xcite , @xmath170 and @xmath171 . we find that the prediction for @xmath98 is very sensitive to the input parameters @xmath177 and @xmath85 . the complete contributions from the top and bottom quarks and the sm gauge bosons , as well as all four scalar fields in the triplet model are included in fig . [ fg : mwmt1 ] and [ fg : mwmh ] . we have also included the box and vertex corrections . in fig . [ fg : mwmt1 ] , we show the prediction in the triplet model for @xmath98 as a function of @xmath49 , allowing @xmath75 , @xmath77 and @xmath76 to vary independently between @xmath178 tev , @xmath179 gev and @xmath180 gev . interestingly , for all scalar masses in the range of @xmath178 tev , the prediction for @xmath98 in the tm model still agrees with the experimental @xmath165 limits . [ fg : mwmh ] shows the prediction for @xmath98 as a function of @xmath75 for various values of @xmath76 and @xmath77 . for small @xmath147 , the lightest neutral higgs boson mass can range from @xmath193 gev to a tev and still satisfy the experimental prediction for @xmath98 this agrees with our conclusion in sec . [ higgstrip ] that to minimize the scalar contribution to @xmath95 , the mass splitting @xmath147 has to be small and that when the mass splitting is small , cancellations can occur between the contributions of the lightest neutral higgs and those of the additional scalar fields . this has new important implications for the higgs searches . we have considered the top quark contribution to muon decay at one loop in the sm and in two models with @xmath0 at tree level : the sm with an addition real scalar triplet and the minimal left - right model . in these new models , because the @xmath137 parameter is no longer equal to one at the tree level , a fourth input parameter is required in a consistent renormalization scheme . these models illustrate a general feature that the @xmath49 dependence in the radiative corrections @xmath95 becomes logarithmic , contrary to the case of the sm where @xmath24 depends on @xmath4 quadratically . one therefore loses the prediction for @xmath49 from radiative corrections . on the other hand , due to cancellations between the contributions to the radiative corrections from the sm higgs and the triplet , a higgs mass @xmath75 as large as a few tev is allowed by the @xmath3 mass measurement . we emphasize that by taking the triplet mass to infinity , one does not recover the sm . this is due to the fact that the triplet scalar field is non - decoupling , and it implies that the one - loop electroweak results can not be split into a sm contribution plus a piece which vanishes as the scale of new physics becomes much larger than the weak scale . this fact has been overlooked by most analyses in the littlest higgs model@xcite , and correctly including the effects of the triplet can dramatically change the conclusion on the viability of the model@xcite . such non - decoupling effect has been pointed out in the two higgs doublet model@xcite , left - right symmetric model@xcite , and the littlest higgs model@xcite . it has _ not _ been discussed before in the model with a triplet . we comment that the non - decoupling observed in these examples do not contradict with the common knowledge that in gut models heavy scalars decouple . these two cases are fundamentally different because in gut models , heavy scalar fields do not acquire vev that break the ew symmetry , while in cases where non - decoupling is observed , heavy scalar fields _ do _ acquire vev that breaks the ew symmetry . the quadratic dependence in scalar masses in the triplet model can be easily understood physically . in sm with only the higgs doublet present , the quadratic scalar mass contribution is protected by the tree level custodial symmetry , and thus the higgs mass contribution is logarithmic at one - loop . this is the well - known screening theorem by veltman@xcite . as the custodial symmetry is broken in the sm at one - loop due to the mass splitting between the top and bottom quarks , the two - loop higgs contribution is quadratic . in models with a triplet higgs , as the custodial symmetry is broken already at the tree level , there is no screening theorem that protects the quadratic scalar mass dependence from appearing . our results demonstrate the importance of performing the renormalization correctly according to the ew structure of the new models . this manuscript has been authored by brookhaven science associates , llc under contract no . de - ac02 - 76ch1 - 886 with the u.s . department of energy . the united states government retains , and the publisher , by accepting the article for publication , acknowledges , a world - wide license to publish or reproduce the published form of this manuscript , or allow others to do so , for the united states government purpose . thanks w. marciano for useful discussions . we summarize below the leading contributions due to the sm top loop to the self - energies of the gauge bosons@xcite , where the definitions of the passarino - veltman functions utilized below are given in@xcite . @xmath194 \nonumber\\ \pi^{ww } ( 0 ) & = & -\frac{3 \alpha}{16\pi s_{\theta}^{2 } } \ , \cdot \ , m_{t}^{2 } \biggl [ 1 + 2 \ln \biggl ( \frac{q^{2}}{m_{t}^{2 } } \biggr ) \biggr ] \\ \pi^{zz } ( m_{z}^{2 } ) & = & -\frac{3 \alpha}{8\pi s_{\theta}^{2 } c_{\theta}^{2 } } \biggl [ \biggl(\biggl ( \frac{1}{2 } - \frac{4}{3}s_{\theta}^{2 } \biggr)^{2 } + \frac{1}{4 } \biggr ) h_{1}(m_{t}^{2})\\ & & -\frac{8}{3 } s_{\theta}^{2 } \biggl ( 1-\frac{4}{3}s_{\theta}^{2 } \biggr ) h_{2}(m_{t}^{2 } ) \biggr]\nonumber \\ \pi_{\gamma\gamma}^{\prime}(0 ) & = & \frac{4\alpha}{9\pi } \ln \biggl(\frac{q^{2}}{m_{t}^{2}}\biggr ) \\ \pi^{\gamma z } ( m_{z}^{2 } ) & = & -\frac{\alpha}{\pi s_{\theta}c_{\theta } } \biggl ( \frac{1}{2 } - \frac{4}{3 } s_{\theta}^{2 } \biggr ) m_{z}^{2 } \biggl [ \frac{1}{3 } \ln \biggl(\frac{q^{2}}{m_{t}^{2}}\biggr ) -2 i_{3}\biggl(\frac{m_{z}^{2}}{m_{t}^{2}}\biggr)\biggr]\end{aligned}\ ] ] where @xmath195 \\ h_{2}(m_{t}^{2 } ) & = & m_{t}^{2 } \biggl [ i_{1}\biggl(\frac{m_{z}^{2}}{m_{t}^{2}}\biggr ) - \ln \biggl(\frac{q^{2}}{m_{t}^{2}}\biggr ) \biggr]\quad .\end{aligned}\ ] ] the integrals are defined as , @xmath196 here @xmath197 is defined in the on - shell scheme ( eq . ( [ swos ] ) ) for the sm and as the effective weak mixing angle ( eq . ( [ swdef ] ) ) for the tm and lr model . the complete contributions to various two - point functions that appear in @xmath95 are given below , where the scalar and fermion contributions are given in ref . , and we have taken the sm gauge boson contributions from ref . . @xmath198 \\ & & + \frac{3 \alpha}{4\pi \hat{s}_{\theta}^{2 } } \biggl [ a_{0}(m_{t}^{2 } ) + m_{b}^{2 } b_{0}(m_{w}^{2},m_{b}^{2},m_{t}^{2 } ) \nonumber\\ & & \qquad -m_{w}^{2 } b_{1}(m_{w}^{2},m_{b}^{2},m_{t}^{2 } ) - 2 b_{22}(m_{w}^{2},m_{b}^{2},m_{t}^{2 } ) \biggr ] \nonumber\\ & & + { \alpha\over 4\pi \hat{s}_{\theta}^{2}}\biggl\ { s_\delta^2h(m_{h^0 } , m_{h^\pm})+c_\delta^2h(m_{h^0 } , m_w)\nonumber \\ & & \qquad + 4c_\delta^2h(m_{k^0 } , m_{h^\pm})+4s_\delta^2h(m_{k^0 } , m_w)\nonumber \\ & & \qquad + s_\delta^2h(m_z , m_{h^\pm})+c_\delta^2h(m_z , m_w)\biggr\ } \nonumber\\ & & + { \alpha\over 4 \pi \hat{s}_{\theta}^{2}}m_w^2 \biggl [ { s_\delta^2 c_\delta^2\over \hat{c}_{\theta}^{2 } } \biggl(b_0(0,m_z , m_{h^\pm})- b_0(m_{w},m_z , m_{h^\pm } ) \biggr ) \nonumber\\ & & \qquad + { ( s_\delta^2-\hat{s}_\theta^2)^2\over \hat{c}_{\theta}^{2 } } \biggl ( b_0(0,m_z , m_w ) - b_0(m_{w},m_z , m_w ) \biggr ) \nonumber \\ & & \qquad + \hat{s}_\theta^2 \biggl(b_0(0,0,m_w ) - b_0(m_w,0,m_w ) \biggr ) \nonumber\\ & & \qquad + c_\delta^2 \biggl ( b_0(0,m_{h^0 } , m_w ) - b_0(m_w , m_{h^0 } , m_w ) \biggr ) \nonumber\\ & & \qquad + 4 s_\delta^2 \biggl(b_0(0,m_{k^0},m_w ) - b_0(m_w , m_{k^0},m_w ) \biggr ) \biggr ] \nonumber \\ & & + \frac{\alpha}{4\pi \hat{s}_{\theta}^{2 } } \biggl [ \hat{c}_{\theta}^{2 } \biggl ( a_{1}(0,m_{z},m_{w } ) - a_{1}(m_w , m_z , m_w ) \biggr ) \nonumber\\ & & \qquad + \hat{s}_{\theta}^{2 } \biggl ( a_{1}(0,0,m_w ) - a_{1}(m_w,0,m_w ) \biggr ) \nonumber\\ & & \qquad - 2\hat{c}_{\theta}^{2 } h(m_z , m_w ) - 2\hat{s}_{\theta}^{2 } h(0,m_w ) \biggr]\nonumber \ ; , \\ & & \nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber \\&&\nonumber\end{aligned}\ ] ] @xmath199 \\ & & + \frac{\alpha}{4\pi \hat{s}_{\theta}\hat{c}_{\theta } } \biggl [ 2(c_{\delta}^{2}-\hat{s}_{\theta}^{2 } + \hat{c}_{\theta}^{2})b_{22}(m_z , m_{h^\pm},m_{h^\pm } ) \nonumber\\ & & \qquad + 2 ( s_{\delta}^2-\hat{s}_{\theta}^2+\hat{c}_{\theta}^2 ) b_{22}(m_z , m_w , m_w ) \nonumber\\ & & \qquad + ( \hat{s}_{\theta}^2-\hat{c}_{\theta}^2-c_{\delta}^2)a(m_{h^\pm } ) + ( \hat{s}_{\theta}^2-\hat{c}_{\theta}^2-s_{\delta}^2)a(m_{w } ) \biggr ] \nonumber\\ & & + \frac{\alpha}{4\pi}\bigl(2m_{w}^{2 } ) \frac{\hat{s}_{\theta}^2-s_{\delta}^2 } { \hat{s}_{\theta}\hat{c}_{\theta } } b_{0}(m_z , m_w , m_w ) \nonumber\\ & & -\frac{\alpha}{4\pi \hat{s}_{\theta}^{2 } } \biggl [ \hat{s}_{\theta}\hat{c}_{\theta } a_{1}(m_z , m_w , m_w ) + 2 \hat{c}_{\theta}\hat{s}_{\theta } a_{2}(m_w ) \nonumber\\ & & \qquad + 2 \hat{s}_{\theta}\hat{c}_{\theta } b_{22}(m_z , m_w , m_w ) \biggr ] \ ; , \nonumber % % % \end{aligned}\ ] ] @xmath200 \ ; , % % % \end{aligned}\ ] ] @xmath201 \ ; , \nonumber\end{aligned}\ ] ] where @xmath202 and @xmath100 is defined in eq . ( [ swdef ] ) . to extract the dependence on the masses of the lightest neutral higgs , @xmath75 , and the extra scalar fields , @xmath203 and @xmath204 , we first note that , in the limit @xmath205 , @xmath206 -\frac{1}{12 } p^{2 } \ln \biggl ( \frac{q^{2}}{m_{1}^{2 } } \biggr ) -\frac{1}{72}\frac{p^{4}}{m_{1}^{2 } } \\ & & + \biggl [ \frac{1}{4 } \ln \biggl(\frac{q^{2}}{m_{1}^{2 } } \biggr ) + \frac{5}{72}\frac{p^{2}}{m_{1}^{2}}\biggr ] \delta m^{2 } + \mathcal{o}\biggl ( ( \delta m^{2})^{2 } , \ ; \biggl(\frac{p^{2}}{m_{1}^{2}}\biggr)^{2 } \biggr)\nonumber \\ & & + ( \mbox{terms with no scalar dependence})\nonumber \nonumber \ ; , \end{aligned}\ ] ] where we have defined @xmath207 and assumed that @xmath208 . using these relations , we then have , @xmath209 @xmath210 on the other hand , in the limit @xmath211 , we get , @xmath212 which gives , @xmath213 % -\frac{1}{12}p^{2 } \ln q^{2 } + \mathcal{o } \biggl ( \biggl(\frac{m_{2}^{2}}{m_{1}^{2}}\biggr ) , \ ; \biggl(\frac{p^{2}}{m_{1}^{2}}\biggr ) \biggr ) \\ & & + \ ; ( \mbox{terms with no scalar dependence } ) \nonumber\end{aligned}\ ] ] @xmath214 the two - point function @xmath44 does not have any scalar dependence , and the function @xmath215 depends on the scalar mass only logarithmically , @xmath216 the scalar dependence in the function @xmath113 is , @xmath217 \\ & = & \frac{\alpha}{4\pi \hat{s}_{\theta } \hat{c}_{\theta } } ( \hat{s}_{\theta}^{2 } -\hat{c}_{\theta}^{2}-c_{\delta}^{2 } ) m_{z}^{2 } \biggl [ \frac{1}{12 } \ln \biggl ( \frac{m_{h^{\pm}}^{2}}{q^{2}}\biggr ) - \frac{1}{72 } \frac{m_{z}^{2}}{m_{h^{\pm}}^{2}}\biggr ] \ ; , \nonumber\end{aligned}\ ] ] thus the dependence is also logarithmic . on the other hand , in the function @xmath114 , the scalar dependence is given by , @xmath218 \ ; . \nonumber\end{aligned}\ ] ] from eqs . ( [ b : eq ] ) and ( [ b0:eq2 ] ) , we know that the contributions from the terms in the square brackets of eq . ( [ square ] ) are logarithmic . thus the only possible quadratic dependence comes from terms in the curly brackets . we consider the following three limits , assuming that all scalar masses are much larger than @xmath98 and @xmath20 : * @xmath118 : in this case , the leading order scalar dependence is given by , @xmath219 \nonumber\\ & & + \frac{5}{72}\biggl [ s_{\delta}^{2}\frac{m_{w}^{2}}{m_{h^{0}}^{2 } } \bigl(m_{h^{\pm}}^{2}-m_{h^{0}}^{2}\bigr ) + 4c_{\delta}^{2 } \frac{m_{w}^{2}}{m_{k^{0}}^{2 } } \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr ) \biggr ] \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] so the dominant scalar contribution to @xmath220 in this case is given by , @xmath221 \nonumber\\ & & \qquad + \frac{5}{72}\biggl [ s_{\delta}^{2}\frac { \bigl(m_{h^{\pm}}^{2}-m_{h^{0}}^{2}\bigr)}{m_{h^{0}}^{2 } } + 4c_{\delta}^{2 } \frac { \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr ] \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] * @xmath222 : in this limit , the leading scalar dependence becomes , @xmath223 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}}{2m_{w}^{2 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) m_{h^{\pm}}^{2 } + \frac{5}{18}c_{\delta}^{2 } \frac{m_{w}^{2}}{m_{k^{0}}^{2 } } \bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr ) \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] the leading scalar contribution to @xmath220 is , @xmath224 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}m_{h^{\pm}}^{2}}{2m_{w}^{4 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) + \frac{5}{18}c_{\delta}^{2 } \frac{\bigl(m_{h^{\pm}}^{2 } - m_{k^{0}}^{2}\bigr)}{m_{k^{0}}^{2 } } \biggr\ } \nonumber \ ; .\end{aligned}\ ] ] * @xmath225 : in this limit , the leading scalar dependence becomes , @xmath226 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2}}{2m_{w}^{2 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) m_{h^{\pm}}^{2 } - c_{\delta}^{2 } \frac{2m_{k^{0}}^{2}}{m_{w}^{2 } } % \biggl(1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{k^{0}}^{2}}\biggr)%\biggr ) m_{h^{\pm}}^{2 } \biggr\ } \ ; , \nonumber\end{aligned}\ ] ] the leading scalar contribution to @xmath220 is thus given by , @xmath227 \nonumber\\ & & -s_{\delta}^{2}\frac{m_{h^{0}}^{2 } m_{h^{\pm}}^{2}}{2m_{w}^{4 } } % \biggl ( 1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{h^{0}}^{2}}\biggr ) % \biggr ) - c_{\delta}^{2 } \frac{2m_{k^{0}}^{2}m_{h^{\pm}}^{2}}{m_{w}^{4 } } % \biggl(1 + \ln \biggl(\frac{m_{h^\pm}^{2}}{m_{k^{0}}^{2}}\biggr)%\biggr ) \biggr\ } \ ; . \nonumber\end{aligned}\ ] ] for the case @xmath228 , make the replacement , @xmath229 . as our second example to show that new physics does not decouple from the sm at one - loop , we consider the left - right ( lr ) symmetric model@xcite which is defined by the gauge group , @xmath230 the minimal left - right symmetric model contains a scalar bi - doublet , @xmath63 , and two @xmath5 triplets , @xmath231 and @xmath232 . we assume that the scalar potential is arranged such that the higgs fields obtain the following vev s : @xmath233 where the quantum numbers of these higgs fields under @xmath58 , @xmath234 and @xmath235 are given inside the parentheses . the vev @xmath236 breaks the @xmath237 symmetry down to @xmath238 of the sm , while the bi - doublet vev s @xmath239 and @xmath240 break the electroweak symmetry ; the vev @xmath241 may be relevant for generating neutrino masses@xcite . after the symmetry breaking , there are two charged gauge bosons , @xmath242 and @xmath243 , two heavy neutral gauge bosons , @xmath244 and @xmath245 , and the massless photon . we will assume that @xmath242 and @xmath244 are the lighter gauge bosons and obtain roughly their sm values after the symmetry breaking . turning off the @xmath58 triplet vev , @xmath246 , and assuming for simplicity that the @xmath247 gauge coupling constants satisfy @xmath248 , there are five fundamental parameters in the gauge / higgs sector , @xmath249 we can equivalently choose @xmath250 as input parameters . the counter term for the weak mixing angle is then defined through these parameters and their counter terms . assuming that the heavy gauge bosons are much heavier than the sm gauge bosons , @xmath251 , then to leading order @xmath252 , the counterterm @xmath253 is given as follows @xcite , @xmath254 where the effective weak mixing angle , @xmath85 , is defined as in eq . ( [ swdef ] ) . to go from the first to the second step in the above equation , we have used the following relation , @xmath255 . in the third line of eq . ( [ ds ] ) , we include only the leading top quark mass dependence . when the limit @xmath256 is taken , @xmath257 approaches zero , and thus the sm result , @xmath258 is _ not _ recovered , which is not what one would naively expect . one way to understand this is that in the left - right model , four input parameters are held fixed , while in the sm three input parameters are fixed . there is thus no continuous limit which takes one from one case ( @xmath83 at tree level ) to the other ( @xmath259 at tree level ) . this discontinuity , which has been pointed out previously@xcite , is closely tied to the fact that the triplet higgs boson is non - decoupling@xcite . due to this non - decoupling effect , even if the triplet vev is extremely small , as long as it is non - vanishing , there is the need for the fourth input parameter . the only exception to this is if there is a custodial symmetry which forces @xmath260 : in this case only the usual three input parameters are necessary . we also note that the contribution of the lightest neutral higgs in this case is given by@xcite , @xmath261 which depends on @xmath262 quadratically , and is suppressed by the heavy gauge boson masses , @xmath263 and @xmath264 . the contributions of the remaining scalar fields also have a similar structure . in this section , we summarize our results on minimization of the scalar potential in the model with a triplet higgs . from the minimization conditions , @xmath265 , we obtain the following conditions , @xmath266 we ues the short hand notaion , @xmath267 . the second derivatives are , @xmath268 if @xmath269 , then there is no mixing between the doublet and the triplet . this requires @xmath270 . a. sirlin , _ phys . rev . _ * d22 * , 971 ( 1980 ) . w. j. marciano and a. sirlin , _ phys . rev . _ * d22 * , 2695 ( 1980 ) [ erratum - ibid . * d31 * , 213 ( 1985 ) ] . a. sirlin and w. j. marciano , _ nucl . _ * b189 * , 442 ( 1981 ) . f. jegerlehner , _ renormalizaing the standard model _ , lecture given at theoretical advanced study institute in elementary particle physics ( tasi 90 ) , boulder , co , june 3 - 29 , 1990 . published in _ boulder tasi 90:476 - 590_. d.m . renormalization of supersymmetric theories _ , lecture given at the theoretical advanced study institute in elementary particle physics ( tasi 97 ) , boulder , co , june 1 - 7 , 1997 . published in _ boulder 1997 : supersymmetry , supergravity and supercolliders _ , 343 - 389 [ hep - ph/9805497 ] . j. r. forshaw , a. sabio vera and b. e. white , _ jhep _ * 0306 * , 059 ( 2003 ) [ arxiv : hep - ph/0302256 ] . g. passarino , _ phys . _ * b247 * , 587 ( 1990 ) . j. f. gunion , r. vega and j. wudka , _ phys . _ * d43 * , 2322 ( 1991 ) . j. c. pati and a. salam , _ phys . rev . _ * d10 * , 275 ( 1974 ) . r. n. mohapatra and j. c. pati , _ phys . _ * d11 * , 566 ( 1975 ) . r. n. mohapatra and j. c. pati , _ phys . rev . _ * d11 * , 2558 ( 1975 ) . g. senjanovic and r. n. mohapatra , _ phys . rev . _ * d12 * , 1502 ( 1975 ) . chen and k. t. mahanthappa , _ phys . rev . _ * d71 * , 035001 ( 2005 ) . chen and k. t. mahanthappa , _ int . j. mod * a18 * , 5819 ( 2003 ) . chen and k. t. mahanthappa , _ aip conf . _ * 721 * , 269 ( 2004 ) . .- c . chen and k. t. mahanthappa , _ phys . rev . _ * d62 * , 113007 ( 2000 ) . chen and k. t. mahanthappa , _ phys . rev . _ * d65 * , 053010 ( 2002 ) . chen and k. t. mahanthappa , _ phys . rev . _ * d68 * , 017301 ( 2003 ) . chen and k. t. mahanthappa , _ phys . _ * d70 * , 113013 ( 2004 ) .

electroweak precision data have been extensively used to constrain models containing physics beyond that of the standard model . when the model contains higgs scalars in representations other than singlets or doublets , and hence @xmath0 at tree level , a correct renormalization scheme requires more inputs than the three commonly used for the standard model case . in such cases , the one loop electroweak results can not be split into a standard model contribution plus a piece which vanishes as the scale of new physics becomes much larger than @xmath1 . we illustrate our results by presenting the dependence of @xmath1 on the top quark mass in a model with a higgs triplet and in the @xmath2 left - right symmetric model . in these models , the allowed range for the lightest neutral higgs mass can be as large as a few tev .

arxiv

a choice of nonconstant meromorphic function @xmath28 on a compact riemann surface @xmath0 realizes @xmath0 as a finite sheeted branched covering of the riemann sphere @xmath29 . _ log - riemann surfaces of finite type _ are certain branched coverings , in a generalized sense , of @xmath30 by a punctured compact riemann surface , namely , which are given by certain transcendental functions of infinite degree . formally a log - riemann surface consists of a riemann surface together with a local holomorphic diffeomorphism @xmath31 from the surface to @xmath30 such that the set of points @xmath19 added to the surface , in the completion @xmath18 with respect to the path - metric induced by the flat metric @xmath32 , is discrete . log - riemann surfaces were defined and studied in @xcite ( see also @xcite ) , where it was shown that the map @xmath31 restricted to any small enough punctured metric neighbourhood of a point @xmath33 in @xmath19 gives a covering of a punctured disc in @xmath30 , and is thus equivalent to either @xmath34 restricted to a punctured disc @xmath35 ( in which case we say @xmath33 is a ramification point of order @xmath1 ) or to @xmath36 restricted to a half - plane @xmath37 ( in which case we say @xmath33 is a ramification point of infinite order ) . a log - riemann surface is said to be of finite type if it has finitely many ramification points and finitely generated fundamental group . we will only consider those for which the set of infinite order ramification points is nonempty ( otherwise the map @xmath31 has finite degree and is given by a meromorphic function on a compact riemann surface ) . in @xcite , @xcite , uniformization theorems were proved for log - riemann surfaces of finite type , which imply that a log - riemann surface of finite type is given by a pair @xmath38 , where @xmath0 is a compact riemann surface , and @xmath31 is a meromorphic function on the punctured surface @xmath16 such that the differential @xmath39 has essential singularities at the punctures of a specific type , namely _ exponential singularities_. given a germ of meromorphic function @xmath40 at a point @xmath41 of a riemann surface , a function @xmath42 with an isolated singularity at @xmath41 is said to have an exponential singularity of type @xmath40 at @xmath41 if locally @xmath43 for some germ of meromorphic function @xmath13 at @xmath41 , while a 1-form @xmath44 is said to have an exponential singularity of type @xmath40 at @xmath41 if locally @xmath45 for some germ of meromorphic 1-form @xmath46 at @xmath41 . note that the spaces of germs of functions and 1-forms with exponential singularity of type @xmath40 at @xmath41 only depend on the equivalence class @xmath47 $ ] in the space @xmath48 of germs of meromorphic functions at @xmath41 modulo germs of holomorphic functions at @xmath41 . thus the uniformization theorems of @xcite , @xcite give us @xmath1 germs of meromorphic functions @xmath49 at the punctures @xmath50 , with poles of orders @xmath3 say , such that near a puncture @xmath51 the map @xmath31 is of the form @xmath52 , where @xmath53 is a germ of meromorphic function near @xmath51 and @xmath28 a local coordinate near @xmath51 . the punctures correspond to ends of the log - riemann surface , where at each puncture @xmath51 , @xmath54 infinite order ramification points are added in the metric completion , so that the total number of infinite order ramification points is @xmath55 . the @xmath54 infinite order ramification points added at a puncture @xmath51 correspond to the @xmath54 directions of approach to the puncture along which @xmath56 so that @xmath57 decays exponentially and @xmath58 converges . in the case of genus zero with one puncture for example , which is considered in @xcite , @xmath31 must have the form @xmath59 where @xmath60 is a rational function and @xmath61 is a polynomial of degree equal to the number of infinite order ramification points . in @xcite , certain spaces of functions and @xmath6-forms on a log - riemann surface @xmath62 of finite type were defined , giving rise to a derham cohomology group @xmath63 . the integrals of the @xmath6-forms considered along curves in @xmath62 joining the infinite ramification points converge , giving rise to a pairing between @xmath63 and @xmath21 , which was shown to be nondegenerate ( @xcite ) . the spaces of functions and @xmath6-forms defined were observed to depend only on the types @xmath49 of the exponential singularities , and so a notion less rigid than that of a log - riemann surface was defined , namely the notion of an _ exp - algebraic curve _ , which consists of a compact riemann surface @xmath0 together with @xmath1 equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions , @xmath2 , \cdots , [ h_n ] \}$ ] , with poles of orders @xmath3 at points @xmath4 . the relevant spaces of functions and @xmath6-forms with exponential singularities at @xmath4 of types @xmath9 , \cdots , [ h_n]$ ] can then be defined as follows : @xmath64 , \cdots , [ h_n ] \ } \\ { { \mathcal o}}_{{{\mathcal h } } } : = & \ { f \in { { \mathcal m}}_{{{\mathcal h } } } \,\mid\ , f \hbox { holomorphic on } s ' \ } \\ \omega_{{{\mathcal h } } } : = & \ { \omega \,\mid\ , \omega \hbox { meromorphic 1-form on } s ' \hbox { with exponential singularities}\\ & \hbox { of types } [ h_1 ] , \cdots , [ h_n ] \ } \\ \omega^0_{{{\mathcal h } } } : = & \ { \omega \in \omega_{{{\mathcal h } } } \,\mid\ , \omega \hbox { holomorphic on } s ' \}.\end{aligned}\ ] ] for @xmath65 ( respectively , @xmath66 ) we can define a divisor @xmath67 ( respectively , @xmath68 ) by @xmath69 if @xmath70 and @xmath71 if @xmath72 , where @xmath13 is a germ of meromorphic function at @xmath10 such that @xmath12 ( respectively , @xmath73 if @xmath70 and @xmath74 if @xmath72 , where @xmath46 is a germ of meromorphic @xmath6-form at @xmath10 such that @xmath15 ) . in @xcite it is shown how to naturally associate to an exp - algebraic curve @xmath75 a degree zero line bundle @xmath76 together with a meromorphic connection @xmath77 with poles at @xmath4 . the connection @xmath6-form of @xmath78 near @xmath10 is given ( with respect to an appropriate local trivialization ) by @xmath79 , so that the pair @xmath80 determines the exp - algebraic curve @xmath81 . there are naturally defined isomorphisms between the space of meromorphic sections of @xmath23 ( respectively , the space of meromorphic @xmath23-valued @xmath6-forms ) and @xmath82 ( respectively , @xmath83 ) , such that a meromorphic section @xmath84 of @xmath23 ( respectively , a meromorphic @xmath23-valued @xmath6-form @xmath46 ) maps to an @xmath65 with the same divisor as @xmath84 ( respectively , an @xmath66 with the same divisor as @xmath46 ) . in particular the space @xmath24 of meromorphic @xmath23-valued @xmath6-forms which are holomorphic on @xmath16 is naturally isomorphic to the space @xmath7 . fixing an @xmath11 inducing a log - riemann surface structure on @xmath0 , with completion @xmath18 , the @xmath6-forms in @xmath7 can be integrated along curves in @xmath21 , giving a map @xmath85 let @xmath25 denote the image of @xmath21 in @xmath86 . then our torelli - type theorem for exp - algebraic curves states that the pair @xmath87 determines the exp - algebraic curve @xmath88 : [ mainthm ] let @xmath89 be two exp - algebraic curves with the same underlying riemann surface @xmath0 , and the same set of punctures @xmath4 . suppose that @xmath90 is nontrivial , that the line bundles @xmath91 are isomorphic and that the induced isomorphism @xmath92 maps @xmath93 to @xmath94 . then @xmath95 . finally , we mention briefly some appearances of functions with exponential singularities in the literature . certain functions with exponential singularities , namely the @xmath1-point _ baker - akhiezer functions _ ( @xcite , @xcite ) , have been used in the algebro - geometric integration of integrable systems ( see , for example , @xcite , @xcite and the surveys @xcite , @xcite , @xcite , @xcite ) . given a divisor @xmath96 on @xmath16 , an @xmath1-point baker - akhiezer function ( with respect to the data @xmath97 ) is a function @xmath42 in the space @xmath98 satisfying the additional properties that the divisor @xmath99 of zeroes and poles of @xmath42 on @xmath16 satisfies @xmath100 , and that @xmath101 is holomorphic at @xmath51 for all @xmath102 . for @xmath96 a non - special divisor of degree at least @xmath13 , the space of such baker - akhiezer functions is known to have dimension @xmath103 . functions and differentials with exponential singularities on compact riemann surfaces have also been studied by cutillas ripoll ( @xcite , @xcite , @xcite ) , where they arise naturally in the solution of the _ weierstrass problem _ of realizing arbitrary divisors on compact riemann surfaces , and by taniguchi ( @xcite , @xcite ) , where entire functions satisfying certain topological conditions ( called structural finiteness " ) are shown to be precisely those entire functions whose derivatives have an exponential singularity at @xmath104 , namely functions of the form @xmath105 , where @xmath106 are polynomials . * acknowledgements . * this work grew out of a visit of the second author to tifr , mumbai . he would like to thank tifr for its hospitality . we recall some basic definitions and facts from @xcite , @xcite , @xcite . a log - riemann surface is a pair @xmath107 , where @xmath0 is a riemann surface and @xmath108 is a local holomorphic diffeomorphism such that the set of points @xmath19 added to @xmath0 in the completion @xmath109 with respect to the path metric induced by the flat metric @xmath32 is discrete . the map @xmath31 extends to the metric completion @xmath62 as a @xmath6-lipschitz map . in @xcite it is shown that the map @xmath31 restricted to a sufficiently small punctured metric neighbourhood @xmath110 of a ramification point is a covering of a punctured disc @xmath111 in @xmath30 , and so has a well - defined degree @xmath112 , called the order of the ramification point ( we assume that the order is always at least @xmath113 , since order one points can always be added to @xmath0 and @xmath31 extended to these points in order to obtain a log - riemann surface ) . a log - riemann surface is of finite type if it has finitely many ramification points and finitely generated fundamental group . for example , the log - riemann surface given by @xmath114 is of finite type ( with the metric @xmath32 it is isometric to the riemann surface of the logarithm , which is simply connected , with a single ramification point of infinite order ) , as is the log - riemann surface given by the gaussian integral @xmath115 , which has two ramification points , both of infinite order , as in the figure below : log - riemann surface of the gaussian integral in @xcite , it is shown that a log - riemann surface of finite type ( which has at least one infinite order ramification point ) is of the form @xmath116 , where @xmath16 is a punctured compact riemann surface @xmath8 and @xmath31 is meromorphic on @xmath16 and @xmath39 has exponential singularities at the punctures @xmath4 . let @xmath49 be the types of the exponential singularities of @xmath39 at the punctures @xmath4 . as described in @xcite , each puncture @xmath51 corresponds to an end of the log - riemann surface where @xmath54 infinite order ramification points are added , @xmath54 being the order of the pole of @xmath117 at @xmath51 . let @xmath33 be an infinite order ramification point associated to a puncture @xmath51 . an @xmath118-ball @xmath119 around @xmath33 is isometric to the @xmath118-ball around the infinite order ramification point of the riemann surface of the logarithm ( given by cutting and pasting infinitely many discs together ) , and there is an argument function @xmath120 defined on the punctured ball @xmath121 . while the function @xmath31 , which is of the form @xmath122 in a punctured neighbourhood of @xmath51 for some meromorphic @xmath6-form @xmath123 , extends continuously to @xmath33 for the metric topology on @xmath62 , in general functions of the form @xmath124 ( where @xmath46 is a @xmath6-form meromorphic near @xmath51 ) do * not * extend continuously to @xmath33 for the metric topology ( @xcite ) . limits of these functions in sectors @xmath125 do exist however and are independent of the sector ; we say that the function is _ stolz continuous _ at points of @xmath19 . define spaces of functions and @xmath6-forms on @xmath62 : @xmath126 we remark that these are simply the spaces @xmath127 defined in the introduction , where @xmath2 , \cdots , [ h_n ] \}$ ] . functions in @xmath98 are stolz continuous at points of @xmath19 taking the value @xmath128 there . the integrals of @xmath6-forms @xmath44 in @xmath129 over curves @xmath130 \to s^*$ ] joining points @xmath131 of @xmath19 converge if @xmath132 is disjoint from the poles of @xmath44 and tends to these points through sectors @xmath133 ( since any primitive of @xmath44 on a sector is stolz continuous ) . the definitions of the above spaces only depend on the types @xmath134 \in { { \mathcal m}}_{p_i}/{{\mathcal o}}_{p_i } \}$ ] of the exponential singularities of the @xmath6-form @xmath39 , which do not change if @xmath39 is multiplied by a meromorphic function . it is natural to define then a structure less rigid than that of a log - riemann surface of finite type . given a punctured compact riemann surface @xmath8 , two meromorphic functions @xmath135 on @xmath16 inducing log - riemann surface structures of finite type are considered equivalent if @xmath136 is meromorphic on the compact surface @xmath0 . an exp - algebraic curve is an equivalence class of such log - riemann surface structures of finite type . it follows from the uniformization theorem of @xcite that an exp - algebraic curve is given by the data of a punctured compact riemann surface and @xmath1 ( equivalence classes of ) germs of meromorphic functions @xmath137 \in { { \mathcal m}}_{p_i}/{{\mathcal o}}_{p_i } \}$ ] with poles at the punctures . we can associate a topological space @xmath138 to an exp - algebraic curve , given as a set by @xmath139 , where @xmath19 is the set of infinite ramification points added with respect to any map @xmath31 in the equivalence class of log - riemann surfaces of finite type , and the topology is the weakest topology such that all maps @xmath140 in the equivalence class extend continuously to @xmath138 . finally , for a meromorphic function @xmath42 on @xmath16 ( respectively , meromorphic @xmath6-form @xmath44 on @xmath16 ) with exponential singularities of types @xmath49 at points @xmath4 we can define a divisor @xmath67 ( respectively , @xmath68 ) by @xmath69 if @xmath70 and @xmath71 if @xmath72 , where @xmath13 is a germ of meromorphic function at @xmath10 such that @xmath12 ( respectively , @xmath73 if @xmath70 and @xmath74 if @xmath72 , where @xmath46 is a germ of meromorphic @xmath6-form at @xmath10 such that @xmath15 ) . note that the divisor @xmath99 can also be defined by @xmath141 , so it follows from the residue theorem applied to the meromorphic @xmath6-form @xmath142 that the divisor @xmath99 has degree zero . let @xmath143 be an exp - algebraic curve , where @xmath0 is a compact riemann surface of genus @xmath13 and @xmath49 are germs of meromorphic functions at points @xmath4 . let @xmath144 be the space of holomorphic @xmath6-forms on @xmath0 . the data @xmath145 defines a degree zero line bundle @xmath76 together with a transcendental section @xmath146 of this line bundle which is non - zero on the punctured surface @xmath16 as follows : solving the mittag - leffler problem locally for the distribution @xmath147 gives meromorphic functions on an open cover such that the differences are holomorphic on intersections , and hence gives an element of @xmath148 . under the exponential this gives a degree zero line bundle as an element of @xmath149 . explicitly this is constructed as follows : let @xmath150 be pairwise disjoint coordinate disks around the punctures @xmath4 and let @xmath151 be an open subset of @xmath16 intersecting each disk @xmath152 in an annulus @xmath153 around @xmath10 such that @xmath154 is an open cover of @xmath0 . define a line bundle @xmath76 by taking the functions @xmath155 to be the transition functions for the line bundle on the intersections @xmath156 . define a holomorphic non - vanishing section of @xmath76 on @xmath16 by : @xmath157 define a connection @xmath77 on @xmath76 by declaring that @xmath158 . then for any holomorphic section @xmath84 on @xmath151 , @xmath159 for some holomorphic function @xmath42 , and @xmath160 , so @xmath77 is holomorphic on @xmath151 . on each disk @xmath152 , letting @xmath161 be the section which is constant equal to @xmath6 on @xmath152 ( with respect to the trivialization on @xmath152 ) , for any holomorphic section @xmath84 on @xmath152 , @xmath162 for some holomorphic function @xmath42 , and @xmath163 , so @xmath164 thus the connection @xmath6-form of @xmath77 with respect to @xmath161 is given by @xmath79 , so @xmath77 is meromorphic on @xmath152 with a single pole at @xmath10 of order @xmath165 . let @xmath166 be the unique section of the dual bundle @xmath167 on @xmath16 such that @xmath168 on @xmath16 . then for any non - zero meromorphic section @xmath84 of @xmath23 , the function @xmath169 is meromorphic on @xmath16 with exponential singularities at @xmath4 of types @xmath49 , and the divisors of @xmath84 and @xmath42 coincide . thus the line bundle @xmath76 has degree zero . in summary we have : [ existence ] for any log - riemann surface of finite type @xmath62 , the line bundle @xmath76 has degree zero and the maps @xmath170 ( respectively , @xmath171 ) are mutually inverse isomorphisms between the spaces of meromorphic sections of @xmath23 and @xmath98 ( respectively , the spaces of meromorphic @xmath23-valued @xmath6-forms and @xmath172 ) preserving divisors . in particular the vector spaces @xmath173 are non - zero . since the isomorphisms above preserve divisors , the spaces @xmath174 correspond to the spaces of meromorphic sections of @xmath23 and meromorphic @xmath23-valued @xmath6-forms which are holomorphic on @xmath16 , both of which are non - empty . the correspondence @xmath175 gives a one - to - one correspondence between exp - algebraic structures on @xmath0 and degree zero line bundles on @xmath0 with meromorphic connections with all poles of order at least two , zero residues , and trivial monodromy . since the connection @xmath6-form of @xmath78 is given by @xmath79 on @xmath152 , all residues of @xmath78 are equal to zero , while the monodromy of @xmath78 is trivial since @xmath176 is a single - valued horizontal section . conversely , given such a meromorphic connection @xmath177 on a degree zero line bundle @xmath178 , if @xmath4 are the poles of @xmath177 and @xmath179 are the connection @xmath6-forms of @xmath177 with respect to trivializations near @xmath4 , then each @xmath180 has zero residue at @xmath10 and pole order at least two , hence there exist meromorphic germs @xmath49 near @xmath4 such that @xmath181 . we obtain an exp - algebraic curve @xmath182 . it is clear for an exp - algebraic curve @xmath26 that @xmath183 , so the correspondences are inverses of each other . finally we remark that by serre duality , the degree zero line bundle @xmath76 , given as an element of @xmath148 , can also be described as an element of @xmath184 using residues , as the linear functional @xmath185 we proceed to the proof of theorem [ mainthm ] . we will need the following theorems from @xcite and @xcite : [ approxn ] let @xmath0 be a compact riemann surface and @xmath188 a closed subset such that @xmath189 has finitely many connected components @xmath190 , and for each @xmath191 let @xmath192 be a point of @xmath193 . then any continuous function @xmath42 on @xmath194 which is holomorphic in the interior of @xmath194 can be uniformly approximated on @xmath194 by functions meromorphic on @xmath0 with poles only in the set @xmath195 . let @xmath196 be two exp - algebraic curves with the same underlying riemann surface @xmath0 and the same set of punctures @xmath4 , and suppose the hypothesis of theorem 1.1 holds , namely the line bundles @xmath91 are isomorphic and the induced isomorphism @xmath197 maps @xmath93 to @xmath94 . since the spaces @xmath198 are isomorphic to the spaces of meromorphic sections of @xmath199 and @xmath200 respectively , there is an induced isomorphism @xmath201 which preserves divisors . we fix non - zero functions @xmath202 which correspond to each other under this isomorphism , and let @xmath203 denote the completions induced by the corresponding log - riemann surface structures . we also fix a meromorphic @xmath6-form @xmath204 on @xmath0 . then the divisor preserving isomorphisms @xmath201 and @xmath205 induced by the isomorphism @xmath206 can be expressed as @xmath207 respectively , where @xmath13 varies over all meromorphic functions on @xmath0 . the hypothesis of theorem [ mainthm ] implies that for any @xmath212 , there is a @xmath213 such that @xmath214 for all meromorphic functions @xmath13 on @xmath0 such that @xmath215 is holomorphic on @xmath16 for @xmath216 . if @xmath208 is a meromorphic function on @xmath0 such that @xmath217 is holomorphic on @xmath16 , then @xmath218 for some meromorphic function @xmath13 on @xmath0 such that @xmath219 is holomorphic on @xmath16 . since the isomorphism @xmath220 is divisor preserving , we have that @xmath221 is also holomorphic on @xmath16 , so for any @xmath212 there is a @xmath222 such that @xmath223 since this is true for all @xmath212 , it follows from theorem [ pairing ] that @xmath224 , so there exists a meromorphic function @xmath225 on @xmath0 such that @xmath226 is holomorphic on @xmath16 and @xmath227 . in this case there exists a closed curve @xmath132 disjoint from the punctures @xmath4 and the poles and zeroes of @xmath235 such that @xmath236 is connected . fix a non - zero meromorphic function @xmath237 on @xmath0 such that @xmath238 ( and hence also @xmath239 ) is holomorphic on @xmath16 . if the meromorphic @xmath6-form @xmath240 ( which is holomorphic outside the punctures and the zeroes and poles of @xmath235 ) is not identically zero , then we can choose a continuous function @xmath241 on @xmath132 such that @xmath242 ( since the @xmath6-form @xmath243 is holomorphic and not identically zero on @xmath132 ) . by theorem [ approxn ] , since @xmath236 is connected and contains @xmath244 , we can choose a meromorphic function @xmath245 on @xmath0 which is holomorphic on @xmath246 and uniformly close enough to @xmath241 on @xmath132 such that @xmath247 . letting @xmath248 , we have that @xmath217 is holomorphic on @xmath16 and @xmath249 , contradicting lemma [ exact ] . it follows that @xmath250 , from which it follows easily that @xmath95 . in this case @xmath251 and we may assume @xmath252 . fix a non - zero polynomial @xmath61 such that @xmath253 are holomorphic on @xmath16 . then by lemma [ exact ] , for all @xmath254 , taking @xmath255 we have @xmath256 from which it follows that the laurent series of @xmath257 around @xmath258 vanishes identically , hence @xmath259 and @xmath95 . in this case @xmath251 and we may assume the single puncture @xmath260 , and that the functions @xmath235 are of the form @xmath261 for some polynomials @xmath262 . in this case it follows from the main theorem of @xcite that the dimension of @xmath263 equals @xmath264 . since @xmath93 and @xmath94 are isomorphic by hypothesis , it follows that @xmath265 say , where @xmath266 since @xmath267 is non - trivial . let @xmath268 . let @xmath269 be a basis for @xmath267 as described in section 4 of @xcite , each @xmath270 being a curve joining a pair of ramification points @xmath271 , where @xmath272 . by hypothesis , for each curve @xmath273 there is a @xmath274 such that @xmath275 for all polynomials @xmath276 . consider the @xmath277 matrix @xmath278 it follows from theorem iii.1.5.1 of @xcite that the @xmath279 @xmath6-forms @xmath280 span @xmath281 , and hence form a basis for @xmath281 . since by theorem [ pairing ] the pairing with @xmath267 is nondegenerate , it follows that the @xmath282 submatrix formed by the first @xmath279 columns of @xmath283 is nonsingular , thus @xmath283 has rank @xmath279 . on the other hand , since @xmath284 , it follows that @xmath285 hence there is a scalar @xmath286 such that @xmath287 , so @xmath288 . it follows from lemma [ exact ] that for any polynomial @xmath276 the @xmath6-form @xmath289 lies in @xmath211 . thus if @xmath290 , then @xmath291 , hence @xmath292 for all polynomials @xmath276 . since all @xmath6-forms in @xmath293 are of the form @xmath294 for some polynomial @xmath61 and any @xmath295 for some polynomial @xmath276 , it follows that @xmath296 is trivial , a contradiction . thus @xmath297 , so @xmath298 and hence @xmath95 .

an exp - algebraic curve consists of a compact riemann surface @xmath0 together with @xmath1 equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions , @xmath2 , \cdots , [ h_n ] \}$ ] , with poles of orders @xmath3 at points @xmath4 . this data determines a space of functions @xmath5 ( respectively , a space of @xmath6-forms @xmath7 ) holomorphic on the punctured surface @xmath8 with exponential singularities at the points @xmath4 of types @xmath9 , \cdots , [ h_n]$ ] , i.e. , near @xmath10 any @xmath11 is of the form @xmath12 for some germ of meromorphic function @xmath13 ( respectively , any @xmath14 is of the form @xmath15 for some germ of meromorphic @xmath6-form ) . for any @xmath14 the completion of @xmath16 with respect to the flat metric @xmath17 gives a space @xmath18 obtained by adding a finite set @xmath19 of @xmath20 points , and it is known that integration along curves produces a nondegenerate pairing of the relative homology @xmath21 with the derham cohomology group defined by @xmath22 . there is a degree zero line bundle @xmath23 associated to an exp - algebraic curve , with a natural isomorphism between @xmath7 and the space @xmath24 of meromorphic @xmath23-valued @xmath6-forms which are holomorphic on @xmath16 , so that @xmath21 maps to a subspace @xmath25 . we show that the exp - algebraic curve @xmath26 is determined uniquely by the pair @xmath27 . [ section ] [ theorem]proposition [ theorem]lemma [ theorem]corollary [ theorem]definition

arxiv

there has been recent progress in understanding physics of strongly correlated electronic systems and their electronic structure near a localization delocalization transition through the development of dynamical mean field theory ( dmft ) @xcite . merging this computationally tractable many body technique with realistic local density approximation ( lda ) @xcite based electronic structure calculations of strongly correlated solids is promising due to its simplicity and correctness in both band and atomic limits . at present , much effort is being made in this direction including the developments of a lda+dmft method anisimovkotliar , lda++ approach @xcite , combined gw and dmft theory @xcite , spectral density functional theory @xcite as well as applications to various systems such as la@xmath0sr@xmath1tio@xmath2 latio3 , v@xmath3o@xmath2 @xcite , fe and ni @xcite , ce @xcite , pu @xcite , transition metal oxides@xcite , and many others . for a review , see ref . . such _ ab initio _ dmft based self consistent electronic structure algorithms should be able to explore all space of parameters where neither dopings nor even degeneracy itself is kept fixed as different states may appear close to the fermi level during iterations towards self consistency . this is crucial if one would like to calculate properties of realistic solid state system where bandwidth and the strength of the interaction is not known at the beginning . it is very different from the ideology of model hamiltonians where the input set of parameters defines the regime of correlations , and the corresponding many body techniques may be applied afterwards . realistic dmft simulations of material properties require fast scans of the entire parameter space to determine the interaction for a given doping , degeneracy and bandwidth via the solution of the general multiorbital anderson impurity model ( aim ) @xcite . unfortunately , present approaches based on either non crossing approximation ( nca ) or iterative perturbation theory ( ipt ) are unable to provide the solution to that problem due to a limited number of regimes where these methods can be applied @xcite . the quantum monte carlo ( qmc ) technique dmft , jarrell is very accurate and can cope with multiorbital situation but not with multiplet interactions . also its applicability so far has been limited either to a small number of orbitals or to unphysically large temperatures due to its computational cost . recently some progress has been achieved using impurity solvers that improve upon the nca approximation rotors , jeschke , haule:2001 , but it has not been possible to retrieve fermi liquid behavior at very low temperatures with these methods in the orbitally degenerate case . as universal impurity solvers have not yet being designed in the past we need to explore other possibilities , and this paper proposes interpolative approach for the self energy in general multiorbital situation . we stress that this is not an attempt to develop an alternative method for solving the impurity problem , but follow up of the ideology of lda theory where approximations were designed by analytical fits @xcite to the quantum monte carlo simulations for homogeneous electron gas @xcite . numerically very expensive qmc calculations for the impurity model display smooth self energies at imaginary frequencies for a wide range of interactions and dopings , and it is therefore tempting to design such an interpolation . we also keep in mind that for many applications a high precision in reproducing the self energies may not be required . one of such applications is , for example , the calculation of the total energy ce , nature , science , nioprl which , as well known from lda based experience , may not be so sensitive to the details of the one electron spectra . as a result , we expect that even crude evaluations of the self energy shapes on imaginary axis may be sufficient for solving many realistic total energy problems , some of which have appeared already @xcite . another point is a computational efficiency and numerical stability . bringing full self consistent loops with respect to charge densities nature and other spectral functions require many iterations towards the convergency which may not need too accurate frequency resolutions at every step . however , the procedure which solves the impurity model should smoothly connect various regions of the parameter space . this is a crucial point if one would like to have a numerically stable algorithm and our new interpolational approach ideally solves this problem . in the calculations of properties such as the low energy spectroscopy and especially transport more delicate distribution of spectral weight is taken place at low energies , and the imaginary part of the analytically continued self energy needs to be computed with a greater precision . here we expect that our obtained spectral functions should be used with care . also , in a few well distinct regimes , such , e.g. , as very near the mott transition , the behavior maybe much more complicated and more difficult to interpolate . for the cases mentioned above extensions of the interpolative methods should be implemented and its beyond the scope of the present work . we can achieve a fast interpolative algorithm for the self energy by utilizing a rational representation . the coefficients in this interpolation can be found by forcing the self energy to obey several limits and constrains . for example , if infinite frequency ( hartree fock ) limit , positions of the hubbard bands , low frequency mass renormalization @xmath4 , mean number of particles @xmath5 as well as the value of the self energy at zero frequency @xmath6 are known from independent calculation , the set of interpolating coefficients is well defined . in this work , we explore the slave boson mean field ( sbmf ) approach @xcite and the hubbard i approximation @xcite to determine the functional dependence of these coefficients upon doping , degeneracy and the strength of the interaction @xmath7 . we verify all trends produced by this interpolative procedure in the regimes of weak , intermediate and strong interactions and at various dopings conditions . these trends are compared with known analytical limits as well as against calculations using the quantum monte carlo method . also , compared with qmc are self energies and spectral functions on both imaginary and real axes for selective values of dopings . they indicate that the sbmf approach can predict such parameters of interpolation as @xmath8and @xmath4 with a good accuracy while the hubbard i method fails in a number of regimes . however , the functional form of the atomic green function which appears within hubbard i can be used to determine positions of atomic satellites which helps to impose additional constraints on our procedure . giving an extraordinary computational speed of this approach we generally find a very satisfactory accuracy in comparisons with the numerically more accurate qmc calculations . if an increased accuracy is desired our method can be naturally extended by imposing more constraints and by implementing more refined impurity solvers other than the ones explored in this work . the paper is organized as follows . in section ii we discuss rational interpolation for the self energy and list the constraints . in section iii we discuss methods for solving anderson impurity model such as the slave boson mean field and the hubbard i approximations which can be used to find these constraints . a brief survey of the qmc method used to benchmark our algorithm is also given . we present numerical comparisons of sbmf and hubbard i techniques against the qmc simulations for such quantities as quasiparticle residue and multiple occupancies . in section iv we report the results of the interpolative method and compare the obtained spectral functions with the qmc . in section v we discuss possible improvements of the algorithm . section vi is the conclusion . to be specific , we concentrate on the anderson impurity hamiltonian @xmath9 , \label{ham}\end{aligned}\]]describing the interaction of the impurity level @xmath10 with bands of conduction electrons @xmath11 via hybridization @xmath12 . @xmath7 is the coulomb repulsion between different orbitals in the @xmath13band . inspired by the success of the iterative perturbation theory @xcite , in order to solve the anderson impurity model in general multiorbital case , we use a rational interpolative formula for the self energy . this can be encoded into a form@xmath14}{\prod\limits_{m=1}^{m}[\omega -p_{m}^{(\sigma ) } ] } , \label{rat}\]]the coefficients @xmath15 , @xmath16 , or , alternatively , the poles @xmath17 , zeroes @xmath18 and @xmath19 in this equation are to be determined . the form ( [ rat ] ) can be also viewed as a continuous fraction expansion but continuous fraction representation will not be necessary for the description of the method . our basic assumption is that only a well distinct set of poles in the rational representation ( [ rat ] ) is necessary to reproduce an overall frequency dependence of the self energy . extensive experience gained from solving hubbard and periodic anderson model within dmft at various ratios of the on site coulomb interaction @xmath7 to the bandwidth @xmath20 shows the appearance of lower and upper hubbard bands as well as renormalized quasiparticle peak in the spectrum of one electron excitations @xcite . it is clear that the hubbard bands are damped atomic excitations and to the lowest order approximation appear as the positions of the poles of the atomic green function . in the @xmath21 symmetry case which is described by the hamiltonian ( [ ham ] ) , these energies are numerated by the number of electrons occupying impurity level , i.e. @xmath22 and the atomic green function takes a simple functional form @xmath23where @xmath24 are the probabilities to find an atom in configuration with @xmath25 electrons while combinatorial factor @xmath26 arrives due to equivalence of all states with @xmath25 electrons in @xmath21 . we can represent the atomic green function ( [ leh ] ) using the rational representation ( [ rat ] ) , i.e. @xmath27}{\prod\limits_{n=1}^{n}[\omega -p_{n}^{(g ) } ] } , \label{gzp}\]]where @xmath28 are all @xmath29 atomic poles , while @xmath30 denote @xmath31 zeroes with @xmath29 being the total degeneracy . the centers of the hubbard bands are thus located at the atomic excitations @xmath32 . using standard expression for the atomic green function @xmath33 $ ] , we arrive to a desired representation for the atomic self energy @xmath34}{\prod\limits_{n=1}^{n-1}[\omega -z_{n}^{(g ) } ] } . \label{sat}\ ] ] using this functional form for finite @xmath35 modifies the positions of poles and zeroes via recalculating probabilities @xmath24 which is equivalent to the famous hubbard i approximation ( discussed in more detail in the next section ) . we now concentrate on the description of the quasiparticle peak which is present in metallic state of the system . for this an extra pole and zero have to be added in eq ( [ sat ] ) . to see this , let us consider the hubbard model for the @xmath21 case where the local green function can be written by the following hilbert transform @xmath36.$ ] if self energy lifetime effects are ignored , the local spectral function becomes @xmath37 $ ] where @xmath38 is the non interacting density of states . the peaks of the spectral functions thus appear as zeroes in eq . ( [ sat ] ) and in order to add the quasiparticle peak , one needs to add one extra zero ( denoted hereafter as @xmath39 to the numerator in eq . ( [ sat ] ) . to make the self energy finite at @xmath40 one has to also add one more pole ( denoted hereafter as @xmath41 ) which should appear in the denominator or eq . ( [ sat ] ) . furthermore , frequently the hartree fock value for the self energy can be computed separately and it is desirable to have a parameter in the functional form ( [ sat ] ) which will allow us to fix @xmath19 . an obvious candidate to be changed is that self energy pole in ( [ sat ] ) which is closest to @xmath42 equal zero . let us denote this parameter as @xmath43 and rewrite the denominator of ( [ sat ] ) as @xmath44 $ ] where the product is now extended over all zeroes of the atomic green functions except the one closest to zero and two extra poles @xmath41 and @xmath43 can control the width of the quasiparticle peak and @xmath45 thus , we arrive to the functional form for the self energy @xmath46}{(\omega -p_{1}^{(\sigma ) } ) ( \omega -p_{2}^{(\sigma ) } ) \prod\limits_{n=1}^{n-2}[\omega -z_{n}^{(g)}]}. \label{szp}\ ] ] we are now ready to list all constrains of our interpolative scheme . to fix the unknown coefficients @xmath47 @xmath48 @xmath49 @xmath50 in eq . ( [ szp ] ) and to write down the linear set of equations for the coefficients @xmath15 , @xmath16 in eq . ( [ rat ] ) . we use the following set of conditions . _ a ) _ _ hartree fock value _ @xmath45 in the limit @xmath40 the self energy takes its hartree fock form @xmath51mean level occupancy _ _ _ _ @xmath52is defined as a sum over all matsubara frequencies for the green s function , i.e.@xmath53where @xmath54defines the impurity green function and @xmath35 is the hybridization function . _ b ) _ _ zero frequency value _ @xmath55 the so called friedel sum rule establishes the relation between the total density and the real part of the self energy at zero frequency@xmath56 _ c ) quasiparticle mass renormalization value _ @xmath57 the slope of the self energy at zero frequency controls the quasiparticle residue , @xmath4 using the following relationship @xmath58formally , constraints ( _ b _ ) and ( _ c _ ) hold for zero temperature only but we expect no significant deviations in many regions of parameters as long as we stay at low enough temperatures . the behavior may be more elaborated in the vicinity of mott transition @xcite . _ d ) _ _ positions of hubbard bands . _ as we discussed , in order that the self energy obeys the atomic limit and places the centers of the hubbard bands at the positions of the atomic excitations , we demand that @xmath59 this condition fixes almost all self energy zeroes @xmath18 in eq.([rat ] ) to the poles @xmath60 however , it alone does not ensure that the weight is correctly distributed among the hubbard bands and that the very distant hubbard bands disappear . for this to occur , distant poles of green function have to be canceled out by nearby zeroes of the green function . it is clear that each pole @xmath28 far from the fermi level has to be accompanied by a nearby zero @xmath30 in order the weight of the pole be small . thus , the self energy has poles at the position of green s function zeroes which can be encoded into the constrain @xmath61^{-1}=0 . \label{zer}\]]we want to keep this property of the self energy for finite @xmath62 and thus demand that self energy diverges ( when lifetime effects are kept , it only reaches a local maximum ) at the zeros of the functional form ( [ leh ] ) of @xmath63 note that the relationship ( [ zer ] ) holds ( approximately ) for frequency @xmath42 larger than the renormalized bandwidth @xmath64 therefore the information about one @xmath65 which lies close to @xmath66 is omitted and replaced by the information about @xmath67 @xmath6 and @xmath4 as it is done by separating @xmath41 and @xmath43 in the denominator of eq . ( szp ) . we can now write down a set of linear equations for all unknown coefficients in the expression ( [ rat ] ) . there is total @xmath68 of parameters @xmath15 and @xmath69 where we can always set @xmath70 the conditions _ a)_,_b),c ) _ give @xmath71according to condition _ d ) _ we can use all @xmath29 poles @xmath28 and @xmath72 zeroes @xmath73 the zero @xmath30 closest to @xmath66 is dropped out . this brings additional @xmath74 equations for the coefficients and makes @xmath75 as the degree of the rational interpolation which are written below @xmath76^{m}-(p_{n}^{(g)}+\mu -\epsilon _ { f})\sum_{m=0}^{n}b_{m}[p_{n}^{(g)}]^{m } & = & 0\text { for } n=1 ... n , \label{co5 } \\ \sum_{m=0}^{n}b_{m}[z_{n}^{(g)}]^{m } & = & 0\text { for } n=1 ... n-2 .\label{co6}\end{aligned}\ ] ] note that while @xmath77 may be rather large , the actual number of poles contributing to the self energy behavior is indeed very small . we can directly see this from eq . ( [ sat ] ) which uses all @xmath29 poles @xmath28 fulfilling eq . ( [ pol ] ) and uses @xmath72 zeroes @xmath30 directly related to @xmath72 poles @xmath78 clearly , when the spectral weight of the atomic excitation becomes small , the corresponding @xmath79 becomes close to @xmath30 and the cancellation occurs . therefore in realistic situations when only the upper and lower hubbard bands have significant spectral weight along with the quasiparticle peak , the actual degree of the polynomial expansion is either two or three . however , it is advantageous numerically and cheap computationally to keep all poles and zeroes in eq . ( [ szp ] ) because the formula automatically distributes spectral weight over all existing hubbard bands . in the limit when @xmath80 the self energy automatically translates to the non interacting one . the atomic poles get close to each other but , most importantly , their spectral weight goes rapidly to zero as it gets accumulated within the quasiparticle band . in the mott insulating regime , the conditions _ b ) _ and _ c ) _ drop out while all poles @xmath28 and zeroes @xmath30 can be used to determine the interpolation . however , in this regime it does not matter whether one of @xmath30 closest to @xmath66 is dropped out or kept , since we can always replace this information by information about @xmath81 therefore the mott transition can be studied without changing the constraints . we thus see that in the insulating case the self energy correctly reproduces the well known result of the hubbard i method where the green function is computed after eq . ( [ imp ] ) with atomic self energy . if the lifetime effects are computed , the parameters @xmath28 and @xmath30 become complex and the hubbard bands will acquire an additional bandwidth . this effect is evident from the simulations using various perturbative or qmc impurity solvers and can be naturally incorporated into the interpolative formulas ( [ rat ] ) or ( [ szp ] ) . however , in practical implementation below we will omit it for illustrative purposes . let us now discuss the quality of interpolation from the perspective of the high frequency behavior for the self energy . the latter can be viewed moments as expansion in terms of the moments @xmath82 , i.e. , @xmath83 . most important for us is to look at highest moments which are given by the hartree fock value , eq . ( [ inf ] ) involving single occupancy matrix @xmath84 , as well as the first moment @xmath85u^{2 } , \label{ms1}\ ] ] containing a double occupancy matrix @xmath86 we see that the interpolation in part relies on the accuracy in computing multiple occupancies which are the functionals of both atomic excitations and the hybridization function . in this regard , using exact atomic green function to find poles @xmath28 and zeroes @xmath30 as part of the constrained procedure may not be as accurate since it would assume the use of _ _ atomic _ _ multiple occupancies which do _ not _ carry information about @xmath87on the other hand , we can also use only a functional form of the atomic green function where the multiple occupancies are computed in a more rigorous manner . in the next section we will show how this can be implemented using the sbmf multiple occupancies which will be found to be in better agreement with the quantum monte carlo data . note that the moments @xmath82 themselves can be used in establishing the constraints for interpolation coefficients . this would involve independent evaluations of @xmath88 , etc . as well as various integrals involving hybridization function @xmath35 . however , we may run into ill defined numerical problem since high frequency information will be used to extract the low frequency behavior . therefore , it is more advantageous numerically to use some poles and zeros of @xmath89 as given by condition _ d ) _ above . we thus see that the interpolational scheme is defined completely once a prescription for obtaining parameters such as @xmath90 @xmath91 as well as poles @xmath28 , and zeroes @xmath30 is given . for this purpose we will test two commonly used methods : sbmf method due to gutzwiller @xcite as described by kotliar and ruckenstein ruck and the well known hubbard i approximation @xcite . we compare these results against more accurate but computationally demanding quantum monte carlo calculations and establish the procedure to extract all necessary parameters . note that once the constraints such as @xmath4 are computed from a given approximate method , some of the quantities such as the total number of particles , @xmath84 , and the value of the self energy at zero frequency , @xmath92 can be computed fully self consistently . they can be compared with their non self consistent values . if the approximate scheme already provides a good approximation for @xmath84 and satisfies the friedel sum rule , the self consistency can be avoided hence accelerating the calculation . indeed we found that inclusion of the self consistency improves the results only marginally except when we are in a close vicinity to the mott transition but here we do not expect that our simple interpolative algorithm is very accurate . we now give the description of the approximate methods for solving the impurity model and then present the comparisons of our interpolative procedure with the qmc calculations . the quantum monte carlo method is a powerful and manifestly not perturbative approach in either interaction @xmath7 or the bandwidth @xmath20 . in the qmc method one introduces a hubbard stratonovich field and averages over it using the monte carlo sampling . this is a controlled approximation using different expansion parameter , the size of the mesh for the imaginary time discretization . unfortunately it is computationally very expensive as the number of time slices and the number of hubbard stratonovich fields increases . also the method works best at imaginary axis while analytical continuation is less accurate and has to be done with a great care . extensive description of this method can be found in ref . . we will use this method to benchmark our calculations using approximate algorithms described later in this section . a fast approach to solve a general impurity problem is the slave boson method @xcite . at the mean field level , it gives the results similar to the famous gutzwiller approximation @xcite . however , it is improvable by performing fluctuations around the saddle point . this approach is accurate as it has been shown recently to give the exact critical value of @xmath7 in the large degeneracy limit at half filling florens . the main idea is to rewrite atomic states consisting of @xmath25 electrons @xmath93 , @xmath94 with help of a set of slave bosons @xmath95 . in the following , we assume @xmath21 symmetric case , i.e. , equivalence between different states @xmath96 for fixed @xmath25 . formulae corresponding to a more general crystal field case are given in appendix b. the creation operator of a physical electron is expressed via slave particles in the standard manner @xcite . in order to recover the correct non interacting limit at the mean field level , the bose fields @xmath97 can be considered as classical values found from minimizing a lagrangian @xmath98 corresponding to the hamiltonian ( [ ham ] ) . two lagrange multipliers @xmath99 and @xmath100 should be introduced in this way , which correspond to the following two constrains : @xmath101 the numbers @xmath102 are similar to the probabilities @xmath24 discussed in connection to the formula for the atomic green function ( leh ) . we thus see the physical meaning of the first constrain which is the sum of probabilities to find atom in any state is equal to one , and the second constrain gives the mean number of electrons coinciding with that found from @xmath103 . a combinatorial factor @xmath104 arrives due to assumed equivalence of all states with @xmath25 electrons . minimization of @xmath98 with respect to @xmath105 leads us to the following set of equations to determine the quantities @xmath105 : @xmath106\psi _ { n}+nbt\sum_{i\omega } \delta ( i\omega ) g_{g}(i\omega ) [ lr\psi _ { n-1}+\psi _ { n}bl^{2}]+\]]@xmath107=0 , \label{guz}\]]where @xmath108 , determines the mass renormalization , and the coefficients @xmath109 , @xmath110 are normalization constants as in refs . . @xmath111 is the total energy of the atom with @xmath25 electrons in @xmath21 approximation . ( [ guz ] ) , along with the constrains ( [ one ] ) , ( [ two ] ) constitute a set of non linear equations which have to be solved iteratively . in practice , we consider eq . ( [ guz ] ) as an eigenvalue problem with @xmath100 being the eigenvalue and @xmath105 being the eigenvectors of the matrix . the physical root corresponds to the lowest eigenvalue of @xmath100 which gives a set of @xmath105 determining the mass renormalization @xmath112 since the matrix to be diagonalized depends non linearly on @xmath105 via the parameters @xmath113 and @xmath114 and also on @xmath99 , the solution of the whole problem assumes the self consistency : i ) we build an initial approximation to @xmath105 ( for example the hartree fock solution ) and fix some @xmath99 , ii ) we solve eigenvalue problem and find new normalized @xmath105 , iii ) we mix new @xmath105 with the old ones using the broyden method @xcite and build new @xmath115 and @xmath114 . steps ii ) and iii ) are repeated until the self consistency with respect to @xmath105 is reached . during the iterations we also vary @xmath99 to obey the constrains . the described procedure provides a stable computational algorithm for solving aim and gives us an access to the low frequency green s function and the self energy of the problem via knowledge of the slope of @xmath116 and the value @xmath117 at zero frequency . , ( b ) dependence of the spectral weight @xmath118 on concentration , and ( c ) density density correlation function , @xmath119 versus filling , @xmath5 , in the two band hubbard model in @xmath120 and @xmath121,title="fig:",scaledwidth=50.0% ] + the described slave boson method gives the following expression for the self energy : @xmath122the impurity green function @xmath123 in this limit is given by the expression @xmath124 as an illustration , we now give the solution of eq . ( [ guz ] ) for non degenerate case ( @xmath125 and at the particle hole symmetry point with @xmath126 . consider a dynamical mean field theory for the hubbard model which reduces the problem to solving the impurity model subject to the self consistency condition with respect to @xmath35 . starting with the semicircular density of states ( dos ) , the self consistency condition is given by eq . ( [ guz])@xmath127 we obtain the following simplifications : @xmath128 , @xmath129 , @xmath130 and @xmath131^{-1}.$ ] the sum @xmath132 appeared in eq . ( [ guz ] ) scales as @xmath133 with the constant @xmath134 being the characteristic of a particular density of states and approximately equal to 0.2 in the semicircular dos case . the self consistent solution of eq . ( [ guz ] ) is therefore possible and simply gives @xmath135 . the mott transition occurs when no sites with double occupancies can be found , i.e. when @xmath136 a critical value of @xmath137 . for @xmath138 , this gives @xmath139 and reproduces the result for @xmath140 known from the qmc calculation within a few percent accuracy . as degeneracy increases , critical @xmath7 is shifted towards higher values @xcite . from numerical calculations we obtained the following values of the critical interactions in the half filled case @xmath141 for @xmath142 ( @xmath143level)@xmath144 @xmath145 for @xmath146 ( @xmath147level)@xmath144 and @xmath148 for @xmath149 ( @xmath13level ) . density density correlation function @xmath119 for local states with @xmath25 electrons is proportional to the number of pairs formed by @xmath25 particles @xmath150 . since the probability for @xmath25 electrons to be occupied is given by : @xmath151 , the physical density density correlator can be deduced from : @xmath152 . similarly , the triple occupancy can be calculated from @xmath153 . let us now check the accuracy of this method by comparing its results with the qmc data . we consider the two band hubbard model in @xmath154 orbitally degenerate case . hybridization @xmath155 satisfies the dmft self consistency condition of the hubbard model on a bethe lattice @xmath156the coulomb interaction is chosen to be @xmath157 which is sufficiently large to open the mott gap at integer fillings . all calculations are done for the temperature @xmath158 . we first compare the average number of electrons vs. chemical potential determined from the slave bosons which is plotted in fig . [ figsbmf](a ) . this quantity is sensitive to the low frequency part of the green function which should be described well by the present method . we see that it reproduces the qmc data with a very high accuracy and only differs by 20 per cent very near the jump of @xmath159 at @xmath160 . the quasiparticle residue @xmath4 versus filling @xmath5 is plotted in fig . [ figsbmf](b ) . the slave boson method gives the fermi liquid and provides estimates for the quasiparticle residue with the overall discrepancy of the order of 30% . in fact , we have performed several additional calculations for other degeneracies ( @xmath161 and @xmath162 ) and for various parameters regimes . the trend to overestimate mass renormalization can be seen in many cases . it disappears only when @xmath7 approaches zero . we need to point out , however , that i ) the extractions of zero frequency self energy slopes from the high temperature qmc is by itself numerically not well grounded procedure , as information for the self energy is known at the matsubara points only , which is then extrapolated to @xmath66 , ii ) other methods for solving impurity model , such as nca or ipt display similar discrepancies and iii ) recent findings @xcite suggest that at least at half filling quasiparticle residues deduced from slave bosons becomes exact when @xmath163 . most importantly for our interpolative method is that the entire functional dependence of @xmath4 vs. filling , interaction and degeneracy is correctly captured . its overall accuracy is acceptable as it is evident from our comparisons of the spectral functions presented in the next section and well within the main goal of our work to provide fast scans of the entire parameter space necessary for simulating real materials . this is important as , for example , for general @xmath13electron material , the qmc method is prohibitively time consuming , but we expect from the sbmf method the results for mass renormalization not worse than 50% for such delicate regime as the vicinity of the mott transition . [ figsbmf](c ) shows the density density correlation function @xmath119 as a function of average occupation @xmath5 . the discrepancy is most pronounced for fillings @xmath164 [ see the inset of fig . [ figsbmf](c ) ] where the absolute values of @xmath119 are rather small . although our slave boson technique captures only the quasiparticle peak , it gives the correlation function in reasonable agreement with the qmc for dopings not too close to the mott transition . now we turn to the hubbard i approximation @xcite which is closely related to the moments expansion method @xcite . consider many body atomic states @xmath165 which in @xmath21 are all degenerate with index @xmath166 numerating these states for a given number of electrons @xmath167 the impurity green function is defined as the average@xmath168and becomes diagonal with all equal elements in @xmath169it is convenient to introduce the hubbard operators @xmath170and represent the one electron creation and destruction operators as follows@xmath171the impurity green function ( [ gim ] ) is given by @xmath172 where the matrix @xmath173 is defined as @xmath174 establishing the equations for @xmath173 can be performed using the method of equations of motion for the @xmath175 operators . performing their decoupling due to hubbard @xcite , carrying out the fourier transformation and analytical continuation to the real frequency axis , and summing over @xmath25 and @xmath176 after ( [ hub ] ) we arrive to the net result @xmath177the @xmath178 can be viewed in the matrix form ( [ hub ] ) with the following definition of a diagonal atomic green function@xmath179with @xmath111 being the total energies of the atom with @xmath25 electrons in @xmath169 the coefficients @xmath24 are the probabilities to find atom with @xmath25 electrons and were already discussed in connection to the formula ( [ leh ] ) for the atomic green function . they are similar to the coefficients @xmath102 introduced within the sbmf method but now found from different set of equations . these numbers are normalized to unity , @xmath180 and are expressed via diagonal elements of @xmath181 as follows : @xmath182their determination in principle assumes solving a non linear set of equations while determining @xmath183 the mean number of electrons can be measured as follows : @xmath184 or as follows @xmath185 the numbers @xmath24 can be also used to find the averages @xmath186 @xmath187 in the way similar to what has been done in the sbmf approach . , ( b ) dependence of the spectral weight @xmath118 on concentration , and ( c ) density density correlation function , @xmath188 versus filling , @xmath5 , in the two band hubbard model in @xmath120 and for @xmath189 . , title="fig:",scaledwidth=50.0% ] + if we neglect by the hybridization @xmath35 in eq . ( [ h1e ] ) , the probabilities @xmath24 become simply statistical weights : @xmath190we thus see that in principle there are several different ways to determine the coefficients @xmath24 , either via self consistent determination ( scw ) , or using statistical formula ( [ stw ] ) , or taking them from sbmf equation ( [ guz ] ) , i.e. setting @xmath191 while still utilizing the functional dependence provided by the hubbard i method . to determine the best procedure let us first consider limits of large and small @xmath7 s . when @xmath192 , @xmath123 is reduced to @xmath193 i.e. the hubbard i method reproduces the atomic limit . setting @xmath194 gives @xmath195^{-1}$ ] , which is the correct band limit . unfortunately , at half filling this limit has a pathology connected to the instability towards mott transition at any interaction strength @xmath7 . to see this , we consider a dynamical mean field theory for the hubbard model . using semicircular density of states , we obtain @xmath196^{-1}g_{at}(\omega ) $ ] and conclude that for any small @xmath7 the system opens a pathological gap in the spectrum . clearly , using hubbard i only , the behavior of the green function at @xmath197 can not be reproduced and the quality of the numbers @xmath24 is at question . this already emphasizes the importance of using the slave boson treatment at small frequencies . ultimately , making the comparisons with the qmc calculations is the best option in picking the most accurate procedure to compute the probabilities @xmath24 . to check the accuracy against the qmc we again consider the two band hubbard model in @xmath120 symmetry as above . the chemical potential@xmath144 mass renormalization and double occupancy are plotted versus filling in fig . [ fighub1 ] . all quantities here were computed with statistical weights after eq . ( [ stw ] ) but we found a similar agreement while utilizing the self consistent determination of @xmath24 after eq . ( [ scw ] ) . we first see that the hubbard i approximation does not give satisfactory agreement with the qmc data for @xmath198 because it misses the correct behavior at low frequencies . the comparisons for @xmath199 plotted in fig . [ fighub1](b ) surprisingly show a relatively good behavior . however , the pathology of this approximation at half filling would predict @xmath200 for any @xmath7 , which is a serious warning not to use it for extracting the quasiparticle weight . [ fighub1](c ) shows @xmath119 as a function of average occupation @xmath5 . as this quantity is directly related to the high frequency expansion one may expect a better accuracy here . however , comparing fig . [ fighub1](c ) and fig . [ figsbmf](c ) , it is clear that the slave boson method gives more accurate double occupancy . this is due to the fact that the density matrix obtained by the slave boson method is of higher quality than the one obtained from the hubbard i approximation . the results of these comparisons suggest that the probabilities @xmath201 provided by the slave boson method is a better way in determining the coefficients @xmath24 in the metallic region of parameters . therefore it is preferable to use these numbers while establishing the equations for the unknown coefficients in the interpolational form ( [ rat ] ) . however , the functional form ( [ atg ] ) of the hubbard i approximation with @xmath202 can still be used as it provides the positions of the poles @xmath28 and zeroes @xmath30 of the atomic green function necessary for the condition _ d ) _ in the previous section . this also ensures accurate high frequency behavior of the interpolated self energy since its moments expressed via multiple occupancies are directly related to @xmath203 interestingly , while more sophisticated qmc approach captures both the quasiparticle peak and the hubbard bands this is not the case for the slave boson mean field method . to obtain the hubbard bands in this method fluctuations need to be computed , which would be very tedious in a general multiorbital situation . however the slave boson method delivers many parameters in a good agreement with the qmc results , and , hence , it can be used to give a functional dependence of the coefficients of the rational approximation . by now the procedure to determine the coefficients is well established . we use the sbmf method to determine @xmath204 @xmath205 as well as poles and zeroes of the atomic green function provided by the sbmf probabilities @xmath102 and by the bare atomic energy levels @xmath206 ( we omit the lifetime effects for simplicity)@xmath127 this generates a set of linear equations for coefficients @xmath207 , @xmath208 in the rational interpolation formula ( [ rat ] ) . in the present section we show the trends our interpolative algorithm gives for the spectral functions in various regions of parameters as well as provide detailed comparisons for some values of doping for both imaginary and real axis spectral functions . the two band hubbard model with semicircular density of states and dmftself consistency condition after ( [ scf ] ) is utilized in @xmath154 symmetry in all cases using the bandwidth @xmath209 and temperature @xmath210 . [ figu2trend ] shows the behavior of the density of states @xmath211 for @xmath212 as a function of the chemical potential @xmath159 computed with respect to the particle hole symmetry point @xmath213 and as a function of frequency @xmath214 the semicircular quasiparticle band is seen at the central part of the figure . its bandwidth is only weakly renormalized by the interactions in this regime . it is half filled for @xmath215 ( i.e. when @xmath216 ) and gets fully emptied when chemical potential is shifted to negative values . several weak satellites can be also seen on this figure which are due to atomic poles . their spectral weight is extremely small in this case and any sizable lifetime effect ( which is not included while plotting this figure ) will smear these satellites out almost completely . while approaching fully emptied ( or fully filled situation ) the spectral weight of the hubbard bands disappears completely and only unrenormalized quasiparticle band remains . it is clear that even without shifting the atomic poles to the complex axis , the numerical procedure of generating the self energy is absolutely stable . and frequency for the two band hubbard model in @xmath120 and at @xmath217.,title="fig:",scaledwidth=50.0% ] + and frequency for the two band hubbard model in @xmath120 and at @xmath217 . , title="fig:",scaledwidth=50.0% ] + and frequency for the two band hubbard model in @xmath120 and at @xmath189.,title="fig:",scaledwidth=50.0% ] + and frequency for the two band hubbard model in @xmath120 and at @xmath189 . , title="fig:",scaledwidth=50.0% ] + this trend can be directly compared with the simulations using a more accurate qmc impurity solver . we present this in fig . [ figu2trendqmc ] for @xmath212 , which shows calculated density of states in the same region of parameters . remarkably that again we can distinguish the renormalized quasiparticle band and very weak hubbard satellites . the hubbard bands appear to be much more diffuse in this figure mainly due to the lifetime effects and partially due to maximum entropy method using for analytical continuation from imaginary to real axis . otherwise the entire picture looks very much like the one on fig . [ figu2trend ] , generated with much less computational effort . [ figu4trend ] gives the same behavior of the density of states for the strongly correlated regime @xmath218 in this case the situation at integer filling is totally different as the system undergoes metal insulator transition . this is seen around the dopings levels with @xmath159 between 0 and -1 and between -3 and -5 where the wight of the quasiparticle band collapses while lower and upper hubbard bands acquire all spectral weight . in the remaining region of parameters both strongly renormalized quasiparticle band and hubbard satellites remain . again , once full filling or full emptying is approached the quasiparticle bands restores its original bandwidth while the hubbard bands disappear . the qmc result for the same region of parameters is given in fig . [ figu4trendqmc ] . again we can distinguish the renormalized quasiparticle band and hubbard satellites as well as the areas of mott insulator and of strongly correlated metal . the hubbard bands appear to be more sharp in this figure which signals on approaching the atomic limit . we now turn to the comparison of the green functions and the self energies obtained using the formulae ( [ imp ] ) , and ( [ rat ] ) respectively against the predictions of the quantum monte carlo method . we will report our comparisons for the two band hubbard model and sets of dopings corresponding to @xmath219 using the value of @xmath220 other tests for different degeneracies , doping levels and the interactions have been performed which display similar accuracy . and @xmath220,scaledwidth=65.0% ] fig . [ figgrn ] shows the comparison between the real and imaginary parts of the green function obtained by the interpolative method with the results of the qmc calculations . as one can see almost complete agreement has been obtained for a wide regime of dopings . the agreement gets less accurate once the half filling is approached , but still very good giving an extraordinary computational speed of the given method compared to qmc . and @xmath220,title="fig:",scaledwidth=65.0% ] + fig . [ figsig ] shows similar comparison between the real and imaginary parts of the self energies obtained by the interpolative and the qmc method . we can see that the self energies exhibit some noise which is intrinsic to stochastic qmc procedure . the values of the self - energies near @xmath221 and @xmath222 are correctly captured with some residual discrepancies are attributed to slightly different chemical potentials used to reproduce given filling within every method . the results at the imaginary axis show slightly underestimated slopes of the self energies within the interpolative algorithm which is attributed to the underestimated values of @xmath4 obtained from the sbmf calculation . ultimately improving these numbers by inclusions of fluctuations beyond mean field will further improve the comparisons . however , even at the present stage of the accuracy all functional dependence brought by the sbmf method quantitatively captures the behavior of the self energy seen from time consuming qmc simulation . . and @xmath220,title="fig:",scaledwidth=65.0% ] + we also made detailed comparisons between calculated densities of states obtained at the real axis using the interpolative method and the qmcalgorithm . the qmc densities of states require an analytical continuation from the imaginary to real axis and were generated using the maximum entropy method . by itself this procedure introduces some errors within the qmc especially at higher frequencies . in fig . [ figdos ] , we show our results for the fillings corresponding to @xmath219 using the value of @xmath189 . one can see the appearance of the quasiparticle band and two hubbard bands distanced by the value of @xmath7 . it can be seen that the interpolative method remarkably reproduces the trend in shifting the hubbard bands upon changing the doping . it automatically holds the distance between them to the value of @xmath7 while this is not always true in the quantum monte carlo method . despite this result , the overall agreement between both methods is very satisfactory . here we would like to discuss possible ways to further improve the accuracy of the method . the inaccuracies are mainly seen in three different quantities : i ) the width of the hubbard bands , ii ) the mass renormalization @xmath223 which is borrowed from the sbmf method , and iii ) the number of electrons @xmath224 extracted from the interpolated impurity green function ( [ imp ] ) . the inaccuracy in the width of the hubbard band is mainly connected to neglecting the lifetime effect . provided it is computed , this will shift the positions of atomic poles onto the complex plane which is in principle trivial to account for within our interpolative algorithm . to improve the accuracy of @xmath223 one can , for example , work out a modified slave boson scheme which will account for the fluctuations around mean field solution . the inaccuracy in @xmath224 is small in many regions of parameters and typically amounts to 510 per cent . we can try to improve this agreement by requirement that @xmath224 obtained via interpolation matches with @xmath225 obtained by the sbmf method . the latter agrees very well with the qmc for a wide region of parameters as it is evident from fig . [ figsbmf](a ) . in reality , our analysis shows that in many cases the discrepancy in @xmath224 is connected with the overestimation of @xmath226 . therefore , points ii ) and iii ) mentioned above are interrelated . the requirement that @xmath227 can be enforced by adjusting the width of the quasiparticle band , and in many regions of parameters this is controlled by @xmath228 however , there are situations when the hubbard band appears in the vicinity of @xmath66 , and changing @xmath223 does not affect the bandwidth@xmath127 to gain a control in those cases it is better to replace the constraints @xmath229 by constraints of fixing the self energies at two frequencies@xmath144 say @xmath230 and @xmath231 where @xmath232 is the frequency of the order of renormalized bandwidth@xmath127 we have found that this scheme brings mass renormalizations which are about 30% smaller than the sbmf ones , and the agreement with the qmc is significantly improved . thus , inaccuracies ii ) and iii ) can be avoided with this very cheap trick . however , we also would like to point out that the condition @xmath227 is essentially non linear as the solution may not exist for all regions of parameters . it is , for example , evident that in such points where @xmath224 is given by a symmetry ( as , e.g. , particle hole symmetry point @xmath233 in the case considered above ) the mass renormalization does not affect the number of electrons . as the philosophy of our approach is to get the best possible fit we are also open to implementing any kinds of _ ad hoc _ renormalizations constants . one of such possibility could be the use of a quasiparticle residue 30 per cent smaller than @xmath234 as @xmath223 should go to unity when @xmath235 , the correction can , for example , be encoded into the formula @xmath236.$ ] we finally would like to remark that the scheme defined by a set of linear equations for the coefficients ( [ co1])([co6 ] ) is absolutely robust as solutions exist for all regimes of parameters such as strength of the interaction , doping and degeneracy . in general , bringing any information on the self energy @xmath237 at some frequency point @xmath238 or its derivative @xmath239 would generate a linear relationship between the interpolation coefficients , thus keeping robustness of the method . on the other hand , fixing such relationships as numbers of electrons brings non linearity to the problem which could lead to multiplicity or non existence of the solutions . it is also clear that by narrowing the regime of parameters , the accuracy of the interpolative algorithm can be systematically increased . to summarize , this paper shows the possibility to interpolate the self energies for a whole range of dopings , degeneracies and the interactions using a computationally efficient algorithm . the parameters of the interpolation are obtained from a set of constraints in the slave boson mean field method combined with the functional form of the atomic green function . the interpolative method reproduces all trends in remarkable agreement with such sophisticated and numerically accurate impurity solver as the qmc method . we also obtain a very good quantitative agreement in a whole range of parameters for such quantities as mean level occupancies , spectral functions and self energies . some residual discrepancies remain which can be corrected provided better algorithms delivering the constraints will be utilized . nevertheless , given the superior speed of the present approach , we have obtained a truly exceptional accuracy times efficiency of the proposed procedure . the work was supported by nsf dmr grants 0096462 , 02382188 , 0312478 , 0342290 and us doe grant no de the authors also acknowledge the financial support from the computational material science network operated by us doe and from the ministry of education , science and sport of slovenia . in the crystal field case we assume that @xmath29fold degenerate impurity level @xmath10 is split by a crystal field onto @xmath240 sublevels @xmath241 . we assume that for each sublevel there is still some partial degeneracy @xmath242 so that @xmath243 in limiting case of @xmath21 degeneracy , @xmath244 , and in non degenerate case , @xmath245 . we need to discuss how a number of electrons @xmath25 can be accommodated over different sublevels @xmath246 . introducing numbers of electrons on each sublevel , @xmath247 , we obtain @xmath248 note the restrictions : @xmath249 @xmath250 and @xmath251 . in @xmath21 case , @xmath252 , and in non degenerate case , @xmath253 , @xmath247 is either 0 or 1 . total energy for the shell with @xmath25 electrons depends on particular configuration @xmath254 @xmath255 . \label{slacrflev}\ ] ] many body wave function is also characterized by a set of numbers @xmath254 , i.e. @xmath256 energy @xmath257 remains degenerate , which can be calculated as product of how many combinations exists to accommodate electrons in each sublevel , i.e. @xmath258 let us further introduce probabilities @xmath259 to find a shell in a given state with energy @xmath260 sum of all probabilities should be equal to @xmath261 , i.e.@xmath262 there are two green functions in gutzwiller method : impurity green function @xmath263 and quasiparticle green function @xmath264 where matrix coefficients @xmath265 represent generalized mass renormalizations parameters . all matrices are assumed to be diagonal and have diagonal elements enumerated as follows : @xmath266 each element in the green function is represented as follows@xmath267@xmath268and determines a mean number of electrons in each sublevel@xmath269the total mean number of electrons is thus : @xmath270 hybridization function @xmath271 is the matrix which is assumed to be diagonal , and it has diagonal elements enumerated as follows : @xmath272 mass renormalizations @xmath273 are determined in each sublevel . the generalization of the non linear equations ( [ guz ] ) has the form@xmath276 \psi _ { n_{1} ... n_{g}}+ \notag \\ & & \sum_{\alpha = 1}^{g}n_{\alpha } [ t\sigma _ { i\omega } \delta _ { \alpha } ( i\omega ) g_{g\alpha } ( i\omega ) ] b_{\alpha } \left [ r_{\alpha } l_{\alpha } \psi _ { n_{1} ... n_{\alpha } -1 ... n_{g}}+b_{\alpha } l_{\alpha } ^{2}\psi _ { n_{1} ... n_{\alpha } ... n_{g}}\right ] + \notag \\ & & \sum_{\alpha = 1}^{g}(d_{\alpha } -n_{\alpha } ) [ t\sigma _ { i\omega } \delta _ { \alpha } ( i\omega ) g_{g\alpha } ( i\omega ) ] b_{\alpha } \left [ r_{\alpha } l_{\alpha } \psi _ { n_{1} ... n_{\alpha } + 1 ... n_{g}}+b_{\alpha } r_{\alpha } ^{2}\psi _ { n_{1} ... n_{\alpha } ... n_{g}}\right ] . \label{slacrfscf}\end{aligned}\ ] ] g. kotliar and s. y. savrasov , in _ new theoretical approaches to strongly correlated systems _ , edited by a. m. tsvelik , ( kluwer academic publishers , the netherlands , 2001 ) , p. 259 , ( available in cond mat/020824 ) ; s. biermann , f. aryasetiawan , and a. georges , phys . lett . * 90 * , 086402 ( 2003 ) . k. held , i. a. nekrasov , g. keller , v. eyert , n. bluemer , a. k. mcmahan , r. t. scalettar , th . pruschke , v. i. anisimov , and d. vollhardt , psi k newsletter # * 56 * ( april 2003 ) , p. 65 ; a. i. lichtenstein , m. i. katsnelson , and g. kotliar , in _ electron correlations and materials properties _ , ed . by a. gonis , n. kioussis and m. ciftan ( kluwer academic , plenum publishers , 2002 ) p. 428

a rational representation for the self energy is explored to interpolate the solution of the anderson impurity model in general orbitally degenerate case . several constrains such as the friedel s sum rule , positions of the hubbard bands as well as the value of quasiparticle residue are used to establish the equations for the coefficients of the interpolation . we employ two fast techniques , the slave boson mean field and the hubbard i approximations to determine the functional dependence of the coefficients on doping , degeneracy and the strength of the interaction . the obtained spectral functions and self energies are in good agreement with the results of numerically exact quantum monte carlo method .

arxiv

the study of nonequilibrium phase transitions is a topic of growing interest due to its application to a variety of complex systems@xmath3 : contact process , domain growth , catalysis , phase separation and transport phenomena . although there is no general theory to account for nonequilibrium model systems , in recent years some progress has been achieved in understanding the stationary states of these systems employing approximate analytical methods and simulations . some rigorous mathematical questions concerning the phase transitions of these complex interacting particle systems can be appreciated in the books of liggett@xmath4 and konno@xmath5 . in this paper we focus our attention on the phase transitions observed in the surface reaction model proposed by ziff , gulari and barshad@xmath6 ( zgb ) , which describes some kinetic aspects of the oxidation of @xmath7 over a catalytic surface . in particular , here we consider a modified version of the zgb model , where we incorporate a random distribution of inert sites on the catalytic surface . the original zgb model is an irreversible lattice model for surface reactions based on the langmuir - hinshelwood mechanism , where the reactants must be adsorbed before reacting . the steps used to describe the zgb model ( a lattice markov process ) are the following : molecules of @xmath0 and @xmath1 from a gaseous phase can be adsorbed onto the sites of a regular square lattice of identical sites . these molecules arrive at the surface according to their partial pressures in the gas mixture , that is , the probability of a @xmath0 molecule arriving is @xmath8 and @xmath9 for the @xmath1 molecule . the @xmath0 molecule requires only a single vacant site to be adsorbed , while the @xmath1 is adsorbed if it finds a nearest - neighbor pair of empty sites . upon adsorption , the @xmath1 molecule dissociates and the two free @xmath10 atoms can react independently . if , after an adsorption step , a nearest - neighbor @xmath11 pair appears on the lattice , they immediately react , forming a @xmath2 molecule that goes to the gas phase , leaving two empty sites on the lattice . therefore , in this adsorption controlled limit , only a single parameter ( @xmath8 ) is sufficient to describe the dynamics of the model . the simulations performed by ziff and co - workers have shown that the system exhibits two phase transitions between active and poisoned states : for @xmath12 , an o - poisoned state is formed , while for @xmath13 the lattice is poisoned by @xmath0 . for @xmath14 a reactive steady - state is found , in which a nonzero number of vacant sites is present in the lattice . at @xmath15 the transition is continuous , whereas at @xmath16 the transition is of the first - order type . using a mean field theory , dickman@xmath17 qualitatively reproduced the phase diagram of the zgb model and showed that , at the level of site approximation , only the first - order transition appears . however , employing the pair approximation , both continuous and first - order transitions are obtained . we are interested on the effects of inert sites on the phase transitions of the zgb model . we have investigated in detail the dependence of the phase transitions on the concentration of inert sites . this problem presents some experimental interest in the automobile industry , where lead particles are deposited over the catalyst during the exhaust of the gases after combustion . this affects the efficiency of the catalytic surface due to the pinning of these lead particles on the surface , forbidding the adsorption of @xmath0 and @xmath1 molecules at the lead positions and reducing the reaction paths . hovi and co - workers@xmath18 , have studied by computer simulations the effect of preadsorbed poison and promoters on the irreversible zgb model . they calculated the coverage of species as a function of the concentration of inert sites for a wide range of values , finding the interesting result that the first - order transition changes to a continuous one at a critical value of the concentration . corts and valencia@xmath19 have also reported some results concerning random impurities distributed over the catalyst , in which they observed the change of the first - order transition into a continuous one as one increases the concentration of impurities . albano@xmath20 simulated the zgb model on incipient percolation clusters ( ipc s ) with a fractal dimension of 1.90 . he showed that both transitions , at @xmath15 and @xmath16 are continuous , and that for an infinite lattice , in which @xmath8 is larger than 0.408 , the reactions stop at finite times because the ipc s are poisoned by pure @xmath0 . casties et al.@xmath21 also performed a monte carlo simulation of the @xmath0 oxidation on probabilistic fractals . they observed a change in the character of the transition at @xmath16 from first order on regular lattices to second order on percolation clusters ( for @xmath22 larger than @xmath23 , which is the percolation threshold on the square lattice ) . in this work we have performed mean - field ( site and pair approximations ) calculations and monte carlo simulations for different values of the concentration of inert sites . the model studied here is a variant of the original zgb model , where inert sites are randomly distributed over the lattice . our approach is close related to that presented by vigil and willmore@xmath24 to study the effects of spatial correlations on the oscillatory behavior of a modified zgb model , where defects are continually added and desorbed from the surface . in their studies , they considered the mean - field site and pair approximations , as well as monte carlo simulations . in the present work we have determined the phase diagram for different concentrations , and the spinodal and transition lines as a function of the concentration of inert sites . we have constructed hysteresis curves to find the critical concentration at which the first - order transition changes into a continuous one . this paper is organized as follows : in sec . ii we present the results obtained within the site approximation ; in sec . iii we introduce the pair approximation equations and show the results obtained using this scheme ; sec . iv presents the results of simulations , and finally , in sec . v we present our conclusions . we take a square lattice as our catalytic surface . a fraction @xmath25 of the sites is randomly distributed over the lattice representing the pinned inert sites . the remaining sites of the lattice can be vacant , or occupied by either @xmath10 atoms or @xmath0 molecules . the zgb model is described by the following steps : @xmath26 where the labels @xmath27 and @xmath28 denote the gaseous phase and an adsorbed reactant on the surface , respectively , and @xmath29 indicates a vacant site . steps ( 1 ) and ( 2 ) indicate the adsorption of the species , whereas the third step is the proper reaction , between distinct species located at adjacent sites of the lattice . in the site approximation the time evolution equations of the concentrations are given by @xmath30 where @xmath31 , @xmath32 and @xmath33 represent , respectively , the coverages of @xmath10 , @xmath0 and blank sites in the lattice . @xmath8 gives the arrival probability of a @xmath0 molecule . in addition , there is the following constraint among the concentrations @xmath34 the steady - state solutions of the above system of equations are given by @xmath35 , that corresponds to a poisoned surface , and @xmath36 inserting eq . ( 7 ) into eq . ( 4 ) we obtain an expression for the steady - state values of the concentration @xmath32 : @xmath37 we exhibit in fig . 1 a typical diagram for the coverages of @xmath0 , @xmath10 and vacant sites obtained for @xmath38 . this diagram was obtained by integrating the equations of motion for the @xmath32 and @xmath31 concentrations , starting from an initial condition in which the number of empty sites is @xmath39 . the site approximation does not give any continuous transition for all values of the concentration of inert sites . this was already pointed out by dickman@xmath17 for the zgb model without inert sites . we observe in fig . 1 , that the limit of stability of the reactive phase is @xmath40 , which corresponds to the spinodal point . therefore , a reactive steady - state is found for all values of @xmath41 . for values of @xmath42 , the system becomes poisoned , with a large amount of @xmath0 and a small concentration of @xmath10 atoms . the presence of @xmath10 atoms in the region @xmath42 is due to the inert sites that can block some oxygen , and also to the simplicity of the site approximation , which does not forbid the formation of @xmath43 nearest - neighbor pairs in the lattice . the tolerance of these @xmath43 pairs also explains the absence of the continuous phase transition , which is observed in the simulations . 2 is a plot of the solutions @xmath32 of eq.(8 ) versus the parameter @xmath44 for different values of concentration @xmath25 of inert sites . we obtain two solutions , which we call @xmath45 and @xmath46 , that join together at the spinodal point . for instance , for @xmath47 the value we find is @xmath48 , which furnishes the value @xmath49 . we also note in fig . 2 that , at the spinodal point , the concentration of @xmath32 molecules remains the same irrespective of the value we choose for @xmath25 . this special value is @xmath50 . then , the net effect of adding @xmath25 is to shift the curves horizontally . in this site approximation , solutions are possible only for values of @xmath51 . this happens because above this value the solution would correspond to the non - physical value @xmath52 . so , the meaning of the two solutions in fig . 2 is the following : the branch @xmath46 represents the stable steady - state solutions whereas the @xmath45 branch gives the unstable solutions . these solutions were obtained after numerical integration of the equations of motion for @xmath32 and @xmath31 , starting from the state described by @xmath39 . for the initial condition @xmath53 and @xmath32 larger than @xmath45 the system evolves to the poisoned state . the initial condition @xmath53 and @xmath32 less than @xmath45 drives the system to the lower stable reactive solution @xmath46 . fig . 2 also shows that , as we approach the spinodal point for any value of @xmath25 , the region of stability becomes narrower . then , we expect that for some value of @xmath41 a first - order transition occurs , that is , the concentration @xmath32 must increase abruptally from a small value ( reactive phase ) to a large value ( poisoned phase ) . unfortunately , we can not use here the usual thermodynamic considerations based on the minimization of a suitable thermodynamic potential . in order to find this first - order transition we adopt the same kinetic criteria employed by dickman@xmath17 , which was borrowed from the work of ziff et al@xmath6 . the phase transition was determined by choosing an initial state where half of the lattice was empty and the other half was completely filled with @xmath0 . in this work we choose as our initial state , to solve the equations of motion for @xmath32 and @xmath31 , the values @xmath54 . it is clear that this choice is not the same as that considered by ziff et al . , because we can not discriminate which sites are empty or not . the phase boundary is defined at the special value @xmath16 where the solution of the equations of motion changes from the reactive to the poisoned state as we vary the value of @xmath8 for the same initial condition , as established above . for @xmath47 we obtain the same value found by dickman . we exhibit in fig . 3 the results obtained for the first - order transition and the spinodal points for @xmath38 . the spinodal was obtained from the initial condition @xmath39 , and the first - order transition from the condition @xmath54 . for this particular value of @xmath38 , we have @xmath55 and @xmath56 . we have considered all values of the concentration of inert sites , and fig . 4 shows the values of @xmath57 ( dashed line ) and @xmath16 ( full line ) as a function of the concentration of inert sites . at the particular value @xmath58 the two lines merge . for values of @xmath59 the transition still remains of the first - order type , although the number of vacant sites that stay in the active state is very small . for instance , for @xmath60 , at the transition point ( @xmath61 ) , the number of vacant sites changes from @xmath62 in the active state , to @xmath63 in the poisoned state . throughout our analysis we considered a given state to be active if the number of vacant sites is larger than @xmath64 . we also exhibit in fig . 5 the number of vacant sites @xmath33 at the active state as a function of the number of inert sites @xmath25 , at the transition and spinodal points . we observe that for all values of @xmath65 , the number of vacant sites at the spinodal point is always larger than that at the transition point . let us consider the application of the pair approximation procedure to this zgb model that includes inert sites . here we introduce the pair probability @xmath66 of a random nearest neighbor pair of sites being occupied by species @xmath67 and @xmath68 . we have the following types of species : @xmath29 , @xmath69 , @xmath70 , and @xmath10 , which represent , respectively , vacant , inert , carbon monoxide , and oxygen . as in the previous treatments @xmath71 we need to consider only the changes that occur at a particular central pair in the lattice . in the table below we display the allowed and forbbiden ( indicated by @xmath72 ) nearest - neighbor pairs in the present model . @xmath73 the next table also exhibits all the possible transitions among pairs . we obtain @xmath74 independent transitions , labelled by numbers in the range @xmath75 . in the table transitions indicated by @xmath72 are prohibited . @xmath76 then , we write the equations relating the probability of each element with the corresponding pair probabilities : @xmath77 the pair probabilities also satisfy the constraint @xmath78 next , we need to write the time evolution equations for the pair probabilities . examining the latter table we can construct the desired equations of evolution . we explicitly write the equations of motion for the pair probabilities @xmath79 . @xmath80 where @xmath81 to @xmath82 are the transition rates . the factors of two arising in the equation of motion for @xmath83 are due to the fact that the pair probabilities @xmath84 and @xmath85 are equal by symmetry . for instance , from the pair @xmath86 we can obtain , with the same probability , the different configurations @xmath87 and @xmath88 . in general , the expressions for the transition rates are lengthy , and we present these transition rates in the appendix . in this pair approximation we can not obtain analytical solutions as we have done in the site approximation . we solved the coupled set of eight nonlinear equations by the fourth - order runge - kutta method , searching for the stationary solutions . we considered the two different initial conditions as in the case of the site approximation . let us first consider the evolution from the initial state where @xmath89 , in which only the pairs @xmath86 , @xmath90 and @xmath91 are present in the lattice at @xmath92 . figure 6 shows the diagram of the model for @xmath93 . for @xmath94 the lattice poisons with oxygen . in the range @xmath95 there is an active region , and for @xmath96 the lattice poisons with @xmath0 . when @xmath97 , we found the same figures obtained by dickman in his pair approximation . for instance , the site and pair approximations give the same value for the spinodal point @xmath57 . however , when we consider some inert sites in the lattice , the spinodal point found in the site approximation is always smaller than that obtained within the pair approximation . for this particular value , @xmath93 , the site approximation yields @xmath98 , whereas @xmath56 is obtained by the pair approximation . the value of @xmath8 at the continuous transition , which now arises in this pair approximation , decreases slightly with increasing values of the concentration of inert sites . we also considered the solutions evolving from an initial condition where half of the free sites ( @xmath89 ) is filled with @xmath0 molecules and the other half left empty . in order to be close to the initial condition used in the simulation , we chose for the initial pair conditions @xmath99 , @xmath100 and @xmath101 , which mimics a division of the lattice into two parts : on one side of the lattice we would have inert sites and @xmath0 molecules and , on the other side , vacant and inert sites . if @xmath97 , we found for the transition between the active and co - poisoned states the value @xmath102 , which agrees with the value found in the simulations . 7 displays the concentration of @xmath0 molecules at the transition point for which @xmath93 . in this pair approximation , the values of @xmath57 and @xmath16 are very close . we also show in fig . 8 the concentration of vacant sites as a function of the concentration of inert sites , at the transition point , and also at the spinodal point . both curves join at @xmath103 , and for @xmath104 , we can not observe any active state . as in the site approximation , an active state is defined only if @xmath105 . then , the calculations performed within the pair approximation give results that are very similar to those obtained by the site approximation , concerning the spinodal and transition points . in addition , it was observed that initial conditions do not affect the point in which the continuous phase transition occurs . in fig . 9 we exhibit the phase diagram for this zgb model with inert sites . the size of the reactive window decreases as we increase the concentration of inert sites . we have plotted the transition line for the first - order transition and for the spinodal line , which gives the limit of stability of the reactive phase . the line separating the active and o - poisoned phases is a continuous transition line . we have performed monte carlo simulations in the zgb model with inert sites in order to check the results we have obtained in the site and pair approximations . the simulations were carried for different values of the concentration of inert sites @xmath106 . for small values of @xmath106 , we considered square lattices of linear size @xmath107 , but for large values of @xmath106 we have used lattices of linear size up to @xmath108 . the first step in the simulation is to randomly distribute the selected fraction @xmath106 of inert sites in the lattice . all simulations then started with a fraction of empty sites equal to @xmath89 . the @xmath0 molecules arrive at the surface with a probability @xmath8 and the @xmath1 molecules with probability @xmath109 . the rules for adsorption and reaction of the species are exactly the same as in the original zgb model@xmath6 . since adsorption of oxygen requires two nearest neighbor empty sites , the effect of the inert sites is to favour the adsorption of @xmath0 relatively to that of @xmath1 molecules . in general , we have taken @xmath110 monte carlo steps ( mcs ) to attain the stationary states , and @xmath110 more to calculate the concentration averages at the stationary states . one mcs is equal to @xmath111 trials of deposition of species , where @xmath112 is the linear size of the lattice . to speed up the simulations we worked with a suitable list of empty sites . we exhibit in fig . 10 the phase diagram of the model in the plane @xmath8 versus @xmath106 . it is similar to that obtained within the pair approximation . however , there is a fundamental difference between the transition line separating the active and co - poisoned phases in both approaches . in the pair approximation the transition line is always of the first - order type , whereas in the simulations there is a critical concentration above which the transition becomes continuous . we have done detailed simulations to find the critical concentration at which the transition becomes continuous . we have found for the critical concentration of inert sites the value @xmath113 . we arrived at this value by looking at the hysteresis loops in the curves of @xmath114 versus @xmath8 for different values of the concentration @xmath106 , as we can see in fig . we proceed as follows : in fig . 11a we fixed the concentration of inert sites at the value @xmath115 and the curve with circles , which is the proper transition curve , was obtained from an initial state where @xmath89 , that is , with a lattice almost empty . the curve with squares was determined from an initial state in which the lattice was almost covered by @xmath0 . we have taken a fraction of only @xmath116 of randomly empty sites over the lattice at the starting time . then , we clearly observe the hysteresis loop at the concentration @xmath106 , which implies that the transition is of first - order . on the other hand , fig . 11b , where the fraction of inert sites is @xmath117 , does not exhibit the hysteresis loop and the transition is clearly a continuous one . the critical value of @xmath113 was obtained analysing the behavior of these curves in the range @xmath118 . as we have pointed out in the introduction , hovi et al.@xmath18 had already observed the change in the nature of this transition as a function of the concentration . the phase boundary separating the active and the o - poisoned phases in fig . 10 is continuous for all values of @xmath106 . we have checked this fact by observing that no hysteresis loop was found for any value of @xmath106 . the width of the active phase decreases with increasing values of @xmath106 . for values of @xmath119 the lattice is poisoned ( absence of empty sites ) with different amounts of @xmath0 and @xmath10 species . due to finite size effects , this value is larger than the value 0.408 found by albano@xmath20 in the limit of very large ipc s . we have also noted that the production rate of @xmath2 molecules attains its maximum value exactly at the first - order transition , for values of @xmath120 . if @xmath121 the maximum production rate of @xmath2 molecules is located inside of the reactive window . this is seen in fig . 12 , where the circles indicate the points where the production rate of @xmath2 is maximum . in the site and pair approximations this maximum occurs always at the phase boundary , irrespective of the value of @xmath106 . 13 shows the production rate @xmath122 of @xmath2 molecules as a function of @xmath106 . as expected , the role of inert sites is also of blocking the reactions over the catalyst . the maximum production rate occurs at a surface free of impurities . we have studied the effects of a random distribution of inert sites on the phase diagram of the zgb model . we determined the time evolution equations for the concentrations of the different species over the catalytic surface within an effective field theory , at the level of site and pair approximations , and also performed monte carlo simulations on the model . we obtained the coverages of the species as function of the deposition rate of @xmath0 and of the concentration of inert sites . in the site and pair approximations we found the transition line and the limit of stability of the reactive phase . in the site approximation , the continuous transition between the o - poisoned and reactive states is absent for any values of the concentration of inert sites . the width of the reactive window exhibits the same behavior , as a function of concentration of inert sites , in both pair approximation and monte carlo simulations . however , the transition between the reactive and co - poisoned phase is always of first - order in the site and pair approximations , whereas monte carlo simulations give a critical point where the transition changes nature . for values of the concentration of inert sites less than the critical value , the transition is first - order and above this value , it changes to a continuous one . the determination of this critical concentration was possible through the analysis of the hysteresis curves for different values of the concentration of inert sites . the production rate of @xmath2 molecules is maximum at the first - order transition , in both site and pair approximations . this is the case in the simulations , but the transition is of the first - order type . when the concentration of inert sites is greater than the critical value , the maximum production rate of @xmath2 molecules moves towards the reactive window . the overall effect of inert sites is to reduce the production of @xmath2 molecules . we would like to thank ron dickman by his many valuable suggestions , and luis g. c. rego by the critical reading of the manuscript . this work was supported by the brazilian agencies capes , cnpq and finep . we present the transition rates in the pair approximation , which we used in section iii to solve the time evolution equations for the pair probabilities . the transition rates are @xmath81 to @xmath82 , which are given by @xmath123 @xmath132 + \frac 13\left ( \frac{p_{vo}}{p_v}\right ) ^2\left ( 1-\frac{p_{oo}}{p_o}\right ) \nonumber \\ & & + \left . \frac 23\frac{p_{vo}}{p_v}\left ( 1-\frac{p_{vo}}{p_v}\right ) \frac{% p_{oo}}{p_o}+\frac 14\left ( \frac{p_{vo}}{p_v}\right ) ^2\frac{p_{oo}}{p_o}% \right\ } \nonumber\end{aligned}\ ] ] @xmath135 + \frac{p_{vc}}{p_c}% \left [ \left ( 1-\frac{p_{vc}}{p_v}\right ) ^2\right . \nonumber \\ & & + \left . \frac{p_{vc}}{p_v}\left ( 1-\frac{p_{vc}}{p_v}\right ) + \frac 13% \left ( \frac{p_{vc}}{p_v}\right ) ^2\right ] \times \left . \left [ \frac{p_{vc}% } { p_v}\left ( 1-\frac{p_{vc}}{p_v}\right ) + \frac 23\left ( \frac{p_{vc}}{p_v}% \right ) ^2\right ] \right\ } \nonumber\end{aligned}\ ] ] @xmath136 \right\ } \nonumber \\ t_{14b } & = & 2(1-y_{co})p_{cd}\frac{p_{vc}}{p_c}\left\ { 2\frac{p_{vv}}{p_v}\left [ \left ( 1-\frac{p_{vc}}{p_v}\right ) \left ( 1-\frac{p_{cd}}{p_d}\right ) + \frac 12\frac{p_{vc}}{p_v}\left ( 1-\frac{p_{cd}}{p_d}\right ) \right . \right . \nonumber \\ & & + \left . \frac 12\left ( 1-\frac{p_{vc}}{p_v}\right ) \frac{p_{cd}}{p_d}+% \frac 13\frac{p_{vc}}{p_v}\frac{p_{cd}}{p_d}\right ] + \frac{p_{vd}}{p_d}% \left ( 1-\frac{p_{vc}}{p_v}\right ) ^2+\frac 22\frac{p_{vc}}{p_v}\frac{p_{vd}% } { p_d } \nonumber \\ & & + \left . \left . \frac 13\left ( \frac{p_{vc}}{p_v}\right ) ^2\right ] \right\ } \nonumber\end{aligned}\ ] ] j. marro and r. dickman , _ nonequilibrium phase transitions in lattice models _ ( cambridige university press , cambridge , 1999 ) . v. privman , ed . , _ nonequilibrium statistical mechanics in one dimension _ ( cambridge university press , cambridge , 1996 ) . liggett , _ interacting particle systems _ ( springer - verlag , new york , 1985 ) . n. konno , _ phase transitions of interacting particle systems _ ( world scientific , singapore , 1994 ) . r. m. ziff , e. gulari , and y. barshad , phys . 56 * , 2553 ( 1986 ) . r. dickman , phys . rev . a * 34 * , 4246 ( 1986 ) . hovi , j. vaari , h. -p . kaukonen and r.m . nieminen , compt . science , * 1 * , 33 ( 1992 ) . joaquin corts and eliana valencia , surf . sci . * 425 * , l357 ( 1999 ) . albano , phys . b * 42 * , 10818 ( 1990 ) . a. casties , j. mai , and w. von niessen , j. chem . phys . * 99 * , 3082 ( 1993 ) . r. dennis vigil and frank t. willmore , phys . e * 54 * , 1225 ( 1996 ) . dickman , b.c.s . grandi , w. figueiredo and r. dickman , phys . e * 59 * , 6361 ( 1999 ) . da costa and w. figueiredo , phys . e * 61 * , 1134 ( 2000 ) .

a random distribution of inert sites is introduced in the ziff - gulari - barshad model to study the phase transitions between active and poisoned states . the adsorption of @xmath0 and @xmath1 molecules is not possible at the position of the inert sites . this model is investigated in the site and pair approximations , as well as through monte carlo simulations . we determine the mean coverages of the elements as a function of the dilution and show that the continuous transition between the active and o - poisoned state is slightly affected by moderate values of dilution in the pair approximation and in the simulations . on the other hand , from the analysis of the hysteresis curves , the transition between the active and co - poisoned states changes from first - order to continuous as one increases the concentration of inactive sites . the observed transition in the site and pair approximations is always of first - order nature . we also found the lines of transition and spinodal points as a function of the concentration of inert sites . finally , the production rate of @xmath2 is calculated as a function of the dilution of sites . pacs number(s ) : 05.70.ln , 05.70.fh , 82.65.jv , 82.20.mj

arxiv

the purpose of this paper is to discuss the hydrodynamic limit for interacting particle systems in the crystal lattice . problems of the hydrodynamic limit have been studied intensively in the case where the underlying space is the euclidean lattice . we extend problems to the case where the underlying space has geometric structures : the _ crystal lattice_. the crystal lattice is a generalization of classical lattice , the square lattice , the triangular lattice , the hexagonal lattice , the kagom lattice ( figure[crystals ] ) and the diamond lattice . before explaining difficulties for this extension and entering into details , we motivate to study these problems . there are many problems on the scaling limit of interacting particle systems , which have their origins in the statistical mechanics and the hydrodynamics . ( see @xcite , @xcite and references therein . ) the hydrodynamic limit for the exclusion process is one of the most studied models in this context . here we give only one example for exclusion processes in the integer lattice , which is a prototype of our results , due to kipnis , olla and varadhan ( @xcite ) . from the view point of physics and mathematics , it is natural to ask for the scaling limit of interacting particle systems evolving in more general spaces and to discuss the relationship between macroscopic behaviors of particles and geometric structures of the underlying spaces . in this paper , we deal with the crystal lattice , which is the simplest extension of the euclidean lattice @xmath0 . although the crystal lattice has periodic global structures , it has inhomogeneous local structures . on the other hand , crystal lattices have been studied in view of discrete geometric analysis by kotani and sunada ( @xcite , @xcite , @xcite , and the expository article @xcite ) . they formulate a crystal lattice as an abelian covering graph , and then they study random walks on crystal lattices and discuss the relationship between asymptotic behaviors of random walks and geometric structures of crystal lattices . in @xcite , they introduce the _ standard realization _ , which is a discrete harmonic map from a crystal lattice into a euclidean space to characterize an equilibrium configuration of crystals . in @xcite , they discuss the relationship between the _ albanese metric _ which is introduced into the euclidean space , associated with the standard realization and the central limit theorem for random walks on the crystal lattice . considering exclusion processes on the crystal lattice , one is interested to ask what geometric structures appear in the case where the interactions depend on the local structures . given a graph , the exclusion process on it describes the following dynamics : particles attempt to jump to nearest neighbor sites , however , they are forbidden to jump to sites which other particles have already occupied . so , particles are able to jump to nearest neighbor vacant sites . then , the problem of the hydrodynamic limit is to capture the collective behavior of particles via the scaling limit . if we take a suitable scaling limit of space and time , then we observe that the density of particles is governed by a partial differential equation as a macroscopic model . here it is necessary to construct a suitable scaling limit for a graph and to know some analytic properties of the limit space . a crystal lattice is defined as an infinite graph @xmath1 which admits a free action of a free abelian group @xmath2 with a finite quotient graph @xmath3 . we construct a scaling limit of a crystal lattice as follows : let @xmath4 be a positive integer . take a finite index subgroup @xmath5 of @xmath2 , which is isomorphic to @xmath6 when @xmath2 is isomorphic to @xmath0 . then we take the quotient of @xmath1 by @xmath5-action : @xmath7 . we call this finite quotient graph @xmath7 the _ @xmath4-scaling finite graph_. the quotient group @xmath8 acts freely on @xmath7 . here we consider exclusion processes on @xmath7 . to observe these processes in the continuous space , we embed @xmath7 into a torus . we construct an embedding map @xmath9 from @xmath7 into a torus by using a harmonic map @xmath10 in the discrete sense in order that the image @xmath11 converges to a torus as @xmath4 goes to the infinity . ( here the convergence of metric spaces is verified by using the gromov - hausdorff topology , however , we do not need this notion in this paper . ) then we obtain exclusion processes embedded by @xmath9 into the torus . in this paper , we deal with the simplest case among exclusion processes : _ the symmetric simple exclusion process _ and its perturbation : _ the weakly asymmetric simple exclusion process_. in the latter case , we obtain a heat equation with nonlinear drift terms on torus as the limit of process of empirical density ( theorem[main ] and examples below ) . we observe that the diffusion coefficient matrices and nonlinear drift terms can be computed by data of a finite quotient graph @xmath3 and a harmonic map @xmath10 . ( see also examples in section [ harmonic ] . ) the hydrodynamic limit for these processes on the crystal lattice is obtained as an extension of the one on @xmath0 . so , first , we review the outline of the proof for @xmath0 , following the method by guo , papanicolaou and varadhan in @xcite . since the lattice @xmath0 is naturally embedded into @xmath12 , the combinatorial laplacian on the scaled discrete torus converges to the laplacian on the torus according to this natural embedding . the local ergodic theorem is the key step of the proof since it enables us to replace local averages by global averages and to verify the derivation of the limit partial differential equation . it is formulated by using local functions on the configuration space and the shift action on the discrete torus . the proof of the local ergodic theorem is based on the one - block estimate and the two - blocks estimate . roughly speaking , the one - block estimate is interpreted as the local law of large numbers and the two - blocks estimate is interpreted as the asymptotic independence of two different local laws of large numbers . second , we look over the outline of the proof for the crystal lattice . there are two main points with regard to the difference between @xmath0 and the crystal lattice , that are the convergence of the laplacian and the local ergodic theorem . although the crystal lattice @xmath1 is embedded into an euclidean space by a harmonic map @xmath10 , the combinatorial laplacian on the image of the @xmath4-scaling finite graph @xmath11 does not converge to the laplacian on the torus straightforwardly . it is proved by averaging each fundamental domain by @xmath2-action because of the local inhomogeneity of the crystal lattice . thus , it is necessary to obtain the local ergodic theorem compatible with the convergence of the laplacian . furthermore , it is also necessary to obtain the local ergodic theorem compatible with the local inhomogeneity of the crystal lattice . for these reasons , we have to modify the local ergodic theorem in the case of crystal lattices . to formulate the local ergodic theorem in the crystal lattice , we introduce the notion of _ @xmath2-periodic local function bundles_. a @xmath2-periodic local function bundle is a family of local functions on the configuration space which is parametrized by vertices periodically . moreover , we introduce two different ways to take local averages of a @xmath2-periodic local function bundle . the first one is to take averages per each fundamental domain as a unit . the second one is to take averages on each @xmath2-orbit . the local ergodic theorem in the crystal lattice is formulated by using @xmath2-periodic local function bundles , two types of local averages and the @xmath13-action on the @xmath4-scaling finite graph @xmath7 . in fact , we use only special @xmath2-periodic local function bundles to handle the weakly asymmetric simple exclusion process . the proof of this local ergodic theorem is also based on the one - block estimate and the two - blocks estimate . proofs of these two estimates are analogous to the case of the discrete torus since we use the fact that the whole crystal lattice is covered by the @xmath2-action of a fundamental domain in the first type of the local average and we restrict to a @xmath2-orbit in the second type of the local average . in this paper , we call the local ergodic theorem the _ replacement theorem _ and prove it in the form of the super exponential estimate . the derivation of the hydrodynamic equation is the same manner as the case of the discrete torus . let us mention related works . interacting particle systems are categorized into the gradient system and the non - gradient system , according to types of interactions . we call the system the gradient system when the interaction term is represented by the difference of local functions . otherwise , we call the system the non - gradient system . we mention a recent work on the non - gradient system by sasada @xcite . the symmetric simple exclusion process is a model of the gradient system . our problems essentially correspond to problems for the gradient system since the hydrodynamic limit for the weakly asymmetric simple exclusion process is reduced to the one for the symmetric simple exclusion process , following @xcite . as for the hydrodynamic limit on spaces other than the euclidean lattice , jara investigates the hydrodynamic limit for zero - range processes in the sierpinski gasket ( @xcite ) . as for the crystal lattice , there is another type of the scaling limit . in @xcite , shubin and sunada study lattice vibrations of crystal lattices and calculate one of the thermodynamic quantities : the specific heat . they derive the equation of motion by taking the continuum limit of the crystal lattice . as a further problem , we mention the following problem : recently , attentions have been payed for interacting particle systems evolving in random environments ( e.g. , @xcite , @xcite and @xcite ) . for example , the quenched invariance principle for the random walk on the infinite cluster of supercritical percolation of @xmath0 with @xmath14 is proved by berger and biskup ( @xcite ) . their argument is based on a harmonic embedding of percolation cluster into @xmath12 . our use of the harmonic map @xmath10 and local function bundles will play a role in the systematic treatment of particle systems in more general random graphs . furthermore , the hydrodynamic limit on the inhomogeneous crystal lattice is considered as the case where the crystal lattice has topological defects . this problem would be interesting in connection with material sciences . this paper is organized as follows : in section [ crystal lattices ] , we introduce the crystal lattice and construct the scaling limit by using discrete harmonic maps . in section [ hydrodynamic limit ] , we formulate the weakly asymmetric simple exclusion process on the crystal lattice and state the main theorem ( theorem [ main ] ) . in section [ replacement theorem ] , we introduce @xmath2-periodic local function bundles and show the replacement theorem ( theorem [ super exponential estimate ] ) . we prove the one - block estimate and the two - blocks estimate . in section [ the proof of the main theorem ] , we derive the quasi - linear parabolic equation , applying the replacement theorem and complete the proof of theorem [ main ] . section [ appendixa ] is appendix;a . we prove some lemmas related to approximation by combinatorial metrics to complete the scaling limit argument . section [ appendixb ] is appendix;b . we refer an energy estimate of a weak solution and a uniqueness result for the partial differential equation to this appendix . _ landau asymptotic notation . _ throughout the paper , we use the notation @xmath15 to mean that @xmath16 as @xmath17 . we also use the notation @xmath18 to mean that @xmath16 as @xmath19 . in this section , we introduce the crystal lattice as an infinite graph and its realization into the euclidean space . let @xmath20 be a locally finite connected graph , where @xmath21 is a set of vertices and @xmath22 a set of all oriented edges . the graph @xmath1 may have loops and multiple edges . for an oriented edge @xmath23 , we denote by @xmath24 the origin of @xmath25 , by @xmath26 the terminus and by @xmath27 the inverse edge of @xmath25 . here we regard @xmath1 as a weighted graph , whose weight functions on @xmath21 and @xmath22 are all equal to one . we call a locally finite connected graph @xmath20 a _ @xmath2-crystal lattice _ if a free abelian group @xmath2 acts freely on @xmath1 and the quotient graph @xmath28 is a finite graph @xmath29 . more precisely , each @xmath30 defines a graph isomorphism @xmath31 and the graph isomorphism is fixed point - free except for @xmath32 . in other words , a @xmath2-crystal lattice @xmath1 is an abelian covering graph of a finite graph @xmath3 whose covering transformation group is @xmath2 . let us construct an embedding of a @xmath2-crystal lattice @xmath1 into the euclidean space @xmath12 of dimension @xmath33 . given an injective homomorphism @xmath34 such that there exits a basis @xmath35 , @xmath36 then we define a harmonic map associated with @xmath37 . fix an injective homomorphism @xmath37 as above . we call an embedding @xmath38 , a _ @xmath37-periodic harmonic map _ if @xmath10 satisfies the followings : _ @xmath10 is @xmath2-periodic _ , i.e. , for any @xmath39 and any @xmath30 , @xmath40 and _ @xmath10 is harmonic _ , i.e. , for any @xmath39 , @xmath41=0 $ ] , where @xmath42 . we note that a @xmath37-periodic harmonic map @xmath10 depends on @xmath37 and call @xmath10 a _ periodic harmonic map _ in short when we fix some @xmath37 . for @xmath43 , we take a lift @xmath44 of @xmath25 , and define @xmath45 . by the @xmath2-periodicity , @xmath46 does not depend on the choices of lifts . for @xmath47 , let us define a @xmath48-matrix by @xmath49 here the matrix is symmetric and positive definite . we call the matrix @xmath50 the _ diffusion coefficient matrix_. _ examples _ 1 . the one dimensional standard lattice . the one dimensional standard lattice @xmath1 which we identify the set of vertices @xmath21 with @xmath51 and the set of ( unoriented ) edges with the set of pairs of vertices @xmath52 . now @xmath51 acts freely on @xmath1 by the additive operation in @xmath51 and the quotient finite graph consists of one vertex and one loop as un oriented graph . when we regard @xmath1 as an oriented graph , we add both oriented edges to @xmath1 and the quotient graph consists of one vertex and two oriented loops ( figure [ one - dim ] ) . + let us choose a canonical injective homomorphism @xmath53 . in our formulation , choose a basis @xmath54 in @xmath55 and define @xmath53 by setting @xmath56 for @xmath57 so that @xmath58 . by identifying the set of vertices of @xmath1 with @xmath51 , we define an embedding map @xmath59 , @xmath60 . this embedding map @xmath10 is a @xmath51-periodic harmonic map . in this case , + 3 . let us give another example of periodic harmonic map for the one dimensional standard lattice @xmath1 . now we define a @xmath51-action on @xmath1 in the following way : for @xmath62 , @xmath39 , define @xmath63 . then this induces a free @xmath51-action on @xmath1 and the quotient graph consists of two vertices and two edges between them as an unoriented graph . let @xmath37 be the injective homomorphism as the same as in example 0a . we define an embedding map @xmath64 by setting @xmath65 , @xmath66 . then @xmath10 is a periodic harmonic map . the image of @xmath10 is not isomorphic to the previous one ( figure [ one - dim2 ] ) . in this case , + in example 0b.,width=487 ] 4 . the square lattice . the square lattice has the standard embedding in @xmath68 and this embedding is shown to be periodic and harmonic in our sense in the following . we identify the set of vertices of the square lattice @xmath1 with @xmath69 and the set of edges with the set of pairs of vertices @xmath70 . now @xmath69 acts freely on @xmath1 by the additive operation in @xmath69 and the quotient graph is the bouquet graph with one vertex and two unoriented loops . when we regard @xmath1 as an oriented graph , we add both oriented edges to @xmath1 and the quotient finite graph is the bouquet graph with one vertex and four oriented loops . + let us choose a canonical injective homomorphism @xmath71 . that is , choose a basis @xmath72 in @xmath68 and define @xmath71 by setting @xmath73 for @xmath74 so that @xmath75 . by identifying the set of vertices of @xmath1 with @xmath69 , we define an embedding map @xmath59 , @xmath76 . this embedding map @xmath10 is a @xmath69-periodic harmonic map . in this case , @xmath77 6 . let us give another example of periodic harmonic map for the square lattice @xmath1 . choose a basis @xmath78 in @xmath68 and define @xmath71 by setting @xmath79 for @xmath74 so that @xmath80 . in the same way as above example 1a , we define an embedding map @xmath59 , @xmath76 . ( figure [ square ] . ) this embedding map @xmath10 is a @xmath69-periodic harmonic map . in this case , @xmath81 + 7 . let us give an example an embedding map @xmath10 which is periodic but not harmonic . we choose an action of @xmath69 on the square lattice @xmath1 in the following way : again , we identify the set of vertices @xmath21 of @xmath1 with @xmath69 . for @xmath82 , @xmath83 , define @xmath84 . then this induces a free @xmath51-action and the quotient graph consists of two vertices , two edges between them and one loop on each vertex ( two loops ) as an unoriented graph . let @xmath37 be the same as in example 1a . we define an embedding map @xmath85 by setting @xmath86 , @xmath87 for @xmath88 . then @xmath89 is periodic but not harmonic since for @xmath90 , @xmath91 = ( 1 , 0)+(-1 , 0 ) + ( 1 , 1/2 ) + ( 1 , -1/2)=(2 , 0 ) \neq ( 0,0)$ ] . the hexagonal lattice . + the hexagonal lattice admits a free @xmath69-action with the quotient graph consisting of two vertices and three edges as an unoriented graph . we define a fundamental subgraph @xmath92 by setting the set of vertices @xmath93 and the set of ( unoriented ) edges @xmath94 . then the hexagonal lattice has a subgraph isomorphic to @xmath92 and is covered by copies of the subgraph translated by the @xmath69-action . choose a basis @xmath95 in @xmath68 and define @xmath71 by setting @xmath79 for @xmath74 so that @xmath80 . we define an embedding map @xmath10 by setting @xmath96 , @xmath97 , @xmath98 and @xmath99 for @xmath88 . ( figure [ hexagonal ] . ) then @xmath10 is a periodic harmonic map . in this case , @xmath100 + 9 . the kagom lattice . + the kagom lattice admits a free @xmath69-action with the quotient graph consisting of three vertices and six edges ( two edges between each pair of vertices ) as an unoriented graph . we define a fundamental subgraph @xmath92 by setting the set of vertices @xmath101 and the set of ( unoriented ) edges @xmath102 . then the kagom lattice has a subgraph isomorphic to @xmath92 and is covered by copies of the subgraph translated by the @xmath69-action . choose a basis @xmath95 in @xmath68 and define @xmath71 as the same as in example 2 . we define an embedding map @xmath10 by setting @xmath103 , @xmath104 , @xmath105 , @xmath106 , @xmath107 for @xmath88 . ( figure [ kagome ] . ) then @xmath10 is a periodic harmonic map . in this case , @xmath100 + _ remark_. the notion of periodic harmonic map on @xmath2-crystal lattice is studied by kotani and sunada and including the standard realization which they introduced in @xcite as a special case . they use harmonic maps to characterize _ equilibrium configurations _ of crystals . in fact , a periodic harmonic map is characterized by a critical map for some discrete analogue of energy functional . the standard realization is not only a critical map but also the map whose energy itself is minimized by changing flat metrics on torus with fixed volume . ( more precisely , see@xcite ) . the existence of periodic harmonic map for every injective homomorphism producing lattices in @xmath12 and the uniqueness up to translation is proved in theorem 2.3 and theorem 2.4 in @xcite . let us construct the scaling limit of the crystal lattice . suppose that @xmath2 is isomorphic to @xmath0 . let @xmath108 be an arbitrary positive integer and @xmath5 the subgroup isomorphic to @xmath6 . the subgroup @xmath5 acts also freely on @xmath1 and its quotient graph @xmath109 is also a finite graph @xmath110 . then @xmath111 acts freely on @xmath7 . we call @xmath7 the _ @xmath4-scaling finite graph_. the map @xmath112 satisfies that @xmath113 for all @xmath39 and all @xmath30 since @xmath10 is @xmath2-equivariant . we have the torus @xmath114 , equipped with the flat metric induced from the euclidean metric . the torus depends on @xmath115 , however , we do not specify it in the following . then the map @xmath116 induces the map @xmath117 we call @xmath9 the _ @xmath4-scaling map_. ( figure[scaling ] . ) @xmath118 -scaling finite graph by a harmonic map in the covering space , width=487 ] next , we observe convergence of the combinatorial laplacian on @xmath7 . since the degrees of @xmath119 might be different , depending on each @xmath120 , we consider average " of the combinatorial laplacian on a fundamental domain . let @xmath121 be the degree of a vertex @xmath119 , i.e. , the cardinality of the set @xmath122 . define the combinatorial laplacian @xmath123 associated with @xmath124 acting on the space of continuous functions @xmath125 by @xmath126,\ ] ] for @xmath127 and @xmath119 . we show that the combinatorial laplacian converges to the laplacian on @xmath128 in the following sense : for every twice continuous derivative functions @xmath129 , for every @xmath130 , for each @xmath131 , take an arbitrary sequence of vertices @xmath132 such that @xmath133 is a lift of @xmath120 and @xmath134 as @xmath17 , then by the taylor formula , @xmath135 since @xmath10 is harmonic , @xmath136 since @xmath137 , the last term is equal to @xmath138 , where @xmath50 is a diffusion coefficient matrix and @xmath139 and @xmath140 . we formulate the symmetric simple exclusion process and the weakly asymmetric simple exclusion process in crystal lattices . as we see below , the former is a particular case of the latter . let @xmath110 be the @xmath4-scaling finite graph of @xmath1 . we denote the configuration space by @xmath141 . we denote the configuration space for the whole crystal lattice @xmath142 by @xmath143 . each configuration is defined by @xmath144 with @xmath145 or @xmath146 and by @xmath147 in the same way . we consider the bernoulli measure @xmath148 and @xmath149 on @xmath150 , @xmath151 , respectively , for @xmath152 . they are defined as the product measures of the bernoulli measure @xmath153 on @xmath154 , where @xmath155 . let @xmath156 be the @xmath157-space of @xmath55-valued functions on @xmath150 . the action of @xmath13 on @xmath7 lifts on @xmath150 by setting @xmath158 for @xmath159 and @xmath119 . the group @xmath13 also acts on @xmath156 by @xmath160 for @xmath161 . for @xmath162 and @xmath163 , we denote by @xmath164 the configuration defined by exchanging the values of @xmath165 and @xmath166 , i.e. , @xmath167 for each @xmath162 , we define the operator @xmath168 by setting @xmath169 . we see that @xmath170 and @xmath171 for @xmath172 . the above notations also indicate corresponding ones for @xmath173 the configuration space on the whole crystal lattice . the symmetric simple exclusion process is defined by the generator @xmath174 acting on @xmath175 as @xmath176 the weakly asymmetric simple exclusion process is defined as a perturbation of the symmetric simple exclusion process . we denote by @xmath177\times \mathbb{t}^{d})$ ] the space of continuous functions with continuous derivatives in @xmath178 $ ] and the twice continuous derivatives in @xmath128 . for each function @xmath179\times \mathbb{t}^{d})$ ] , the weakly asymmetric simple exclusion process on @xmath7 is defined by the generator @xmath180 acting on @xmath175 as @xmath181 where @xmath182.\ ] ] here @xmath183 is the @xmath4-scaling map . the meaning of the perturbation is as follows : we introduce a `` small '' drift depending on space and time in particles . in the original process , a particle jumps with rate @xmath184 from @xmath24 to @xmath26 ( @xmath25 is an edge ) at time @xmath185 , while in the perturbed process , a particle jumps approximately with rate @xmath186 therefore , the external field which is now @xmath187 gives a small asymmetry of the order @xmath188 in the jump rate . notice that we obtain the symmetric simple exclusion process when @xmath189 is constant . _ remark _ the weakly asymmetric simple exclusion process which we introduced here does not include the well - studied case where for one dimensional lattice , the external field is @xmath190 for some constant @xmath191 and its limit equation produces the viscous burgers equation ( e.g. , @xcite ) . this process corresponds to the case with @xmath192 which we do not treat here . let @xmath193 , z_{n})$ ] be the space of paths which are right continuous and have left limits for some arbitrary fixed time @xmath194 . for a probability measure @xmath195 on @xmath150 , we denote by @xmath196 the distribution on @xmath193 , z_{n})$ ] of the continuous time markov chain @xmath197 generated by @xmath198 with the initial measure @xmath195 . the main theorem is stated as follows : [ main ] let @xmath199 $ ] be a measurable function . if a sequence of probability measures @xmath195 on @xmath150 satisfies that @xmath200=0,\ ] ] for every @xmath201 and for every continuous functions @xmath202 , then for every @xmath203 , @xmath204=0,\ ] ] for every @xmath201 and for every continuous functions @xmath202 , where @xmath205 is the unique weak solution of the following quasi - linear parabolic equation : @xmath206 here we define @xmath207 for @xmath43 . we give examples corresponding to ones in section [ harmonic ] . _ examples _ 1 . the one dimensional standard lattice . + for the embedding in example 0a , we recover the equation in theorem 3.1 in @xcite : @xmath208 + for the embedding in example 0b . , we have the following equation : @xmath209 2 . the square lattice . + for the square lattice and its embedding in example 1a . , we have the following equation : @xmath210 + for the square lattice and its embedding in example 1b . , we have the following equation : + @xmath211 3 . the hexagonal lattice , the kagom lattice . + for the hexagonal lattice , the kagom lattice and their embeddings in example 2 . and 3 , we have the following same equation : @xmath212 in this section , we formulate the replacement theorem and give its proof . the replacement theorem is given by the form of super exponential estimate and follows from the one - block estimate and the two blocks estimate . for our purpose , we introduce local function bundles which describe the local interactions of particles and the two types of local averages for local function bundles . a local function bundle @xmath213 on @xmath214 is a function @xmath215 , which satisfies that for each @xmath216 there exists @xmath217 such that @xmath218 depends only on @xmath219 . here @xmath220 is the graph distance in @xmath1 . we say that a local function bundle @xmath221 is @xmath2-periodic if it holds that @xmath222 for any @xmath30 , @xmath39 and @xmath147 . here we give examples of @xmath2-periodic local function bundles on @xmath223 . we use the first one and the third one later . _ examples _ * if we define @xmath221 by for @xmath39 and @xmath147 @xmath224 then @xmath213 is a @xmath2-periodic local function bundle on @xmath223 . * if we define @xmath221 by for @xmath39 and @xmath147 @xmath225 then @xmath213 is a @xmath2-periodic local function bundle on @xmath223 . * fix @xmath226 . if we define @xmath227 by @xmath228 then @xmath213 is a @xmath2-periodic local function bundle on @xmath223 . note that a @xmath2-periodic local function bundle @xmath221 induces a map @xmath229 for large enough @xmath4 in the natural way . to abuse the notation , we indicate the induced map by the same character @xmath213 . first , for @xmath230 , we define the @xmath231-ball by @xmath232 here @xmath233 is the word metric appearing in section [ approximation ] . we regard that @xmath234 via the covering map when @xmath4 is large enough for @xmath231 . for a local function bundle @xmath215 , we define the local average of @xmath213 on blocks @xmath235 by for @xmath39 , @xmath236b(d_{x_{0}},r)\right|}\sum_{z \in [ x]b(d_{x_{0}},r ) } f_{z } : z \to \mathbb{r},\ ] ] where @xmath237 $ ] is a unique element @xmath30 such that @xmath238 and @xmath239b(d_{x_{0}},r)\right|$ ] denotes the cardinality of the set , which is equal to @xmath240 . note that @xmath241 for every @xmath242 . as a special case , we define for @xmath147 and @xmath39 , @xmath243b(d_{x_{0}},r)\right|}\sum_{z \in [ x]b(d_{x_{0}},r ) } \eta_{z } : z \to \mathbb{r}.\ ] ] second , we define the local average of @xmath213 on each @xmath2-orbit , @xmath244 by for @xmath39 , @xmath245 note that @xmath246 and @xmath247 are @xmath2-periodic when @xmath213 is @xmath2-periodic . if a local function bundle @xmath213 is @xmath2-periodic and @xmath4 is large enough , then @xmath248 induce the functions on @xmath150 in the natural way . to abuse the notation , we indicate the induced maps by the same characters @xmath248 . for a local function bundle @xmath215 , for @xmath39 , let us define @xmath249 $ ] , the expectation with respect to the bernoulli measure @xmath149 . the following estimate allows us to replace the local averages of the local function bundle by the global averages of the empirical density . we call the following theorem the replacement theorem . we prove it in the form of the super exponential estimate . [ super exponential estimate](super exponential estimate of the replacement theorem ) fix @xmath194 . for any @xmath2-periodic local function bundles @xmath215 , for every @xmath242 and for every @xmath201 , it holds that @xmath250 where @xmath251 note that for every @xmath242 , @xmath252 @xmath253 . we denote by @xmath254 the corresponding distribution on @xmath193 , z_{n})$ ] of continuous time markov chain @xmath197 generated by @xmath255 with the initial measure @xmath195 . furthermore , we denote by @xmath256 the corresponding distribution on @xmath193 , z_{n})$ ] of continuous time markov chain @xmath197 generated by @xmath255 with the initial measure @xmath257 , i.e. , an equilibrium measure . we denote by @xmath258 the expectation with respect to @xmath196 , by @xmath259 the one with respect to @xmath254 and by @xmath260 the one with respect to @xmath256 . for a probability measure @xmath261 on some probability space , we also denote by @xmath262 the expectation with respect to @xmath261 . by the following proposition , we reduce the super exponential estimate for @xmath196 to the one for @xmath254 . [ radon - nikodym ] there exists a constant @xmath263 such that @xmath264 to simplify the notation , put @xmath265 for @xmath119 . to calculate the radon - nikodym derivative : @xmath266 \\ & = \exp\bigg[\sum_{x \in v_{n}}h(t , x)\eta_{x}(t)-\sum_{x \in v_{n}}h(0 , x)\eta_{x}(0 ) -\int_{0}^{t}\bigg\{\sum_{x \in v_{n}}\frac{\partial h}{\partial t}(t , x)\eta_{x}(t ) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \frac{n^{2}}{4}\sum_{e \in e_{n}}\big(\eta_{oe}(1-\eta_{te})e^{h(t , te)-h(t , oe ) } + \eta_{te}(1-\eta_{oe})e^{h(t , oe)-h(t , te ) } \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\eta_{oe}(1-\eta_{te } ) - \eta_{te}(1-\eta_{oe})\big)\bigg\}dt\bigg ] \\ & = \exp\bigg[\sum_{x \in v_{n}}h(t , x)\eta_{x}(t)-\sum_{x \in v_{n}}h(0 , x)\eta_{x}(0 ) -\int_{0}^{t}\big\{\sum_{x \in v_{n}}\frac{\partial h}{\partial t}(t , x)\eta_{x}(t ) \\ & + \frac{n^{2}}{2}\sum_{x \in v_{n}}\sum_{e \in e_{n , x}}\eta_{x}\left(e^{h(t , te)-h(t , oe)}-1\right ) - \frac{n^{2}}{4}\sum_{e \in e_{n}}\left(e^{h(t , te)-h(t , oe)}+e^{h(t , oe)-h(t , te)}-2\right)\eta_{oe}\eta_{te}\big\}dt\bigg ] . \\\end{aligned}\ ] ] by the inequality @xmath267 for @xmath268 , we have that @xmath269 \\ & -n^{2}\sum_{e \in e_{n , x}}\left(h(t , te)- h(t , oe)\right ) - \frac{1}{2}n^{2}\sum_{e \in e_{n , x}}\left(h(t , te)- h(t , oe)\right)^{2}\big| \\ & \le \frac{1}{6}n^{2}\sum_{e \in e_{n , x}}\left|h(t , te)- h(t , oe)\right|^{3}e^{\left|h(t , te)- h(t , oe)\right|},\end{aligned}\ ] ] and thus , since @xmath10 is harmonic , @xmath270 \\ & = \frac{1}{2}\sum_{e \in e_{n , x}}\sum_{i , j=1}^{d}\frac{\partial^{2}h}{\partial x_{i } \partial x_{j}}(t , x)v_{i}(e)v_{j}(e ) + \frac{1}{2}\sum_{e \in e_{n , x}}\left(\nabla_{{\bf v}(e)}h(t , x)\right)^{2 } + o_{n}.\end{aligned}\ ] ] furthermore , @xmath271 hence , there exists a constant @xmath263 depending only on @xmath189 and @xmath272 such that @xmath273 is bounded from above by @xmath274 . it completes the proof . the super exponential estimate for @xmath254 induces the one for @xmath196 since it holds that for any borel sets @xmath275 , z_{n})$ ] , @xmath276 . furthermore , it is enough to prove the super exponential estimate for @xmath256 since for any borel sets @xmath275 , z_{n})$ ] , @xmath277 . let @xmath278 be the spaces of probability measures on @xmath279 , respectively . define @xmath280 , @xmath281 the @xmath282-bernoulli measure on @xmath150 , @xmath151 , respectively . here we introduce a functional on @xmath283 , which is the dirichlet form for the density function . for @xmath284 , put the density @xmath285 . the dirichlet form of @xmath286 is defined by @xmath287 note that @xmath288 . _ the functional @xmath289 is also called the @xmath290-functional in the different literatures . let us define the subset of the space of probability measures on @xmath150 by for @xmath291 , @xmath292 the proof of the super exponential estimate for @xmath256 is reduced to the following : [ local ergodic theorem ] for every @xmath291 and every @xmath242 , @xmath293 first , we prove theorem[super exponential estimate ] by using therem[local ergodic theorem ] . fix any @xmath294 . by the above argument , it is enough to show that for any @xmath2-periodic local function bundles @xmath215 , for every @xmath242 and for every @xmath201 , @xmath295 where @xmath251 by the chebychev inequality , for every @xmath296 and every @xmath201 , @xmath297 now , the operator @xmath298 acting on @xmath299 is self - adjoint for all @xmath296 and all @xmath242 . let @xmath300 be the largest eigenvalue of @xmath298 . by using the feynman - kac formula , @xmath301 therefore , it is suffice to show that for every @xmath296 , @xmath302 since ( [ eigenvalue ] ) implies that @xmath303 and we obtain the theorem by taking @xmath304 to the infinity . by the variational principle , the largest eigenvalue @xmath300 is represented by the following : @xmath305 see @xcite appendix 3 for more details . denote the average of @xmath261 by @xmath13-action by @xmath306 then @xmath307 is @xmath13-invariant so that @xmath308=\mathbb{e}_{\widetilde \mu}\left|\widetilde f_{x , k } - \langle f_{x}\rangle(\overline \eta_{x_{0 } , \epsilon n})\right|.\ ] ] the functional @xmath309 is also @xmath13-invariant , i.e. , @xmath310 for @xmath284 , @xmath311 . thus , it is suffice to consider @xmath13-invariant measures @xmath261 to estimate the largest eigenvalue @xmath300 . furthermore , it is suffice to consider the case where @xmath261 satisfies @xmath312 there exists a constant @xmath313 depending on @xmath213 such that @xmath314 , thus we reduce @xmath284 to every @xmath13-invariant measure satisfying that for every @xmath315 , @xmath316 this shows that we can reduce to @xmath317 for every @xmath318 , and thus it is enough to show for every @xmath315 and every @xmath242 , @xmath319 to obtain ( [ eigenvalue ] ) . this follows from theorem[local ergodic theorem ] . it completes the proof . in this section , we prove the one - block estimate . we regard a probability measure @xmath261 on @xmath150 as one on @xmath151 by periodic extension . let @xmath320 be the covering map by @xmath5-action , and define the periodic inclusion @xmath321 by @xmath322 . we identify @xmath261 on @xmath150 with its push forward by @xmath323 . on the other hand , we identify an @xmath5-invariant probability measure on @xmath151 with a probabiltiy measure on @xmath150 . first , for a finite subgraph @xmath324 of x , we define the restricted state space @xmath325 and the @xmath282-bernoulli measure by @xmath326 on @xmath327 . let us define the operator acting on @xmath328 by @xmath329 . for @xmath330 , @xmath331 stands for the restriction of @xmath261 on @xmath327 and @xmath332 its density . we also define the corresponding dirichlet form of @xmath333 by @xmath334 for large enough @xmath4 , we regard @xmath335 as a subgraph of @xmath7 by taking a suitable fundamental domain in @xmath21 for @xmath5-action . by the convexity of the dirichlet form , @xmath336 by putting @xmath337 . the one - block estimate is stated as follows : [ the one - block estimate](the one - block estimate . ) for every @xmath2-periodic local function bundles , @xmath215 , every @xmath242 and every @xmath315 , @xmath338 for any probability measure @xmath339 , we apply the above argument by setting @xmath335 as @xmath340 and @xmath341 . since @xmath342 and @xmath343 are @xmath13-invariant , @xmath344 since @xmath339 and @xmath345 , it holds that @xmath346 we note that @xmath347 is compact with respect to the weak topology , and thus @xmath348 has a subsequence which convergences to some @xmath261 in @xmath347 . let @xmath349 be the set of all limit points of @xmath350 in @xmath347 . by the above argument , @xmath351 for all @xmath352 . therefore , since @xmath353 for every @xmath352 by the definition of @xmath354 , we obtain that @xmath355 for every @xmath356 and every @xmath357 . this shows that random variables @xmath358 are exchangeable under @xmath261 . by the de finetti theorem , there exists a probability measure @xmath359 on @xmath360 $ ] such that @xmath361 . since @xmath362 it is enough to show that @xmath363}\mathbb{e}_{\nu_{\rho}}\left| \widetilde f_{x , k } - \langle f_{x } \rangle \left(\overline \eta_{x_{0 } , k}\right ) \right|=0,\ ] ] for every @xmath2-invariant local function bundles @xmath213 . by the definition of the @xmath2-periodic local function bundle , there exists a constant @xmath364 such that for every @xmath39 , @xmath365 depends on at most @xmath366b(d_{x_{0}},l)\}$ ] . therefore we obtain that there exists a constant @xmath313 depending only on @xmath213 such that @xmath367\right]^{2}\le c(f)\cdot \frac{l^{d}}{k^{d } } \to 0 \ \ \ \ \ \text{as $ k \to \infty$}.\ ] ] note that @xmath368 $ ] for every @xmath242 since the bernoulli measure @xmath149 is @xmath13-invariant . in addition , we also obtain that there exists a constant @xmath369 not depending on @xmath370 , @xmath371 finally , since @xmath372 is a polynomial with respect to @xmath370 , in particular , uniformly continuous on @xmath360 $ ] , we obtain that @xmath373 } \mathbb{e}_{\nu_{\rho}}\left| \widetilde f_{x , k } - \langle f_{x } \rangle(\overline \eta_{x_{0 } , k } ) \right|=0 \ \ \ \ \ \text{for every $ x \in d_{x_{0}}$}.\ ] ] this concludes the theorem @xmath374 in this section , we prove the two - blocks estimate . we identify a probability measure on @xmath150 with its periodic extension on @xmath151 in the same manner as section[section of the one - block estimate ] . ( the two - blocks estimate)[the two - blocks estimate ] for every @xmath315 , @xmath375 let us denote by @xmath376 the space of probability measures on @xmath377 . we define the map for @xmath30 , @xmath378 by @xmath379 , @xmath147 . for @xmath330 , we define the push forward of @xmath261 by @xmath380 via @xmath381 . let us denote by @xmath382 the set of all limit points of @xmath383 in @xmath384 as @xmath17 and by @xmath385 the set of all limit points of @xmath386 as @xmath387 . we put @xmath388 , where @xmath389 stands for a copy of @xmath390 . then , it holds that @xmath391 note that @xmath392 we introduce two types of generators acting on @xmath393 and the corresponding dirichlet forms . the first one is used for treating two different states at the same time independently . the second one is used for treating exchanges of particles between two different states . as in section[section of the one - block estimate ] , we define a subgraph @xmath394 of @xmath1 by setting @xmath395 and the operator acting on @xmath328 by @xmath396 for @xmath397 , we denote by @xmath398 the restriction of @xmath261 on @xmath399 . define the dirichlet form of @xmath400 by @xmath401 let us introduce the notation which describes exchanges of states for @xmath402 . for @xmath403 , @xmath404 is the configuration obtained by exchanging values @xmath405 and @xmath406 , i.e. , @xmath407 is defined by setting @xmath408 @xmath409 moreover , for @xmath410 , we define @xmath411 . we define the operator acting on @xmath412 by @xmath413 the corresponding dirichlet form of @xmath400 is defined by @xmath414 we prove two lemmas needed later . the first one is easy to show , so we omit the proof . [ quasi - iso ] there exist constants @xmath415 such that for all @xmath30 , @xmath416 where @xmath220 is the graph distance of @xmath1 . for @xmath417 and for @xmath163 , @xmath418 is the configuration obtained by exchanging two values @xmath405 and @xmath419 , i.e. , @xmath420 and moreover for @xmath417 , we define the operator @xmath421 for @xmath422 . these notations also indicates ones for @xmath151 . the second one is the following : [ path ] for every @xmath13-periodic functions @xmath422 and every @xmath423 , @xmath424 for @xmath425 , there exists a path @xmath426 such that @xmath427 and @xmath428 . define a sequence of edges @xmath429 by setting @xmath430 . for @xmath163 , let us define @xmath431 , inductively . we note that @xmath432 . then we have that @xmath433 if @xmath434 is @xmath13-periodic , then for each @xmath435 , @xmath436 therefore @xmath437 it completes the proof . let us define the subset of @xmath384 by for a constant @xmath438 , @xmath439 then we have the following lemma . [ inclusion ] there exists a constant @xmath438 such that @xmath440 as in section[section of the one - block estimate ] , we define a subgraph @xmath441 of @xmath1 by setting @xmath442 for @xmath443 . we take large enough @xmath444 for the diameter of @xmath335 , @xmath443 , so that @xmath445 for @xmath446 and @xmath447 for @xmath448 by taking a suitable fundamental domain in @xmath21 by @xmath5-action . take @xmath449 . for @xmath450 , if we define @xmath451 by @xmath452 and @xmath453 then @xmath454 . let us consider the operator acting on @xmath328 by @xmath455 for @xmath456 , we denote the density of @xmath457 by @xmath458 . then @xmath459 for any @xmath456 , we put @xmath337 . by the convexity of the dirichlet form and the @xmath13-invariance of @xmath37 , @xmath460 since @xmath461 , @xmath462 therefore @xmath463 for any @xmath464 . furthermore @xmath465 for any @xmath466 by the continuity of the functional @xmath467 . for @xmath456 , by the convexity of the dirichlet form , @xmath468 by the @xmath13-invariance of @xmath456 and by lemma[path ] , @xmath469 by lemma[quasi - iso ] and the definition of @xmath456 , for all @xmath30 such that @xmath448 , @xmath470 by setting @xmath471 and the continuity of the @xmath467 , for every @xmath466 , we have that @xmath472 . it concludes that @xmath473 . let us prove theorem[the two - blocks estimate ] . denote by @xmath474 the set of all limit points of @xmath475 as @xmath19 . for every @xmath476 , it holds that @xmath477 and @xmath478 by the continuity of the functionals @xmath467 and @xmath479 . these show that for any @xmath480 , @xmath481 and thus for any @xmath482 and for any @xmath483 , @xmath484 , @xmath485 and @xmath486 , i.e. , @xmath487 is exchangeable on @xmath377 . by the de finetti theorem there exists a probability measure @xmath359 on @xmath488 $ ] such that @xmath489}\nu_{\rho}\otimes \nu_{\rho}\lambda(d\rho)$ ] . as in the proof of theorem[the one - block estimate ] , @xmath490}\mathbb{e}_{\nu_{\rho}}\left|\overline \eta_{x_{0 } , k } - \rho \right|^{2}=0 $ ] , therefore , by the triangular inequality , @xmath491}\mathbb{e}_{\nu_{\rho}\otimes \nu_{\rho}}\left|\overline \eta_{x_{0 } , k}-\overline \eta_{x'_{0 } , k}\right| \\ & \le 2 \sup_{\rho \in [ 0 , 1]}\mathbb{e}_{\nu_{\rho}}\left|\overline \eta_{x_{0 } , k}- \rho \right| \to 0 \ \ \ \ \ \text{as $ k \to \infty$}.\end{aligned}\ ] ] finally , by lemma[inclusion ] , @xmath473 for some @xmath438 and thus , @xmath492 this completes the theorem @xmath493 let us prove theorem[local ergodic theorem ] by using the one - block estimate theorem[the one - block estimate ] and the two - blocks estimate theorem[the two - blocks estimate ] . first , we note that there exist positive constants @xmath494 and @xmath495 such that for any @xmath147 , @xmath496 and thus there exists a constant @xmath497 depending on @xmath498 and @xmath443 such that for any @xmath147 , @xmath499 uniformly . then , since @xmath261 is @xmath2-invariant as a probability measure on @xmath151 , @xmath500 where the last inequality comes from the fact that for any @xmath501 it holds that @xmath502 . applying theorem[the two - blocks estimate ] , we have that @xmath503 for every @xmath2-periodic local function bundles @xmath213 , @xmath504 is uniformly continuous on @xmath488 $ ] . therefore , @xmath505 furthermore , applying the one - block estimate theorem[the one - block estimate ] , for every @xmath242 , @xmath506 it completes the proof of theorem[local ergodic theorem ] . in this section , we prove theorem[main ] . let @xmath507 be the @xmath4-scaling map . we define the empirical density by @xmath508 where @xmath509 is the delta measure at @xmath510 . the empirical density is the measure valued process . we denote by @xmath511 \times \mathbb{t}^{d})$ ] the space of continuous functions with continuous derivatives in @xmath178 $ ] and twice continuous derivatives in @xmath128 . for every @xmath512 \times \mathbb{t}^{d})$ ] , we define @xmath513 to abuse the notation , we denote the inner product in @xmath514 by @xmath515 let us define the process as follows : @xmath516 where @xmath517 and @xmath518 where @xmath519 here @xmath520 are martingales with respect to the filtration @xmath521 , where @xmath522 and it holds that @xmath523 then we have the following lemma by applying the doob inequality . [ doob ] @xmath524=0.\ ] ] for @xmath525 \times \mathbb{t}^{d})$ ] and for each @xmath162 , we have that @xmath526\left[j_{s}\left(\phi_{n}(te)\right)-j_{s}\left(\phi_{n}(oe)\right)\right].\end{aligned}\ ] ] by the regularity of @xmath527 and the compactness of @xmath128 , we note that there exists a constant @xmath528 depending only on @xmath527 , such that uniformly , @xmath529 thus @xmath530^{2 } \le n^{2}|e_{n}|c'\frac{c(j)^{2}}{|v_{n}|^{2}n^{2 } } = \frac{c''}{|v_{n}|},\ ] ] where @xmath531 is a constant such that @xmath532 and @xmath533 . then we obtain that @xmath534 applying the doob inequality for the right continuous martingale @xmath535 , @xmath536 \le 4 \mathbb{e}_{n}^{h}\left|m_{n}(t)\right|^{2},\ ] ] we conclude that @xmath537=0 $ ] . we denote by @xmath538 the space of nonnegative borel measures with the total measure less than or equal to one on @xmath128 , endowed with the weak topology . since @xmath128 is a compact metric space , the space of continuous functions @xmath125 with the supremum norm is separable . fix a dense countable subset @xmath539 of @xmath125 , then the weak topology of @xmath540 is given by the distance @xmath541 by @xmath542 for @xmath543 where @xmath544 . we note that @xmath540 with the weak topology is compact . define the space of paths in @xmath540 by @xmath545 , \mathcal{m}):=\left\{\xi_{\cdot } : [ 0 , t ] \to \mathcal{m } \ \text{$\xi$ is right continuous with left limits.}\right\},\ ] ] equipped with the skorohod topology . for a given process @xmath546 \to \mathcal{m}$ ] such that @xmath547 , \mathcal{m})\right)=1 $ ] , we denote by @xmath548 the distribution of @xmath549 on @xmath550 , \mathcal{m})$ ] . then we show that the sequence @xmath551 has a subsequential limit . the following proposition gives a sufficient condition for this . see @xcite section 4 , theorem 1.3 for the proof . [ relative compactness ] if for every @xmath552 , @xmath553 , and every @xmath201 , @xmath554 then there exists a subsequence @xmath555 and a probability measure @xmath556 on @xmath550 , \mathcal{m})$ ] such that @xmath557 weakly converges to @xmath556 as @xmath558 . the next proposition claims that each subsequential limit @xmath556 in proposition[relative compactness ] is absolutely continuous with respect to lebesgue measure on the torus for each time @xmath185 , and its density has the value in @xmath488 $ ] a.e . the proof is the same as in @xcite section 4 , pp.57 , so we omit the proof . [ absolute continuity ] all limit points @xmath556 of @xmath551 are concentrated on trajectories of absolutely continuous measures with respect to the lebesgue measure for each time @xmath185 , i.e. , there exists a borel set @xmath559 , \mathcal{m})$ ] such that @xmath560 and for every @xmath561 and every @xmath562 $ ] , @xmath563 is absolutely continuous with respect to the lebesgue measure @xmath564 . moreover , the density @xmath565 satisfies that @xmath566 , @xmath564-a.e . to simplify the notation , we put @xmath567 for @xmath568 , \mathbb{t}^{d})$ ] , respectively . then we have : @xmath569 \cdot ( -\eta_{oe}\eta_{te}+\eta_{oe})(j(t , te)- j(t , oe ) ) \\ & = \frac{n^{2}}{2|v_{n}|}\sum_{x \in v_{n}}\sum_{e \in e_{n , x}}\left[\left\{\exp\left(h(t , te ) - h(t , oe)\right)-1\right\}\cdot \left(j(t , te ) - j(t , oe)\right)\eta_{oe}+\left(j(t , te)- j(t , oe)\right)\eta_{oe}\right ] \\ & -\frac{n^{2}}{4|v_{n}|}\sum_{e \in e_{n}}\left\{\exp\left(h(t , te ) - h(t , oe)\right)- \exp\left(h(t , oe)-h(t , te)\right)\right\}\cdot \eta_{oe}\eta_{te}(j(t , te)- j(t , oe)).\end{aligned}\ ] ] for @xmath23 , we denote the directional derivative along @xmath46 by @xmath570\times \mathbb{t}^{d}$}.\ ] ] applying the inequality @xmath571 for @xmath268 , by the regularity of @xmath189 and by the compactness @xmath128 , there exists a constant @xmath318 not depending on each point of @xmath178\times \mathbb{t}^{d}$ ] such that for every @xmath4 and for every @xmath162 , @xmath572 @xmath573 @xmath574 by the convergence of the combinatorial laplacian in section[scaling limit ] , we have that @xmath575 hence , @xmath576 we replace @xmath196 by @xmath548 regarding the empirical density @xmath549 as the measure on @xmath550,\mathcal{m})$ ] . let us prove the following lemma : [ equi - continuity ] for every @xmath127 and for every @xmath201 , @xmath577=0.\ ] ] for every continuous functions @xmath578 , it holds that @xmath579 since by ( [ n2l ] ) there exists a constant @xmath580 such that for large enough @xmath4 , @xmath581 uniformly , we obtain that by the chebychev inequality and by the triangular inequality , for every @xmath201 , for every @xmath582 and for large enough @xmath4 , @xmath583 \le ( 1/\delta)\mathbb{e}_{n}^{h}\left[c \gamma + 2 \sup_{0 \le t \le t } \left|m_{n}(t)\right|\right].\ ] ] then by lemma[doob ] , @xmath584\le c\frac{\gamma}{\delta}.\ ] ] therefore for every @xmath201 , @xmath585=0.\ ] ] it completes the proof . we have the following estimate : there exists a constant @xmath586 depending only on @xmath189 and @xmath527 such that for every @xmath162 and for every @xmath587 , @xmath588 thus , by the regularity of @xmath189 and @xmath527 , putting @xmath589 for @xmath162 , @xmath590 here we regard @xmath591 as a local function bundle independent of states and denote by @xmath592 the local average . by the uniform continuity of the twice derivative of @xmath527 , putting @xmath593 it holds that @xmath594 here we also regard @xmath595 as a local function bundle . by the above argument , we obtain that @xmath596 here @xmath597 is the local function bundle appearing in the third example in section[local function bundles ] and @xmath598 its local average . by applying theorem[super exponential estimate ] and by the continuity of @xmath599 , @xmath600 and @xmath601 on the compact space @xmath178 \times \mathbb{t}^{d}$ ] , it holds that for every @xmath562 $ ] and for every @xmath201 , @xmath602 @xmath603 and @xmath604 here we use @xmath605 for every @xmath119 in the third estimate above . by the triangular inequality , it holds that for every @xmath562 $ ] and for every @xmath201 , @xmath606 applying the convergence of the combinatorial laplacian in section [ scaling limit ] , we have that @xmath607 recall that @xmath608 by lemma[doob ] and by the chebychev inequality , for every @xmath201 , @xmath609 furthermore , by the triangular inequality , we have that for every @xmath610 and for every @xmath562 $ ] , @xmath611 by lemma[characteristic function ] , we replace @xmath612 by @xmath613 and the summation for @xmath311 by the integral . since by lemma[equi - continuity ] and by proposition[relative compactness ] , the sequence @xmath614 is relatively compact in the weak topology , for a limit point @xmath556 there exists a subsequence @xmath615 weakly converging to @xmath556 . by proposition[absolute continuity ] , the empirical density @xmath616 concentrates on an absolutely continuous trajectory @xmath617 as @xmath558 . by the assumption of theorem[main ] , we replace @xmath618 by @xmath619 , and then we have that for every @xmath201 and for every @xmath620 $ ] , @xmath621 here we replace @xmath46 for @xmath226 by @xmath46 for @xmath43 by the @xmath2-invariance of @xmath622 . by the lebesgue dominated convergence theorem as @xmath19 and by the triangular inequality , we have that for every @xmath201 and for every @xmath620 $ ] , @xmath623 this shows @xmath624 concentrates on @xmath370 , which is a weak solution of the quasi - linear parabolic equation ( [ pde1 ] ) . furthermore , @xmath370 has finite energy by lemma[energy ] . by the uniqueness result of the weak solution lemma[uniqueness ] in section[appendixb ] , we conclude that the limit point @xmath556 of @xmath551 is unique and @xmath549 concentrates on @xmath625 as @xmath4 goes to the infinity . that is , for every @xmath201 , @xmath626 where @xmath627 is the skorohod distance in @xmath193 , \mathcal{m})$ ] . in particular , since @xmath194 is arbitrary , it follows that for every @xmath203 , for every @xmath201 and for every continuous functions @xmath127 , @xmath628=0.\ ] ] it completes the proof of theorem[main ] . take a @xmath51-basis @xmath629 of @xmath2 and identify @xmath2 with @xmath0 . define the standard generator system of @xmath2 by @xmath630 . we introduce the length function associated with @xmath631 , @xmath632 by @xmath633 for @xmath30 . then the map @xmath634 induces the metric in @xmath2 , which is called the word metric associated with @xmath631 . let @xmath635 be the image of @xmath629 by the natural homomorphism @xmath636 . then @xmath637 generates @xmath13 . the length function associated with @xmath638 , @xmath639 is also defined in the same way . to abuse the notation , we denote the word metric in @xmath13 associated with @xmath638 by the same symbol @xmath233 . we define an @xmath640-norm in @xmath641 by @xmath642 for @xmath643 and the distance @xmath644 in @xmath12 by @xmath645 for @xmath646 . denote by @xmath647 the induced metric in @xmath128 from @xmath648 . fix @xmath649 and a fundamental domain @xmath650 such that @xmath651 and @xmath652 is connected in the following sense : for any @xmath653 there exist a path @xmath654 in @xmath22 such that @xmath655 and @xmath656 are all in @xmath652 . this kind of set @xmath652 always exists if we take a spanning tree in @xmath3 and its lift in @xmath1 . to abuse the notation , we denote by @xmath657 the images of @xmath658 by the covering map , respectively . we also fix a fundamental domain @xmath659 which is identified with @xmath660 . to abuse the notation , we denote by @xmath661 the image of @xmath662 by the covering map . we define the map @xmath663:v \to \gamma$ ] as follows : for @xmath39 , there exists a unique element @xmath30 such that @xmath238 since @xmath2 acts on @xmath1 freely . define @xmath237:=\sigma$ ] . since there exists a constant @xmath664x_{0})-\phi(x)\|_{1}$ ] by the @xmath2-perodicity , we have that @xmath665x_{0})-\phi([z]x_{0})\|_{1 } -2c_{0 } \le \|\phi(x)-\phi(z)\|_{1 } \le \|\phi([x]x_{0})-\phi([z]x_{0})\|_{1 } + 2c_{0}\ ] ] for any @xmath666 . furthermore , since @xmath667x_{0})- \phi([z]x_{0})\|_{1}=\left|[x]-[z]\right|$ ] , we have that @xmath668-[z]\right| -2c_{0 } \le \|\phi(x)-\phi(z)\|_{1 } \le \left|[x ] -[z]\right| + 2c_{0}.\ ] ] as in section[harmonic ] , suppose that we have an injective homomorphism @xmath669 such that @xmath670 take a fundamental parallelotope @xmath671 . in the similar way to the above , we define the map @xmath663:\gamma \otimes \mathbb{r } \to \gamma$ ] as follows : for @xmath672 , there exists a unique element @xmath30 such that @xmath673 . define @xmath674:=\sigma$ ] . since there exists a constant @xmath675x_{0})- { \bf x}\|_{1}$ ] by the @xmath2-periodicity , we have that @xmath676x_{0})-\phi([{\bf z}]x_{0})\|_{1 } -2c_{1 } \le \|{\bf x } - { \bf z}\|_{1 } \le \|\phi([{\bf x}]x_{0})-\phi([{\bf z}]x_{0})\|_{1 } + 2c_{1}\ ] ] for any @xmath677 . furthermore , we have that @xmath678-[{\bf z}]\right| -2c_{1 } \le \|{\bf x}-{\bf z}\|_{1 } \le \left|[{\bf x } ] -[{\bf z}]\right| + 2c_{1}.\ ] ] let us define an @xmath679-ball in @xmath128 of the center @xmath680 by @xmath681 . we show the following : [ ball approximation ] there exists a constant @xmath682 depending only on @xmath679 such that for any @xmath680 and for any @xmath108 , @xmath683 here @xmath684 stands for the volume of a borel set @xmath272 and @xmath685 the cardinality of a set @xmath686 . for any @xmath680 , take a lift @xmath687 then @xmath688x_{0})- \tilde { \bf z}\|_{1 } \le c_{1}$ ] . for sufficiently small @xmath689 , take a lift @xmath690 of @xmath691 . again , from the above argument , it holds that @xmath692p \subset \widetilde b^{1}_{\tilde{\bf z}}(n\epsilon ) \subset \cup_{|\sigma| \le \epsilon n + 2c_{1}}\sigma [ \tilde{\bf z}]p$ ] . note that @xmath693 , and thus @xmath694p\right)\right| \le vol\left(p \right)(2^{d}/d!)\left((\epsilon n + 2c_{1})^{d } -(\epsilon n -2c_{1})^{d}\right)$ ] . since @xmath695p\right ) / vol\left(p\right)= \left| \cup_{|\sigma| \le \epsilon n}\sigma [ \tilde{\bf z}]d_{x_{0}}\right| / |v_{0}|$ ] and @xmath696 , it concludes that there exists a constant @xmath697 depending only on @xmath679 such that @xmath698d_{x_{0 } } \right|}{n^{d}|v_{0}|}\right| \le \frac{c(\epsilon)}{n}.\ ] ] the cardinality of the set @xmath699 is invariant under translation . it completes the proof . let us define a measure @xmath261 on @xmath128 by @xmath700 . let @xmath701 be a characteristic function defined by @xmath702 for the empirical density @xmath703 on @xmath128 , @xmath163 , then we have the following lemma . [ characteristic function ] there exists a constant @xmath704 depending only on @xmath689 , such that for any @xmath163 and any @xmath705 , @xmath706d_{x_{0}}\right|}\sum_{x \in \cup_{|\sigma|\le \epsilon n}\sigma [ z]d_{x_{0 } } } \eta_{x}\right| \le \frac{c(\epsilon)}{n},\ ] ] where @xmath707 take a lift @xmath708 of @xmath705 and a lift @xmath709 of @xmath710 . in the similar way to the proof of lemma[ball approximation ] , we obtain that @xmath711d_{x_{0 } } \subset \left\{x \in v \ \bigg| \ \left\|\frac{1}{n}\phi(x ) - \frac{1}{n}\phi(\tilde z)\right\|_{1 } \le \epsilon \right\ } \subset \bigcup_{|\sigma|\le \epsilon n + 2c_{0}}\sigma [ \tilde z]d_{x_{0}}.\ ] ] furthermore , we take a lift @xmath712 of @xmath163 , then it holds that @xmath713 and @xmath714d_{x_{0 } } } \widetilde \eta_{x}\right| \le \frac{1}{|v_{n}|}\sum_{x \in \cup_{\epsilon n - 2c_{0 } \le |\sigma|\le \epsilon n + 2c_{0}}\sigma [ \tilde z]d_{x_{0}}}\widetilde \eta_{x}.\ ] ] the last term is bounded by @xmath715 , and thus there exists a constant @xmath716 depending only on @xmath679 such that @xmath714d_{x_{0 } } } \widetilde \eta_{x}\right| \le \frac{c_{1}(\epsilon)}{n}.\ ] ] by lemma[ball approximation ] and @xmath717d_{x_{0 } } \right|=\left|\cup_{|\sigma| \le \epsilon n}\sigma d_{x_{0 } } \right|=\left|\cup_{|\sigma| \le \epsilon n}\sigma[z]d_{x_{0 } } \right|$ ] , @xmath718d_{x_{0 } } \right|}\right| & \le \frac{1}{|v_{n}|\mu\left(b^{1}_{\phi_{n}(z)}(\epsilon)\right)}\frac{1}{\left|\cup_{|\sigma| \le \epsilon n}\sigma[z]d_{x_{0 } } \right|}\frac{c(\epsilon)}{n } |v_{n}| \\ & \le \frac{c_{2}(\epsilon)}{n^{d+1}},\end{aligned}\ ] ] where @xmath719 is a constant depending only on @xmath679 . finally , @xmath720d_{x_{0}}\right|}\sum_{x \in \cup_{|\sigma|\le \epsilon n}\sigma [ z]d_{x_{0 } } } \eta_{x}\right| & \le \frac{1}{\mu\left(b^{1}_{\phi_{n}(z)}(\epsilon)\right)}\frac{c_{1}(\epsilon)}{n } + \frac{c_{2}(\epsilon)}{n^{d+1}}|v_{0}|(\epsilon n)^{d } \\ & \le \frac{c_{3}(\epsilon)}{n},\end{aligned}\ ] ] where @xmath721 is a constant depending only on @xmath679 . it completes the proof . [ energy ] suppose that @xmath722 is a sequence of probability measures on @xmath193,\mathcal{m})$ ] . for any limit point @xmath723 of @xmath722 , @xmath723-a.s . there exists a measurable function @xmath205 such that @xmath724 , @xmath370 has @xmath725\times \mathbb{t}^{d}),\ ] ] for @xmath726 , and satisfies @xmath727 for every @xmath728\times \mathbb{t}^{d})$ ] and @xmath726 . for fixed @xmath729 , we define a lattice @xmath730 whose vertex set @xmath731 is the subset of @xmath732 in the following : @xmath731 is the orbit of @xmath390 by @xmath13 , i.e. , @xmath733 . define @xmath734 the set of oriented edges @xmath213 such that @xmath735 for some @xmath736 . then @xmath13 acts on @xmath737 naturally . a configuration @xmath738 on @xmath7 induces the one on @xmath737 by restriction . we use the same symbol @xmath738 for this restriction . for @xmath739\times \mathbb{t}^{d})$ ] and for @xmath740 , we define @xmath741,\mathbb{r})$ ] , @xmath742 and @xmath743 for any @xmath284 , we have that @xmath744 we use the cauchy - schwarz inequality in the last inequality . by using the argument in the proof of lemma[path ] , there exists a constant @xmath580 such that @xmath745 since @xmath746 and @xmath747 , we get @xmath748 note that @xmath749 for large enough @xmath4 . consider for @xmath296 the self - adjoint operator @xmath750 and suppose @xmath751 to be the largest eigenvalue of this operator . by the variational formula @xmath752 by the simple inequality for @xmath753 , @xmath754 , the last formula is bounded by @xmath755 . we put @xmath756 . on the other hand , @xmath757 where @xmath758 is the directional derivative along @xmath759 and @xmath760 . by the entropy inequality , @xmath761 since @xmath762 for some constant @xmath531 , by the feynman - kac formula , we obtain @xmath763 \times \mathbb{t}^{d})}^{2 } + |v_{0}|c'.\ ] ] for a limit point of @xmath764 , @xmath723 , we get @xmath765 \le 16ct \|j\|_{l^{2}([0,t ] \times \mathbb{t}^{d})}^{2}+|v_{0}|c'.\ ] ] denote a countable dense subset of @xmath766\times \mathbb{t}^{d})$ ] by @xmath767 , we also get the following estimate : @xmath768 \le |v_{0}|c'.\ ] ] see @xmath769 pp.107 , section 5.7 for details . therefore for almost all @xmath370 , there exists @xmath770 such that for every @xmath728\times \mathbb{t}^{d})$ ] , @xmath771 that is , @xmath772 this implies the linear functional @xmath773\times \mathbb{t}^{d } ) \to \mathbb{r}$ ] defined by @xmath774 is extended on @xmath775\times\mathbb{t}^{d})$ ] . by the riesz representation theorem , there exists @xmath776\times\mathbb{t}^{d})$ ] such that @xmath777 for every @xmath728\times \mathbb{t}^{d})$ ] and every @xmath778 . this yields lemma[energy ] . [ uniqueness ] for any @xmath779\times \mathbb{t}^{d})$ ] , a weak solution of the quasi - linear partial differential equation @xmath780 with the measurable initial value @xmath199 $ ] , of bounded energy , i.e , @xmath781 is unique . first , the author would like to thank professor tsuyoshi kato for his constant encouragement and helpful suggestions . he wishes to express his gratitude to professors motoko kotani , nobuaki sugimine , satoshi ishiwata and makiko sasada for their valuable advice and helpful discussions . second , he would like to thank professor yukio nagahata for his helpful comments on this subject and the background of the hydrodynamic limit . third , he would like to thank doctors sei - ichiro kusuoka and makoto nakashima for their advices on the presentation of this paper . fourth , he would like to thank professor kazumasa kuwada for his valuable comments on the earlier version of the manuscript and his encouragement . the author could not write up this paper without a great deal of his advice . fifth , the author partially carried out this work at riken center for developmental biology in kobe . he would like to thank professors hiroki r. ueda and yohei koyama for their interest in this topic and their encouragement . sixth , the author partially carried out this work at max planck institute for mathematics in the sciences in leipzig . he would like to thank professors jrgen jost and nihat ay for helpful discussions . the author is supported by the research fellowships of the japan society for the promotion of science for young scientists . 99 berger , n. , biskup , m. : quenched invariance principle for simple random walk on percolation clusters . prob . theory related fields * 137 * , 83 - 120 ( 2007 ) bertini , l. , landim , c. , mourragui , m. : dynamical large deviations for the boundary driven weakly asymmetric exclusion process . * 37 * ( 6 ) , 2357 - 2403 ( 2009 ) faggionato , a. : random walks and exclusion processes among random conductances on random infinite clusters : homogenization and hydrodynamic limit . electron . j. probab . * 13 * , no . 73 , 2217 - 2247 ( 2008 ) gonalves , p. , jara , m. : density fluctuations for a zero - range process on the percolation cluster . comm . in probab . * 14 * , 382 - 395 ( 2009 ) guo , m.z . , papanicolaou , g.c . , varadhan , s.r.s . : nonlinear diffusion limit for a system with nearest neighbor interactions . * 118 * , 31 - 59 ( 1988 ) jara , m. : hydrodynamic limit for a zero - range process in the sierpinski gasket . comm . math . * 288 * , 773 - 797 ( 2009 ) kipnis , c. , landim , c. : scaling limits of interacting particle systems . grundlehren der mathematischen wissenschaften * 320 * , springer - verlag , berlin ( 1999 ) kipnis , c. , olla , s. , varadhan , s. r. s. : hydrodynamics and large deviation for simple exclusion processes . comm . pure appl . xlii , 115 - 137 ( 1989 ) kotani , m. , sunada , t. : albanese maps and off diagonal long time asymptotics for the heat kernel . phys . * 209 * , 633 - 670 ( 2000 ) kotani , m. , sunada , t. : standard realizations of crystal lattices via harmonic maps . trans . * 353 * , 1 - 20 ( 2001 ) kotani , m. , sunada , t. : large deviation and the tangent cone at infinity of a crystal lattice . math . z. * 254 * , 837 - 870 ( 2006 ) koukkous , a. : hydrodynamic behavior of symmetric zero - range processes with random rates . stochastic process appl . * 84 * 297 - 312 ( 1999 ) sasada , m. : hydrodynamic limit for two - species exclusion processes . appl . * 120 * , issue 4 , 494 - 521 ( 2010 ) shubin , m. , sunada , t. : geometric theory of lattice vibrations and specific heat , pure appl . math . q. * 2 * no . 3 , 745 - 777 ( 2006 ) spohn , h. : large scale dynamics of interacting particles , texts and monograph in physics , springer verlag , heidelberg ( 1991 ) sunada , t. : crystals that nature might miss creating . notices of the ams , * 55 * no . 2 , 208 - 215 ( 2008 )

we investigate the hydrodynamic limit for weakly asymmetric simple exclusion processes in crystal lattices . we construct a suitable scaling limit by using a discrete harmonic map . as we shall observe , the quasi - linear parabolic equation in the limit is defined on a flat torus and depends on both the local structure of the crystal lattice and the discrete harmonic map . we formulate the local ergodic theorem on the crystal lattice by introducing the notion of local function bundle , which is a family of local functions on the configuration space . the ideas and methods are taken from the discrete geometric analysis to these problems . results we obtain are extensions of ones by kipnis , olla and varadhan to crystal lattices .

arxiv

the study of harmonic maps was initiated by f. b. fuller , j. nash and j. h. sampson @xcite while the first general result on the existence of harmonic maps is due to eells - sampson @xcite . harmonic maps are extrema ( critical points ) of the energy functional defined on the space of smooth maps between riemannian ( pseudo - riemannian ) manifolds . the trace of the second fundamental form of such maps vanishes . more precisely , let @xmath0 , @xmath1 be pseudo - riemannian manifolds and @xmath2 denotes the sections of the tangent bundle @xmath3 of @xmath4 , that is , the space of vector fields on @xmath4 . then _ @xmath5 of a smooth map @xmath6 is defined by the formula @xmath7 where @xmath8 is the volume measure associated to the metric @xmath9 and the _ energy density _ @xmath10 of @xmath11 is the smooth function @xmath12 given by @xmath13 for each @xmath14 . in the above equation @xmath15 is a linear map @xmath16 therefore it can be considered as a section of the bundle @xmath17 where @xmath18 is the pullback bundle having fibres @xmath19 , @xmath14 and @xmath20 is the pullback metric on @xmath21 . if we denote by @xmath22 and @xmath23 the levi - civita connections on @xmath24 and @xmath25 respectively , then the second fundamental form of @xmath11 is the symmetric map @xmath26 defined by @xmath27 for any @xmath28 . where @xmath29 is the pullback of the levi - civita connection @xmath23 of @xmath25 to the induced vector bundle @xmath30 . the section @xmath31 , defined by @xmath32 is called the _ tension field _ of @xmath11 and a map is said to be harmonic if its tension field vanishes identically ( see @xcite ) . if we consider @xmath33 a smooth two - parameter variation of @xmath11 such that @xmath34 and let @xmath35 be the corresponding variational vector fields then @xmath36 the _ hessian _ of a harmonic map @xmath11 is defined by : @xmath37 the index of a harmonic map @xmath38 is defined as the dimension of the tangent subspace of @xmath39 on which the hessian @xmath40 is negative definite . a harmonic map @xmath11 is said to be _ stable _ if morse index ( _ i.e. _ , the dimension of largest subspace of @xmath41 on which the hessian @xmath40 is negative definite ) of @xmath11 is zero and otherwise , it is said to be _ unstable _ ( see @xcite ) . for a non - degenerate point @xmath42 , we decompose the space @xmath43 into its _ vertical space _ @xmath44 and its _ horizontal space _ @xmath45 , that is , @xmath46 , so that @xmath47 . the map is said to be horizontally conformal if for each @xmath14 either the rank of @xmath48 is zero ( that is , @xmath49 is a critical point ) , or the restriction of @xmath48 to the horizontal space @xmath50 is surjective and conformal ( here @xmath49 is a regular point ) @xcite . the premise of harmonic maps has acknowledged several important contributions and has been successfully applied in computational fluid dynamics ( cfd ) , minimal surface theory , string theory , gravity and quantum field theory ( see @xcite ) . most of works on harmonic maps are between riemannian manifolds @xcite . the harmonic maps between pseudo - riemannian manifolds behave differently and their study must be subject to some restricted classes of pseudo - riemannian manifolds @xcite . this paper is organized as follows . in sect . [ prem ] , the basic definitions about almost para - hermitian manifolds , almost paracontact manifolds and normal almost paracontact manifolds are given . in sect . [ paraholo ] , we define and study paraholomorphic map . we prove that the tension field of any @xmath51-paraholomorphic map between almost para - hermitian manifold and para - sasakian manifold lies in @xmath52 . [ parap ] deals with parapluriharmonic map in which we obtain the necessary and sufficient condition for a @xmath53-paraholomorphic map between para - sasakian manifolds to be @xmath54-parapluriharmonic and give an example for its illustrations . a smooth manifold @xmath55 of dimension @xmath56 is said to be an almost product structure if it admits a tensor field @xmath57 of type @xmath58 satisfying : @xmath59 in this case the pair @xmath60 is called an almost product manifold . an almost para - complex manifold is an almost product manifold @xmath61 such that the eigenbundles @xmath62 associated with the eigenvalues @xmath63 of tensor field @xmath57 have the same rank @xcite . an almost para - hermitian manifold @xmath64 is a smooth manifold endowed with an almost para - complex structure @xmath57 and a pseudo - riemannian metric @xmath65 compatible in the sense that @xmath66 it follows that the metric @xmath65 has signature @xmath67 and the eigenbundles @xmath62 are totally isotropic with respect to @xmath65 . let @xmath68 be an orthonormal basis and denote @xmath69 : @xmath70 for @xmath71 and @xmath72 for @xmath73 . the fundamental @xmath74-form of almost para - hermitian manifold is defined by @xmath75 and the co - differential @xmath76 of @xmath77 is given as follows @xmath78 an almost para - hermitian manifold is called para - khler if @xmath79 @xcite . a @xmath80 smooth manifold @xmath81 of dimension @xmath82 is said to have a triplet @xmath83-structure if it admits an endomorphism @xmath84 , a unique vector field @xmath85 and a @xmath86-form @xmath87 satisfying : @xmath88 where @xmath89 is the identity transformation ; and the endomorphism @xmath84 induces an almost paracomplex structure on each fibre of @xmath90 the contact subbundle , _ i.e. _ , eigen distributions @xmath91 corresponding to the characteristic values @xmath63 of @xmath84 have equal dimension @xmath92 . + from the equation ( [ eta ] ) , it can be easily deduced that @xmath93 this triplet structure @xmath83 is called an almost paracontact structure and the manifold @xmath81 equipped with the @xmath83-structure is called an almost paracontact manifold ( see also @xcite ) . if an almost paracontact manifold admits a pseudo - riemannian metric @xmath94 satisfying : @xmath95 where signature of @xmath94 is necessarily @xmath96 for any vector fields @xmath97 and @xmath98 ; then the quadruple @xmath99 is called an almost paracontact metric structure and the manifold @xmath81 equipped with paracontact metric structure is called an almost paracontact metric manifold . with respect to @xmath94 , @xmath87 is metrically dual to @xmath85 , that is @xmath100 also , equation ( [ gphi ] ) implies that @xmath101 further , in addition to the above properties , if the structure-@xmath99 satisfies : @xmath102 for all vector fields @xmath103 on @xmath81 , then the manifold is called a paracontact metric manifold and the corresponding structure-@xmath99 is called a paracontact structure with the associated metric @xmath94 @xcite . for an almost paracontact metric manifold , there always exists a special kind of local pseudo - orthonormal basis @xmath104 ; where @xmath105 @xmath85 and @xmath106 s are space - like vector fields and @xmath107 s are time - like . such a basis is called a @xmath84-basis . hence , an almost paracontact metric manifold @xmath108 is an odd dimensional manifold with a structure group @xmath109 , where @xmath110 is the para - unitary group isomorphic to @xmath111 . an almost paracontact metric structure-@xmath99 is para - sasakian if and only if @xmath112 from eqs . , and , it can be easily deduced for a para - sasakian manifold that @xmath113 in particular , a para - sasakian manifold is @xmath114-paracontact @xcite . on an almost paracontact metric manifold , one defines the @xmath115-tensor field @xmath116 by @xmath117 - 2\,d\eta\otimes\xi,\end{aligned}\ ] ] where @xmath118 $ ] is the nijenhuis torsion of @xmath84 . if @xmath116 vanishes identically , then we say that the manifold @xmath81 is a normal almost paracontact metric manifold @xcite . the normality condition implies that the almost paracomplex structure @xmath57 defined on @xmath119 by @xmath120 is integrable . here @xmath97 is tangent to @xmath81 , @xmath121 is the coordinate on @xmath122 and @xmath123 is a @xmath124 function on @xmath119 . now we recall the following proposition which characterized the normality of almost paracontact metric @xmath125-manifolds : @xcite for an almost paracontact metric @xmath125-manifold @xmath126 , the following three conditions are mutually equivalent * @xmath126 is normal , * there exist smooth functions @xmath127 on @xmath126 such that @xmath128 * there exist smooth functions @xmath127 on @xmath126 such that @xmath129 where @xmath22 is the levi - civita connection of the pseudo - riemannian metric @xmath94 . the functions @xmath127 appearing in eqs . ( [ nablaxphiy ] ) and ( [ nablaxxi ] ) are given by @xmath130 a normal almost paracontact metric @xmath125-manifold is called paracosymplectic if @xmath131 and para - sasakian if @xmath132 @xcite . one can look structure preserving mapping between almost para - hermitian and almost paracontact manifolds as analogous of the well - known holomorphic mappings in complex geometry @xcite . + let @xmath133 , @xmath1 be almost paracontact metric manifolds and @xmath64 be an almost para - hermitian manifold . then a smooth map * @xmath134 is @xmath135-paraholomorphic map if @xmath136 . for such a map @xmath137 . * @xmath138 is @xmath139-paraholomorphic map if @xmath140 . here * @xmath142 is @xmath53-paraholomorphic map if @xmath143 . in particular , @xmath144 and @xmath145 . when @xmath15 interwines the structures upto a minus sign , we say about @xmath135-anti paraholomorphic , @xmath139-anti paraholomorphic and @xmath53-anti paraholomorphic mappings . now , we prove the following result . let @xmath11 be a smooth @xmath53-paraholomorphic map between para - sasakian manifolds @xmath146 , @xmath1 . then @xmath147 where @xmath148 . since @xmath15 has values in @xmath149 so that @xmath150 and @xmath151 have values in @xmath149 . thus , we have @xmath152 in the last equality , we have used . on the other hand , we obtain @xmath153 from eqs . and , we have @xmath154 let @xmath155 be a local orthonormal frame for @xmath156 . taking the trace in and using the fact that @xmath157 is symmetric , we have . this completes the proof . following the proof of the above proposition , we can give the following remarks : for a para - sasakian manifold @xmath158 and a para - hermitian manifold @xmath64 . if * @xmath134 be a @xmath135-paraholomorphic map then we have @xmath159 where @xmath160 . + * @xmath138 be a @xmath139-paraholomorphic map then we have @xmath161 where @xmath162 . [ t1 ] let @xmath11 be a @xmath135-paraholomorphic map between para - sasakian manifold @xmath163 and para - k@xmath164hler manifold @xmath64 . then @xmath11 is harmonic . let @xmath165 be a local orthonormal adapted basis on @xmath166 , then from eqs . and , we have @xmath167 ( since for a @xmath135-paraholomorphic map @xmath137 ) . it follows by the use of equation that @xmath168 as @xmath55 is a para - k@xmath169hler manifold . therefore , @xmath170 and @xmath11 is harmonic . this completes the proof of the theorem . for @xmath1 , let @xmath171 be real distributions , respectively , on para - sasakian manifolds @xmath172 of rank @xmath173 then it admits globally defined @xmath86-form @xmath174 such that @xmath175 . clearly , @xmath176 , where @xmath177 is the real distribution of rank one defined by @xmath178 @xcite . now , we prove : [ hk1 ] for any @xmath139-paraholomorphic map @xmath11 between almost para - hermitian manifold @xmath64 and para - sasakian manifold @xmath179 , the tension field @xmath180 . before going to proof of this theorem , we first prove the following proposition : for an almost para - hermitian manifold @xmath64 , we have @xmath181\bigg\}\end{aligned}\ ] ] where @xmath182 is a local orthonormal frame on @xmath183 . it is straightforward to calculate @xmath184-j(\nabla_{e'_{i}}e'_{i})+j(\nabla_{je'_{i}}je'_{i})\big\ } \ ] ] and the result follows from and . this completes the proof . _ proof of theorem [ hk1 ] . _ since @xmath185 , @xmath186 therefore for any local orthonormal frame @xmath182 on @xmath183 , we obtain by using eqs . , , and that @xmath187 employing eq . , the above equation reduces to @xmath188\bigg),\xi_{1}\bigg).\end{aligned}\ ] ] reusing eq . in , we get @xmath189 which shows that @xmath180 . this completes the proof of the theorem . by the consequence of the above theorem we can state the following result as a corollary of the theorem [ hk1 ] . let @xmath64 and @xmath163 be para - k@xmath164hler and para - sasakian manifolds respectively . then for any @xmath139-paraholomorphic map @xmath138 , the tension field @xmath180 . in this section we define the notion of @xmath54-parapluriharmonic map which is similar to the notion of @xmath84-pluriharmonic map between almost contact metric manifold and riemannian manifold , for @xmath84-pluriharmonic map see : @xcite . a smooth map @xmath11 between almost paracontact metric manifold @xmath179 and pseudo - riemannian manifold @xmath190 is said to be @xmath54-parapluriharmonic if @xmath191 where the second fundamental form @xmath192 of @xmath11 is defined by . in particular , @xmath193 for any tangent vector @xmath97 . we recall that @xmath137 for a @xmath135-paraholomorphic map and @xmath200 is para - k@xmath201hler , and that from eq . for any vectors @xmath202 tangent to @xmath203 , we have @xmath204 using equation for a given map , we obtain @xmath205 replacing @xmath98 by @xmath206 and employing eqs . and , the above equation reduces to @xmath207 by the virtue of the fact that @xmath157 is symmetric , we obtain from above equation that @xmath208 the above expresion implies that @xmath11 is @xmath54-parapluriharmonic and thus harmonic from the proposition [ pro1 ] . this completes the proof of the theorem . let @xmath198 be a normal almost paracontact metric 3-manifold with @xmath210constant , @xmath199 be a para - k@xmath164hler manifold and @xmath211 be a smooth @xmath135-paraholomorphic map . then @xmath212 is paracosymplectic manifold . since @xmath11 is a @xmath53-paraholomorphic map then for all @xmath214 there exists a function @xmath123 on @xmath215 such that @xmath216 for any @xmath217 , we have from eqs . , and that @xmath218 from above equation and the fact that @xmath157 is symmetric , we obtain that @xmath219 replacing @xmath98 by @xmath220 in above expression and using eqs . and , we find @xmath221 this implies that @xmath222 if and only if @xmath213 . this completes the proof of the theorem . let @xmath223 be @xmath125-dimensional manifolds with standard cartesian coordinates . define the almost paracontact structures @xmath224 respectively on @xmath225 by @xmath226 where @xmath227 , @xmath228 , @xmath229 , @xmath230 , @xmath231 and @xmath232 . by direct calculations , one verifies that the nijenhuis torsion of @xmath233 for @xmath1 vanishes , which implies that the structures are normal . let the pseudo - riemannian metrics @xmath234 are prescribed respectively on @xmath225 by @xmath235=\begin{bmatrix}-x&0&0\\0&x^4+x&x^2\\0&x^2&1\end{bmatrix},\ \left[g_2\left(e_s',e_t'\right)\right]=\begin{bmatrix}v^4+v&0&v^2\\0&-v&0\\v^2&0&1\end{bmatrix},\end{aligned}\ ] ] for all @xmath236 . for the levi - civita connections @xmath237 with respect to metrics @xmath238 respectively , we obtain @xmath239 @xmath240 from above expressions and equation , we find @xmath241 , @xmath242 . hence the @xmath203 and @xmath243 are para - saakian manifolds with invariant distributions @xmath244 and @xmath245 respectively . let @xmath246 be a mapping defined by @xmath247 . then @xmath143 , _ i_.e . , @xmath11 is a @xmath53-paraholomorphic map between para - sasakian manifolds . for any @xmath248 and @xmath249 , it is not hard to see that @xmath250 , @xmath251 and @xmath252 thus theorem [ main ] is verified . : harmonic maps in general relativity and quantum field theory , in : gauduchon , p. , editor , harmonic mappings , twistors , and @xmath253-models ( luminy , 1986 ) . advanced series in math . physics * 4 * , 270 - 305 , world scientific publishing , singapore ( 1988 ) .

the purpose of this paper is to study the harmonicity of maps to or from para - sasakian manifolds . we derive the condition for the tension field of paraholomorphic map between almost para - hermitian manifold and para - sasakian manifold . the necessary and sufficient condition for a paraholomorphic map between para - sasakian manifolds to be parapluriharmonic are shown and a non - trivial example is presented for its illustrations . dst / inspire/03/2014/001552 . ]

arxiv

the star formation rate ( sfr ) of galaxies sets the rate at which galaxies grow and evolve and is the one of the most important measures for understanding the hierarchical build - up of our universe over cosmic time . large scale simulations , however , have shown that unregulated star formation leads to an overabundance of high mass galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? therefore some process ( or processes ) must be able to stop , or `` quench , '' star formation before the galaxy grows to be too big . the answer seems to lie in supermassive black holes ( smbh ) which nearly all massive galaxies harbor in their centers . smbhs grow through accretion of cold material ( active galactic nuclei ; agn ) , and the huge loss of gravitational energy of the cold material is converted into radiation that is evident across the whole electromagnetic spectrum and manifests itself as a bright point source in the nucleus of galaxies . the agn can deposit this energy into the ism of its host galaxy through jets ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) or powerful outflows that either heat the gas or remove it altogether , i.e. `` feedback '' processes ( e.g * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? . indirect evidence of this `` feedback '' has been observed through the simple , scaling relationships between the mass of the smbh and different properties of the host galaxy such as the stellar velocity dispersion in the bulge , the bulge mass , and the bulge luminosity ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the relative tightness of these relationships suggests a strong coevolution of the host galaxy and smbh . much debate remains however as to the exact mechanism of agn feedback and whether or not it plays a dominant role in the overall evolution of galaxies especially in light of new observations at both low and high @xmath5 that seem to deviate from the well - established relationships ( see * ? ? ? * for a detailed review ) . evidence for agn feedback though should also manifest itself in the sfr of its host galaxy , therefore much work has also focused on the so - called starburst - agn connection ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the problem lies in determining accurate estimates of the sfr in agn host galaxies . well - calibrated indicators , such as h@xmath6 emission and uv luminosity , are significantly , if not completely , contaminated by the central agn . many studies therefore turn to the infrared ( ir ) regime ( @xmath7 @xmath1 m ) where dust re - emits the stellar light from young stars . dust fills the interstellar medium ( ism ) of galaxies and plays an important part in the heating and cooling of the ism and the general physics of the galaxy . while dust contributes very little to the overall mass of a galaxy ( @xmath8 ) , the radiative output , mainly in the infrared ( ir ) regime , can , on average , constitute roughly half of the bolometric luminosity of the entire galaxy @xcite , although there is an enormous range in the fraction . dust efficiently absorbs optical and uv emission and re - radiates it in the mid- and far - infrared ( mir , fir ) depending on the temperature as well as grain size @xcite . recently formed o and b stars produce the majority of the optical and uv light in galaxies , therefore measuring the total ir light from dust provides insights into the current ( @xmath9 myr ) star formation rate ( sfr ) ( e.g. * ? ? ? * ) , although for very passive galaxies where the current sfr is much lower than it was earlier , ir emission can be an overestimate due to dust heating by an older stellar population.(e.g . * ) however , dust is also the key component in obscuring our view of agn . dust heated by the agn is thought to primarily live in a toroidal - like structure that encircles the agn and absorbs its radiative output for certain lines of sight . the dusty torus is used to explain the dichotomy of agn into seyfert 1 ( sy 1 ) and seyfert 2 ( sy 2 ) within a unified model @xcite . like o and b stars in star - forming regions , the agn outputs heavy amounts of optical and uv light , and like dust in the ism the dusty torus absorbs and re - emits this as ir radiation . spectral energy distribution ( sed ) models @xcite as well as observations @xcite suggest the torus mainly emits in the mir ( @xmath10 @xmath1 m ) with the flux density dropping rapidly in the fir ( @xmath11 @xmath1 m ) . further the sed for stellar dust re - radiation peaks in the fir @xcite , making the fir the ideal waveband to study star - formation in agn host galaxies . space - based telescopes such as the _ infrared astronomical satellite _ ( iras ; * ? ? ? * ) , _ spitzer space telescope _ @xcite , and _ infrared space observatory _ @xcite greatly expanded our knowledge of the ir universe and provided a window into the fir properties of galaxies . but , before the launch of the _ herschel space observatory _ @xcite , the fir sed was limited to @xmath12 @xmath1 m , except for studies of small samples of the brightest galaxies using ground - based instruments such as _ scuba _ ( e.g. * ? ? ? * ; * ? ? ? herschel _ with the spectral and photometric imaging receiver ( spire ; * ? ? ? * ) has pushed into the submillimeter range with observations in the 250 , 350 , and 500 @xmath1 m wavebands , probing the rayleigh - jeans tail of the modified blackbody that accurately describes the broadband fir sed of galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? these wavebands are crucial for measuring dust properties ( i.e. temperature and mass ) as @xcite and @xcite show . further , @xcite found that fir and submillimeter data are important for estimating the sfr of agn host galaxies . recent studies , such as @xcite and @xcite , focusing on the dust and star - forming properties of agn have shown the power of long wavelength _ herschel _ data to better constrain the sfr , dust mass , and dust temperature in agn host galaxies . @xcite analyzed the ir seds of low redshift ( @xmath13 ) , quasi - stellar objects ( qsos ) broadly finding most of the fir emission can be attributed to thermally - heated dust . @xcite looked at the ir seds of 24 @xmath14 m selected agn at slightly higher redshift ( @xmath15 ) around galaxy clusters finding a strong correlation between the agn and star - forming luminosity which could be due to their shared correlation with galaxy stellar mass . both studies , however , rely on agn selection using different wavebands ( optical vs. mid - infrared ) and generally probe the higher agn luminosity population . therefore , we have assembled a large ( @xmath16 ) , low redshift ( @xmath0 ) sample of agn selected using ultra - hard x - ray observations with the _ swift_/_burst alert telescope _ ( bat ) and imaged each one with _ this sample focuses on moderate luminosity seyfert galaxies ( @xmath17 ) . in @xcite , we presented the pacs data of the _ herschel_-bat agn which provided photometry at 70 and 160 @xmath1 m . in this paper , we complete the fir sed of the bat agn with the creation and analysis of the spire images . we focus on the overall luminosity distributions at the spire wavebands as well as the spire colors ( @xmath2 and @xmath3 ) to determine the likely heating sources of cold dust in agn host galaxies . we also look for correlations with a proxy for the bolometric agn luminosity to potentially reveal any indication that agn heated dust is contributing to the fir sed . this paper sets us up for a complete study of the mid - far ir sed to fully explore the star - forming properties of agn host galaxies and reveal the global starburst - agn connection in the nearby universe ( shimizu et al , in preparation ) . throughout this paper we assume a @xmath18cdm cosmology with @xmath19 km s@xmath20 mpc@xmath21 , @xmath22 , and @xmath23 . luminosity distances for each agn were calculated based on their redshift and assumed cosmology , except for those with @xmath24 where we referred to the _ _ extragalactic distance database__. swift_/bat @xcite operates in the 14195 kev energy range , continuously monitoring the sky for gamma - ray bursts . this constant monitoring has also allowed for the most complete all - sky survey in the ultra - hard x - rays . to date , bat has detected 1171 sources at @xmath25 significance corresponding to a sensitivity of @xmath26 ergs s@xmath20 @xmath27 @xcite . over 700 of those sources have been identified as a type of agn ( seyfert , blazar , qso , etc . ) we selected our sample of 313 agn from the 58 month _ swift_/bat catalog @xcite , imposing a redshift cutoff of @xmath0 . all different types of agn were chosen only excluding blazars / bl lac objects which most likely introduce complicated beaming effects . to determine their agn type , for 252 sources we used the classifications from the bat agn spectroscopic survey ( koss et al , in preparation ) which compiled and analyzed optical spectra for the _ swift_/bat 70 month catalog @xcite . seyfert classification was determined using the standard scheme from @xcite and @xcite . for the remaining 61 agn we used the classifications provided in the 70 month catalog . in total the sample contains 30 sy 1 , 30 sy 1.2 , 79 sy 1.5 , 1 sy 1.8 , 47 sy 1.9 , 121 sy 2 , 4 liners , and 1 unclassified agn . for the purpose of broad classification , in the rest of this paper we choose to classify all sy 1 - 1.5 as sy 1 s , and all sy 1.8 - 2 as st 2 s . in table [ tbl : bat_info ] we list the entire _ herschel_-bat sample along with positions and redshifts taken from the _ nasa / ipac extragalactic database _ ( ned ) . selection of agn by ultra - hard x - rays provides multiple advantages over other wavelengths . due to their high energy , ultra - hard x - rays easily pass through compton - thin gas or dust in the line of sight providing a direct view of the agn . using optical or mid - infrared selection can be problematic due to contamination by the host galaxy . also , ultra - hard x - rays are unaffected by any type of absorption by material obscuring the agn provided it is optically thin to compton scattering ( @xmath28 ) which is a concern for hard x - rays in the 2 - 10 kev energy range . numerous studies have been done on the bat sample in the past that span nearly the entire electromagnetic spectrum . @xcite and @xcite used _ spitzer_/irs spectra to study the mid - infrared properties of the bat agn . @xcite and @xcite studied the x - ray spectral properties for a subsample , while @xcite looked at the optical host galaxy properties and @xcite analyzed the optical spectra . along with these , many of the bat agn are detected at radio wavelengths with the first @xcite and nvss @xcite survey as well . one key ingredient missing though is the far - infrared ( fir ) where emission from ultraviolet - heated dust peaks . [ cols="<,<,<,^,^,<,^,^ " , ] while all the relationships show some amount of correlation with very low ( @xmath29 1% ) probabilities of occurring by chance , the strongest ones occur between wavelengths that are nearest each other . the 160 vs. 250 @xmath1 m and 250 vs. 350 @xmath1 m correlations have a correlation coefficient @xmath30 . this makes sense within the context of multiple temperature components . photometry from nearby wavelengths should be produced from closely related temperature components . the weak correlation between 70 and 500 @xmath1 m indicates the emission in these wavebands does not originate from closely related processes . 70 @xmath1 m emission comes from much hotter and smaller dust grains than 500 @xmath1 m and several processes could provide an explanation . since this is an agn sample , there could be a strong contribution from agn heated dust at 70 @xmath1 m , whereas at 500 @xmath1 m , agn related emission would likely be negligible . this is supported by our findings in @xcite where we showed that the 70 @xmath1 m luminosity is weakly correlated with agn luminosity . further , in @xcite we found that the bat agn morphologies at 70 @xmath1 m were concentrated in the nucleus potentially indicating an agn contribution . the weak correlation , however , can also be explained if non - star - forming processes also contribute to the 500 @xmath1 m emission . while in non - agn galaxies , the majority of 70 @xmath1 m emission is most likely due to small , stochastically heated dust grains around hii regions , @xmath31 @xmath1 m emission is likely produced by the heating of larger dust grains in the diffuse ism by older stars ( e.g. * ? ? ? therefore , the disconnect between the stellar populations would produce significant scatter in the correlation between 70 and 500 @xmath1 m . a third possibility is that synchrotron radiation produced by radio jets associated with agn can contribute to the fir , especially the longest wavelengths as seen in some radio - loud galaxies @xcite . this non - thermal emission would be completely unrelated to the thermal emission at 70 @xmath1 m , thereby producing a weaker correlation between the luminosities at those wavebands . in a later section we will show there are indeed some radio - loud sources in our sample where synchrotron emission dominates the spire emission , although the fraction of sources is quite low . when we break the sample down into sy 1 s and 2 s we do not find much difference between the correlation coefficients . this shows that sy 1 s and 2 s are not different in terms of their overall fir emission and the same processes are likely producing the fir emission . sy 1 s do show a slightly weaker correlation between the _ herschel _ luminosities especially the ones involving 500 @xmath1 m . this is likely due to the fact that most radio - loud agn are classified as sy 1 s so synchrotron emission is contributing strongest at 500 @xmath1 m compared to the other wavebands . correlations between each spire waveband luminosity and the bat 14195 kev luminosity . blue circles in the left column represent sy 1 s . red squares in the right column are sy 2 s . sources with gray arrows indicate @xmath32 upper limits . ] ultra - hard x - ray luminosity directly probes the current strength of the agn because it likely originates very close to the smbh . the 14195 kev luminosity then provides an unambiguous measure of the agn power especially for compton - thin sources . if we want to determine whether the agn contributes in any way to the fir luminosity , the first check would be to correlate the 14195 kev luminosity with each waveband s luminosity . @xcite ran correlation tests for the pacs wavebands finding a weak , but statistically significant correlation between the 70 and 160 @xmath1 m luminosity and the 14195 kev luminosity for sy 1 s but not for sy 2 s . using the same methods as we did to measure strengths of the correlations between each _ herschel _ luminosity , we measured the correlation between each spire and 14195 kev luminosity . the last lines of each section of table [ tab : wave_corrs ] lists the results of the correlation tests and figure [ fig : lum_spire_bat ] plots the correlations with gray arrows indicating upper limits . for the agn sample as a whole , no significant correlation exists between the spire and 14195 kev luminosity . all of the @xmath33 , after accounting for the partial correlation with distance , are below 0.1 with @xmath34 either at or above 5% . however , when we break the sample up into sy 1 s and sy 2 s and redo the correlation tests , we find a very weak correlation between the 250 and 350 @xmath1 m luminosity and ultra - hard x - ray luminosity for sy 1 s only ( @xmath35 and 0.10 ) . sy 2 s @xmath33 are consistent with no correlation with @xmath36 for all three wavebands . this continues the trend with what was found in @xcite where only sy 1 s were found to have a weak correlation between the bat luminosity and the pacs waveband luminosities . the partial correlation coefficients were @xmath37 and @xmath38 for sy 1s and @xmath39 and @xmath40 for sy 2 s at 70 and 160 @xmath1 m respectively ( see table 3 of * ? ? ? * ) * * * * . we note however that except for the 70 @xmath1 m waveband , none of the correlation coefficients are @xmath41 away from a null correlation coefficient . so even though @xmath42 , these are all quite weak correlations between the _ herschel _ wavebands and bat luminosity for sy 1 s . at 500 @xmath1 m , the correlation completely disappears . as we discuss in @xcite , this extends the trend observed in the mir where strong correlations have been measured between the 9 , 12 , and 18 @xmath1 m luminosities and the bat luminosity @xcite but moving towards longer wavelengths the correlation degrades rapidly as shown in @xcite for 90 @xmath1 m emission . clearly then , at long wavelengths ( @xmath43 @xmath1 m ) , emission from dust unrelated to the agn dominates most galaxies . however , we must still explain why sy 1 s retain a weak correlation while sy 2 s do not . @xcite discussed in detail several theories for why sy 1 s would show a different correlation between the _ herschel _ luminosities and bat luminosity . these included an intrinsically different bat luminosity distribution for sy 1 s and sy 2 s and the addition of compton - thick ( ct ) agn in the sy 2 sample . several authors have found that the sy 2 luminosity function breaks at a significantly lower luminosity than for sy 1s ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . at low bat luminosity , then , there are more sy 2 s than sy 1 s as is evident in figure [ fig : lum_spire_bat ] . @xcite showed that at low agn luminosity the correlation between sfr and agn luminosity flattens . this can be explained one of two ways : 1 . ) only at high agn luminosity is there a direct connection between star formation and agn activity . 2 . ) at high agn luminosity , the ir - related agn emission overwhelms any star - forming related ir emission even at long wavelengths . regardless of the physical reason , the flattening of the sf - agn relationship at low luminosity could explain the correlation differences seen between sy 1s and sy 2s since sy 2s are preferentially found at lower luminosity than sy 1s . @xcite tested this for the pacs wavebands and found that only using high luminosity objects did not improve the x - ray - to - ir correlation for sy 2 s . we repeated this test with the spire luminosities and limited the samples to only agn with bat luminosity greater than @xmath44 ergs s@xmath20 . for both sy 1 s and sy 2 s the correlations become insignificant , likely because of the reduction in number of sources used in the analysis . it is then inconclusive whether or not a difference in intrinsic agn luminosity is the cause of the differences in correlations between x - ray and ir luminosity for sy 1 s and sy 2 s . the other possibility is that ct sources are contaminating the sy 2 sample . this would have an effect if the high column density ( @xmath45 @xmath27 ) material obscuring the agn scatters 14 - 195 kev photons out of our line sight causing a lower measured bat luminosity . @xcite identified 44 either confirmed ct agn or likely ct agn based on x - ray hardness ratios in our sample . we removed these likely ct sources and redid the correlation tests , finding no difference from before just as @xcite found . therefore , it is unlikely that ct sources are the cause of the difference between the sy 1 and sy 2 correlations . given the inconclusiveness of the first test limiting the sample to high luminosity objects , we can only speculate about the reason for the difference in correlations . however , @xcite did find that restricting the sample to high luminosity objects increased the strength of the correlation for sy 1 s but not sy 2 s in the pacs wavebands . it is possible then that either a direct physical link between the sfr and agn luminosity that is only evident in high luminosity agn or increased contamination of the agn to the ir sed is causing the relatively stronger correlation in sy 1s but not sy 2s . what is conclusive is that the spire emission from the agn host galaxies on average is not strongly contaminated by agn - related emission given the small values for the correlation coefficients even for sy 1 s . while in the previous sections , we examined the absolute luminosities of each spire waveband and the correlations between each other and other wavebands ( pacs and bat ) , in this section we examine the spire colors ( i.e. flux ratios ) . colors in general provide measures of the shape of the sed . different objects and mechanisms produce significantly different sed shapes across the same wavelength regime , therefore colors can be used to separate distinct populations from each other especially when groups display the same absolute brightnesses . we investigate two colors , @xmath2 and @xmath3 , that probe the rayleigh jeans tail of a modified blackbody if the dominant process producing the emission is cold dust . figure [ fig : hist_colors ] plots the kde of the two colors . the top row compares the distribution of the colors ( @xmath2 on the left and @xmath3 on the right ) from the bat agn and hrs samples . while the hrs galaxies are local like the bat agn , one major difference is the stellar mass distribution . the hrs sample contains more low stellar mass galaxies while the bat agn are strictly found in galaxies with stellar mass ( @xmath46 ) values above @xmath47 m@xmath48 @xcite . as @xcite show , fir colors can be affected by the physical properties of the galaxy , especially the colors probing the cold dust such as the ones we are investigating here . therefore , we broke the hrs sample into two groups , a high mass group ( @xmath49 m@xmath48 ) and low mass one ( @xmath49 m@xmath48 ) indicated in figure [ fig : hist_colors ] by the solid and dashed green lines . stellar masses for the hrs were obtained from @xcite . we also plot the theoretical color of the modified blackbody with a dust temperature of 20 k and emissivity ( @xmath50 ) of 2.0 and 1.5 , values typical of normal , star - forming galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . the hrs high mass group and bat agn display nearly identical color distributions for both colors whereas the hrs low mass group is skewed toward lower colors . results of a k - s test show that the hrs high mass group and bat agn colors are drawn from the same parent population with a @xmath51 and 22% for @xmath2 and @xmath3 respectively . on the other hand the hrs low mass group colors are significantly different from the bat agn with @xmath52 values much less than 1% . this is consistent with what was found in @xcite , who showed that the spire colors for the hrs sample were affected by the metallicity of the galaxy with metal rich galaxies displaying larger flux ratios and a higher @xmath50 than metal poor ones . given the strong , positive relationship between metallicity and @xmath46 ( e.g. * ? ? ? * ) , this is exactly in line with what is seen in figure [ fig : hist_colors ] . the hrs high mass group and bat agn display colors closer to the ones expected for a modified blackbody with @xmath53 while the low mass hrs group are closer to @xmath54 . in figure [ fig : color - color ] we plot both colors together for the hrs and bat agn . nearly all of the hrs galaxies are concentrated along a main locus as well as many of the bat agn . we also plot the expected colors for a modified blackbody with varying temperature between 10 and 60 k and an emissivity of either 2.0 ( green line and squares ) or 1.5 ( purple line and diamonds ) . each square or diamond represents an increase of 5 k starting at 10 k in the lower left . the main locus for both samples is clearly aligned with a modified blackbody with temperatures between 1530 k. @xcite fit the fir sed of the hrs sample using a single temperature modified blackbody finding exactly this range of temperatures and an average emissivity of 1.8 . further these values are consistent with dust in the milky way , andromeda , and other nearby galaxies @xcite . in the bottom rows , we compare sy 1 s and sy 2 s . based on the results of our analysis in section [ sec : det_rate_lum_dist ] , we would expect sy 1 s and sy 2 s to show the same distribution of colors . indeed this is the case as both distributions in both colors peak at nearly the same values and have nearly the same spread . k - s tests reveal the colors for the two seyfert types are drawn from the same parent population with @xmath55 for @xmath2 and @xmath56 for @xmath3 . while the bulk of the spire colors are very similar between the hrs and bat , and the two seyfert types , one noticeable difference is a distinct bump in the color distribution around 0.75 . this bump is absent in the hrs sample and mainly is made up of sy 1s . with both flux ratios less than one , this indicates a monotonically rising sed that is in stark contrast with the rapidly declining sed characteristic of a modified blackbody . the equation for a modified blackbody is @xmath57 where @xmath58 is the standard planck blackbody function with a temperature of @xmath59 . the bump seen in figure [ fig : hist_colors ] is very evident in figure [ fig : color - color ] as a separate population in the lower left - hand corner . specifically 6 bat agn and one hrs galaxy occupy the region of color - color space where @xmath60 and @xmath61 . based on the theoretical curves , these exceptional colors can not be explained as either a different temperature or emissivity . rather an entirely different process is producing the fir emission in these galaxies and since the colors indicate essentially a rising sed , we suspected synchrotron radiation as the likely emission mechanism with its characteristic increasing power law shape with wavelength . further there seems to be a horizontal spread in the distribution of the bat agn in figure [ fig : color - color ] that is clearly not evident in the hrs . also this effect is not seen figure [ fig : hist_colors ] and the kdes because it only becomes evident when analyzing the two colors together . both samples span the same range of colors , however their distribution in color - color space is different . this is characterized by a large group of bat agn above and to the left of the main locus and @xmath62 line ( purple ) as well as a smaller group of agn below and to the right of the main locus and @xmath63 line ( green ) . the latter group can be explained simply from a decrease in temperature and increase in emissivity up to a beta value of 3.0 ( cyan line in figure [ fig : hist_colors ] ) , indicating the prevalence of large amounts of cold dust . the former group could be explained by a decrease in the emissivity closer to around values of 1.0 ( gray line ) , however this would require the dust temperature to increase to values above 60 k , not typical of regular star - forming galaxies . rather these high temperatures ( 70100 k ) are near the expected temperatures for dust heated by the agn , which show characteristic peaks in their sed between 2040 @xmath1 m @xcite . if the agn is affecting the colors of these sources more than the ones on the main locus then there should be some correlation between the offset from the main locus and an indicator of agn strength such as x - ray luminosity . to quantify the offset from the main locus , we fit the sed of all of the sources in figure [ fig : color - color ] using a modified blackbody ( eq . [ eq : mod_blackbody ] ) with a fixed emissivity of 2.0 to measure the excess or deficiency of observed 500 @xmath1 m emission compared to the model . with the emissivity fixed at 2.0 , there are only two free parameters , the dust temperature and normalization . we fit the sources within a bayesian framework using uniform priors for the logarithm of the normalization and dust temperature and a standard gaussian likelihood function . to sample the posterior probability density function , we use the ` emcee ` package @xcite that implements the affine - invariant ensemble sampler for markov chain monte carlo ( mcmc ) originally proposed by @xcite . for the model fitting , we only use 160 , 250 , and 350 @xmath1 m flux densities . we exclude the 500 @xmath1 m data point because our aim is to compare the expected 500 @xmath1 m emission from the model with the observed one and do not want the fitting influenced by the observed emission . we also exclude the 70 @xmath1 m flux density because it can be dominated by emission from hotter dust heated by young stars in dense star - forming regions or the agn itself @xcite . each sample from the mcmc chain contains values for the parameters of the modified blackbody that are likely given the posterior distribution . from all of these parameters , we calculated 40000 modeled 500 @xmath1 m emission and `` excess '' using the following equation : @xmath64 @xmath4 then represents a fractional excess ( or deficiency ) as compared to the model emission . a deficiency would be indicated by a negative value for @xmath4 . the final excess value associated with the source is then determined as the median of all of the excess values . in figure [ fig : color - color_excess ] we plot the same color - color diagram as in figure [ fig : color - color ] with each point colored by its measured @xmath4 . in general , points with low values of the @xmath3 color show high values of @xmath4 and vice versa for high values of the @xmath3 color . points along the main locus are scattered around @xmath65 . thus , @xmath4 can quantify a source s distance from the main locus and allows us to study possible causes for this excess emission at 500 @xmath1 m . we first measure the correlation between @xmath4 and radio loudness . agn historically have been classified into two groups based on how bright their radio emission is compared to another waveband , usually optical . these groups are `` radio - loud '' and `` radio - quiet '' agn with the former group showing bright radio emission and the latter faint radio emission relative to the optical or x - ray emission @xcite . while originally radio - loud and radio - quiet agn seemed to form a dichotomy , the consensus now seems to be that there is a broad distribution of radio - loudness rather than a bimodality @xcite . further , the original radio loudness parameter , @xmath66 which measured the ratio of the radio to optical luminosity , was shown to underestimate the radio loudness especially for low - luminosity seyfert galaxies @xcite . rather @xmath67 which measures the nuclear radio to x - ray luminosity ratio was confirmed to be a better radio - loudness indicator given x - rays are less affected by obscuration and contamination from the host galaxy . therefore , for the bat agn , we use @xmath68 to measure the radio - loudness with @xmath69 and @xmath70 . for @xmath71 we first cross - correlated the bat agn with the first and nvss databases which provide 1.4 ghz flux densities over all of the northern sky . first flux densities were preferred over nvss due to the much better angular resolution ( 5 `` vs. 45 '' ) . since _ swift_/bat was an all - sky survey , nearly half of the bat agn were not included in either first or nvss . for these southern sources we turned to the sydney university molonglo sky survey ( sumss ; * ? ? ? * ) which surveyed the southern sky at 843 mhz . finally , for the remaining sources missing radio data , we performed a literature search and found 5 ghz fluxes from various other studies @xcite . to convert all flux densities to 1.4 ghz , we assumed a power - law spectrum , @xmath72 , that is typical for synchrotron emission . figure [ fig : excess_vs_rx ] plots @xmath73 against @xmath68 to test our hypothesis that the excess 500 @xmath1 m emission is related to the radio loudness of the agn . in the left panel we plot all of the sources together to show the full range of @xmath73 . indeed , the six agn with the largest values of @xmath73 exhibit high values of radio loudness ( @xmath74 ) . these six agn are hb 890241 + 622 , 2masx j23272195 + 1524375 , 3c 111.0 , 3c 120 , pictor a , and pks 2331 - 240 and all are well known radio - loud agn . they correspond to the six sources in figures [ fig : color - color ] and [ fig : color - color_excess ] that lie in the lower left hand corner . further , the lone hrs galaxy seen in figure [ fig : color - color ] among the six bat agn is the radio galaxy m87 , whose jets and radio activity have been studied extensively . based on this , we prescribe color cutoffs that can easily separate radio - loud agn from radio - quiet agn and normal star - forming galaxies : @xmath60 and @xmath61 ( see dashed lines in figure [ fig : color - color_excess ] ) . while radio - loudness can explain the most extreme values of @xmath73 , it does not explain the more moderate ones . in the right panel of figure [ fig : excess_vs_rx ] , we zoom in on the agn with @xmath75 . visually there does not appear to be any strong correlation between @xmath68 and @xmath73 and the spearman rank correlation coefficient between them is -0.15 , weak and in the opposite sense of what would be expected if synchrotron emission was contaminating the 500 @xmath1 m emission . to explore even further , we analyzed the correlations between @xmath73 and two agn - related indicators , the _ swift_/bat luminosity , @xmath76 , and the 3.4 to 4.6 @xmath1 m flux ratio ( @xmath77 ) . the 3.4 ( @xmath78 ) and 4.6 ( @xmath79 ) @xmath1 m fluxes for the bat agn were obtained from the _ wide - field infrared survey explorer _ ( _ wise _ ; * ? ? ? * ) allwise catalog accessed through the _ nasa / ipac _ infrared science archive ( irsa ) . details of the compilation of _ wise _ fluxes for the bat agn will be available in an upcoming publication ( shimizu et al . in preparation ) . @xcite showed that @xmath76 can be used as a measure of the intrinsic bolometric luminosity of the agn , unaffected by host galaxy contamination or line - of - sight absorption . @xmath77 has been shown to be an effective discriminator between agn - dominated and normal star - forming galaxies that has both high reliability and completeness @xcite in magnitude units for selecting agn . in flux units this changes to @xmath80 . @xcite also show that as the fraction of emission coming from the host galaxy increases @xmath77 increases as well making it a good measure of the relative contribution of the agn to the infrared luminosity . [ fig : excess_vs_agn ] shows the relationships between both @xmath81 ( left panel ) and @xmath77 ( right panel ) with @xmath73 after removing the six radio - loud agn . both parameters display noticeable correlations with @xmath73 with @xmath81 positively correlated and @xmath77 negatively correlated . we calculated spearman rank correlation coefficients finding values of 0.49 and -0.49 respectively . pearson correlation coefficients are 0.30 and -0.50 respectively reflecting the more linear relationship between @xmath73 and @xmath77 than the one between @xmath73 and @xmath81 . all correlations have a probability of a null correlation less than 0.01% . in the right panel we also plot the @xcite cutoff for agn - dominated galaxies where values to the left of this line indicate agn - dominated colors . both panels indicate that the strength of the agn in the host galaxy is possibly having an effect on the spire colors . a stronger agn in relation to the host galaxy is causing deviations from a standard modified blackbody in the form of a small but noticeable 500 @xmath1 m offset . without longer wavelength data , however , its impossible to determine the exact cause of the 500 @xmath1 m excess so we can only speculate . submillimeter excess emission has been observed in a number of objects including dwarf and normal star - forming galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) as well as the small and large magellanic clouds @xcite and even our own milky way @xcite . various explanations have been proposed including the presence of a very cold ( @xmath82 k ) component @xcite , grain coagulation that causes the emissivity to increase for colder temperatures @xcite , fluctuations in the cosmic microwave background @xcite , and an increase in magnetic material in the ism @xcite . while all of these explanations are certainly still possible to explain the excess seen in the bat agn , they lack any direct connection to the strength of the agn . further , a key result from all of the previous work is that the submillimeter excess is more prevalent in very metal - poor galaxies ( @xmath83 ) . all of the bat agn reside in high stellar mass galaxies @xcite and given the mass - metallicity relationship @xcite should also be quite metal rich . rather , we speculate the excess is related to radio emission more closely associated with the agn itself . several studies of the radio properties of agn have revealed a millimeter excess around 100 ghz @xcite that is likely due to either an inverted or flat sed between cm and mm wavelengths . because @xcite found the excess mainly in low luminosity agn similar to sgr a * , they invoked advection dominated accretion flows ( adaf ) that produce compact nuclear jets to explain the inverter or flat seds . however the sample of @xcite was composed of x - ray bright agn including high eddington ratio ( @xmath84 , a measure of the accretion rate relative to the eddington limit ) sources where an adaf is unlikely . @xcite instead use the radio - to - x - ray luminosity ratio to argue that the high - frequency radio emission originates near the x - ray corona of the accretion disk given the ratio s similarity to that found for stellar coronal mass ejections ( e.g * ? ? ? * ) as well as the compact nature of the radio emission . magnetic activity around the accretion disk in the core of the agn would then be responsible for the excess and if magnetic activity increases with @xmath84 , this could explain the relationship seen with @xmath81 as well as @xmath77 . this strengthens the need for a more comprehensive survey of agn in the mm wavelength range as it could clearly reveal interesting physics possibly occurring near the accretion disk . we have produced the _ herschel_/spire maps for 313 agn selected from the _ swift_/bat 58 month catalog in three wavebands : 250 , 350 , and 500 @xmath1 m . combined with the pacs photometry from @xcite , the spire flux densities presented in this paper form the complete fir seds for a large , nearby , and relatively unbiased sample of agn . we used two methods for measuring the flux densities : timeline fitting for point sources and aperture photometry for extended and undetected sources . we summarize below the results of our statistical analysis and comparison to the _ herschel _ reference survey sample of normal star - forming galaxies . * sy 2s are detected at a higher rate than sy 1s , and after accounting for upper limits , sy 2 s have slightly higher spire luminosities than sy 1 s . however the effect is small and indicates that on average , the global fir properties of agn are independent of orientation . * using a partial correlation survival analysis to account for the luminosity - distance effect and upper limits , we find all of the _ herschel _ luminosities are correlated with each other suggesting the process ( or processes ) producing the emission from 70500 @xmath1 m is connected . luminosities with the smallest wavelength difference ( i.e. 160 and 250 @xmath1 m ) are much more correlated than pairs further apart ( i.e. 70 and 500 @xmath1 m ) , in agreement with different temperature components associated with different wavebands . while this could point to the agn affecting the shorter wavebands more than the longer ones and increasing the scatter , it can also be explained by an increased contribution from older stellar populations to the emission at longer wavelengths . * none of the spire luminosities are well correlated with the 14195 kev luminosity , a proxy for the bolometric agn luminosity . the agn , in general , is unlikely to be strongly affecting either the 250 , 350 , or 500 @xmath1 m emission , however sy 1s do show a very weak correlation at 250 and 350 @xmath1 m . removing ct sources does not improve the correlation for sy 2 s . it remains to be seen what the exact explanation is for the difference in correlations between sy 1s and sy 2s but possible explanations include a direct link between star - formation and agn luminosity that is evident only at high luminosity or increased contamination by the agn . * we compared the spire colors , @xmath2 and @xmath3 , with the colors of the hrs galaxies . the bat agn have statistically similar spire color distributions as the high stellar mass ( @xmath85 m@xmath48 ) hrs galaxies . this further emphasizes that on average , the fir emission of agn host galaxies is likely produced by cold dust in the ism heated by stellar radiation just as in normal star - forming galaxies without an agn . * we did find anomalous colors for 6 bat agn with @xmath60 and @xmath61 . the fir seds for these agn are dominated by synchrotron emission from a radio jet rather than thermally heated dust . * another group of agn with less anomalous colors but still removed from the main locus were analyzed by fitting the seds with a modified blackbody and calculating a 500 @xmath1 m excess . we found the 500 @xmath1 m excess is not related to radio loudness , but is well correlated with the 14195 kev luminosity and @xmath77 ( 3.4/4.6 @xmath1 m ) color from _ wise_. we speculate this is possibly related to the millimeter excess emission recently seen in agn caused by coronal emission above the accretion disk . future work will focus on combining the photometry from @xcite and this paper as well as archival data to perform detailed sed modeling to investigate the local starburst - agn connection and the agn contribution to the fir . we thank the anonymous referee whose comments and suggestions contributed and improved the quality of this paper . the herschel spacecraft was designed , built , tested , and launched under a contract to esa managed by the herschel / planck project team by an industrial consortium under the overall responsibility of the prime contractor thales alenia space ( cannes ) , and including astrium ( friedrichshafen ) responsible for the payload module and for system testing at spacecraft level , thales alenia space ( turin ) responsible for the service module , and astrium ( toulouse ) responsible for the telescope , with in excess of a hundred subcontractors . spire has been developed by a consortium of institutes led by cardiff university ( uk ) and including univ . lethbridge ( canada ) ; naoc ( china ) ; cea , lam ( france ) ; ifsi , univ . padua ( italy ) ; iac ( spain ) ; stockholm observatory ( sweden ) ; imperial college london , ral , ucl - mssl , ukatc , univ . sussex ( uk ) ; and caltech , jpl , nhsc , univ . colorado ( usa ) . this development has been supported by national funding agencies : csa ( canada ) ; naoc ( china ) ; cea , cnes , cnrs ( france ) ; asi ( italy ) ; mcinn ( spain ) ; snsb ( sweden ) ; stfc , uksa ( uk ) ; and nasa ( usa ) . this research has made use of the nasa / ipac extragalactic database ( ned ) , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . 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 .

we present far - infrared ( fir ) and submillimeter photometry from the _ herschel _ _ space observatory s spectral and photometric imaging receiver _ ( spire ) for 313 nearby ( @xmath0 ) active galactic nuclei ( agn ) . we selected agn from the 58 month _ swift _ burst alert telescope ( bat ) catalog , the result of an all - sky survey in the 14195 kev energy band , allowing for a reduction in agn selection effects due to obscuration and host galaxy contamination . we find 46% ( 143/313 ) of our sample is detected at all three wavebands and combined with our pacs observations represents the most complete fir spectral energy distributions of local , moderate luminosity agn . we find no correlation between the 250 , 350 , and 500 @xmath1 m luminosities with 14195 kev luminosity , indicating the bulk of the fir emission is not related to the agn . however , seyfert 1s do show a very weak correlation with x - ray luminosity compared to seyfert 2s and we discuss possible explanations . we compare the spire colors ( @xmath2 and @xmath3 ) to a sample of normal star - forming galaxies , finding the two samples are statistically similar , especially after matching in stellar mass . but a color - color plot reveals a fraction of the _ herschel_-bat agn are displaced from the normal star - forming galaxies due to excess 500 @xmath1 m emission ( @xmath4 ) . our analysis shows @xmath4 is strongly correlated with the 14195 kev luminosity and 3.4/4.6 @xmath1 m flux ratio , evidence the excess is related to the agn . we speculate these sources are experiencing millimeter excess emission originating in the corona of the accretion disk . [ firstpage ] galaxies : active galaxies : seyfert galaxies : photometry infrared : galaxies methods : data analysis

arxiv

* samaniego : * it s a real pleasure to be back at florida state , myles . i spent my first postdoctoral year in the statistics department here , and i have many fond memories . though we ve been friends for over 35 years , there are many details of your life and career that i m looking forward to hearing more about . let s start somewhere near the beginning . i know that you began your college career at carnegie mellon as an engineering major . can you tell me how you got interested in statistics ? * hollander : * i came to carnegie mellon , it wascarnegie tech when i entered in 1957 , with the aim of becoming a metallurgical engineer , but all the engineering students took more or less the same curriculum , including calculus , chemistry , english , history of western civilization . as the year progressed i found i liked math and chemistry the best so near the end of the year , i went to see the heads of metallurgy and math . the metallurgy chair was informative but laid back and said it was my decision . the math chair , david moscovitz , was much more enthusiastic . he said , `` hollander , we want you . '' well , i was only 17 , impressionable , and i liked being wanted so i became a math major . i did nt encounter a formal course in statistics until my junior year . that year , morrie degroot ( who had come to carnegie the same year i did1957he with a ph.d . from the university of chicago ) taught a course that i really enjoyed . it was based on mood s `` introduction to the theory of statistics . '' degroot wrote some encouraging comments on a couple of my exams and i began thinking i might become a statistician . then in my senior year , i took two more excellent statistics courses from ed olds . olds at that point was a senior faculty member who had actually done some work on rank correlation but was , i think , more known for his consulting with nearby industry , westinghouse , u.s . steel and others . in the afternoon he taught a statistical theory course from cramr s `` mathematical methods in statistics . '' in the evening he taught a course on quality control . i liked the juxtaposition of beautiful theory that could also be useful in an important applied context . i would say those three courses , those two teachers , sealed the deal for me . carnegie wanted me to stay on and do my ph.d . there in the math department but the lure of california , palo alto , stanford s statistics department , was too great , so i headed west . * samaniego : * let me ask a quick question about the books you mentioned . cramr is even today thought of as a very high - level book mathematically . it s surprising that it was used in an undergraduate course . * hollander : * in retrospect it is surprising but olds taught a beautiful course and it helped me later on in my studies . i still have the book in my library and i look at it from time to time . * samaniego : * i see it and it s clearly well worn . * samaniego : * you were attracted to math and science in your early years . was that your main focus in high school ? * hollander : * i was on an academic track in high school and studied mostly math and science . i attended an excellent public high school , erasmus hall , in the heart of the flatbush avenue section of brooklyn . it was a three - block walk from my apartment house . naturally , i also took other types of courses , english , social studies , history , mechanical drawing , and spanish . math was my best subject and that seemed fortunate for a kid who wanted to be an engineer . * samaniego : * how did a kid from brooklyn end up choosing to go to a private college in pittsburgh ? i suppose that once the dodgers left town , you felt free to leave , too . * hollander : * i could have stayed in brooklyn and gone to brooklyn college , thereby saving a lot of money . i could have stayed in new york state and gone to rensselaer polytechnic institute , where several of my close friends chose to go . i wanted something different , and pittsburgh , despite its reputation then as a smoggy city , due to the steel industry , appealed to me . that the dodgers were leaving brooklyn the same time i was ( 1957 was their last season in ebbets field and also my senior year of high school ) did nt affect my thinking . i did get to see them play a few times at forbes field in pittsburgh during my years at carnegie . forbes field was actually a short walk from carnegie and you could enter the ball game for free after the seventh inning . * samaniego : * tell me about your parents and their influence on your academic development . * hollander : * my mom and dad were committed to education , wanted me to go to college , and worked hard to make it happen . my dad had one year of college . he was at brooklyn polytechnic institute in the 19271928 academic year majoring in civil engineering . then the following year the depression hit and my father , as the oldest of three siblings , went to work to help support his family . he never got back to college . my dad went on to open a sequence of haberdashery stores , mostly selling pants and shirts , in the boroughs of manhattan , queens and brooklyn . my mother did not have college training but worked as a bookkeeper , mostly for a firm that managed parking lots throughout the city . they both left early in the morning and came back at dinner time . i was a latch - key kid before the term became popular . i lived on the first floor of an apartment house on linden boulevard , directly across the street from a branch of the brooklyn public library . the library was a good place to study and in my senior year i would thumb through books on engineering . civil , mechanical , electrical , aeronautical were the popular areas but metallurgy appealed to me : the chemistry labs , blast furnaces , protective masks , etc . i looked for schools that offered it and i also thought that by applying to a less popular field , i would increase my chances of being accepted , and getting a scholarship . * samaniego : * i know you had scholarship support from the ladish forging company while at carnegie mellon , and also worked for them in the summers . what was the work like ? did it play a role in your decision to go to graduate school ? * hollander : * when i switched from metallurgy to math at the end of my freshman year , i contacted the ladish forging company . they said that was fine , they would still support me , which i obviously appreciated . then in the summer of my junior and senior years i worked for them in cudahy , wisconsin . i estimated the costs of drop forgings using the costs of materials , the geometrical shapes of the parts , labor costs . i did some of that each summer and also wrote some programs in basic for the ibm 1401 . my supervisor told me on the parts i estimated for which the company was low bidder , the company lost money . i was biased low . but he said it was fine because the workers needed the work . ladish actually wanted me to work for them after graduation but i wanted to study statistics and my heart was set on stanford . ladish was nt my last position in the private sector . in the summers of 19621963 , after my first and second years of grad school , i worked for the sylvania reconnaissance laboratories in mountain view . there i did get to use some of the material i was learning at stanford , particularly markov chains and stochastic processes . in the summer of my junior year , i had an internship at the presbyterian medical center in san francisco . gerry chase and i rode the southern pacific railroad from palo alto to san francisco two or three times a week and worked on medical data . nevertheless , even though i liked these summer jobs , as my years in graduate school increased my inclination to join the private sector decreased . * samaniego : * your graduate study at stanford heavily impacted your career choices and the statistical directions you have taken . tell me about your cohort of students at stanford . * hollander : * it was a terrifically talented cohort . brad efron , howie taylor , joe eaton , carl morris , grace wahba , barry arnold , jim press , paul holland , jean donio , galen shorack , gerry chase , and many more . i should really name them all . we were all excited about the material . we wanted to learn what our professors taught and we wanted to learn how to do it ourselves . we were very cooperative and friendly among ourselves . i have many memories , howie taylor working on ( and talking about ) a probability problem at the blackboard in our office in cedar hall , carl morris and i talking about pitman efficiency at a blackboard in an empty classroom in sequoia hall and carl shedding light on what was going on , barry arnold and i discussing a mathematical statistics problem in cedar many , many such instances . brad efron was a senior student to our group who interacted with us and helped us in many ways , including discussing geometrical interpretations of theorems . we typically took the qualifying exams in the middle of our third year . to help us prepare , we would each choose a topic and write a 1012-page focused summary with solutions to problems , theorems , key ideas . i did one on nonparametrics , howie taylor did one on advanced probability , and so forth . we put the summaries together , made copies and passed them amongst ourselves . when we took our orals we were pumped , prepared , and , to the extent that one can be for such a momentous test , we were confident . also , of course , we were nervous . my exam committee was lincoln moses , rupert miller , charles stein and gerry lieberman and i see them sitting there today just as i am looking at you and i remember most of the questions to this day . * samaniego : * give me an example of a question that was asked . * hollander : * well , lincoln moses asked about nonparametric tests for dispersion and i decided to mention one of his rank tests . then gerry lieberman turned to lincoln and said in mock surprise , `` lincoln , you have a test ? '' they were close friends so gerry could tease him in this way but lincoln was nt particularly happy about my answer and then he threw a tough question at me about the asymptotic distribution of the kolmogorov smirnov statistic . charles stein asked me about decision theory and i was ready for that . i went to the blackboard and outlined the framework of a decision theory problem just like he did at the beginning of many of his lectures . * samaniego : * he did nt ask any testy inadmissibility questions , did he ? * hollander : * i had covered the blackboard and used a lot of time but he did ask about the relationship between admissibility and invariance . it had been covered in his course so i was ready for it . * samaniego : * which faculty members at stanford had the greatest influence on you , personally and professionally ? * hollander : * lincoln ellsworth moses had the greatest influence . i was lucky at the start because my first ta assignment in fall quarter , 1961 , was to be a grader in the elementary decision theory course he was teaching out of chernoff and moses . he gave the main lectures and five or six tas graded papers and met with sections to go over homework . i got to know lincoln through this activity and he also encouraged me to attend the biostatistics seminar that he and rupert miller were giving in the medical school . i would also be invited to his home in los trancos woods and got to know his wife jean and their children . i was close to him throughout and after he married mary lou coale , glee and i remained very close with them . beginning in the fall of 1963 , lincoln taught a two - quarter course on nonparametric statistics . it was a beautiful contemporary sequence and there was lots of nonparametric research in that period , particularly by erich lehmann and joe hodges at berkeley , lincoln , rupert miller , vernon johns at stanford . lincoln named me the ta for that course even though i was taking it at the same time . there i was , grading the papers of my really talented fellow students , like joe eaton and so i had to be good . i was determined to excel , to be one of the best if not the best in the class . later , motivated by this course , i wrote a thesis on nonparametrics under lincoln s direction . lincoln became my role model , the statistician i most admired and tried to emulate . he showed me how to be a professional , the joy of statistics , and the great pleasure of being a university professor . in my career i have tried to do for my students what lincoln did for me . * samaniego : * what you say about lincoln moses rings very true . from my own few interactions with him , and from things i ve heard about him over the years , he was both a fine teacher and scholar and a true gentleman . tell me about your interactions with other stanford faculty . * hollander : * i was also strongly influenced by other professors from whom i took courses . rupert miller via the biostatistics seminar , ingram olkin through the problems seminar he co - taught with shanti gupta , who was visiting in 1961 ( they started out assigning problems in cramr s book and that was a break for me as a beginning student because i had seen most of the problems at carnegie ) . ingram also taught multivariate analysis which i also took . i took charles stein s decision theory sequence and manny parzen s time series sequence . kai - lai chung taught the advanced probability sequence . they were all dedicated to their subjects , made them come alive , each had his own style , and each was at the top of his game . then on two sabbaticals at stanford , working in the medical center , i became friendly with bill and jan brown and reinforced my friendship with rupert and barbara miller . bill and jan became the godparents to our children . one special bond that existed between rupert and barbara and glee and me : jennifer ann miller and layne q hollander were delivered the same day , october 29 , 1964 , at stanford hospital , and glee and barbara shared the same hospital room for three or four days . over the years , i ve grown closer to ingram through the various international conferences on reliability that you and i have attended and to manny through his work with the nonparametrics section of asa . * samaniego : * all of the people that you ve mentioned have written very good books in probability or statistics . i m wondering , since you ve co - authored three books yourself , whether these people and the way they wrote influenced you ? * hollander : * i did put a high premium on clear writing in the three books i ve co - authored . i think the person who influenced me the most in that regard was frank proschan , who insisted on clear writing . when i took the course on stochastic processes , it was based on manny s notes ( his book was not yet out ) and it was taught by don gaver . when i took ingram s multivariate analysis course , he used his notes and ted anderson s book . i used rupert s book on multiple comparisons for research , but i did nt take that subject as a course . kai - lai used the notes that would become his beautiful book on advanced probability . certainly manny , ingram , rupert and kai - lai wrote in clear , captivating ways . * samaniego : * you met your wife glee at stanford and the two of you were married in the memorial church on the stanford campus . many of your friends feel that your bringing glee into the extended statistics community is your greatest contribution to the field ! tell me how you met glee and how you managed to persuade her to marry you . ( laughs ) * hollander : * i was sitting in my office on the second floor of ventura hall at stanford in october , 1961 . it s a spacious office and even though it had four desks , only two students would come regularly , jon kettenring and me . ( a year later pat suppes would take over that office . ) i was working on a hard problem and i paused to look out the window . i saw a young girl walking briskly , determined , in high heels , with blond hair , bouncing along with remarkable energy ( past ventura , maybe to the computer center ) . a california girl ! clearly i could never even approach a person like that . she passed out of my view and i went back to my homework , probably a waiting time problem in stochastic processes . the expected waiting time for me to approach the girl i had just seen was no doubt infinite . eight months later , in june , 1962 , my friend heinz , an engineering student from germany , and i decided to go on a double date . we decided to meet on a friday night at el rancho , a restaurant on el camino real , in palo alto . in addition to dinner , el rancho also had a dance floor and a lively band . when i arrived i realized that heinz s date was unmistakably the girl i had seen when gazing from my ventura hall office in the fall glee . the evening was going well and i was totally enthralled by glee , her brightness , her wit , her energy , her enthusiasm , her bounce . after about an hour the band played `` it s cherry pink and apple blossom white''a cha cha . i asked my date to dance but she said she did nt cha cha . i mustered the courage to ask glee . she said , `` i ll try . '' of course she was and still is a great dancer and i was on cloud nine . i thought i d made a good impression . a week later i called her on the phone and said , `` hi glee , it s myles hollander . '' she said , `` who ? '' obviously i did not impress her as much as she had impressed me . clearly i needed to go into high gear . i took her sailing on san francisco bay . i took her horseback riding in the foothills behind stanford . i took her skiing at heavenly valley . eventually my persistence triumphed . we hit it off over a period of about a year , and got married at stanford memorial church on the stanford campus in august , 1963 . we went on to have two fine sons , layne q and bart q , who , with their wives , tracy and catherine , also gave us five wonderful grandchildren taylor , connor , andrew , robert and caroline . glee earned her ph.d . at fsu in an excellent clinical psychology program and worked in private practice , and also at florida state hospital in chattahoochee . i like to say it all started with the cha cha and we re still dancing after all these years ! * samaniego : * on the statistical front , you published a major portion of your thesis in a pair of _ annals _ papers . what was the main focus of this work ? * hollander : * my thesis was devoted to rank tests for ordered alternatives in the two - way layout . lincoln moses , in his nonparametric sequence in the third year of my graduate work , had covered ordered alternatives in the one - way layout and that suggested to me some ideas for randomized blocks . i proposed a test based on a sum of overlapping signed rank statistics that is not strictly distribution - free but can be made asymptotically distribution - free . kjell doksum at berkeley was also working on closely related problems at the same time and in the end our two papers were published adjacently in the 1967 _ annals _ ( doksum , @xcite ; hollander , @xcite ) . in my thesis i also pointed out a certain multiple comparison procedure , thought by peter nemenyi ( nemenyi , @xcite ) to be distribution - free , was not , but could be made asymptotically distribution - free . i published the asymptotically distribution - free multiple comparison procedure in the 1966 _ annals _ ( hollander , @xcite ) . * samaniego : * you ve written quite a few papers on classical nonparametric testing problems . give us an idea of the range of problems you have worked on in this area . * hollander : * in my early years at fsu i wrote nonparametric papers on bivariate symmetry , regression , uncorrelated nonparametric statistics , and did a little more on ordered alternatives . i also worked with my first ph.d . student , ron randles , on a paper that was decision - theoretic rather than nonparametric . we developed @xmath0-minimax procedures for selection procedures and it was published in the 1971 _ annals _ ( randles and hollander , @xcite ) . ron took my class in nonparametrics and even though his thesis was not nonparametric in character , he did excellent work , went on to be a leader in nonparametrics and set a very high bar for my subsequent ph.d . students . thus in the beginning i was working on my own and with students . that was the way the senior leaders in the department , ralph bradley and richard savage , wanted it . work on your own , prove your mettle , and move away from your thesis topic . later on , when i began to collaborate with frank proschan and jayaram sethuraman , two great statisticians , my scope of topics vastly increased and my research got better ! whenever i received an offer or feeler from another place , i had to ponder whether i could find and establish working relationships with such superb collaborators at the next stop . i alwaysdoubted it . * samaniego : * your research over the years has been distinctly nonparametric , including , of course , interesting and important contributions to bayesian nonparametrics . you and your doctoral student , ramesh korwar , were the first to develop inference procedures for the hyperparameter of ferguson s dirichlet process , establishing the foundations for an empirical bayes treatment of nonparametric estimation . i see that it s an interest you ve sustained up to the present time . how did you get interested in this latter problem area ? * hollander : * my interest in the dirichlet process arose from tom ferguson s seminal paper ( ferguson , @xcite ) . that was the principal motivation . i had obtained a preprint before its publication . i had read some earlier papers at stanford on bayesian nonparametrics but ferguson s paper was the most tractable , the most promising . i ca nt remember the exact timing but i went to a bayesian nonparametric conference at ohio state where tom was the principal speaker . he was also aware of some of the results by ramesh korwar and me and mentioned them in his lectures . his wonderful lectures got me further fired up and i went on to do more bayesian nonparametrics with ramesh , and then later with two more of my ph.d . students , greg campbell and bob hannum , and more recently with sethu ( campbell and hollander , @xcite ; hannum , hollander and langberg , @xcite ; hannum and hollander , @xcite ; sethuraman and hollander , @xcite ) . * samaniego : * which ideas or results in your bayesian nonparametric papers seem to have had the most impact ? * hollander : * ramesh korwar and i had several interesting results in our 1973 paper in the _ annals of probability _ ( korwar and hollander , @xcite ) . we showed that when the parameter @xmath1 of the dirichlet process is nonatomic and @xmath2-additive , @xmath3 can be estimated from a sample from the process . the estimator we devised is @xmath4 , where @xmath5 is the number of distinct observations in the sample . we proved that estimator converges almost surely to @xmath6 where @xmath1 is a finite nonnull measure on a space @xmath7 that comes equipped with a @xmath8-field of subsets . we also showed in the nonatomic and @xmath8-additive case , that given @xmath5 , the @xmath5 distinct sample values are i.i.d . with distribution @xmath9 . this result has been used by others . for example , in an _ annals _ paper doksum and lo ( doksum and lo , @xcite ) considered bayes procedures when @xmath10 is chosen by a dirichlet prior and used the result to study consistency properties of posterior distributions . another result that ramesh and i had in that 1973 paper gave the joint distribution of the indicators that tell if the @xmath11th observation is distinct from the previous @xmath12 . the indicators are independent , but not identically distributed , bernoulli random variables . diaconis and freedman ( diaconis and freedman , @xcite ) used this result in their study of inconsistent bayes estimators of location . in our 1976 _ annals _ paper ( korwar and hollander , @xcite ) ramesh and i used the dirichlet process to define a sequence of empirical bayes estimators of a distribution function . one interesting consequence of that paper was a result reminiscent of the famous james stein result on the inadmissibility of multivariate @xmath13 when the dimension is @xmath14 . ramesh and i showed that if there are at least three distribution functions to be estimated , one could do better than estimating each distribution by its sample distribution . in a 1981 _ annals of probability _ paper ( hannum , hollander and langberg , @xcite ) bob hannum , naftali langberg , and i studied the distribution of a random functional @xmath15 of a dirichlet process . we related the cumulative distribution of that functional evaluated at @xmath16 , say , to the distributions of random variables @xmath17 and we obtained the characteristic function of @xmath17 . it has been surprising and gratifying to see some recent uses of this result . for example , it is used ( cifarelli and melilli , @xcite ) to study the distribution of the variance functional . the result is also used ( regazzini , guglielmi and di nunno , @xcite ) to study the probability distribution of the variance of a dirichlet measure and the probability distribution of the mean of a dirichlet measure . thus the result is getting a little play in the italian school . * samaniego : * you ve been at florida state for 42 years ! i d like to ask you about your extensive and fruitful collaborations with some of your colleagues here . tell me about your first joint paper with frank proschan . it was , i believe , one of the first papers in which tests were developed to detect particular nonparametric ( nbu ) alternatives to the exponential distribution . * hollander : * frank came here in 1971 from the boeing research labs . he was very open , very dedicated to his research . our offices were close and we became friends . one day he walked into my office and said , `` let s write a paper . '' i said , `` great . '' i was excited he asked . his main area was reliability and mine was nonparametric statistics , so we aimed to work in the intersection , namely nonparametric methods in reliability . the first paper we wrote covered our nbu ( new better than used ) test ( hollander and proschan , @xcite ) . the test is based on a @xmath18-statistic , partially reminiscent of the wilcoxon whitney statistic . we enjoyed working on it and there was a mild surprise . in calculating the probability that the statistic assumes its maximum value , the fibonacci sequence pops up . the sequence had not arisen in frank s longer research experience , nor in my shorter one . it is nice to have a mild connection with a famous pre - renaissance mathematician . i believe the paper stimulated more research in testing and estimation for the various nonparametric classes arising naturally in reliability , including more research avenues for us . * samaniego : * in a subsequent paper , you and frank discovered an interesting new context in which the total - time - on - test statistic arose . i m sure that was a pleasant surprise . * hollander : * frank and i wrote a testing paper on mean residual life ( hollander and proschan , @xcite ) that was published in _ biometrika_. we considered the decreasing mean residual life class , the new better than used in expectation class , and their duals . we defined measures of dmrlness and nbueness based on @xmath10 , plugged in the empirical for @xmath10 , and used those plug - in statistics as test statistics , standardized to make them scale - invariant . in the nbue case , we obtained the total - time - on - test statistic . up to that time it had been viewed as a test of exponentiality versus ifr or ifra alternatives . we showed its consistency class contained the larger set of nbue distributions , thus broadening its interpretation and applicability . large nonparametric classes of life distributions captured our attention for awhile . for example , we co - directed our student frank guess on a project where we defined new classes relating to a trend change in mean residual life . in our 1986 _ annals _ paper ( guess , hollander and proschan , @xcite ) we considered the case where the change point is known . later ( kochar , loader and hawkins , @xcite ) procedures were given for the situation where the change point is unknown . * samaniego : * on a personal level , what was it like to collaborate with frank proschan ? give us a feeling for his sense of humor , his work ethic , the `` reliability club '' and his overall influence on you . * hollander : * frank , as you know , had a deadpan sense of humor . he would often remind me of the comedian fred allen who was very funny but never cracked a smile , never laughed at his own jokes . when he gave a lecture frank would adroitly use transparencies , and there was always a parallel processing taking place , the material in the lecture , and humorous asides . he was dedicated to his research . he would come to the office very early , work for a few hours , go to the university pool for a swim , go home for lunch , then come back and work again . we both would come in on saturday mornings , talk about what we wanted to show , go back to our offices , try to get a result , write up the progress , then put a copy in the other person s mailbox . this went back and forth . some mornings we would come in and do this without talking face to face . the results would accumulate , and then we would have a paper . later , frank started the reliability club which met on saturday mornings to present and discuss topics on reliability . many students , several faculty and visitors would attend , and it would lead to dissertations , joint work , research grants , more papers . i had the habit of working some weekends ( including some sunday nights ; glee and i lived very close to campus then , about an 8-minute drive ) before frank arrived but frank solidified it and showed me i was not crazy doing it ( or else we were both crazy ) . without trying or fully realizing it , frank s style and work ethic became a part of mine . * samaniego : * you ve nicely integrated the parallel processing of material and humor into your own presentation strategy . * hollander : * frank , i ve always tried to be funny . it s both a strength and a weakness . i like to make people laugh but every once in a while it s not the time to be funny . over the years i ve become better at resisting the temptation to try to say something funny . but i still like to make witty remarks . i like to present to people the notion that statisticians have pizazz . * samaniego : * i ve found myself that in teaching our subject , a little bit of well - timed humor not the stand - up comedy type but the things that actually have something to do with the material we are talking about helps people stay aboard ; most people listen and enjoy it . * hollander : * at this point your advice on how to teach gets much higher marks than mine because you have just won an outstanding award at the university of california , davis . i wo nt even mention the figure here ; otherwise people will come by your house at night and break in . * samaniego : * well , myles , i ve always enjoyed your presentation style and have probably stolen more than i care to acknowledge from the talks i ve heard you give . * samaniego : * you ve written a good many papers with jayaram sethuraman . what would you consider to be the highlights of that work ? * hollander : * even before sethu and i worked together , i would go to his office for consulting . he is a brilliant statistician and he can often point you in a direction that will help , or lead you to a breakthrough when you are stuck . his entire career he has been doing that for all those wise enough to seek his assistance . my first paper with sethu is also joint with frank . it is the dt ( decreasing in transposition ) paper ( hollander , proschan and sethuraman , @xcite ) and is certainly a highlight . it is a paper on stochastic comparisons which yields many monotonicity results . among the applications in that paper were power inequalities for many rank tests . al marshall and ingram olkin later changed the dt term to ai ( arrangements increasing ) . in a later paper ( hollander and sethuraman , @xcite ) , sethu and i gave a solution to a problem posed to us by sir maurice kendall during his short visit to tallahassee in 1976 . it was `` how should one test if two groups of judges , each giving a complete ranking to a set of @xmath19 objects , agree , that is , have a common opinion ? '' we proposed a conditionally distribution - free test using the wald wolfowitz statistic . * samaniego : * tell me about your more recent work with sethu . * hollander : * sethu and i have , on and off , been working on repair models in reliability for the last 15 years . our interest was sparked by your groundbreaking paper with lyn whitaker ( whitaker and samaniego , @xcite ) in which you developed what is now called the whitaker samaniego estimator of the distribution @xmath10 of the time to first repair in imperfect repair models . with brett presnell , we considered the problem in a counting process framework ( hollander , presnell and sethuraman , @xcite ) and also developed a simultaneous confidence band for @xmath10 as well as a wilcoxon - type two - sample test in the repair context . many other important parameters , such as the expected time between repairs , depend on @xmath10 and the nature of the repair process , so the problem of estimating @xmath10 is important . five years later , with cris dorado ( dorado , hollander and sethuraman , @xcite ) we proposed a very general repair model that contains most of the models in the literature . we also introduced the notion of life supplements or boosts , so not only could the repairman move the effective age of the system to a point better than , say , minimal repair , he could also boost the residual life . recently we finished a paper on bayesian methods for repair models ( sethuraman and hollander , @xcite ) . for example , if you put a dirichlet prior on @xmath10 in , say , the imperfect repair model , and take two observations , the posterior distribution of @xmath10 is no longer dirichlet . thus there is , for these complicated repair processes which induce dependencies , a need for a broader class of priors which are conjugate . we introduced partition - based priors and showed they form a conjugate class . beyond repair models , we believe this new method for putting priors on distributions has potential in many other areas . * samaniego : * one of my favorites among your papers is a jasa paper you wrote with chen and langberg on the fixed - sample - size properties of the kaplan meier estimator . it was based on a simple but very clever idea . can you describe that work and how it came about ? * hollander : * i was interested in the kme s exact bias and its exact variance . brad efron ( efron , @xcite ) , in his fundamental article on the two - sample problem for censored data , had given bounds on the bias . proportional hazards provided a clean way to get exact results . earlier , allen ( allen , @xcite ) proved that when the cumulative hazard function of the censoring distribution is proportional to that of the survival distribution , the variables @xmath20 and the indicator function @xmath21 are independent , where @xmath22 is the time to failure , @xmath23 is the time to censorship . in his 1967 paper , efron used this result for obtaining efficiencies for his generalized wilcoxon statistic in the case when the censoring and survival distributions are exponential , and he thanked jayaram sethuraman for bringing the result to his attention . in the kme setting we ( chen , hollander and langberg , @xcite ) obtained an exact expression for moments of the kme by conditioning on @xmath24 and using allen s result . getting exact results in this setting was a natural consequence of my interest in rank order probabilities . erich lehmann really planted the seed with his famous work on the power of rank tests ( lehmann , @xcite ) where he obtained exact powers against what are now called lehmann alternatives . my natural tendency is to first try hard to get exact results , then move to asymptotics . * samaniego : * you ve done extensive joint work with some of your doctoral students . perhaps your collaboration with edsel pea is the most varied and most productive . tell me a little about that work . * hollander : * edsel is an amazingly dynamic and energetic researcher . he loves to do research and his enthusiasm is infectious . he is also very talented . we have worked on a broad range of problems . we started ( hollander and pea , @xcite ) with obtaining exact conditional randomization distributions for various tests used to compare treatments in clinical trials that use restricted treatment assignment rules , such as the biased coin design . we have also worked on confidence bands and goodness - of - fit tests in censored data settings . for example , in our 1992 jasa paper we ( hollander and pea , @xcite ) defined agoodness - of - fit test for randomly censored data that reduces to pearson s classical test when there is no censoring . we considered the simple null hypothesis and later li and doss ( li and doss , @xcite ) extended it to the composite case . thus , although not ideal , there are secondary gains in not solving the more general problem straight out . you inspire others and your paper gets cited . edsel and i have also worked on interesting reliability models . for example , in hollander and pea ( @xcite ) we used a markovian model to describe and study system reliability for systems or patients subject to varying stresses . as some parts fail , more stresses or loads may be put on the still - functioning parts . we use the failure history to incorporate the changing degrees of loads and stresses on the components . shortly after that ( hollander and pea , @xcite ) we addressed the problem about how a subsystem s performance in one environment can be used to predict its performance in another environment . another idea that may attract some interest is our class of models proposed in 2004 in the _ mathematical reliability _ volume ( pea and hollander , @xcite ) . we introduced a general class of models for recurrent events . the class includes many models that have been proposed in reliability and survival analysis . our model simultaneously incorporates effects of interventions after each event occurrence , effects of covariates , the impact of event recurrences on the unit , and the effect of unobserved random effects ( frailties ) . edsel and his colleagues and students have been studying asymptotic properties of the estimators and also applying them to various data sets . * samaniego : * tell me about your three books . * hollander : * the nonparametric books with doug , the first and second editions , were very successful ( hollander and wolfe , @xcite , @xcite ) . one important feature of these books are the real examples from diverse fields . it helped us broaden our audience beyond statisticians . doug and i also taught a short course for about nine years , mid70s to 80 s , at the george washington university continuing engineering education center . the audience at those courses consisted mainly of people in government and industry so again , in a way , we were bringing the nonparametric ideas and techniques to a different audience . wiley has sought a third edition , but doug and i have not yet committed to it . bill brown and i began writing the medical statistics book ( brown and hollander , @xcite ) in 1972 when i was on sabbatical at stanford . we also featured real examples and it was adopted at many medical schools . i also used it for many years at fsu for a basic course on statistics in the natural sciences . wiley always wanted a second edition , but bill and i never got around to it . wiley is now going to publish the original book as a paperback in its wiley classics library series . the book _ the statistical exorcist _ with frank proschan ( hollander and proschan , @xcite ) was great fun to write . the book consisted of vignettes that treated a variety of problems . we wrote in a way to explain to the readers what statistics does , rather than give a formulaic approach on how to do statistics . in fact , we did nt use any mathematical formulas or symbols . one interesting feature is the cartoons , about half of which were drawn by frank and pudge s daughter virginia and half drawn by glee . frank and i described the scenes and supplied the captions and ginny ( virginia ) and glee did the drawings . we also opened the vignettes with epigraphs , relating to statistics , from novels . some of the epigraphs are real and some were created by frank and me . in an appendix we informed the reader which ones were from our imagination . for a text , however , students found it difficult without a few formulas upon which to hang their hats , for example , when to multiply probabilities , when to add , and so on . marcell dekker also wanted a second edition and it is not beyond the realm of possibility . this semester i ve been teaching an advanced topics course . the material was an eclectic mixture of survival analysis and reliability theory where i focused on some of the parallels between the two subjects . the course title is `` nonparametric methods in reliability and survival analysis . '' whenever i look at the syllabus , it occurs to me that the material would make a good monograph . the problem is that most books on reliability are not big sellers although some are beautiful and informative . when i write , i do it not so much to make a few extra dollars , but to be read and thus a potentially large audience is the draw . * samaniego : * did any specific examples in _ the statistical exorcist _ come out of your joint research with frank ? * hollander : * some of the subject matter was motivated by the joint research . for example , we had vignettes on reliability which are unusual in an elementary book . we also had vignettes on nonparametric statistics , so the vignettes were influenced to some extent by our favorite subjects . * samaniego : * myles , _ the statistical exorcist _ is , i would say , unique in the field as an introduction to statistical thinking . the book is distinctive in a variety of ways including its general content , the humor of its cartoons and epigraphs and even the titles of some of its sections . there is one entitled , `` a tie is like kissing your sister . '' tell me about that section . * hollander : * there was a time when college football games could end in ties ; that time has long passed and now they play extra sessions to determine a winner . but the conventional wisdom of most coaches was that a tie was no good . it leaves everybody frustrated and unhappy , the players and fans on both teams . some coach coined the phrase `` a tie is like kissing your sister . '' which meant , you love your sister but you do nt get much satisfaction out of kissing her . the vignette considered an optimal strategy for near the end of the game , taking into account the chance of making an extra point ( one - point ) play , the chance of making a two - point play and the relative value of winning the game versus the relative value of tying the game . * samaniego : * with all these activities , plus your teaching and the mentoring of your graduate students , one would think that there might have been little time for other responsibilities . but , in fact , you served for nine years as the chair of your department . what were the main challenges you encountered as chair , and what achievements are you proudest of ? * hollander : * a chair obviously has many priorities : the faculty , the students , the staff , the administration . they are all important and you have to serve and contribute to the well - being of each . however , in my mind the top priority is to recruit well , get the best people possible . then everything desirable follows : a stronger curriculum , research grants , better students , and so forth . in my first term , 19781981 , my most significant hire was ian mckeague who , in 1979 , came from unc , chapel hill . he stayed 25 years , participated in grants , became an expert in survival analysis , and served a three - year term as chair . we co - directed jie yang on a topic on confidence bands for survival functions and have two papers that emanated from that work and related work on quantile functions with gang li ( hollander , mckeague and yang , @xcite ; li , hollander , mckeague and yang , @xcite ) . in my second and third terms , 19992005 , among the tenure - earning people i hired , flori bunea , from u. washington , eric chicken from purdue , dan mcgee from university of south carolina medical school , and marten wegkamp from yale , seem the most likely to contribute and hopefully stay at fsu for a long time . each filled an important gap in our curriculum , taught new courses , got involved with grants . i recruited dan as a senior biostatistician and he has been a driving force in establishing our new m.s . and ph.d . programs in biostatistics . he also succeeded me as chair . * samaniego : * tell me a bit about how you tried to broaden the department s focus and reach . what are some aspects beyond recruiting ? * hollander : * in fall 1999 i called ron randles , who was statistics chair at the university of florida ( uf ) at that time , and suggested we create an fsu - uf biannual statistics colloquium series . ron liked the idea and after getting approval from our faculties it began and continues today . the idea is that it provides the opportunity for the recent appointees of each faculty to get some outside exposure by giving a talk in the other department . thus in one semester uf comes to tallahassee and a uf person talks , and the next semester fsu goes to gainesville and an fsu person gives the colloquium talk . i also hope it leads to some joint research . some people have had discussions , but to my knowledge it hasnt happened yet . when i was chair , i was a mentor to all of our students , many of whom i recruited . i tried to teach them how to become professionals . i helped them get summer jobs and of course wrote reference letters for them . when i was younger i played intramural basketball and softball with some of them . i ve gone to some of their weddings . many students still stay in close touch with me . of course you do nt have to be the chairman to engage in these mentoring activities , but as chair one gets many opportunities to give extra advice at , for example , orientation and frequent student visits to the chair s office . here s a chair s story that goes into the highlight category . ron hobbs , an m.s . graduate of our department in 1967 , and his wife carolyn hobbs , who earned a b.s . in recreation studies from fsu in 1965 , endowed a chair in our department . it worked like this . each year for six consecutive years , ron and carolyn contributed $ 100,000 . then after six years , the state contributed $ 400,000 . then the university had one million dollars to help support the chair . one year , in early december , ron attended a meeting with me in my office and handed me an envelope with roughly $ 100,000 worth of america on line shares of stock . i thought for a moment , there s a delta jet with connecting flights to hawaii leaving in about an hour . i could promptly turn the envelope over to the university s chief fundraiser at the time , pat martin , who was also attending the meeting . or i could excuse myself , take the envelope with me ostensibly to return in a moment with the shares in a more carefully labeled envelope , and instead catch that jet.@xmath25 later that morning i noticed the sky was blue and clear as the engines roared and we took off to the west . * samaniego : * myles , that could be the beginning of the next great american novel ! * samaniego : * you served as the editor of the theory and methods section of jasa in 19941996 . i know this is an extremely labor - intensive job . you seemed to thrive on the experience . what did you enjoy most about it ? * hollander : * i had a great board of associate editors , including you , and you gave me the luxury of three reviews per paper . i liked working with the board . i also enjoyed reading the submissions one year i had 503!and the reviews . i tried to encourage authors , and with the reviews , improve the papers . even if a paper was declined , i wanted the disappointed author to feel his / her paper was treated with respect and got a fair shake . i helped to get a page increase and in some of my issues i had over 30 papers in theory and methods . i also increased the t&m acceptance rate to around 30% . i suspect it is significantly lower now . it was just a great experience . many nights and weekends i would bring a stack of folders home . if an ae was very tardy , i threatened to send in a swat team or toss him in a dark cellar until i received the reviews . one of my main goals was to make the papers readable and understandable . i insisted the authors write for the readers . i believe that was a mark of my editorship and your editorship , frank , as well . i enjoyed being jasa editor and a jasa ae before the editorship . i served on the boards of paul switzer , ray carroll and ed wegman , learned a lot from them , and was grateful for the opportunities . i ve also continued with editorial activities after my jasa term ended . in 1993 , the first volume of the _ journal of nonparametric statistics _ , founded by ibrahim ahmad , appeared and i have been a board member since then , with one break . in 1995 , mei - ling lee launched _ lifetime data analysis _ , and i have been a board member since the beginning . both of those journals publish important papers and the profession should be , and i believe is , grateful to ibrahim and mei - ling for their visions and dedicated work . * samaniego : * in 2003 , you received the noether senior scholar award for your work in nonparametric statistics . that must have been extremely satisfying . what do you see as the important open problems that current and future researchers in this area might wish to focus on ? * hollander : * the noether award is very special to me . the list of awardees consists of distinguished people with major accomplishments in nonparametrics and i am very grateful for the honor . the awardees thus far are erich lehmann , bob hogg , pranab sen , me , tom hettmansperger , manny parzen , brad efron and peter hall . stephen hawking , the great physicist , says you can not predict the great innovations in the future ; that s partially why they are termed great innovations . if , however , dennis lindley is correct about this being a bayesian century , and it seems to be going in that direction , then i would like nonparametrics to play a major role . thus i would wish for new , important innovations in bayesian nonparametrics . in my department we have at least three faculty members , anuj srivastava , victor patrangenaru and wei wu , working in image analysis , target recognition , face recognition and related areas . i would like to see nonparametric developments in these areas which are obviously important in many arenas including medical diagnoses and national security . as a field , i m glad we are pushing hard in interdisciplinary work , and it s good for our future role in science . it s valuable for the quality of research in the outside areas with which we participate and for scientific research overall . i m hopeful statisticians will contribute significantly to many of the important open questions in other fields and many already do . in academic settings , it s critical that university administrations recognize the importance of strong statistical support raising the quality of research . i want to be surprised in the future but , like hawking says , it s hard to guess at the surprises . what do you think , frank ? * samaniego : * in the 20th century , especially from say , 1940 to 1990 , the mathematical aspects of statistics were emphasized in both teaching and research . mathematical statistics was prime . the power of computation changed that considerably . then , applied problems , real applications with large and complex data sets , changed it even more . today , there are areas like data mining that are of great interest and importance but havent yet been mathematized . i wonder if it s just too early to mathematize challenging problems like these . i m guessing that some sort of theory of optimality , some sense of what s good and what s better than something else , will be part of the future development of these evolving problem areas . it s just simply too hard to do this with tools we have available now . * hollander : * it is true that you can do a lot of things now with computer - intensive methods and not worry about getting the exact results . it s a little reminiscent of when karl pearson was classifying curves . there are a lot of data - based methods , but the mathematical foundations may have to be solidified . i think now that we are pushing applied stuff , computer - intensive methods , we can get results relatively easily , for example , nonparametrics with bootstrapping and bayesian methods with mcmc . we may have to go back a little bit and shore up some of the methods , study their performance and properties as you suggest . but i think that will be considered only by theoretical statisticians . the computer - intensive surge is of course going to keep rolling , yield many new discoveries , and is great for the field . * samaniego : * you have many collateral interests , not the least of which is baseball . you once told me that you were as pleased with your published letters to the editor of _ sports illustrated _ as you were with many of your professional accomplishments . tell me about your interest in the dodgers and in sports in general . * hollander : * i was just kidding about the importance of the si letters . getting a statistical paper published is much more satisfying and represents a long - term and dedicated effort . however , the letters arose this way . my friend bob olds , a psychiatrist in st . augustine , used to live in tallahassee and write columns for the local newspaper . his future wife , ann , took a few classes from me when she was an undergraduate at fsu . bob sent a few letters to si and they were not published . he is a wonderful writer , much better than me , but just for fun i submitted two and , surprisingly , both were accepted . the first was about dodger pitcher fernando valenzuela during a period of fernandomania in la . the second was a comparison of the stanford and florida state marching bands . the latter was prompted by that bizarre play in november , 1982 , at the cal stanford big game . you may recall that the stanford band prematurely went on the field near the end of the game thinking stanford had won and they inadvertently ended up as blockers on cal s game - ending touchdown . my interest in the dodgers came about naturally during my childhood in brooklyn . during my summers in high school most of my friends were away at what was then called sleep - away camp . my parents could have afforded to send me , and i wanted to go , but i was an only child and they liked having me around . so i had summer jobs in the city and then on weekends , and on some evenings , i would walk to ebbets field , sit in the bleachers or the grandstand , and watch the bums , as they were affectionately called . this was the era of jackie robinson who displayed tremendous courage when he broke the color line in baseball . branch rickey , the dodgers general manager at the time , also deserves a lot of credit for giving robinson the opportunity . i enjoyed talking baseball to strangers at the game , seeing afro - americans and caucasians get along , and i loved the teamwork on the field . i ve lived my life with respect for people from all walks of life , from different backgrounds and cultures , and the dodgers played a role in teaching me that . during my years as chair , i tried to instill the same kind of teamwork in the department . i liked playing sports , mostly basketball , baseball and tennis . in my childhood , on the streets of brooklyn , i played city sports like punchball and stickball . i also played basketball in schoolyards and baseball at the parade ground in brooklyn . i played some tennis in high school but did nt get reasonably skilled at it until the early 70s . * samaniego : * one of the things that i ve noticed about you over the years is that you and glee like to go down to vero beach to see some spring training games . how long has that tradition been going on ? * hollander : * i would say it dates back to the 70s , almost the time we first came to tallahassee . we came to tallahassee in 1965 . we used to go to see the dodgers . it was a different era . we could actually go up to them and talk to them and chat about baseball , whereas today they re much more isolated . there are fences . i had some good conversations with players over the years . i remember once we went to vero beach and the game was rained out . it was a game against boston . fernando valenzuela was practicing with his pitching coach , ron perranoski . they were tossing the ball on a practice field so glee and i went up to them and started talking to them and they also posed for pictures . we have many pictures from those years . one with our sons layne and bart and hall - of - fame dodger pitcher sandy koufax is here on the office wall . * samaniego : * has any of your work involved sports in statistics ? * hollander : * i havent done serious sports statistics like the type that interests the sports statistics section of the asa . in the early 70s , however , woody woodward , who had been a player on fsu s baseball team , came to my office for help on the design and analysis of a study on different methods of rounding first base . i helped him and it became part of his master s thesis . later doug and i put the example in our nonparametrics book . in appreciation for the consulting , woodward sent me a baseball glove from spring training when he was a member of the cincinnati reds . i used it when i played intramural and city league softball at fsu and i still take it with me to spring training games and major league games , hoping to catch a foul ball . * samaniego : * i m visiting fsu on the occasion of a conference honoring your contributions to statistics and your department and university and commemorating your upcoming retirement . i know that you re looking forward to spending more time with family . i m sure your sons and your grandkids will soak up plenty of your freed - up time . any special plans ? * hollander : * you re right . glee and i do want to spend more time with our sons layne , and his children taylor and connor , and bart , his wife catherine , and their children andrew , robert and caroline . one set lives in plantation , florida , one in amherst , massachusetts . that will prompt some traveling . also , glee has siblings in hilton head , south carolina and spokane , washington and i have family in la , so we will get around . i also hope to go to a few statistical events . i love the international travel to conferences . you and i often attend the ones featuring reliability with the usual reliability club , ingram olkin , nozer sinpurwalla , allan sampson , nancy fluornoy , henry block , edsel pea , mark brown , phil boland , jim lynch , joe glaz , nikolaos limnios , misha nikulin , many more . glee and i own a beach house at alligator point , florida . it s about an hour drive from our home in tallahassee . we expect to be there a lot , walk on the beach , take bike rides to the western end of the point where there is a bird sanctuary , read novels , and so forth . * samaniego : * i ve got to believe that you have at least one more book in you . do you hope to do some writing once you are officially retired ? * hollander : * possibly i ll write a book . realistically , i think it s more likely i ll stay involved by writing a paper every now and then and recycling back to fsu from time to time to teach . lincoln moses said , `` there are no facts for the future . '' despite being a statistician , i ca nt predict . * samaniego : * you ve had a long and productive career as a research statistician . looking back , what would you say is your `` signature '' result ? * hollander : * i ll interpret the word `` signature '' literally and take the opportunity to say i greatly enjoyed the work we did together on your elegant concept of signatures in reliability theory during my sabbatical visit to uc davis in spring , 2006 ( hollander and samaniego , @xcite ) . for comparison of two coherent systems , each having i.i.d . components with a common distribution @xmath10 , we suggested the distribution - free measure @xmath26 where @xmath22 is the lifelength of system 1 and @xmath23 is the lifelength of system 2 . we found a neat way to calculate the measure directly in terms of the systems signatures and probabilities involving order statistics . among other things , we resolved the noncomparability issues using stochastic ordering , hazard rate ordering and likelihood ratio ordering that you ( kochar , murkerjee and samaniego , @xcite ) encountered for certain pairs of systems . in the bigger picture , my signature career quest was to promote nonparametric statistics , bring it into other areas , get more people to use it , and get students to study the subject and make contributions to the field . * samaniego : * it seems that , over the period of your career , nonparametric methods have become more and more important and pervasive . there is no question that your work has helped that direction significantly . * hollander : * thank you , frank . when i look in journals there are a lot of papers that are nonparametric in nature and the adjective nonparametric does not appear in the titles . it s just a natural way to start a problem now , letting the underlying distributions be arbitrary . * samaniego : * you ve worked with some of the legendary figures in our discipline including ralph bradley , frank wilcoxon , richard savage . these colleagues , and others , have played important roles in your professional evolution . how did the general environment at florida state help shape your career ? * hollander : * i came to florida state because of ralph , frank and richard . they were three luminaries in nonparametric statistics and i wanted to do nonparametrics research . frank and i shared an office ; he and his wife feredericka and glee and i became friends , but he died three months after i arrived . i never did research with frank , ralph or richard . but i was close to them . ralph and his wife marion and richard and his wife jo ann were always friendly to glee and me . although i did nt write with richard , up to the time he left for yale in 1973 he carefully read each one of my technical reports and often made valuable suggestions . i gave the bradley lectures at the university of georgia in 1999 , and after ralph passed away , i was asked by his family to deliver a eulogy at his memorial service in athens which i did , with pleasure . the environment at fsu was dedicated to research and i liked that . i came to a place where that was the top priority . also i came in 1965 , only six years after the department was founded by ralph , so there was the excitement of building . as it turned out , i was there when the first ph.d . graduated and thus far i have seen all of our ph.d . students graduate . * samaniego : * my recollection is that there s a famous quote attributed to you about the discipline of statistics . tell me about it . * hollander : * the saying is : `` statistics means never having to say you re certain . '' i saw the movie `` love story '' in 1971 . it was a big hit . it was based on a book of the same title by erich segal . i read the book after i saw the movie . as the title indicates , it s a love story . a wealthy harvard law student , oliver barrett , falls for a poor radcliffe girl , jennifer cavilleri , and eventually they marry . at one point , after a spat , oliver apologizes and jenny replies , `` love means never having to say you re sorry . '' in statistics we give type - i and type - ii error probabilities , confidence coefficients , confidence bands , false discovery rates , posterior probabilities and so forth , but we hedge our bets . we assess the uncertainty . with the movie fresh in my mind i transformed segal s phrase to `` statistics means never having to say you re certain . '' * samaniego : * thanks , myles . this excursion has been most enjoyable ! * hollander : * frank , we have had a long friendship that has stood the test of a continental divide between us . i look forward to its future pleasures . thank you for the conversation . it was highly enjoyable and i m grateful for the opportunity to interact in this manner and offer my musings . myles hollander and frank samaniego thank pamela mcghee , candace ooten and jennifer rivera for carefully transcribing the conversation . hollander , m. and samaniego , f. j. ( 2008 ) . the use of stochastic precedence in the comparison of engineered systems . in _ proceedings of the 2007 international conference on mathematical methods in reliability_. to appear . pea , e. and hollander , m. ( 2004 ) . models for recurrent phenomena in survival analysis and reliability . in _ mathematical reliability an expository perspective _ ( t. mazzuchi , n. singpurwalla , and r. soyer , eds . ) 105123 kluwer , norwell , massachusetts .

myles hollander was born in brooklyn , new york , on march 21 , 1941 . he graduated from carnegie mellon university in 1961 with a b.s . in mathematics . in the fall of 1961 , he entered the department of statistics , stanford university , earning his m.s . in statistics in 1962 and his ph.d . in statistics in 1965 . he joined the department of statistics , florida state university in 1965 and retired on may 31 , 2007 , after 42 years of service . he was department chair for nine years 19781981 , 19992005 . he was named professor emeritus at florida state upon retirement in 2007 . hollander served as editor of the _ journal of the american statistical association , theory and methods _ , 19941996 , and was an associate editor for that journal from 1985 until he became _ theory and methods _ editor - elect in 1993 . he also served on the editorial boards of the _ journal of nonparametric statistics _ ( 19931997 ; 20032005 ) and _ lifetime data analysis _ ( 19942007 ) . hollander has published over 100 papers on nonparametric statistics , survival analysis , reliability theory , biostatistics , probability theory , decision theory , bayesian statistics and multivariate analysis . he is grateful for the generous research support he has received throughout his career , most notably from the office of naval research , the u.s . air force office of scientific research , and the national institutes of health . myles hollander has received numerous recognitions for his contributions to the profession . he was elected fellow of the american statistical association ( 1972 ) and the institute of mathematical statistics ( 1973 ) , and became an elected member of the international statistical institute ( 1977 ) . at florida state university he was named distinguished researcher professor ( 1996 ) , he received the professorial excellence award ( 1997 ) , and in 1998 he was named the robert o. lawton distinguished professor , an award made to only one faculty member per year and the university s highest faculty honor . myles hollander was the ralph a. bradley lecturer at the university of georgia in 1999 , and in 2003 he received the gottfried e. noether senior scholar award in nonparametric statistics from the american statistical association . he was the buckingham scholar - in - residence at miami university , oxford , ohio in september , 1985 , and had sabbatical visits at stanford university ( 19721973 ; 19811982 ) , the university of washington ( 19891990 ) and the university of california at davis ( spring , 2006 ) . the following conversation took place in myles hollander s office at the department of statistics , florida state university , tallahassee , on april 19 , 2007 . .

arxiv

the fundamental group of the complement of plane curves is a very important topological invariant , which can be also computed for line arrangements . we list here some applications of this invariant . chisini @xcite , kulikov @xcite and kulikov - teicher @xcite have used the fundamental group of complements of branch curves of generic projections in order to distinguish between connected components of the moduli space of smooth projective surfaces , see also @xcite . moreover , the zariski - lefschetz hyperplane section theorem ( see @xcite ) states that @xmath3 where @xmath4 is an hypersurface and @xmath5 is a generic 2-plane . since @xmath6 is a plane curve , the fundamental groups of complements of curves can be used also for computing the fundamental groups of complements of hypersurfaces in @xmath7 . a different need for fundamental groups computations arises in the search for more examples of zariski pairs @xcite . a pair of plane curves is called _ a zariski pair _ if they have the same combinatorics ( to be exact : there is a degree - preserving bijection between the set of irreducible components of the two curves @xmath8 , and there exist regular neighbourhoods of the curves @xmath9 such that the pairs @xmath10 are homeomorphic and the homeomorphism respects the bijection above @xcite ) , but their complements in @xmath11 are not homeomorphic . for a survey , see @xcite . it is also interesting to explore new finite non - abelian groups which serve as fundamental groups of complements of plane curves in general , see for example @xcite . an arrangement of lines in @xmath12 is a union of copies of @xmath13 in @xmath12 . such an arrangement is called _ real _ if the defining equations of the lines can be written with real coefficients , and _ complex _ otherwise . note that the intersection of the affine part of a real arrangement with the natural copy of @xmath14 in @xmath12 is an arrangement of lines in the real plane . for real and complex line arrangements @xmath0 , fan @xcite defined a graph @xmath15 which is associated to its multiple points ( i.e. points where more than two lines are intersected ) : given a line arrangement @xmath0 , the graph @xmath15 of multiple points lies on @xmath0 . it consists of the multiple points of @xmath0 , with the segments between the multiple points on lines which have at least two multiple points . note that if the arrangement consists of three multiple points on the same line , then @xmath15 has three vertices on the same line ( see figure [ graph_gl](a ) ) . if two such lines happen to intersect in a simple point ( i.e. a point where exactly two lines are intersected ) , it is ignored ( and the lines are not considered to meet in the graph theoretic sense ) . see another example in figure [ graph_gl](b ) ( note that this definition gives a graph different from the graph defined in @xcite ) . fan @xcite proved some results concerning the projective fundamental group : [ fan ] let @xmath0 be a complex arrangement of @xmath16 lines and@xmath17 be the set of all multiple points of @xmath0 . suppose that @xmath18 , where @xmath19 is the first betti number of the graph @xmath15 ( hence @xmath18 means that the graph @xmath15 has no cycles ) . then : @xmath20 where @xmath21 is the multiplicity of the intersection point @xmath22 and @xmath23 . in @xcite , similar results were achieved for the affine and projective fundamental groups by different methods . fan @xcite has conjectured that the inverse implication is also correct , i.e. if the fundamental group @xmath24 can be written as a direct sum of free groups and infinite cyclic groups , then the graph @xmath15 has no cycles . in an unpublished note , fan @xcite shows that if the fundamental group of the affine complement is a free group , then the arrangement consists of parallel lines . recently , eliyahu , liberman , schaps and teicher @xcite proved fan s conjecture completely . these results motivate the following definition : let @xmath25 be a fundamental group of the affine or projective complements of some line arrangement with @xmath16 lines . we say that @xmath25 has _ a conjugation - free geometric presentation _ if @xmath25 has a presentation with the following properties : * in the affine case , the generators @xmath26 are the meridians of lines at some far side of the arrangement , and therefore the number of generators is equal to @xmath16 . * in the projective case , the generators are the meridians of lines at some far side of the arrangement except for one , and therefore the number of generators is equal to @xmath27 . * in both cases , the relations are of the following type : @xmath28 where @xmath29 is an increasing subsequence of indices , where @xmath30 in the affine case and @xmath31 in the projective case . note that for @xmath32 we get the usual commutator . note that in usual geometric presentations of the fundamental group , most of the relations have conjugations ( see section [ mt ] ) . based on the last definition , fan s result yields that if the graph associated to the arrangement is acyclic , then the corresponding fundamental group has a conjugation - free geometric presentation . the following natural problem arises : which line arrangements have a fundamental group which has a conjugation - free geometric presentation ? the aim of this paper is to attack this problem . the importance of this family of arrangements is that the fundamental group can be read directly from the arrangement or equivalently from its incidence lattice ( where the _ incidence lattice _ of an arrangement is the partially - ordered set of non - empty intersections of the lines , ordered by inclusion , see @xcite ) without any computation . hence , for this family of arrangements , the incidence lattice determines the fundamental group of the complement . we start with the easy fact that there exist arrangements whose fundamental groups have no conjugation - free geometric presentation : the fundamental group of the affine ceva arrangement ( also known as the _ braid arrangement _ , appears in figure [ ceva ] ) has no conjugation - free geometric presentation . this fact was checked computationally by a package called _ testisom _ @xcite , which looks for isomorphisms ( or proves a non - isomorphism ) between two given finitely - presented group . note that the ceva arrangement is the minimal arrangement ( with respect to the number of lines ) with this property . our main result is : the fundamental group of following family of real arrangements have a conjugation - free geometric presentation : an arrangement @xmath0 , where @xmath15 is a union of disjoint cycles of any length , has no line with more than two multiple points , and the multiplicities of the multiple points are arbitrary . we also give the exact group structure ( by means of a semi - direct product ) of the fundamental group for an arrangement of @xmath1 lines whose graph is a cycle of length @xmath2 ( i.e. a triangle ) , where all the multiple points are of multiplicity @xmath2 : let @xmath0 be the real arrangement of @xmath1 lines , whose graph consists of a cycle of length @xmath2 , where all the multiple points are of multiplicity @xmath2 . moreover , it has no line with more than two multiple points . then : @xmath33 where @xmath34 is the free product . as mentioned above , for the family of arrangements with a conjugation - free geometric presentation of the fundamental group , the incidence lattice of the arrangement determines its fundamental group . there are some well - known families of arrangements whose lattice determines the fundamental group of its complement - the families of _ nice _ arrangements ( jiang - yau @xcite ) and _ simple _ arrangements ( wang - yau @xcite ) . it is interesting to study the relation between these families and the family of arrangements whose fundamental groups have conjugation - free geometric presentations , since for the latter family , the lattice determines the fundamental group of the complement too . we have the following remark : the fundamental group of the arrangement @xmath35 ( appears in figure [ a_n ] ) has a conjugation - free geometric presentation ( this fact was checked computationally ) , but this arrangement is neither nice nor simple . it will be interesting to find out whether our family of arrangements is broader than the family of simple arrangements , or whether there exists a simple arrangement whose fundamental group has no conjugation - free geometric presentation . [ rem_dehornoy ] it is worth to mention that conjugation - free geometric presentations are complemented positive presentations ( defined by dehornoy @xcite , see also @xcite ) . some initial computations show that in general conjugation - free geometric presentations are not complete ( since the cube condition is not satisfied for some triples of generators ) . nevertheless , we do think that there exist conjugation - free geometric presentations which are complete and hence have all the good properties induced by the completeness ( see the survey @xcite ) . we will discuss this subject in a different paper . the paper is organized as follows . in section [ mt ] , we give a quick survey of the techniques we are using throughout the paper . in section [ length_three ] , we show that the fundamental group of a real arrangement whose graph has a unique cycle of length 3 has a conjugation - free geometric presentation . in this section , we also deal with the exact structure of the fundamental group of a real arrangement whose graph consists of a cycle of length @xmath2 , where all the multiple points have multiplicity @xmath2 . section [ length_n ] deals with the corresponding result for a real arrangement whose graph has a unique cycle of length @xmath16 . we also generalize this result for the case of arrangements whose graphs are a union of disjoint cycles . in this section , we present the computation of the fundamental group of the complement of real line arrangements . this is based on the moishezon - teicher method @xcite and the van kampen theorem @xcite . some more presentations and algorithms can be found in @xcite . if the reader is familiar with this algorithm , he can skip this section . to an arrangement of @xmath36 lines in @xmath14 one can associate a _ wiring diagram _ @xcite , which holds the combinatorial data of the arrangement and the position of the intersection points . a wiring diagram is a collection of @xmath36 wires ( where a _ wire _ in @xmath37 is a union of segments and rays , homeomorphic to @xmath38 ) . the induced wiring diagram is constructed by choosing a new line ( called the _ guiding line _ ) , which avoids all the intersection points of the arrangement , such that the projections of intersection points do not overlap . then , the @xmath36 wires are generated as follows . start at the @xmath39 end of the line with @xmath36 parallel rays , and for every projection of an intersection point , make the corresponding switch in the rays , as in figure [ latowd ] . to a wiring diagram , one can associate a list of _ lefschetz pairs_. any pair of this list corresponds to one of the intersection points , and holds the smallest and the largest indices of the wires intersected at this point , numerated locally near the intersection point ( see @xcite and @xcite ) . for example , in the wiring diagram of figure [ wdtolp ] , the list of lefschetz pairs is ( we pass on the intersection points from right to left ) : @xmath40}},{{\left[{2},{4}\right]}},{{\left[{1},{2}\right]}},{{\left[{4},{5}\right]}},{{\left[{2},{3}\right]}},{{\left[{3},{4}\right]}},{{\left[{4},{5}\right]}},{{\left[{2},{3}\right ] } } ) .\ ] ] let @xmath41 be a closed disk in @xmath14 , @xmath42 a set of @xmath36 points , and @xmath43 . let @xmath44 be the group of all diffeomorphisms @xmath45 such that @xmath46 is the identity and @xmath47 . the action of such @xmath48 on the disk applies to paths in @xmath41 , which induces an automorphism on @xmath49 . the _ braid group _ , @xmath50 $ ] , is the group @xmath44 modulo the subgroup of diffeomorphisms inducing the trivial automorphism on @xmath49 . an element of @xmath51 $ ] is called a _ braid_. for simplicity , we will assume that @xmath52 , and that @xmath53 . choose a point @xmath54 ( for convenience we choose it to be below the real line ) . the group @xmath55 is freely generated by @xmath56 , where @xmath57 is a loop starting and ending at @xmath58 , enveloping the @xmath59th point in @xmath60 . the set @xmath61 is called a _ geometric base _ or _ g - base _ of @xmath55 ( see figure [ fig_1 ] ) . let @xmath62}},\dots,{{\left[{a_p},{b_p}\right]}})$ ] be a list of lefschetz pairs associated to a real line arrangement @xmath63 with @xmath36 lines . the of the complement of the arrangement is a quotient group of @xmath55 . there are @xmath64 sets of relations , one for every intersection point . in each point , we will compute an object called a _ skeleton _ , from which the relation is computed . in order to compute the skeleton @xmath65 associated to the @xmath59th intersection point , we start with an _ initial skeleton _ corresponding to the @xmath59th lefshetz pair @xmath66}}$ ] which is presented in figure [ fig3 ] , in which the points correspond to the lines of the arrangement and we connect by segments adjacent points which correspond to a local numeration of lines passing through the intersection point . 0.7 cm to the initial skeleton , we apply the lefschetz pairs @xmath67 } } , \cdots,{{\left[{a_1},{b_1}\right]}}$ ] . a lefschetz pair @xmath68}}$ ] acts by rotating the region from @xmath69 to @xmath70 by @xmath71 counterclockwise without affecting any other points . for example , consider the list @xmath72}},{{\left[{2},{4}\right]}},{{\left[{4},{5}\right]}},{{\left[{1},{3}\right]}},{{\left[{3},{4}\right]}})$ ] . let us compute the skeleton associated to the 5th point . the initial skeleton for @xmath73}}$ ] is given in figure [ fig4](a ) . by applying @xmath74}}$ ] and then @xmath75}}$ ] , we get the skeleton in figure [ fig4](b ) . then , applying @xmath76}}$ ] yields the skeleton in figure [ fig4](c ) , and finally by acting with @xmath77}}$ ] we get the final skeleton in figure [ fig4](d ) . 4.5 cm from the final skeletons we compute the relations , as follows . we first explain the case when @xmath78}}$ ] corresponds to a simple point , @xmath79 . then the skeleton is a path connecting two points . let @xmath41 be a disk circumscribing the skeleton , and let @xmath60 be the set of points . choose an arbitrary point on the path and pull it down , splitting the path into two parts , which are connected in one end to @xmath80 and in the other to the two end points of the path in @xmath60 . the loops associated to these two paths are elements in the group @xmath81 , and we call them @xmath82 and @xmath83 . the corresponding elements commute in the of the arrangement s complement . figure [ av_bv ] illustrates this procedure . now we show how to write @xmath82 and @xmath83 as words in the generators @xmath84 of @xmath85 . we start with the generator corresponding to the end point of @xmath82 ( or @xmath83 ) , and conjugate it as we move along @xmath82 ( or @xmath83 ) from its end point on @xmath60 to @xmath58 as follows : for every point @xmath86 which we pass from above , we conjugate by @xmath87 when moving from left to right , and by @xmath88 when moving from right to left . for example , in figure [ av_bv ] , @xmath89 and so the induced relation is : @xmath90 one can check that the relation is independent of the point in which the path is split . for a multiple intersection point of multiplicity @xmath91 , we compute the elements in the group @xmath81 in a similar way , but the induced relations are of the following type : @xmath92 we choose an arbitrary point on the path and pull it down to @xmath58 . for each of the @xmath91 end points of the skeleton , we generate the loop associated to the path from @xmath58 to that point , and translate this path to a word in @xmath93 by the procedure described above . in the example given in figure [ av_bv_mul ] , we have : @xmath94 , @xmath95 and @xmath96 , so the relations are @xmath97 in this section , we prove the following proposition : [ triangle - prop ] the fundamental group of a real affine arrangement without parallel lines , whose graph which has a unique cycle of length @xmath2 and has no line with more than two multiple points , has a conjugation - free geometric presentation . in the first subsection we present the proof of proposition [ triangle - prop ] . the second subsection will be devoted to studying the group structure of the fundamental group of the simplest arrangement of this family . for simplicity , we will assume that all the multiple points have the same multiplicity @xmath98 , but the same argument will work even if the multiplicities are not equal . by rotations and translations , one can assume that we have a drawing of an arrangement which has a unique cycle of multiple points of length 3 and has no line with more than two multiple points , as in figure [ arrange_mult_3 ] . we can assume it due to the following reasons : first , one can rotate a line that participates in only one multiple point as long as it does not unite with a different line ( by results 4.8 and 4.13 of @xcite ) . second , moving a line that participates in only one multiple point over a different line ( see figure [ triangle - line ] ) is permitted in the case of a triangle due to a result of fan @xcite that the family of configurations with @xmath1 lines and three triple points is connected by a finite sequence of smooth equisingular deformations . each of the blocks 1,4,5 contains simple intersection points of two pencils . in block 1 , one can assume that all the intersections of any horizontal line are adjacent , without intervening points from the third pencil . in blocks 4 and 5 , one can assume that all the intersections of any vertical line are adjacent ( in block 4 , the vertical lines are those with positive slopes ) . all the intersection points of block 5 are to the left of all the intersection points of block 4 . hence , we get the list of lefschetz pairs as in table [ tab1 ] ( we put a double line to separate between the pairs related to different blocks ) . @xmath99 & & 2n(n-1)+3 & [ n , n+1 ] \\ 2 & [ n-1,n ] & & 2n(n-1)+4 & [ n-1,n ] \\ \vdots & \vdots & & \vdots & \vdots \\ n & [ 1,2 ] & & n(2n-1)+2 & [ 1,2 ] \\ \hline n+1 & [ n+1,n+2 ] & & n(2n-1)+3 & [ n+1,n+2 ] \\ n+2 & [ n , n+1 ] & & n(2n-1)+3 & [ n , n+1 ] \\ \vdots & \vdots & & \vdots & \vdots \\ 2n & [ 2,3 ] & & n(2n)+2 & [ 2,3 ] \\ \hline \vdots & \vdots & & \vdots & \vdots\\ \hline ( n-2)n+1 & [ 2n-2,2n-1 ] & & ( 3n-1)(n-1)+3 & [ 2n-2,2n-1 ] \\ ( n-2)n+2 & [ 2n-3,2n-2 ] & & ( 3n-1)(n-1)+4 & [ 2n-3,2n-2 ] \\ \vdots & \vdots & & \vdots & \vdots\\ ( n-1)n & [ n-1,n ] & & 3n(n-1)+2 & [ n-1,n ] \\ \hline\hline ( n-1)n+1 & [ n,2n ] & & 3n(n-1)+3 & [ n,2n ] \\ \hline\hline ( n-1)n+2 & [ 2n,3n ] & & & \\ \hline\hline ( n-1)n+3 & [ 2n-1,2n ] & & & \\ ( n-1)n+4 & [ 2n-2,2n-1 ] & & & \\ \vdots & \vdots & & & \\ ( n-1)(n+1)+2 & [ n+1,n+2 ] & & & \\ \hline ( n-1)(n+1)+3 & [ 2n,2n+1 ] & & & \\ ( n-1)(n+1)+4 & [ 2n-1,2n ] & & & \\ \vdots & \vdots & & & \\ ( n-1)(n+2)+2 & [ n+2,n+3 ] & & & \\ \hline \vdots & \vdots & & & \\ \hline ( 2n-1)(n-1)+3 & [ 3n-2,3n-1 ] & & & \\ ( 2n-1)(n-1)+4 & [ 3n-3,3n-2 ] & & & \\ \vdots & \vdots & & & \\ 2n(n-1)+2 & [ 2n,2n+1 ] & & & \\ \hline \end{array}\ ] ] by the moishezon - teicher algorithm ( see section [ mt ] ) , we get the following skeletons : * for point @xmath91 , where @xmath100 , the corresponding final skeleton appears in figure [ braid1 - 6](a ) , where @xmath101 and @xmath102 . * for point @xmath103 , the corresponding final skeleton appears in figure [ braid1 - 6](b ) . * for point @xmath104 , the corresponding final skeleton appears in figure [ braid1 - 6](c ) . * for point @xmath91 , where @xmath105 , the corresponding final skeleton appears in figure [ braid1 - 6](d ) , where @xmath106 and @xmath107 . * for point @xmath91 , where @xmath108 , the corresponding final skeleton appears in figure [ braid1 - 6](e ) , where @xmath109 and @xmath110 . * for point @xmath111 , the corresponding final skeleton appears in figure [ braid1 - 6](f ) . before we proceed to the presentation of the fundamental group , we introduce one notation : instead of writing the relations ( where @xmath22 are words in a group ) : @xmath112 we will sometimes write : @xmath113 $ ] . by the van kampen theorem ( see section [ mt ] ) , we get the following presentation of the fundamental group of the line arrangement s complement : generators : @xmath114 + relations : + 1 . @xmath115=e$ ] , where @xmath101 and@xmath102 . 2 . @xmath116 $ ] . 3 . @xmath117 $ ] . @xmath118=e$ ] where @xmath106 and @xmath107 . 5 . @xmath119=e$ ] where @xmath109 and @xmath110 . @xmath120 $ ] . now , we show that all the conjugations in relations ( 1),(2),(3 ) and ( 4 ) can be simplified . we start with relations ( 1 ) , and then relations ( 2 ) . we continue to relations ( 4 ) and we finish with relations ( 3 ) . we start with the first set of relations : for @xmath121 , we get that for all @xmath101 we have : @xmath122=e$ ] . now , we proceed to @xmath123 . for @xmath124 , we get : @xmath125=e$ ] . by the relation @xmath126=e$ ] , it is simplified to @xmath127=e$ ] . in this way , we get that for @xmath123 , we have : @xmath128=e$ ] for @xmath101 . by increasing @xmath129 one by one , we get that all the conjugations disappear and we get @xmath130=e$ ] , where @xmath101 and @xmath102 , as needed . relations ( 2 ) can be written as : @xmath131 @xmath132 @xmath133 by the simplified version of relations ( 1 ) , we can omit all the generators @xmath134 . hence we get : @xmath135 as needed . we proceed to relations ( 4 ) . we start with @xmath136 . taking @xmath124 , we get : @xmath137=e.\ ] ] by relations ( 6 ) , we have : @xmath138 , and hence we get : @xmath139=e.\ ] ] for @xmath140 , we get : @xmath141=e.\ ] ] by relations ( 6 ) again and the simplified version of relations ( 1 ) , we get : @xmath142=e.\ ] ] using the simplified relation @xmath143=e$ ] , we get @xmath144=e$ ] . in the same way , we get that for @xmath136 and @xmath145 , we have : @xmath146=e$ ] . we continue to @xmath147 . taking @xmath124 , we have : @xmath148=e.\ ] ] by the simplified version of relations ( 1 ) , we can omit all the generators @xmath149 . hence we get : @xmath150=e.\ ] ] by relations ( 5 ) , we can omit @xmath151 too , and therefore : @xmath152=e$ ] . for @xmath140 , we have : @xmath153=e.\ ] ] by the simplified version of relations ( 1 ) , we can omit all the generators @xmath149 . hence we get : @xmath154=e.\ ] ] by relations ( 5 ) , we can omit @xmath151 too , and therefore : @xmath155=e.\ ] ] by @xmath152=e$ ] , we get @xmath156=e$ ] . in the same way , we get that for @xmath147 and @xmath145 , we get : @xmath157=e$ ] . in the same way , by increasing @xmath129 one by one , we will get that for all @xmath158 and @xmath145 , we get : @xmath130=e$ ] as needed . relations ( 3 ) can be written : @xmath159 @xmath160 @xmath161 by relations ( 1 ) , we can omit the generators @xmath134 , so we get : @xmath162 @xmath163 @xmath164 by relations ( 2 ) , we can omit also the generators @xmath165 in order to get : @xmath166 hence , we get the following simplified presentation : generators : @xmath114 + relations : 1 . @xmath130=e$ ] , where @xmath101 and @xmath102 . 2 . @xmath167 $ ] . 3 . @xmath168 $ ] . @xmath130=e$ ] where @xmath106 and @xmath107 . @xmath119=e$ ] where @xmath109 and @xmath110 . 6 . @xmath120 $ ] . therefore , we have a conjugation - free geometric presentation , and hence we are done . cohen and suciu @xcite give the following presentation of @xmath169 , which is known @xcite to be the fundamental group of the complement of the affine ceva arrangement ( see figure [ ceva ] ) : @xmath170 the actions of the automorphisms @xmath171 and @xmath172 are defined as follows : @xmath173 + @xmath174 @xmath175 + @xmath176 @xmath177 + @xmath178 notice that if we rotate clockwise the lowest line in the affine ceva arrangement ( figure [ ceva ] ) , we get an arrangement @xmath0 whose graph consists of a unique cycle of length @xmath2 , where all the multiple points are of multiplicity @xmath2 . by a simple check , the effect of this rotation is the addition of the commutator relation @xmath179=e$ ] to the presentation of the group . hence , we get that the actions of the automorphisms @xmath171 and @xmath172 are changed as follows : @xmath173 + @xmath174 @xmath180 @xmath177 + @xmath178 this is the presentation of the group : @xmath181 , where @xmath34 is the free product . to summarize , we get the following result : let @xmath0 be the arrangement of @xmath1 lines without parallel lines whose graph is a unique cycle of length @xmath2 , where all the multiple points are of multiplicity @xmath2 . then : @xmath33 it is interesting to check how this proposition can be generalize to arrangements whose graphs are cycles of length @xmath182 . in this section , we show that the fundamental group of a real affine arrangement whose graph is a unique cycle of any length and has no line with more than two multiple points , has a conjugation - free geometric presentation . at the end of this section , we generalize this result to arrangements whose graphs are unions of disjoint cycles . we start by investigating the case of a cycle of length @xmath183 and then we generalize it to any length . in figure [ cycle_multiple5 ] , we present a real arrangement whose graph is a cycle of @xmath183 multiple points ( note that any real arrangement whose graph is a unique cycle of @xmath183 multiple points and has no line with more than two multiple points , can be transferred to this drawing by rotations , translations and equisingular deformations ) . based on figure [ cycle_multiple5 ] , we get the list of lefschetz pairs presented in table [ tab2 ] . @xmath184 & 2 & & 13 & [ 6,7 ] & 2 & & 25 & [ 8,9 ] & 2 \\ 2 & [ 5,6 ] & 2 & & 14 & [ 7,8 ] & 2 & & 26 & [ 6,7 ] & 2 \\ 3 & [ 7,8 ] & 2 & & 15 & [ 3,4 ] & 2 & & 27 & [ 5,6 ] & 2 \\ 4 & [ 6,7 ] & 2 & & 16 & [ 4,5 ] & 2 & & 28 & [ 7,8 ] & 2 \\ 5 & [ 4,5 ] & 2 & & 17 & [ 5,6 ] & 2 & & 29 & [ 6,7 ] & 2 \\ 6 & [ 3,4 ] & 2 & & 18 & [ 6,7 ] & 2 & & 30 & [ 4,6 ] & 3 \\ 7 & [ 5,6 ] & 2 & & 19 & [ 4,6 ] & 3 & & 31 & [ 3,4 ] & 2 \\ 8 & [ 4,5 ] & 2 & & 20 & [ 3,4 ] & 2 & & 32 & [ 4,5 ] & 2 \\ 9 & [ 2,3 ] & 2 & & 21 & [ 4,5 ] & 2 & & 33 & [ 2,3 ] & 2 \\ 10 & [ 1,2 ] & 2 & & 22 & [ 8,9 ] & 2 & & 34 & [ 1,2 ] & 2 \\ 11 & [ 2,4 ] & 3 & & 23 & [ 7,8 ] & 2 & & 35 & [ 2,4 ] & 3 \\ 12 & [ 4,6 ] & 3 & & 24 & [ 9,10 ] & 2 & & & & \\ \hline\hline \end{array}\ ] ] by the moishezon - teicher algorithm ( see section [ mt ] ) , one can compute the skeletons of the braid monodromy . after the computation , one should notice that actually we can group the intersection points into blocks according to their braid monodromies ( see figure [ cycle_multiple5_block ] ) , since the structure of the skeletons is similar . following this observation , we can deal with each block separately . so , we get the following sets of skeletons : * quadruples of type q1 : see figure [ quadruple_case1](a ) for @xmath185 . * quadruples of type q2 : see figure [ quadruple_case2](a ) for @xmath186 , @xmath187 , @xmath188 . * a triple of type t1 : see figure [ triple_case1](a ) . * triples of type t2 : see figure [ triple_case2 ] for @xmath189 . * a triple of type t3 : see figure [ triple_case3](a ) . * a triple of type t4 : see figure [ triple_case4](a ) . now we pass to the general case . one can draw an arrangement of @xmath190 lines whose graph is a unique cycle of length @xmath16 and has no line with more than two multiple points in a similar way to the way we have drawn the arrangement of @xmath191 lines whose graph is a cycle of length @xmath183 . hence , one can compute the braid monodromy of the general arrangement in blocks similar to what we have done in the case of @xmath192 : * quadruples of type q1 : for @xmath193 , see figure [ quadruple_case1](b ) . * quadruples of type q2 : for @xmath194 , @xmath187 , @xmath195 , see figure [ quadruple_case2](b ) . * a triple of type t1 : see figure [ triple_case1](b ) . * triples of type t2 : for @xmath196 , see figure [ triple_case2 ] . * a triple of type t3 : see figure [ triple_case3](b ) . * a triple of type t4 : see figure [ triple_case4](b ) . by the van - kampen theorem ( see section [ mt ] ) , we get the following presentation of the fundamental group of the complement of the arrangement : generators : @xmath197 + relations : * from quadruples of type q1 : 1 . @xmath198=e$ ] where @xmath193 2 . @xmath199=e$ ] where @xmath193 3 . @xmath200=e$ ] where @xmath193 4 . @xmath201=e$ ] where @xmath193 * from quadruples of type q2 : for @xmath194 , @xmath187 , @xmath195 : 1 . @xmath202=e$ ] 2 . @xmath203=e$ ] 3 . @xmath204=e$ ] 4 . @xmath205=e$ ] * from the triple of type t1 : 1 . @xmath206=e$ ] 2 . @xmath207=e$ ] 3 . @xmath208 * from triples of type t2 : 1 . @xmath209 where @xmath196 2 . @xmath210=e$ ] where @xmath196 3 . @xmath211=e$ ] where @xmath196 * from the triple of type t3 : 1 . @xmath212 2 . @xmath213=e$ ] 3 . @xmath214=e$ ] * from the triple of type t4 : 1 . @xmath215=e$ ] 2 . @xmath216=e$ ] 3 . @xmath217 we now show that all the conjugations can be simplified , and hence we have a conjugation - free geometric presentation for the fundamental group . we have conjugations in the relations coming from triples of points and quadruples of points . we start with the relations which correspond to triples of points . the conjugation in relation ( 3 ) of the triple of type t1 can be simplified using relations ( 1 ) and ( 2 ) of the triple of type t1 . the conjugation in relation ( 1 ) of triples of type t2 can be simplified using relations ( 2 ) and ( 3 ) of the corresponding triples of type t2 . the conjugation in relation ( 3 ) of the triple of type t3 can be simplified using relation ( 3 ) of the triple of type t4 . the conjugation in relation ( 2 ) of the triple of type t3 can be simplified using relations ( 1 ) and ( 2 ) of the triple of type t4 . the conjugation in relation ( 1 ) of the triple of type t3 can be simplified using relations ( 2)(3 ) of the triple of type t3 and relations ( 1)(3 ) of the triple of type t4 . we continue to the relations induced by to the quadruples of type q1 . by the first two relations of the quadruples of type q1 , one can easily simplify the conjugations which appear in the last two relations of the quadruples of type q1 . so we get that the relations correspond to the quadruples of type q1 can be written without conjugations . now , we pass to the relations correspond to the quadruples of type q2 . we start with @xmath218 : we have the following relations : 1 . @xmath219=e$ ] 2 . @xmath220=e$ ] 3 . @xmath221=e$ ] 4 . @xmath222=e$ ] by relations ( 2 ) and ( 3 ) of triple t2 ( for @xmath223 ) , we have the relations @xmath224=e$ ] and @xmath225=e$ ] . hence , the conjugations in relation ( d ) are canceled and we get @xmath226=e$ ] . by the same relations and the simplified version of relation ( d ) , we get the following relation from relation ( a ) : @xmath227=e$ ] . substituting @xmath223 in relation ( 1 ) of triple t2 yields @xmath228 by this relation , relation ( c ) becomes @xmath229=e$ ] , and relation ( b ) becomes @xmath230=e$ ] . the same argument holds for any @xmath231 , where @xmath232 and @xmath233 . hence , one can simplify the conjugations in these cases . now , we pass to the case where @xmath234 and @xmath233 . let @xmath235 . we have the following relations : 1 . @xmath236=e$ ] 2 . @xmath237=e$ ] 3 . @xmath238=e$ ] 4 . @xmath239=e$ ] by relations ( 2 ) and ( 3 ) of triple t2 ( for @xmath223 ) , we have the relations @xmath224=e$ ] and @xmath225=e$ ] . hence , relations ( a ) and ( d ) become : 1 . @xmath240=e$ ] 2 . @xmath241=e$ ] by relations ( a ) and ( d ) above , we get @xmath242=e$ ] , and therefore we also get @xmath243=e$ ] . substituting @xmath223 in relation ( 1 ) of triple t2 yields @xmath228 by this relation , relations ( b ) and ( c ) become : 1 . @xmath244=e$ ] 2 . @xmath245=e$ ] by relations ( b ) and ( c ) above , we get @xmath246=e$ ] and hence @xmath247=e$ ] . it is easy to show by a simple induction that we can simplify the conjugations for any @xmath231 , where @xmath248 and @xmath233 . the remaining case is @xmath249 . we start with @xmath250 . we have the following relations : + \(a ) @xmath251=e$ ] + ( b ) @xmath252=e$ ] + ( c ) @xmath253=e$ ] + ( d ) @xmath254=e$ ] we will show that these conjugations can be simplified . by relation ( 3 ) of triple t4 and relation ( 3 ) of triple t3 , relations ( c ) and ( d ) can be written as : + ( c ) @xmath255=e$ ] + ( d ) @xmath256=e$ ] by relation ( 3 ) of triple t2 for @xmath257 , we have @xmath258=e$ ] , and hence relation ( c ) becomes @xmath259=e$ ] . by relation ( 1 ) of triple t2 for @xmath257 , we have : @xmath260 and then relation ( d ) becomes : @xmath261=e$ ] . now , we simplify relation ( a ) . again , by relations ( 2 ) and ( 3 ) of triple t2 for @xmath257 , we have @xmath258=e$ ] and @xmath262=e$ ] , and hence : @xmath263=e\ ] ] by relation ( 3 ) of triple t4 , we have : @xmath264=e.\ ] ] by relations ( 1 ) and ( 2 ) of triple t4 , we have : @xmath265=e.\ ] ] by relations ( 1 ) of triple t3 , we finally have : @xmath266=e$ ] . now , we simplify relation ( b ) . by relation ( 3 ) of triple t4 , we have : @xmath267=e.\ ] ] by relations ( 1 ) and ( 2 ) of triple t4 , we get : @xmath268=e.\ ] ] by relations ( 2 ) and ( 3 ) of triple t3 , we get : @xmath269=e.\ ] ] finally , by relation ( 1 ) of triple t2 for @xmath257 , we have @xmath260 so we get : @xmath270=e$ ] . by similar tricks , one can simplify the conjugations for all the cases where @xmath249 and @xmath271 . hence , we have a presentation based on the topological generators without conjugations in the relations , and hence we are done . the above proof is based on the fact that the multiplicity of each multiple point is @xmath2 . we now explain why it can be generalized to any multiplicity . in case of higher multiplicities , the quadruples from the previous case will be transformed to a block of @xmath272 simple points . it can be easily checked that all the conjugations can be simplified in this case . moreover , the triples from the previous case will be transformed into blocks similar to the blocks we had in the case of a cycle of length @xmath2 ( see proposition [ triangle - prop ] ) , and in this case too , it can be easily checked that all the conjugations can be simplified , and hence we have shown that arrangements whose graph is a unique cycle and have no line with more than two multiple points , have a conjugation - free geometric presentation . using the following decomposition theorem of oka and sakamoto @xcite , we can generalize the result from the case of one cycle to the case of a union of disjoint cycles : * ( oka - sakamoto ) * let @xmath273 and @xmath274 be algebraic plane curves in @xmath275 . assume that the intersection @xmath276 consists of distinct @xmath277 points , where @xmath278 are the respective degrees of @xmath273 and @xmath274 . then : @xmath279 hence , we have the following result : if the graph of the arrangement @xmath0 is a union of disjoint cycles of any length and the arrangement has no line with more than two multiple points , then its fundamental group has a conjugation - free geometric presentation . we would like to thank patrick dehornoy , uzi vishne and eran liberman for fruitful discussions . we owe special thanks to an anonymous referee for many useful corrections and advices and for pointing out the connection between our presentations and dehornoy s positive presentations ( remark [ rem_dehornoy ] ) . m. eliyahu , e. liberman , m. schaps and m. teicher , _ characterization of line arrangements for which the fundamental group of the complement is a direct product _ , alg . topo . , to appear . [ available online : http://arxiv.org/abs/0810.5533 ] . d. garber , _ on the connection between affine and projective fundamental groups of line arrangements and curves _ , singularits franco - japonaises ( j .- p . brasselet and t. suwa , eds . ) , sminaires & congrs * 10 * , 6170 ( 2005 ) . d. garber and m. teicher , _ the fundamental group s structure of the complement of some configurations of real line arrangements _ , complex analysis and algebraic geometry , edited by t. peternell and f .- o . schreyer , de gruyter , 173 - 223 ( 2000 ) . kulikov and m. teicher , _ braid monodromy factorizations and diffeomorphism types _ , izv . nauk ser . mat . * 64*(2 ) , 89120 ( 2000 ) [ russian ] ; english translation : izv . 64*(2 ) , 311341 ( 2000 ) . b. moishezon and m. teicher , _ braid group technique in complex geometry ii : from arrangements of lines and conics to cuspidal curves _ , in algebraic geometry , lect . notes in math . * 1479 * , 131180 ( 1990 ) .

we introduce the notion of a _ conjugation - free geometric presentation _ for a fundamental group of a line arrangement s complement , and we show that the fundamental groups of the following family of arrangements have a conjugation - free geometric presentation : a real arrangement @xmath0 , whose graph of multiple points is a union of disjoint cycles , has no line with more than two multiple points , and where the multiplicities of the multiple points are arbitrary . we also compute the exact group structure ( by means of a semi - direct product of groups ) of the arrangement of @xmath1 lines whose graph consists of a cycle of length @xmath2 , and all the multiple points have multiplicity @xmath2 .

arxiv

in recent time people are too much interested to find some flavor symmetry in order to generate mass and mixing pattern of fermions . continuous symmetry like @xmath9 @xcite , @xmath10 @xcite symmetry and most popular discrete symmetry , @xmath11 exchange symmetry ( @xmath12@xcite have got some success to describe mass and mixing pattern in leptonic sector . to avoid mass degeneracy of @xmath13 and @xmath14 under @xmath15 symmetry , e. ma and g. rajasekaran in @xcite have introduced first time the @xmath1 symmetry . after this paper , a lot of work have done with this symmetry @xcite-@xcite . after introduction of tri - bi maximal mixing pattern ( @xmath16 , @xmath17 , @xmath18)@xcite , people have tried to fit this mixing pattern through the @xmath1 symmetry . in an well motivated extension of the standard model through the inclusion of @xmath1 discrete symmetry tri - bi maximal mixing pattern comes out in a natural way in the work of altarelli and feruglio @xcite . more precisely , the leptonic mixing arises solely from the neutrino sector since the charged lepton mass matrix is diagonal . the model @xcite also admits hierarchical masses of the three charged leptons whereas the neutrino masses are quasi - degenerate or hierarchical . although the model gives rise to @xmath19 ( @xmath20 ) which is consistent with the chooz - palo verde experimental upper bound ( @xmath21 at 3@xmath22 ) , however , the non - zero and complex value of @xmath0 leads to the possibility to explore _ violation in the leptonic sector which is the main goal of many future short and long baseline experiments . within the framework of @xmath23 model , non - zero @xmath0 is generated either through the radiative correction @xcite or due to the introduction of higher dimensional mass terms @xcite . generation of non zero complex @xmath0 and possibility of non - zero cp violation has been extensively studied in @xcite for the proposed model of altarelli - feruglio @xcite with explicit soft breaking of @xmath1 symmetry @xcite . in the model @xcite the authors showed that the tri - bi maximal mixing pattern is also generated naturally in the framework of see - saw mechanism with @xmath23 symmetry . exact tri - bi maximal pattern forbids at low energy cp violation in leptonic sector . the textures of mass matrices in @xcite could not generate lepton asymmetry also . in the present work , we investigate the generation of non - zero @xmath0 through see saw mechanism by considering a small perturbation in @xmath2 , the dirac neutrino mass matrix , keeping the same texture of the right - handed majorana neutrino mass matrix as proposed in ref.@xcite . at first , we have studied in detail perturbation of @xmath2 by adding a small parameter at different entries of @xmath2 and see the variations of three mixing angles in terms of other model parameters considering all of them real . we extend our analysis to the complex case for a suitable texture . we study detailed phenomenology of neutrino mass and mixing including cp violation at low energy , neutrinoless double beta decay and leptogenesis . our approach to get nonzero @xmath0 is minimal as we break @xmath1 symmetry explicitly by single parameter in single element of @xmath2 . generation of cp violation at low energy as well as high energy is also minimal as we consider only one parameter complex . we consider the model proposed in @xcite , which gives rise to diagonal @xmath2 and @xmath24 ( the charged lepton mass matrix ) along with a competent texture of @xmath25 and after see - saw mechanism and diagonalisation gives rise to tri - bimaximal mixing pattern . the model consists of several scalar fields to generate required vacuum alignment to obtain tri - bimaximal mixing . in table i. , we have listed the scalar fields and their vev s and representation content under all those symmetries . .list of fermion and scalar fields used in this model , @xmath26 . [ cols="^,^,^,^",options="header " , ] the model is fabricated in such a way that after spontaneous breaking of @xmath1 symmetry , the @xmath15 symmetry remains on the neutrino sector and the charged lepton sector is invariant under @xmath27 symmetry . consider the lagrangian of the model @xcite , @xmath28 after spontaneous symmetry breaking , the charged lepton mass matrix comes out diagonal with @xmath29 , @xmath30 , and @xmath31 . the neutrino sector gives rise to the following dirac and majorana matrices @xmath32 where @xmath33 , @xmath34 . the structure of light neutrino mass matrix can be obtained from see - saw formula : @xmath35 where , @xmath36 this is clear from eq.[ssf ] that @xmath37 is the diagonalising matrix for light neutrino mass matrix @xmath38 . the form of @xmath37 is in eq.[tbmix ] which is nothing but the so called tribimaximal mixing matrix . from eq.[ssf ] we have the eigenvalues of @xmath38 : @xmath39 from eq.[tbmix ] we have the mixing angles @xmath16 , @xmath40 and @xmath18 and from eq.[a4ev ] we get the solar and atmospheric mass squared differences as @xmath41 where @xmath42 , @xmath43 and all parameters are real . from the experiments we know @xmath3 is positive and dictates either @xmath44 or @xmath45 . if @xmath44 , then it has to be small in order to generate small value of @xmath3 provided @xmath46 is not too small as @xmath3 . but small positive @xmath47 corresponds to same order of magnitude of @xmath3 and @xmath4 which is not acceptable according to the experimental results . now @xmath44 only acceptable for @xmath48 and hierarchy of @xmath3 and @xmath4 obtained with the singular nature of @xmath4 as in eq.[a4msd ] near @xmath49 . this corresponds to normal hierarchical mass spectrum . again for @xmath50 , @xmath45 is the physical region . this region of @xmath47 makes @xmath51 which is so called inverted ordering of neutrino mass pattern . again @xmath52 should take small value in order to generate small value of @xmath3 . for one complex parameter @xmath53 , we can write the mass differences in the following form @xmath54 in the complex case , positivity of @xmath3 can be obtained either with @xmath44 and @xmath55 or with @xmath56 and @xmath57 . for the first case with @xmath48 and with @xmath58 one can have normal hierarchical mass spectrum . for the second case hierarchy will be inverted and @xmath59 have to be small . in both case @xmath47 should take the value such that the @xmath60 range also satisfy . the mixing pattern is tri - bi maximal eq.[tbmix ] and it is independent to the fact whether the parameters are real or complex . in this mixing pattern @xmath61 and non - zero complex @xmath0 is a basic requirement to see the non - zero dirac cp violation . now we concentrate on the issue of leptogenesis in this model . the decay of right handed heavy majorana neutrinos to lepton(charged or neutral ) and scalar(charged or neutral ) generate non - zero lepton asymmetry if i ) c and cp are violated , ii)lepton number is violated and iii ) decay of right handed neutrinos are out of equilibrium . we are in the energy scale where @xmath1 symmetry is broken but the sm gauge group remains unbroken . so , the higgs scalars both charged and neutral are physical . the cp asymmetry of decay is characterized by a parameter @xmath62 which is defined as @xmath63 spontaneous @xmath1 symmetry breaking generates right handed neutrino mass and the mass matrix @xmath25 obtained is shown in eq . we need to diagonalize @xmath25 in order to go into the physical basis ( mass basis ) of right handed neutrino . this form of @xmath25 gives the diagonalising matrix in the tri - bi maximal form @xmath37 in eq.[tbmix ] : @xmath64 however , the eigenvalues are not real . we need to multiply one diagonal phase matrix @xmath65 with @xmath37 . hence , diagonalising matrix @xmath66 relates the flavor basis to eigen basis of right handed neutrino : @xmath67 in this basis the couplings of @xmath68 with leptons and scalars are modified and it will be : @xmath69 at the tree level there there are no asymmetry in the decay of right handed neutrinos . due to the interference between tree level and one loop level diagrams , the asymmetry is generated . there are vertex diagram and self energy diagram to contribute to the asymmetry @xcite . the vertex contribution is : @xmath70 \label{vertex}\end{aligned}\ ] ] and the self energy part is : @xmath71 where @xmath72 and @xmath73 the key matrix , whose elements are necessary to calculate leptogenesis , is @xmath74 . in this model @xmath2 is diagonal and proportional to identity . hence , @xmath74 matrix is real diagonal and it is also proportional to identity matrix and it is independent of the form of @xmath75 . the terms for decay asymmetry generated by `` i '' th generation of right handed neutrino @xmath76 for both vertex and self energy contributions are proportional to @xmath77 ( where @xmath78 ) as in eq.[vertex ] and eq . all off - diagonal elements of @xmath74 are zero . so , decay of all three generation of right handed majorana neutrinos could not generate lepton asymmetry . so , in this model of @xmath1 symmetry tri - bi maximal mixing pattern is not compatible with the low energy dirac cp violation as well as high energy cp violation . in order to obtain non - zero @xmath79 , low energy dirac cp violation and leptogenesis we need to break the @xmath1 symmetry through not only spontaneously but also explicitly introducing some soft @xmath1 symmetry breaking ( soft in the sense that the breaking parameter is small to consider @xmath1 as an approximate symmetry ) terms in the lagrangian . we consider minimal breaking of @xmath1 symmetry through a single parameter in a single element of @xmath2 keeping @xmath25 unaltered as @xmath80 we introduce the breaking by small dimensionless parameter @xmath81 to the @xmath82 element of dirac type yukawa term for neutrino . after spontaneous @xmath83 symmetry breaking it modifies only one element @xmath82 of @xmath2 of neutrino . there are nine possibilities to incorporate the breaking parameter @xmath81 in @xmath2 . we know that after spontaneous @xmath1 symmetry breaking , a residual @xmath15 symmetry appears in neutrino sector . there is a special feature of @xmath15 symmetry which ensures one @xmath84 and one maximal @xmath85 mixing angles . there is one task to check whether our newly introduced explicit breaking term can break @xmath15 symmetry or not . this is important because we need non - zero @xmath79 . we have seen that in one case out of the nine possibilities , residual @xmath15 symmetry remains invariant . this is @xmath86 case . in other cases @xmath15 symmetry is broken and one expect non - zero @xmath79 from those cases . primarily , we consider that all parameters are real . we want to study the mixing pattern and want to see its deviation from tri - bi maximal pattern considering experimental value of mass squared differences of neutrinos . we explore all nine cases including @xmath86 case . although @xmath86 case could not generate non - zero @xmath79 , however , we want to see whether this breaking can reduce the tri - bi maximal value of @xmath87 ( @xmath88 ) to its best fit value ( @xmath89 ) or not along with the special feature @xmath90 and @xmath91 . here , we explicitly demonstrate the procedure for a single case and for the other cases expressions for eigenvalues and mixing angles are given in apendix . \(i ) breaking at 22 element : in this case , the structure of @xmath2 is given by @xmath92 and after implementation of see - saw mechanism keeping the same texture of @xmath25 , three light neutrino mass eigenvalues come out as @xmath93 and the three mixing angles come out as @xmath94\nonumber\\ \sin\theta_{13 } & = & { \frac{\epsilon}{3 } } \left({\frac{d}{{\sqrt 2}a } } -{\frac{{\sqrt 2}d}{4a - 2d}}\right ) \label{angel22 } \end{aligned}\ ] ] assuming a relationship between the parameters @xmath95 and @xmath96 as @xmath97 we rewrite in a convenient way the above three mixing angles as @xmath98 \qquad\sin\theta_{13 } = \frac{\e k(1-k)}{3{\sqrt 2}(2-k ) } \label{mangel22 } \end{aligned}\ ] ] and the mass - squared differences are @xmath99\nonumber\\ \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-2\e(2k^2 - 4k-1)\right ] \label{masd22}\end{aligned}\ ] ] where @xmath100 . defining the ratio @xmath101 in terms of mass - squared differences we get @xmath102 } { \left[3k(2-k)-2\e(2k^2 - 4k-1)\right ] } \label{r22 } \end{aligned}\ ] ] which in turn determines the parameter @xmath81 as @xmath103 } { \left[(k-1)^2(2k^2 + 4k+1)+ r(k+1)^2(2k^2 - 4k-1)\right ] } \label{ep22 } \end{aligned}\ ] ] similarly , we have evaluated all other possible cases which we have listed in the appendix . plot of @xmath87 with respect to @xmath47 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] plot of @xmath106 with respect to @xmath47 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] plot of @xmath107 with respect to @xmath47 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] now the @xmath108 parameter is determined in terms of @xmath101 and @xmath47 and we substitute it to the expressions for mixing angles . thus it is possible to explore all three mixing angles @xmath87 , @xmath106 and @xmath107 in terms of @xmath101 and @xmath47 . particularly , the deviation from tri - bimaximal mixing depends only on @xmath101 and @xmath47 . for the best fit values of the solar and atmospheric mass squared differences ( @xmath109 ) , we have shown in fig . [ th12 ] , fig . [ th23 ] and fig . [ th13 ] the variations of @xmath87 , @xmath106 , @xmath107 verses @xmath47 , respectively . we have studied all nine possible cases and shown in the plots . first of all , non - zero value of @xmath107 is obtained if we allow @xmath1 symmetry breaking terms explicitly in any one of the 12 , 13 , 21 , 31 , 22 , 33 element of the dirac neutrino mass matrix and those are the cases of our interest . in the present analysis , we have shown that non - zero @xmath107 is generated in a softly broken @xmath1 symmetric model which leads to deviation from the tri - bimaximal mixing . in @xmath1 symmetric model @xmath107 is zero because of residual @xmath15 symmetry in neutrino sector after spontaneous breaking of @xmath1 symmetry . apart from the explicit breaking of @xmath1 symmetry at 11 element , @xmath15 is broken for all 12 , 13 , 21 , 31 , 22 , 33 , 23 , 32 cases . furthermore , perturbation around 23 , 32 elements also lead to zero value of @xmath107 at the leading order although @xmath15 symmetry is broken , non - zero value is generated if we consider higher order terms of @xmath110 which are too tiny and hence , discarded from our analysis . we include 11 case for completeness which preserves @xmath15 symmetry and hence generates @xmath111 and @xmath91 . it only shifts @xmath87 from the tri bimaximal value , but it can not be able to go towards the best fit value of solar angle , @xmath112 . if @xmath113 , then we get @xmath114 , and thereby , the value of @xmath107 is very small also @xmath87 will hit the exact tri - bimaximal value in some cases . the effect of variation on the mixing angles around @xmath115 are asymmetric . for some cases ( for example 23 , 32 ) @xmath106 changes very fast in the @xmath116 region . so , we explore the mixing angles with the range @xmath117 . we choose the most feasible cases in which perturbation is applied around 12 , 13 elements , because in those cases , variation of @xmath47 encompasses the best - fit values of @xmath87 and @xmath106 . although , in the 21 , 31 cases , the value of @xmath106 touches the best - fit value @xmath85 , however , @xmath87 far apart from the best - fit value . in order to achieve large @xmath107 , we have to choose the 21 , 31 cases , but we have to allow the variation of @xmath87 around as large as @xmath118 . in case of 22 , 33 , the structure of @xmath2 is still diagonal and also we can get larger @xmath107(upto @xmath119 ) and also @xmath87 is within @xmath120 , however , @xmath106 will reach @xmath121 value . in summary , we have shown that non - zero @xmath107 is generated in a @xmath1 symmetric model which leads to deviation from the tri - bimaximal mixing through see - saw mechanism due to the incorporation of an explicit @xmath1 symmetry breaking term in @xmath2 . the breaking is incorporated through a single parameter @xmath108 and we have investigated the effect of such breaking term in all nine elements of @xmath2 . some of them generates still zero value of @xmath107 and rest of the others generated non - zero @xmath107 . we expressed all three mixing angles in terms of one model parameter and showed the variation of all three mixing angles with the model parameter @xmath47 . we find breaking through 12 and 13 elements of @xmath2 are most feasible in view of recent neutrino experimental results . in this section , we consider one of the parameter is complex and out of all nine cases as mentioned earlier , we investigate one suitable case arises due to 13 element perturbation . this is one of the suitable positions of breaking justified from real analysis . again this extension is minimal to generate non - zero cp violation because we consider only one parameter complex . we take @xmath95 as complex : @xmath53 . hence , the form of @xmath2 and @xmath25 under explicit @xmath1 symmetry breaking with complex extension are : @xmath122 using the see - saw mechanism we get the light neutrino mass matrix as @xmath123 we need to diagonalize the @xmath38 to obtain the masses and mixing angles . the eigenvalues are same as we have in the real case and only difference is that the @xmath95 is complex now . we explicitly write down the complex phase in the mass matrix . the obtained eigenvalues are : @xmath124 where we keep terms upto first order in @xmath81 . now with @xmath42 , @xmath125 and keeping term upto first order in @xmath81 we get the three neutrino mass squared as @xmath126 using those expressions we get the mass squared differences and their ratio which are , @xmath127 and @xmath128 . \label{rc13 } \end{aligned}\ ] ] the mixing angles are obtained from diagonalisation of @xmath38 . we solve the equations of the form @xmath129 . these @xmath130 will give the columns of the diagonalising unitary matrix @xmath131 . throughout our calculation we assume that breaking parameter @xmath81 is small . we have the nonzero @xmath132 which is proportional to @xmath81 . so , the values of @xmath133 and @xmath134 will give the solar and atmospheric mixing angles , respectively . the expressions for the mixing angles come out as @xmath135 plot of @xmath87 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in 13 element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values.,height=302 ] @xmath137 @xmath138^{1/2 } \label{t13c13}. \end{aligned}\ ] ] from the expression of mixing angles it is clear that the deviations from tri - bi maximal are first order in @xmath81 . the independent parameters in this model are @xmath96 , @xmath95 , @xmath139 , @xmath81 and @xmath136 . alternatively the independent parameters are @xmath140 , @xmath141 , @xmath47 , @xmath81 and @xmath136 ( where @xmath142,@xmath125 , @xmath143 ) . in the above analysis of light neutrino mass and mixing , scale @xmath140 did not appear explicitly . we have four well measured observable which are @xmath3 , @xmath4 , @xmath87 and @xmath106 , and , thus , in principle it is possible to determine four parameters @xmath141 , @xmath47 , @xmath81 and @xmath136 and we are able to predict the other less known observable such as angle @xmath107 , cp violating parameter @xmath144 etc . it is difficult to get inverse relations of those observable . from the expression of @xmath101 in eq . [ rc13 ] we easily obtain the expression for @xmath81 as plot of @xmath106 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in 13 element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] plot of @xmath107 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] @xmath145}{4\left[r(1+k^2 - 2k\cos\phi)(1+k^2 + 2k\cos\phi)+(2+k^2 + 2k\cos\phi)(1+k^2 - 2k\cos\phi)\right]}.\nonumber\\ \label{epc13 } \end{aligned}\ ] ] now using the relation of @xmath4 with the parameters eq . [ msdc13 ] we get the expression for @xmath46 : @xmath146 where @xmath81 is in the form of eq . [ epc13 ] . thus , @xmath81 and @xmath46 depend on the parameters @xmath47 , @xmath136 and experimentally known @xmath101 . extraction of @xmath47 and @xmath136 from other two known mixing angles is little bit difficult . rather we have plotted @xmath87 and @xmath106 with respect to @xmath47 and @xmath136 and obtain the restriction on the parameter space of @xmath47 and @xmath136 . from the expression of @xmath147 in eq . [ t12c13 ] we are seeing that there is a factor @xmath59 in the denominator . for @xmath116 there will be a @xmath136 for which the quantity @xmath59 becomes zero . hence , we should keep @xmath45 . again the factor @xmath59 should be small to ensure that @xmath81 is also small . it justifies our whole analysis because we consider first order perturbation as we considered symmetry @xmath1 remains approximate . we consider the range @xmath148 and @xmath149 . from fig . [ th12c13 ] we see that @xmath87 changes from the tri - bi maximal value @xmath150 to @xmath151 . near @xmath152 it crosses the best fit value @xmath112 . we have plotted @xmath106 in fig . [ th23c13 ] . the variation of @xmath106 is from @xmath153 to @xmath154 for the same range of @xmath47 and @xmath136 . the best fit value @xmath155 is remain within range of variation and it is in the low @xmath47 low @xmath136 region . the plot of @xmath107 is in fig . [ th13c13 ] . value of @xmath107 remains within @xmath156 for the same range of @xmath47 and @xmath136 and the model predicts @xmath107 is very small but non - zero . question may arise whether such small value of @xmath107 can generate observable cp violation or not . keeping all those constraints in view next we explore the parameter space of cp violation parameter @xmath157 . the parameter @xmath157 defined as @xcite @xmath158 } { \delta m^2_{21}\delta m^2_{31}\delta m^2_{32 } } \label{jcp1}\end{aligned}\ ] ] where @xmath159 , @xmath160 is dirac phase . this @xmath161 is associated with cp violation in neutrino oscillation and is directly related to dirac phase of mixing matrix . plot of @xmath157 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in 13 element of @xmath2 in the unit of @xmath162 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] plot of @xmath163 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] the from eq . ( [ mnuc ] ) we can express @xmath38 matrix in terms of @xmath81 and @xmath141 and @xmath47 and @xmath136 . with the expressions for @xmath81 and @xmath141 we can easily obtain the @xmath38 and hence @xmath164 completely in terms of @xmath47 and @xmath136 . hence , similar to the mixing angles @xmath144 will be also only function of @xmath47 and @xmath136 . we have plotted @xmath144 in fig . [ fjcp1 ] where the values are normalized by a factor @xmath162 . for the same range of @xmath47 and @xmath136 the model predicts @xmath144 up to the order of @xmath165 which is appreciable to observe through the forthcoming experiments . inverting the expression of @xmath144 , the phase @xmath160 is extracted in terms of @xmath47 and @xmath136 and it is plotted in fig . [ dcp13 ] . we see that the value of @xmath160 is large upto @xmath166 and , thereby , compensates small @xmath107 effect in @xmath144 and makes it observable size . one important discussion we have to make about the range of @xmath136 and @xmath47 . one can ask why we are keeping ourselves small range of those parameters where larger @xmath136 can enhance the @xmath107 and @xmath144 . we have studied that the larger value of @xmath136 and also @xmath47 in negative become responsible for breaking the analytic bound @xmath167 . so , we keep ourselves in shrinked parameter space which keep @xmath87 and @xmath106 in exceptionally good values according to the experiment and also can able to generate observable cp violation instead of small @xmath107 . another thing we want to point out that negative value of @xmath136 equally acceptable as far as it is small . it could not change mixing angles because their expressions depend on @xmath136 through @xmath168 and @xmath169 . only @xmath144 and @xmath160 will change in sign which are unsettled according to the experiments . at the end we want to check whether the range of @xmath47 and @xmath136 can satisfy the double beta decay bound @xmath170 ev . in our model expression for this quantity is : @xmath171 and it will be also only function of @xmath47 and @xmath136 . we plot this in fig . [ mee13 ] and it remains well below the experimental upper bound . again we want to discuss about the mass pattern . throughout our whole analysis in real as well as complex case we keep @xmath3 and @xmath4 to their best fit value and take the negative sign of @xmath4 . it corresponds to so called inverted ordering of neutrino mass . it is the feature near @xmath115 . it is necessary to keep @xmath172 . why we so fond of this region of @xmath47 instead of region @xmath173 which can give the normal hierarchical mass spectrum . the reason is that the inverted ordering corresponds to the light neutrino mass scale @xmath174 where @xmath175 for normally ordered mass spectrum . so , from the point of view of observable cp violation , it is inevitable to choose larger value of @xmath141 because @xmath176 . so , inverted hierarchical mass spectrum compatible with the observable cp violation . now we extend our study of this model to leptogenesis . we want to see whether we can have appreciable leptogenesis compatible with baryon asymmetry for the same parameter space @xmath47 and @xmath136 after successful low energy data analysis for the feasible value of scale @xmath140 . plot of @xmath177 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values . , height=302 ] after successful predictions of low energy neutrino data we want to see whether this model can generate non - zero lepton - asymmetry with proper size and sign to describe baryon asymmetry . we keep the same right handed neutrino mass matrix @xmath25 as before . the change only appear in dirac type yukawa coupling and hence in @xmath2 also . the diagonalisation of @xmath25 gives @xmath178 and hence the masses of the right handed neutrino are @xmath179 and the phases are @xmath180 the explicit form of diagonalising matrix is @xmath181 where the expressions for the phases are given in eq . ( [ majp ] ) . the @xmath2 matrix is no longer diagonal after explicit breaking of @xmath1 symmetry . in the mass basis of right handed neutrino the modified dirac mass term is @xmath182 . hence the relevant matrix for describing leptogenesis is : @xmath183 to calculate lepton asymmetry as in eq . ( [ vertex ] ) and eq . ( [ self ] ) we need to calculate following quantities from matrix @xmath74 : @xmath184 calculating @xmath72 from eq . [ mrev ] , @xmath185 from eq . ( [ hc13 ] ) and taking @xmath77 from eq . ( [ imh ] ) we calculate the self energy part of lepton asymmetry from eq . ( [ self ] ) and vertex part of lepton asymmetry from eq . ( [ vertex ] ) . adding both we obtain the following decay asymmetry of right handed neutrinos for all three generations @xmath186.\nonumber\\ \label{las1 } \end{aligned}\ ] ] @xmath187 \label{las2 } \end{aligned}\ ] ] @xmath188\nonumber\\ \label{las3 } \end{aligned}\ ] ] cp asymmetry parameters @xmath62 are related to the leptonic asymmetry parameters through @xmath189 as @xcite @xmath190 where @xmath191 is the lepton number density , @xmath192 is the anti - lepton number density , @xmath193 is the entropy density , @xmath194 is the dilution factor for the cp asymmetry @xmath62 and @xmath195 is the effective number of degrees of freedom @xcite at temperature @xmath196 . value of @xmath195 in the sm with three right handed majorana neutrinos and one extra higgs doublet is @xmath197 . the baryon asymmetry @xmath198 produced through the sphaleron transmutation of @xmath189 , while the quantum number @xmath199 remains conserved , is given by @xcite @xmath200 where @xmath201 is the number of fermion families and @xmath202 is the number of higgs doublets . the quantity @xmath203 in eq . ( [ barasym ] ) for sm with two higgs doublet . now we introduce the relation between @xmath198 and @xmath204 , where @xmath204 is the baryon number density over photon number density @xmath205 . the relation is @xcite @xmath206 where the zero indicates present time . now using the relations in eqs.([leptasym],[barasym ] , [ yetar ] ) , @xmath203 and @xmath207 we have @xmath208 this dilution factor @xmath194 approximately given by @xcite @xmath209 where @xmath210 is the decay width of @xmath211 and @xmath212 is hubble constant at @xmath196 . their expressions are @xmath213 where @xmath214 , @xmath215gev and @xmath216gev . thus we have @xmath217 for our model @xmath218 , @xmath219 and @xmath220 are @xmath221 plot of baryon asymmetry @xmath222 in unit of @xmath223 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values and have plotted for mass scale of right handed neutrino @xmath224 gev , height=302 ] plot of baryon asymmetry @xmath222 in unit of @xmath223 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values and have plotted for mass scale of right handed neutrino @xmath225 gev , height=302 ] plot of baryon asymmetry @xmath222 in unit of @xmath223 with respect to @xmath47 and @xmath136 for the breaking of @xmath1 in `` 13 '' element of @xmath2 . we keep @xmath104 and @xmath105 to their best fit values and have plotted for mass scale of right handed neutrino @xmath226 gev , height=302 ] we combine all three plots of baryon asymmetry for three right handed neutrino mass scale along with the wmap value of baryon asymmetry @xmath7 which is the plane surface.,height=302 ] where @xmath227 , @xmath228 . using eq . ( [ k1k2k3 ] ) into eq . ( [ kppa ] ) and from the expression of @xmath62 we can say that apart from logarithmic factor , @xmath229 and @xmath230 . so , baryon asymmetry will be independent of @xmath141 and @xmath231 . they only appear through logarithmic factor in @xmath194 . we consider @xmath232 . substituting @xmath141 from eq . ( [ m013c ] ) , @xmath81 from eq . ( [ epc13 ] ) and considering @xmath140 some specific value into the expressions of @xmath62 , @xmath194 we can have baryon asymmetry as function of @xmath47 and @xmath136 only . in fig . [ basymp1 ] , fig . [ basymp2 ] and fig . [ basymp3 ] we have plotted @xmath204 as function of @xmath47 and @xmath136 in the unit of @xmath223 for three @xmath140 values , @xmath224 gev , @xmath225 gev and @xmath226 gev rspectively . we have seen that the experimental value of @xmath233 is obtainable in our model within the same range of @xmath234 and @xmath136 as in the low energy case . to see more explicitly the baryon asymmetry plots we combine all three plots along with the observed baryon asymmetry value @xmath7 which corresponds the plane surface in fig . [ basym0 ] . the observed wmap value of the baryon asymmetry curve intersect the lower curve ( for @xmath224 gev ) near the boundary of @xmath234 and @xmath136 variation . so , lower value of @xmath140 could not generate observable baryon asymmetry . in the intersection region @xmath235 and @xmath236 . if we allow that much of variation of @xmath87 and @xmath106 , we can have large low energy cp violation as well as baryon asymmetry with proper size and sign with @xmath224 gev which is near the upper bound of right handed neutrino mass scale for generation of lepton as well as baryon asymmetry . if we more relax , we can easily see that the intersection of experimental and theoretical curve for @xmath225 gev and @xmath226 gev is in the lower value of @xmath47 and @xmath136 where well known neutrino mixing angles are more closer to their best fit value . let us give a close look to the plot of @xmath204 in fig . [ basymp1 ] , fig . [ basymp2 ] and fig . [ basymp3 ] near very low @xmath136 and @xmath237 region . this is the region where @xmath238 become very large . from the expression of @xmath239 and @xmath240 it is clear that @xmath239 and @xmath240 both are singular at @xmath241 . this corresponds to equality of the masses @xmath242 or @xmath243 . this singularity can be avoided considering finite decay width of right handed neutrinos . we can able to maximize @xmath239 and @xmath240 and hence @xmath204 using resonant condition @xmath244 . but , as we have already obtained the observed baryon asymmetry without resonance , it is not necessary to think about so finely tuned condition . again in the region where the resonant condition is applicable , the @xmath144 is miserably small to observe through any experiments . contour plot of baryon asymmetry @xmath222 , @xmath87 and , @xmath106 in @xmath136-@xmath47 plane for @xmath104 and @xmath105 to their best fit values and for mass scale of right handed neutrino @xmath224 gev.,height=226 ] contour plot of baryon asymmetry @xmath222 , @xmath87 and , @xmath106 in @xmath136-@xmath47 plane for @xmath104 and @xmath105 to their best fit values and for mass scale of right handed neutrino @xmath225 gev.,height=226 ] contour plot of baryon asymmetry @xmath222 , @xmath87 and , @xmath106 in @xmath136-@xmath47 plane for @xmath104 and @xmath105 to their best fit values and for mass scale of right handed neutrino @xmath226 gev.,height=226 ] we end our analysis with the help of three contour plots of baryon asymmetry @xmath245 for three scales @xmath224 gev , @xmath225 gev and , @xmath226 gev in @xmath136-@xmath47 plane . we insert the contours of @xmath106 and @xmath87 and manage to find the intersection of three contours for some reasonable value of @xmath106 and @xmath87 . case(i ) : @xmath224 gev , from fig . [ fig : contplot1 ] , we are seeing that three contours @xmath246 , @xmath247 and , @xmath245 are intersecting at a point ( @xmath248 , @xmath249 ) in @xmath136-@xmath47 plane . so , the mixing angles are within nearly @xmath156 variation about the best fit values . obtained @xmath47 , @xmath136 value gives @xmath250 , @xmath251 , @xmath252 , @xmath253 ev , @xmath254 ev , @xmath255 and @xmath256 . case ( ii ) : @xmath225 gev , now from fig . [ fig : contplot2 ] , we have the intersection of the contour @xmath245 with the contours @xmath257 , @xmath258 at ( @xmath259 , @xmath260 ) in @xmath136-@xmath47 plane . so , @xmath87 and @xmath106 are more closer to their best fit values ( nearly @xmath261 deviation from their best fit value ) . at this point we have @xmath262 , @xmath263 , @xmath264 , @xmath265 ev , @xmath266 ev , @xmath267 and @xmath268 . case ( iii ) : @xmath226 gev , from the fig . [ fig : contplot3 ] , the contours @xmath245 , @xmath269 and , @xmath270 have crossed in @xmath136-@xmath47 plane at ( @xmath271 , @xmath272 ) . so , higher scale of right handed neutrino mass helps to have very good value of @xmath87 and @xmath106 . at the intersection point , we get @xmath273 , @xmath274 , @xmath275 , @xmath276 ev , @xmath277 ev , @xmath278 and @xmath279 . the small value of @xmath81 compatible with all experimental results . so , we can demand that @xmath1 is an approximate symmetry . in this model everything is determinable in terms of parameter @xmath47 and @xmath136 . well measured quantities fix the value of those parameters . so , value of the rest of the physical quantities ( some of them not so well measured in experiment like @xmath107 , @xmath280 and some of them yet to measure in experiment like @xmath144 , @xmath160 and the majorana phases also ) are obtainable in this model . question may arise whether we can have any relations among the phases in this model or not . first of all , there are two kinds phases , low energy and high energy phases . high energy phases are responsible to generate the lepton asymmetry . the low energy phases are responsible for determining low energy leptonic cp violation . low energy phases are in two type , one is the lepton no preserving cp violating phase @xmath160 and another two are the lepton no breaking cp violating phases . in general , all phases are independent , meaning that there are no correlation among the phases between high and low energy sector , and also phases inside a particular sector are not correlated . for three generations of neutrinos , there are three key phases which are responsible for leptogenesis . those are phases in @xmath281 , @xmath282 , and @xmath283 . from matrix @xmath74 given in eq . ( [ hc13 ] ) , we have @xmath284 it leads to the relation , @xmath285 for a given @xmath47 and @xmath136 the @xmath286 and @xmath287 are known from eq . ( [ majp ] ) . hence @xmath288 , @xmath289 and @xmath290 are individually determinable phases . but , in this model values of those high energy phases follow the relation given in eq . ( [ hprl ] ) . now , let us give a fresh look to the leptonic mixing matrix . in eq . ( [ mnuc ] ) we have given the tri - bimaximal rotated form of the neutrino mass matrix . keeping terms upto first order in @xmath81 we obtain the diagonalising matrix in the following form , @xmath291 where the @xmath292 , @xmath293 , @xmath294 and the associated phases are completely known function of @xmath47 and @xmath136 . an additional phase matrix @xmath295 is needed to make masses of light neutrino real , from eq . ( [ msevc ] ) we have the phase matrix @xmath296 to obtain the ckm form of mixing matrix we need to rotate @xmath297 by two diagonal phase matrix , let @xmath298 and @xmath299 . so , we have @xmath300 now the with @xmath107 small we can write @xmath301 and more six relations . but these three are sufficient for our discussions . the phases associated to @xmath297 elements , like @xmath302 , @xmath303 , and @xmath304 associated to @xmath305 , @xmath306 and @xmath307 respectively , are completely determinable in terms of @xmath47 , @xmath136 using functional form of @xmath81 , @xmath292 , @xmath293 , @xmath294 and their associates phases . now from the eq . ( [ ckm ] ) we obtain the following phase relations @xmath308 now the form of total mixing matrix is , @xmath309 this phase part in the parenthesis can be absorbed to charged lepton fields and the remaining part gives the leptonic mixing matrix of the form @xmath310 , where the @xmath311 and @xmath312 are the two majorana phases of leptonic mixing matrix . from eq . ( [ ffmv ] ) and using relations in eq . ( [ prl1 ] ) and eq . ( [ phh ] ) , we have the majorana phases , @xmath313 where @xmath314 and @xmath315 are known function of @xmath47 and @xmath136 . in eq . ( [ majp2 ] ) we have the relation of low energy and high energy phases . so , in our model we have correlation among the cp violating phases . we have shown that non - zero @xmath0 is generated in a softly broken @xmath1 symmetric model through see - saw mechanism incorporating single parameter perturbation in @xmath2 in single element . first , we have studied all possible nine cases to explore the mixing angles considering all model parameters real . the extent of @xmath107 investigated , keeping the experimental values of present solar and atmospheric mixing angles . among all nine possible texture of @xmath2 some of them generates non - zero @xmath107 . out of those non - zero @xmath107 generating textures of @xmath2 we find that breaking at 12 and 13 elements are encompassing the best values of @xmath87 and @xmath106 . however , the reach of @xmath107 in those cases are around @xmath261 . considering one of the parameter complex we extend our analysis with one of the most suitable texture of @xmath2 with breaking at 13 element . we have calculated mixing angles and neutrino mass squared differences in terms four model parameters ( @xmath141 , @xmath81 , @xmath47 , @xmath136 ) . we restrict model parameters utilising the well measured quantities @xmath3 , @xmath4 , @xmath87 and @xmath106 and we have obtained @xmath107 ( upto @xmath156 ) and large @xmath144 ( @xmath316)and @xmath317 well below the present experimental upper bound . in addition to that a large @xmath160 is also obtained . further study on leptogenesis is also done and the present wmap value of baryon asymmetry is obtained for a right handed neutrino mass scale @xmath8 gev . in our model , we have seen that the phases responsible for the leptogenesis are correlated . we also find out the relations among low energy cp violating phases and the lepton asymmetry phases . small @xmath1 symmetry breaking parameter @xmath81 , is sufficient to describe the all low energy neutrino data and high energy cp violation ( leptogenesis ) . so , @xmath1 symmetry is an approximate symmetry . here we consider breaking of @xmath1 symmetry in all other entries of @xmath2 . case ( i ) is already discussed in the text . 0.1 in ( ii)breaking at 11 element : in this case @xmath2 is given by @xmath318 the mass eigenvalues are @xmath319 and the three mixing angles come out as @xmath320 the solar and atmospheric mass differences and their ratio are @xmath321\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-4\e(k-1)^2\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+4\e(k^2 + 2k-1)\right ] } { \left[3k(2-k)-4\e(k-1)^2\right ] } \label{msd11 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath322 } { \left[r(k+1)^2+k^2 + 2k-1\right ] } \label{ep11 } \end{aligned}\ ] ] iii ) breaking at 33 element : in this case , the structure of @xmath2 is given by @xmath323 mass eigenvalues are @xmath324 and the angels are @xmath325 \quad \sin\theta_{13 } = -\frac{\e k(1-k)}{3{\sqrt 2}(2-k ) } \label{angel33 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath326\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-2\e(2k^2 - 4k-1)\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+2\e(2k^2 + 4k+1)\right ] } { \left[3k(2-k)-2\e(2k^2 - 4k-1)\right ] } \label{msd33 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath327 } { \left[(k-1)^2(2k^2 + 4k+1)+ r(k+1)^2(2k^2 - 4k-1)\right ] } \label{ep33 } \end{aligned}\ ] ] iv ) breaking at 12 element : in this case , the structure of @xmath2 is given by @xmath328 mass eigenvalues are @xmath329 and the three mixing angles come out as @xmath330 \quad \sin\theta_{13 } = -\frac{\e}{3\sqrt 2}\frac{k^2-k-3}{(k-2)}\nonumber\\ \label{angel12 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath331\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-4\e(1-k)^2)\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+4\e(k^2 + 2k+2)\right ] } { \left[3k(2-k)-4\e(1-k)^2\right ] } \label{msd12 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath332 } { \left[(k^2 + 2k+2)+ r(k+1)^2\right ] } \label{ep12 } \end{aligned}\ ] ] v ) breaking at 13 element : in this case , the structure of @xmath2 is given by the mass eigenvalues are @xmath329 and the three mixing angles come out as @xmath334\nonumber\\ \sin\theta_{13}&= & -\frac{\e}{3\sqrt 2}\frac{k^2-k-3}{(k-2 ) } \label{angel13 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath331\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-4\e(1-k)^2)\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+4\e(k^2 + 2k+2)\right ] } { \left[3k(2-k)-4\e(1-k)^2\right ] } \label{msd13 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath332 } { \left[(k^2 + 2k+2)+ r(k+1)^2\right ] } \label{ep13 } \end{aligned}\ ] ] vi ) breaking at 23 element : in this case , the structure of @xmath2 is given by @xmath335 the mass eigenvalues are @xmath336 and the three mixing angles come out as @xmath337 \quad\quad \sin\theta_{13 } = 0 \label{angel23 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath326\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-2\e(2k^2 - 4k+5)\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+2\e(2k^2 + 4k+1)\right ] } { \left[3k(2-k)-2\e(2k^2 - 4k+5)\right ] } \label{msd23 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath327 } { \left[r(1+k)^2(2k^2 - 4k+5)+ ( k-1)^2(2k^2 + 4k+1)\right ] } \label{ep23 } \end{aligned}\ ] ] vii ) breaking at 21 element : in this case , the structure of @xmath2 is given by @xmath338 the mass eigenvalues are @xmath329 and the three mixing angles come out as @xmath339\nonumber\\ \sin\theta_{13 } & = & \frac{\epsilon}{3\sqrt 2}\frac{(3+k^2 - 4k ) } { ( 2-k ) } \label{angel21 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath331\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-4\e(1-k)^2\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+4\e(k^2 + 2k+2)\right ] } { \left[3k(2-k)-4\e(1-k)^2\right ] } \label{msd21 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath340 } { \left[r(1+k)^2+k^2 + 2k+2 \right ] } \label{ep21 } \end{aligned}\ ] ] viii ) breaking at 31 element : in this case , the structure of @xmath2 is given by @xmath341 the mass eigenvalues are @xmath329 and the three mixing angles come out as @xmath342\nonumber\\ \sin\theta_{13 } & = & -\frac{\e}{3\sqrt{2 } } \frac{3+k^2 - 4k}{2-k } \label{angel31 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath331\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-4\e(1-k)^2\right]\nonumber\\ & & r = \frac{\ds}{\da}= \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+4\e(k^2 + 2k+2)\right ] } { \left[3k(2-k)-4\e(1-k)^2\right ] } \label{msd31 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath332 } { \left[r(1+k)^2+k^2 + 2k+2 \right ] } \label{ep31 } \end{aligned}\ ] ] ix ) breaking at 32 element : in this case , the structure of @xmath2 is given by @xmath343 the mass eigenvalues are @xmath336 and the three mixing angles come out as @xmath344 \quad \sin\theta_{13 } = 0 \label{angel32 } \end{aligned}\ ] ] the solar and atmospheric mass differences and their ratio are @xmath326\nonumber\\ & & \da = \frac{m_0 ^ 2}{3(1-k)^2}\left [ 3k(2-k)-2\e(2k^2 - 4k+5)\right]\nonumber\\ & & r = \frac{\ds}{\da } = \frac{(k -1)^2}{(k+1)^2 } \frac{\left[3k(k+2)+2\e(2k^2 + 4k+1)\right ] } { \left[3k(2-k)-2\e(2k^2 - 4k+5)\right ] } \label{msd32 } \end{aligned}\ ] ] the obtained expression for @xmath81 in terms of model parameter @xmath47 and experimentally known @xmath101 : @xmath327 } { \left[r(1+k)^2(2k^2 - 4k+5)+(1-k)^2 ( 2k^2 + 4k+1 ) \right ] } \label{ep32 } \end{aligned}\ ] ] 99 s. t. petcov , phys . b * 110 * , 245 ( 1982 ) ; for more recent studies see , e.g. , r. barbieri _ et al . _ , jhep * 9812 * , 017 ( 1998 ) ; a. s. joshipura and s. d. rindani , eur . phys . j. c * 14 * , 85 ( 2000 ) ; r. n. mohapatra , a. perez - lorenzana and c. a. de sousa pires , phys . b * 474 * , 355 ( 2000 ) ; q. shafi and z. tavartkiladze , phys . b * 482 * , 145 ( 2000 ) . l. lavoura , phys . rev . d * 62 * , 093011 ( 2000 ) ; w. grimus and l. lavoura , phys . d * 62 * , 093012 ( 2000 ) ; t. kitabayashi and m. yasue , phys . rev . d * 63 * , 095002 ( 2001 ) ; a. aranda , c. d. carone and p. meade , phys . d * 65 * , 013011 ( 2002 ) ; k. s. babu and r. n. mohapatra , phys . b * 532 * , 77 ( 2002 ) ; h. j. he , d. a. dicus and j. n. ng , phys . b * 536 * , 83 ( 2002 ) h. s. goh , r. n. mohapatra and s. p. ng , phys . b * 542 * , 116 ( 2002 ) ; g. k. leontaris , j. rizos and a. psallidas , phys . b * 597 * , 182 ( 2004 ) . l. lavoura and w. grimus , jhep * 0009 * , 007 ( 2000 ) ; hep - 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we have shown that non - zero @xmath0 is generated in a see - saw type softly broken @xmath1 symmetric model through a single parameter perturbation in @xmath2 in a single element . we have explored all possible 9 cases to study the neutrino mixing angles considering the best fitted values of @xmath3 and @xmath4 with all parameters real . we have extended our analysis for the complex case and demonstrated large low energy cp violation ( @xmath5 ) and @xmath6 in addition to mixing and mass pattern . we have also investigated leptogenesis and for a reasonable choice of model parameters compatible with low energy data , wmap value of baryon asymmetry @xmath7 is obtained for right handed neutrino mass scale @xmath8 gev . we have obtained a relation among the phases responsible for leptogenesis and have shown that those phases also have correlations with low energy cp violating phases . pacs number(s ) : 14.60.pq , 11.30.hv , 98.80.cq

arxiv

organic molecular crystals , namely crystals composed of organic molecules held together by weak van der waals forces , are emerging as excellent candidates for fabricating nanoscale devices . these have potential application in electronics and optoelectronics in particular in areas such as solar energy harvesting , surface photochemistry , organic electronics and spintronics @xcite . a feature common to such class of devices is that they are composed from both an organic and inorganic component , where the first forms the active part of the device and the second provides the necessary electrical contact to the external circuitry . clearly the electronic structure of the interface between these two parts plays a crucial role in determining the final device performance and needs to be understood carefully . in particular it is important to determine how charge transfers between the organic and the inorganic component and the energies at which the transfer takes place . this is a challenging task , especially in the single - molecule limit . upon adsorption on a substrate , the electron addition and removal energies of a molecule change value from that of their gas phase counterparts . this is expected since , when the molecule is physisorbed on a polarisable substrate , the removal ( addition ) of an electron from ( to ) the molecule gives rise to a polarisation of the substrate . the image charge accumulated on the substrate in the vicinity of the molecule alters the addition or removal energy of charge carriers from the molecule . a common way to calculate the addition and removal energies is to use a quasiparticle ( qp ) description . within the qp picture , one ignores the effects of relaxation of molecular orbitals due to addition or removal of electrons and consequently takes the relative alignment of the metal fermi level , @xmath1 , with either the lowest unoccupied molecular orbital ( lumo ) and highest occupied molecular orbital ( homo ) of a molecule as removal energy . this effectively corresponds to associate the electron affinity and the ionization potential respectively to the lumo and homo of the molecule . the adequacy of the qp description then depends on the level of theory used to calculate the energy levels of the homo and lumo . if the theory of choice is density functional theory ( dft ) @xcite , then a number of observations should be made . firstly , it is important to note that except for the energy of the homo , which can be rigorously interpreted as the negative of the ionization potential @xcite , in general the kohn - sham orbitals can not be associated to qp energies . this is , however , commonly done in practice and often the kohn - sham qp levels provide a good approximation to the true removal energies , in particular in the case of metals . for molecules unfortunately the situation is less encouraging with the local and semi - local approximations of the exchange and correlation functional , namely the local density approximation ( lda ) and the generalized gradient approximation ( gga ) , performing rather poorly even for the homo level . such situation is partially corrected by hybrid functionals @xcite or by functional explicitly including self - interaction corrections @xcite , and extremely encouraging results have been recently demonstrated for range separated functionals @xcite . the calculation of the energy levels alignment of a molecule in the proximity of a metal , however , presents additional problems . in fact , the formation of the image charge , although it is essentially a classical electrostatic phenomenon , has a completely non - local nature . this means that unless a given functional is explicitly non - local it will in general fail in capturing such effect . the most evident feature of such failure is that the position of the homo and lumo changes very little when a molecule approaches a metallic surface @xcite . such failure is typical of the lda and gga , and both hybrid and self - interaction corrected functionals do not improve much the situation . a possible solution to the problem is that of using an explicit many - body approach to calculate the qp spectrum . this is for instance the case of the gw approximation @xcite , which indeed is capable of capturing the energy levels renormalization due to the image charge effect @xcite . the gw scheme , however , is highly computationally demanding and can be applied only to rather small systems . this is not the case for molecules on surfaces , where the typical simulation cells have to include several atomic layers of the metal and they should be laterally large enough to contain the image charge in full . this , in addition to the gw necessity to compute a significant fraction of the empty states manifold , make the calculations demanding and it is often not simple even to establish whether convergence has been achieved . in this paper we approach the problem of evaluating the charge transfer energies of an organic molecule physisorbed on an inorganic substrate with the help of a much more resource - efficient alternative , namely constrained density functional theory ( cdft ) @xcite . in cdft , one transfers one electron from the molecule to the substrate ( and vice versa ) and calculates the difference in energy with respect to the locally charge neutral configuration ( no excess of charge either on the molecule or the substrate ) @xcite . as such cdft avoids the calculation of a qp spectrum , which is instead replaced by a series of total energy calculations for different charge distributions this approach is free of any interpretative issues and benefits from the fact that even at the lda level the total energy is usually an accurate quantity . finally , it is important to remark that , for any given functional , cdft is computationally no more demanding than a standard dft calculation , so that both the lda and the gga allow one to treat large systems and to monitor systematically the approach to convergence . here we use the cdft approach to study the adsorption of molecules on a 2-dimensional ( 2d ) metal in various configurations . it must be noted that in contrast to a regular 3d metal , in a 2d one the image charge induced on the substrate is constrained within a one - atom thick sheet . this means that electron screening is expected to be less efficient than in a standard 3d metal and the features of the image charge formation in general more complex . in particular we consider here the case of graphene , whose technological relevance is largely established @xcite . most importantly for our work , recently graphene has been used as template layer for the growth of organic crystals @xcite . it is then quite important to understand how such template layer affects the level alignment of the molecules with the metal . as a model system we consider a simple benzene molecule adsorbed on a sheet of graphene . this has been studied in the past @xcite , so that a good description of the equilibrium distance and the corresponding binding energy of the molecule in various configurations with respect to the graphene sheet are available . furthermore , a @xmath2 study for some configurations exists @xcite , so that our calculated qp gap can be benchmarked . our calculations show that the addition and removal energies decrease in absolute value as the molecule is brought closer to the graphene sheet . such decrease can be described with a classical electrostatic model taking into account the true graphene dielectric constant . as it will be discussed , a careful choice of the substrate unit cell is necessary to ensure the inclusion of the image charge , whose extension strongly depends on the molecule - substrate distance . we also reveal that the presence of defects in the graphene sheet , such as a stone - wales one , does not significantly alter the charge transfer energies . in realistic situations , _ e.g. _ at the interface between a molecular crystal and an electrode , a molecule is surrounded by many others , which might alter the level alignment . we thus show calculations , where neighboring molecules are included above , below and in the same plane of the one under investigation . interestingly , our results suggest that the charge transfer states are weakly affected by the presence of other molecules . in order to find the ground state energy of a system , kohn - sham dft minimises a universal energy functional @xmath3=\sum_{\sigma}^{\alpha,\beta}\sum_{i}^{n_\sigma}\langle \phi_{i \sigma}|-\frac{1}{2}\nabla^{2}|\phi_{i \sigma}\rangle+\int d\mathbf{r}v_n(\mathbf{r})\rho(\mathbf{r})+j[\rho]\\ + e_\mathrm{xc}[\rho^{\alpha},\rho^{\beta}]\ : , \end{split}\ ] ] where @xmath4 , @xmath5 and @xmath6 denote respectively the hartee , exchange - correlation ( xc ) and external potential energies . the kohn - sham orbitals , @xmath7 , for an electron with spin @xmath8 define the non - interacting kinetic energy @xmath9 , while @xmath10 is the total number of electrons with spin @xmath8 . the electron density , is then given by @xmath11 . in contrast to regular dft , in cdft one wants to find the ground state energy of the system subject to an additional constraint of the form @xmath12 where @xmath13 is a weighting function that describes the spatial extension of the constraining region and @xmath14 is the number of electrons that one wants to confine in that region . in our case @xmath15 is set to 1 inside a specified region and zero elsewhere . in order to minimise @xmath16 $ ] subject to the constraint , we introduce a lagrange multiplier @xmath17 and define the constrained functional @xcite @xmath18=e[\rho]+v_\mathrm{c}\left(\sum_{\sigma}\int w_\mathrm{c}^{\sigma}(\mathbf{r})\rho^{\sigma}(\mathbf{r})d\mathbf{r}-n_\mathrm{c}\right)\ ] ] now the task is that of finding the stationary point of @xmath19 $ ] under the normalization condition for the kohn - sham orbitals . this leads to a new set of kohn - sham equations @xmath20\phi_{i\sigma}\\ = \epsilon_{i\sigma}\phi_{i\sigma}\ : , \end{split}\ ] ] where @xmath21 is the exchange and correlation potential . equation ( [ equ4 ] ) does not compute @xmath17 , which remains a parameter . however , for each value of @xmath17 it produces a unique set of orbitals corresponding to the minimum - energy density . in this sense we can treat @xmath19 $ ] as a functional of @xmath17 only . it can be proved that @xmath19 $ ] has only one stationary point with respect to @xmath17 , where it is maximized @xcite . most importantly the stationary point satisfies the constraint . one can then design the following procedure to find the stationary point of @xmath19 $ ] : ( i ) start with an initial guess for @xmath22 and @xmath17 and solve eq . ( [ equ4 ] ) ; ( ii ) update @xmath17 until the constraint eq . ( [ equ2 ] ) is satisfied ; ( iii ) start over with the new @xmath17 and a new set of @xmath23s . here we use cdft to calculate the charge transfer energy between a benzene molecule and a graphene sheet . for any given molecule - to - substrate distance , @xmath24 , we need to perform three different calculations : 1 . a regular dft calculation in order to determine the ground state total energy @xmath25 and the amount of charge on each subsystem ( _ i.e. _ on the molecule and on the graphene sheet ) 2 . a cdft calculation with the constraint that the graphene sheet contains one extra electron and the molecule contains one hole . this gives the energy @xmath26 . 3 . a cdft calculation with the constraint that the graphene sheet contains one extra hole and the molecule one extra electron . this gives the energy @xmath27 . the charge transfer energy for removing an electron from the molecule and placing it on the graphene sheet is then @xmath28 . similarly , that for the transfer of an electron from the graphene sheet to the molecule is @xmath29 . since in each run the cell remains charge neutral , there is no need here to apply any additional corrections . however , we have to keep in mind that this method is best used when the two subsystems are well separated so that the amount of charge localized on each subsystem is a well defined quantity . in our calculations we use the cdft implementation @xcite for the popular dft package siesta@xcite , which adopts a basis set formed by a linear combination of atomic orbitals ( lcao ) . the constrain is introduced in the form of a projection over a specified set of basis orbitals and in particular use the lowdin projection scheme . throughout this work we adopt double - zeta polarized basis set with an energy cutoff of 0.02 ry . the calculations are done with norm - conserving pseudopotential and the lda is the exchange - correlation functional of choice . a mesh cutoff of 300 ry has been used for the real - space grid . we impose periodic boundary conditions with different cell - sizes and the @xmath30-space grid is varied in accordance with the size of the unit cell . for instance , an in plane 5@xmath315 @xmath30-grid has been used for a 13@xmath3113 graphene supercell . we begin this section with a discussion on the equilibrium distance for a benzene molecule adsorbed on graphene . this is obtained by simply minimizing the total energy difference @xmath32 , where @xmath33 is the total energy for the cell containing benzene on graphene , while @xmath34 ( @xmath35 ) is the total energy of the same cell when only the benzene ( graphene ) is present . this minimization is performed for two different orientations of the benzene molecule with respect to the graphene sheet : the _ hollow _ ( h ) configuration , in which all the carbon atoms of the benzene ring are placed exactly above the carbon atoms of graphene , and the _ stack _ ( s ) configuration , in which alternate carbon atoms of the benzene molecule are placed directly above carbon atoms of the graphene sheet [ see fig . [ fig : figure1](a , b ) ] . for the h configuration we find an equilibrium distance of 3.4 , while for the s one this becomes 3.25 . these results are in fair agreement with another lda theoretical study @xcite ( predicting 3.4 and 3.17 respectively for for the h and s orientations ) . note that a more precise evaluation of such distances requires the use of van der waals corrected functionals . this exercise , however , is outside the scope of our work and here we just wish to establish that the equilibrium distance is large enough for our constrain to remain well defined . it can also be noted that the equilibrium distance of 3.6 obtained with a vdw - df study @xcite is not very different from our lda result . , for different unit cell sizes of graphene sheet . the results are presented for two different molecule - to - graphene distances : 3.4 and 6.8 .,scaledwidth=45.0% ] we then study the dependence of the charge transfer energies on the size of the graphene unit cell used . this is achieved by looking at the charge transfer gap , @xmath36 , as a function of the unit cell size at various molecule - to - graphene distances ( see fig . [ fig : figure2 ] ) . when the molecule is very close to the graphene sheet , after transferring an electron , the image charge is strongly attracted by the oppositely charged molecule and thereby remains highly localized . however , as the molecule moves away from the substrate , the attraction reduces since the coulomb potential decays with distance , resulting in a delocalization of the image charge . this will eventually spread uniformly all over the graphene sheet in the limit of an infinite distance . if the unit cell is too small , the image charge will be artificially over - confined , resulting in an overestimation of @xmath37 and @xmath38 and , as a consequence , of the charge transfer energies . this effect can be clearly seen in fig . [ fig : figure2 ] , where we display the variation of the charge transfer energies as a function of the cell size . clearly , for the shorter distance ( 3.4 corresponding to the average equilibrium distance ) , the energy gap converges for supercells of about 10@xmath3110 ( 10@xmath3110 graphene primitive cells ) . at the larger distance of 6.8 the same convergence is achieved for a 13@xmath3113 supercell . next we compute the charge transfer energies as a function of the distance between the sheet and the molecule . in order to compare our results with the gap expected in the limit of an infinite distance , we need to evaluate first the ionization potential , @xmath39 , and the electron affinity , @xmath40 , of the isolated benzene molecule . this is also obtained in terms of total energy differences between the neutral and the positively and negatively charged molecule , namely with the @xmath41scf method . this returns a quasiparticle energy gap , @xmath42 , of 11.02 ev , in good agreement ( within 4.5% ) with the experimental value @xcite . likewise we also determine the fermi level ( @xmath43 ) of graphene , which is found to be 4.45 ev . in fig . [ fig : figure3](a ) we show the change in the charge transfer energy gap with the distance of the benzene from the graphene sheet for the h configuration . as expected , when the molecule is close to the surface , there is a considerably large attraction between the image charge and the opposite charge excess on the molecule , resulting in an additional stabilization of the system and a reduction in magnitude of @xmath44 and @xmath37 . hence , in such case the charge transfer energies have a reduced magnitude and the charge transfer gap is smaller than that in the gas phase . then , as the molecule moves away from the graphene sheet , the charge transfer energies increase and so does the charge transfer energy gap until it eventually reaches the value corresponding to the homo - lumo gap of the isolated molecule in the limit of an infinite distance . in figs . [ fig : figure3](b ) , ( c ) , ( d ) and ( e ) we show the excess charge - density , @xmath45 , in different parts of the system after transferring one electron for two different molecule - to - graphene distances . the excess charge - density @xmath45 is defined as @xmath46 , where @xmath47 and @xmath48 are respectively the charge densities of the system before and after the charge transfer . thus the portion of @xmath45 localized on the graphene sheet effectively corresponds to the image charge profile . clearly , due to the stronger coulomb attraction , the image charge is more localized for @xmath49 than for @xmath50 . at equilibrium for the s configuration , @xmath51 , the charge transfer energy gap is calculated to be 8.91 ev , which is in good agreement ( within 4% ) with the gap obtained by @xmath52 @xcite . in table [ table : configuration ] , for the purpose of comparison , we have listed the charge transfer energies and charge transfer gaps for two different heights , 3.4 and 6.8 , and in different configurations . the most notable feature is that for the case of a pristine graphene substrate the specific absorption site plays little role in determining the charge transfer levels alignment . in general actual graphene samples always display lattice imperfections @xcite . in order to determine the effect of such structural defects on the ct energies , we consider a reference system where a stone - wales ( sw ) defect ( in which a single c - c bond is rotated by 90@xmath53 ) is present in the graphene sheet . we have then calculated @xmath54 for two different positions of the molecule with respect to the defect on the sheet , namely the @xmath55 position , in which the molecule is placed right above the defect and the @xmath56 position , in which it is placed above the sheet far from the defect ( see fig . [ fig : figure1 ] ) . our findings are listed in tab . [ table : configuration ] , where we report the charge transfer energies for both the configurations , assuming the molecule is kept at the same distance from the graphene sheet . from the table it is evident that the structural change in graphene due to presence of such defect does not alter the charge transfer energies of the molecule . this is because the image charge distribution on graphene is little affected by presence of the sw defect . in addition , the density of states ( dos ) of graphene remains almost completely unchanged near its fermi energy after introducing such defect as can be seen in fig . [ figure4 ] , which shows that the partial density of states ( pdos ) of the atoms forming the sw defect has no significant presence near the fermi level . thus , after the charge transfer , the electron added to ( or removed from ) the graphene sheet has the same energy that it would have in the absence of the defect , i.e. it is subtracted ( added ) from a region of the dos where there is no contribution from the sw defect . in this context , it is noteworthy that a @xmath52 study @xcite has concluded that altering the structure of pristine graphene by introducing dopant ( which raises the fermi level of graphene by 1 ev ) also has minor effect on the qp gap of benzene , reducing it by less than 3% . .@xmath57 , @xmath58 and @xmath59 for various configurations of a benzene molecule on pristine and defective graphene . h and s denote adsorption of benzene on graphene in the _ hollow _ and _ stack _ configuration , respectively . @xmath55 and @xmath56 correspond to adsorption on graphene with sw defect , with the former corresponding to adsorption exactly on top of the defect and the latter corresponding to adsorption away from the site of the defect . the configurations @xmath60 and @xmath61 both correspond to adsorption of two benzene molecules in _ hollow _ configuration- one at height 3.4 and another at a height 6.8 . while in @xmath60 , the ct is calculated for the lower molecule , in @xmath61 , the ct is calculated for the upper one . finally @xmath62 represents the case in which we have a layer of non - overlapping benzenes adsorbed on graphene and one is interested in calculating the ct energy for one of them , which is placed in the _ hollow _ configuration . [ cols="^,^,^,^,^",options="header " , ] in real interfaces between organic molecules and a substrate , molecules usually are not found isolated but in proximity to others . it is then interesting to investigate the effects that the presence of other benzene molecules produce of the charge transfer energies of a given one . to this end we select three representative configurations . in the first one , @xmath60 , the graphene sheet is decorated with two benzene molecules , one at 3.4 while the other is placed above the first at 6.8 from the graphene plane . we then calculate the charge transfer energies of the middle benzene ( the one at 3.4 from the sheet ) . the excess charge on different parts of the system ( image charge ) , after transferring one electron to the sheet , is displayed in fig . [ fig : he](a ) and fig . [ fig : he](b ) . the second configuration , @xmath61 , is identical to the first one but now we calculate the charge transfer energies of the molecule , which is farther away from the graphene sheet , namely at a distance of 6.8 . for this configuration , the excess charge after a similar charge transfer is shown in fig . [ fig : he](c ) and fig . [ fig : he](d ) . in the third configuration , @xmath62 , we arrange multiple benzene molecules in the same plane . the molecules are in close proximity with each other although their atomic orbitals do not overlap . charge transfer energies are then calculated with respect to one benzene molecule keeping the others neutral and an isovalue plot for similar charge transfer is shown in fig . [ fig : he](e ) and fig . [ fig : he](f ) . the charge transfer energies calculated for these three configurations are shown in tab . [ table : configuration ] . if one compares configurations where the molecule is kept at the same distance from the graphene plane , such as the case of h(@xmath49 ) , @xmath60 and @xmath62 or of h(@xmath50 ) and @xmath61 , it appears clear that the presence of other molecules has some effect on the charge transfer energies . in particular we observe than when other molecules are present both @xmath57 and @xmath58 get more shallow , i.e. their absolute values is reduced . interestingly the relative reduction of @xmath57 and @xmath58 depends on the details of the positions of the other molecules ( e.g. it is different for @xmath60 and @xmath62 ) but the resulting renormalization of the homo - lumo gap is essentially identical [ about 25 mev when going from h(@xmath49 ) to either @xmath60 or @xmath62 ] . this behaviour can be explained in terms of a simple classical effect . consider the case of @xmath60 for example . when one transfers an electron from the middle benzene to the graphene sheet the second benzene molecule , placed above the first , remains neutral but develops an induced charge dipole . the moment of such dipole points away from the charged benzene and lowers the associated electrostatic potential . importantly , also the potential of graphene will be lowered . however , since the potential generated by an electrical dipole is inversely proportional to the square of distance , the effect remains more pronounced at the site of the middle benzene than at that of the graphene sheet . a similar effect can be observed for an electron transfer from the graphene sheet to the middle benzene and for the @xmath62 configuration . in the case of @xmath61 , the system comprising the topmost benzene ( from which we transfer charge ) and the graphene plane can be thought of as a parallel - plate capacitor . the work , @xmath64 , done to transfer a charge @xmath65 from one plate to the other is @xmath66 where @xmath67 is the capacitance , which in turn is proportional to the dielectric constant of the medium enclosed between the plates . hence , at variance with the case of h(d=6.8 ) , the space in between the molecule and the graphene sheet is occupied by a molecule with finite dielectric constant and not by vacuum . this results in a reduction of @xmath64 , so that the charge transfer energies for @xmath61 are smaller than those for h(d=6.8 ) . finally , we show that our calculated energy levels alignment can be obtained from a classical electrostatic model . if one approximates the transferred electron as a point charge and the substrate where the image charge forms as an infinite sheet of relative permittivity @xmath68 then , for a completely planar distribution of the bound surface charge , the work done by the induced charge to take an electron from the position of the molecule ( at a distance @xmath24 ) to infinity is @xmath69 hence , this electrostatic approximation predicts that the presence of the substrate lowers the @xmath70 of the molecule by @xmath71 with respect to the corresponding gas - phase value . however , the actual image charge is not strictly confined to a 2d plane but instead spills out over the graphene surface . we can account for such non - planar image charge distribution by introducing a small modification to the above expression @xcite and write the lumo at a height @xmath24 as @xmath72 where @xmath73 is the distance between the centre of mass of the image charge and the substrate plane and @xmath74 is the gas - phase lumo ( the electron affinity ) . a similar argument for the homo level shows an elevation of same magnitude due to the presence of the substrate . in fig . [ fig : classical_plot ] we plot the charge transfer energies and show that they compare quite well with the curves predicted by the classical model by using an effective dielectric constant of 2.4 for graphene @xcite . when drawing the classical curves we have used an approximate value , @xmath75 , which provides an excellent estimate for smaller distances , @xmath24 . it is worth noting that for larger distances , though the actual value of @xmath73 should be much less , the overall effect of @xmath73 is very small and almost negligible . in the same graph , we have also plotted the classical curves corresponding to benzene on a perfectly metallic ( @xmath76 ) surface . this shows that the level renormalization of benzene for physisorption on graphene is significantly different from that on a perfect metal , owing to the different screening properties of graphene . ( circles ) and @xmath77 ( squares ) calculated for different molecule - to - substrate distances . the cdft results are seen to agree well with the classically calculated curve given in red . the horizontal lines mark the same quantities for isolated an molecule ( gas - phase quantities ) . the continuous black line shows the position of the classically calculated level curve for adsorption on a perfect metal @xmath76.,scaledwidth=44.0% ] we have used cdft as implemented in the siesta code to calculate the energy levels alignment of a benzene molecule adsorbed on a graphene sheet . in general the charge transfer energies depend on the distance between the molecule and the graphene sheet , and this is a consequence of the image charge formation . such an effect can not be described by standard kohn - sham dft , but it is well captured by cdft , which translates a quasi - particle problem into an energy differences one . with cdft we have simulated the energy level renormalization as a function of the molecule - to - graphene distance . these agree well with experimental data for an infinite separation , where the charge transfer energies coincide with the ionization potential and the electron affinity . furthermore , an excellent agreement is also obtained with @xmath0 calculations at typical bonding distances . since cdft is computationally inexpensive we have been able to study the effects arising from bonding the molecule to a graphene structural defect and from the presence of other benzene molecules . we have found that a stone - wales defect does not affect the energy level alignment since its electronic density of state has little amplitude at the graphene fermi level . in contrast the charge transfer energies change when more then a molecule is present . all our results can be easily rationalized by a simple classical electrostatic model describing the interaction of a point - like charge and a uniform planar charge distribution . this , at variance to the case of a perfect metal , takes into account the finite dielectric constant of graphene . this work is supported by the european research council , quest project . computational resources have been provided by the supercomputer facilities at the trinity center for high performance computing ( tchpc ) and at the irish center for high end computing ( ichec ) . additionally , the authors would like to thank dr . ivan rungger and dr . a. m. souza for helpful discussions . 29ifxundefined [ 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 @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1002/cphc.200700177 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrev.136.b864 [ * * , ( ) ] link:\doibase 10.1103/physrevb.18.7165 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.88.165112 [ * * , ( ) ] @noop * * , ( ) @noop ( ) \doibase http://dx.doi.org/10.1016/j.apsusc.2010.07.069 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.96.146107 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.88.235437 [ * * , ( ) ] @noop * * , ( ) \doibase http://dx.doi.org/10.1016/0022-1902(81)80486-1 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( )

constrained density functional theory ( cdft ) is used to evaluate the energy level alignment of a benzene molecule as it approaches a graphene sheet . within cdft the problem is conveniently mapped onto evaluating total energy differences between different charge - separated states , and it does not consist in determining a quasi - particle spectrum . we demonstrate that the simple local density approximation provides a good description of the level aligmnent along the entire binding curve , with excellent agreement to experiments at an infinite separation and to @xmath0 calculations close to the bonding distance . the method also allows us to explore the effects due to the presence of graphene structural defects and of multiple molecules . in general all our results can be reproduced by a classical image charge model taking into account the finite dielectric constant of graphene .

arxiv

the first step in describing the interaction between many particles is to determine their pair potential or the forces among a single pair . if the governing physical equations are linear ( like for gravity or electrostatics ) , this approach yields a quantitatively reliable description of the physical system considered , based on the linear superposition principle . however , if nonlinearities are present , linear superposition of pair potentials is no longer accurate and nonadditivity gives rise to many - body effects . these latter effects can lead , e.g. , to a strengthening or weakening of the total force acting on a particle surrounded by more than a single other one , a change of sign of that force , or the appearance of stable or unstable configurations . many - body effects appear in rather diverse systems such as nuclear matter , superconductivity @xcite , colloidal suspensions @xcite , quantum - electrodynamic casimir forces @xcite , polymers @xcite , nematic colloids @xcite , and noble gases with van der waals forces acting among them @xcite . each of these systems is characterized by a wide range of time and length scales . integrating out the degrees of freedom associated with small scales ( such as the solvent of colloidal solutes or polymers ) for fixed configurations of the large particles , generates effective interactions among the latter , which are inherently not pairwise additive . this is the price to be paid for achieving a reduced description of a multicomponent system . driven by these effective interactions the large particles of the system may exhibit collective behavior of their own ( like aggregation or phase separation , see refs . @xcite and @xcite and references therein ) , which can be described much easier if it is governed by pair potentials . in order to be able to judge whether this ansatz is adequate one has to check the relative magnitude of genuine many - body forces . in this paper we assess the quantitative influence of such many - body effects on critical casimir forces ( ccfs ) @xcite . these long - ranged forces arise as a consequence of the confinement of the order parameter fluctuations in a critical fluid @xcite . they have been analyzed paradigmatically by studying the effective interaction between a _ single _ colloidal particle and a homogeneous @xcite or inhomogeneous @xcite container wall as well as between _ two isolated _ colloidal particles @xcite upon approaching the critical point of the solvent . here we add one sphere to the sphere - wall configuration , which is the simplest possibility to study many - body forces . ( the wall mimics a third , very large sphere . ) in order to be able to identify the latter ones one has to resort to a theoretical scheme which allows one to compute the forces between individual pairs and the three - body forces on the same footing . since these forces are characterized by universal scaling functions , which depend on the various geometrical features of the configuration and on the thermodynamic state , we tackle this task by resorting to field - theoretic mean field theory ( mft ) , which captures the universal scaling functions as the leading contribution to their systematic expansion in terms of @xmath1 spatial dimensions . experience with corresponding previous studies for simple geometries tells that this approach does yield the relevant qualitative features of the actual universal scaling functions in @xmath2 ; if suitably enhanced by renormalization group arguments these results reach a semi - quantitative status . we point out that even within this approximation the numerical implementation of this corresponding scheme poses a severe technical challenge . thus at present this approach appears to be the only feasible one to explore the role of many - body critical casimir forces within the full range of their scaling variables . accordingly we consider the standard landau - ginzburg - wilson hamiltonian for critical phenomena of the ising bulk universality class , which is given by @xmath3 = \int_v{\rm d}^d\mathbf{r } \left\ { \frac{1}{2}\left ( \nabla\phi \right)^2 + \frac{\tau}{2}\phi^2 + \frac{u}{4!}\phi^4 \right\ } ~,\ ] ] with suitable boundary conditions ( bcs ) . in the case of a binary liquid mixture near its consolute ( demixing ) point , the order parameter @xmath4 is proportional to the deviation of the local concentration of one of the two species from the critical concentration . @xmath5 is the volume accessible to the fluid , @xmath6 is proportional to the reduced temperature @xmath7 , and the coupling constant @xmath8 stabilizes the statistical weight @xmath9 in the two - phase region , i.e. , for @xmath10 . close to the bulk critical point @xmath11 the bulk correlation length @xmath12 diverges as @xmath13 , where @xmath14 in @xmath2 and @xmath15 in @xmath16 , i.e. , within mft @xcite . the two non - universal amplitudes @xmath17 are of molecular size ; they form the universal ratio @xmath18 for @xmath2 and @xmath19 for @xmath16 @xcite . the bcs reflect the generic adsorption preference of the confining surfaces for one of the two components of the mixture . for the critical adsorption fixed point @xcite , the bc at each of the confining surfaces is either @xmath20 or @xmath21 , to which we refer as @xmath22 or @xmath23 , respectively . within mft the equilibrium order parameter distribution minimizes the hamiltonian in eq . for the aforementioned bcs , i.e. , @xmath24/\delta\phi = 0 $ ] . far from any boundary the order parameter approaches its constant bulk value @xmath25 for @xmath10 or @xmath26 for @xmath27 . @xmath28 is a non - universal bulk amplitude and @xmath29 ( for @xmath16 ) is a standard critical exponent . in the following we consider the reduced order parameter @xmath30 . the remainder of this paper is organized as follows . in sec . [ section_the_system ] we define the system under consideration and the scaling functions for the ccfs as well as the normalization scheme . in sec . [ section_results ] we present the numerical results obtained for the universal scaling functions of the ccfs , from which we extract and analyze the many - body effects . in sec . [ section_conclusion ] we summarize our results and draw some conclusions . we study the normal and the lateral ccfs acting on two colloidal particles immersed in a near - critical binary liquid mixture and close to a homogeneous , planar substrate . we focus on the critical concentration which implies the absence of a bulk field conjugate to the order parameter [ see eq . ] . the surfaces of the colloids and of the substrate are considered to exhibit a strong adsorption preference for one of the two components of the confined liquid leading to @xmath31 bcs . the forces are calculated using the full three - dimensional numerical analysis of the appropriate mft as given by eq . . specifically , we consider two three - dimensional spheres of radii @xmath32 and @xmath33 with bcs @xmath34 and ( @xmath35 ) , respectively , facing a homogeneous substrate with bc @xmath36 at sphere - surface - to - substrate distances @xmath37 and @xmath38 , respectively ( see fig . [ system_sketch ] ) . the coordinate system @xmath39 is chosen such that the centers of the spheres are located at @xmath40 and @xmath41 so that the distance between the centers , projected onto the @xmath42-axis , is given by @xmath43 . the bcs of the whole system are represented by the set @xmath44 , where @xmath45 , @xmath35 , and @xmath46 can be either @xmath47 or @xmath48 . it is important to point out that we discuss colloidal particles with the shape of a hypercylinder @xmath49 where @xmath50 are the semiaxes ( or radii ) of the hypercylinder and @xmath51 , @xmath52 , @xmath53 . if @xmath54 and @xmath55 , the hypercylinder reduces to a hypersphere . the generalization of @xmath56 to values larger than 3 is introduced for technical reasons because @xmath57 is the upper critical dimension for the relevance of the fluctuations of the order parameter . these fluctuations lead to a behavior different from that obtained from the present mft which ( apart from logarithmic corrections @xcite ) is valid in @xmath16 . we consider two hypercylinders in @xmath16 with @xmath58 and @xmath59 . the two colloids are taken to be parallel along the fourth dimension with infinitely long hyperaxes in this direction . considering hypercylinders , which are translationally invariant along the @xmath60axis , allows us to minimize @xmath61 $ ] numerically using a three dimensional finite element method in order to obtain the spatially inhomogeneous order parameter profile @xmath62 for the geometries under consideration ( see fig . [ system_sketch ] ) . and @xmath33 immersed in a near - critical binary liquid mixture ( not shown ) and close to a homogeneous , planar substrate at @xmath63 . the two colloidal particles with bcs @xmath34 and @xmath64 are located at sphere - surface - to - substrate distances @xmath37 and @xmath38 , respectively . the substrate exhibits bc @xmath36 . the lateral distance between the centers of the spheres along the @xmath42-direction is given by @xmath43 , while the centers of both spheres lie in the plane @xmath65 . in the case of four spatial dimensions the figure shows a cut of the system , which is invariant along the fourth direction , i.e. , the spheres correspond to parallel hypercylinders with one translationally invariant direction , which is @xmath66 . ] in the case of an upper critical demixing point of the binary liquid mixture at the critical concentration , @xmath67 corresponds to the disordered ( i.e. , mixed ) phase of the fluid , whereas @xmath68 corresponds to the ordered ( i.e. , phase separated ) phase . the meaning of the sign is reversed for a lower critical point . in the following we assume an upper critical point . the normal ccf @xmath69 acting on sphere @xmath70 in the presence of sphere @xmath71 ( @xmath72 and @xmath73 ) along the @xmath74-direction takes the scaling form @xmath75 where @xmath76 , @xmath77 , @xmath78 , @xmath79 , and @xmath80 ( i.e. , @xmath81 for @xmath27 and @xmath82 for @xmath10 ) . equation describes the singular contribution to the normal force emerging upon approaching @xmath11 . @xmath83 is the force per length of the hypercylinder due to its extension in the translationally invariant direction . in the spirit of a systematic expansion in terms of @xmath84 around the upper critical dimension we study the scaling functions @xmath85 within mft as given by eq . for hypercylinders in @xmath16 , which captures the correct scaling functions in @xmath16 up to logarithmic corrections occurring in @xmath86 @xcite , which we do not take into account here . since mft renders the leading contribution to an expansion around @xmath16 , geometrical configurations with small @xmath87 , @xmath88 , or @xmath89 are not expected to be described reliably by the present approach due to the dimensional crossover in narrow slit - like regions , which is not captured by the @xmath84 expansion . the colloidal particles will also experience a lateral ccf @xmath90 , for which it is convenient to use the scaling form @xmath91 where @xmath92 ( i.e. , @xmath93 for @xmath27 and @xmath94 for @xmath10 ) . note that the choice of @xmath95 as the scaling variable does not depend on the type of particle the force acts on . equation also describes the singular contribution to the lateral force near @xmath11 . the total ccf acting on particle @xmath70 is @xmath96 where @xmath97 and @xmath98 are the unit vectors pointing in @xmath42- and @xmath74-direction , respectively . due to symmetry all other components of the ccf are zero . as a reference configuration we consider a single spherical colloid of radius @xmath99 with bc @xmath100 at a surface - to - surface distance @xmath101 from a planar substrate with bc @xmath36 . this colloid experiences ( only ) a normal ccf @xmath102 in the following we normalize the scaling functions @xmath103 and @xmath104 by the amplitude @xmath105 of the ccf acting at @xmath11 on a single colloid for @xmath106 bcs at a surface - to - surface distance @xmath107 . accordingly , in the following we consider the normalized scaling functions @xmath108 with @xmath109 and @xmath110 . experimentally it can be rather difficult to obtain @xmath105 . a standard alternative way to normalize is to take the more easily accessible amplitude @xmath111 for the ccf at @xmath11 between two parallel plates with @xmath106 bcs , which is given within mft by ( see ref . @xcite and references therein ) @xmath112 ^ 4}{u } \simeq -283.61 u^{-1 } ~,\ ] ] where @xmath113 is the elliptic integral of the first kind . within mft the amplitude @xmath105 can be expressed in terms of @xmath111 : @xmath114 equation allows for a practical implementation of the aforementioned normalization , which eliminates the coupling constant @xmath115 , which is unknown within mft . we calculate the normal and lateral forces directly from the numerically determined order parameter profiles @xmath62 by using the stress tensor which , within the ginzburg - landau approach , is given by @xcite @xmath116 ~,\ ] ] with @xmath117 . the first index of the stress tensor denotes the direction of a force , the second index denotes the direction of the normal vector of the surface upon which the force acts . therefore one has @xmath118 where @xmath119 is a hypersurface enclosing particle @xmath70 , @xmath120 is the @xmath121-th component ( to be summed over ) of its unit outward normal , and @xmath122 is the length of the @xmath123-dimensional hyperaxis of @xmath124 . in particular we focus on the normal and lateral ccfs acting on colloid ( 2 ) for the configuration shown in fig . [ system_sketch ] , for @xmath125 , and with the binary liquid mixture at its critical concentration . in the following analysis we consider colloid ( 1 ) to be fixed in space at a sphere - surface - to - substrate distance @xmath126 , equally sized colloids ( i.e. , @xmath127 ) , and fixed @xmath128 bc for the substrate . we proceed by varying either the vertical ( @xmath74-direction ) or the horizontal ( @xmath42-direction ) position of colloid ( 2 ) by varying either @xmath38 or @xmath129 , respectively . we also consider different sets of bcs for the colloids . in the following results the numerical error is typically less than @xmath130 , unless explicitly stated otherwise . in fig . [ d2_z ] we show the behavior of the normalized [ eq . ] scaling function @xmath131 of the normal ccf acting on colloid ( 2 ) with @xmath132 bcs close to a homogeneous substrate with @xmath128 bc and in the presence of colloid ( 1 ) with @xmath133 bc . the scaling functions are shown as functions of the scaling variable ratio @xmath134 , i.e. , for @xmath27 . the various lines correspond to distinct fixed values of @xmath135 as the sphere - surface - to - substrate distance in units of the sphere radius . thus fig . [ d2_z ] shows the temperature dependence of the normal ccf on colloid 2 for three different values of @xmath38 and for colloid ( 1 ) fixed in space . from fig . [ d2_z ] ( a ) one can see that , for colloid ( 1 ) with @xmath136 bc and for any given value of @xmath137 , the scaling function of the normal ccf acting on colloid ( 2 ) with @xmath138 bc changes sign upon varying @xmath135 . due to the change of sign of @xmath139 , for any value of @xmath137 there is a certain value @xmath140 at which the normal ccf acting on colloid ( 2 ) vanishes . for sphere - surface - to - substrate distances sufficiently large such that @xmath141 , colloid ( 2 ) is pushed away from the substrate due to the dominating repulsion between the two colloids in spite of the attraction by the substrate , whereas for @xmath142 it is pulled to the substrate due to the dominating attraction between it and the substrate this implies that , in the absence of additional forces , levitation of colloid ( 2 ) ( i.e. , zero total normal force ) at height @xmath143 is not stable against perturbations of the sphere - surface - to - substrate distance . on the other hand , upon varying temperature , any distance @xmath38 can become a stable levitation position for colloid ( 2 ) with @xmath144 bc in the presence of colloid ( 1 ) with @xmath133 bc [ see fig . [ d2_z ] ( b ) , according to which each scaling function corresponding to a certain value of @xmath38 exhibits a zero so that , at this zero , increasing ( decreasing ) @xmath38 at fixed temperature leads to an attraction ( repulsion ) to ( from ) the substrate ] . in this case the attraction between the two colloids is dominating for large sphere - surface - to - substrate distances @xmath135 , while the repulsion between colloid ( 2 ) and the substrate dominates for small values of @xmath135 . of the normal ccf acting on colloid ( 2 ) with bcs @xmath145 in ( a ) and @xmath144 in ( b ) . the scaling functions are shown for @xmath27 as functions of the scaling variable ratio @xmath146 for three fixed values of the scaling variable @xmath77 : @xmath147 ( black lines ) , 1.5 ( red lines ) , and 2 ( green lines ) , while @xmath148 for all curves in ( a ) and ( b ) so that @xmath149 . for @xmath33 fixed the three curves correspond to three different vertical positions of colloid ( 2 ) with colloid ( 1 ) fixed in space . as expected , the forces become overall weaker upon increasing @xmath135 . @xmath150 @xmath151 implies that the colloid is attracted to ( repelled from ) the substrate along the @xmath74-direction . ] figure [ d2_x ] shows the behavior of the normalized scaling function @xmath152 of the lateral ccf acting on colloid ( 2 ) in the presence of colloid ( 1 ) having the same bc , i.e. , @xmath153 . the scaling functions are shown as functions of the scaling variable ratio @xmath154 . from figs . [ d2_x ] ( a ) and ( b ) one can infer that @xmath155 . therefore colloid ( 2 ) is always attracted towards colloid ( 1 ) which has the same bc . hence the substrate does not change the sign of the lateral ccf as compared with the attractive lateral ccf in the absence of the confining substrate . however , the shapes of the scaling functions for @xmath156 bcs [ fig . [ d2_x ] ( a ) ] differ from the ones for @xmath157 bcs [ fig . [ d2_x ] ( b ) ] ; without the substrate , they are identical . in the former case and in contrast to the latter one , the scaling functions exhibit minima above @xmath11 , which is reminiscent of the shape of the corresponding scaling functions in the absence of the substrate . of the lateral ccf acting on colloid ( 2 ) facing a homogeneous substrate with @xmath128 bc and in the presence of colloid ( 1 ) with @xmath158 bcs . the scaling functions are shown for @xmath27 as functions of the scaling variable ratio @xmath154 for three fixed values of the scaling variable @xmath77 : @xmath147 ( black curves ) , 1.5 ( red curves ) , and 2 ( green curves ) , while @xmath148 for all curves in ( a ) and ( b ) so that @xmath149 . for @xmath33 fixed the three curves correspond to three different vertical positions of colloid ( 2 ) with colloid ( 1 ) fixed in space . as expected , the forces become overall weaker upon increasing @xmath135 . @xmath159 implies that colloid ( 2 ) is attracted towards colloid ( 1 ) . two different sets of @xmath160 bcs are considered : @xmath156 in ( a ) and @xmath157 in ( b ) . ] in fig . [ l_x ] we show the results obtained for the normalized scaling functions @xmath161 of the lateral ccf acting on colloid ( 2 ) . in fig . [ l_x](a ) the scaling function is shown as function of the scaling variable ratio @xmath154 ; the black , red , and green curves correspond to @xmath162 , @xmath163 , and @xmath164 , respectively . in the absence of the substrate , the ccf between two colloids with opposite bcs is repulsive . however , as shown in fig . [ l_x](a ) , in the presence of the substrate with @xmath128 bc , there is a change of sign in the scaling function of the lateral ccf . this implies that the lateral ccf acting between the two colloids changes from being attractive to being repulsive ( or reverse ) upon decreasing ( increasing ) the reduced temperature . thus temperature allows one to control both the strength and the sign of the lateral ccf in the case of two colloids with opposite bcs being near a wall . the at first sight unexpected lateral attraction between two colloids with opposite bcs in the presence of the substrate ( i.e. , @xmath165 ) can be understood as follows . in the absence of the two colloids , the order parameter profile @xmath166 is constant along any path within a plane @xmath167 because in this case @xmath168 . since the substrate area is much larger than the surface areas of the colloids , one can regard the immersion of these colloidal spheres as a perturbation of this profile . in figs . [ l_x](a ) and ( b ) , the region within which the scaling function is negative ( corresponding to an attractive force ) indicates that under these circumstances [ i.e. , when the colloids are sufficiently away from each other ; see fig . [ l_x](b ) ] the perturbation generated by the presence of the spheres decreases upon decreasing the lateral distance between them . this causes the colloids to move towards each other in order to weaken the perturbation by reducing its spatial extension ; this amounts to an attraction , i.e. , @xmath165 . on the other hand , when they are sufficiently close to each other the pairwise interaction between the two colloids dominates and the total lateral ccf is positive ( i.e. , repulsive ) . in fig . [ l_x](c ) we show how the equilibrium lateral distance @xmath169 measured in units of @xmath170 varies as function of temperature , i.e. , @xmath170 for fixed @xmath99 . of the lateral ccf acting on colloid ( 2 ) . both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . [ system_sketch ] ] . the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye the four data points are connected by straight lines.,title="fig : " ] of the lateral ccf acting on colloid ( 2 ) . both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . [ system_sketch ] ] . the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye the four data points are connected by straight lines.,title="fig : " ] of the lateral ccf acting on colloid ( 2 ) . both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . [ system_sketch ] ] . the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye the four data points are connected by straight lines.,title="fig : " ] in order to determine the preferred arrangement of the colloids , we have also analyzed the direction of the total ccf @xmath180 acting on colloid ( 2 ) [ see eq . ] for several spatial configurations and bcs . for @xmath156 bcs we have found that the colloids tend to aggregate laterally in such a way that several particles with @xmath22 bc , facing a substrate with the same bc , can be expected to form a monolayer on the substrate . on the other hand , for the case of @xmath157 bcs , we have found that the colloids can be expected to aggregate on top of each other so that a collection of colloids with such bcs is expected to form three - dimensional sessile clusters . these tendencies become more pronounced upon approaching the critical point ( see fig . [ direction_resulting_force_d2 ] ) . similar results have been found by soyka et al . @xcite in experiments using chemically patterned substrates . for them the authors have found indeed that colloids with @xmath23 bc distributed over those parts of the substrate with the same bc [ which is equivalent to @xmath156 bcs ] aggregate and form a single layer . moreover , they have found that colloids distributed over parts of the substrate with opposite bc [ corresponding to @xmath157 bcs ] form three - dimensional clusters . and @xmath157 bcs , with @xmath181 . the black rectangles represent the substrate and blue circles represent colloid ( 1 ) while black , red , and green circles represent colloid ( 2 ) with @xmath182 , and 2 , respectively . the centers of all colloids lie in the plane @xmath65 . ] we have determined the many - body force acting on particle @xmath183 by subtracting from the total force @xmath184 [ see eq . ] the sum of the pairwise forces acting on it , i.e. , the colloid - colloid ( cc ) and the colloid - substrate ( cs ) forces . accordingly the many - body ccf @xmath185 acting on colloid ( @xmath70 ) is given by ( see fig . [ system_sketch ] ) @xmath186 where @xmath187 with @xmath188 is the pairwise colloid - colloid force ( acting on colloid 2 @xmath23 or 1 @xmath22 with 2 having the larger @xmath42-coordinate ) expressed in terms of the absolute value @xmath189 of the force between two colloids at surface - to - surface distance @xmath190 in free space . @xmath191 is the ccf between the substrate and a single colloid @xmath183 . we have studied both the normal @xmath192 and the lateral @xmath193 many - body ccfs which are characterized by corresponding scaling functions [ compare eqs . and ] : @xmath194 and @xmath195 in fig . [ mb_l_z ] we show the normalized [ see eqs . and ] scaling functions @xmath196 of the many - body normal ccf acting on colloid ( 2 ) . this figure reveals similar results for @xmath156 [ fig . [ mb_l_z ] ( a ) ] and @xmath197 [ fig . [ mb_l_z ] ( b ) ] bcs . in these cases , each mb scaling function exhibits both a maximum and at least one minimum , the former one appearing for smaller values of the scaling variable @xmath198 ( i.e. , at temperatures closer to @xmath11 ) . for a certain range of temperatures close to @xmath11 , as the distance @xmath129 between the colloids increases , the many - body normal ccf changes from attractive to repulsive . this shows that for each temperature within this range there is a lateral distance @xmath199 for which the many - body contribution to the normal force acting on colloid ( 2 ) is zero . this means that under such conditions the sum of pairwise forces provides a quantitatively reliable description of the total force acting on colloid ( 2 ) . for temperatures sufficiently far from @xmath11 , the many - body normal ccf is always attractive with a monotonic dependence on @xmath198 . here , as in figs . [ d2_z ] - [ l_x ] , the ccfs decay exponentially for @xmath200 . as expected , the many - body effects are more pronounced if the colloids are closer to each other and/or closer to the substrate . indeed for situations in which the colloids are close to each other [ see , e.g. , the black curves in figs . [ mb_l_z ] ( a ) and ( b ) ] we have found that when the normal many - body ccf reaches its maximal strength , corresponding to the minimum of the scaling function @xmath201 at @xmath202 , the relative contribution of the many - body ccf reaches @xmath0 of the strength of the total normal ccf . for larger distances between the colloids [ see , e.g. , the green curves in figs . [ mb_l_z ] ( a ) and ( b ) ] this relative contribution is smaller ( around @xmath203 for @xmath204 and @xmath205 for @xmath206 ) . of the many - body normal ccf acting on colloid ( 2 ) for @xmath207 and @xmath208 . the scaling functions are shown as functions of the scaling variable ratio @xmath209 for the sets of bcs @xmath156 in ( a ) and @xmath197 in ( b ) . the black , red , and green lines correspond to @xmath210 , and @xmath211 , respectively . figures [ d2_z](a ) and [ mb_l_z](b ) allow a direct comparison between the full ccf and the corresponding many - body contribution ( note the different scales of the ordinates . ) ] we are not aware of results for the quantum - electrodynamic casimir interactions which are obtained along the same lines as our ccf analysis above . nonetheless , in order to assess the significance of our results we compare them with the results in ref . @xcite , which is the closest comparable study which we have found in the literature . therein the authors study theoretically two dielectric spheres immersed in ethanol while facing a plate . depending on the kind of fluid and on the materials of the spheres and of the plate as well as on the distances involved , also the quantum - electrodynamic casimir force can be either attractive or repulsive . it is well known @xcite that for two parallel plates with permittivities @xmath212 and @xmath213 separated by a fluid with permittivity @xmath214 and without further boundaries , the quantum - electrodynamic casimir force is repulsive if @xmath215 within a suitable frequency range . in ref . @xcite it is stated that this also holds for two spheres immersed in a fluid . the authors of ref . @xcite analyze the effect of nonadditivity for the above system by studying the influence of an additional , adjacent substrate on the equilibrium separation @xmath56 between two nanometer size dielectric spheres . to this end , they consider two spheres of different materials with the same radii @xmath207 and the same surface - to - plate distances @xmath216 and analyze how the lateral equilibrium distance @xmath217 between the spheres depends on @xmath101 . by comparing the equilibrium distance @xmath217 with that in the absence of the substrate , @xmath218 , they find that @xmath217 increases or decreases ( depending on the kind of materials of the spheres ) by as much as @xmath219 as the distance from the plate varies between @xmath220 and @xmath221 . they also find that `` the sphere - plate interaction changes the sphere - sphere interaction with the same sign as @xmath101 becomes smaller '' , which means that if the sphere - plate force is repulsive ( attractive ) , the sphere - sphere one will become more repulsive ( attractive ) upon decreasing the distance from the plate @xmath101 . by construction these changes are genuine many - body contributions . in the case of two chemically different spheres , the sign of the many - body force contribution ( i.e. , whether it is attractive or repulsive ) agrees with the sign of the stronger one of the two individual sphere - plate interactions . in fig . [ comparing ] we show schematically the system considered in ref . @xcite [ ( a ) and ( b ) ] and the system considered here [ ( c ) , ( d ) , and ( e ) ] . for the quantum - electrodynamic casimir effect , the dielectric spheres are represented by circles of equal radii , with the green one corresponding to a polystyrene sphere and the red one to a silicon sphere . the semi - infinite plates are represented by gray and yellow rectangles for teflon and gold , respectively . the whole configuration is immersed in ethanol which , for simplicity , is not shown in the figure . the dashed arrows indicate the direction of the strongest of the two pairwise sphere - substrate forces , while the solid arrows indicate the direction of the lateral many - body force . the directions of the arrows in figs . [ comparing ] ( a ) and ( b ) are chosen as to illustrate the findings in ref . @xcite , according to which the sign of the _ lateral _ many - body force is the same as the one of the strongest _ normal _ pairwise sphere - plate force : attractive in ( a ) and repulsive in ( b ) . also in the case of the critical casimir forces , depicted in figs . [ comparing ] ( c ) , ( d ) , and ( e ) , we represent the colloids by circles and the laterally homogeneous semi - infinite substrate by rectangles . the orange filling represents the @xmath22 bc while the blue filling represents the @xmath23 bc . again , the dashed arrows indicate the direction of the stronger one of the two pairwise ( normal ) colloid - substrate forces , while the solid arrows indicate the direction of the lateral many - body contribution to the ccf . as one can infer from fig . [ mb_x ] , the _ lateral _ many - body ccf acting on colloid ( 2 ) is always attractive for the given geometrical configuration , regardless of the bcs . and alluding to the system studied in ref . @xcite , the circles represent the projections of dielectric spheres with equal radii , the green one corresponding to polystyrene and the red one to silicon ; the rectangles represent semi - infinite plates with their surfaces perpendicular to the @xmath222 plane , the gray and the yellow one being teflon and gold , respectively . the system is immersed in ethanol , which is not indicated in the figure . in ( a ) and ( b ) each dashed arrow indicates the direction of the stronger one of the two corresponding pairwise forces between the dielectric spheres and the plate , which turns out to determine the direction of the many - body lateral force acting on the spheres : if the stronger one of the two pairwise forces is attractive [ repulsive ] , the lateral many - body force will also be attractive [ repulsive ] ( see ref . also in the case of the critical casimir interaction [ ( c ) , ( d ) , and ( e ) ] , the circles and rectangles represent projections of spherical colloids and of homogeneous substrates , respectively : orange and blue indicate @xmath22 and @xmath23 bcs , respectively . in ( c ) and ( d ) the pairwise normal forces between each of the two spheres and the substrate are equal : attractive in ( c ) and repulsive in ( d ) . in ( e ) the two pairwise normal forces have opposite directions with the repulsive one being the stronger one @xcite . the corresponding dashed arrows have the same meaning as in ( a ) and ( b ) . according to fig . [ mb_x ] the many - body lateral ccfs are attractive for all three cases ( c ) , ( d ) , and ( e ) . the comparison shows that the systems in ( a ) and ( c ) behave similarly . however , the behavior of system ( b ) has no counterpart for ccfs [ see ( d ) and ( e ) ] . in this figure all surface - to - surface distances equal the sphere radius , which in our notation corresponds to @xmath223 . ] we can also compare our results with those for two atoms close to the surface of a planar solid body . mclachlan @xcite has tackled this problem by treating the solid as a uniform dielectric . by using the image method he derived an expression for the many - body corrections to the pairwise interaction energies , i.e. , the atom - atom ( london ) and the atom - surface energies , in order to obtain the total interaction energy between the two atoms close to the surface . qualitatively , he found that the leading contribution of the many - body correction leads to a repulsion if the atoms are side by side , i.e. , at equal surface - to - substrate distances . rauber et al . @xcite used mclachlan s approach to study the electrodynamic screening of the van der waals interaction between adsorbed atoms and molecules and a substrate . the latter plays a role which is `` analogous to that of the third body in the three - body interaction between two particles embedded in a three - dimensional medium '' . the van der waals interaction between the two atoms at equal distances from the substrate is altered by the presence of the solid substrate and this perturbation is given by @xcite @xmath224 where @xmath225 is the distance between the atoms , @xmath121 is the height above the image plane , which is the same for both atoms , and @xmath226 . the coefficients are given by @xmath227 and @xmath228 with @xmath229 / \left [ \epsilon(i\zeta ) + 1 \right ] ~,\ ] ] where @xmath230 is an imaginary frequency , @xmath231 is the polarizability of the atoms ( with the dimension of a volume ) , and @xmath232 is the dielectric function of the solid ( i.e. , the substrate ) . the lateral force due to the perturbation potential given by eq . , which is the analogue of the many - body contribution to the lateral ccf , follows from differentiating @xmath233 with respect to @xmath225 : @xmath234 ~.\end{gathered}\ ] ] in figure [ mclachan_fig ] we plot the lateral force given by eq . as function of the distance @xmath225 between the two atoms for several ( equal ) distances @xmath121 above the substrate . we use the values provided in ref . @xcite for the coefficients @xmath235 and @xmath236 : @xmath237 ev@xmath238()@xmath239 and @xmath240 ev@xmath238()@xmath239 , which correspond to ne , and @xmath241 ev@xmath238()@xmath239 and @xmath242 ev@xmath238()@xmath239 , which correspond to ar . as one can infer from fig . [ mclachan_fig ] , the many - body contribution to the lateral van der waals force is always repulsive and , as the two atoms approach the substrate , its strength increases . on the other hand , in the case of the many - body contribution to the lateral ccf , we have found that it is attractive for all bcs considered , if the surface - to - surface distances between the spheres and the sphere - surface - to - substrate distances are equal to each other and to the radius of the spheres ( see fig . [ mb_x ] ) . further , we can _ quantitatively _ compare our results with those from refs . @xcite and @xcite . to this end , we assign values to the geometrical parameters characterizing the configuration of the two atoms close to the substrate and compare the results of refs . @xcite and @xcite with those for similar configurations in our model . for example , estimating the many - body contribution to the lateral van der waals force for a configuration of two atoms close to a substrate corresponding to the configuration associated with the black curve the black curve corresponds to a function of temperature but there is no temperature dependence of the van der waals force between the atoms . therefore the comparison has to be carried out by choosing a certain value of @xmath243 . in fig . [ mb_x ] ( i.e. , @xmath244 in the case of the ccf and @xmath245 in the case of the two atoms ) , one obtains from eq . a value for the relative contribution of the many - body force to the lateral force which corresponds to ca . this is comparable with the relative contribution of the many - body force to the lateral ccf sufficiently close to @xmath11 , although in the case of the van der waals force it is repulsive ( fig . [ mclachan_fig ] ) while in the case of the ccf ( fig . [ mb_x ] ) it is attractive . considering the decay of the many - body contribution to the normal ccf as function of the surface - to - surface distance @xmath129 between the spheres ( for @xmath216 ) , we can compare the corresponding decay of the normal many - body force @xmath246 given by the potential @xmath233 in eq . . for small separations @xmath225 between the atoms the many - body contribution to the normal van der waals force increases as @xmath247 , while for large separations it decays as @xmath248 . by analyzing the data shown in fig . [ mb_l_z ] one finds that for @xmath249 the scaling function of the many - body contribution to the normal ccf decays slower than @xmath250 . this means that in this temperature regime the many - body contribution to the normal ccf is much more long ranged than the corresponding contribution to the normal van der waals force in the case of two atoms close to a surface . for fixed @xmath225 the many - body contribution to the lateral van der waals force decays as @xmath251 upon increasing the distance of both atoms from the substrate whereas the normal force on a single atom decays as @xmath252 . as a final remark we point out that we have not found a completely stable configuration for the two colloids ( fig . [ system_sketch ] ) : whenever there is a stable position in the horizontal ( vertical ) direction , the force is nonzero in the vertical ( horizontal ) direction . for example , consider a vertical path with @xmath253 in figs . [ d2_z](b ) and [ d2_x](b ) . from the first one can see that along this path the normal ccf changes from being repulsive to being attractive as the sphere - surface - to - substrate distance for colloid ( 2 ) is increased , implying that there is a vertical position of colloid ( 2 ) in which the normal ccf is zero . however , according to fig . [ d2_x](b ) the lateral ccf is always attractive regardless of the vertical position of colloid ( 2 ) . this means that there is a configuration which is stable only in the normal direction . accordingly , a dumbbell configuration with a rigid thin fiber between the two colloids can levitate over the substrate . whether this configuration is stable with respect to a vertical tilt remains as an open question . of the lateral many - body ccf acting on colloid ( 2 ) for @xmath207 and @xmath254 . the scaling function is shown as function of the scaling variable ratio @xmath255 . the black , red , and green lines correspond to the bcs @xmath157 , @xmath197 , and @xmath156 , respectively . for all three bcs the many - body contribution is not monotonic as function of temperature . quantitatively the green , red , and black curves here should be compared with the black curves in figs . [ d2_x](a ) , [ d2_x](b ) , and [ l_x](a ) , respectively . however , for the data shown in _ this _ figure the error bars ( not shown ) due to limits of the numerical accuracy are between 10@xmath256 and 15@xmath256 . this is the main reason why we refrain from showing what would be an instructive plot such as @xmath257 as a function of @xmath258 for various values of @xmath259 and @xmath170 ( as we did in fig . [ l_x ] ) , which would allow for a direct comparison with the case of atoms . ] , derived from the expression for the excess potential given in refs . @xcite and @xcite [ eq . ] . the forces are plotted as functions of the lateral separation @xmath225 between the two atoms for several equal vertical distances @xmath121 of the atoms from the substrate . the curves correspond to two sets of values for the coefficients @xmath235 and @xmath236 in eq . @xcite , corresponding to ne ( solid lines ) and ar ( dashed lines ) . ] we now turn our attention to the case in which the colloids are vertically aligned with respect to a planar , homogeneous substrate , i.e. , when their centers have the same coordinates in both the @xmath42 and the @xmath260 direction ( see fig . [ vertical_sketch ] ) . we focus on the normal ccf acting on colloid ( 1 ) when the system is immersed in a near - critical binary liquid mixture at its critical concentration . as before we consider @xmath31 bcs corresponding to a strong adsorption preference for one of the two components of the confined liquid . in particular , we consider two three - dimensional spheres of radii @xmath32 and @xmath33 with bcs @xmath34 and ( @xmath35 ) , respectively , facing a laterally homogeneous substrate with bc @xmath36 . colloid ( 1 ) is positioned at a sphere - surface - to - substrate distance @xmath37 and colloid ( 2 ) is at a surface - to - surface distance @xmath261 from colloid ( 1 ) ( see fig . [ vertical_sketch ] ) . the coordinate system @xmath39 is chosen such that the centers of the spheres are located at @xmath262 and @xmath263 so that the distance between the centers , along the @xmath74-axis , is given by @xmath264 . as before , the bcs of the system as a whole are represented by the set @xmath44 , where @xmath45 , @xmath35 , and @xmath46 can be either @xmath47 or @xmath48 . and @xmath33 immersed in a near - critical binary liquid mixture ( not shown ) and close to a homogeneous , planar substrate at @xmath63 . the colloidal particle ( 1 ) with bc @xmath34 is located vertically at the sphere - surface - to - substrate distance @xmath37 , whereas the colloidal particle ( 2 ) with bc @xmath64 is located vertically at the surface - to - surface distance @xmath261 between the spheres . the substrate exhibits bc @xmath36 . the vertical distance between the centers of the spheres along the @xmath74-direction is given by @xmath264 , while the centers of both spheres lie on the vertical axis @xmath265 . in the case of four spatial dimensions the figure shows a three - dimensional cut of the system , which is invariant along the fourth direction , i.e. , the spheres correspond to parallel hypercylinders with one translationally invariant direction , which is @xmath66 [ see eq . ] . ] the normal ccf @xmath266 acting on colloid ( 1 ) along the @xmath74-direction takes the scaling form @xmath267 where @xmath268 ( i.e. , @xmath269 for @xmath27 and @xmath270 for @xmath10 ) , @xmath76 , @xmath271 , and @xmath272 ; @xmath273 . equation describes the singular contribution to the normal force emerging upon approaching @xmath11 . @xmath274 is the force on a hypercylinder divided by its extension in the translationally invariant direction [ see eq . ] . we use the same reference system as the one described by eq . in order to normalize the scaling function defined in eq . according to @xmath275 we calculate the many - body contribution to the normal ccf acting on particle ( 1 ) @xmath276 by subtracting from the total force the sum of the pairwise forces acting on it , i.e. , the colloid - colloid and the colloid - substrate forces [ see eq . ] . this many - body force takes the scaling form @xmath277 in fig . [ mb_vertical_z ] we show the normalized [ see eqs . , , and ] scaling functions @xmath278 of the many - body contribution to the normal ccf acting on colloid ( 1 ) as functions of the scaling variable ratio @xmath279 for two spherical colloids of the same size ( @xmath207 ) . keeping the surface - to - surface distance between the spheres fixed at @xmath280 , we vary the sphere - surface - to - substrate distance @xmath37 for several bcs : @xmath156 in ( a ) , @xmath281 in ( b ) , @xmath197 in ( c ) , and @xmath157 in ( d ) . from figs . [ mb_vertical_z](a ) and ( d ) one can infer that if the colloids have symmetric bcs , the scaling function of the many - body normal ccf acting on colloid ( 1 ) is negative ( i.e. , it is directed towards the substrate ) for any value of @xmath198 . on the other hand , for non - symmetric bcs between the colloids [ figs . [ mb_vertical_z](b ) and ( c ) ] , the many - body contribution to the normal ccf acting on colloid ( 1 ) is positive for any value of @xmath198 . the apparent change of sign in figs . [ mb_vertical_z](b ) and ( c ) is likely to be an artifact occurring within the error bars due to numerical imprecision . the relative contribution of the many - body ccf to the total force is between 10@xmath256 and 15@xmath256 . we point out that this configuration with the the two colloids vertically aligned with respect to the substrate allows for a wide range of interesting aspects which will be further explored in future works . we have investigated critical casimir forces ( ccfs ) for a system composed of two equally sized spherical colloids ( @xmath207 ) immersed in a near - critical binary liquid mixture and close to a laterally homogeneous substrate with @xmath128 boundary condition ( bc ) ( see fig . [ system_sketch ] ) . by denoting the set of bcs of the system as @xmath44 , where @xmath282 corresponds to the bc at colloid @xmath283 and @xmath46 to the bc at the substrate , we have first focused on the total normal and lateral forces acting on one of the colloids [ labeled as `` colloid ( 2 ) '' ] for several geometrical configurations of the system and various combinations of bcs at the colloids . both the normal and the lateral forces are characterized by universal scaling functions [ eqs . and , respectively ] , which have been studied in the one - phase region of the solvent as functions of @xmath284 and @xmath285 . @xmath129 is the surface - to - surface distance between the two colloids , and @xmath178 is the bulk correlation length of the binary mixture in the mixed phase . we have used mean - field theory together with a finite element method in order to calculate the order parameter profiles , from which the stress tensor renders the normalized scaling functions associated with the ccfs . for the scaling function of the total normal ccf acting on colloid ( 2 ) with @xmath145 bc , in the presence of colloid ( 1 ) with @xmath286 , we have found ( fig . [ d2_z](a ) ) that the scaling function changes sign for a fixed value of @xmath284 as the distance @xmath38 between colloid ( 2 ) and the substrate increases , signaling the occurrence of an unstable mechanical equilibrium configuration of a vanishing normal force . for the total normal ccf acting on colloid ( 2 ) with @xmath144 bc and in the presence of colloid ( 1 ) with @xmath133 , we have found ( fig . [ d2_z](b ) ) that the force changes sign upon changing the temperature . for this combination of bcs , the equilibrium configuration of colloid ( 2 ) is stable in the normal direction . without a substrate , at the critical composition of the solvent the ccf between two @xmath22 spheres is identical to the one between two @xmath23 ones . this degeneracy is lifted by the presence of the substrate as one can infer from the comparison of the scaling functions for the lateral ccfs for @xmath156 and @xmath157 bcs [ see figs . [ d2_x](a ) and ( b ) , respectively ] . in the first case , the shape of the scaling function resembles that of the two colloids far away from the substrate , with a minimum at @xmath287 . in the second case this minimum at @xmath288 disappears . these substrate - induced changes are more pronounced if the two spheres are close to the substrate ( fig . [ d2_x ] ) . without a substrate the ccf between spheres of opposite bcs is purely repulsive . in the presence of a substrate the corresponding lateral ccf for @xmath197 bcs turns attractive for large lateral distances @xmath129 ( fig . [ l_x ] ) , which is a pure many - body effect . we have also studied the direction of the total ccf acting on colloid ( 2 ) for various spatial configurations and bcs in order to assess the preferred arrangement of the colloids . for @xmath156 bcs we have found that they tend to aggregate laterally . in this case a collection of colloids with @xmath22 bcs , facing a substrate with the same bc , are expected to form a monolayer on the substrate . for the situation of @xmath157 bcs , we have found that the colloids are expected to aggregate on top of each other . this indicates that a set of several colloids with such bcs are expected to form three - dimensional clusters . these tendencies are enhanced upon approaching the critical point ( fig . [ direction_resulting_force_d2 ] ) . by calculating the pairwise colloid - colloid and colloid - substrate forces and subtracting them from the total force , we have determined the pure many - body contribution to the force acting on colloid ( 2 ) . for the scaling functions associated with the normal many - body ccfs we have found the interesting feature of a change of sign at fixed temperature upon varying the lateral position of colloid ( 2 ) ( fig . [ mb_l_z ] ) . this implies that , for a given temperature , there is a lateral position where the normal many - body ccf is zero , in which case the sum of pairwise forces provides a quantitatively reliable description of the interactions of the system . as expected we have found that the contribution of the many - body ccfs to the total force is large if the colloid - colloid and colloid - substrate distances are small , as well as if the binary liquid mixture is close to its critical point . we have compared our results with corresponding ones for quantum - electrodynamic casimir interactions . to this end we have referred to the results in ref . @xcite for two dielectric spheres immersed in ethanol and facing a plate . these authors analyze the influence of the distance @xmath289 from the plate on the equilibrium separation @xmath217 between the spheres , which are subject to quantum - electrodynamic casimir forces . they find that the lateral many - body force is attractive ( repulsive ) if the stronger one of the two normal sphere - plate forces is attractive ( repulsive ) [ figs . [ comparing](a ) and ( b ) ] . on the other hand , in the case of ccfs we have found that for a configuration in which the surface - to - surface distance between the colloids is equal to the sphere - surface - to - substrate ones and equal to the radius of the spheres , the many - body contribution to the lateral ccf is always attractive , regardless of the bcs ( fig . [ mb_x ] ) . we have also compared our results with the corresponding ones for the case of two atoms close to the planar surface of a solid body . in this respect we have referred to the mclachlan model @xcite for the many - body contribution to the van der waals potential [ see eq . and fig . [ mclachan_fig ] ] and the results from ref . @xcite . from this comparison we have found that if the two atoms are fixed at the same distance from the surface of the solid body , the normal many - body contribution to the total van der waals force decays with the atom - atom distance @xmath225 as @xmath248 for large atom - atom distances . this decay is much faster than the decay we estimate for the many - body contribution to the normal ccf , which within a suitable range appears to be slower than @xmath250 . furthermore , we have found that the many - body contribution to the lateral van der waals force is repulsive while the corresponding many - body ccf is attractive regardless of the set of bcs . finally we have considered the configuration in which the two colloids are vertically aligned with respect to the substrate ( fig . [ vertical_sketch ] ) . we have calculated the many - body contribution to the normal ccf acting on colloid ( 1 ) for two spherical colloids of the same size ( @xmath207 ) keeping the sphere - surface - to - surface distance @xmath280 fixed ( fig . [ mb_vertical_z ] ) . we have varied the sphere - surface - to - substrate distance @xmath37 for several bcs and have found that if the colloids have the same bcs , the many - body contribution to the normal ccf is directed towards the substrate [ figs . [ mb_vertical_z](a ) and ( d ) ] , whereas for colloids with opposite bcs , the many - body contribution to the normal ccf is directed away from the substrate [ figs . [ mb_vertical_z](b ) and ( c ) ] . we have found that the contribution of the many - body ccf to the total force is between 10@xmath256 and 15@xmath256 . t.g.m . would like to thank s. kondrat for valuable support with the computational tools used to perform the numerical calculations . s.d . thanks m. cole for providing ref . @xcite .

within mean - field theory we calculate the scaling functions associated with critical casimir forces for a system consisting of two spherical colloids immersed in a binary liquid mixture near its consolute point and facing a planar , homogeneous substrate . for several geometrical arrangements and boundary conditions we analyze the normal and the lateral critical casimir forces acting on one of the two colloids . we find interesting features such as a change of sign of these forces upon varying either the position of one of the colloids or the temperature . by subtracting the pairwise forces from the total force we are able to determine the many - body forces acting on one of the colloids . we have found that the many - body contribution to the total critical casimir force is more pronounced for small colloid - colloid and colloid - substrate distances , as well as for temperatures close to criticality , where the many - body contribution to the total force can reach up to @xmath0 .

arxiv

molecular hydrogen is the most abundant molecule in the universe and the main constituent of regions where stars are forming . h@xmath2 plays an important role in the chemistry of the interstellar medium , and its formation governs the transformation of atomic diffuse clouds into molecular clouds . because of the inefficient gas phase routes to form h@xmath2 , dust grains have been recognized to be the favored habitat to form h@xmath2 molecules ( @xcite , @xcite ) . the sticking of h atoms onto surfaces has received considerable attention because this mechanism governs the formation of h@xmath2 , but also other molecules that contain h atoms . the sticking of h atoms onto dust grains can also be an important mechanism to cool interstellar gas ( @xcite ) . in the past decades , a plethora of laboratory experiments and theoretical models have been developed to understand how h@xmath2 forms . as h atoms arrive on dust surfaces , they can be weakly ( physisorbed ) or strongly ( chemisorbed ) bound to the surface . the sticking of h in the physisorbed state ( @xcite , @xcite , @xcite ; @xcite ) and in the chemisorbed state ( @xcite ; @xcite ; @xcite ) has been highlighted by several experiments on different types of surfaces ( amorphous carbon , silicates , graphite ) . in the ism , dust grains are mainly carbonaceous or silicate particles with various sizes and represent an important surface for the formation of h@xmath2 . however , a large part ( @xmath3 50@xmath4 ) of the available surface area for chemistry is in the form of very small grains or pahs ( @xcite ) . these pahs are predicted to have characteristics similar to graphite surfaces : however , once the first h atom is chemisorbed on the basal plane , subsequent adsorptions of h atoms in pairs appear to be barrierless for the para dimer and with a reduced barrier for the ortho dimer ( @xcite ) . h@xmath2 can then form by involving a pre - adsorbed h atom in monomer ( @xcite ; @xcite ; @xcite ; @xcite ) or in a para - dimer configuration ( @xcite ) . however , while these routes represent efficient paths to form h@xmath2 , the inefficient sticking of h atoms in monomers constitutes an important obstacle to enter the catalytic regime for h@xmath2 formation . this results in a very low h@xmath2 formation efficiency on graphitic / pah surfaces ( @xcite ) . the hydrogenation on the pah edges has been identified as an important route to form h@xmath2 in the ism ( @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) . density functional theory calculations have shown that the first hydrogenation of neutral coronene is associated with a barrier ( @xmath360 mev ) but that subsequent hydrogenation barriers vanish ( @xcite ) . recently , coronene films exposed to h / d atoms at high temperature , were studied by means of ir spectroscopy ( @xcite ) and mass spectrometry ( @xcite ) . these measurements showed that neutral pahs , when highly hydrogenated , are efficient catalysts for the formation of h@xmath2 , and confirmed the high h@xmath2 formation rate attributed to pahs in pdrs ( @xcite ) . pah cations , which are usually present at lower extinction a@xmath5 , and therefore reside at the surfaces of pdrs , also represent an important route to form h@xmath2 ( @xcite ; @xcite ) . the addition of the first h atom is predicted to be barrierless . this reaction is exothermic but the product should be stabilized by ir emission . a second h atom can react with the already adsorbed h to form h@xmath2 without a barrier ( @xcite ; @xcite ) . in this letter , we study experimentally the hydrogenation of coronene cations in the gas phase through exposure to hydrogen atoms . by using mass spectrometry , we show that odd hydrogenation states of coronene cations predominantly populate the mass spectrum . our results highlight the fact that the further hydrogenation of pah cations is associated with a barrier if the number already attached h atoms is odd , and no barrier if this number is even . this alternanting barrier - no barrier occurence seems to remain with increasing hydrogenation . these results suggest that pah cations can also enjoy highly hydrogenated states in the interstellar medium , and acts as catalysts for h@xmath2 formation . in this pilot experiment we show the feasibility of studying the hydrogenation of pahs in the gas phase . for this purpose , we use a setup designed to study molecular ions in a radiofrequency ion trap . time - of - flight mass spectrometry of the trap content is used to identify the changes in mass of the coronene cations and therefore deduce their respective degrees of hydrogenation . the experiments have been performed using a home - built tandem - mass spectrometer shown schematically in figure [ fig : setup ] ( @xcite ) . a beam of singly charged coronene radical cations ( [ c@xmath6h@xmath7@xmath8 , m / z 300 ) was extracted from an electrospray ion source . the ions were phase - space compressed in an radiofrequency ( rf ) ion funnel and subsequently in an rf quadrupole ion guide . mass selection was accomplished by using an rf quadrupole mass filter . accumulation of the ions took place in a three dimensional rf ion trap ( paul trap ) . a he buffer gas at room temperature was used to collisionally cool the trapped cations . exposure to gas - phase atomic hydrogen for variable periods of time led to multiple hydrogen adsorption on the coronene cations . an electric extraction field was then applied between the trap end - caps to extract the trapped hydrogenated cations into a time - of - flight ( tof ) mass spectrometer with resolution m/@xmath9 m @xmath3 200 . to obtain mass spectra of sufficient statistics , typically a couple of hundred tof traces were accumulated . electrospray ionization allows to gently transfer ions from the liquid phase into the gas phase . inspired by the method of @xcite we have run the ion source with a solution consisting of 600 @xmath10l of saturated solution of coronene in methanol , 350 @xmath10l of hplc grade methanol and 50 @xmath10l of 10 mm solution of @xmath11 solution in methanol . in the liquid phase , electron transfer from a coronene molecule to a silver ion leads to formation of the required radical cation . the trapped ions are exposed to hydrogen atoms produced from h@xmath2 by a slevin type source which has been extensively used in crossed beam experiments ( @xcite,@xcite ) . while in the earlier work the dissociation fractions were determined by means of electron impact excitation or heii line emission , we now use charge removal ( captured ionization ) and dissociation induced by 40 kev he@xmath12 . for these processes the cross sections are well - known ( @xcite ) . in this way we determine a hydrogen dissociation fraction of @xmath13 . the temperature of the h beam is around room temperature ( @xmath325 mev ) . coronene ions are exposed to a constant flux of h atoms for different periods of time before their degree of hydrogenation is determined by means of mass spectrometry . the irradiation time is varied from 1.0 up to 30 s to study the time - dependence of coronene hydrogenation . the data obtained from our experiment are a series of mass spectra of hydrogenated coronene cations as a function of h exposure time . some of the spectra are shown in fig.[fig : rawdata ] . fig.[fig : rawdata](a ) shows the mass spectrum of the native m / z=300 coronene cations . a similar , thus unchanged , mass spectrum is obtained ( not shown in this article ) if we irradiate coronene cations with molecular hydrogen . this means that molecular hydrogen does not stick to coronene cations at room temperature . after turning on the hydrogen source and exposing the coronene cations to the atomic hydrogen beam for 1.0 s ( fig.[fig : rawdata ] , ( b ) ) , the peak at @xmath14 shifts to 301 , which means that the trap content main constituent is ( c@xmath6h@xmath15+h)@xmath8 . for increasing irradiation time ( fig.[fig : rawdata](c ) t= 2 s , ( d ) 3 s , ( e ) 4 s and ( f ) 4.75 s ) , the peak at @xmath16=301 disappears progressively while a peak at @xmath17 and then at @xmath18 ( for t = 4.75 s see fig.[fig : rawdata](f ) ) appears , which indicates the addition of 3 and 5 hydrogen atoms , respectively . at longer exposure time ( fig.[fig : longdata](a ) t @xmath315 s ) , the @xmath16=303 peak dominates the signal , and a peak at @xmath16=305 appears . at even longer irradiation times ( fig.[fig : longdata](b ) t @xmath330 s ) , the peak @xmath16=305 dominates and peaks at @xmath16=307 and 309 appear . these peaks clearly show the evolution of the hydrogenation states of coronene cations with h irradiation time . our results show that the most important peaks measured in the mass spectrum shift from lower masses to higher masses with increasing h exposure time . in order to follow the evolution of the first hydrogenated state of coronene cation ( c@xmath6h@xmath15+h)@xmath8 ( corh@xmath8 ) to the second ( c@xmath6h@xmath15 + 2h)@xmath8 ( corh@xmath19 ) , third ( corh@xmath20 ) and fourth ( corh@xmath21 ) hydrogenated states , we use a simple model that describes this evolution : @xmath22 @xmath23 @xmath24 @xmath25 hydrogenation of corh@xmath26@xmath8 follows an arrhenius expression where a@xmath27 is the prefactor and e@xmath27 is the barrier , while hydrogenation of corh@xmath28 follows the same expression with a prefactor a@xmath26 and no barrier . k@xmath29 is the boltzmann constant and t the temperature of the h beam ( t@xmath325 mev ) . in these equations we do not include abstraction , meaning that the time evolution of the contribution of each state is governed entirely by hydrogenation . this assumption is made in order to derive the first barriers of hydrogenation . abstraction can be neglected in the conditions of our experiments for low exposure times . this is supported by previous experiments where the cross section for addition of hydrogen to neutral coronene is predicted to be 20 times that for abstraction ( @xcite ) . further support is drawn from a kinetic chemical model we developed , which shows that abstraction has to be very low compared to hydrogenation to be able to mimic the experimental results ( boschman et al . in prep ) . however , for long h exposure time we expect the hydrogenation degree of the coronene cations to reach a steady state which will allow us to derive the contribution of abstraction relative to addition , and therefore derive the h@xmath0 formation rate due to pah cations . it should also be kept in mind that in the conditions of our experiments , the h atoms are at room temperature meaning that they cross the barriers for abstraction ( 10 mev , @xcite ) and addition ( 40 - 60 mev , @xcite ) with similar ease . under interstellar conditions , however , the abstraction will dominate by 8 orders of magnitude ( at 20 k ) because of the barrier differences . the first hydrogenation is expected to take place at the outer edge carbon atom ( @xcite ) . this state provides more conformational freedom to the four neighbouring outer edge carbon atoms , ensuring a preference for the second hydrogenation to take place at one of those four carbon atoms . the third hydrogenation will preferentially take place at the outer edge carbon next to the second h atom . again , the forth h atom can be bound to one of the four neighbouring outer edge carbon atoms , and the fifth sticks on the neighboring outer edge carbon . this scenario of h atoms sticking preferentially on outer edge carbons next to already adsorbed atoms is described in @xcite . the contribution of every peak is determined by fitting our data with gaussians with identical widths ( see fig.[fig : fit](a ) ) . the ratios between different hydrogenation states as function of time are reported in fig.[fig : fit](b ) . it appears that the ratio between the contribution of the first ( corh@xmath8 ) and the second ( corh@xmath19 ) hydrogenation state does not evolve with time for short time scales @xmath30 . also , the ratio between the third ( corh@xmath20 ) and the forth ( corh@xmath21 ) hydrogenation state shows identical behaviour after t@xmath31 2s @xmath32 . before this exposure time the n@xmath33 and n@xmath34 signals are very weak , and the ratio is uncertain . we can therefore assume that for these measurements @xmath35 and @xmath36 . the expression for the corh@xmath37 to corh@xmath38 as well as for the corh@xmath39 to corh@xmath40 energy barriers can then be written as : @xmath41 @xmath42 from these expressions we derive the energy barrier e@xmath2 as 72@xmath16 mev and e@xmath43 as 43@xmath18 mev , as shown in fig.[fig : fit](c ) . this shows that hydrogenation barriers are decreasing with increasing hydrogenation . however , our results also show that odd hydrogenated states dominate the mass spectrum even for high degrees of hydrogenation ( fig.[fig : longdata ] ) . this highlights the presence of a barrier - no barrier alternation from one hydrogenated state to another , up to high hydrogenation states . so our results indicate that even if the hydrogenation barriers decrease for the first hydrogenations , they do not vanish completely and remain at higher hydrogenation states . the barriers derived in our study are similar to the one calculated by @xcite for neutral coronene . this means that the first hydrogenations of coronene cations should be comparable to the hydrogenation of neutral coronene . however , for higher degree of hydrogenation we show that these barriers still exist , while the calculations from @xcite predict that these barriers vanish after a few hydrogenations . recent mass spectrometric measurements of coronene films exposed to h / d atoms do not show preferences for even or odd hydrogenation states of neutral coronene ( @xcite ) . however , these measurements are not very sensitive to barrier heights well bellow 100 mev , since the experiments were performed with atoms at beam temperature of 170 mev . in pdrs exposed to uv fields less than few hundreds g@xmath44 , the spatial distribution of h@xmath2 and pahs does correlate ( @xcite , @xcite , @xcite ) contrary to what is seen in the presence of strong uv fields ( @xcite , @xcite ) . the h@xmath2 formation rates have been derived for several pdrs exposed to various uv radiation fields . these rates can be explained by the contribution of pahs to the formation of h@xmath2 ( @xcite ) . depending on the uv intensity , the pahs observed can either be pah cations , that are present in regions at low visual exctinctions a@xmath5 , or neutral pahs , which are located at higher extinctions . work by @xcite and @xcite has shown that high - uv and high density pdrs ( n@xmath45@xmath3110@xmath46 @xmath47 and g@xmath44@xmath31100 , g@xmath48 ) can maintain a @xmath3 30@xmath4 cationic fraction upto a few mag in a@xmath5 . more relevant to this work , @xcite have studied low - uv pdrs ( g@xmath44@xmath49100 ) , and followed the pah charge balance for different densities , uv radiation fields and metallicities . they found that pah cations dominate over neutrals and anions for a@xmath5@xmath492 mag . the h@xmath2 formation rates observed in pdrs exposed to different uv fields can therefore be partly attributed to neutral and cationic pahs . our results show that the hydrogenation processes of neutral and cationic pahs are similar and should contribute similarly to the formation of h@xmath2 . further experimental investigations will allow us to derive the h@xmath2 formation rate for pah cations . we have investigated the addition of hydrogen atoms to coronene cations in the gas phase and observed increasing hydrogenation with h exposure time . our results show that odd hydrogenated states dominate the mass spectrum , which evidences the presence of a barrier for the further hydrogenation of odd hydrogenation states . the first hydrogen sticks to the coronene cations without a barrier ( @xcite , @xcite ) . the second and forth hydrogenations are associated with barriers of about 72 @xmath1 6 mev and 43 @xmath1 8 mev , while the third and fifth hydrogenation are barrierless . these barriers are similar to the one calculated for neutral coronene ( @xcite ) . our results indicate that superhydrogenated pah cations ( @xcite ) should also be found in the interstellar medium , and be important catalysts for the formation of h@xmath2 , as it is the case for their neutral counterparts . l. b. and s. c. are supported by the netherlands organization for scientific research ( nwo ) . g.r . recognizes the funding by the nwo dutch astrochemistry network . we would like to thank the anonymous referee for the helpful comments .

molecular hydrogen is the most abundant molecule in the universe . a large fraction of h@xmath0 forms by association of hydrogen atoms adsorbed on polycyclic aromatic hydrocarbons ( pahs ) , where formation rates depend crucially on the h sticking probability . we have experimentally studied pah hydrogenation by exposing coronene cations , confined in a radiofrequency ion trap , to gas phase atomic hydrogen . a systematic increase of the number of h atoms adsorbed on the coronene with the time of exposure is observed . odd coronene hydrogenation states dominate the mass spectrum up to 11 h atoms attached . this indicates the presence of a barrier preventing h attachment to these molecular systems . for the second and fourth hydrogenation , barrier heights of 72 @xmath1 6 mev and 40 @xmath1 10 mev , respectively are found which is in good agreement with theoretical predictions for the hydrogenation of neutral pahs . our experiments however prove that the barrier does not vanish for higher hydrogenation states . these results imply that pah cations , as their neutral counterparts , exist in highly hydrogenated forms in the interstellar medium . due to this catalytic activity , pah cations and neutrals seem to contribute similarly to the formation of h@xmath0 .

arxiv

as is well known , the atom or atoms in the atomic clock are passive they do not `` tick''so the clock needs an active oscillator in addition to the atom(s ) . in designing an atomic clock to realize the second as a measurement unit in the international system of units ( si ) , one encounters two problems : ( a ) the resonance exhibited by the atom or atoms of the clock varies with the details of the clock s construction and the circumstances of its operation ; in particular the resonance shifts depending on the intensity of the radiation of the atoms by the oscillator . ( b ) the oscillator , controlled by , in effect , a knob , drifts in relation to the knob setting . problem ( a ) is dealt with by introducing a wave function parametrized by radiation intensity and whatever other factors one deems relevant . the si second is then `` defined '' by the resonance that `` would be found '' at absolute zero temperature ( implying zero radiation ) . for a clock using cesium 133 atoms , this imagined resonance is declared by the general conference of weights and measures to be 9 192 631 770 hz , so that the si second is that number of cycles of the radiation at that imagined resonance @xcite . to express the relation between a measured resonance and the imagined resonance at 0 k , a wave function is chosen . problem ( b ) is dealt with by computer - mediated feedback that turns the knob of the oscillator in response to detections of scattering of the oscillator s radiation by the atom(s ) of the clock , steering the oscillator toward an aiming point . a key point for this paper is that the wave function incorporated into the operation of an atomic clock can never be unconditionally known . the language of quantum theory reflects within itself a distinction between ` explanation ' and ` evidence ' . for explanations it offers the linear algebra of wave functions and operators , while for evidence it offers probabilities on a set of outcomes . outcomes are subject to quantum uncertainty , but uncertainty is only the tip of an iceberg : how can one `` know '' that a wave function describes an experimental situation ? the distinction within quantum theory between linear operators and probabilities implies a gap between any explanation and the evidence explained . @xcite : [ prop : one ] to choose a wave function to explain experimental evidence requires reaching beyond logic based on that evidence , and evidence acquired after the choice is made can call for a revision of the chosen wave function . because no wave function can be unconditionally known , not even probabilities of future evidence can be unconditionally foreseen . here we show implications of the unknowability of wave functions for the second as a unit of measurement in the international system ( si ) , implications that carry over to both digital communications and to the use of a spacetime with a metric tensor in explaining clock readings at the transmission and reception of logical symbols . clocks that generate universal coordinated time ( utc ) are steered toward aiming points that depend not only on a chosen wave function but also on an hypothesized metric tensor field of a curved spacetime . like the chosen wave function , the hypothesis of a metric tensor is constrained , but not determined , by measured data . guesses enter the operations of clocks through the computational machinery that steers them . taking incoming data , the machinery updates records that determine an aiming point , and so involves the writing and reading of records . the writing must take place at a phase of a cycle distinct from a phase of reading , with a separation between the writing and the reading needed to avoid a logical short circuit . in sec . [ sec : turing ] we picture an explanation used in the operation of a clock as a string of characters written on a tape divided into squares , one symbol per square . the tape is part of a turing machine modified to be stepped by a clock and to communicate with other such machines and with keyboards and displays . we call this modified turing machine an _ open machine_. the computations performed by an open machine are open to an inflow numbers and formulas incalculable prior to their entry . because a computer cycles through distinct phases of memory use , the most direct propagation of symbols from one computer to another requires a symbol from one computer to arrive during a suitable phase of the receiving computer s cycle . in sec . [ sec : phasing ] we elevate this phase dependence to a principle that defines the _ logical synchronization _ necessary to a _ channel _ that connects clock readings at transmission of symbols to clock readings at their reception recognizing the dependence of logic - bearing channels on an interaction between evidence and hypotheses about signal propagation engenders several types of questions , leading to a _ discipline of logical synchronization _ , outlined in sec . [ sec : patterns ] . the first type of question concerns patterns of channels that are possible aiming points , as determined in a blackboard calculation that assumes a theory of signal propagation . [ sec : typei ] addresses examples of constraints on patterns of channels under various hypotheses of spacetime curvature , leading to putting `` phase stripes '' in spacetime that constrain channels to or from a given open machine . an example of a freedom to guess an explanation within a constraint of evidence is characterized by a subgroup of a group of clock adjustments , and a bound on bit rate is shown to be imposed by variability in spacetime curvature . [ sec : adj ] briefly addresses the two other types of questions , pertaining not to _ hypothesizing _ possible aiming points ` on the blackboard ' , but to _ using _ hypothesized aiming points , copied into feedback - mediating computers , for the steering of drifting clocks . after discussing steering toward aiming points copied from the blackboard , we note occasions that invite revision of a hypothesized metric tensor and of patterns of channels chosen as aiming points . computer - mediated feedback , especially as used in an atomic clock , requires logic open to an inflow of inputs beyond the reach of calculation . to model the logic of a computer that communicates with the other devices in a feedback loop , we modify a turing machine to communicate with external devices , including other such machines . the turing machine makes a record on a tape marked into squares , each square holding one character of an alphabet . operating in a sequence of ` moments ' interspersed by ` moves ' , at any moment the machine scans one square of the tape , from which it can read , or onto which it can write , a single character . a move as defined in the mathematics of turing machines consists ( only ) of the logical relation between the machine at one moment and the machine at the next moment @xcite , thus expressing the logic of a computation , detached from its speed ; however , in a feedback loop , computational speed matters . let the moves of the modified turing machine be stepped by ticks of a clock . a step occurs once per a period of revolution of the clock hand . this period is adjustable , on the fly . we require that the cycle of the modified turing machine correspond to a unit interval of the readings of its clock . to express communication between open machines as models of computers , the modified turing machine can receive externally supplied signals and can transmit signals , with both the reception and the transmission geared to the cycle of the machine . in addition , the modified turing machine registers a count of moments at which signals are received and moments at which signals are transmitted . at a finer scale , _ the machine records a phase quantity in the cycle of its clock , relative to the center of the moment at which a signal carrying a character arrives . _ we call such a machine an _ open machine_. an open machine can receive detections and can command action , for instance the action of increasing or decreasing the frequency of the variable oscillator of an atomic clock . calculations performed on an open machine communicating with detectors and actuators proceed by moves made according to a rule that can be modified from outside the machine in the course of operation . these calculations respond to received influences , such as occurrences of outcomes underivable from the contents of the machine memory , when the open machine writes commands on a tape read by an external actuator . the wider physical world shows up in an open machine as both ( 1 ) unforeseeable messages from external devices and ( 2 ) commands to external devices . we picture a real - time computer in a feedback loop as writing records on the tape of an open machine . the segmentation into moments interspersed by moves is found not just in turing machines but in any digital computer , which implies the logical result of any computation is oblivious to variations in speed at which the clock steps the computer . + corollary 2.1 . _ no computer can sense directly any variation in its clock frequency . _ although it can not directly sense variation in the tick rate of its clock , the logic of open machine stepped by an atomic clock can still control the adjustment of the clock s oscillator by responding to variations in the detection rate written moment by moment onto its turing tape . a flow of unforeseeable detections feeds successive computations of results , each of which , promptly acted on , impacts probabilities of subsequent occurrences of outcomes , even though those subsequent outcomes remain unforeseeable . the computation that steers the oscillator depends not just on unforeseeable inputs , but also on a steering formula encoded in a program . * remarks * : 1 . to appreciate feedback , take note that a formula is distinct from what it expresses . for example a formula written along a stretch of a turing tape as a string of characters can contain a name @xmath0 for wave function as a function of time variable @xmath1 and space variables . the formula , containing @xmath0 , once written , just `` sits motionless , '' in contrast to the motion that the formula expresses . 2 . although unchanged over some cycles of a feedback loop , a feedback loop operates in a larger context , in which steering formulas are subject to evolution . sooner or later , the string that defines the action of an algorithm , invoking a formula , is apt to be overwritten by a string of characters expressing a new formula . occasions for rewriting steering formulas are routine in clock networks , including those employed in geodesy and astronomy . logical communication requires clocking . the reading of a clock of an open machine @xmath2an @xmath2-reading has the form @xmath3 where @xmath4 indicates the count of cycles and @xmath5 is the phase within the cycle , with the convention that @xmath6 . we define a channel from @xmath2 to @xmath7 , denoted @xmath8 , as a set of pairs , each pair of the form @xmath9 . the first member @xmath3 is an @xmath2-reading at which machine @xmath2 can transmit a signal and @xmath10 is a @xmath7-reading at which the clock of machine @xmath7 can register the reception of the signal . define a _ repeating channel _ to be a channel @xmath8 such that @xmath11)(\exists m , n , j , k ) ( m+\ell j.\phi_{a,\ell},n+\ell k.\phi_{b,\ell } ) ] \in \abr,\ ] ] for theoretical purposes , it is convenient to define an _ endlessly repeating channel _ for which @xmath12 ranges over all integers . again for theoretical purposes , on occasion we consider channels for which the phases are all zero , in which case we may omit writing the phases . because they are defined by local clocks without reference to any metric tensor , channels invoke no assumption about a metric or even a spacetime manifold . for this reason evidence from the operation of channels is independent of any explanatory assumptions involving a manifold with metric and , in particular , is independent of any global time coordinate , or any `` reference system '' @xcite . thus clock readings at the transmission and the reception of signals can prompt revisions of hypotheses about a metric tensor field . a record format for such evidence was illustrated in earlier work @xcite , along with the picturing of such records as _ occurrence graphs_. from the beating of a heart to the bucket brigade , life moves in phased rhythms . for a symbol carried by a signal from an open machine @xmath2 to be written into the memory of an open machine @xmath7 , the signal must be available at @xmath7 within a phase of the cycle of @xmath7 during which writing can take place , and the cycle must offer room for a distinct other phase . we elevate engineering commonplace to a principle pertaining to open machines as follows . [ prop : three ] a logical symbol can propagate from one open machine to another only if the symbol arrives within the writing phase of the receiving machine ; in particular , respect for phasing requires that for some positive @xmath13 any arrival phase @xmath14 satisfy the inequality @xmath15 prop . [ prop : three ] serves as a fixed point to hold onto while hypotheses about signal propagation in relation to channels are subject to revision . we call the phase constraint on a channel asserted by ( [ eq : main ] ) _ logical synchronization_. for simplicity and to allow comparing conditions for phasing with conditions for einstein synchronization , we take the engineering liberty of allowing transmission to occur at the same phase as reception , so that both occur during a phase interval satisfying ( [ eq : main ] ) . the alternative of demanding reception near values of @xmath16 can be carried out with little extra difficulty . + * remarks : * 1 . note that @xmath14 in the proposition is a phase of a cycle of a variable - rate clock that is _ not _ assumed to be in any fixed relation to a proper clock as conceived in general relativity . indeed , satisfying ( [ eq : main ] ) usually requires the operation of clocks at variable rates . the engineering of communications between computers commonly detaches the timing of a computer s receiver from that of the computer by buffering : after a reception , the receiver writes into a buffer that is later read by the computer@xcite . in analyzing open machines we do without buffering , confining ourselves to character - by - character phase meshing as asserted in prop . [ prop : three ] , which offers the most direct communication possible . given the definition of a channel and the condition ( [ eq : main ] ) essential to the communication of logical symbols , three types of questions arise : * type i : * what patterns of interrelated channels does one try for as aiming points ? * type ii : * how can the steering of open machines be arranged to approach given aiming points within acceptable phase tolerances ? * type iii : * how to respond to deviations from aiming points beyond tolerances ? such questions point the way to exploring what might be called a _ discipline of logical synchronization_. so far we notice two promising areas of application within this discipline : 1 . provide a theoretical basis for networks of logically synchronized repeating channels , highlighting 1 . possibilities for channels with null receptive phases as a limiting case of desirable behavior , and 2 . circumstances that force non - null phases . 2 . explore constraints on receptive phases imposed by gravitation , as a path to exploring and measuring gravitational curvature , including slower changes in curvature than those searched for by the laser gravitational wave observatory @xcite . answers to questions of the above types require hypotheses , if only provisional , about signal propagation . for this section we assume that propagation is described by null geodesics in a lorentzian 4-manifold @xmath17 with one or another metric tensor field @xmath18 , as in general relativity . following perlick @xcite we represent an open machine as a timelike worldline , meaning a smooth embedding @xmath19 from a real interval into @xmath17 , such that the tangent vector @xmath20 is everywhere timelike with respect to @xmath18 and future - pointing . we limit our attention to worldlines of open machines that allow for signal propagation between them to be expressed by null geodesics . to say this more carefully , we distinguish the _ image _ of a worldline as a submanifold of @xmath17 from the worldline as a mapping . consider an open region @xmath21 of @xmath17 containing a smaller open region @xmath22 , with @xmath21 containing the images of two open machines @xmath2 and @xmath7 , with the property that every point @xmath23 of the image of @xmath2 restricted to @xmath22 is reached uniquely by one future - pointing null geodesic from the image of @xmath7 in @xmath21 and by one past - pointing null geodesic from the image of @xmath7 in @xmath21 . we then say @xmath2 and @xmath7 are _ radar linkable _ in @xmath22 . we limit our attention to open machines that are radar linkable in some spacetime region @xmath22 . in addition we assume that the channels preserve order ( what is transmitted later arrives later ) . indeed , we mostly deal with open machines in a gently curved spacetime region , adequately described by fermi normal coordinates around a timelike geodesic . for simplicity and to allow comparing conditions for phasing with conditions for einstein synchronization , we take the liberty of allowing transmission to occur at the same phase as reception , so that both occur during a phase interval satisfying ( [ eq : main ] ) . the perhaps more realistic alternative of demanding reception near values of @xmath16 can be carried out with little difficulty . to develop the physics of channels , we need to introduce three concepts : \(1 ) we define a _ group of clock adjustments _ as transformations of the readings of the clock of an open machine . as it pertains to endlessly repeating channels , a group @xmath24 of clock adjustments consists of functions on the real numbers having continuous , positive first derivatives . group multiplication is the composition of such functions , which , being invertible , have inverses . to define the action of @xmath24 on clock readings , we speak ` original clock readings ' as distinct from adjusted readings an adjustment @xmath25 acts by changing every original reading @xmath26 of a clock @xmath2 to an adjusted reading @xmath27 . as we shall see , clock adjustments can affect echo counts . \(2 ) to hypothesize a relation between the @xmath2-clock and an accompanying proper clock , one has to assume one or another metric tensor field @xmath18 , relative to which to define proper time increments along @xmath2 s worldline ; then one can posit an adjustment @xmath28 such that @xmath29 where @xmath30 is the reading imagined for the accompanying proper clock when @xmath2 reads @xmath26 . \(3 ) we need to speak of positional relations between open machines . for this section we assume that when an open machine @xmath7 receives a signal from any other machine @xmath2 then @xmath7 echoes back a signal to @xmath2 right away , so the echo count @xmath31 defined in sec . [ sec : phasing ] involves no delay at @xmath7 . in this case , evidence in the form of an echo count becomes explained , under the assumption of a metric tensor field @xmath18 , as being just twice the radar distance @xcite from @xmath2 to the event of reception by @xmath7 . questions of type i concern constraints on channels imposed by the physics of signal propagation . here we specialize to constraints on channels imposed by spacetime metrics , constraints obtained from mathematical models that , while worked out so to speak on the blackboard , can be copied onto turing tapes as aiming points toward which to steer the behavior of the clocks of open machines . questions of types ii and iii are deferred to the sec . [ sec : adj ] . we begin by considering just two machines . assuming an hypothetical spacetime @xmath32 , suppose that machine @xmath2 is given as a worldline parametrized by its clock readings : what are the possibilities and constraints for an additional machine @xmath7 with two - way repeating channels @xmath8 and @xmath33 with a constant echo count ? we assume the idealized case of channels with null phases , which implies integer echo counts . for each @xmath2-tick there is a future light cone and a past light cone . with tick events indicated ; ( b ) light cones associated to ticks of @xmath2 ; ( c ) ticks of @xmath34 and @xmath35 at light cone intersections corresponding to @xmath36 . , width=470 ] the future light cone from an @xmath2-reading @xmath37 has an intersection with the past light cone for the returned echo received at @xmath38 . [ fig:3 ] illustrates the toy case of a single space dimension in a flat spacetime by showing the two possibilities for a machine @xmath7 linked to @xmath2 by two - way channels at a given constant echo count . in each solution , the clocking of @xmath7 is such that a tick of @xmath7 occurs at each of a sequence of intersections of outgoing and incoming light cones from and to ticks of @xmath2 . note that the image of @xmath7 , and not just its clock rate , depend on the clock rate of @xmath2 . determination of the tick events for @xmath7 leaves undetermined the @xmath7 trajectory between ticks , so there is a freedom of choice . one can exercise this freedom by requiring the image of @xmath7 to be consistent with additional channels of larger echo counts . a clock adjustment of @xmath2 of the form @xmath39 for @xmath40 a positive integer increases the density of the two - way channel by @xmath40 and inserts @xmath41 events between successive @xmath7-ticks , thus multiplying the echo count by @xmath40 . as @xmath40 increases without limit , @xmath7 becomes fully specified . turning to two space dimensions , the image of @xmath7 must lie in a tube around the image of @xmath2 , as viewed in a three - dimensional space ( vertical is time ) . so any timelike trajectory within the tube will do for the image of @xmath7 . for a full spacetime of 3 + 1 dimensions , the solutions for the image of @xmath7 fall in the corresponding `` hypertube . '' the argument does not depend on flatness and so works for a generic , gently curved spacetime in which the channels have the property of order preservation . and @xmath7 freely chosen ; ( b ) light signals lacing @xmath2 and @xmath7 define tick events ; ( c ) interpolated lacings of light signals added to make @xmath36.,width=470 ] a different situation for two machines arises in case only the image of @xmath2 s worldline is specified while its clocking left to be determined . in this case the image of @xmath7 can be freely chosen , after which the clocking of both @xmath2 and @xmath7 is constrained , as illustrated in fig . [ fig:4 ] for the toy case of flat spacetime with 1 space dimension . to illustrate the constraint on clocking , we define a `` lacing '' of light signals to be a pattern of light signals echoing back and forth between two open machines as illustrated in fig . [ fig:4 ] ( b ) . for any event chosen in the image of @xmath2 , there is a lacing that touches it . in addition to choosing this event , one can choose any positive integer @xmath40 to be @xmath31 , and choose @xmath41 events in the image of @xmath2 located after the chosen event and before the next @xmath2-event touched by the lacing of light signals . the addition of lacings that touch each of the @xmath41 intermediating events corresponds to a repeating channel @xmath8 with echo count @xmath42 , along with a repeating channel @xmath33 with the same echo count @xmath43 . this construction does not depend on the dimension of the spacetime nor on its flatness , and so works also for a curved spacetime having the property of order preservation . evidence of channels as patterns of clock readings leaves open a choice of worldlines for its explanation . in the preceding example of laced channels between open machines @xmath2 and @xmath7 , part of this openness can be reflected within analysis by the invariance of the channels under a subgroup of the group of clock adjustments that `` slides the lacings , '' as follows . suppose that transmissions of an open machine @xmath2 occur at given values of @xmath2-readings . we ask about clock adjustments that can change the events of a worldline that correspond to a given @xmath2-reading . if a clock adjustment @xmath28 takes original @xmath2-readings @xmath26 to a revised @xmath2-readings @xmath27 , transmission events triggered by the original clock readings become triggered when the re - adjusted clock exhibits the _ same readings_. as registered by original readings , the adjusted transmission occurs at @xmath44 . based on this relation we inquire into the action of subgroups of @xmath45 on the readings of the clocks of two open machines @xmath2 and @xmath7 . in particular , there is a subgroup @xmath46 that expresses possible revisions of explanations that leave invariant the repeating channels with constant echo count @xmath40 . an element @xmath47 is a pair of clock adjustments that leaves the channels invariant , and such a pair can be chosen within a certain freedom . for the adjustment @xmath28 one is free to : ( a ) assign an arbitrary value to @xmath48 ; and ( b ) , if @xmath49 , then for @xmath50 , choose the value of @xmath51 at will , subject to the constraints that @xmath52 and @xmath53 is less than the original clock reading for the re - adjusted first echo from @xmath54 . with these choices , @xmath55 is then constrained so that each lacing maps to another lacing . the condition ( a ) slides a lacing along the pair of machines ; the condition ( b ) nudges additional lacings that show up in the interval between a transmission and the receipt of its echo . in this way a freedom to guess within a constraint is expressed by @xmath56 . moving to more than two machines , we invoke the _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * definition : * an _ arrangement of open machines _ consists of open machines with the specification of some or all of the channels from one to another , augmented by proper periods of the clock of at least one of the machines . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ( without specifying some proper periods , the scale of separations of one machine from another is open , allowing the arrangement to shrink without limit , thus obscuring the effect of spacetime curvature . ) although gentle spacetime curvature has no effect on the possible channels linking two open machines , curvature does affect the possible channels and their echo counts in some arrangements of five or more machines , so that the possible arrangements are a measure spacetime curvature . the way that spacetime curvature affects the possible arrangements of channels is analogous to the way surface curvature in euclidean geometry affects the ratios of the lengths of the edges of embedded graphs . the effect on ratios shows up in mappings from graphs embedded in a plane to their images on a sphere . for example , a triangle can be mapped from a plane to a generic sphere , in such a way that each edge of the triangle is mapped to an arc of the same length along a great circle on the sphere . the same holds for two triangles that share an edge , as illustrated in fig . [ fig : sphere ] , panel ( a ) ; however , the gauss curvature of the sphere implies that the complete graph on 4 vertices generically embedded in the plane , shown in panel ( b ) , can not be mapped so as to preserve all edge lengths . the property that blocks the preservation of edge ratios is the presence of an edge in the plane figure that can not be slightly changed without changing the length of at least one other edge ; we speak of such an edge as `` frozen . '' in a static spacetime , which is all we have so far investigated , a generic arrangement of 4 open machines , is analogous to the triangle on the plane in that a map to any gently curved spacetime can preserve all the echo counts . [ prop : nine]assume four open machines in a static spacetime , with one machine stepped with a proper - time period @xmath57 , and let @xmath40 be any positive integer . then , independent of any gentle riemann curvature of the spacetime , the four open machines can be arranged , like vertices of a regular tetrahedron , to have six two - way channels with null phases , with all echo counts being @xmath58 . _ proof : _ assuming a static spacetime , choose a coordinate system with all the metric tensor components independent of the time coordinate , in such a way that it makes sense to speak of a time coordinate distinct from space coordinates ( for example , in a suitable region of a schwarzschild geometry ) . let@xmath59 denote the machine with specified proper period @xmath60 , and let @xmath61 , @xmath62 , and @xmath63 denote the other three machines . for @xmath64 , @xmath65 , we prove the possibility , independent of curvature , of the channels @xmath66 require that each of four machines be located at some fixed spatial coordinate . because the spacetime is static , the coordinate time difference between a transmission at @xmath59 and a reception at any other vertex @xmath67 ( a ) is independent of the value of the time coordinate at transmission and ( b ) is the same as the coordinate time difference between a transmission at @xmath67 and a reception at @xmath59 . for this reason any one - way repeating channel of the form ( [ eq : vs ] ) can be turned around to make a channel in the opposite direction , so that establishing a channel in one direction suffices . for transmissions from any vertex to any other vertex , the coordinate - time difference between events of transmission equals the coordinate time difference between receptions . a signal from a transmission event on @xmath59 propagates on an expanding light cone , while an echo propagates on a light cone contracting toward an event of reception on @xmath59 . under the constraint that the echo count is @xmath58 , ( so the proper duration from the transmission event to the reception event for the echo is @xmath68 ) , the echo event must be on a 2-dimensional submanifold a sphere , defined by constant radar distance @xmath69 of its points from @xmath59 with transmission at a particular ( but arbitrary ) tick of @xmath59 . in coordinates adapted to a static spacetime , this sphere may appear as a `` potatoid '' in the space coordinates , with different points on the potatoid possibly varying in their time coordinate . the potatoid shape corresponding to an echo count of @xmath58 remains constant under evolution of the time coordinate . channels from @xmath59 to the other three vertices involve putting the three vertices on this potatoid . put @xmath61 anywhere on the `` potatoid '' . put @xmath62 anywhere on the ring that is intersection of potatoid of echo count @xmath58 radiated from @xmath61 and that radiated from @xmath59 . put @xmath63 on an intersection of the potatoids radiating from the other three vertices . + q.e.d . according to prop [ prop : nine ] the channels , and in particular the echo counts possible for a complete graph of four open machines in flat spacetime are also possible for a spacetime of gentle static curvature , provided that three of the machines are allowed to set their periods not to a fixed proper duration but in such a way that all four machines have periods that are identical in coordinate time . the same holds if fewer channels among the four machines are specified . but for five machines , the number of channels connecting them matters . five open machines fixed to space coordinates in a static spacetime are analogous to the 4 vertices of a plane figure , in that an arrangement corresponding to an incomplete graph on five vertices can have echo counts independent of curvature , while a generic arrangement corresponding to a complete graph must have curvature - dependent relations among its echo counts . [ prop:9.5 ] assuming a static spacetime , consider an arrangement of five open machines obtained by starting with a tetrahedral arrangement of four open machines with all echo counts of @xmath58 as in prop . [ prop : nine ] , and then adding a fifth machine : independent of curvature , a fifth open machine can be located with two - way channels having echo counts of @xmath58 linking it to any three of the four machines of tetrahedral arrangement , resulting in nine two - way channels altogether . _ proof : _ the fifth machine can be located as was the machine @xmath63 , but on the side opposite to the cluster @xmath59 , @xmath61 , @xmath62 . + q.e.d . in contrast to the arrangement of 9 two - way channels , illustrated in fig . [ fig:5pt ] ( a ) consider an arrangement of 5 open machines corresponding to a complete graph on five vertices , with has ten two - way channels , as illustrated in fig . [ fig:5pt ] ( b ) . for five open machines in a generic spacetime , not all of the ten two - way channels can have the same echo counts . instead , channels in a flat spacetime as specified below can exist with about the simplest possible ratios of echo counts . label five open machines , @xmath70 , @xmath71 , @xmath72 , @xmath34 , and @xmath35 . take @xmath34 to be stepped by a clock ticking at a fixed proper period @xmath57 , letting the other machines tick at variable rates to be determined . let @xmath73 be any machine other than @xmath34 . for a flat spacetime it is consistent for the proper periods of all 5 machines to be @xmath57 , for the echo counts @xmath74 to be @xmath75 and for the echo counts @xmath76 to be @xmath77 , leading to twenty channels , conveniently viewed as in fig . [ fig:5pt ] ( b ) as consisting of ten two - way channels . [ prop : ten ] consider 5 open machines each fixed to space coordinates in a static curved spacetime in which the machines are all pairwise radar linkable , with 10 two - way channels connecting each machine to all the others ; then : 1 . allowing for the periods of the machines other than @xmath34 to vary , it is consistent with the curvature for all but one of the ten two - way channels to have null phases and echo counts as in a flat spacetime , but at least one two - way channel must have a different echo count that depends on the spacetime curvature . 2 . suppose @xmath4 of the 10 two - way links are allowed to have non - zero phases . if the spacetime does not admit all phases to be null , in generic cases the least possible maximum amplitude of a phase decreases as @xmath4 increases from 1 up to 10 . the periods of the clocks of the open machines can be taken to be the coordinate - time interval corresponding to the proper period @xmath57 at @xmath34 . _ proof : _ reasoning as in the proof of prop . [ prop : nine ] with its reference to a static spacetime shows that the same echo counts are possible as for flat spacetime _ with the exception _ that at least one of the two - way channels must be free to have a different echo count . for @xmath78 , similar reasoning shows that allowing @xmath79 machines vary in echo count allows reduction in the maximum variation from the echo counts in a flat spacetime , compared to the case in which only @xmath4 machines are allowed to vary in echo count . + q.e.d . adding the tenth two - way channel to an arrangement of five open machines effectively `` freezes '' all the echo counts . to define `` freezing '' as applied to echo counts , first take note an asymmetry in the dependence of echo counts on clock rates . consider any two machines @xmath2 and @xmath7 ; unlike echo count @xmath80 , which @xmath7 can change by changing it clock rate , the echo count @xmath31 is insensitive to @xmath7 s clock rate . an echo count @xmath31 will be said to be _ to _ @xmath7 and _ from _ @xmath2 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * definition : * an arrangement of open machines is _ frozen _ if it has an echo count to a machine @xmath7 that can not be changed slightly without changing the length of another echo count to @xmath7 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the property of being frozen is important because of the following . whether or not a frozen arrangement of open machines is consistent with an hypothesized spacetime depends on the weyl curvature of the spacetime . for example , think of the 5 open machines as carried by 5 space vehicles coasting along a radial geodesic in a schwarzschild geometry . in this example the variation of echo counts with curvature is small enough to be expressed by non - null phases of reception . in fermi normal coordinates centered midway between the radially moving open machine @xmath34 and @xmath35 one has the metric with a curvature parameter @xmath81 , where @xmath82 is the schwarzschild radial coordinate to the origin of the fermi normal coordinates , @xmath83 is the radial distance coordinate from from the center point between @xmath34 and @xmath35 , and @xmath84 and @xmath85 are transverse to the radial direction along which @xmath86 coasts@xcite . the speed of light is @xmath87 . we make the adiabatic approximation which ignores the time dependence of @xmath82 , so that in calculations to first order in curvature we take advantage of the ( adiabatically ) static spacetime by locating open machines at fixed values of @xmath88 . the metric is symmetric under rotation about the ( radially directed ) @xmath83-axis . let @xmath34 and @xmath35 be located symmetrically at positive and negative values , respectively , of the @xmath83-axis , and let @xmath89 , @xmath70 , and @xmath71 be located on a circle in the plane @xmath90 . with the five machines so located , the coordinate - time difference between transmissions is then the same as the coordinate - time difference between receptions , and the coordinate - time delay in one direction equals that in the opposite direction ( as stated in the proof of prop [ prop : nine ] ) . we construct seven two - way channels as above with null phases and show that the remaining 3 two - way channels can have the equal phases , but that this phase must be non - null with a curvature dependent amplitude @xmath91 . [ prop : eleven ] under the stated conditions , if @xmath92 is small enough so that + @xmath93 , then @xmath94 for a fixed separation @xmath95 between @xmath34 and @xmath35 , an adiabatic change in curvature imposes a constraint on bit rate possible for the channels , stemming from a lower bound on clock periods . suppose the cluster of 5 open machines is arranged so that the proper radar distance @xmath95 from @xmath34 to @xmath35 is 6,000 km , suppose the cluster descends from a great distance down to a radius of @xmath96 km from an earth - sized mass @xmath97 kg . for simplicity , assume that the positions and clock rates are continually adjusted to maintain null phases for all but the three channels @xmath98 . because @xmath99 , prop . [ prop : eleven ] implies @xmath100 , which with ( [ eq : main ] ) implies that @xmath101 . substituting the parameter values , one finds that for the phases for the channels @xmath98 to satisfy ( [ eq : main ] ) , it is necessary that @xmath102 s. if an alphabet conveys @xmath103 bits / character , the maximum bit rate for all the channels in the 5-machine cluster is @xmath104 bits / s . turning from type i to questions of type ii , we look at how the preceding `` blackboard modeling '' of clocks , expressed in the mathematical language of general relativity , get put to work when models are encoded into the open machines that manage their own logical synchronization . for questions of type ii ( and type iii ) both models that explain or predict evidence and the evidence itself , pertaining to physical clocks , come into play . models encoded into computers contribute to the steering of physical clocks in rate and relative position toward an aiming point , generating echo counts as evidence that , one acquired , can stimulate the guessing of new models that come closer to the aiming point . to express the effect of quantum uncertainty on logical synchronization , specifically on deviations from aiming points , one has to bring quantum uncertainty into cooperation with the representation of clocks by general - relativistic worldlines . this bringing together hinges on distinguishing evidence from its explanations . timelike worldlines and null geodesics in explanations , being mathematical , can have no _ mathematical _ connection to physical atomic clocks and physical signals . to make such a connection them one has to invoke the logical freedom to make a guess . within this freedom , one can resort to quantum theory to explain deviations of an atomic clock from an imagined proper clock , represented as a worldline , without logical conflict . because of quantum uncertainty and for other reasons , if an aiming point in terms of channels and a given frequency scale is to be reached , steering is required , in which evidence of deviations from the aiming point combine with hypotheses concerning how to steer @xcite . to keep things simple , consider a case of an aiming point with null phases , involving two open machines @xmath2 and @xmath7 , as in the example of sec . [ sec : typei ] , modeled by a given worldline @xmath2 with given clock readings @xmath26 , where @xmath7 aims to maintain two - way , null - phase channel of given @xmath105 . for this @xmath7 registers arriving phases of reception and adjusts its clock rate and its position more or less continually to keep those phases small . deviations in position that drive position corrections show up not directly at @xmath7 but as phases registered by @xmath2 , so the steering of machine @xmath7 requires information about receptive phases measured by @xmath2 . the knowledge of the deviation in position of @xmath7 at @xmath106 can not arrive at @xmath7 until its effect has shown up at @xmath2 and been echoed back as a report to @xmath7 , entailing a delay of at least @xmath80 , hence requiring that machine @xmath7 predict the error that guides for @xmath80 prior to receiving a report of the error . that is , steering deviations by one open machine are measured in part by their effect on receptive phases of other open machines , so that steering of one machine requires information about receptive phases measured by other machines , and the deviations from an aiming point must increase with increasing propagation delays that demand predicting further ahead . as is clear from the cluster of five machines discussed in sec . [ sec : typei ] , the aiming - point phases can not in general all be taken to be zero . for any particular aiming - point phase @xmath107 there will be a deviation of a measured phase quantity @xmath91 given by @xmath108 whatever the value of @xmath107 , adjustments to contain phases within tolerable bounds depends on phase changes happening only gradually , so that trends can be detected and responded to on the basis of adequate prediction ( aka guesswork ) . + * remarks : * 1 . unlike cycle counts of open machines , which we assume are free of uncertainty , measured phases and deviations of phases from aiming points are quantities subject to uncertainty . for logic to work in a network , transmission of logical symbols must preserve sharp distinctions among them ; yet the maintenance of sharp distinctions among transmitted symbols requires responses to fuzzy measurements . 2 . the acquisition of logical synchrony in digital communications involves an unforeseeable waiting time , like the time for a coin on edge to fall one way or the other @xcite . aiming points are not forever , and here we say a few words about questions of type iii , in which an aiming point based on a hypothesized metric tensor appears unreachable , and perhaps needs to be revised . we have so far looked at one or another manifold with metric @xmath32 as some given hypothesis , whether explored on the blackboard or coded into an open machine to serve in maintaining its logical synchronization . in this context we think of @xmath32 as `` given . '' but deviations of phases outside of tolerances present another context , calling for revising a metric tensor field . in this context one recognizes that a metric tensor field is hypothesized provisionally , to be revised as prompted by deviations outside allowed tolerances in implementing an aiming point . drawing on measured phases as evidence in order to adjust a hypothesis of a metric tensor is one way to view the operation of the laser interferometer gravitational - wave observatory ( ligo ) @xcite . while ligo sensitivity drops off severely below 45 hz , the arrangement of five open machines of prop . [ prop : ten ] has no low - frequency cutoff , and so has the potential to detect arbitrarily slow changes in curvature . 99 b. n. taylor and a. thompson , eds , _ the international system of units ( si ) _ , nist special publication 330 , 2008 edition , national institutes of science and technology . j. m. myers and f. h. madjid , `` a proof that measured data and equations of quantum mechanics can be linked only by guesswork , '' in s. j. lomonaco jr . and h.e . brandt ( eds . ) _ quantum computation and information _ , contemporary mathematics series , vol . 305 , american mathematical society , providence , 2002 , pp . . f. h. madjid and j. m. myers , `` matched detectors as definers of force , '' ann . physics * 319 * , 251273 ( 2005 ) . j. m. myers and f. h. madjid , `` ambiguity in quantum - theoretical descriptions of experiments , '' in k. mahdavi and d. koslover , eds . , _ advances in quantum computation _ , contemporary mathematics series , vol . 482 ( american mathematical society , providence , i , 2009 ) , pp . 107123 . j. m. myers and f. h. madjid , `` what probabilities tell about quantum systems , with application to entropy and entanglement , '' in a. bokulich and g. jaeger , eds . , _ quantum information and entanglement _ , cambridge university press , cambridge uk , pp . 127150 ( 2010 ) . a. m. turing , `` on computable numbers with an application to the entscheidungsproblem , '' proc . london math . soc . , series 2 , * 42 * , 230265 ( 193637 ) . m. soffel et al . , `` the iau resolutions for astrometry , celestial mechanics , and metrology in the relativistic framework : explanatory supplement , '' the astronomical journal , * 126 * , 26872706 ( 2003 ) . j. m. myers and f. h. madjid , `` rhythms essential to logical communication , '' in quantum information and computation ix , e. donkor , a. r. pirich , and h. e. brandt , eds , proceedings of the spie , * 8057 * , pp . 80570n112 ( 2011 ) . j. m. myers and f. h. madjid , `` rhythms of memory and bits on edge : symbol recognition as a physical phenomenon , '' arxiv:1106.1639 , 2011 . h. meyr and g. ascheid , _ synchronization in digital communications _ , wiley , new york , 1990 . the ligo scientific collaboration ( http://www.ligo.org ) `` ligo : the laser interferometer gravitational - wave observatory , '' rep . prog . phys . * 72 * , 076901 ( 2009 ) v. perlick , `` on the radar method in general - relativistic spacetimes , '' in h. dittus , c. lmmerzahl , and s. turyshev , eds . , _ lasers , clocks and drag - free control : expolation of relativistic gravity in space _ , ( springer , berlin , 2008 ) ; also arxiv:0708.0170v1 . f. k. manasse and c. w. misner , j. math phys . , `` fermi normal coordinates and some basic concepts in differential geometry , '' * 4 * , 735745 ( 1963 ) . t. e. parker , s. r. jefferts , and t. p. heavner , `` medium - term frequency stability of hydrogen masers as measured by a cesium fountain , '' 2010 ieee international frequency control symposium ( fcs ) , pp . 318323 ( ( available at http://tf.boulder.nist.gov/general/pdf/2467.pdf ) j. levine and t. parker , `` the algorithm used to realize utc(nist ) , '' 2002 ieee international frequency control symposium and pda exhibition , pp . 537542 ( 2002 )

a clock steps a computer through a cycle of phases . for the propagation of logical symbols from one computer to another , each computer must mesh its phases with arrivals of symbols from other computers . even the best atomic clocks drift unforeseeably in frequency and phase ; feedback steers them toward aiming points that depend on a chosen wave function and on hypotheses about signal propagation . a wave function , always under - determined by evidence , requires a guess . guessed wave functions are coded into computers that steer atomic clocks in frequency and position clocks that step computers through their phases of computations , as well as clocks , some on space vehicles , that supply evidence of the propagation of signals . recognizing the dependence of the phasing of symbol arrivals on guesses about signal propagation elevates ` logical synchronization . ' from its practice in computer engineering to a dicipline essential to physics . within this discipline we begin to explore questions invisible under any concept of time that fails to acknowledge the unforeseeable . in particular , variation of spacetime curvature is shown to limit the bit rate of logical communication .

arxiv

during past few years research in areas of wireless ad - hoc networks and wireless sensor networks ( wsns ) are escalated . ieee 802.15.4 is targeted for wireless body area networks ( wbans ) , which requires low power and low data rate applications . invasive computing is term used to describe future of computing and communications [ 1 - 3 ] . due to these concepts , personal and business domains are being densely populated with sensors . one area of increasing interest is the adaptation of technology to operate in and around human body . many other potential applications like medical sensing control , wearable computing and location identification are based on wireless body area networks ( wbans ) . main aim of ieee 802.15.4 standard is to provide a low - cost , low power and reliable protocol for wireless monitoring of patient s health . this standard defines physical layer and mac sub layer . three distinct frequencies bands are supported in this standard . however , 2.4 ghz band is more important . this frequency range is same as ieee 802.11b / g and bluetooth . ieee 802.15.4 network supports two types of topologies , star topology and peer to peer topology . standard supports two modes of operation , beacon enabled ( slotted ) and non - beacon enabled ( unslotted ) . medium access control ( mac ) protocols play an important role in overall performance of a network . in broad , they are defined in two categories contention - based and schedule - based mac protocols . in contention - based protocols like carrier sense multiple access with collision avoidance ( csma / ca ) , each node content to access the medium . if node finds medium busy , it reschedules transmission until medium is free . in schedule - based protocols like time division multiple access ( tdma ) , each node transmits data in its pre - allocated time slot . this paper focuses on analysis of ieee 802.15.4 standard with non - beacon enabled mode configure in a star topology . we also consider that sensor nodes are using csma / ca protocol . to access channel data . in literature , beacon enabled mode is used with slotted csma / ca for different network settings . in [ 1 ] , performance analysis of ieee 802.15.4 low power and low data rate wireless standard in wbans is done . authors consider a star topology at 2.4 ghz with up to 10 body implanted sensors . long - term power consumption of devices is the main aim of their analysis . however , authors do not analyze their study for different data rates . an analytical model for non - beacon enabled mode of ieee 802.15.4 medium access control ( mac ) protocol is provided in [ 2 ] . nodes use un - slotted csma / ca operation for channel access and packet transmission . two main variables that are needed for channel access algorithm are back - off exponent ( be ) and number of back - offs ( nb ) . authors perform mathematical modeling for the evaluation statistical distribution of traffic generated by nodes . this mathematical model allows evaluating an optimum size packet so that success probability of transmission is maximize . however , authors do not analyze different mac parameters with varying data rates . authors carry out an extensive analysis based on simulations and real measurements to investigate the unreliability in ieee 802.15.4 standard in [ 3 ] . authors find out that , with an appropriate parameter setting , it is possible to achieve desired level of reliability . unreliability in mac protocol is the basic aspect for evaluation of reliability for a sensor network . an extensive simulation analysis of csma / ca algorithm is performed by authors to regulate the channel access mechanism . a set of measurements on a real test bed is used to validate simulation results . a traffic - adaptive mac protocol ( tamac ) is introduced by using traffic information of sensor nodes in [ 4 ] . tamac protocol is supported by a wakeup radio , which is used to support emergency and on - demand events in a reliable manner . authors compare tamac with beacon - enabled ieee 802.15.4 mac , wireless sensor mac ( wisemac ) , and sensor mac ( smac ) protocols . important requirements for the design of a low - power mac protocol for wbans are discussed in [ 5 ] . authors present an overview to heartbeat driven mac ( h - mac ) , reservation - based dynamic tdma ( dtdma ) , preamble - based tdma ( pb - tdma ) , and body mac protocols , with focusing on their strengths and weaknesses . authors analyze different power efficient mechanism in context of wbans . at the end authors propose a novel low - power mac protocol based on tdma to satisfy traffic heterogeneity . authors in [ 6 ] , examine use of ieee 802.15.4 standard in ecg monitoring and study the effects of csma / ca mechanism . they analyze performance of network in terms of transmission delay , end - to - end delay , and packet delivery rate . for time critical applications , a payload size between 40 and 60 bytes is selected due to lower end - to - end delay and acceptable packet delivery rate . in [ 7 ] , authors state that ieee 802.15.4 standard is designed as a low power and low data rate protocol with high reliability . they analyze unslotted version of protocol with maximum throughput and minimum delay . the main purpose of ieee 802.15.4 standard is to provide low power , low cost and highly reliable protocol . physical layer specifies three different frequency ranges , 2.4 ghz band with 16 channels , 915 mhz with 10 channels and 868 mhz with 1 channel . calculations are done by considering only beacon enabled mode and with only one sender and receiver . however , it consumes high power . as number of sender increases , efficiency of 802.15.4 decreases . throughput of 802.15.4 declines and delay increases when multiple radios are used because of increase in number of collisions . a lot of work is done to improve the performance of ieee 802.15.4 and many improvements are made in improving this standard , where very little work is done to find out performance of this standard by varying data rates and also considering acknowledgement ( ack ) and no ack condition and how it affects delay , throughput , end - to - end delay and load . we get motivation to find out the performance of this standard with parameters load , throughput , delay and end to end delay at varying data rates . ieee 802.15.4 is proposed as standard for low data rate , low power wireless personal area networks ( wpans ) [ 1],[2 ] . in wpans , end nodes are connected to a central node called coordinator . management , in - network processing and coordination are some of key operations performed by coordinator . the super - frame structure in beacon enabled mode is divided into active and inactive period . active period is subdivided into three portions ; a beacon , contention access period ( cap ) and contention free period ( cfp ) . in cfp , end nodes communicate with central node ( coordinator ) in dedicated time slots . however , cap uses slotted csma / ca . in non - beacon enabled mode , ieee 802.15.4 uses unslotted csma / ca with clear channel assessment ( cca ) for channel access . in [ 2 ] , ieee 802.15.4 mac protocol non - beacon enabled mode is used . nodes use un - slotted csma / ca operation for channel access and packet transmission . two main variables that are needed for channel access algorithm are back off exponent ( be ) and number of back offs ( nb ) . nb is the number of times csma / ca algorithm was required to back off while attempting channel access and be is related to how many back off periods , node must wait before attempting channel access . operation of csma / ca algorithm is defined in steps below : @xmath0 nb and be initialization : first , nb and be are initialized , nb is initialized to 0 and be to macminbe which is by default equal to 3 . + @xmath1 random delay for collision avoidance : to avoid collision algorithm waits for a random amount of time randomly generated in range of @xmath2 , one back off unit period is equal to @xmath3 with @xmath4s + @xmath5 clear channel assessment : after this delay channel is sensed for the unit of time also called cca . if the channel is sensed to be busy , algorithm goes to step 4 if channel is idle algorithm goes to step 5 . + @xmath6 busy channel : if channel is sensed busy then mac sub layer will increment the values of be and nb , by checking that be is not larger than @xmath7 . if value of nb is less than or equal to @xmath8 , then csma / ca algorithm will move to step 2 . if value of nb is greater than @xmath8 , then csma / ca algorithm will move to step 5 `` packet drop '' , that shows the node does not succeed to access the channel . + @xmath9 idle channel : if channel is sensed to be idle then algorithm will move to step 4 that is `` packet sent '' , and data transmission will immediately start . fig . 1 illustrates aforementioned steps of csma / ca algorithm , starting with node has some data to send . csma / ca is a modification of carrier sense multiple access ( csma ) . collision avoidance is used to enhance performance of csma by not allowing node to send data if other nodes are transmitting . in normal csma nodes sense the medium if they find it free , then they transmits the packet without noticing that another node is already sending packet , this results in collision . csma / ca results in reduction of collision probability . it works with principle of node sensing medium , if it finds medium to be free , then it sends packet to receiver . if medium is busy then node goes to backoff time slot for a random period of time and wait for medium to get free . with improved csma / ca , request to send ( rts)/clear to send ( cts ) exchange technique , node sends rts to receiver after sensing the medium and finding it free . after sending rts , node waits for cts message from receiver . after message is received , it starts transmission of data , if node does not receive cts message then it goes to backoff time and wait for medium to get free . csma / ca is a layer 2 access method , used in 802.11 wireless local area network ( wlan ) and other wireless communication . one of the problems with wireless data communication is that it is not possible to listen while sending , therefore collision detection is not possible . csma / ca is largely based on the modulation technique of transmitting between nodes . csma / ca is combined with direct sequence spread spectrum ( dsss ) which helps in improvement of throughput . when network load becomes very heavy then frequency hopping spread spectrum ( fhss ) is used in congestion with csma / ca for higher throughput , however , when using fhss and dsss with csma / ca in real time applications then throughput remains considerably same for both . 2 shows the timing diagram of csma / ca . @xmath10 data transmission time @xmath11 , backoff slots time @xmath12 , acknowledgement time @xmath13 are given by equation 2 , 3 , and 4 respectively[2 ] . + [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^ " , ] [ tab : addlabel ] @xmath14 the following notations are used : + + @xmath15 @xmath16 @xmath17 + @xmath18 @xmath19 @xmath20 @xmath21 + @xmath22 @xmath23 @xmath19 + @xmath24 @xmath25 @xmath19 + @xmath26 @xmath27 @xmath28 + @xmath29 @xmath20 @xmath30 @xmath31 + @xmath32 @xmath20 @xmath33 @xmath31 + @xmath34 @xmath20 @xmath35 @xmath36 @xmath37 @xmath38 + @xmath39 @xmath20 @xmath33 @xmath40 + @xmath41 @xmath20 @xmath42 @xmath43 @xmath44 + @xmath45 @xmath46 @xmath47 @xmath42 @xmath43 @xmath48 + @xmath49 @xmath50 in csma / ca mechanism , packet may loss due to collision . collision occurs when two or more nodes transmits the data at the same time . if ack time is not taken in to account then there will be no retransmission of packet and it will be considered that each packet has been delivered successfully . the probability of end device successfully transmitting a packet is modeled as follows[3 ] . @xmath51 where , @xmath52 is the number of end devices that are connected to router or coordinator . be is the backoff exponent in our case it is 3 . @xmath53 is the probability of transmission success at a slot . @xmath54 is the probability of end device successfully allocated a wireless channel . general formula for @xmath55 is given by equation 8 . probability of time delay caused by csma / ca backoff exponent is estimated as in [ 7 ] . maximum number of backoff is 4 . value of be=3 has been used in following estimation and we estimate by applying summation from 3 to 5 . @xmath56 is the probability of time delay event . @xmath57 @xmath58 expectation of the time delay is obtained as from [ 7 ] . @xmath59 and @xmath60 are taken from equations 7 and 8 respectively . @xmath61=p({e_{a}|e_{b}})\nonumber\\ \nonumber\\ = \frac{\sum_{n=0}^{7 } n\frac{1}{2_{be}}p + \sum _ { n=8}^{15 } n\frac{1}{2_{be}}p + \sum _ { n=16}^{31 } n\frac{1}{2_{be}}p}{\sum_{n=0}^{2^{be-1 } } n\frac{1}{2_{be}}p{1-p}^{be-2}}\end{aligned}\ ] ] statistical data of throughput , load , end - to - end delay and delay of ieee 802.15.4 at varying data rates is shown in table i. it shows different values of delay , throughput , end - to - end delay and load recorded at different time . load at all data rates and at all time intervals remains same . start time for simulation is kept at 0 seconds and stop time is kept to infinity . load in all three data rates at different time intervals remains same as shown in table i. there is very small difference between delay and end - to - end delay . at 20 kbps maximum delay of 2145 seconds is recorded with maximum throughput of 4352 bits / sec at 60 min . at 40kbps maximum delay of 380 seconds and minimum delay 2.5 seconds is recorded . throughput of 8805 ( bits / sec ) is the highest throughput recorded on 60 min . in case of 250 kbps delay remains very small , near to negligible where as throughput matches load with 10388 ( bits / sec ) . beacon order & 6 + superframe order & 0 + maximum routers & 5 + maximum depth & 5 + beacon enabled network & disabled + route discovery time & 10(sec ) + network parameter are given in table ii . non beacon mode is selected in our analysis and beacon order is kept at 6 . due to non - beacon enabled mode superframe order is not selected . maximum routers or nodes that can take part in simulation is 5 , each having tree depth of 5 . discovery time that is needed by each router to discover route is 10 sec . minimum backoff exponent & 3 + maximum number of backoff & 5 + channel sensing duration & 0.1 ( sec ) + data rates & 20 , 40 , 250 kbps + packet reception power & -85 ( dbm ) + transmission band & 2.4 ( mhz ) + packet size & 114 bytes + packet interarrival time & 0.045(sec ) + transmit power & 0.05 ( w ) + ack wait duration & 0.05 ( sec ) + number of retransmissions & 5 + simulation parameters of 802.15.4 with its value are shown in table iii . minimum be is kept at 3 with maximum no . of back - off to 5 . default settings of 802.15.4 are used in this simulation . packet reception power is kept at -85 dbm with transmitting power of 0.5 watt(w ) . in ack enabled case , ack wait duration is kept at 0.05 sec with no of retransmissions to 5 . in no ack case these parameters are disabled . 114 bytes is the packet size with interarrival time of 0.045 sec . transmission band used in this simulation is 2.4 ghz . simulations have been performed at varying data rates of 20 , 40 , 250 kbps . simulations for both ack and non ack cases have also been performed . opnet modeler is the simulator used for simulations . simulations are executed for one hour with update interval of 5000 events . graphs are presented in overlaid statistics . overlaid means that , graphs of each scenario has been combined with each other . data of graphs are averaged over time for better results . personal area network identity ( pan i d ) is kept at default settings , coordinator automatically assigns pan i d to different personal area networks if they are attached . we consider non beacon mode for our simulations . using non - beacon enabled mode improves the performance and changing different parameters affects performance of 802.15.4 . csma / ca values are kept to default with minimum backoff exponent to 3 and having maximum backoff of 5 . changing these parameters does not affect its performance . we perform simulations with ack and non ack . in non ack there is only delay due to node waiting while sensing medium , there is no delay due to ack colliding with packets . in ack case there is collision for packets going towards receiver and ack packet coming from receiver at same time . delay in ack is more as compare to non ack case . we use standard structure of ieee 802.15.4 with parameters shown in table ii . in this section , performance of default mac parameters of ieee 802.15.4 standard non beacon enabled mode . simulations are performed considering 10 sensor nodes environment with coordinator collecting data from all nodes . fig 3 , 4 , 5 and 6 show graphical representation of performance parameters of 802.15.4 . delay represents the end - to - end delay of all the packets received by 802.15.4 macs of all wpans nodes in the network and forwarded to the higher layer . load represents the total load in ( bits / sec ) submitted to 802.15.4 mac by all higher layers in all wpans nodes of the network . load remains same for all the data rates . throughput represents the total number of bits in ( bits / sec ) forwarded from 802.15.4 mac to higher layers in all wpans nodes to the network . end - to - end delay is the total delay between creation and reception of an application packet . delay , load and throughput are plotted as function of time . as load is increasing , there is increase in throughput and delay . when load becomes constant , throughput also becomes constant , however , delay keeps on increasing . delay in 802.15.4 occurs due to collision of data packets or sometimes nodes keeps on sensing channel and does not find it free . when node senses medium and find it free , it sends packet . at same time , some other nodes are also sensing the medium and find it free , they also send data packets and thus results in collision . collision also occurs due to node sending data packet and at same time coordinator sending ack of successfully receiving packet and causing collision . when ack is disabled this type of collision will not occur . delay , throughput and load is analyzed at 40 kbps in fig 4 . with increase in load , there is increase in throughput and delay , however , it is less as compared to 20kbps , this is due to increase in data rate of 802.15.4 . increase in bit transfer rate from 20 to 40kbps causes decrease in delay and hence increases throughput . fig 5 shows behavior of 802.15.4 load , throughput and delay at 250kbps data rate . delay is negligible at this data rate , with throughput and load showing same behavior . delay approaching zero shows that , at 250 kbps data rate there are less chances of collision or channel access failure . ieee 802.15.4 performs best at this data rate compared to 20 and 40 kbps . at same time end - to - end delay of ieee 802.15.4 at varying data rates of 20 , 40 and 250 kbps are shown in fig 6 . this figure shows that end to end delay for 20 kbps data rate is higher than 40 kbps and 250 kbps . minimum end - to - end delay is found at 250 kbps data rate . at 250 kbps , more data can pass at same time with less collision probability hence having minimum delay and at 20 kbps , less data transfers at same time causing more end to end delay . statistical data of end - to - end delay is shown in table i , which shows end to end variation with change in time . fig 7 shows the delay , throughput , load and end - to - end delay of ieee 802.11.4 at 20 kbps data rate with and without ack . load remains same in both cases . there is no collision because of ack packets due to which packets once send are not sent again . there is decrease in delay and increase in ack due to less collision . end - to - end delay performs same as delay . ieee 802.15.4 performs better with non ack other than ack due to decrease in collision probability in no ack compared to ack case . delay , throughput , load and end - to - end delay with and without ack at 40 kbps are presented in fig 8 . there is considerable difference between the analysis in ack and without ack case . delay is reduced to negligible at low value of @xmath62 in no ack case due to reason that , at this data rate there is no collision therefor , delay is nearly zero . as there is no collision and channel sensing time is also low , this increase throughput and load in non ack case , as compared to ack . 9 shows analysis with ack and no ack cases of delay , throughput , load and end to end delay at 250 kbps , at this high data rates load and throughput in both cases becomes equal to each other and data is sent in first instant to coordinator by nodes . delay in both cases nearly equal to zero , which shows that , there is very less collision at this high data rates and channel sensing time is also very low . end to end delay slightly differs from delay in no ack case . in this paper , performance of ieee 802.15.4 standard with non - beacon enabled mode is analyzed at varying data rates . we have evaluated this standard in terms of load , delay , throughput and end - to - end delay with different mac parameters . we have also analyzed performance with ack enabled mode and no ack mode . we considered a full size mac packet with payload size of 114 bytes for data rates 20 kbps , 40 kbps and 250 kbps . it is shown that better performance in terms of throughput , delay , and end - to - end delay is achieved at higher data rate of 250kbps . ieee 802.15.4 performs worse at low data rates of 20kbps . performance of this standard improves with increase in data rate . in future research work , we will investigate the performance of ieee 802.15.4 in wbans by changing frequency bands on different data rates . we also intend to examine the effect of changing inner structure of mac layer in ieee 802.15.4 . 1 f. timmons , n @xmath63 @xmath64 . , `` analysis of the performance of ieee 802.15.4 for medical sensor body area networking '' , sensor and ad hoc communications and networks , 2004 . c. buratti and r. verdone @xmath63 @xmath64 . , `` performance analysis of ieee 802.15.4 non beacon - enabled mode '' , ieee transaction on vehicular technology , vol . 58 , no . 7 , september 2009 . anastasi , g @xmath63 @xmath64 . , `` the mac unreliability problem in ieee 802.15.4 wireless sensor networks '' , mswim09 proceedings of the 12th acm international conference on modeling , analysis and simulation of wireless and mobile systems , october 2009 . s. ullah , k. s. kwak @xmath63 @xmath64 . , `` an ultra - low power and traffic - adaptive medium access control protocol for wireless body area network '' , j med syst , doi 10.1007/s10916 - 010 - 9564 - 2 . s. ullah , b.shen , s.m.r . islam , p. khan , s. saleem and k.s . kwak @xmath63 @xmath64 . , `` a study of medium access control protocols for wireless body area networks '' . x. liang and i. balasingham @xmath63 @xmath64 . , `` performance analysis of the ieee 802.15.4 based ecg monitoring network '' . b. latre , p.d . mil , i. moerman , b. dhoedt and p. demeester @xmath63 @xmath64 . , `` throughput and delay analysis of unslotted ieee 802.15.4 '' , journal of networks , vol . 1 , no . 1 , may 2006 .

ieee 802.15.4 standard is designed for low power and low data rate applications with high reliability . it operates in beacon enable and non - beacon enable modes . in this work , we analyze delay , throughput , load , and end - to - end delay of non - beacon enable mode . analysis of these parameters are performed at varying data rates . evaluation of non beacon enabled mode is done in a 10 node network . we limit our analysis to non beacon or unslotted version because , it performs better than other . protocol performance is examined by changing different medium access control ( mac ) parameters . we consider a full size mac packet with payload size of 114 bytes . in this paper we show that maximum throughput and lowest delay is achieved at highest data rate . ieee 802.15.4 , throughput , delay , end - to - end , load

arxiv

multiple optimised parameter estimation and data compression ( moped ; @xcite ) is a patented algorithm for the compression of data and the speeding up of the evaluation of likelihood functions in astronomical data analysis and beyond . it becomes particularly useful when the noise covariance matrix is dependent upon the parameters of the model and so must be calculated and inverted at each likelihood evaluation . however , such benefits come with limitations . since moped only guarantees maintaining the fisher matrix of the likelihood at a chosen point , multimodal and some degenerate distributions will present a problem . in this paper we report on some of the limitations of the application of the moped algorithm . in the cases where moped does accurately represent the likelihood function , however , its compression of the data and consequent much faster likelihood evaluation does provide orders of magnitude improvement in runtime . in @xcite , the authors demonstrate the method by analysing the spectra of galaxies and in @xcite they illustrate the benefits of moped for estimation of the cmb power spectrum . the problem of `` badly '' behaved likelihoods was found by @xcite for the problem of light transit analysis ; nonetheless , the authors present a solution that still allows moped to provide a large speed increase . we begin by introducing moped in section 2 and define the original and moped likelihood functions , along with comments on the potential speed benefits of moped . in section 3 we introduce an astrophysical scenario where we found that moped did not accurately portray the true likelihood function . in section 4 we expand upon this scenario to another where moped is found to work and to two other scenarios where it does not . we present a discussion of the criteria under which we believe moped will accurately represent the likelihood in section 5 , as well as a discussion of an implementation of the solution provided by @xcite . full details of the moped method are given in @xcite , here we will only present a limited introduction . we begin by defining our data as a vector , @xmath0 . our model describes @xmath0 by a signal plus random noise , @xmath1 where the signal is given by a vector @xmath2 that is a function of the set of parameters @xmath3 defining our model , and the true parameters are given by @xmath4 . the noise is assumed to be gaussian with zero mean and noise covariance matrix @xmath5 , where the angle brackets indicate an ensemble average over noise realisations ( in general this matrix may also be a function of the parameters @xmath6 ) . the full likelihood for @xmath7 data points in @xmath0 is given by @xmath8^{\textrm{t } } \mathcal{n}(\btheta)^{-1 } [ { \bf x}-{\bf u}(\btheta)]\right\}}.\end{aligned}\ ] ] at each point , then , this requires the calculation of the determinant and inverse of an @xmath9 matrix . both scale as @xmath10 , so even for smaller datasets this can become cumbersome . moped allows one to eliminate the need for this matrix inversion by compressing the @xmath7 data points in @xmath0 into @xmath11 data values , one for each parameters of the model . additionally , moped creates the compressed data values such that they are independent and have unit variance , further simplifying the likelihood calculation on them to an @xmath12 operation . typically , @xmath13 so this gives us a significant increase in speed . a single compression is done on the data , @xmath0 , and then again for each point in parameter space where we wish to compute the likelihood . the compression is done by generating a set of weighting vectors , @xmath14 ( @xmath15 ) , from which we can generate a set of moped components from the theoretical model and data , @xmath16 note that the weighting vectors must be computed at some assumed fiducial set of parameter values , @xmath17 . the only choice that will truly maintain the likelihood peak is when the fiducial parameters are the true parameters , but obviously we will not know these in advance for real analysis situations . thus , we can choose our fiducial model to be anywhere and iterate the procedure , taking our likelihood peak in one iteration as the fiducial model for the next iteration . this process will converge very quickly , and may not even be necessary in some instances . for our later examples , since we do know the true parameters we will use these as the fiducial ( @xmath18 ) in order to remove this as a source of confusion ( all equations , however , are written for the more general case ) . note that the true parameters , @xmath4 , will not necessarily coincide with the peak @xmath19 of the original likelihood or the peak @xmath20 of the moped likelihood ( see below ) . the weighting vectors must be generated in some order so that each subsequent vector ( after the first ) can be made orthogonal to all previous ones . we begin by writing the derivative of the model with respect to the @xmath21th parameter as @xmath22 . this gives us a solution for the first weighting vector , properly normalised , of @xmath23 the first compressed value is @xmath24 and will weight up the data combination most sensitive to the first parameter . the subsequent weighting vectors are made orthogonal by subtracting out parts that are parallel to previous vectors , and are normalized . the resulting formula for the remaining weighting vectors is @xmath25 @xmath26 where @xmath27 . weighting vectors generated with equations and form an orthnomal set with respect to the noise covariance matrix so that @xmath28 this means that the noise covariance matrix of the compressed values @xmath29 is the identity , which significantly simplifies the likelihood calculation . the new likelihood function is given by @xmath30 where @xmath31 represents the compressed data and @xmath32 represents the compressed signal . this is a much easier likelihood to calculate and is time - limited by the generation of a new signal template instead of the inversion of the noise covariance matrix . the peak value of the moped likelihood function is not guaranteed to be unique as there may be other points in the original parameter space that map to the same point in the compressed parameter space ; this is a characteristic that we will investigate . moped implicity assumes that the covariance matrix , @xmath33 , is independent of the parameters . with this assumption , a full likelihood calculation with @xmath7 data points would require only an @xmath34 operation at each point in parameter space ( or @xmath35 if @xmath33 is diagonal ) . in moped , however , the compression of the theoretical data is an @xmath36 linear operation followed by an @xmath12 likelihood calculation . thus , one loses on speed if @xmath33 is diagonal but gains by a factor of @xmath37 otherwise . for the data sets we will analyze , @xmath38 . we begin , though , by assuming a diagonal @xmath33 for simplicity , recognizing that this will cause a speed reduction but that it is a necessary step before addressing a more complex noise model . one can iterate the parameter estimation procedure if necessary , taking the maximum likelihood point found as the new fiducial and re - analyzing ( perhaps with tighter prior constraints ) ; this procedure is recommended for moped in @xcite , but is not always found to be necessary . moped has the additional benefit that the weighting vectors , @xmath39 , need only to be computed once provided the fiducial model parameters are kept constant over the analysis of different data sets . computed compressed parameters , @xmath40 , can also be stored for re - use and require less memory than storing the entire theoretical data set . in order to demonstrate some of the limitations of the applicability of the moped algorithm , we will consider a few test cases . these originate in the context of gravitational wave data analysis for the _ laser interferometer space antenna _ ( _ lisa _ ) since it is in this scenario that we first discovered such cases of failure . the full problem is seven - dimensional parameter estimation , but we have fixed most of these variables to their known true values in the simulated data set in order to create a lower - dimensional problem that is simpler to analyse . we consider the case of a sine - gaussian burst signal present in the lisa detector . the short duration of the burst with respect to the motion of lisa allows us to use the static approximation to the response . in frequency space , the waveform is described by ( @xcite ) @xmath41 here @xmath42 is the dimensionless amplitude factor ; @xmath43 is the width of the gaussian envelope of the burst measured in cycles ; @xmath44 is the central frequency of the oscillation being modulated by the gaussian envelope ; and @xmath45 is the central time of arrival of the burst . this waveform is further modulated by the sky position of the burst source , @xmath46 and @xmath47 , and the burst polarisation , @xmath48 , as they project onto the detector . the one - sided noise power spectral density of the lisa detector is given by ( @xcite ) @xmath49 where @xmath50s is the light travel time along one arm of the lisa constellation , @xmath51hz@xmath52 is the proof mass acceleration noise and @xmath53hz@xmath52 is the shot noise . this is independent of the signal parameters and all frequencies are independent of each other , so the noise covariance matrix is constant and diagonal . this less computationally expensive example allows us to show some interesting examples . we begin by taking the one - dimensional case where the only unknown parameter of the model is the central frequency of the oscillation , @xmath44 . we set @xmath54 and @xmath55s ; we then analyze a @xmath56s segment of lisa data , beginning at @xmath57s , sampled at a @xmath58s cadence . for this example , the data was generated with random noise ( following the lisa noise power spectrum ) at an snr of @xmath59 with @xmath60hz ( thus @xmath61hz for moped ) . the prior range on the central frequency goes from @xmath62hz to @xmath63hz . @xmath64 samples uniformly spaced in @xmath44 were taken and their likelihoods calculated using both the original and moped likelihood functions . the log - likelihoods are shown in figure [ fig : likecomp ] . note that the absolute magnitudes are not important but the relative values within each plot are meaningful . both the original and moped likelihoods have a peak close to the input value @xmath65 . for the chosen template.,width=312 ] we see , however , that in going from the original to moped log - likelihoods , the latter also has a second peak of equal height at an incorrect @xmath44 . to see where this peak comes from , we look at the values of the compressed parameter @xmath66 as it varies with respect to @xmath44 as shown in figure [ fig : yf_vs_f ] . the true compressed value peak occurs at @xmath67hz , where @xmath68 . however , we see that there is another frequency that yields this exact same value of @xmath66 ; it is at this frequency that the second , incorrect peak occurs . by creating a mapping from @xmath44 to @xmath66 that is not one - to - one , moped has created the possibility for a second solution that is indistinguishable in likelihood from the correct one . this is a very serious problem for parameter estimation . interestingly , we find that even when moped fails in a one - parameter case , adding a second parameter may actually rectify the problem , although not necessarily . if we now allow the width of the burst , @xmath43 , to be a variable parameter , there are now two orthognal moped weighting vectors that need to be calculated . this gives us two compressed parameters for each pair of @xmath44 and @xmath43 . each of these may have its own unphysical degeneracies , but in order to give an unphysical mode of equal likelihood to the true peak , these degeneracies will need to coincide . in figure [ fig : ytruecontours ] , we plot the contours in @xmath69 space of where @xmath70 as @xmath6 ranges over @xmath44 and @xmath43 values . we can clearly see the degeneracies present in either variable , but since these only overlap at the one location , near to where the true peak is , there is no unphysical second mode in the moped likelihood . and @xmath71 as they vary over @xmath44 and @xmath43 . the one intersection is the true maximum likelihood peak.,width=312 ] hence , when we plot the original and moped log - likelihoods in figure [ fig : fqlikes ] , although the behaviour away from the peak has changed , the peak itself remains in the same location and there is no second mode . adding more parameters , however , does not always improve the situation . we now consider the case where @xmath43 is once again fixed to its true value and we instead make the polarisation of the burst , @xmath48 , a variable parameter . there are degeneracies in both of these parameters and in figure [ fig : ytruecontours3 ] we plot the contours in @xmath72-space where the compressed values are each equal to the value at the maximum moped likelihood point . these two will necessarily intersect at the maximum likelihood solution , near the true value ( @xmath73 hz and @xmath74 rad ) , but a second intersection is also apparent . this second intersection will have the same likelihood as the maximum and be another mode of the solution . however , as we can see in figure [ fig : fpslikes ] in the left plot , this is not a mode of the original likelihood function . moped has , in this case , created a second mode of equal likelihood to the true peak . and @xmath75 values as they vary as functions of @xmath44 and @xmath48.,width=312 ] for an even more extreme scenario , we now fix to the true @xmath48 and allow the time of arrival of the burst @xmath45 to vary ( we also define @xmath76 ) . in this scenario , the contours in @xmath77-space where @xmath70 are much more complicated . thus , we have many more intersections of the two contours than just at the likelihood peak near the true values and moped creates many alternative modes of likelihood equal to that of the original peak . this is very problematic for parameter estimation . in figure [ fig : ytruecontours2 ] we plot these contours so the multiple intersections are apparent . figure [ fig : ftlikes ] shows the original and moped log - likelihoods , where we can see the single peak becoming a set of peaks . and @xmath75 values as they vary as functions of @xmath44 and @xmath45 . we can see many intersections here other than the true peak.,width=312 ] what we can determine from the previous two sections is a general rule for when moped will generate additional peaks in the likelihood function equal in magnitude to the true one . for an @xmath11-dimensional model , if we consider the @xmath78-dimensional hyper - surfaces where @xmath70 , then any point where these @xmath11 hyper - surfaces intersect will yield a set of @xmath6-parameter values with likelihood equal to that at the peak near the true values . we expect that there will be at least one intersection at the likelihood peak corresponding to approximately the true solution . however , we have shown that other peaks can exist as well . the set of intersections of contour surfaces will determine where these additional peaks are located . this degeneracy will interact with the model s intrinsic degeneracy , as any degenerate parameters will yield the same compressed values for different original parameter values . unfortunately , there is no simple way to find these contours other than by mapping out the @xmath79 values , which is equivalent in procedure to mapping the moped likelihood surface . the benefit comes when this procedure is significantly faster than mapping the original likelihood surface . the mapping of @xmath79 can even be performed before data is obtained or used , if the fiducial model is chosen in advance ; this allows us to analyse properties of the moped compression before applying it to data analysis . if the moped likelihood is mapped and there is only one contour intersection , then we can rely on the moped algorithm and will have saved a considerable amount of time , since moped has been demonstrated to provide speed - ups of a factor of up to @xmath80 in @xcite . however , if there are multiple intersections then it is necessary to map the original likelihood to know if they are due to degeneracy in the model or were created erroneously by moped . in this latter case , the time spent finding the moped likelihood surface can be much less than that which will be needed to map the original likelihood , so relatively little time will have been wasted . if any model degeneracies are known in advance , then we can expect to see them in the moped likelihood and will not need to find the original likelihood on their account . one possible way of determining the validity of degenerate peaks in the moped likelihood function is to compare the original likelihoods of the peak parameter values with each other . by using the maximum moped likelihood point found in each mode and evaluating the original likelihood at this point , we can determine which one is correct . the correct peak and any degeneracy in the original likelihood function will yield similar values to one another , but a false peak in the moped likelihood will have a much lower value in the original likelihood and can be ruled out . this means that a bayesian evidence calculation can not be obtained from using the moped likelihood ; however , the algorithm was not designed to be able to provide this . the solution for this problem presented in @xcite is to use multiple fiducial models to create multiple sets of weighting vectors . the log - likelihood is then averaged across these choices . each different fiducial will create a set of likelihood peaks that include the true peaks and any extraneous ones . however , the only peaks that will be consistent between fiducials are the correct ones . therefore , the averaging maintains the true peaks but decreases the likelihood at incorrect values . this was tested with 20 random fiducials for the two - parameter models presented and was found to leave only the true peak at the maximum likelihood value . other , incorrect , peaks are still present , but at log - likelihood values five or more units below the true peak . when applied to the full seven parameter model , however , the snr threshold for signal recovery is increased significantly , from @xmath81 to @xmath82 . the moped algorithm for reducing the computational expense of likelihood functions can , in some examples , be extremely useful and provide orders of magnitude of improvement . however , as we have shown , this is not always the case and moped can produce erroneous peaks in the likelihood that impede parameter estimation . it is important to identify whether or not moped has accurately portrayed the likelihood function before using the results it provides . some solutions to this problem have been presented and tested , pg s phd is funded by the gates cambridge trust . feroz f , gair j , graff p , hobson m p , & lasenby a n , cqg , * 27 * 7 pp . 075010 ( 2010 ) , arxiv:0911.0288 [ gr - qc ] . gupta s & heavens a f , mnras * 334 * 167 - 172 ( 2002 ) , arxiv : astro - ph/0108315 . heavens a f , jimenez r , & lahav o , mnras * 317 * 965 - 972 ( 2000 ) , arxiv : astro - ph/9911102 . protopapas p , jimenez r , & alcock a , mnras * 362 * 460 - 468 ( 2005 ) , arxiv : astro - ph/0502301 .

we investigate the use of the multiple optimised parameter estimation and data compression algorithm ( moped ) for data compression and faster evaluation of likelihood functions . since moped only guarantees maintaining the fisher matrix of the likelihood at a chosen point , multimodal and some degenerate distributions will present a problem . we present examples of scenarios in which moped does faithfully represent the true likelihood but also cases in which it does not . through these examples , we aim to define a set of criteria for which moped will accurately represent the likelihood and hence may be used to obtain a significant reduction in the time needed to calculate it . these criteria may involve the evaluation of the full likelihood function for comparison . [ firstpage ] methods : data analysis methods : statistical

arxiv

the cosmological constant problem ( ccp ) is a biggest puzzle in particle physics @xcite , and consists of several pieces . the first piece is that the vacuum energy density @xmath4 can be the cosmological constant @xmath5 and various sources of @xmath4 exist , e.g. , a zero point energy of each particle and potential energies accompanied with phase transitions such as the breakdown of electroweak symmetry via higgs mechanism and the chiral symmetry breaking due to quark condensations . the second one is that @xmath4 can receive large radiative corrections including a cutoff scale . the third one is that the experimental value of @xmath5 is estimated as @xmath6gev@xmath7 where @xmath8 is the newton constant , and @xmath9 . ] from the observation that the expansion of present universe is accelerating @xcite . the energy density defined by @xmath10 is referred as @xmath11dark energy density , and its existence has been a big mystery . the pieces of puzzle are not fitted in the framework of the einstein gravity and the standard model of particle physics . because a fundamental theory including gravity has not yet been established , it would be meaningful to give a suggestion based on an effective description of various experimental results concerning the vacuum energy . it might be necessary to introduce a principle , assumptions and/or a framework beyond common sense of an accepted physics , and then the ccp is replaced by the problem to disclose the essence of new staffs . in this article , we study physics on the ccp in the framework of effective field theory and suggest that a dominant part of dark energy can originate from gravitational corrections of vacuum energy under the assumption that the classical gravitational fields do not couple to a large portion of the vacuum energy effectively , in spite of the coupling between graviton and matters at a microscopic level . the content of our article is as follows . in the next section , we explain the pieces of puzzle on the ccp and their implications . in sect . 3 , we give an effective description for physics concerning the ccp and predict that gravitational corrections of vacuum energy can be a candidate of dark energy . in the last section , we give conclusions and discussions . _ the vacuum energy density @xmath4 can be the cosmological constant @xmath5 . _ the energy - momentum tensor of perfect fluids is given by @xmath12 where @xmath13 , @xmath14 and @xmath15 are an energy density , a pressure and a four - velocity of fluids , respectively . the energy - momentum tensor of vacuum is of the form , @xmath16 where @xmath17 is a constant . from ( [ t - pf ] ) and ( [ t - v ] ) , we obtain the relations , @xmath18 where @xmath19 and @xmath20 are an energy density and a pressure of vacuum , respectively . the vacuum has a negative pressure with @xmath21 . the einstein equation is given by @xmath22 where @xmath23 is a bare cosmological constant . from ( [ t - v ] ) and ( [ e - eq ] ) , the cosmological constant is given by @xmath24 , effectively . _ there can be various sources of @xmath4 . _ the first one is a zero point energy of each particle . for a relativistic bosonic particle with a mass @xmath25 , the energy density @xmath26 and the pressure @xmath27 due to zero point fluctuations are given by @xmath28 respectively . here @xmath29 is a momentum of particle . the second one is the energy density from the higgs potential after the breakdown of electroweak symmetry , and its absolute value is estimated as @xmath30 where @xmath31gev ) is the vacuum expectation value on the neutral component of higgs doublet . the third one is the energy density accompanied with the chiral symmetry breaking due to quark condensations , and its absolute value is estimated as @xmath32 where @xmath33 is the qcd scale . _ the vacuum energy density can receive large radiative corrections including a cutoff scale . _ the zero point energy density is calculated by using an effective potential at the one - loop level , and it naively contains quartic , quadratic and logarithmic terms concerning an ultra - violet ( uv ) cutoff parameter @xmath34 . by imposing the relativistic invariance of vacuum ( [ rel - v ] ) on @xmath26 and @xmath27 , @xmath26 in ( [ zero - point ] ) should be of the form , @xmath35 up to some finite terms . note that the terms proportional to @xmath36 and @xmath37 do not satisfy @xmath38 , and they can be regarded as artifacts of the regularization procedure . after the subtraction of logarithmic divergence , @xmath26 is given by @xmath39 where @xmath40 is a renormalization point . for the higgs boson , its zero point energy is estimated as @xmath41 where we use @xmath42gev for the higgs boson mass and take @xmath43gev corresponding to the temperature of present universe @xmath44k . _ the experimental value of cosmological constant is estimated as @xmath45gev@xmath7 , from the observation that the expansion of our present universe is accelerating . _ the vacuum energy density of universe is theoretically given by @xmath46 where @xmath47 is the zero point energy density due to a particle labeled by @xmath48 and the ellipsis stands for other contributions containing unknown ones from new physics . from ( [ rho - higgs ] ) , ( [ rho - qcd ] ) and ( [ rho - zero - higgs ] ) , we have the inequalities , @xmath49 where @xmath50 is a dark energy density defined by @xmath10 . there is a possibility that the magnitude of @xmath4 becomes the 4-th power of the terascale through a cancellation among various contributions from a higher energy physics based on a powerful symmetry such as supersymmetry . because supersymmetry can not work to reduce @xmath4 close to @xmath50 , an unnatural fine - tuning is most commonly required to realize @xmath50 . from ( [ rho - th ] ) and ( [ rho - ineq ] ) , we have a puzzle that consists of unfitted pieces . nature proposes us a big riddle @xmath51why is the observed vacuum energy density so tiny compared with the theoretical one? and a big mystery @xmath51what is an identity of dark energy density? to uncover a clue of ccp and probe into an identity of @xmath50 , let us start with the question whether @xmath4 in ( [ rho - th ] ) exists in physical reality , it gravitates or the classical gravitational field feels @xmath4 . in the absence of gravity , the vacuum energy from matters @xmath52 itself is not observed directly because there is a freedom to shift the origin of energy . here , matters mean various fields including radiations such as photon except for graviton . for instance , the zero point energy of free fields is removed by taking a normal ordering in the hamiltonian . only energy differences can be physically meaningful , as suggested by the casimir effect . in the presence of gravity , if @xmath4 gravitates , the motion of the planets in our solar system can be affected by @xmath4 @xcite . from the non - observation of such an effect for mercury , we have a constraint , @xmath53 as seen from ( [ rho - higgs ] ) , ( [ rho - qcd ] ) and ( [ rho - zero - higgs ] ) , the existence of @xmath54 , @xmath47 ( for particles heavier than 10ev ) and/or @xmath55 threatens the stability of our solar system . hence , it seems to be natural to suppose that the classical gravitational field does not feel a large portion of @xmath52 . in contrast , the ratios of the gravitational mass to the inertial mass stay for heavy nuclei , and hence it is reasonable to conclude that the equivalence principle holds with accuracy at the atomic level and the gravitational field couples to every process containing radiative corrections , accompanied by an emission and/or an absorption of matters . more specifically , the external matter - dependent part of energies must gravitate in both macroscopic and microscopic world . now , let us move to the next step , as phenomenological ingredients of ccp are already on the table . first , based on a standpoint that the einstein gravity is a classical effective theory , physics on the ccp can be described by @xmath56,~~ \rho_{\rm de } \equiv \varlambda_{\rm c(exp)}/(8 \pi g ) = 2.4 \times 10^{-47 } { \rm gev}^4 , \label{scl}\end{aligned}\ ] ] where @xmath57 is the ricci scalar made of the classical gravitational field @xmath58 , @xmath59 , @xmath60 is the lagrangian density of matters as classical objects and @xmath60 does not contain a constant term . because an effective theory is , in general , an empirical one , it would not be so strange even if it can not answer the questions why a large portion of @xmath52 does not gravitate and what the identity of dark energy is . those questions remain as subjects in a fundamental theory . second , we explain why it is difficult to derive ( [ scl ] ) in the framework of ordinary quantum field theory , starting from the action , @xmath61 , \label{s}\end{aligned}\ ] ] where @xmath62 is the ricci scalar made of the graviton @xmath63 , @xmath64 , and @xmath65 and @xmath66 are the lagrangian densities for the standard model particles and other particles beyond the standard model , respectively . the amplitudes representing the coupling between gravitons and the vacuum energy are evaluated by calculating green s functions , @xmath67 for example , on the background minkowski spacetime , two - point function is written as @xmath68 where @xmath69 is the quantum part of graviton in the interaction picture and @xmath70 is the interaction hamiltonian density . the transition amplitude is obtained by removing the propagators on the external lines . the vacuum expectation value @xmath71 corresponds to the vacuum energy density , and then a large cosmological constant term is derived after the identification of classical gravitational fields for external gravitons . here , external gravitons mean gravitons in real states represented by wave functions . third , to reconcile ( [ scl ] ) and ( [ g2 ] ) , we need a radical idea and take a big assumption that _ the classical gravitational fields do not couple to a large portion of the vacuum energy effectively , in spite of the coupling between graviton and matters at a microscopic level . _ we expect that it stems from unknown features of external gravitons . for example , if a kind of exclusion principle works , as a bold hypothesis , that _ external gravitons can not take the same place in the zero total four - momentum state _ , external gravitons would not feel the vacuum energy . however , if it holds in the strong form , we would arrive at undesirable conclusions such as the violation of the equivalence principle for external matter fields with the zero total four - momentum , the vanishing scattering amplitudes among only gravitons and the absence of dark energy . to improve them , we need another assumption such that _ the exclusive attribute of external gravitons is violated by the coupling of external matter fields ( at the same point and/or different ones ) , gravitons with derivatives or internal gravitons . _ here , internal gravitons mean gravitons in virtual states represented by propagators . note that the exclusion principle is merely an example of reasoning to justify the first assumption . the point of the second one is that _ external gravitons can couple to gravitational corrections of vacuum energy involving internal gravitons . _ under the above assumptions , we give a conjecture on a candidate of dark energy for the case that the minkowski spacetime is taken as a background one , i.e. , @xmath72 . in this case , the full propagator of graviton is proportional to @xmath73 . here , @xmath74 is the gravitational scale ( the reduced planck scale ) defined by @xmath75gev ) . using the propagator , we obtain the gravitational corrections of @xmath76 , @xmath77 at one - loop level . by replacing @xmath76 into @xmath52 , we obtain the zero point energy density of @xmath78 , @xmath79 after the subtraction of logarithmic divergence . if @xmath80 dominates @xmath50 , the magnitudes of @xmath81 and @xmath52 are estimated as @xmath82 and @xmath83 respectively . here we take @xmath43gev . then , we have a conjecture that physics around the terascale is relevant to the dark energy of our universe . if the zero point energy of some scalar field dominates @xmath52 , such scalar field has a mass of @xmath84tev and become a candidate of dark matter called @xmath11wimp ( weakly interacting massive particle ) . if superpartners appear around the terascale , they can produce zero point energies of @xmath84tev@xmath7 . in this case , we obtain an interesting scenario that a vacuum acquires an energy of @xmath84tev@xmath7 from zero point energies of dark matter and/or superpartners , it does not gravitate directly and the zero point energy of graviton becomes a source of dark energy . let us study the evolution of @xmath50 in the case with @xmath85 and @xmath86 . if we identify @xmath40 with a temperature of the universe , @xmath50 is not constant but varying logarithmically such that @xmath87 , \label{rho - de}\end{aligned}\ ] ] where @xmath44k and @xmath88 @xmath89 is a scale factor of the universe ( the present one ) . here , we also use the fact that the temperature is inversely proportional to the radius of our universe . the evolution of several energy densities are depicted in figure [ f1 ] . , @xmath90 and @xmath50 , respectively . here , @xmath91 and @xmath90 are energy densities of radiations and non - relativistic matters ( except for dark matter ) , respectively.,title="fig:",width=377 ] - in the appendix , we derive ( [ rho - de ] ) and show that the logarithmically changing energy density is described by the equation of state @xmath92 . if @xmath50 evolves as ( [ rho - de ] ) , there are several possibilities for physics beyond the terasclae . first one is that there is no sensitive physics to contribute the vacuum energy beyond the terascale , i.e. , no processes associated with a huge vacuum energy such as the breakdown of grand unified symmetry and no superheavy particles generating huge zero point energies . second one is that there is a higher energy physics , but a miraculous cancellation can occur among various contributions based on a powerful symmetry such as supersymmetry . then , the vacuum energies can be diminished in supersymmetric grand unified theories and/or supergravity theories , and the vacuum energy due to inflaton can vanish at the end of inflation in early universe . third one is that there survives a huge vacuum energy originated from a higher energy physics in the absence of a powerful mechanism , but even virtual gravitons do not couple to such a vacuum energy beyond the terascale . then , another graviton could be required to realize a higher energy physics . finally , we pursue a last possibility in the presence of the zero point energy of inflaton with a mass of @xmath93gev . we consider the action , @xmath94 , \label{tildes}\end{aligned}\ ] ] where @xmath95 is a coupling constant , @xmath96 is the ricci scalar made of another graviton @xmath97 , @xmath98 , and @xmath99 is the lagrangian density of inflaton . we assume that the ordinary graviton does not couple to inflaton directly . in the same way as the ordinary graviton , we obtain the zero point energy of @xmath97 , @xmath100 where @xmath101 . if @xmath102 dominates @xmath50 with @xmath103gev@xmath7 , the magnitude of @xmath104 is estimated as @xmath105gev . we have studied physics on the ccp in the framework of effective field theory and suggested that a dominant part of dark energy can originate from gravitational corrections of vacuum energy of @xmath84tev@xmath7 , under the following assumptions . * the graviton @xmath78 couples to the potential of matter fields with the strength of @xmath106 , and couples to the vacuum energy of matters with the same strength in the virtual state . * the classical gravitational fields do not couple to a large portion of the vacuum energy effectively . * external gravitons can couple to gravitational corrections of vacuum energy involving internal gravitons . the second one is beyond our comprehension , because it is difficult to understand it in the framework of ordinary local quantum field theory . however , if the ccp is a highly non - trivial problem that can not be solved without a correct theory of gravity in a proper manner , it would not be so strange that it is not explained from the present form of quantum gravity theory . it ease a major bottleneck of the ccp if realized with unknown features of gravity at a more fundamental level , and hence we have tried to step boldly from common sense . as an example of reasoning , we have presented a kind of exclusion principle that external gravitons can not take the same place in the zero total four - momentum state , unless they couple to external matter fields ( at the same point and/or different ones ) , gravitons with derivatives or internal gravitons . it may provide a useful hint to disclose physics behind the ccp . the universe dominated by our dark energy might cause instability , because it does not satisfy the energy condition such as @xmath107 for perfect fluids . a phenomenologically viable model must be fulfilled the requirements that matters are stable and a time scale of instability should be longer than the age of universe . if our speculation were correct , the ccp is replaced by the challenge to construct a microscopic theory of gravity compatible with the above assumptions . we might need a new ingredient or a novel formalism . it would be important to pursue much more features of graviton and the relationship between the classical gravitational field and the quantum one . in such a case , the theory of fat graviton may provide a helpful perspective and clue . we would like to thank prof . t. inami , prof . k. izumi and prof . m . ho for useful comments on the bimetric theory and the area - metric theory . according to them , we have returned the title and contents of our article to the original one . we discuss the evolution of dark energy ( [ rho - de ] ) . the effective potential including contributions of graviton at one - loop level is given by @xmath108 where @xmath76 is the potential of matters . from the feature that @xmath109 is independent of the renormalization point @xmath40 , i.e. , @xmath110 , we obtain the relation , @xmath111 from ( [ dv - eff ] ) , we find that the magnitude of @xmath112 is much less than @xmath52 if @xmath52 is much less than @xmath113 . then , the zero point energy density @xmath114 given by ( [ gr - loop ] ) varies on @xmath40 almost logarithmically . hence , the relation ( [ rho - de ] ) is derived after @xmath114 is regarded as the dark energy density @xmath50 and @xmath40 is identified with the temperature , that is proportional to the radius of our universe . the potential @xmath76 changes after the incorporation of zero point energies of particles and the breakdown of symmetry , and contains @xmath4 given in ( [ rho - th ] ) at present . from the above observation , the magnitude of @xmath76 is almost constant except for the period of the early universe . next , we show that the logarithmically changing energy density is described by the van der waals type equation of state , @xmath115 where @xmath116 is a positive constant . the energy conservation @xmath117 is rewritten as @xmath118 where @xmath88 is the radius of our universe . from ( [ eq - state ] ) and ( [ drho ] ) , we derive the differential equation @xmath119 and obtain the solution , @xmath120 where @xmath121 is a constant . by comparing ( [ rho - de ] ) with ( [ rho - b ] ) , @xmath116 and @xmath121 are determined as @xmath122 , \label{b - b0}\end{aligned}\ ] ] respectively . it is pointed out that , for models of dark energy characterized by @xmath123 , the system can be unstable if @xmath124 , which is realized by a negative kinetic term . although our effective theory of dark energy is different from the case with @xmath124 , @xmath14 and @xmath13 yielding ( [ eq - state ] ) do not satisfy the condition @xmath107 derived from various energy conditions and might cause instability . it is not clear whether they are phenomenologically viable or not , because it depends on the details of model at a microscopic level .

we study physics concerning the cosmological constant problem in the framework of effective field theory and suggest that a dominant part of dark energy can originate from gravitational corrections of vacuum energy , under the assumption that the classical gravitational fields do not couple to a large portion of the vacuum energy effectively , in spite of the coupling between graviton and matters at a microscopic level . our speculation is excellent with terascale supersymmetry . dark energy from gravitational corrections .45em 1.5 cm yugo abe@xmath0 , masaatsu horikoshi@xmath1 and yoshiharu kawamura@xmath1 1.5em @xmath2_graduate school of science and engineering , shimane university , _ + matsue 690 - 8504 , japan + @xmath3_department of physics , shinshu university , _ + matsumoto 390 - 8621 , japan 4.5em = 1.0em = 1.0em = 0.5em = 0.5em

arxiv

spectroscopic studies of hyperfine manifolds in alkalies , such as measurements of energy separations , have benefitted by the high precision of the experimental techniques available to interrogate atoms @xcite . their hydrogen - like structure makes interpretation of experimental results straightforward in terms of electromagnetic fields generated by the valence electron and nuclear moments . precise measurements in higher excited states accessible through two - step transitions@xcite have appeared in recent years . this has renewed interest in improving calculations in other states where theoretical methods such as many - body perturbation theory ( mbpt ) ( see for example the recent book of w. r. johnson @xcite ) are yet to be tested against experimental results . precise measurements in excited states , beyond the first one , have several experimental complications . standard spectroscopic techniques rely on the high population of atoms in the ground state to guarantee a good signal to noise ratio of the fluorescence or absorption of the atomic sample . in two - step transitions this is no longer the case . the amount of population transferred to the intermediate level , for reasonable powers of the lasers , tends to be small , and detectors at the desired frequency might no be readily available . we present in this paper two - color modulation transfer spectroscopy as a tool for studies of atomic properties of higher excited states . the method consist of two lasers ( pump and probe ) counter - propagating through a thermal vapour . before being directed to the interaction region , one of the lasers is modulated . the first step of the transition _ i.e. _ the pump , connects the ground state to a resonant intermediate state while the probe scans over the desired energy manifold . we monitor the absorption of the pump laser as a function of probe laser detuning . the non - linear interaction of the lasers burns a hole " in the atomic ground state population . the generated spectra presents sub - doppler peaks ( sometimes called lamb - bennett dips ) corresponding to the atomic resonances with the trademark sidebands at their side . this technique overcomes the two main inconveniences of direct absorption of the probing laser _ i.e. _ low signal to noise ratio and non - availability of detectors at the desired wavelength . we present two ladder systems in @xmath0rb to illustrate the main features of the technique and two different applications of the modulation . we select the @xmath1 and the @xmath2 ladder transitions to illustrate their different uses . the amplitude of the probe laser is modulated for the first system while the second system has its pump frequency modulated . the frequency modulation of the pump laser and good signal to noise ratio allows us to lock the probe laser to the @xmath3 excited atomic resonance . in this case the probe laser remains modulation free . this is highly desired since the electronic modulation of the laser itself can carry unwanted effects such as sidebands at higher or lower frequencies as well as bandwidth problems . the method we are presenting is , of course , not limited to these two cases and can be extended to other atomic levels . the organization of the paper is as follows : section ii contains the theoretical model , section iii explains the experimental setup and results , section iv has a summary of the precise measurements using this method , and section v presents the conclusions . we start with a three level model that can show some of the qualitative features of the experimental spectra . we use a density matrix formalism to describe a three level atom in ladder configuration interacting with two lasers , one of which has sidebands . we model our system as doppler - free ignoring zeeman sublevels to keep it tractable . the experimental situation is more complex and for quantitative analysis it is necessary to take into account those same effects that we are ignoring . figure [ figure energy levels theory ] shows our theoretical model . we treat two cases . fig [ figure energy levels theory ] ( a ) is a ladder type system with an amplitude modulated probe ( amp ) . fig ( b ) presents the same system except it has a frequency modulated pump ( fmp ) . the intermediate and last levels are coupled by a single laser with three frequencies : a carrier and two sidebands separated form the carrier by @xmath4 ( in mhz ) . we represent the amplitude of the carrier by a rabi frequency @xmath5 and the sidebands by a modulation depth @xmath6 . the ground and intermediate states are coupled by @xmath7 . the detuning of the carrier between levels @xmath8 and @xmath9 is zero in the model as it is for our experiment and we let the detuning between levels @xmath9 and @xmath10 vary as @xmath11 . the total population is normalized to unity . [ figure energy levels theory ] ( b ) follows the same nomenclature except that the sidebands arise from frequency modulation and they appear in the pump laser @xmath7 . for the fmp systems the sidebands have the appropriate sign difference . we have a set of nine linear equations for the slowly varying elements of the density matrix @xmath12 after using the rotating wave approximation with the sidebands rotating - one clockwise , one counter clockwise - at a frequency @xmath4 . the equations are : @xmath13\sigma_{nm}~+}\\ & & \frac{i}{2}\sum_{k}(\alpha_{nk}\sigma_{km}-\sigma_{nk}\alpha_{km})=\dot{\sigma}_{nm}~for~n\neq m,\nonumber\end{aligned}\ ] ] where @xmath14 is the transition frequency , and @xmath15 is the laser frequency connecting the levels . the damping rate is given by : @xmath16 and @xmath17 for the fmp system and @xmath18 for the amp system . the time dependence of the rabi frequency makes the standard approach of obtaining the steady state solution of the system not feasible . instead , we use a floquet basis expansion of the density matrix @xcite to solve our system of equations . we replace each of the slowly rotating elements of the density matrix by : @xmath19 where @xmath20 is the fourier amplitude of the component oscillating at @xmath21 . the system is now a series of @xmath22 coupled equations for some large @xmath23 that have to be solved recursively . it is necessary to set @xmath24 for some @xmath23 to cut off the infinite number of coupled equations . by solving the equations in terms of their predecessors we can extract @xmath25 . . the parameters are ( in units of @xmath26 ) : @xmath27 , @xmath28 , @xmath29 , @xmath30 , and @xmath31,width=283 ] . the parameters are ( in units of @xmath26 ) : @xmath32 , @xmath33 , @xmath34 , @xmath35 , and @xmath36,width=283 ] for our experiment we are interested in the terms @xmath37 , @xmath38 , and @xmath39 which are proportional to the absorption of the first laser carrier and sidebands , respectively . we plot the absolute value of the imaginary part as a function of @xmath11 to recover the absorption . this is necessary to take into account the square - law nature of the photodiode . our three level model reproduces the resonance features of the absorption observed as the second excitation goes into resonance for both amp and fmp systems ( see fig . [ figure absorption am ] and fig . [ figure absorption fm ] , respectively ) . the demodulation of the fmp signal yields the expected error - like feature shown in fig . [ figure demodulated absorption ] . . the parameters are ( in units of @xmath26 ) : @xmath32 , @xmath33 , @xmath40 , @xmath41 , and @xmath36,width=283 ] the size of the sidebands in our model depends on the modulation index ( separation from resonance and strength ) , as well as the specific decay rates of the levels which set up the rabi frequencies @xmath42 in the amp and fmp systems . figure [ figure experimental setup pump ] and fig . [ figure experimental setup probe ] present block diagrams of the fmp and amp systems , respectively . a coherent 899 - 01 ti : sapphire laser with a linewidth of less than 100 khz is the pump laser in both cases . a small amount of laser power from the pump laser is frequency modulated by a small bandwidth electro - optical modulator at @xmath4315 mhz and sent to a glass cell filled with rubidium at room temperature to lock the laser frequency to the @xmath44 crossover line of the @xmath45 and @xmath46 hyperfine levels for the fmp system and to the on resonance @xmath47 transition of the @xmath48 level for the amp system at 795 nm with a pound - drever - hall lock . level @xmath8 in the amp system corresponds to the lower hyperfine state of the 5s@xmath49 level ( @xmath45 ) while @xmath9 is the highest hyperfine state of the 5p@xmath49 level ( @xmath50 ) of @xmath0rb . the decay rate between the two levels is @xmath51 5.7 mhz @xcite . we simplify the hyperfine states of the @xmath52 level to just one level with decay rate @xmath53 3.5 mhz @xcite . for the fmp system , the probe laser is an sdl diode laser with a linewidth of 5 mhz at 776 nm . the lasers overlap inside an independent rubidium glass cell at room temperature wrapped in @xmath54-metal in lin - perp - lin polarization configuration . their @xmath55 power diameter of the laser beams is 1 mm . we scan the probe laser over the @xmath56 level hyperfine manifold and observe the absorption of the pump laser as a function of the probe laser detuning using a fast photodetector . we send the signal to a bias - t and record the dc and demodulated ac components with a wavesurfer digital oscilloscope with an 8-bit resolution from lecroy . we keep the power of the pump laser and the modulation depth fixed to a value of 100 @xmath54w and @xmath57 , respectively . we change the power of the probe beam and observe its influence on the spectra . it is possible to observe the resonant features of the @xmath56 hyperfine manifold with little as 100 @xmath54w of probe power . higher probe power increases the signal size and the width of the features . playing with the polarization and powers we also observe eit features @xcite . we restrict ourselves to a space parameter where these very narrow features are absent . figure [ figure absorption with sidebands ] ( fmp ) and [ figure absorption ] ( fmp ) show typical experimental traces of the absorption of the 780 nm laser . the spectrum has been offset to zero transmission for convenience . the first of these , fig . [ figure absorption with sidebands ] , has the dc component of the absorption with the sidebands appearing on both sides of the main resonances . no doppler background is observed for any of the experimental conditions explored , showing that this is a doppler free spectrum . [ figure absorption ] ( a ) shows the lower hyperfine states of the @xmath56 level manifold with no sidebands for clarity . fig . [ figure absorption](b ) has the demodulated ac component of the absorption . the dashed lines identify the error - like features with their corresponding hyperfine levels . we use this spectrum to stabilize the frequency of the probe laser . level in @xmath0rb . it presents the main resonances as well as the indicated sidebands ( sb ) . the power of the probe and pump beam are 4.3 mw and 100 @xmath54w , respectively . , width=283 ] resonances in @xmath0rb of ( a ) absorption without sidebands and ( b ) demodulated absorption of 780 nm laser as a function of detuning of the 776 nm laser . the power of the probe and pump beam are 4.3 mw and 100 @xmath54w , respectively.,width=283 ] we monitor the laser frequency of the probe beam using a coherent confocal fabry - perot cavity with a free spectral range of 1.5 ghz to test the performance of the laser lock . [ figure error signal ] shows the fringe - side transmission of the probe laser through the cavity . we monitor the behavior of the laser before and after it has been locked . the reduction of the frequency excursions is quite evident as the laser is locked to the atomic resonance . under normal experimental conditions we have observed locking times of 30 minutes , and a significant reduction of the rms noise of more than a factor of seven . excited atomic transition using fmp.,width=283 ] a thick glass plate splits into two the main beam at 795 nm in the amp system before entering an independent rubidium vapor glass cell inside a three layered magnetic shield . a grating narrowed diode laser at 1.324 @xmath54 m ( from here on referred to as 1.3 @xmath54 m laser ) with a linewidth better than 500 khz excites the second transition . we scan the frequency of the 1.3 @xmath54 m laser over the hyperfine manifold of the @xmath52 level . a fiber - coupled semiconductor amplifier increases the power of the 1.3 @xmath54 m laser before it goes to a large bandwidth ( @xmath4310 ghz ) electro - optic modulator ( eom ) that generates the sidebands . the power of each 795 nm beam is approximately 10 @xmath54w with a diameter of 1 mm . we operate in the low intensity regime to avoid power broadening , differential ac stark shifts and line splitting effects such as the autler - townes splitting . both beams are circularly polarized by a @xmath58 waveplate . the counter propagating 1.3 @xmath54 m laser beam with a power of 4 mw and approximately equal diameter overlaps one of the 795 nm beams . after the glass cell an independent photodiode detects each 795 nm beam . the outputs of the detectors go to a differential amplifier to reduce common noise . a digital oscilloscope records the output signal for different values of modulation . [ figure whole scan ] shows an absorption spectrum of the 795 nm laser as a function of the detuning of the 1.3 @xmath54 m laser that shows the signature sidebands of the technique . [ figure linear regression ] shows a plot of the distance between the sidebands as a function of the modulation of the 1.3 @xmath54 m laser . the sidebands that appear on the absorption spectra provide _ in situ _ calibration for the energy spacing of the hyperfine splittings . this effectively translates a measurement of energy spacings from the optical region to a much easier measurement in the radio frequency range . we observe a rich atomic behavior such as eit and reversal of the peaks that depends on the power of the lasers , relative polarization and magnetic field intensity ( see for example fig [ figure bad absorption ] ) . this points towards a stringent control on experimental parameters for precision studies of energy separations . , f=1 and f=2 hyperfine states of @xmath0rb with sidebands . the big sideband belongs to the f=1 peak . the small feature on the side of the f=2 peak corresponds to the second sideband of the f=1 peak . the glass cell is in a magnetic field of 0.37 g.,width=283 ] m laser for @xmath0rb . the arrow point to the value of the modulation that corresponds to the overlap of the sidebands and half the hyperfine splitting of the @xmath52 level hyperfine splitting.,width=283 ] table [ table theory and experiment ] shows the values of the magnetic dipole constants using relativistic mbpt @xcite with single double ( sd ) and single double partial triple ( sdpt ) wave functions and values extracted from measurements of the hyperfine splitting in other electronic states currently in the literature for @xmath59=1/2 @xcite . we have not been able to find in the literature values for higher levels with adequate precision to include them in the figure . the agreement of the theory with the experiment , for @xmath59=1/2 levels , is well within the 1% level . the sdpt relativistic wave functions do indeed improve the accuracy of the calculations of the single double wave functions . .single double ( sd ) and partial triple ( sdpt ) excitation calculated from _ ab intio _ mbpt in ref . @xcite and experiment magnetic dipole constants for the first @xmath59=1/2 levels in @xmath60rb . ( adapted from ref . @xcite ) . [ cols="<,^,^,^",options="header " , ] the accuracy of the 6@xmath61 measurement is high enough to extract a hyperfine anomaly @xcite in an excited state , which shows that the effect is independent of the n state of the level , as originally predicted by bohr and weiskopf @xcite . we have presented two - color modulation transfer spectroscopy as reliable and simple method for studies of atomic properties in excited states . the characteristic sidebands appearing the spectra have the two - fold utility of working as an _ in situ _ ruler for measurements of energy separations or to lock the frequency of a laser to an excited transition . the good quality of the data presented is due to monitoring of the absorption of the pump beam instead of direct absorption of the probe beam . the absorption of the pump beam ( or lack thereof ) , is always guaranteed since a vast amount of atoms are always in the ground state and even small changes _ i.e. _ excitation to the last step of the transition , will be noticeable even for small powers of the pump beam . in addition , the spectra does not show a doppler background due to the lack of an equilibrium thermal population in the intermediate state . it is the hope that the method will stimulate studies of atomic properties of excited states and further push the experimental precision and theoretical work in excited atomic states . work supported by nsf . a.p.g . would like to thank e. gomez for discussions on the subject of this article and p. barberis for help on the theory of three level atoms .

we present two - color modulation transfer spectroscopy as a tool for precision studies of atomic properties of excited states . the bi - colored technique addresses a narrow set of velocity groups of a thermal atomic vapour using a two - step transition to burn a hole " in the velocity distribution . the resulting spectrum presents sub - doppler linewidths , good signal to noise ratio and the trademark sidebands that work as an _ in situ _ ruler for the energy spacing between atomic resonances . the spectra obtained can be used for different applications such as measurements of energy splittings or stabilization of laser frequencies to excited atomic transitions . = 10000

arxiv

the physics potential of forward proton tagging at the lhc has attracted a great deal of attention in recent years @xcite . a main focus of interest is the central exclusive production ( cep ) process , @xmath7 , in which the protons remain intact and the central system @xmath8 is separated from the outgoing protons by a large rapidity gap . to a very good approximation , @xmath8 is constrained to be in a colour singlet , @xmath9 , state . observation of any particle , such as a standard model higgs boson , in the central exclusive channel would therefore provide a direct observation of its quantum numbers . furthermore , by detecting the outgoing protons and measuring their energy loss accurately , it is possible to measure the mass of the centrally produced particle regardless of its decay products @xcite . because of these unique properties , it has been proposed that forward proton detectors should be installed either side of the interaction points of atlas and/or cms . the fp420 collaboration has proposed to install detectors in the region 420 m from the interaction points @xcite . these detectors would allow the detection of central systems in the approximate mass range 70 gev @xmath10 gev . proposals also exist to upgrade the capabilities of atlas and cms to detect protons in the 220 m region @xcite . these detectors , when used in conjunction with 420 m detectors , would extend the accessible mass range well beyond @xmath11 gev . in this paper , we focus on the central exclusive production of the standard model ( sm ) higgs boson and a supersymmetric ( mssm ) higgs boson , with @xmath12 gev . the cep process is shown schematically in figure [ centralexclusive ] . for this mass region , the dominant decay channel of the higgs boson is to @xmath13 , which is very difficult to observe in conventional higgs searches at the lhc because of the large qcd background . this is not the case in central exclusive production due to the @xmath9 selection rule , which suppresses the leading order central exclusive @xmath13 background by a factor of @xmath14 , where @xmath15 is the mass of the @xmath13 di - jet system . as we shall see , this renders the @xmath13 decay channel observable at the lhc in certain scenarios if appropriate proton tagging detectors are installed . the structure of the paper is as follows . firstly , we give a brief overview of the proposed forward detector upgrades at 220 m and 420 m at atlas and cms , including a simulation of the acceptance of the detectors . we then discuss the predicted signal cross sections and survey the background processes . taking a 120 gev standard model higgs boson as the benchmark , we perform a simulated analysis including an estimation of the detector acceptance and smearing effects and level 1 trigger strategies . the analysis is then extended to the mssm for a particular choice of parameters . , there are two properties of the proposed forward detector systems that are critical to this analysis ; the acceptance of the detectors in the mass range of interest and the ability of the forward detectors to correctly associate the detected outgoing protons with a higgs boson candidate event measured in the central detector . this matching is critical at high luminosities , where the large number of proton - proton collisions per bunch crossing ( often referred to as pile - up ) leads to a high probability that forward protons from single diffractive or double pomeron ( dpe ) collisions not associated with a higgs candidate event will enter the forward detectors during the same bunch crossing . the proposed forward detectors aim to associate particular protons with the higgs candidate event by making a measurement of the outgoing proton time - of - flight ( tof ) from the interaction vertex to the detectors . the difference in the arrival times of the protons on opposite sides of the central detector , @xmath16tof , allows a vertex measurement to be made under the assumption that the detected pair of protons originate from a single hard interaction . this vertex can then be matched with the vertex of a candidate higgs event reconstructed using the central detector alone . the current design goal of the forward detectors is to achieve a 10ps accuracy in the tof measurement @xcite , which translates into a vertex measurement accurate to 2.1 mm . the use of fast - timing measurements is discussed in more detail in section [ olap ] . it has been suggested that the central detector could also be used provide a third timing measurement @xcite , which could allow for an improved rejection of pile - up events . we discuss the effect of this possibility in section [ sec : results ] . the acceptance of the forward detectors is governed by the distance of the active edge of the detector from the beam , which determines the smallest measurable energy loss of the outgoing protons , and the aperture of the lhc beam elements between the interaction point and the forward detectors ; protons that lose too much momentum will be absorbed by beam elements , imposing an upper limit on the measurable momentum loss of the protons . the distance of the active edge of the detector from the beam depends primarily on the beam size at each detector location . previous estimates @xcite have assumed that the closest distance , @xmath17 , is given by @xmath18 where @xmath19 is the gaussian beam size at the detector location and 0.5 mm is a constant term that accounts for the distance from the sensitive edge of the detector to the bottom edge of the window . the beam size @xmath20 is approximately 250 @xmath21 m at 420 m and 100 @xmath21 m at 220 m , leading to a distance of closest approach of 3 mm for detectors at 420 m and 1.5 mm for detectors at 220 m . it is likely that the detectors will begin operation at a larger distance from the beam , at least until the detectors and machine background conditions are well understood @xcite . figure [ forwardaccept ] ( a ) shows the acceptance for events in which both outgoing protons are detected at 420 m around ip1 ( atlas ) , as a function of the mass of the central system for different detector distances from the beam . the protons are generated using the exhume monte carlo @xcite and tracked through the lhc beam lattice using the fptrack program with version 6.500 of the lhc optics @xcite . the central system mass , @xmath15 , is calculated from the forward proton momenta , @xmath22 where the @xmath23 are the fractional momentum losses of the protons and @xmath24 is the centre - of - mass energy of the collision . the acceptance for a 120 gev higgs boson is independent of the distance of approach of the detectors from the beam up to approximately 7 mm in the case where both protons are detected at 420 m . this is consistent with the findings of @xcite . the situation is very different for events in which one proton is tagged at 220 m and the other at 420 m , as shown in figure [ forwardaccept ] ( b ) . in this case , the acceptance is increased when either detector is moved closer to the beam . for a 120 gev higgs boson , with 420 m detectors at 5 mm and 220 m detectors at 2 mm , the acceptance is 28% if both protons are tagged at 420 m ( symmetric tag ) with an additional 16% acceptance if one proton is tagged at 220 m and one at 420 m ( asymmetric ) . moving the 420 m detectors inwards to 3 mm and the 220 m detectors to 1.5 mm increases the asymmetric acceptance by up to a factor of three , as shown in figure [ forwardaccept ] ( b ) . figures [ forwardaccept ] ( a ) and ( b ) also demonstrate the increasing importance of 220 m detectors as the mass of the central system increases . at ip5 ( cms ) the symmetric acceptance is identical to that at ip1 . for the asymmetric tags , however , the acceptance is worse by a factor of @xmath25 across the mass range of interest . this is caused by the horizontal ( rather than vertical ) plane of the crossing angle of the beams at ip5 @xcite . in this paper we concentrate on the measurement around ip1 ( atlas ) . central exclusive signal and background events are simulated with parton showering and hadronisation effects using exhume v1.3.4 @xcite . exhume contains a direct implementation of the khoze , martin and ryskin ( kmr ) calculation of the central exclusive production process @xcite . the cross section for the cep of a standard model higgs boson decaying to @xmath27 as a function of the higgs mass is shown in figure [ sigmamh ] . the rapid decrease in cross section at @xmath28 gev occurs because the primary decay channel changes from @xmath13 to @xmath29 . for masses above @xmath30 gev , it is expected that the higgs boson should be observed in the @xmath29 channel with a luminosity of 30 fb@xmath4 using forward proton tagging @xcite . the primary uncertainties in the predicted cross section come from two sources ; the parton distribution functions ( pdf ) and the soft survival ( often termed rapidity gap survival ) factor . the cep cross sections are relatively sensitive to the pdfs because the derivative of the gluon density enters to the fourth power . figure [ sigmamh ] shows the cross section prediction for three different pdf choices , cteq6 m , mrst2002nlo and cteq6l . the cross section for the cep of a standard model higgs boson of mass @xmath31 gev decaying into b - quarks varies from 1.86 fb ( mrst2002nlo ) to 7.38 fb(cteq6l1 ) . the spread of a factor of @xmath32 is consistent with the findings of @xcite . for the purposes of this analysis , we chose cteq6 m as our default pdf as it lies between the two extremes . there is some justification for choosing an nlo pdf because the kmr calculation contains a nlo qcd k - factor ( 1.5 ) for sm higgs boson production . the soft survival factor , @xmath33 , is the probability that there are no additional hard scatters in a single @xmath34 collision and is expected to vary from process to process . for cep processes , we take the exhume default of @xmath35 at lhc energies . until very recently , there was a consensus that @xmath33 should be between @xmath36 and @xmath37 for the cep of a higgs boson at the lhc @xcite . two very recent studies have predicted a lower value @xcite and it remains to be seen whether a new theoretical consensus can be reached . in any case , @xmath33 will be measurable in early lhc data . , for three different proton parton distribution functions.[sigmamh],scaledwidth=50.0% ] the mssm contains three neutral higgs bosons - two scalar and one pseudo - scalar . the @xmath38 , parity - even selection rules strongly suppress cep of the pseudo - scalar . this can be advantageous in areas of mssm parameter space where the pseudo - scalar is almost degenerate in mass to one ( or both ) of the scalar higgs bosons , since cep would provide a clean and complimentary measurement of the mass of the scalar only @xcite , and allow nearly - degenerate higgs bosons to be distinguished @xcite . furthermore , at large values of tan@xmath39 , the cross section for the cep of the scalar higgs bosons can be strongly enhanced relative to the cep of a sm higgs boson of the same mass @xcite . we choose a point in parameter space defined by the @xmath40 scenario @xcite with @xmath41 gev and tan@xmath42 , resulting in the scalar higgs boson having a mass of 119.5 gev and a decay width of 3.3 gev . with this choice of parameters , the lightest scalar higgs boson has an enhanced cep cross section and is almost degenerate in mass with the pseudo - scalar . the cep cross section for the lightest higgs boson decaying to b - quarks in this scenario is predicted to be @xmath43 fb . the uncertainty on this prediction is the same as that for the sm higgs boson . the backgrounds to central exclusive higgs production can be broken down into three categories ; central exclusive di - jet production , double pomeron exchange and overlap . the most difficult backgrounds to deal with are the central exclusive ( cep ) di - jet backgrounds . central exclusive @xmath13 production is suppressed at leading order by the @xmath45 selection rule , but is still present and forms an irreducible continuum background beneath the higgs mass peak . central exclusive glue - glue production has a much larger cross section than @xmath13 production since it is not suppressed . it contributes to the background when the gluon jets are mis - identified as b - jets ( 1.3% probability for each mis - tag at atlas - see section [ detector ] ) . central exclusive @xmath46 events do not contribute to the background for two reasons . firstly , @xmath46 production is suppressed with respect to the @xmath13 process by @xmath47 . secondly , the @xmath46 background is further suppressed by a factor of @xmath48 relative to the @xmath13 cross section assuming that the probability of mis - identifying a c - quark is approximately 0.1 for a b - tag efficiency of 0.6 @xcite . higher order cep di - jet backgrounds such as the 3-jet final state @xmath49 process have been studied in @xcite . it is expected that these backgrounds will be lower than the lo @xmath50 and @xmath27 backgrounds after experimental cuts , but they should be implemented into the exhume monte carlo and a full study performed before definitive conclusions can be drawn . double pomeron exchange ( dpe ) is defined as the process @xmath51 , where @xmath52 is a central system produced by pomeron - pomeron fusion ( @xmath53 ) . the pomeron is assigned a partonic structure and so there are always pomeron remnants accompanying the hard scatter as shown in figure [ dpepomeron ] . the relevant background processes are separated into @xmath13 and @xmath54 , where @xmath55 represents light - quark and gluons jets . the dpe events are simulated using the pomwig v2.0 event generator @xcite , which implements the diffractive parton distribution functions measured by the h1 collaboration @xcite . we use h1 2006 fit b , although we discuss the effect of using different diffractive pdfs . we choose the pomwig option to treat the valence partons in the pomeron as gluons . pomwig is normalised to the h1 data , and therefore does not account for the soft survival factor . for hard double pomeron events , we take @xmath35 . pomwig is also capable of generating scatters involving sub - leading ( non - diffractive ) exchanges ( @xmath56 ) . we do not include this contribution because the cross section for @xmath57 @xmath13 events is @xmath58 0.034 fb for @xmath59 , @xmath60 gev and @xmath61-quark @xmath62 gev . this is negligible compared to the cross section for @xmath63 @xmath13 , which is @xmath5865 fb in the same kinematic region . system via double pomeron exchange ( dpe ) . [ dpepomeron],scaledwidth=50.0% ] overlap events are defined as a coincidence between an event that produces a central system of interest , @xmath52 , and one or more diffractive events ( single diffractive @xmath64 or dpe ) which produce protons in the acceptance range of a forward detector in the same bunch crossing . on average there will be 3.5 interactions per bunch crossing at low instantaneous luminosity ( 10@xmath65 @xmath3 s@xmath4 ) and 35 interactions at high instantaneous luminosity ( 10@xmath66 @xmath3 s@xmath4 ) . we investigate three types of overlap event ; [ p][x][p ] , [ pp][x ] and [ px][p ] . the square brackets specify the interaction to which a part of the overlap event belongs - in this notation both the cep and dpe events would be [ pxp ] . the cross section , @xmath67 , for the overlap background may be estimated by @xmath68 } \ , \left [ \ , \sum_{n=3}^{\infty } \frac{\lambda^{n } e^{-\lambda}}{n ! } \ , p_{2[p]}\left(n-1\right ) \ , + \ , \sum_{n=2}^{\infty } \frac{\lambda^{n } e^{-\lambda}}{n ! } \ , p_{[pp ] } \left(n-1\right ) \right ] \nonumber \\ & & + \ , \ , \sigma_{[px ] } \ , \sum_{n=2}^{\infty } \frac{\lambda^{n } e^{-\lambda}}{n ! } \ , p_{[p ] } \left(n-1\right ) , \end{aligned}\ ] ] where @xmath69}$ ] is the inclusive ( @xmath70 ) di - jet cross section , @xmath71 is the average number of @xmath34 interactions per bunch crossing and @xmath72 is the actual number of interactions in a specific bunch crossing . because the actual number of interactions is not fixed , we sum over all possible numbers and weight each configuration by a poisson distribution . in the first term , @xmath73}\left(n\right)$ ] is the probability that , given @xmath74 interactions , there are at least two events that produce a forward proton ( one on each side of the interaction point ) by any mechanism . this is dominated by soft single diffractive events @xmath64 . @xmath73}\left(n\right)$ ] is given by a trinomial distribution , i.e. @xmath75}\left(n\right ) = \sum_{r+q=2}^{n } \sum_{q=1}^{r+q-1 } \frac{n!}{\left(n-\left[r+q\right]\right ) ! \ , r ! \ , q ! } \ , \left ( f_{[p]}^{+}\right)^{r } \ , \left(f_{[p]}^{-}\right)^{q } \ , \left ( 1 - f_{[p]}^{+ } - f_{[p]}^{-}\right ) ^{n - r - q}\ ] ] where , for example , @xmath76}^{+}$ ] is the fraction of events at the lhc that produce a forward proton in the @xmath77 direction and within the forward detector acceptance . in the second term , @xmath78}(n)$ ] is defined as the probability that there is at least one event that contains an outgoing proton on each side of the interaction point within the acceptance of the forward detectors ( dominated by the soft double pomeron process @xmath79 ) . @xmath78}(n)$ ] is a binomial distribution that utilises the event fraction @xmath80}$ ] . note that double pomeron events can also contribute to the first term in equation [ olapxs ] , in the case where only one of the outgoing protons falls within the detector acceptance . the third term deals with a two - fold coincidence between a single diffractive di - jet event ( @xmath64 ) , which produces a proton within the forward detector acceptance and a hard central diffractive system @xmath52 that mimics the signal , and an overlap event that produces a proton on the opposite side . @xmath81}$ ] is the total single diffractive di - jet cross section ( @xmath64 ) , i.e the outgoing proton can be on either side of the interaction point . @xmath82}(n)$ ] is defined as the probability that there is at least one event with a forward proton in the forward detector acceptance on the opposite side of the ip to the single diffractive proton from the hard event . this is defined in a similar way to @xmath78}(n)$ ] , but using the event fraction @xmath76}$ ] . for small proton momentum losses , the cross section for events at the lhc that contain a forward proton is dominated by the single diffractive cross section , @xmath83 @xcite ; @xmath84 where @xmath85 is the total cross section at the lhc , @xmath86 is the soft - survival factor for soft single diffraction ( 0.087 ) , @xmath87 is the pomeron trajectory , @xmath88 is the squared 4-momentum transfer at the proton vertex , @xmath89 is the pomeron nucleon coupling and @xmath90 is the triple pomeron vertex . the value of @xmath76}$ ] is calculated by integrating equation [ sdxs ] using standard monte carlo techniques , for a specific acceptance range in @xmath91 and @xmath88 . the cross section for the [ p][x][p ] di - jet events , for parton @xmath92gev and @xmath93 , @xmath94 is shown in figure [ olaplumidep ] ( a ) . the cross section increases by two orders of magnitude from low to high luminosity . at larger values of @xmath91 , there will be an additional contribution from non - diffractive ( i.e. reggeon exchange ) events . this contribution is estimated using the pythia @xcite and phojet @xcite event generators . the predictions for the diffractive and non - diffractive contributions to @xmath76}$ ] are compared to equation [ sdxs ] in table [ fractions ] . as expected , the diffractive contribution dominates at small @xmath91 . the non - diffractive contribution becomes increasingly important at higher @xmath91 , but remains smaller than the diffractive contribution for @xmath95 . phojet predicts a higher fraction of non - diffractive events than pythia and also predicts that a large fraction of forward protons are due to dpe events . in this analysis we use the prediction for single diffraction given by equation [ sdxs ] for @xmath96 ( which would give hits only at 420 m ) , and take the non - diffractive contribution to be negligible . for @xmath97 , we add the pythia prediction for the non - diffractive component , since this agrees more closely with theoretical expectations that the non - diffractive fraction of @xmath34 collisions at the lhc in this kinematic range should be between 1.0% and 1.7% @xcite . .the fraction of events at the lhc that produce a forward proton on one side of the interaction point in a specific kinematic range . the pythia and phojet event generators are compared to the single diffractive cross section given in equation [ sdxs ] . sd labels the outgoing proton from single diffractive scatters and nd labels the protons produced from non - diffractive scatters . dpe labels double pomeron exchange events . [ fractions ] [ cols="^,^,^,^,^,^ " , ] we have shown that the central exclusive production of the lightest scalar higgs boson , for the choice of mssm parameter space described in section [ scenarios ] , can be observed with a significance of at least @xmath1 in the @xmath13 decay channel within 3 years of data taking at the lhc , if suitable proton tagging detectors are installed around atlas and cms and the current predictions for the cep cross section are correct . we have evaluated the most important backgrounds and shown that they can be rejected with high efficiency using a set of exclusivity variables . we have also shown that a fraction of b - jet events can be retained by the currently forseen level 1 trigger hardware , if the trigger strategies we outline are adopted . whilst we have only considered a particular choice of parameters in detail , the general conclusions should hold for a wide range of scenarios . as a general rule , if the cep cross section for the production of the higgs boson is greater than @xmath98 fb , and the higgs boson decays predominantly to b - quarks , then the analysis presented here should apply if the decay width is not too large . the analysis will also apply to any new particle that decays predominantly to b - quarks . it is worth speculating what the future experimental strategy might be if a higgs sector such as the @xmath0 scenario of the mssm is discovered at the lhc or tevatron , and the cep channel proves to be observable with the forward detector configurations currently proposed . the largest loss of cep signal events comes from the limited acceptance of the proton detectors ( between @xmath99 and @xmath100 depending on detector configuration ) and the l1 trigger efficiency , which is at best around @xmath101 at high luminosity for the strategies we consider in this paper . if a hardware upgrade of the l1 trigger systems of atlas and cms were to increase the trigger latency such that the 420 m detector signals could be included , then a trigger efficiency of close to @xmath102 could be achieved . figure [ upgrade ] ( a ) shows a typical mass fit , with 300 fb@xmath4 of data taken at high luminosity using 420 m detectors alone ( i.e. symmetric events ) , assuming 100% trigger efficiency . the significance of this signal is approximately @xmath103 . if it is assumed that , in addition , fast timing detector improvements can be made as described in section [ mssm ] , then the significance rises to nearly @xmath104 , as shown in figure [ upgrade ] ( b ) . in this case , a measurement could be made within 100 fb@xmath4 . combining the symmetric and asymmetric analyses increases the significance such that a 5@xmath105 measurement could be achieved within 30 fb@xmath4 . the mass of the lightest scalar higgs can be measured with an accuracy of better than @xmath106 gev . for the @xmath0 scenario considered here , the width of the lightest higgs is not much larger than the mass resolution of the proton detectors , and therefore the extraction of the higgs width from the fits is marginal . in other scenarios with widths in excess of @xmath107 gev , however , a direct measurement of the width would also be possible . we remind the reader that observation of any particle in the cep channel would provide a direct measurement of the quantum numbers of the particle . furthermore , if the pseudo - scalar higgs is close in mass to the lightest scalar higgs , then cep would provide an unambiguous separation of the two states since the pseudo - scalar can not be produced . finally , with such a trigger strategy in addition to improved fast timing to reduce the overlap backgrounds , the sm higgs may be obervable in the b - jet channel . we would like to thank mike albrow , michele arneodo , andrew brandt , albert deroeck , jeff forshaw , valery khoze , henri kowalski , paul newman , will plano , misha ryskin , marek tasevsky and chris tevlin for interesting discussions and suggestions throughout this project . this work was funded in the uk by stfc and the royal society . a. bialas and p. v. landshoff , phys . b * 256 * ( 1991 ) 540 . j. r. cudell and o. f. hernandez , nucl . b * 471 * ( 1996 ) 471 [ arxiv : hep - ph/9511252 ] . v. a. khoze , a. d. martin and m. g. ryskin , eur . j. c * 23 * ( 2002 ) 311 [ arxiv : hep - ph/0111078 ] . b. cox , j. r. forshaw and b. heinemann , phys . b * 540 * ( 2002 ) 263 [ arxiv : hep - ph/0110173 ] . a. de roeck , v. a. khoze , a. d. martin , r. orava and m. g. ryskin , eur . j. c * 25 * ( 2002 ) 391 [ arxiv : hep - ph/0207042 ] . a. b. kaidalov , v. a. khoze , a. d. martin and m. g. ryskin , eur . j. c * 33 * , 261 ( 2004 ) [ arxiv : hep - ph/0311023 ] . b. e. cox , j. r. forshaw , j. s. lee , j. monk and a. pilaftsis , phys . rev . d * 68 * ( 2003 ) 075004 [ arxiv : hep - ph/0303206 ] . b. e. cox , aip conf . * 753 * ( 2005 ) 103 [ arxiv : hep - ph/0409144 ] . j. r. forshaw , arxiv : hep - ph/0508274 . m. boonekamp , c. royon and r. peschanski , nucl . a * 755 * , 599 ( 2005 ) . j. r. ellis , j. s. lee and a. pilaftsis , phys . rev . d * 71 * ( 2005 ) 075007 [ arxiv : hep - ph/0502251 ] . p. j. bussey , t. d. coughlin , j. r. forshaw and a. d. pilkington , jhep * 0611 * , 027 ( 2006 ) [ arxiv : hep - ph/0607264 ] . v. a. khoze , a. d. martin and m. g. ryskin , arxiv:0705.2314 [ hep - ph ] . v. a. khoze , a. d. martin and m. g. ryskin , phys . b * 650 * , 41 ( 2007 ) [ arxiv : hep - ph/0702213 ] . s. n. white , `` on the correlation of subevents in the atlas and cms / totem experiments , '' arxiv:0707.1500 [ hep - ex ] . v. avati and k.osterberg , _ totem forward measurements : leading proton acceptance _ , in hera and the lhc proceedings part b , arxiv : hep - ph/0601013 . r. appleby , r. m. jones and f. roncarolo , the proceedings of pac07 , albuquerque , us , june 25 - 29 2007 . s. alekhin _ et al . _ , `` survival probability of large rapidity gaps '' , p221 , `` hera and the lhc - a workshop on the implications of hera for lhc physics : proceedings part a '' , arxiv : hep - ph/0601012 . e. gotsman , e. levin and u. maor , arxiv:0708.1506 [ hep - ph ] . l. frankfurt , c. e. hyde - wright , m. strikman and c. weiss , phys . d * 75 * ( 2007 ) 054009 [ arxiv : hep - ph/0608271 ] . m. strikman , private communication a. b. kaidalov , v. a. khoze , a. d. martin and m. g. ryskin , eur . j. c * 31 * ( 2003 ) 387 [ arxiv : hep - ph/0307064 ] . s. heinemeyer , v. a. khoze , m. g. ryskin , w. j. stirling , m. tasevsky and g. weiglein , arxiv:0708.3052 [ hep - ph ] . m. s. carena , s. heinemeyer , c. e. m. wagner and g. weiglein , eur . j. c * 26 * ( 2003 ) 601 [ arxiv : hep - ph/0202167 ] . atlas : detector and physics performance technical design report , volume 1 , p317 - 346 , 1999 . cern - lhcc-99 - 14 a. aktas _ et al . _ [ h1 collaboration ] , eur . j. c * 48 * ( 2006 ) 749 [ arxiv : hep - ex/0606003 ] . a. aktas _ et al . _ [ h1 collaboration ] , eur . j. c * 48 * ( 2006 ) 715 [ arxiv : hep - ex/0606004 ] . v. a. khoze , a. d. martin and m. g. ryskin , phys . b * 643 * , 93 ( 2006 ) [ arxiv : hep - ph/0609312 ] . t. sjostrand , l. lonnblad and s. mrenna , arxiv : hep - ph/0108264 . r. engel and j. ranft , phys . d * 54 * ( 1996 ) 4244 [ arxiv : hep - ph/9509373 ] . v. a. khoze and m. g. ryskin , private communication . g. corcella _ et al . _ , arxiv : hep - ph/0210213 . j. m. butterworth , j. r. forshaw and m. h. seymour , z. phys . c * 72 * ( 1996 ) 637 [ arxiv : hep - ph/9601371 ] . m. grothe et al . , `` triggering on forward physics , '' cern - 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[ hep - ph/0601013 ]

a detailed study is presented of the search for higgs bosons in the b - decay channel in the central exclusive production process at the lhc . we present results for proton tagging detectors at both 220 m and 420 m around atlas or cms . we consider two benchmark scenarios ; a standard model ( sm ) higgs boson and the @xmath0 scenario of the minimal supersymmetric standard model ( mssm ) . detector acceptance , smearing and event trigger strategies are considered . we find that the sm higgs will be challenging to observe in the b - jet channel without improvements to the currently proposed experimental configuration , but a neutral scalar mssm higgs boson could be observable in the b - jet channel with a significance of @xmath1 or greater within three years of data taking at all luminosities between @xmath2 @xmath3 s@xmath4 and @xmath5 @xmath3 s@xmath4 , and at @xmath6 or greater after three years in certain scenarios .

arxiv

zero - field hall effect in chiral @xmath0-wave superconductors ( scs ) has drawn much attention in literature recently . @xcite because of the nature of broken time reversal ( @xmath5 ) symmetry , a nonzero hall conductivity can be possible in a chiral @xmath0-wave sc . indeed , it has already been shown that spontaneous hall effect could arise from the intrinsic angular momentum of cooper pairs @xcite as well as from the spontaneous surface current . @xcite more recently , hall conductivity due to impurity effect @xcite or to multiband sc structure @xcite was also studied , which could give possible explanation to the observed polar kerr effect in the superconducting state of sr@xmath6ruo@xmath7 . @xcite in this work , we address the zero - field hall effect in a chiral @xmath0-wave sc originating from another mechanism , namely the vortex dynamics near kosterlitz - thouless ( kt ) transition . in two - dimensional ( 2d ) superfluid ( sf ) or sc films , quantized vortices are realized as topological defects in the condensates , whose dynamics has been one of the key ingredients in understanding 2d phase transition phenomena . @xcite a few decades ago , kosterlitz and thouless @xcite suggested a static theory to relate a phase transition observed in superfluid @xmath8he film @xcite to vortex - antivortex pair unbinding process across a transition temperature @xmath9 . in this picture , the logarithmic vortex - antivortex interaction is screened by smaller pairs and is renormalized to @xmath10 of its bare value @xmath11 for temperature @xmath12 . the length - dependent dielectric constant @xmath13 is used to describe the static screening of pair interaction . when @xmath14 , there exists a finite pair size @xmath15 such that the interaction becomes vanishingly small . consequently , the pair unbinds and free vortices emerge ; superfluidity is then destroyed . soon after that , ambegaokar , halperin , nelson , and siggia ( ahns ) @xcite combined this static theory with hall and vinen s dynamical description of vortex motion @xcite to give an analysis of the dynamical effect on the phase transition . concisely speaking , the renormalization process in the static theory @xcite is truncated by vortex dynamics with a characteristic length @xmath16 instead of going to its completion . @xcite here @xmath17 is the diffusivity of vortex movement and @xmath18 is the driving frequency . this results in broadening transition observed in @xmath8he sf films @xcite as well as in charged fermi systems such as high - temperature scs . @xcite it is instructive to explore any physical consequence stemming from this kt transition in a broken @xmath5 symmetry state . possible experimental candidates of broken @xmath5 symmetry state could be superconducting sr@xmath6ruo@xmath7 ( ref . ) or @xmath19he - a phase thin film , @xcite in which pairing of chiral @xmath0-wave type is expected . indeed , in literature some theoretical works have been done to investigate new features specific to scs with pairing of this type near kt transition . @xcite in this work , we consider a 2d @xmath0-wave pairing state with @xmath20-vector @xmath21 where @xmath22 is the unit vector normal to film surface , @xmath23 and @xmath24 denote the @xmath25 and @xmath26 component of the relative momentum @xmath27 of a cooper pair , and @xmath28 is the fermi momentum . assuming isotropic fermi surface , two kinds of pairing fields can be obtained : @xmath29 and @xmath30 with asymptotic behavior at large @xmath31 being @xmath32 , @xmath33 , @xmath34 , and @xmath35 . @xcite here @xmath36 is the spatial coordinate and @xmath37 is the modulus of the energy gap in the bulk . from these pairing fields , we can identify two types of integer vortices called @xmath38vortex and @xmath39vortex respectively . because of spontaneously broken @xmath5 symmetry , these two types of vortices are not equivalent . @xcite in particular , their hall and vinen coefficients @xcite do not share the same value , i.e. , @xmath40 and @xmath41 ( see section ii a ) . this results in a nonzero convective " term in a vortex pair polarization fokker - planck equation in addition to the conventional diffusive terms , while in its @xmath42-wave counterpart such convective motion does not enter the dynamics . @xcite the relative strength of convection is quantified by a convective ratio @xmath43 in this paper . it is due to such distinct feature that pair polarization transverse to the driving force field becomes possible even without applied magnetic field . a nonzero vortex - dynamics - induced hall conductivity @xmath44 then follows naturally . the main result of this work is that in the bound pair dynamics description , we obtain nonvanishing hall conductivity @xmath44 and ac conductivity @xmath45 near the kt transition . one of the interesting features in the hall conductivity is that strong positive peak and sign changes in @xmath46 are observed at suitable frequency region above the transition temperature , as well as above @xmath9 in temperature domain at fixed frequencies . on the other hand , @xmath47 is shown to have similar features as in ahns s results . we note that the shapes of two length - dependent response functions @xmath48 and @xmath49 , which corresponds respectively to the longitudinal and transverse response of bound pairs with separation @xmath50 to external perturbation with frequency @xmath18 , play a determining role on the behavior of @xmath44 and @xmath45 . we also discuss the contribution of free vortex motion and the resulting total conductivity tensor . the paper is organized as follows : in section ii , we generalize ahns s vortex dynamics in the chiral @xmath0-wave context . to describe the vortex - antivortex bound pair dynamics , the above - mentioned response functions @xmath48 and @xmath49 are derived from the fokker - planck equation governing the pair motion . together with the free vortex contribution , we arrive at a matrix dielectric function @xmath51 which describes the total screening effect under time - dependent perturbation . in section iii , we investigate the frequency and temperature dependence of the conductivity tensor @xmath52 constructed from @xmath51 , treating the bound pair and the free vortex contribution separately . the behavior of total conductivity @xmath4 is also discussed . a summary and remark are given in section iv . finally , we discuss analytic expression of @xmath3 , @xmath53 , and @xmath54 in opposite limit of the convective ratio @xmath55 and @xmath56 in appendix a. to construct a matrix dielectric function , we consider a neutral sf film system resembling that employed in ahns s dynamical theory , with film thickness of order the superconducting coherence length and linear dimension @xmath57 ( @xmath58 ) along @xmath25 ( @xmath26 ) direction . @xmath57 is very large and @xmath58 is large but finite . vortex core motion relative to the local superfluid velocity leads to a magnus force @xmath59 @xcite @xmath60 in the above equation , @xmath61 and @xmath62 are the velocity of the @xmath63-th vortex core and the local superfluid flow excluding the diverging self - field of the @xmath63-th vortex respectively . @xmath64 is the vorticity of the @xmath63-th vortex . @xmath65 is the bare areal superfluid mass density , which is defined as the three - dimensional superfluid density integrated across the film thickness . @xmath66 is the mass of the constituting particle , which is equal to the mass of a cooper pair . the sf film is driven by a vibrating substrate . a vortex core moving relatively to the substrate experiences a vorticity - dependent drag force @xmath67 @xcite @xmath68 where @xmath69 is the moving substrate velocity . @xmath70 and @xmath71 denote the vorticity - dependent drag coefficients originating from interactions with the substrate and with thermally excited quasiparticles and collective modes . here @xmath72 ( @xmath73 ) @xmath74 ( @xmath75 ) for @xmath76 or @xmath77 ( @xmath78 ) for @xmath79 . these quantities have been obtained for a three - dimensional clean @xmath42-wave sf / sc with isotropic fermi surface , @xcite and are related to the relaxation time @xmath80 for the caroli - degennes - matricon mode @xcite in the vortex core . due to broken @xmath5 symmetry , the relaxation time @xmath81 for the mode in the @xmath39vortex and @xmath38vortex core , and thus the values of their drag coefficients , are different , resulting in the vorticity - dependent drag force @xmath67 on the vortex core . these drag coefficients for 2d clean sf / scs with cylindrical fermi surface can be inferred from the three - dimensional result easily . when two forces balance @xmath82 , the @xmath63-th vortex velocity is expressed by @xmath83 where @xmath84 here @xmath85 with @xmath86 being the boltzmann constant . @xmath87 are fluctuating gaussian noise sources incorporated to bring the vortices to equilibrium . their components satisfy @xmath88 . from ref . , the local superfluid velocity @xmath89 is related to the spatial average superfluid velocity @xmath90 and the positions of individual vortices @xmath91 by @xmath92 . here @xmath93 is the green function satisfying @xmath94 and boundary condition @xmath95 on the edges . the function @xmath96 is localized in a region around @xmath97 with radius of the order of coherence length ( we could say that the function @xmath96 is a delta function in the coarse - grained scale ) . far away from the edges , @xmath98 . the spatial average of @xmath99 turns out to be zero and thus @xmath100 represents the spatial average of @xmath89 . finally , the time evolution of @xmath100 obeys @xmath101 reflecting the fact that the average superfluid velocity in the @xmath25 direction changes by @xmath102 when a vortex with @xmath103 moves across a strip with width @xmath104 . in a charged system , if we follow kopnin s description , @xcite the driving force on the vortex due to a transport current @xmath105 is a lorentz force @xmath106/c $ ] where @xmath107 and @xmath108 is the electron charge . this force is balanced by the force from environment @xmath109 . if we set @xmath110 in eqs . ( [ fd ] ) and ( [ one_vortex ] ) , the results derived in a neutral system can be carried over to a charged system by the translation @xmath111 and @xmath112 . we consider the polarization of a test vortex - antivortex pair whose constituting vortices interact via a screened logarithmic interaction . the pair is under the influence of an infinitesimal oscillating external field @xmath113 . the langevin equation for their relative coordinate @xmath114 can be obtained by subtracting eq . ( [ one_vortex ] ) from each other for opposite vorticity @xmath115 here @xmath116 and @xmath117 . @xmath118 is one - half the dimensionless potential energy of the pair and the gaussian noise now satisfies @xmath119 . the potential energy is given by @xmath120 in the above equation , the first term on the right hand side describes the logarithmic interaction screened by the kosterlitz dielectric constant @xmath121 . @xmath122 in the second term is related to the energy required to create a pair with separation @xmath123 . in the last term , @xmath124 has dimension of velocity and acts as the perturbation . in the integration limit @xmath123 is a length scale related to the size of a vortex core , and @xmath50 is the pair separation . we can see in eq . ( [ pair_eom ] ) that in addition to the conventional diffusive terms depending on @xmath125 , a convective term proportional to @xmath126 also enters the dynamical equation . while the strength of the former is proportional to the _ average _ of @xmath127 , that of the latter is related to the _ difference _ of @xmath128 between opposite vorticity . we emphasize here that such a convective pair motion is one of the special features for a system with unequal opposite vortices , and is thus absent in an @xmath42-wave sf / sc since @xmath129 in that case . given this nonzero @xmath126 , the pair polarization is tilted away from the direction of the force field @xmath130 , and has both longitudinal and transverse components even without applied magnetic field . the fokker - planck equation corresponding to eq . ( [ pair_eom ] ) is given by @xmath131 where @xmath132 is the density of pairs per unit area of separation . we take the time - independent state @xmath133 to be @xmath134 where @xmath135 . now we follow the standard procedure , @xcite letting @xmath136 and keeping terms to first order in @xmath137 . in frequency space , we obtain @xmath138 \nonumber\\ & & + 2 \overline c \nabla \cdot \left [ \left ( \frac{m^\ast}{\hbar } \hat z \times \delta \boldsymbol e \gamma_0 \right ) - \delta \gamma \hat z \times \nabla u_0 \right ] + 2 \overline d \nabla^2 \delta \gamma.\end{aligned}\ ] ] we employ the expansion @xmath139 , where @xmath140 is the angle measured from @xmath141 to @xmath142 in anti - clockwise sense . only @xmath143 with @xmath144 and @xmath145 are coupled to the external field . we define a convective ratio @xmath146 , which measures the relative strength between convection and diffusion , and introduce an ansatz @xmath147 @xmath148 . together with an approximation @xcite @xmath149 ( which is valid near the transition ) and a change of variable @xmath150 , the equations of @xmath151 in the ansatz corresponding to angular momentum @xmath152 are given by @xmath153 we are then able to write down the change of distribution function @xmath154 in terms of @xmath155 explicitly @xmath156 , \label{d_gamma}\end{aligned}\ ] ] where @xmath157 , \quad\quad \mathcal{g}_\perp(r ) = -\frac{i}{2}\left [ g_+ ( r,\omega ) - g_- ( r,\omega ) \right ] , \label{g_g}\end{aligned}\ ] ] and @xmath158 . in eq . ( [ d_gamma ] ) , @xmath159 takes the role of pair - size - dependent response longitudinal to the driving field @xmath137 and @xmath160 is the transverse response function . in the limit of vanishing @xmath43 , @xmath159 reduces to the @xmath42-wave sf / sc result @xmath161 while @xmath160 becomes identically zero . in eq . ( [ g_g ] ) , we find that @xmath162 depend on two quantities @xmath163 which describe pair motion with angular momentum @xmath164 respectively . ( left column ) and @xmath160 ( right column ) are plotted as a function of @xmath165 for @xmath166 ( a ) @xmath167 , ( b ) @xmath168 , and ( c ) @xmath169 respectively . the real and imaginary part are indicated by blue dashed lines and red solid lines respectively . @xmath159 is qualitatively similar to the @xmath42-wave sf / sc result . @xmath160 is a new feature with a peak in the real part and a dip - recouping shape in the imaginary part around @xmath170.,scaledwidth=80.0% ] we describe @xmath162 first . in fig . [ mathcalg ] we plot @xmath159 and @xmath160 as a function of @xmath165 for fixed frequency @xmath171 for ( a ) @xmath172 , ( b ) @xmath173 , ( c ) @xmath174 . blue dashed lines and red solid lines represent their real and imaginary parts respectively . in the left column of fig . [ mathcalg ] , the longitudinal response function @xmath159 is qualitatively similar to the @xmath42-wave sf / sc result @xmath175 even for finite @xmath43 . it has a step - function - like real part and a delta - function - like imaginary part concentrating near @xmath170 ( @xmath170 marks the position where @xmath176 changes sign ; see appendix a ) . the response function @xmath177 is a new feature in this model . in the right column of fig . [ mathcalg ] , @xmath178 $ ] has a peak structure . this means that neither smaller nor larger pair gives response in transverse direction . only pairs with characteristic pair size @xmath179 can give rise to the hall effect . besides , @xmath180 $ ] shows negative dip shape for @xmath181 , and then recoups when @xmath182 . it turns out in later section that such a dip - recouping antisymmetric shape in @xmath176 about @xmath170 plays a determining role in many features of the hall conductivity . these features become more significant when @xmath43 increases . ( left column ) and @xmath183 ( right column ) corresponding to angular momentum @xmath152 motion are plotted as a function of @xmath165 for @xmath166 ( a ) @xmath167 , ( b ) @xmath168 , and ( c ) @xmath169 respectively . the real and imaginary part are indicated by blue dashed lines and red solid lines respectively . asymmetry between the two columns becomes more prominent when @xmath43 increases.,scaledwidth=80.0% ] in eq . ( [ g_g ] ) , we can see that @xmath184 depends on the average of @xmath163 , i.e. , the average response with @xmath185 and @xmath186 angular momentum , and @xmath187 is related to the _ difference _ between them . this means that any asymmetry between @xmath152 motion gives rise to nonzero transverse response . in fig . [ capitalg ] we plot @xmath163 as a function of @xmath188 using the same set of @xmath18 and @xmath43 as in fig . [ mathcalg ] . again , blue dashed lines and red solid lines represent their real and imaginary parts respectively . we observe that asymmetry between @xmath152 motion grows with increasing @xmath43 . for small @xmath43 in fig . [ capitalg](a ) , the motion described by @xmath189 and @xmath190 is only slightly asymmetric , and they look like the familiar curve @xmath191 . when @xmath43 increases in fig . [ capitalg](b ) and ( c ) , the asymmetry between the left and right column becomes more and more significant . it is worth noting that , when we subtract from each other the real part of the functions for opposite @xmath192 , we can obtain the dip - recouping shape of @xmath193 . equation ( [ eqt_for_g ] ) can be solved exactly to give the length - dependent response functions , but here we can employ approximate solution to @xmath194 in a manner similar to ref . by neglecting @xmath195 and @xmath196 in eq . ( [ eqt_for_g ] ) . then @xmath197 become @xmath198 where @xmath199 is a factor to fit the exact curve . it is selected to be @xmath200 when @xmath201 and is of order @xmath202 for a wide range of @xmath43 . from eqs . ( [ app_g1 ] ) and ( [ app_g2 ] ) we can see that for a small convective ratio @xmath203 , there are poles given by @xmath204 . this means that the pair size with which a pair gives a strong response differs from the standard result @xmath205 by a length of order @xmath206 for motion with @xmath152 , creating the asymmetry demonstrated in fig . [ capitalg ] . for a large convective ratio @xmath207 , such pole is removed for @xmath145 therefore , a broad and flat curve is expected [ fig . [ capitalg](c ) right column ] . in this subsection , we introduce a susceptibility matrix @xmath208 for the bound pair polarization @xmath209 whose components are defined as @xmath210 they are so defined that the real parts of @xmath211 and @xmath212 are positive when @xmath213 . using eqs . ( [ d_gamma ] ) and ( [ chib ] ) , we arrive at the expression @xmath214 the infinitesimal external field redistributes the pair polarization by an amount @xmath215 according to @xmath216 . the definition in eq . ( [ chib ] ) means the external field tilts the polarization in clockwise direction when @xmath213 . it should also be noted that the integration are performed from @xmath217 to @xmath15 where @xmath15 is a coherence length with behavior @xmath218 for @xmath12 and @xmath219 $ ] for @xmath14 . @xmath220 is a non - universal positive constant of order unity . from eq . ( [ chibb ] ) it is now clear that while the longitudinal polarization @xmath221 takes the role of the function @xmath222 in ahns s theory , the transverse polarization @xmath223 has no simple analogy in @xmath42-wave sf / scs and is specific to systems with finite convective ratios . integrating the two pair - size - dependent response functions with the weight function @xmath224 gives the susceptibility matrix elements from the bound pair contribution . as for the free vortex contribution , from eq . ( [ one_vortex ] ) we can obtain the equation for polarization @xmath225 for plasma of free vortices under the spatial averaged driving field @xmath226 . the total free vortex density is given by @xmath227 with @xmath228 being the total number of free vortices and @xmath229 being the area of the film . in frequency space , we have @xmath230.\end{aligned}\ ] ] if we define the susceptibility for free vortices as @xmath231 , we have @xmath232 where @xmath233 . on the other hand , the free vortex density is related to the coherence length by @xmath234 where @xmath235 is a positive constant of order unity . @xcite together with the bound pair contribution , the total dielectric function can be obtained as @xmath236 and its inverse reads @xmath237 where @xmath238 $ ] and @xmath239 $ ] . we are now in position to present our results on the hall conductivity and power dissipation due to this dielectric function . it was discussed that the conductivity tensor @xmath4 in a charged system was related to the inverse dielectric function @xmath240 discussed above . @xcite we can understand the relation by considering the total current under the influence of a driver coil electric field @xmath241 . @xcite the vortex - modified total current @xmath242 is related to the field by @xmath243 where @xmath244 is the sheet kinetic inductance . this relation is generalized to our model with a matrix dielectric function @xmath245 . if we use the sign convention that the normal state electron hall conductivity with magnetic field pointing in @xmath22 direction is positive , @xmath52 and @xmath246 is related by @xmath247 in particular , we may write down @xmath248 , @xmath249 , and @xmath250 . among these expressions , the first two are associated with the real and imaginary part of the hall conductivity , while the last one is related to the power dissipation @xmath251 because @xmath252 . in the following subsections , we investigate the frequency and temperature dependence of @xmath253 , @xmath254 , and @xmath255 respectively . as a function of ( a ) @xmath256 and ( b ) @xmath257 . the other parameters used are @xmath172 , @xmath258 and @xmath259 . in ( a ) , above the transition temperature , @xmath260 has peak structures and sign reversals . @xmath261 has a sharp increase when frequency increases . in ( b ) , @xmath260 shows peak structure and sign reversal above @xmath262 and @xmath261 has a peak . the features broaden and move to higher temperature when frequency increases.,scaledwidth=80.0% ] we consider bound pair contribution to the hall conductivity first by ignoring the terms @xmath263 and @xmath264 in @xmath240 . plots of the negative real part ( blue circles ) and imaginary part ( red squares ) of @xmath265 versus ( a ) @xmath256 and ( b ) @xmath266 are presented in fig . [ hall ] . here , @xmath267 is the scaled frequency , and @xmath257 is the reduced temperature . in this and the following figures , we use the renormalization group flow equations @xmath268 and @xmath269,@xcite with initial conditions @xmath270/(1+t)$ ] and @xmath258 related to the bare interaction strength @xmath11 and chemical potential @xmath271 respectively . at transition temperature , @xmath272 which is obtained numerically . here , it suffices to notice that @xmath273 is given by @xmath274 where @xmath275 is the renormalized interaction strength , and @xmath276 is related to the the number of bound pairs found with pair size @xmath277 . also , we use the convective ratio @xmath172 in the plots for illustration purpose . we first focus on the discussion of the frequency dependence of @xmath265 at some fixed temperatures in fig . [ hall](a ) . in the low temperature phase @xmath278 , both @xmath261 and @xmath260 are positive and increase steadily with frequency . when temperature increases to the high temperature phase @xmath279 , @xmath260 shows a positive - valued peak followed by sign reversal at higher frequency indicated by the red horizontal double - headed bar . meanwhile , @xmath261 increases sharply around the peak of @xmath260 . these features move to higher frequency side when temperature increases further ( @xmath280 ) . for temperature dependence at fixed frequencies in fig . [ hall](b ) , the sign reversal and peak structure in @xmath260 are also observed when temperature varies above @xmath9 . meanwhile , @xmath261 shows a simple peak structure across @xmath9 . when frequency increases , the structure broadens and moves to higher temperature region . sign anomaly in hall conductivity has been observed in various superconducting systems such as high - temperature scs , @xcite and it is known that single vortex upstream " motion or change of sign of charge carriers could give rise to such phenomenon.@xcite here in our model the sign change in @xmath260 stems from the vortex - antivortex pair unbinding process . for a vortex - antivortex pair moving downstream " with same speed , total transverse electric field is canceled and no hall effect can be observed . now that the constituting vortices are not equivalent to each other , they can respond differently under the influence of transport current , and a net transverse electric field follows . under transport current @xmath281 with convective ratio @xmath213 . the black solid arrow is the transport current @xmath281 . the blue and red solid arrow represent the pair polarization of pair with size @xmath181 and @xmath182 respectively . the dashed arrows show the electric field generated by pair motion in the respective cases . pairs with size @xmath181 contribute to positive @xmath260 and vice verse . the net @xmath282 determines the overall sign of @xmath260 . , scaledwidth=80.0% ] from the dip - recouping shape of @xmath283 $ ] in fig . [ mathcalg ] , the dip ( recouping ) region with pair size @xmath181 ( @xmath182 ) contributes to positive ( negative ) @xmath260 , since @xmath284/r$ ] . the function @xmath285 is proportional to the number of pairs present in a ring with radius @xmath50 and width @xmath286 . therefore , the sign of @xmath260 depends on whether there are more pairs with pair size smaller or larger than @xmath170 . an interesting point to mention is that , upstream vortex motion is not required for negative @xmath260 ; instead , it is the asymmetric pair motion in opposite angular directions in fig . [ capitalg ] that brings about the hall anomaly . we interpret such result to be that the direction of electric field @xmath282 generated by a vortex pair is also pair - size - dependent . this interpretation is illustrated in fig . [ polarization ] . the pair with @xmath181 polarizes in a such a way that @xmath282 points in the direction giving positive hall signal and vice verse . the final sign of @xmath260 depends on the net @xmath282 caused by all the pairs . versus @xmath188 for @xmath287 , @xmath288 , and @xmath289 . the vertical arrow marks @xmath170 , the position where @xmath290 changes sign . @xmath291 . at @xmath292 , @xmath293 is indicated by the red double - headed arrow . three representative situations concerning the sign issue of @xmath260 are @xmath294 increasing at @xmath170 , @xmath294 decreasing at @xmath170 , and @xmath295.,scaledwidth=80.0% ] the weight function @xmath294 , which describes the probability of a pair having pair size @xmath277 , is plotted in fig . [ y2 ] at different temperatures . the other parameters used are @xmath296 , @xmath172 , and @xmath258 . for @xmath287 , @xmath294 is slightly decreasing at @xmath170 , so the dip - recouping shape in @xmath283 $ ] almost cancels each other in the integral in eq . ( [ chibb ] ) , giving a weak positive @xmath260 . if the temperature increases , for example @xmath297 , @xmath294 becomes increasing at @xmath170 . more pairs are present in the recouping region and the sign change of @xmath260 thus occurs . if the temperature increases further to @xmath298 , the coherence length @xmath15 characterizing the maximum pair size becomes comparable to @xmath170 . as a result , only the dip part is integrated because the recouping part lies outside of the integration limit in eq . ( [ chibb ] ) . a strong positive - valued peak then replaces the sign reversal . in summary , temperature controls the pair size distribution , which then determines the sign of @xmath260 . since the positive slope of @xmath294 at @xmath179 only occurs at the high temperature phase , the claim that the sign reversal of @xmath260 is related to kt transition is justified . to close this subsection , we discuss the hall conductivity in the static limit . for low temperature phase @xmath278 , hall conductivity diverges in zero - frequency limit . log - log plots of @xmath260 and @xmath261 versus @xmath299 using linear fitting show that @xmath300 and @xmath301 with both @xmath302 and @xmath303 greater than zero but smaller than unity for the temperature range @xmath304 down to @xmath305 and frequency range @xmath306 to @xmath307 . this mean that @xmath308 decreases slower than @xmath18 when frequency approaches zero , and as a result @xmath309 diverges at @xmath310 . as a function of ( a ) @xmath256 and ( b ) @xmath257 . @xmath172 and @xmath258 . in ( a ) , @xmath311 increases with frequency when @xmath278 and is suppressed at small frequency when @xmath312 . in ( b ) , @xmath311 has peak structure around @xmath9 . the peak broadens and moves to higher temperature when frequency increases.,scaledwidth=80.0% ] in this subsection , we discuss the frequency and temperature dependence of @xmath3 . again , we consider bound pair contribution here . in fig . [ diss ] , plots of @xmath311 versus ( a ) @xmath256 and ( b ) @xmath266 are presented . the other parameters used are the same as those in fig . [ hall ] . in fig . [ diss](a ) , @xmath311 increases steadily with frequency at low temperature phase @xmath278 . when @xmath312 , it is suppressed at small frequency . in fig . [ diss](b ) , @xmath311 first increases with temperature , and is suppressed above @xmath9 , resulting in a peak structure . at higher frequency , the peak is broadened and shifts to higher temperature . these features can be understood under the standard ahns theory : above @xmath9 , the coherence length @xmath15 becomes finite , and the integration limits can not cover the peak structure in @xmath313 at large @xmath188 . thus , @xmath311 is suppressed at low frequency for @xmath312 in fig . [ diss](a ) , and above some temperatures in fig . [ diss](b ) . indeed , it is not surprising that the result behaves similarly to its @xmath42-wave counterpart when we notice that the shape of @xmath314 is qualitatively similar to that of @xmath315 even for finite @xmath43 . in zero - frequency limit in the low temperature phase , @xmath316 decreases slower than @xmath18 when frequency approaches zero and thus the @xmath317 diverges . this is found in log - log plots of @xmath311 versus @xmath18 that @xmath318 with @xmath319 but smaller than unity using linear fitting from @xmath304 to @xmath305 and @xmath320 to @xmath307 . however , we can not interpret it to be the divergence of power dissipation because the dissipation also depends on the magnitude of the electric field in the superconducting bulk which is supposed to be vanishingly small in the static limit . as a function of ( a ) @xmath256 and ( b ) @xmath266 . the red squares and the blue circles represent the real part and imaginary part of @xmath321 respectively . @xmath322 and @xmath323 for illustration . the other parameters used are the same as those in fig . [ hall].,scaledwidth=80.0% ] as a function of ( a ) @xmath256 and ( b ) @xmath266 . the other parameters used are the same as those in fig . [ totalhall].,scaledwidth=80.0% ] having discussed the result due to bound pair dynamics , we study the total contribution from both bound pair and free vortex dynamics in this subsection . in order to discern the contribution of bound pair and free vortex dynamics to the total hall conductivity and power dissipation , we plot the total conductivity tensor @xmath321 and @xmath324 in figs . [ totalhall ] and [ totaldiss ] as a function of ( a ) @xmath256 and ( b ) @xmath266 , as well as the @xmath325 , @xmath254 , and @xmath253 due purely to the free vortex motion in fig . [ figfree ] . we can compare these figures with those which take only the bound pair dynamics into account ( figs . [ hall ] and [ diss ] ) . for the sake of illustration , we use @xmath322 and @xmath323 for the weight of free vortex contribution . for the frequency dependence in figs . [ totalhall ] and [ totaldiss ] at temperature @xmath279 , we can see free vortex signal emerging at small frequency region . if the temperature increases further at @xmath280 , the free vortex signal can merge with and outweigh the bound pair signal [ compared with figs . [ hall ] and [ figfree](a ) @xmath280 ] . for @xmath278 ( not shown ) , since there is no free vortex contribution , we expect the curves are the same as those in low temperature phase in figs . [ hall ] and [ diss](a ) . as for the temperature dependence in figs . [ totalhall ] and [ totaldiss](b ) , we can see for @xmath320 the free vortex signal appears at temperature very close to the bound pair signal and they almost merge into each other . when the frequency increases ( @xmath327 ) , this extra strong signal moves to higher temperature and becomes distinguishable from the bound pair signal . the magnitude of the free vortex signal can be seen in fig . [ figfree ] . as discussed in the previous two subsections , @xmath328 , @xmath329 , and @xmath330 diverge when @xmath310 . however , it is only true in low temperature phase . in the case where @xmath312 , free vortex contribution dominates in the static limit @xmath331 . with such condition , we have @xmath332 which is purely imaginary . from eq . ( [ free ] ) , the diagonal part of @xmath333 and the off - diagonal part of @xmath334 are given by @xmath335 $ ] and @xmath336 $ ] respectively , and therefore both of them are proportional to @xmath337 . this shows that @xmath338 is purely real , constant in frequency , and diverges when @xmath339 . as a result , in the high temperature phase , both the hall conductivity and dissipation are protected from divergence in the zero - frequency limit but have strong signal when approaching @xmath9 from above . , @xmath254 , and @xmath253 as a function of ( a ) @xmath256 and ( b ) @xmath266 . the green triangles represent @xmath325 . the blue circles and the red squares represent the negative real part and imaginary part of @xmath265 respectively . the insets show the full plot range of @xmath325 . the other parameters used are the same as those in fig . [ totalhall].,scaledwidth=80.0% ] finally , fig . [ figfree ] shows @xmath325 , @xmath254 , and @xmath253 as a function of ( a ) @xmath256 and ( b ) @xmath266 including only the free vortex but not the bound pair contribution . the frequency and temperature range are chosen to match those in figs . [ totalhall ] and [ totaldiss ] . by eq . ( [ free ] ) , we see that important features in the figure appear around the crossover from @xmath340 to the opposite limit @xmath341 . in fig . [ figfree](a ) on the right panel , when observing from small frequency region @xmath340 , we see that all three quantities increase with frequency . when frequency increases further , @xmath253 changes sign and then all the quantities decrease in magnitude with frequency , which means that it starts to be inefficient for the free vortex to respond to the perturbation when frequency is high . in fig . [ figfree](b ) , similar features can be observed in the temperature domain . besides , we can see that the main features in the curves emerge at a temperature closer to @xmath9 for small frequency ( left panel ) than for high frequency ( right panel ) . although the sign change in @xmath253 can also be observed in free vortex picture , it does not arise from the response function @xmath342 mentioned before . instead , it is attributed to the pole structure @xmath343 . in this work , we have generalized ahns s vortex dynamics in the chiral @xmath0-wave superconducting state . the behavior of the conductivity tensor near kt transition is investigated . we show that the hall conductivity can be nonzero arising from the vortex - antivortex pair unbinding process or from free vortex motion even in the absence of magnetic field . power dissipation is also predicted in this dynamical picture . in low temperature phase , both the @xmath44 and @xmath344 diverge in the static limit . but in high temperature phase , the contribution from free vortex motion gives finite results in the static limit . we can distinguish the bound pair and free vortex contribution to the total conductivity tensor by comparing the figures from different contribution . sign reversal and strong positive peak in @xmath46 are illustrated in some suitable frequency close to transition , as well as above the transition temperature at fixed frequencies . on the other hand , the @xmath47 behaves in a fashion similar to that of the @xmath42-wave case . in bound pair description , the induced hall conductivity is strongly influenced by the dip - recouping shape of transverse response function @xmath290 , which stems from the asymmetric angular response of pairs to the driving field . the convective term in the fokker - planck equation , which originates from the broken @xmath5 symmetry nature of a chiral @xmath0-wave sc , is essential to this asymmetry . all these results depend solely on the convective ratio @xmath43 , and are valid even without applied magnetic field . in our work , temperature dependence of the drag coefficients is omitted . theoretical calculation of the drag coefficients has been performed microscopically for scs @xcite and for sf @xmath19he ( refs . ) at temperatures considerably lower than the superconducting transition temperature by the use of quasiclassical theory . generalization of such calculation in chiral @xmath0-wave pairing state around @xmath9 will be useful to estimate the strength of the convective ratio . indeed , if we use the formulas of drag coefficients which only take into account the caroli - degennes - matricon mode at very low temperature @xmath345 $ ] and @xmath346 $ ] , @xcite we can arrive at the result @xmath347 and @xmath348 , which results in @xmath201 . however , we believe that the effect of temperature and delocalized excitations can contribute to the convective ratio . in this appendix , we apply certain approximation to obtain analytic expression of @xmath311 , @xmath260 , and @xmath261 in the bound pair dynamics picture . we first obtain analytic expression for susceptibility matrix elements @xmath211 and @xmath212 by employing an approximate solution @xmath349 and taking @xmath350 to be a constant function of @xmath50 . we rewrite eq . ( [ chibb ] ) in the form @xmath351 , \label{chi_para0 } \\ \chi_\text{b}^{\perp } & = & -i\int_{a_0}^{\xi_+ } dr \frac{d \widetilde \epsilon}{dr } \left [ \left ( \frac { g_+ - g_-}{2 } \right ) + ix_0 \left ( \frac { g_+ + g_-}{2 } \right ) \right ] , \label{chi_perp0}\end{aligned}\ ] ] and further approximate the two functions appearing in the above integrands by @xmath352 \nonumber\\ & & + i \left[r i_2 \delta(r - r_2 ) - x_0 r i_1 \delta(r - r_1)\right ] , \\ \frac { g_+ - g_-}{2 } & \approx & r i_1 \delta(r - r_1 ) + i r i_3 \delta(r - r_3 ) \nonumber\\ & & -ix_0 \left [ \left ( \frac { 1 } { 1+x_0 ^ 2}\right ) \theta ( r_4 - r ) + \text{sgn}(x_0 ) r i_1 \delta(r - r_1 ) \right],\end{aligned}\ ] ] where @xmath353 ^ 2 } - \frac{1 } { 1 + [ \omega r^2 / ( \lambda \overline d)+ x_0]^2 } \right ) \nonumber\\ & = & \frac { x_0 \pi } { 4 + 4 x_0 ^ 2 } , \\ i_2 & \equiv & \frac{1}{2 } \int_0^\infty dr \frac{1}{r } \left ( \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)- x_0]^2 } + \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)+ x_0]^2 } \right ) \nonumber\\ & = & \frac { \pi } { 4 } , \\ i_3 & \equiv & \frac{1}{2 } \int_0^\infty dr \frac{1}{r } \left ( \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)- x_0]^2 } - \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)+ x_0]^2 } \right ) \nonumber\\ & = & \frac{1}{4 } \left [ \cot^{-1 } \frac{2 x_0}{-1+x_0 ^ 2 } + \frac{\pi}{2 } \text{sgn}(x_0 ) \right].\end{aligned}\ ] ] in the above expression , @xmath354 , @xmath355 , @xmath356 , and @xmath357 are some characteristic lengths defined as follows : @xmath354 , @xmath355 , and @xmath356 are the peak position of the integrand of @xmath358 , @xmath359 , and @xmath360 respectively . @xmath357 is chosen such that @xmath361 ^ 2 \ } = 1/ [ 2(1 + x_0 ^ 2 ) ] $ ] . they are given respectively by @xmath362 , @xmath363 , and @xmath364 . one more characteristic length worth mentioning is the pair size @xmath365 at which @xmath290 changes its sign . within the approximation described above , we find that @xmath366 . then we can obtain the quantity @xmath367 mentioned in the main text . finally , using @xmath368 , we obtain analytic expression for the components of the susceptibility matrix for @xmath369 @xmath370 where some shorthand notations are introduced : @xmath371 and @xmath372 . from the expression above , we notice that @xmath373 ( @xmath374 ) is ( anti- ) symmetric in @xmath43 as it should be . we can also identify the contribution from the dip and recouping part of @xmath290 to be @xmath375 and @xmath376 in the imaginary part of eq . ( [ chi_perp ] ) . [ [ analytic - expression - of - im - epsilon_parallel-1-imepsilon_perp-1-and - re - epsilon_perp-1 ] ] analytic expression of @xmath311 , @xmath260 , and @xmath261 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ in this subsection , @xmath311 , @xmath260 , and @xmath261 for nonzero @xmath43 are calculated in two limiting cases @xmath377 and @xmath378 . for @xmath377 , we take @xmath379 , @xmath380 , and @xmath381 at the opposite limit @xmath382 , we take @xmath383 , @xmath384 , and @xmath385 besides , we neglect the @xmath43 dependence of @xmath386 and @xmath387 in the expansion . such approximation is valid for @xmath388 and all temperature near @xmath9 because @xmath389 is almost independent of @xmath43 . it is also valid for @xmath382 in low temperature phase because the renormalization almost go to completion at large @xmath43 and @xmath387 can be treated as independent of @xmath390 . but it is not valid for @xmath382 in high temperature phase for two reasons : first , @xmath387 is increasing at large @xmath43 and second , @xmath389 are very likely to be greater than @xmath15 so that the integrations in eqs . ( [ chi_para0 ] ) and ( [ chi_perp0 ] ) give the trivial results @xmath391 and @xmath392 . expanding using small @xmath43 , the imaginary part of @xmath393 is approximately given by @xmath394 where we have used @xmath395 in the last step . the presence of the second term acts as a suppression of dissipation when @xmath43 is switched on . if we neglect the correction term proportional to @xmath396 , the expression reduces to the @xmath42-wave sf / scs result . in the opposite limit @xmath378 , the imaginary part of @xmath393 to the leading order in the smallness of @xmath397 is given by @xmath398 assuming @xmath399 . surprisingly , the leading order of @xmath311 at large @xmath43 has the same form as its small @xmath43 counterpart . the only difference is that the characteristic length is changed from @xmath400 to @xmath401 . this can be understood if we recall that the shape of @xmath402 has no qualitative change for finite @xmath43 . following similar procedure we can obtain analytic expression for @xmath403 and @xmath404 . for small @xmath43 , we obtain @xmath403 to the first order in @xmath43 @xmath405 } { ( \widetilde \epsilon_4 ^ 2 + k_0 ^ 2 \pi^8 y_4 ^ 4)^2 } \nonumber\\ & \approx & x_0 \widetilde \epsilon_4^{-2 } k_0 \pi^4 \left [ \left(y_1 ^ 2 - y_4 ^ 2\right ) + 4 \widetilde \epsilon_4^{-1 } k_0 \pi^3 y_4 ^ 4 \right].\end{aligned}\ ] ] we have used the condition @xmath406 in the last line . from the expression , the behavior depends on whether @xmath407 or @xmath408 dominates . from numerical results , @xmath407 dominates at a wide range of temperature in the low temperature phase . only when approaching the transition temperature from below and away from the vicinity @xmath409 , the @xmath410 becomes more significant . on the other hand , @xmath411 we have used @xmath412 in the second line . for large @xmath43 , we assume @xmath413 and obtain to the leading order @xmath414 and @xmath415 it has to be mentioned that , although we have used the condition @xmath406 extensively , it is only valid in the temperature range @xmath416 . above such temperature , numerical results show that @xmath417 can be greater than @xmath386 at small frequency region .

we discuss hall effect and power dissipation in chiral @xmath0-wave superconductors near kosterlitz - thouless transition in the absence of applied magnetic field . in bound pair dynamics picture , nonzero hall conductivity emerges when vortex - antivortex bound pair polarization has a component transverse to the direction of external perturbation . such effect arises from the broken time reversal symmetry nature of a chiral @xmath0-wave superconducting state and does not require an applied magnetic field . a frequency - dependent matrix dielectric function @xmath1 is derived to describe the screening effect due to the pair polarization . quantities related to the hall conductivity and power dissipation , denoted as @xmath2 and @xmath3 , are investigated in frequency and temperature domain . the imaginary part of the former can show peak structure and sign reversal as a function of frequency close to transition temperature , as well as in the temperature domain at various fixed frequencies . the latter shows peak structure near transition temperature . these features are attributed to pair - size - dependent longitudinal and transverse response function of bound pairs . consequences due to free vortex dynamics and the resulting total conductivity tensor @xmath4 are also discussed .

arxiv

the measurement of the rotation curves ( rcs ) of disk galaxies is a powerful tool to investigate the nature of dark matter ( dm ) , including its content relative to the baryonic components and their distributions . in particular , dwarf galaxies are good candidates to reach this aim as their kinematics are generally dominated by the dark component , down to small galactocentric radii @xcite . this leads to a reliable measurement of the dynamical contribution of the dm to the rc and hence of its density profile . therefore , a dwarf galaxy like the orion dwarf provides us with an important test as to whether dm density profiles arising in @xmath3 cold dark matter ( @xmath3cdm ) numerical simulations @xcite are compatible with those detected in actual dm halos around galaxies . let us comment that nfw profile arises from pure n - body dm simulations . it is well known that , as effect of the baryonic infall in the cosmological dm halos and of the subsequest process of stellar disk formation , shallower profiles of the dm halo may arise ( see @xcite ) . recent studies of the rcs of dwarf galaxies have tested the nfw scenario . it is now clear that kinematic data are better fitted by a dm halo with a constant density core ( e.g. @xcite ) , than by one that is centrally peaked . one specific example is ddo 47 , whose velocity field is clearly best fitted if the dm halo is cored ; moreover , its ( small ) detected non - circular motions can not account for the discrepancy between data and the nfw predictions @xcite . + the present investigation examines the dm content of the orion dwarf galaxy . this nearby system harbors an extended disk , and thus provides us with an important test of the above paradigm . as we show below , the orion dwarf is one of the few known galaxies whose kinematics _ unambiguously _ point towards a cored profile . this system is thus critically important for investigating the nature of the dm particle and of the evolution of dm halos . + mond accounts for the evidence that rcs of spiral galaxies are inconsistent with the corresponding distribution of the luminous matter @xcite . rather than postulating the existence of a dark halo made by massive collisionless elementary particles , this scenario advocates that the gravitational force at low accelerations leaves the standard newtonian regime to enter a very different one . historically mond has generally been successful in reproducing the rcs of spiral galaxies with only the ( observed ) luminous matter ( e.g. @xcite ) . however , cases of tension between data and the mond formalism do exist @xcite . + it is important to stress that in order to derive the dm density profile or to test the mond formalism , we must know the distribution of the ordinary baryonic components , as well as have reliable measurements of the gas kinematics . for the orion dwarf , 21-cm surface brightness and kinematics have recently been published @xcite : their analysis provides a high quality , high resolution rc , that , in addition , can be easily corrected for asymmetric drift and tested for non - circular motions . this galaxy is a very useful laboratory in that a simple inspection of the rc ensures us that it shows a large mass discrepancy at all radii . moreover , the baryonic components are efficiently modeled ( i.e. , no stellar bulge is evident and the stellar disk shows a well - behaved exponential profile , see @xcite ) . the distance to the galaxy , which is critical for an unambiguous test of mond @xcite , is estimated to be [email protected] mpc @xcite . it is important to stress that the distance of the orion dwarf remains a significant source of uncertainty . @xcite estimate the distance using the brightest stars method . the intrinsic uncertainty in this technique may allow a distance ambiguity much larger than the formal errors estimated by @xcite , because in their work this method yields a scatter as large as @xmath5 in distance . finally , the system s inclination ( 47@xmath1 ) is kinematically measured ( see section ( [ 3.1 ] ) ) and is high enough to not affect the estimate of the circular velocity . the properties described above make the orion dwarf galaxy an attractive candidate to determine the underlying gravitational potential of the galaxy . this paper is organized as follows . in sec . 2 we present the stellar surface photometry . in sec . 3 , the surface density and kinematics data are presented and discussed ; we also provide the analysis of possible non - circular motions of the neutral gas . in sec . 4 we model the rc in the stellar disk using a cored / cusped halo framework . in sec . 5 we test the orion kinematics against the mond formalism . our conclusions are given in sec . following the discussion in @xcite , the underlying stellar mass in the orion dwarf is estimated using the near - infrared ( ir ) photometry ( j and bands ) presented by @xcite . those authors find ( j@xmath6 ) @xmath0 @xmath20.80 and a total magnitude of @xmath210.90 . when comparing to models ( see below ) we assume that the color difference between k and is negligible ; further , we assume [email protected] @xcite . accounting for extinction , the total k - band luminosity of the orion dwarf is @xmath83.5@xmath910@xmath10 l@xmath11 . the mass of the stellar component was estimated by @xcite to be ( [email protected])@xmath910@xmath12 . the stellar surface brightness profile is well fitted by an exponential thin disk , with a scale length of @xmath13= 25 @xmath4 1 ( equivalent to 1.33 @xmath4 0.05 kpc at the adopted distance ) . moreover , there are no departures from an exponential profile that would be indicative of a prominent central bulge . spectral line imaging was acquired with the _ very large array _ and presented in @xcite . we refer the reader to that work for a full discussion of the data handling , and we summarize salient details here . the final data cubes have a circular beam size of 20 , with a 3@xmath14 column density sensitivity of n@xmath15= 1.5 @xmath910@xmath16@xmath17 . the first three moment maps ( i.e. the integrated intensity , the velocity field , and the velocity dispersion ) are shown in figure ( [ figcap3 ] ) . + the neutral gas disk of the orion dwarf shows rich morphological and kinematic structure at this physical resolution . the outer disk contains tenuous gas , but column densities rise above the 5@xmath910@xmath18 @xmath17 level at intermediate radii . there is plentiful high - column density ( @xmath1910@xmath20 @xmath17 ) throughout the disk . the more or less parallel iso - velocity contours at inner radii are indicative of linear rotation ( although almost certainly not solid body ) and the curving of the outer contours suggests that the outer rotation curve has a fairly constant velocity . the outer disk contours show no evidence for a decrease in rotational velocity at large radii . in the central regions of the disk , however , some `` holes '' or `` depressions '' manifest a pronounced kink in these contours ( consider the contours at 370@xmath420 ) . the intensity weighted velocity dispersion averages to @xmath87 - 8 throughout the disk , although the innermost regions show dispersions above 10 . + the total flux integral , proportional to the disk mass , was found to be [email protected] jykm s@xmath21 , a value somewhat lower than the single - dish flux measure of [email protected] jykm s@xmath21 by @xcite ; the difference may arise from the lack of short interferometric spacings that provide sensitivity to diffuse structure . the total mass is found to be m@xmath15 @xmath0 ( [email protected])@xmath910@xmath22 . after applying the usual 35% correction for helium and molecular material , we adopt m@xmath23 @xmath0 ( [email protected])@xmath910@xmath22 as the total gas mass . + in figure ( [ reswri ] ) we plot the 10 /20 resolution surface density , throughout the gas disk . a simple fit ( valid out to the last measured point and for the scope of this work ) yields : _ m_/pc^2 , where @xmath24 is in kpc . the related fitting uncertainty on @xmath25 is about @xmath26 . figure ( [ reswri ] ) shows that the surface density rises from the center of the galaxy , reaches a maximum , and then declines exponentially . at the last measured point , i.e. out to @xmath8 7 kpc , the profile has almost ( though not completely ) reached the edge of the disk and rapidly converges to zero . note that , in newtonian gravity , the outer gaseous disk contributes in a negligible way to the galaxy total gravitational potential . the channel maps of the orion dwarf provide evidence of well - ordered rotation throughout the disk ( see cannon et al . the intensity - weighted - mean velocity field ( figure ( [ figcap3]b ) ) exhibits symmetric structure in the outer disk . twisted iso - velocity contours at inner radii coincide with the holes near the centre of the disk . the disk is therefore dominated by circular motion . the rc of the galaxy was derived by fitting a tilted ring model to the intensity - weighted - mean velocity field using the gipsy task rotcur . the routine carries out a least - squares fit to @xmath27 , the line of sight - velocity . to derive the best - fitting model , an iterative approach was adopted in which the various combinations of the parameters were fitted . the final rc was extract by fixing all other parameters . the receding and approaching sides of the galaxy were fitted separately . the best fitting parameters are @xmath28 , @xmath29 , @xmath30 km / s , and ( @xmath31,@xmath32 ) = ( 05:45:01.66 , 05:03:55.2 ) for the dynamical centre . we have realized that the inclination is not dependent on the radius , and the fit is shown in figure ( [ inclination ] ) . its weighted value is @xmath33 . notice that because the errors reported by gipsy / rotcur include only errors on the fits and systematics are not included , the @xmath34 error estimate comes from attempting the rotcur fits in various orders ( e.g. , holding each variable fixed in turn ) . + the resulting rc is shown in figure ( [ asymm ] ) . notice that in this object the disk inclination is determined kinematically and therefore it is quite accurate . no result of this paper changes by adopting different values of @xmath35 , inside the quoted errorbar . the second - order moment map for the galaxy is shown in figure ( [ figcap3]c ) . throughout most of the disk , the velocity dispersion is roughly constant at @xmath14 @xmath36 7@xmath42 km / s , with a more complex behaviour near the galaxy centre and at the outermost radii . this velocity dispersion estimate allows us to derive the asymmetric drift correction to the rc yielded by the tilted ring model . the observed rotation velocity , @xmath37 , is related to the circular velocity @xmath38 via v^2_c(r)= v^2_rot(r ) - ^2(r ) . [ circ ] from an examination of figure ( [ asymm ] ) it is clear that the @xmath37 and @xmath38 profiles differ by less than 1@xmath39 . throughout this paper , we use the latter for the purposes of mass modelling . we notice that in very small dwarfs this correction is not negligible ( @xmath40 ) and it introduces an uncertainty in the analysis , e.g. @xcite . in summary , the orion dwarf rc has a spatial resolution of 0.26 kpc ( i.e. 0.2 @xmath13 ) , and extends out to 5.1 r@xmath41 . the uncertainties on the rc are few km / s and the error on the rc slope @xmath42 . is the circular velocity given by eq . ( [ circ ] ) a proper estimate of the gravitational field ? to further investigate the presence of non - circular motions within the disk that jeopardize the kinematics , we carried out a harmonic decomposition of the intensity - weighted velocity field to search for any significant non - circular components . this test is necessary in that the undetected presence of non - circular motions can lead to incorrect parametrization of the total mass distribution . + following @xcite , the line - of - sight velocities from the velocity field are decomposed into harmonic components up to order @xmath43 according to @xmath44 where @xmath45 is the systemic velocity , @xmath46 and @xmath47 are the magnitudes of the harmonic components , @xmath48 the harmonic number , and @xmath49 the azimuthal angle in the plane of the galaxy . the gipsy task reswri was used to carry out the decomposition by fitting a purely circular model to the velocity field , subtracting it from the data , and then determining from the residual the magnitudes of the non - circular components . the tilted ring model fitted by reswri had its kinematic centre fixed to that of the purely circular tilted ring model used to derive the rc above . the position angles and inclinations were fixed to constant values of 20 and @xmath50 , respectively . the parameters of the best - fitting model are shown in figure ( [ reswri4 ] ) . adjacent points are separated by a beam width in order to ensure that they are largely independent of one another . we argue that because the standard tilted ring model has fewer free parameters than the model incorporating the higher order fourier components , it is not as essential to space the points on the rotation curve by a full beam width . then in this model only @xmath51 points are considered instead of the @xmath52 points used in fitting the rc . + at inner radii the inferred non - circular motions are not negligible , but this is almost certainly due to the fact that the distribution over this portion of the disk is irregular , being dominated by the large central under - densities . the harmonic components of the outer disk are , instead , reliable and demonstrate the gas flow to be dominated by circular kinematics . the circular velocity so obtained well matches that found by means of the tilted ring model presented above . the amplitudes of @xmath53 and @xmath54 are too small to hide a cusp inside an apparently solid body rc ( as suggested by @xcite ) . these results provide further decisive support for the use of @xmath37 of the orion dwarf as a tracer of its mass distribution . we model the orion dwarf as consisting of two `` luminous '' components , namely the stellar and the gaseous disks , embedded in a dark halo . the stellar component is modelled as an exponential thin disk @xcite with a scale length of 1.33 kpc . any bulge component is assumed to be negligible in terms of mass . the dynamical contribution of the gas to the observed rc is derived from the total intensity map . a scaling factor of 1.33 is incorporated to account for the presence of helium and other elements . for the dark halo we consider two different parametrizations of the mass distribution : an nfw profile @xcite and the cored profile of the halo universal rotation curve ( urch ) @xcite . it is well known that the nfw profile is one outcome in numerical simulations of cold dark matter structure formation , whereas the cored profile ( an empirical result ) , by design , fits the broad range of rc shapes of spiral galaxies . the rc is modelled as the quadrature sum of the rcs of the individual mass components : @xmath55 for the cored halo parametrization we adopt the urch profile : v^2_urch(r)&=&6.4 , where the disk mass , the core radius @xmath56 and the central halo density @xmath57 are free parameters . it is evident that this model yields a total rc that fits the data extremely well ( see figure ( [ urcnfw ] ) left panel ) , with best - fitting parameters of @xmath58 kpc , m@xmath59 m@xmath60 and @xmath61g/@xmath62 . more accurate statistics is not necessary ; the mass model predicts all the @xmath63 data points within their observational uncertainty . notice that the derived value of the disk mass agrees with the photometric estimate discussed above . the corresponding virial mass and radius of the dm halo are m@xmath64 m@xmath60 ( see eq . 10 in @xcite ) and r@xmath65 kpc @xcite , respectively . we note that the orion dwarf has a mass 20 times smaller than that of the milky way , with the dm halo dominating the gravitational potential at all galactocentric radii . the baryonic fraction is @xmath66 , while the gas fraction is @xmath67 . the rc for the nfw dark matter profile is v^2_nfw(r)=v^2_vir , where @xmath68 , @xmath69^{-1}$ ] and @xmath70 is the concentration parameter ( see @xcite ) . we fitted the rc of the orion dwarf by adjusting m@xmath71 and m@xmath72 . the resulting best - fit values are m@xmath73 m@xmath60 and m@xmath74 m@xmath60 , but since @xmath75 , i.e. the fit is unsuccessful , the best - fit values of the free parameters and those of their fitting uncertainties do not have a clear physical meaning . we plot the results in the right panel of figure ( [ urcnfw ] ) . the nfw model , at galactocentric radii r@xmath762 kpc , overestimates the observed circular velocity ( see figure ( [ urcnfw ] ) right panel ) . an alternative to newtonian gravity was proposed by @xcite to explain the phenomenon of mass discrepancy in galaxies . it was suggested that the true acceleration @xmath77 of a test particle , at low accelerations , is different from the standard newtonian acceleration , @xmath78 : a= , where @xmath79 is an interpolation function and @xmath80 cm s@xmath81 is the critical acceleration at which the transition occurs ( see @xcite ) . for this we adopt the following form of the interpolation function ( see @xcite ) : ( a / a_0)=. in this framework the circular velocity profile can be expressed as a function of @xmath82 and of the standard newtonian contribution of the baryons to the rc , v@xmath83=(v@xmath84+v@xmath85)@xmath86 , obtaining for it v^2_mond = v^2_bar(r ) ( ) , [ mond ] ( see @xcite ) ( [ mond ] ) shows that in the mond framework the resulting rc is similar to the no - dm standard newtonian one , with an additional term that works to mimic and substitute for the dm component @xcite . no result of this paper changes by adopting the `` standard '' mond interpolation function ( see @xcite ) . the best - fitting mond mass model is shown in figure ( [ mond1 ] ) . the model total rc ( cyan line ) completely fails to match the observations . we fix the stellar mass m@xmath41 at m@xmath87 . if we let the disk mass becomes higher , covering the mass range estimated in @xcite , the fit is not even able to reproduce the rc at inner radii . note that in the mond formalism , the distance of the galaxy and the amount of gaseous mass are both crucial in deriving the model rc . to quantify the discrepancy of these observations with the mond formalism , note that only if the orion dwarf were 1.9 times more distant than the current estimate we would obtain a satisfactory fit to the rc ( see figure ( [ mond2 ] ) ) . the orion dwarf galaxy is representative of a population of dwarfs with a steep inner rc that gently flattens at the edge of the gas disk . the observed kinematics imply the presence of large amounts of dm also in the central regions . we have used new observations of the orion dwarf to analize its kinematics and derive the mass model . the derived rc is very steep and it is dominated by dm at nearly all galactocentric radii . baryons are unable to account for the observed kinematics and are only a minor mass component at all galactic radii . we have used various mass modeling approaches in this work . using the nfw halo , we find that this model fails to match the observed kinematics ( as occurs in other similar dwarfs ) . we show that non - circular motions can not resolve this discrepancy . then we modelled the galaxy by assuming the urch parametrization of the dm halo . we found that this cored distribution fits very well the observed kinematics . orion is a typical dwarf showing a cored profile of the dm density and the well - known inability of dm halo cuspy profiles to reproduce the observed kinematics . finally , we find that the mond model is discrepant with the data if we adopt the literature galaxy distance and gas mass . the kinematic data can be reproduced in the mond formalism if we allow for significant adjustments of the distance and/or value of the gas mass . let us point out that the present interferometric observations may miss some of the objects flux , although this may be limited in that the cubes do not have significant negative bowls . obviously , for bigger values of the mass , the distance at which the baryon components would well fit the data will also somewhat decrease . + it is worth stressing that there is a galaxy distance ( albeit presently not - favoured ) for which mond would strike an extraordinary success in reproducing the observed kinematics of the orion dwarf . the orion dwarf has a favorable inclination , very regular gas kinematics , a small asymmetric drift correction , a well - understood baryonic matter distribution , and a large discrepancy between luminous and dynamical mass . all of these characteristics make this system a decisive benchmark for the mond formalism and a promising target for further detailed studies . of particular value would be a direct measurement of the distance ( for example , infrared observations with the hubble space telescope would allow a direct distance measurement via the magnitude of the tip of the red giant branch ) . the authors would like to thank the referee , gianfranco gentile , for his very fruitful comments that have increased the level of presentation of the paper .

dwarf galaxies are good candidates to investigate the nature of dark matter , because their kinematics are dominated by this component down to small galactocentric radii . we present here the results of detailed kinematic analysis and mass modelling of the orion dwarf galaxy , for which we derive a high quality and high resolution rotation curve that contains negligible non - circular motions and we correct it for the asymmetric drift . moreover , we leverage the proximity ( d @xmath0 5.4 kpc ) and convenient inclination ( 47@xmath1 ) to produce reliable mass models of this system . we find that the universal rotation curve mass model ( freeman disk @xmath2 burkert halo @xmath2 gas disk ) fits the observational data accurately . in contrast , the nfw halo + freeman disk @xmath2 gas disk mass model is unable to reproduce the observed rotation curve , a common outcome in dwarf galaxies . finally , we attempt to fit the data with a modified newtonian dynamics ( mond ) prescription . with the present data and with the present assumptions on distance , stellar mass , constant inclination and reliability of the gaseous mass , the mond `` amplification '' of the baryonic component appears to be too small to mimic the required `` dark component '' . the orion dwarf reveals a cored dm density distribution and a possible tension between observations and the canonical mond formalism . [ firstpage ] dark matter ; galaxy : orion dwarf ; mass profiles

arxiv

the trihydrogen dication , @xmath1 , which consists of three protons and one electron , is among the simplest coulomb systems . its stability has been studied intensely in the sixties and early seventies . in a series of articles , conroy @xcite investigated the potential energy surfaces of the electronic ground state and the lowest excited states at linear and isosceles triangular configurations . he employed a variational approach in which the electronic trial wavefunction is expanded around the center of the nuclear charges . analyzing the contour plots conroy concluded that @xmath1 is not stable . schwartz and schaad @xcite , and somorjai and yue @xcite , who reported single - point calculations of the system @xmath2 at the supposed equilibrium equilateral triangular configuration of @xmath3 , did not address the stability problem . to assess conroy s results , berkowitz and stocker @xcite searched for this ion through charge stripping experiments on @xmath4 . they could not find evidence of stable @xmath1 . later , the issue was reconsidered also from the theoretical side , by shoucri and darling @xcite , who examined equilateral configurations with the variational linear combination of atomic orbitals ( lcao ) method , and by hernndes and carb @xcite , who studied two particular configurations with a more compact variational approach and obtained total energy values below those published before . no bound state has been determined in these calculations . johnson and poshusta @xcite reported another single - point calculation in the context of gaussian basis set optimization for some one - electron systems . about twenty years later ackermann _ et al . _ @xcite revisited the question about the existence of @xmath1 using the finite element method which provided much higher accuracy than previously achieved . the problem of the stability of @xmath1 was treated keeping the nuclear charge as a continuous parameter . critical values of the charges for the existence of stable or metastable equilateral triangular configurations were obtained as @xmath5 and @xmath6 , respectively . the authors excluded the possibility of stable @xmath1 in the electronic ground state . however , the explicit electronic energy data are reported only for one particular equilateral triangular configuration at the triangle size @xmath7 . in conclusion , accurate _ ab initio _ results on the basis of which the non - existence of @xmath1 can be demonstrated are scarce and not that convincing . this question is thus addressed once again in the present study . one of the motivations of our study is related to a fact that @xmath1 in equilateral triangular configuration may exist as metastable state in a magnetic field @xmath8 g @xcite . we study a coulomb system of one electron and three protons @xmath2 which form an equilateral triangle of size @xmath9 . the protons are assumed to be infinitely massive according to the born - oppenheimer approximation at zero order . the schrdinger equation for the system is written as @xmath10\psi({\mathbf r } ) = e\psi({\mathbf r } ) \ , \ ] ] where @xmath11 is the electron momentum , @xmath12 and @xmath13 are the distances from each proton to the electron and @xmath9 is the interproton distance , see figure [ trian ] . atomic units are used throughout ( @xmath14=@xmath15=@xmath16=1 ) , although energies are expressed in rydbergs ( ry ) . our goal is to study the stability of the molecular ion @xmath1 . if such an ion exists , it implies the existence of the ground state of the system @xmath2 . based on symmetry arguments it seems evident that the optimal geometry of @xmath2 in the case of existence of a bound state is the equilateral triangle . two methods are used to explore the system : ( i ) variational with physically relevant trial functions ( see e.g. @xcite ) which we will call _ specialized _ and ( ii ) _ standard _ variational based on using standard gaussian trial functions as implemented in _ ab initio _ quantum chemistry packages such as molpro @xcite . both methods lead to highly accurate quantitative results for total energy versus the size of the triangle . in the first variational approach , a trial function is taken in a form of linear superposition of six basis functions @xmath17 where @xmath18 are linear parameters . each function @xmath19 is chosen in such a way as to describe different physical characteristics of the system . in general , @xmath19 has the form of a symmetrized product of three coulomb orbitals @xmath20 let us give a brief description of each of them : @xmath21 : : : all @xmath22 s are chosen to be equal to @xmath23 , @xmath24 it is a heitler - london type function . this corresponds to _ coherent _ interaction between the electron and all protons . supposedly , it describes the system at small interproton distances and , probably , the equilibrium configuration . it might be verified a posteriori . @xmath25 : : : two @xmath22 s are equal to zero and the remaining one is set to be equal to @xmath26 , @xmath27 it is a hund - mulliken type function . this function possibly describes the system at large distances , where essentially the electron interacts with only one proton at a time thus realizing _ incoherent _ interaction . @xmath28 : : : one @xmath22 is equal to zero , two others are different from zero and equal to each other and to @xmath29 , @xmath30 it is assumed that this function describes the system @xmath31 plus proton when a triangle is of a sufficiently small size . in fact , it is the heitler - london function of @xmath31 symmetrized over protons . @xmath32 : : : one @xmath22 is equal to zero and two others are different from each other being equal to @xmath33 , respectively , @xmath34 it is assumed that this function describes the system @xmath31 plus one proton . in fact , it is the guillemin - zener function of @xmath31 symmetrized over protons . if @xmath35 , the function @xmath32 is reduced to @xmath28 . if @xmath36 , the function @xmath32 is reduced to @xmath25 . hence @xmath32 is a non - linear interpolation between @xmath25 and @xmath28 . it has to describe intermediate interproton distances . @xmath37 : : : two @xmath22 s are equal but the third one is different , @xmath38 it describes a `` mixed '' state of three hydrogen atoms . if @xmath39 , the function @xmath37 is reduced to @xmath21 . if @xmath40 , the function @xmath37 is reduced to @xmath25 . if @xmath41 , the function @xmath37 is reduced to @xmath28 . hence @xmath37 is a non - linear interpolation between @xmath21 , @xmath25 and @xmath28 . as function ( [ psi4 ] ) this is a type of guillemin - zener function and should describe intermediate interproton distances . @xmath42 : : : all @xmath22 s are different , @xmath43 this is a general non - linear interpolation of all functions @xmath44 . the total number of the parameters of the function ( [ trial ] ) is equal to 15 , where five of them are linear ones . note that @xmath45 can be fixed at @xmath46 . + in standard _ ab initio _ calculations , @xmath47 is most commonly expanded in terms of gaussian basis functions @xcite @xmath48 centered at atoms @xmath49 , @xmath50 whose coefficients @xmath51 are then determined variationally . the basis functions @xmath52 themselves are built up by primitive gaussians @xcite @xmath53 with contraction coefficients @xmath54 held fixed . our calculations were performed using the hartree - fock code implemented in the molpro suite of programs @xcite with the correlation consistent cc - pv6z and modified mcc - pv7z basis sets @xcite . the cc - pv6z basis set contains 91 contracted gaussians per atom , with @xmath55 quantum numbers up to @xmath56 , i.e. @xmath57 $ ] , yielding a total of 273 basis functions . the mcc - pv7z basis includes @xmath58 functions , leading to 140 contracted gaussians per atom , @xmath59 $ ] , or 420 basis functions in total . calculations were carried out for a range of equilateral triangular configurations using @xmath60 symmetry . in this point group , the lowest electronic state is @xmath61 . the total number of contracted gaussians of this symmetry is 168 for the cc - pv6z basis set and 255 for the mcc - pv7z basis set , respectively . the cc - pv6z results , which are not reported here explicitly , have been generated to assess the accuracy of this type of calculations . judging from such a comparison , we estimate the accuracy of the mcc - pv7z data to about @xmath62 over a large range of distances , deteriorating somewhat at short distances where the basis functions tend to become linearly dependent . in framework of the specialized variational method ( i ) some numerical computations were made . the minimization routine minuit @xcite from the cern - lib library was used as well as d01fcf routine from the nag - lib @xcite for three - dimensional numerical integration . numerical values of the total energy @xmath63 of the system @xmath2 for different values of the interproton distance @xmath9 were obtained , see table [ tmedel ] . the results from the molpro calculation with a huge standard - type basis set ( mcc - pv7z ) are given for comparison . a problem of the standard apprach is its slow convergence with respect to the angular momentum quantum number @xmath55 , requiring the use of large basis sets . the method based on the specially tailored trial function , eq . ( [ trial ] ) , leads to systematically lower variational energy values with considerably less terms . it should be noted that this method relies on a careful optimization of non - linear parameters . .variational results obtained with the specialized method ( [ trial ] ) and with a standard quantum chemistry method molpro for the total energy @xmath63 as a function of the internuclear distance @xmath9 for the system @xmath64 in the equilateral triangular geometry . for @xmath65 , in @xcite @xmath66 . [ tmedel ] [ cols="^ , > , > " , ] as a conclusion we have to state that the total energy @xmath63 as a function of the internuclear distance @xmath9 does not indicate either to a minimum or even slight non - adiabatic irregularities for finite @xmath9 . thus , the molecular ion @xmath1 does not exist in equilateral triangular configuration in the born - oppenheimer approximation . hmc expresses his deep gratitude to j.c . lpez vieyra for the valuable comments and for their interest in the present work , avt thanks universite libre de bruxelles for the hospitality extended to him where this work was completed . this work was supported in part by fenomec and papiit grant * in121106 * ( mexico ) . turbiner and j.c . lopez vieyra , _ phys.rev . * a66 * _ , 023409 ( 2002 ) h .- j . werner , p. j. knowles , r. lindh , f. r. manby , m. schtz , p. celani , t. korona , g. rauhut , r. d. amos , a. bernhardsson , a. berning , d. l. cooper , m. j. o. deegan , a. j. dobbyn , f. eckert , c. hampel , g. hetzer , a. w. lloyd , s. j. mcnicholas , w. meyer , m. e. mura , a. nicklass , p. palmieri , r. pitzer , u. schumann , h. stoll , a. j. stone , r. tarroni , and t. thorsteinsson , molpro , version 2006.1 , a package of ab initio programs ( 2006 ) , see http://www.molpro.net

it is shown that the molecular ion @xmath0 does not exist in a form of the equilateral triangle . to this end , a compact variational method is presented which is based on a linear superposition of six specially tailored trial functions containing non - linear parameters . careful optimization of a total of fifteen parameters gives consistently lower variational results for the electronic energy than can be obtained with standard methods of quantum chemistry even with huge basis sets as large as mcc - pv7z . _ dedicated to professor rudolf zahradnik on the occasion of his 80th birthday _

arxiv

recent investigations of the large scale distribution of galaxies in the sloan digital sky survey ( sdss ; @xcite ) have revealed a complex relationship between the properties of galaxies , ( such as color , luminosity , surface brightness , and concentration ) and their environments ( @xcite ) . these and other investigations using the sdss ( @xcite ) and the two - degree field galaxy redshift survey ( @xcite ) have found that galaxy clustering is a function both of star formation history and of luminosity . for low luminosity galaxies , clustering is a strong function of color , while for luminous galaxies clustering is a strong function of luminosity . for red galaxies , clustering is a non - monotonic function of luminosity , peaking at both high and low luminosities . although galaxy clustering correlates also with surface brightness and concentration , @xcite and @xcite show that galaxy environment is independent of these properties at fixed color and luminosity . thus , color and luminosity measures of star formation history appear to have a more fundamental relationship with environment than do surface brightness and concentration measures of the distribution of stars within the galaxy . some of the investigations above have explored the scale dependence of these relationships . studies of the correlation function , such as @xcite and @xcite , can address this question , but do not address directly whether the density on large scales is related to galaxy properties _ independent _ of the relationships with density on small scales . if only the _ masses _ of the host halos of galaxies strongly affect their properties , then we expect no such independent relationship between galaxy properties and the large scale density field . thus , it is important to examine this issue in order to test the assumptions of the `` halo model '' description of galaxy formation and of semi - analytic models that depend only on the properties of the host halo ( _ e.g. _ , @xcite ) . recent studies of this question have come to conflicting conclusions . for example , @xcite have concluded from their analysis of sdss and 2dfgrs galaxies that the equivalent width of h@xmath4 is a function of environment measured on scales of 1.1 @xmath2 mpc and 5.5 @xmath2 mpc independently of each other . on the other hand , @xcite find that at fixed density at scales of 1 @xmath2 mpc , the distribution of d4000 ( a measure of the age of the stellar population ) is not a strong function of density on larger scales . here we address the dependence on scale of the relative bias of sdss galaxies . section [ data ] describes our data set . section [ results ] explores how the relationship between the color , luminosity , and environments of galaxies depends on scale . section [ bluefrac ] resolves the discrepancy noted in the previous paragraph between @xcite and @xcite , finding that only small scales are important to the recent star formation history of galaxies . section [ summary ] summarizes the results . where necessary , we have assumed cosmological parameters @xmath5 , @xmath6 , and @xmath7 km s@xmath8 mpc@xmath8 with @xmath9 . the sdss is taking @xmath10 ccd imaging of @xmath11 of the northern galactic sky , and , from that imaging , selecting @xmath12 targets for spectroscopy , most of them galaxies with @xmath13 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? automated software performs all of the data processing : astrometry @xcite ; source identification , deblending and photometry @xcite ; photometricity determination @xcite ; calibration @xcite ; spectroscopic target selection @xcite ; spectroscopic fiber placement @xcite ; and spectroscopic data reduction . an automated pipeline called idlspec2d measures the redshifts and classifies the reduced spectra ( schlegel et al . , in preparation ) . the spectroscopy has small incompletenesses coming primarily from ( 1 ) galaxies missed because of mechanical spectrograph constraints ( 6 percent ; * ? ? ? * ) , which leads to a slight under - representation of high - density regions , and ( 2 ) spectra in which the redshift is either incorrect or impossible to determine ( @xmath14 percent ) . in addition , there are some galaxies ( @xmath15 percent ) blotted out by bright galactic stars , but this incompleteness should be uncorrelated with galaxy properties . for the purposes of computing large - scale structure and galaxy property statistics , we have assembled a subsample of sdss galaxies known as the nyu value added galaxy catalog ( nyu - vagc ; @xcite ) . one of the products of that catalog is a low redshift catalog . here we use the version of that catalog corresponding to the sdss data release 2 ( dr2 ) . the low redshift catalog has a number of important features which are useful in the study of low luminosity galaxies . most importantly : 1 . we have checked by eye all of the images and spectra of low luminosity ( @xmath16 ) or low redshift ( @xmath17 ) galaxies in the nyu - vagc . most significantly , we have trimmed those which are `` flecks '' incorrectly deblended out of bright galaxies ; for some of these cases , we have been able to replace the photometric measurements with the measurements of the parents . for a full description of our checks , see @xcite . 2 . for galaxies which were shredded in the target version of the deblending , the spectra are often many arcseconds away from the nominal centers of the galaxy in the latest version of the photometric reductions . we have used the new version of the deblending to decide whether these ( otherwise non - matched spectra ) should be associated with the galaxy in the best version . we have estimated the distance to low redshift objects using the @xcite model of the local velocity field ( using @xmath18 ) , and propagated the uncertainties in distance into uncertainties in absolute magnitude . for the purposes of our analysis below , we have matched this sample to the results of @xcite , who measured emission line fluxs and equivalent widths for all of the sdss spectra . below , we use their results for the h@xmath4 equivalent width . the range of distances we include is @xmath19 @xmath2 mpc , making the sample volume limited for galaxies with @xmath20 . the total completeness - weighted effective area of the sample , excluding areas close to tycho stars , is 2220.9 square degrees . the catalog contains 28,089 galaxies . @xcite have investigated the luminosity function , surface brightness selection effects , and galaxy properties in this sample . we will be studying the environments of galaxies as a function of their luminosity and color below . to give a sense of the morphological properties of galaxies with various luminosities and colors , figure [ color_mag ] shows galaxies randomly selected in bins of color and luminosity . each image is 40 @xmath2 kpc on a side . red , high luminosity galaxies are classic giant ellipticals . lower luminosity red galaxies tend to be more flattened and less concentrated . blue , high luminosity galaxies have well - defined spiral structure and dust lanes . lower luminosity blue galaxies have less well - defined bulges and fewer spiral features . in order to evaluate the environments of galaxies in our sample , we perform the following procedure . first , for each given galaxy in the sample , we count the number of other galaxies @xmath21 with @xmath22 outside a projected radius of 10 @xmath2 kpc and within some outer radius @xmath23 , which we will vary below , and within @xmath24 km s@xmath8 in the redshift direction . this trace catalog is volume - limited within @xmath25 . in order to make a more direct comparison to @xcite , we will also use a trace catalog containing only galaxies with @xmath26 . second , we calculate the mean expected number of galaxies in that volume as : @xmath27 where @xmath28 is the sampling fraction of galaxies in the right ascension ( @xmath4 ) and declination ( @xmath29 ) direction of each point within the volume . we perform this integral using a monte carlo approach , distributing random points inside the volume with a density modulated by the sampling fraction @xmath28 . in order to calculate the mean density around galaxies in various classes , we will simply calculate : @xmath30 as the density with respect to the mean . when one calculates the mean density around galaxies , it is necessary to have a fair sample of the universe . for the most luminous galaxies in our sample ( @xmath31 ) the sample is volume - limited out to our redshift limit of @xmath32 and constitutes the equivalent of a 60 @xmath2 mpc radius sphere , which constitutes a fair sample for many purposes ( @xmath33cdm predicts a variance in such a sphere to be about 0.13 ) . however , the lower luminosity galaxies can only be seen in the fraction of this volume which is nearby , and below a certain luminosity the sample is no longer fair . for example , consider figure [ check_rho_converge ] , which shows the cumulative mean density around galaxies with @xmath34 in spheres of larger and larger radius around the milky way . the mean overdensity does not converge until a volume which corresponds to approximately @xmath35 . thus , it is not really safe to evaluate the mean density around galaxies that are too low luminosity to be observed out that far in redshift , which is to say , less luminous than @xmath36 . however , for the moment let us consider figure [ biden_all ] . the greyscale and contours show the mean density relative to the mean as a function of color and luminosity , using a projected radius of @xmath37 @xmath2 mpc . the mean is calculated in a sliding box with the width shown . if the sliding box contains fewer than 20 galaxies , the result is ignored and colored pure white . here we show the results for the entire sample . our statistical uncertainties are well - behaved down to about @xmath38 , but we are likely to be cosmic variance limited for @xmath39 , as indicated by the solid vertical line . thus , the apparent decline in the mean overdensity for red galaxies lower luminosity than @xmath40 is probably spurious . despite that limitation , we note that there is a strong relationship between environment and color even at @xmath41 . we note in passing that we can still use the variation of the density within @xmath42 to study the properties of galaxies as a function of density down to low luminosity . just because the _ mean _ density of galaxies in that volume has not converged does _ not _ imply that there is insufficient variation of density to study the variation of galaxy properties with environment . for our fair sample of galaxies with @xmath43 , figure [ biden_scales ] shows the dependence of overdensity on luminosity and color for six different projected radii : 0.2 , 0.5 , 1 , 2 , 4 , and 6 @xmath2 mpc . we only show results for @xmath44 , for which we have a fair sample . obviously , the density contrast decreases with scale ; on the other hand , the qualitative form of the plot does not change . our results remain similar to those shown in @xcite and @xcite . the results here demonstrate that the environments of low luminosity , red galaxies do continue to become denser as absolute magnitude increases down to absolute magnitudes of @xmath45 ( about two magnitudes less luminous than explored by our previous work ) . figure [ biden_ratios ] shows the ratio of the overdensity @xmath29 at each scale relative to that at the largest scale of @xmath46 @xmath2 mpc . this ratio is a measure of the steepness of the cross - correlation between galaxies of a given color and absolute magnitude with _ all _ galaxies in our volume - limited sample ( @xmath47 ) . interestingly , the contours in steepness are qualitatively similar to the contours in overdensity in figure [ biden_scales ] . this similarity implies that for each class of galaxy , the strength of the correlation on large scales always is associated with a _ steeper _ correlation function . another way of looking at similar results is to ask , as a function of environment , what fraction of galaxies are blue . we split the sample into `` red '' and `` blue '' galaxies using the following , luminosity - dependent cut : @xmath48 blue galaxies thus have @xmath49 . we then sort all the galaxies with @xmath50 into bins of density on three different scales : @xmath51 , @xmath52 , and @xmath53 @xmath2 mpc . in each bin we calculate the fraction of blue galaxies . figure [ lowz_fracblue ] shows this blue fraction as a function of density . in all cases , the blue fraction declines as a function of density , as one would expect based on figure [ biden_scales ] above , and from the astronomical literature ( a highly abridged list of relevant work would include @xcite ) . if we divide the sample into bins of luminosity , we find that higher luminosities have smaller blue fractions ( of course ) but that the dependence of blue fraction on density does not change . the question naturally arises : _ which _ scales are important to the process of galaxy formation ? is the local environment within 0.5 @xmath2 mpc the only important consideration ? or is the larger scale environment also important ? for example , consider figure [ denvden ] , which shows the conditional dependence of the three density estimators at the three scales on each other . the diagonal plots simply show the distribution within our sample of each density estimator . the off - diagonal plots show the conditional distribution of the quantity on the @xmath54-axis given the quantity on the @xmath55-axis . as an example , the lower right panel shows @xmath56 shows the fraction of blue galaxies as a function of two density estimates : one with @xmath51 @xmath2 mpc and one with @xmath57 @xmath2 mpc . in this case it is clear that the blue fraction is a function of both densities . that is , even at a fixed density on scales of @xmath37 @xmath2 mpc , the density outside that radius matters to the blue fraction ; in addition , at a fixed density on scales of @xmath52 @xmath2 mpc , the distribution of galaxies within that radius appears to affect the blue fraction as well . on the other hand , consider figure [ lowz_fracblue2_1.0 - 6.0 ] , which is the same as figure [ lowz_fracblue2_0.5 - 1.0 ] , but now showing the densities at scales of @xmath57 and @xmath53 @xmath2 mpc . in figure [ lowz_fracblue2_1.0 - 6.0 ] the contours are vertical , indicating that the density between @xmath52 and @xmath53 @xmath2 mpc has very little effect on galaxy properties . at a fixed value of the density at the smaller scale , the larger scale environment appears to be of little importance . @xcite found that these contours were not vertical when he looked at the fraction of galaxies for which the h@xmath4 equivalent width was @xmath58 . their result appears in conflict with that of the previous paragraph . on the other hand , the emission lines measure a more recent star formation rate than does the color ; it is possible in principle that the more recent star formation rate depends more strongly on large - scale environment . to rule out this possibility , figure [ lowz_frachalpha2_1.0 - 6.0 ] shows the same result as figure [ lowz_fracblue2_1.0 - 6.0 ] , but now showing the fraction of galaxies with h@xmath4 equivalent widths ( as measured by @xcite ) greater than 4 . again , for strong emission line fraction as for the blue fraction , the smaller scales are important , but the 6 @xmath2 mpc scales are not , in contradiction with @xcite . why , then , did @xcite conclude that large scales _ were _ important ? there are a number of differences between our study and theirs . first , their contouring method differs ; instead of measuring the blue fraction in bins of fixed size , at each point they measure the star - forming fraction among the nearest 500 galaxies in the plane of @xmath59 and @xmath60 . we have found that this procedure creates a _ slight _ bias in the contouring in the sense that near the edges of the distribution vertical contours will become diagonal . however , this effect is not strong enough to explain the differences between our results and those of @xcite . second , to estimate the density in their sample they used a spherical gaussian filter , whereas here we use the overdensity in cones . we have not investigated what effect this difference has . finally , they use tracer galaxies with a considerably lower mean density than ours . their effective absolute magnitude limit is @xmath61 ; such galaxies have a mean density of @xmath62 @xmath63 mpc@xmath64 . our tracers ( @xmath22 ) have a mean density of @xmath65 @xmath63 mpc@xmath64 , almost six times higher . figure [ lowz_frachalpha2_m-20.5_1.0 - 6.0 ] shows our results when we restrict our tracer sample to @xmath61 . the contours in this figure are very diagonal , similar to the results of @xcite . this result suggests that one of two possible mechanisms are causing the differences between our results and those of @xcite . first , the higher luminosity galaxies with @xmath61 might be yielding fundamentally different information about the density field than our lower luminosity tracers . second , the lower mean density of the galaxies with @xmath61 might be effectively introducing `` noise '' in the measurement on small scales . remember that the large scale and small scale densities are intrinsically correlated . so if the small scale measurement is noisy enough , the higher signal - to - noise ratio large scale measurement could actually be adding extra information about the environment on _ small _ scales . such an effect would make the contours in figure [ lowz_frachalpha2_m-20.5_1.0 - 6.0 ] diagonal . we have performed a simple test to distinguish these possibilities , which is to remake figure [ lowz_frachalpha2_1.0 - 6.0 ] using the low luminosity tracers ( @xmath66 ) but subsampling them to the same mean density as the high luminosity tracers ( @xmath67 ) . this test yields diagonal contours , meaning one can understand the diagonal contours of figure [ lowz_frachalpha2_m-20.5_1.0 - 6.0 ] and of @xcite as simply reflecting the low signal - to - noise ratio of the density estimates . we explore the relative bias between galaxies as a function of scale , finding the following . 1 . the dependence of mean environment on color persists to the lowest luminosities we explore ( @xmath68 ) . red , low luminosity galaxies tend to be in overdense regions , down at least to @xmath69 . this result extends those found by @xcite and @xcite towards lower luminosities by about 2 magnitudes . 3 . at any given point of color and luminosity , a correlation function with a stronger amplitude implies correlation function with a steeper slope . 4 . in regions of a given overdensity on small scales ( @xmath70 @xmath2 mpc ) , the overdensity on large scales ( @xmath46 @xmath2 mpc ) does not appear to relate to the recent star formation history of the galaxies . the last point above deserves elaboration . first , it contradicts the results of @xcite . we have found that their results are probably due to the low mean density of the tracers they used . this explanation underscores the importance of taking care when using low signal - to - noise quantities . galaxy environments are difficult to measure , in the sense we use tracers that do not necessarily trace the `` environment '' perfectly , meaning neither with low noise nor necessarily in an unbiased manner . we claim here that our higher density of tracers marks an improvement over previous work , but it is worth noting the limitations of assuming that the local galaxy density fairly and adequately represents whatever elements of the environment affect galaxy formation . second , if the galaxy density field is an adequate representation of the environment , the result has important implications regarding the physics of galaxy formation . in simulations whose initial conditions are constrained by cosmic microwave background observations and galaxy large - scale structure observations , virialized dark matter halos do not extend to sizes much larger than @xmath71@xmath72 @xmath2 mpc . thus , our results are consistent with the notion that only the masses of the host halos of the galaxies we observe are strongly affecting the star formation of the galaxies . in addition , @xcite find that only the star formation histories , not the azimuthally - averaged structural parameters , are directly related to environment . for these reasons , it is likely that we can understand the process of galaxy formation by only considering the properties of the host dark matter halos . our results therefore encourage the `` halo model '' description of galaxy formation and the pursuit of semi - analytic models which depend only on the properties of the host halo ( _ e.g. _ , @xcite ) . thanks to eric bell and george lake for useful discussions during this work . thanks to guinevere kauffmann for encouraging us to pursue this question . thanks to christy tremonti and jarle brinchmann for the public distribution of their measurements of sdss spectra . funding for the creation and distribution of the sdss has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium ( arc ) for the participating institutions . the participating institutions are the university of chicago , fermilab , the institute for advanced study , the japan participation group , the johns hopkins university , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , university of pittsburgh , princeton university , the united states naval observatory , and the university of washington . , m. , eke , v. , miller , c. , lewis , i. , bower , r. , couch , w. , nichol , r. , bland - hawthorn , j. , baldry , i. k. , baugh , c. , bridges , t. , cannon , r. , cole , s. , colless , m. , collins , c. , cross , n. , dalton , g. , de propris , r. , driver , s. p. , efstathiou , g. , ellis , r. s. , frenk , c. s. , glazebrook , k. , gomez , p. , gray , a. , hawkins , e. , jackson , c. , lahav , o. , lumsden , s. , maddox , s. , madgwick , d. , norberg , p. , peacock , j. a. , percival , w. , peterson , b. a. , sutherland , w. , & taylor , k. 2004 , , 348 , 1355

we investigate the relationship between the colors , luminosities , and environments of galaxies in the sloan digital sky survey spectroscopic sample , using environmental measurements on scales ranging from @xmath0 to @xmath1 @xmath2 mpc . we find : ( 1 ) that the relationship between color and environment persists even to the lowest luminosities we probe ( @xmath3 ) ; ( 2 ) at luminosities and colors for which the galaxy correlation function has a large amplitude , it also has a steep slope ; and ( 3 ) in regions of a given overdensity on small scales ( 1 @xmath2 mpc ) , the overdensity on large scales ( 6 @xmath2 mpc ) does not appear to relate to the recent star formation history of the galaxies . of these results , the last has the most immediate application to galaxy formation theory . in particular , it lends support to the notion that a galaxy s properties are related only to the mass of its host dark matter halo , and not to the larger scale environment .

arxiv

transverse spin fluctuations are gapless , low - energy excitations in the broken - symmetry state of magnetic systems possessing continuous spin - rotational symmetry . therefore at low temperatures they play an important role in diverse macroscopic properties such as existence of long - range order , magnitude and temperature - dependence of the order parameter , nel temperature , spin correlations etc . specifically in the antiferromagnetic ( af ) ground state of the half - filled hubbard model transverse spin fluctuations are important both in two and three dimensions , where antiferromagnetic long - range order ( aflro ) exists at @xmath4 . in the strong - coupling limit @xmath5 , where spin fluctuations are strongest , they significantly reduce the zero - temperature af order parameter in two dimensions to nearly @xmath6 of the classical ( hf ) value.@xcite similarly the nel temperature in three dimensions is reduced to nearly @xmath7 of the mean - field result @xmath8 , for the equivalent @xmath9 quantum heisenberg antiferromagnet ( qhaf).@xcite recently there has also been interest in spin fluctuations due to defects , disorder and vacancies in the quantum antiferromagnet . in the random-@xmath3 model , where @xmath3 is set to zero on a fraction @xmath10 of sites , the lattice - averaged af order parameter appears to be enhanced for small @xmath10 , as seen in qmc calculations,@xcite presumably due to an overall suppression of quantum spin fluctuations . on the other hand spin fluctuations are enhanced by strong disorder in the hubbard model with random on - site energies . in the strong disorder regime , overlap of the two hubbard bands leads to formation of essentially empty and doubly - occupied sites , which act like spin vacancies.@xcite the problem of spin vacancies in the quantum antiferromagnet is also relevant to the electron - doped materials like @xmath11 , where spin - dilution behavior is observed.@xcite while the problem of magnon energy renormalization due to spin vacancies has been addressed recently,@xcite these methods are limited to the low - concentration limit , and the vacancy - induced enhancement in transverse spin fluctuations has not been studied in the whole range of vacancy concentration . in this paper we describe a new method for evaluating transverse spin correlations and quantum spin - fluctuation corrections about the hf - level broken - symmetry state , in terms of magnon mode energies and spectral functions obtained in the random phase approximation ( rpa ) . the method is applicable in the whole @xmath0 range of interaction strength , and is illustrated with three applications involving the af ground state of the half - filled hubbard model ( i ) the pure model in @xmath12 , ( ii ) spin vacancies in the strong coupling limit in @xmath13 , and ( iii ) low-@xmath3 impurities in @xmath13 . this method for obtaining quantum correction to sublattice magnetization solely in terms of transverse spin correlations is parallel to the spin - wave - theory ( swt ) approach,@xcite and differs from the method involving self - energy corrections.@xcite the rpa approach has been demonstrated earlier to properly interpolate between the weak and strong coupling limits for the spin - wave velocity.@xcite by going beyond the rpa level within a systematic inverse - degeneracy expansion scheme , which preserves the spin - rotational symmetry and hence the goldstone mode order by order , it was also shown that in the strong coupling limit identical results are obtained for all quantum corrections , order by order , as from the swt approach for the qhaf.@xcite a renormalized rpa approach has also been devised recently to obtain the magnetic phase diagram for the three dimensional hubbard model in the whole @xmath0 range , and the @xmath14 vs. @xmath3 behaviour was shown to properly interpolate between the weak and strong coupling limits.@xcite the method is based on a convenient way to perform the frequency integral in order to obtain spin correlations from spin propagators , and we illustrate it here for transverse spin correlations . we write the time - ordered , transverse spin propagator for sites @xmath15 and @xmath16 , @xmath17|\psi_{\rm g}\rangle$ ] at the rpa level in frequency space as , @xmath18=\frac{[\chi^0(\omega)]}{1-u[\chi^0(\omega ) ] } = \sum_n \frac{\lambda_n(\omega)}{1-u\lambda_n(\omega ) } @xmath19 and @xmath20 are the eigenvectors and eigenvalues of the @xmath21 $ ] matrix . here @xmath21_{ij}=i\int ( d\omega'/2\pi ) g_{ij}^{\uparrow}(\omega')g_{ji}^{\downarrow}(\omega'-\omega)$ ] is the zeroth - order , antiparallel - spin particle - hole propagator , evaluated in the self - consistent , broken - symmetry state from the hf green s functions @xmath22 . spin correlations are then obtained from , @xmath23_{ij}\ ; e^{-i\omega ( t - t ' ) } \nonumber \\ & = & \pm \sum_n \frac{\phi_n ^i ( \omega_n)\phi_n ^j ( \omega_n ) } { u^2 ( d\lambda_n / d\omega)_{\omega_n } } e^{-i\omega_n ( t - t ' ) } \ ; , \end{aligned}\ ] ] where the collective mode energies @xmath24 are obtained from @xmath25 , and @xmath20 has been taylor - expanded as @xmath26 near the mode energies to obtain the residues . for convergence , the retarded ( advanced ) part of the time - ordered propagator @xmath27 , having pole below ( above ) the real-@xmath28 axis , is to be taken for @xmath29 ( @xmath30 ) . the frequency integral is conveniently replaced by an appropriate contour integral in the lower or upper half - plane in the complex-@xmath28 space for these two cases , respectively , which results in eq . we first illustrate this method for the half - filled hubbard model in two and three dimensions on square and simple - cubic lattices , respectively . in this case it is convenient to use the two - sublattice representation due to translational symmetry , and we work with the @xmath32 matrix @xmath33 $ ] in momentum space , which is given in terms of eigensolutions of the hf hamiltonian matrix.@xcite the @xmath34-summation is performed numerically using a momentum grid with @xmath35 and 0.05 , in three and two dimensions , respectively . equal - time , same - site transverse spin correlations are then obtained from eq . ( 2 ) by summing over the different @xmath36 modes , using a momentum grid with @xmath37 and @xmath38 in three and two dimensions , respectively . we consider @xmath39 , so that the retarded part is used , with positive mode energies . from spin - sublattice symmetry , correlations on a and b sublattice sites are related via @xmath40 . thus the transverse spin correlations are obtained from magnon amplitudes on a and b sublattices , and from eq . ( 2 ) we have @xmath41 from the commutation relation @xmath42=2s^z$ ] , the difference @xmath43 , of transverse spin correlations evaluated at the rpa level , should be identically equal to @xmath44 . this is becasue both the rpa and hf approximations are o(1 ) within the inverse - degeneracy expansion scheme@xcite in powers of @xmath45 ( @xmath46 is the number of orbitals per site ) , and therefore become exact in the limit @xmath47 , when all corrections of order @xmath45 or higher vanish . this is indeed confirmed as shown in figs . 1 and 2 . the deviation at small @xmath3 is because of the neglect in eq . ( 2 ) of the contribution from the single particle excitations across the charge gap , arising from the imaginary part of @xmath48 in eq . the sum @xmath49 yields a measure of transverse spin fluctuations about the hf state , and in the strong coupling for spin @xmath50 , one obtains @xmath51.@xcite using the identity , @xmath52 in this limit , the sublattice magnetization @xmath53 is then obtained from , @xmath54^{1/2 } .\ ] ] to order @xmath55 , this yields the correction to the sublattice magnetization of @xmath56=0.156 $ ] and 0.393 for @xmath9 in three and two dimensions , respectively . this same result at one loop level was also obtained from a different approach in terms of the electronic spectral weight transfer,@xcite and is in exact agreement with the swt result.@xcite as @xmath57 is the maximum ( classical ) spin polarization in the z - direction , and therefore also the maximum eigenvalue of the local @xmath58 operator , therefore for arbitrary @xmath3 , the hf magnitude @xmath57 plays the role of the effective spin quantum number , @xmath50 . the sublattice magnetization @xmath59 is therefore obtained from @xmath60 , where the first - order , quantum spin - fluctuation correction @xmath61 is obtained from eq . ( 4 ) with @xmath62 , @xmath63 in the strong coupling limit @xmath2 , @xmath64 for @xmath9 , so that the spin - fluctuation correction simplifies to , @xmath65 . for a site on the b sublattice , where @xmath66 , we have @xmath67 . the @xmath3-dependence of sublattice magnetization @xmath59 is shown in figs . 1 , and 2 for @xmath13 and @xmath68 , respectively . in both cases it interpolates properly between the weak and strong coupling limits , approaching the swt results 0.607 and 0.844 , respectively , as @xmath2 . a comparison of the @xmath59 vs. @xmath3 behaviour with earlier results is presented in fig . 3 for the well studied @xmath13 case . earlier studies have employed a variety of methods including the variational monte carlo ( vmc),@xcite self - energy corrections ( se),@xcite quantum monte carlo ( qmc),@xcite functional - integral schemes,@xcite the generalized linear spin - wave approximation ( glswa),@xcite and mapping of low - energy excitations to those of a qhaf with @xmath3-dependent , extended - range spin couplings.@xcite in addition to the two - sublattice basis , we have also used the full site representation in the strong coupling limit , in order to illustrate the scaling of the quantum correction with system size . here the @xmath69 matrix is evaluated and diagonalized in the site basis for finite lattices . results for lattice sizes with @xmath70 are shown in fig . 4 . a quadratic least - square fit is used to extrapolate to infinite system size , which yields @xmath71 , in agreement with the result from eq . the site representation has also been used to obtain transverse spin fluctuations for the problem of spin vacancies in the af in the limit @xmath2 . as mentioned already this method is applicable in the whole range of vacancy concentration , and allows determination of the critical vacancy concentration at which the af order parameter vanishes . for the vacancy problem we consider the following hamiltonian on a square lattice with nearest - neighbor ( nn ) hopping , @xmath72 where the hopping terms @xmath73 if sites @xmath15 or @xmath16 are vacancy sites , and @xmath74 otherwise . thus , for a vacancy on site @xmath15 , all hopping terms @xmath75 connecting @xmath15 to its nn sites @xmath16 are set to zero . the vacancy site is thus completely decoupled from the system . half - filling is retained by having one fermion per remaining site . we consider the @xmath2 limit , where the local moments are fully saturated , and the vacancy problem becomes identical to the spin - vacancy problem in the qhaf . this is also equivalent to the problem of non - magnetic impurities in the af in the limit of the impurity potential @xmath76.@xcite the structure of the @xmath48 matrix in the host af , and the modification introduced by spin vacancies has been considered earlier in the context of static impurities.@xcite as the goldstone mode is preserved , it is convenient to work with the matrix @xmath77 $ ] , inverse of which yields the spin propagator in eq . ( 1 ) near the mode energies . when expressed in units of @xmath78 , where @xmath79 is the af gap parameter , and @xmath80 , it has the following simple structure for the host af : @xmath81 vacancies introduce a perturbation in @xmath82 due to absence of hopping between the vacancy and nn sites , and we take @xmath83 to refer to this vacancy - induced perturbation , so that @xmath84 . if site @xmath15 is a vacancy site and @xmath16 the nn sites , then the matrix elements of @xmath85 are , @xmath86 and the magnitude of @xmath87 is irrelevant since the vacancy site @xmath15 is decoupled . thus @xmath88 , reflecting the decoupling of the vacancy , and the static part of the diagonal matrix elements @xmath89 on nn sites are reduced by 1/4 . for @xmath90 vacancies on nn sites , the static part is @xmath91 . this ensures that the goldstone mode is preserved . thus if a spin on site @xmath16 were surrounded by a maximum of four vacancies on nn sites , then the static part vanishes , and @xmath92 , representing an isolated spin , which yields a @xmath93 pole in the transverse spin propagator . this is an isolated single - spin cluster , and with increasing vacancy concentration , larger isolated spin clusters are formed . as these are decoupled from the remaining system , their spin - fluctuation contributions are not included , as the op vanishes for finite spin clusters . when @xmath94 exceeds the percolation threshold @xmath95 , the fraction of macroscopically large spin clusters in the system vanishes , and therefore no aflro is possible for @xmath96 . for a given vacancy concentration and system size , the appropriate number of vacancies are placed randomly across the lattice , and the matrix @xmath82 constructed accordingly . exact diagonalization of @xmath82 is carried out , and the eigensolutions are used to compute the transverse spin correlations from eq . ( 2 ) . the quantum , spin - fluctuation correction is then obtained from the strong - coupling limit of eq . the transverse spin correlation @xmath97 is averaged over all spins within the a sublattice . as mentioned already , only the spins in the macroscopic cluster spanning the whole lattice are considered , and contributions from spins in isolated spin clusters are excluded . configuration averaging over several realizations of the vacancy distribution is also carried out . the quantum spin - fluctuation correction vs. vacancy concentration is shown in fig . 5 for three lattice sizes , @xmath98 . best fits are obtained with an expression including a cubic term , @xmath99 . for the three lattice sizes @xmath1000.236 , 0.260 and 0.278 , respectively , and as shown in fig . 3 , it extrapolates to 0.39 as @xmath101 . the coefficient of the linear term is found to be nearly independent of system size , @xmath102 . and the cubic term @xmath103 takes values approximately 3.6 , 4.0 , and 4.4 for the three lattice sizes , and extrapolates to about 6.5 as @xmath101 . with these coefficients , the spin - fluctuation correction @xmath61 is nearly 1 for @xmath104 . beyond @xmath104 , the percolation limit , there is no single , macroscopically large spin cluster left in the system . therefore the point where the af order parameter vanishes and aflro is destroyed due to spin fluctuations nearly coincides with the percolation threshold . this is in agreement with results from series expansion@xcite and quantum monte carlo simulations@xcite of the qhaf with spin vacancies . we now consider a quenched impurity model with a random distribution of impurity sites characterized by a local coulomb interaction @xmath105 for the host sites . with h and i referring to the sets of host and impurity sites respectively , we consider the following hubbard model in the particle - hole symmetric form at half - filling and on a square lattice , @xmath106 the motivations for studying this impurity model are threefold . in view of the observed _ enhancement _ of magnetic order at low concentration of impurities,@xcite we shall analyze the suppression of quantum spin fluctuations to examine whether this is due to a local suppression at the low-@xmath3 sites . the rpa evaluation of transverse spin correlations is also extended to the case of site - dependent interactions . furthermore , at half - filling this model also provides a simplistic representation for magnetic impurity doping in an af . this may appear contradictory in view of the apparently nonmagnetic ( @xmath107 ) nature of the impurity sites . however , this feature is expressed only away from half - filling.@xcite the atomic limit @xmath108 provides a convenient starting point for further discussions . in the particle - hole symmetric form of eq . ( 9 ) , since not only local interaction terms , but the on - site energy terms are also modified ( from @xmath109 to @xmath110 ) at the impurity sites , therefore the energy levels for added hole and particle are @xmath111 and @xmath112 for host and impurity sites respectively . to order @xmath113 , this impurity model therefore canonically maps to the following @xmath9 heisenberg model @xmath114 where in the first term @xmath80 is the conventional exchange coupling between neighboring host spins , and @xmath115 is the exchange coupling between impurity spins and neighboring host spins . in writing eq . ( 10 ) we have assumed the dilute impurity limit , and discounted the possibility of two impurity spins occupying nn positions , in which case the impurity - impurity exchange coupling will be @xmath116 . therefore , in the strong correlation limit , this random-@xmath3 model also describes magnetic impurities in the af within an impurity - spin model . the magnetic - impurity doping is characterized by identical impurity and host spins @xmath117 , but with different impurity - host exchange coupling @xmath118 . both cases @xmath119 or @xmath120 are possible , and in this paper we have considered the two cases : ( i ) @xmath121 so that @xmath122 , and @xmath123 so that @xmath124 . while this model is easily generalized to other magnetic - impurity models represented by locally modified impurity - host hopping terms @xmath125 , and/or different impurity energy levels , in fact , the essential features are already contained here as the impurity exchange coupling @xmath126 is the relevant quantity in determining the spin fluctuation behavior . we recast the rpa expression for the transverse spin propagator in a form suitable for site - dependent interactions . in terms of a diagonal interaction matrix @xmath127 $ ] , with elements @xmath127_{ii}=u_i $ ] , the local coulomb interaction at site @xmath15 , the time - ordered transverse spin propagator at the rpa level can be rewritten , after simple matrix manipulations , as @xmath18= \frac{[\chi^{0}(\omega ) ] } { 1 - [ u ] [ \chi^{0}(\omega ) ] } = \frac{1}{[a(\omega ) ] } - \frac{1}{[u]}\ ] ] where @xmath128=[u ] - [ u][\chi^0 ( \omega)][u]$ ] is a symmetric matrix . as @xmath127 $ ] is non - singular , the singularities in @xmath129 $ ] , which yield the magnon modes , are then given completely by the vanishing of the eigenvalues of the matrix @xmath130 $ ] . in terms of @xmath131 and @xmath132 , the eigenvalues and eigenvectors of the matrix @xmath130 $ ] , we have @xmath128^{-1 } = \sum_n \lambda_n(\omega)^{-1 } magnon - mode energies @xmath24 are then given by @xmath133 , and in analogy with eq . ( 2 ) the transverse spin correlations are obtained from , @xmath134 as we are interested in the dilute behaviour , we examine the correction to sublattice magnetization due to two impurities , one on each sublattice for symmetry . since @xmath135 , corrections to both @xmath136 and @xmath61 are expressed in the dilute limit ( impurity concentration @xmath94 ) as @xmath137 , and @xmath138 . the overall @xmath139 behavior therefore depends on the relative magnitude of the coefficients @xmath140 and @xmath141 . for @xmath142 and @xmath143 we find , at the hf level , that @xmath144 and @xmath145 , both quite independent of system size . the ( site - averaged ) spin fluctuation correction @xmath61 is obtained from eq . ( 5 ) with and without impurities , and the impurity contribution extracted . for @xmath143 we find a net _ reduction _ in @xmath61 . divided by @xmath146 , the impurity concentration , this yields the coefficient @xmath141 defined above , and also the per - impurity contribution to the total spin fluctuation correction over the whole lattice . the spin - fluctuation correction @xmath61 for the pure case , and the per - impurity contribution @xmath147 are shown in fig . 6 for different lattice sizes , along with least - square fits . it is seen that in the infinite size limit , the per - impurity reduction is nearly 0.2 , which is more than half of the correction per site in the pure case ( 0.35 ) . thus , there is a substantial reduction in the averaged spin fluctuation correction due to the low@xmath1 impurities . as the two coefficients @xmath140 and @xmath141 are very nearly the same , the sublattice magnetization @xmath148 shows negligible concentration dependence . thus the ( relatively small ) reduction in the hf value due to the low-@xmath3 impurities is almost fully compensated by the ( relatively substantial ) reduction in the spin fluctuation correction . to first order in @xmath94 , we thus find that there is nearly no loss of af order due to the low-@xmath3 impurities . as mentioned earlier , even a slight enhancement in the af order was recently seen for the case @xmath149.@xcite we next examine the site - dependence of the local spin fluctuation corrections @xmath150 near the impurities . table i shows that spin fluctuation is actually _ enhanced _ on the low-@xmath3 impurity sites . the suppression of @xmath150 in the vicinity more than compensates for this local enhancement , resulting in an overall reduction on the average . on the other hand , for @xmath151 , we find that the correction is suppressed on the high-@xmath3 impurity site , while it is enhanced on the average . thus , to summarize , when the impurity spin is coupled more strongly ( weakly ) , the spin - fluctuation correction is enhanced ( suppressed ) locally at the impurity site , but the average correction to sublattice magnetization is suppressed ( enhanced ) . this local enhancement can be understood in terms of the correlations @xmath152 as follows . since the impurity spin is more strongly coupled to the neighboring spins , the nn matrix elements @xmath153 are enhanced . this puts more magnon amplitude @xmath154 on the impurity site , so that from eq . ( 2 ) the transverse spin correlations @xmath152 , and therefore the spin - fluctuation correction , are enhanced for low-@xmath3 impurities . the overall decrease in the averaged fluctuation correction , however , is due to the stiffening of the magnon spectrum in the important low - energy sector , following from the increased average spin coupling . .the local spin - fluctuation corrections @xmath155 for a @xmath156 system ( @xmath142 ) , with two impurities at ( 11,4 ) and ( 4,14 ) . for the two cases @xmath143 ( @xmath157 ) and @xmath151 ( @xmath158 ) , quantum corrections are enhanced / suppressed locally at the impurity sites ( indicated in boldfaces ) , but are suppressed / enhanced on the average . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^ " , ] 2 while these results also follow from the impurity - spin picture , the small charge gap at the impurity site does have an impact on the magnon spectrum . within the localized spin picture , the highest - energy magnon mode corresponding to a local spin deviation at the impurity site would cost energy @xmath159 . however , the highest energy in the magnon spectrum is actually seen to be @xmath160 , which is substantially smaller than @xmath161 . this shows the compression effect of the low charge gap ( @xmath162 ) on the magnon spectrum.@xcite in conclusion , using a convenient numerical method for evaluating transverse spin correlations at the rpa level , quantum spin - fluctuation corrections to sublattice magnetization are obtained for the half - filled hubbard model in the whole @xmath0 range . results in two and three dimensions are shown to interpolate properly between both the weak and strong correlation limits , and approach the swt results as @xmath2 . the method is readily extended to other situations of interest involving defects in the af , such as vacancies / impurities / disorder . numerical diagonalization for finite lattices , along with finite - size scaling tested with the pure hubbard model , allows for exact treatment of defects at the rpa level . this is illustrated with a study of the defect - induced enhancement / suppression in transverse spin fluctuations for spin vacancies and low-@xmath3 impurities in two dimensions . while the quantum spin fluctuation correction to sublattice magnetization is sharply enhanced by spin vacancies , it is strongly suppressed by the low-@xmath3 impurities , although the fluctuation correction is enhanced at the low@xmath1 sites . helpful conversations with d. vollhardt and m. ulmke , and support from the alexander von humboldt foundation through a research fellowship at the universitt augsburg are gratefully acknowledged .

a numerical method is described for evaluating transverse spin correlations in the random phase approximation . quantum spin - fluctuation corrections to sublattice magnetization are evaluated for the antiferromagnetic ground state of the half - filled hubbard model in two and three dimensions in the whole @xmath0 range . extension to the case of defects in the af is also discussed for spin vacancies and low@xmath1 impurities . in the @xmath2 limit , the vacancy - induced enhancement in the spin fluctuation correction is obtained for the spin - vacancy problem in two dimensions , for vacancy concentration up to the percolation threshold . for low-@xmath3 impurities , the overall spin fluctuation correction is found to be strongly suppressed , although surprisingly spin fluctuations are locally enhanced at the low@xmath1 sites . 2

arxiv

in recent years , the study of nonequilibrium dynamics in quantum field theory has received much attention in various areas of physics , and particularly in cosmology . the work has been driven largely by inflation @xcite , the most successful known mechanism for explaining the large - scale homogeneity and isotropy of the universe _ and _ the small - scale inhomogeneity and anisotropy of the universe @xcite . with observations for the first time able to directly test the more detailed predictions of specific inflationary models , the efforts in understanding inflation and its dynamics have redoubled . one area of particular interest is the dynamics of multi - field models of inflation in which the inflaton is coupled to another dynamical field during inflation . these models can lead to a variety of features unavailable in the case of a single field . such multi - field scenarios include the well known hybrid inflation models @xcite . on top of the dynamics during inflation , the subsequent phase of energy transfer between the inflaton and other degrees of freedom leading to the standard picture of big bang cosmology has been the subject of intense study . the inflaton may decay through perturbative processes @xcite as well as non - perturbative parametric amplification @xcite . the latter can lead to explosive particle production and very efficient reheating of the universe . hybrid inflation and reheating models share an important common thread . they both involve the coupling of two or more dynamical , interacting scalar fields ( or higher spin fields @xcite ) . an important aspect of such systems is the possibility of mixing between the fields . in ref . @xcite for example the classical inflaton decay is investigated for a two field model by solving the non - linear equations of motions on a grid . in ref . @xcite , the authors treat the problem of coupled quantum scalar and fermion fields at the tree level . due to the small couplings involved in inflationary cosmology , such a tree level analysis is useful in a variety of physical situations . however , hybrid models as well as the dynamics of reheating typically include processes such as spinodal decomposition @xcite and parametric amplification which require one to go beyond the tree level by including quantum effects either in a perturbative expansion or by means of non - perturbative mean field techniques such as the hartree approximation or a large-@xmath0 expansion @xcite . going beyond tree level brings in the issue of renormalization . the problem of renormalization of time evolution equations in single field models was understood several years ago . in one of the first papers in this field , cooper and mottola showed in 1987 ( ref . @xcite , that it is possible to find a renormalization procedure which leads to counter terms independent of time and initial conditions of the mean field . they used a wkb expansion in order to extract the divergences of the theory . in a later paper cooper et al . also discussed a closely related adiabatic method in order to renormalize the @xmath1-theory in the large n approximation . also boyanovsky and de vega , ref . @xcite , used a wkb method in order to renormalize time dependent equations in one - loop order , later on boyanovsky et al . @xcite investigated a @xmath1 model in the large - n approximation and the hartree approximation , too . in 1996 baacke et al . , ref . @xcite , proposed a slightly different method in order to extract the divergences of the theory , which enabled them to use dimensional regularization . in contrast to the wkb ansatz this method can be extended for coupled system , which was demonstrated in ref . this procedure will be used also in this paper . we work in the context of a closed time path formalism @xcite appropriate to following the time - dependent evolution of the system . in this formalism , the _ in_-vacuum plays a predominant role , as quantities are tracked by their _ in - in _ expectation values ( in contrast to the _ in - out _ formalism of scattering theory ) . we construct the _ in_-vacuum by diagonalizing the mass matrix of the system at the initial time @xmath2 . however , because of the time - dependent mixing , a system initially diagonalized in this way will generally not be diagonalized at later times . one approach to this problem , taken in ref . @xcite , is to diagonalize the mass matrix at each moment in time through the use of a time - dependent rotation matrix . the cost of doing so is the appearance of time derivatives of the rotation matrix into the kinetic operators of the theory . while such a scheme is in principle workable beyond the tree level , the modified kinetic operators introduce complications into the extraction of the fluctuation corrections as well as the divergences that are to be removed via renormalization . we take an alternative approach where the mass matrix is allowed to be non - diagonal for all times @xmath3 and account for the mixing by expanding each of the fields in terms of _ all _ of the _ in_-state creation and annihilation operators . the cost of doing so in an @xmath0-field system is the need to track @xmath4 complex mode functions representing the fields instead of the usual @xmath0 . however , this allows standard techniques to be used to properly renormalize the system . for the two - field systems common in inflationary models , this effective doubling of the field content adds a relatively minor cost . for simplicity and clarity , we will work in minkowski space time and in a one - loop approximation . extensions both to friedmann - robertson - walker spacetimes and to simple non - perturbative schemes such as the hartree approximation , while more complicated than the present analysis , present no fundamental difficulties . we note that minkowski space time is a good approximation in the latter stages of certain hybrid inflation models , and it will also allow comparison with much of the original reheating literature @xcite which often neglects the effects of expansion , allowing us to directly determine the role played by the mixing of the fields in the dynamics . the outline of the paper is as follows . we begin by considering the lagrangian for @xmath0 coupled scalar fields and set up our formalism for the quantization of the system . this is followed by an outline of the renormalization procedure . we then provide a summary of the results for the two - field case . we demonstrate the formalism with two examples : a simple reheating model and a hybrid inflation model motivated by supersymmetry . in the reheating model we investigate two relevant regimes discussed in detail in the literature @xcite , viz . , the narrow resonance regime and the broad resonance regime these different regimes occur depending on the choice of initial conditions . usually in these models the mixing effects of the fields were neglected by choosing a vanishing initial value for one of the mean fields : we are now able to treat the full system and to investigate these mixing effects . for this purpose we concentrate on studying the behavior of the fluctuation integrals for the different fields and the time - dependent mixing angle . depending on the regime , as the mean fields evolve , the effects of the mixing can be quite different . in the narrow resonance regime the mixing angle is very small and plays a sub - dominant role , whereas in the broad resonance regime the mixing effects are very important . therefore , we emphasize that neglecting the mixing could lead to incomplete results . supersymmetric hybrid models are a special realization of general hybrid inflationary models ( see e.g. refs . @xcite ) . based on a softly broken supersymmetry potential , the special feature of these models is the occurrence of only one coupling constant , whereas in nonsupersymmetric hybrid models there are at least two different couplings . thus , in the supersymmetric case there is only one natural frequency of oscillation for the mean fields as long as fluctuations are neglected . this leads to efficient particle production during the preheating stage in the early universe . however , we show below that , by taking into account the fluctuations and investigating the full mixed system , this feature of supersymmetric hybrid models can be lost in some regimes . this is because the effective mass corrections for the two fields are different in these regimes , which leads to a chaotic trajectory for the renormalized field equations of motion in a phase space which mimics the situation of a nonsupersymmetric hybrid model . it appears , then , that supersymmetric hybrid models can lose some of their attractiveness compared to general hybrid models . we work with the following lagrangian for real scalar fields @xmath5 with @xmath6 : @xmath7 = \sum_{i=1}^n \frac12 \partial_\mu \phi_i(x ) \partial^\mu \phi_i(x ) - v[\phi_i(x ) ] \ ; , \ ] ] where the potential is @xmath8 note that @xmath9 , @xmath10 , and @xmath11 are symmetric in each index , but are generally non - diagonal resulting in the mixing of the different fields . in what follows , subscripts and superscripts @xmath12 run over the values @xmath13 and we use a convention in which summation is assumed over repeated lowered indices , but not raised indices . we will expand each field about their expectation values ( taken to be space translation invariant ) : @xmath14 expanding the equations of motion and keeping terms to quadratic order in the fluctuations yields a one loop approximation . the equations of motion for the zero modes @xmath15 are determined via the tadpole condition . we have @xmath16 to this order , the fluctuations obey the equation @xmath17 with the mass matrix @xmath18 as indicated in the introduction , the complication that arises is not the fact that the mass matrix ( [ mij ] ) contains mixing between the various fields , rather that the mixing changes with time as the @xmath15 evolve according to ( [ phiieq ] ) . this means that if we diagonalize the mass matrix at one time , it will not generally be diagonal at any other time . nonetheless , it is most convenient to quantize in terms of a diagonal system at the initial time @xmath2 . we define the matrix @xmath19 and the corresponding fluctuation fields @xmath20 where @xmath21 is an orthogonal rotation matrix . @xmath22 is diagonal at the initial time : @xmath23 without summation over the raised index @xmath24 . the @xmath25 obey the equations of motion @xmath26 we quantize the system by defining a set of creation and annihilation operators @xmath27 and @xmath28 where @xmath29 corresponds to the _ in_-state quanta of frequency @xmath30 as the mixing changes in time , each of the fields @xmath25 is expanded in terms of all of the _ in_-state operators . we have @xmath31 \ ; .\ ] ] the initial conditions for the @xmath4 complex mode functions are @xmath32 it is convenient to define the fluctuation integrals @xmath33 from which it is straightforward to determine the contributions appearing in the zero mode equations ( [ phiieq ] ) : @xmath34 it will also prove convenient to introduce the rotated couplings @xmath35 which allows us to write the zero mode equations as @xmath36 while the mode functions obey the equations @xmath37 in addition to the equations of motion , it is useful to have an expression for the energy density of the system . this is particularly true when one completes numerical simulations of the system , since energy conservation is a powerful check of the accuracy of the simulations . after once again decomposing the fields into their expectation values and fluctuations , the energy density to one loop order is @xmath38 where we ve defined the integrals @xmath39 the mode integrals in the equation of motion defined by eq . ( [ xijfluct ] ) and in the energy density defined by eqs . ( [ xidotfluct ] ) , ( [ xigradfluct ] ) are divergent and have t be regulated , allowing for a renormalization of the theory . we require a method of extracting the divergent terms appearing in the mode integrals , a nontrivial task , since the mode equations vary in time and they are coupled . our aim is now , to find counter terms , which are independent of the initial value of the mean fields in order to formulate a finite theory . the correct choice of the initial condition for the fluctuations guarantees that the theory is renormalizable . one way to extract the divergences of the mode integrals is due to a wkb method which allows for a high momentum expansion of the mode functions . however , when the fields are coupled , as in the present case , the usual formulation of the wkb expansion runs into difficulties which are yet to be resolved . an alternative method has been developed @xcite which relies on a formal perturbative expansion in the effective masses and time derivatives of the masses of the fields . as such , it results in a series expansion of the mode functions in powers of @xmath40 and @xmath41 , etc . the first few terms in the series include the divergent parts of the integrals that are to be removed via renormalization . we begin by introducing the following ansatz for the mode functions : @xmath42 the first term on the right hand side anticipates a quadratic divergence in the quantities @xmath43 . we define the following potential @xmath44 the equations of motion for the mode functions eqs . ( [ mod ] ) can be written in a suggestive form with the help of eqs . ( [ ansatz0],[ansatz1 ] ) @xmath45 the terms on the right hand side of this expression are treated as perturbations to write the @xmath46 order by order in @xmath47 , with the initial conditions @xmath48 . to first order in @xmath47 , we have the equations of motion : @xmath49 the corresponding integral solutions for the real part of the @xmath50 s are : @xmath51 while the imaginary part is of order @xmath52 and does not contribute to the divergences @xcite . using these results , we find quadratic and logarithmic divergences : @xmath53 which must be removed via some renormalization procedure while also providing finite corrections to the parameters of the theory . to make the renormalization scheme explicit , we adopt dimensional regularization . we define the following divergent integrals @xmath54 where @xmath55 is an arbitrary renormalization point and @xmath56 carries the infinite contributions . in dimensional regularization @xmath57 is given by @xmath58 the infinite part of @xmath59 is found to be simply @xmath60 this leads to mass and coupling constant counter terms of the following form @xmath61 \ ; , \\ \delta g_{ijk } & = & \frac32 i_{-3}(\mu ) g_{ilm } \lambda_{lmjk } \ ; , \\ \label{dell } \delta \lambda_{ijkl } & = & \frac32 i_{-3}(\mu ) \lambda_{ijmn } \lambda_{mnkl } \ ; . \end{aligned}\ ] ] it is important to notice , that these counterterms are independent of the initial conditions of the mean fields @xmath15 . in addition to these counterterms , there are finite corrections of the parameters coming from the finite parts of the integrals ( [ xixjdiv ] ) : @xmath62 from this , we extract the following finite contributions to the couplings and mass : @xmath63 \ ; , \\ \delta m_{ij } & = & - \frac{1}{32\pi^2 } \left [ \lambda_{ijkk } d^k + \left(g_{ikl}g_{klj } + \lambda_{ijkl } m_{kl}\right ) \ln \frac{d^k}{\mu^2 } \right ] \ ; , \\ \delta g_{ijk } & = & - \frac{1}{8\pi^2 } g_{ilm}\lambda_{lmjk } \ln \frac{d^m}{\mu^2 } \ ; , \\ \label{dellf } \delta \lambda_{ijkl } & = & - \frac{3}{32\pi^2 } \lambda_{ijmn } \lambda_{mnkl } \ln \frac{d^m}{\mu^2 } \ ; .\end{aligned}\ ] ] these finite corrections are also contributing to the energy density . in addition we find a finite part due to the cosmological constant renormalization . the full , finite equations of motion become @xmath64 = 0 \ ; . \label{phiieqxren}\end{aligned}\ ] ] the two - field case is often encountered , and the physical applications we present in the next section are both in this category . it is therefore worthwhile to pause to look at a few details of such systems . we begin with a system of two real scalar fields @xmath65 and @xmath66 @xmath67 with the potential @xmath68 this lagrangian has the same form as that studied in the preceding section with the identifications @xmath69 the remaining components of @xmath11 are determined by the fact that it is symmetric in each of its indices . the mass matrix @xmath70 is @xmath71 for two fields , the orthogonal rotation matrix can be written in terms of a single mixing angle @xmath72 . the matrix has the form @xmath73 where the mixing angle is determined by the @xmath2 mass matrix @xmath70 , eq . ( [ m_ij ] ) , through the relation @xmath74 \ ; . \label{thetaeqn}\ ] ] the eigenvalues of @xmath75 are the diagonal elements of the matrix @xmath76 , eq . ( [ dij ] ) , at the initial time : @xmath77 with the values @xmath78 \ ; , \\ d^2 & = & \frac12 \left [ { \cal m}_{11}(0 ) + { \cal m}_{22}(0 ) - \sqrt{\left({\cal m}_{22}(0 ) - { \cal m}_{11}(0 ) \right)^2 + 4 { \cal m}^2_{12}(0 ) } \right ] \ ; .\end{aligned}\ ] ] for general times , the mass matrix for the fields @xmath79 and @xmath80 , writing @xmath81 and @xmath82 , is @xmath83 the zero mode equations , before renormalization , read @xmath84 where @xmath85 the total energy density of the system including the fluctuations can be expressed as @xmath86 now we have to formulate finite equations of motion and a finite energy density . we adopt the renormalization procedure of section [ renorm ] for the @xmath0 field case . by using the identifications ( [ ident ] ) it is to derive the appropriate counterterms for the two field case from eqs . ( [ dela])-([dell ] ) . we find in particular @xmath87 of course we get also similar to eqs . ( [ delaf])-([dellf ] ) in the @xmath0 field case finite corrections to the masses and couplings of the form @xmath88 where @xmath89 as a result of these finite corrections eqs . ( [ fin0]-[fin4 ] ) , the total lagrangian eq . ( [ lag ] ) is also modified . this is exactly the renormalized lagrangian which we needed . we also find an additional finite contribution to the classical lagrangian given by @xmath90\right \}\ , , \nonumber \\\end{aligned}\ ] ] and , the final zero mode equations for @xmath91 and @xmath92 are given by @xmath93 and , @xmath94 after writing down the finite zero mode equations of motion we also have to renormalize the energy density . again by using the ansatz ( [ ansatz0 ] ) we can extract the divergent terms of the fluctuation integrals in eq . ( [ energy0 ] ) . in addition to the quadratic and logarithmic divergences we find a quartic divergence . this leads to a counter term which acts as a cosmological constant and has the form @xmath95 altogether the divergent part of the energy density reads : @xmath96 this expression leads of course to the same counter terms we found for the equations of motion , and therefore also to the same finite corrections to couplings and masses . therefore it is straightforward to formulate a finite energy expression . now , we are in a position to discuss the physical applications of our problem . this we shall do in the next section , but first , we introduce one more quantity that is convenient in discussing the degree to which the mixing plays a role in the dynamics . to better understand how the system evolves in time , it is useful to have a measure of how much the fields @xmath91 and @xmath92 mix at each moment , and how this mixing evolves with the system . to provide us with a measure of the mixing , we introduce a time - dependent mixing angle @xmath97 , which is defined in terms of the time - dependent mass matrix @xmath98 , eq . ( [ m_ij ] ) : @xmath99 \ ; . \label{thetateqn}\ ] ] after setting up the technical framework , we are now in a position to investigate some relevant cosmological multi - field models for inflation . we begin our analysis with a simple two - field model often used for studying the phase of parametric amplification . ( this phase occurs just after the completion of inflation in chaotic inflationary models @xcite . ) this model provides a useful context to analyze the effects due to field mixing . next we turn our attention to a supersymmetric hybrid inflationary model , which is of particular interest in cosmology . as discussed in the literature ( see for example ref . @xcite ) particle production ( and hence reheating ) in these models is much more efficient compared to the nonsupersymmetric hybrid models . until now the mixing effects in these models have not been treated fully including back reaction effects of the quantum fluctuations in the mean field approximation . this approximation does not take rescattering processes into account and therefore we can not address the problem of thermalization . the reheating phase in chaotic inflationary models is characterized by two different regimes , which depend on the chosen initial conditions : the first is the narrow resonance regime , while the second is the case of broad resonance . in order to investigate these regimes we examine two different parameter sets , where only the initial values for the zero modes are varied . we find significant differences in the behavior of the fluctuation integrals as well as the mixing angle in the two regimes . to make the analysis as simple as possible , we set @xmath100 as well as @xmath101 . for the remaining parameters , we set @xmath102 , which just acts as a unit of mass , and @xmath103 . with these parameters , the case usually studied in the literature , @xmath104 , does not introduce any mixing between the fields since the off - diagonal elements of the @xmath91-@xmath92 mass matrix are proportional to @xmath92 . however , taking @xmath104 may not always be the case . for instance , @xmath92 field could take a large vacuum expectation value during inflation , provided @xmath92 is treated as a field other than the inflaton . for the purpose of illustration we may consider a non - zero initial condition for @xmath92 which is of order its effective mass @xmath105 at the beginning of the preheating stage and examine the consequences . the initial condition for @xmath91 in the narrow resonance regime is fixed by the condition @xmath106 ( remember , @xmath107 is fixed to be @xmath108 ) and for the broad resonance regime by @xmath109 . if we take @xmath110 to be approximately the planck scale as appropriate to the end of inflation , this would correspond to @xmath111 and @xmath112 for the two respective cases . note that these parameters are chosen to depict the phenomena of interest during a time scale that can be accurately simulated . the results are , in any case , representative of two field mixing in the narrow and broad resonance regimes regardless of the precise parameter values in any particular model . 1 shows the log of the three fluctuation integrals @xmath113 , @xmath114 and @xmath115 for the narrow resonance case . these are seen to be dominated by the exponential growth of @xmath115 , while the other contributions grow more slowly . therefore , the evolution is characterized by production of @xmath80 particles . we now turn to the broad resonance regime , where things look quite different . here , each of the fluctuation integrals grows rapidly as shown in fig . 2 . significant mixing of the species occurs along with copious particle production . the behavior of the fluctuation integrals is consistent with the behavior of the time - dependent mixing angle @xmath116 . here , the mixing plays a sub - dominant role in the narrow resonance regime ( fig.3 ) with the mixing angle remaining near zero . this means that @xmath80 predominantly corresponds to the @xmath92 field , such that the process is one of @xmath92 particle production , which is as expected . concentrating on the time - dependent mixing angle in the broad resonance regime , fig . 4 , significant mixing between the fields is observed . the rapid variation in the mixing angle indicates that mixing between the fields plays a very important role in the evolution of preheating in the broad resonance regime . the large influence this has on the behavior of the system is clear from the evolution of the zero mode components @xmath117 in fig . 5 and @xmath118 in fig . we now consider a hybrid inflationary model where the finite coupling of two fields plays an interesting role in the termination of slow roll inflation @xcite . the particular model we study is based on softly broken supersymmetry @xcite with the potential @xmath119 the field @xmath120 plays the role of an inflaton during inflation while the field @xmath0 is trapped in a false vacuum @xmath121 . the inflaton rolls down the potential along the @xmath120 direction to reach a critical value @xmath122 . once @xmath120 reaches its critical value , the effective squared mass for the @xmath0 field becomes negative and consequently it rolls down from the false vacuum to its global minimum through the mechanism of spinodal instability @xcite . thus , inflation comes to an end and both the fields begin oscillations around their respective minima given by @xmath123 this is the onset of the preheating stage , which has been discussed in the literature @xcite . the difference between the supersymmetric hybrid potential and non - supersymmetric hybrid potentials lies in different coupling constants . in eq . ( [ pot1 ] ) , there is only single coupling parameter @xmath124 , while in the non - supersymmetric version there can be at least two different coupling constants . the above potential , except for the @xmath120 mass term , can be derived very easily from the superpotential for f - term spontaneously supersymmetry breaking : @xmath125 the appearance of a mass term for @xmath120 in eq . ( [ pot1 ] ) is due to the presence of soft supersymmetry breaking . its presence is essential for slow roll inflation to produce adequate density perturbations and also to provide a correct tilt in the power spectrum @xcite . the height of the potential during inflation is given by @xmath126 . similar potentials to eq . ( [ pot1 ] ) can also be derived from d - term supersymmetry breaking as discussed in refs . @xcite . in these models the critical value @xmath127 and the height of the potential energy are related to the fayet - illipoulus term coming from an anomalous u(1 ) symmetry . as in any inflationary model , hybrid inflation is constrained by cobe @xcite . this imposes the bound @xmath128 where @xmath129 is one of the slow roll parameters which determines the slope of the power spectral index @xcite . for our purpose we fix it to be @xmath130 . in order to discuss the details of the physics we mention here the equivalence between eq . ( [ pot1 ] ) and eq . ( [ pot ] ) . this helps us to establish direct relationship with our earlier analysis : @xmath131 notice , that @xmath132 is negative . an interesting feature of the hybrid model is that irrespective of the values of the parameters @xmath124 , @xmath127 , and @xmath133 , as long as they satisfy the cobe constraints , the behavior of the mean fields follow a common pattern once they begin to oscillate @xcite . first of all , the mass term for the @xmath120 field , @xmath133 , becomes less dominant compared to the effective frequency for the two fields , which is given by the effective mass for the two fields during oscillations @xmath134 hence , there is a single natural frequency of oscillation , thanks to supersymmetry . since the masses of the fields are the same at the global minima , there exists a particular solution of the equations of motion for the @xmath120 and @xmath0 fields . their trajectory follows a straight line towards their global minima : @xmath135 the maximum amplitude attained by the @xmath120 field is @xmath136 , while the other field attains a larger amplitude @xmath137 . ( we remind readers that @xmath138 corresponds to the point where the effective mass for @xmath0 field changes its sign . ) this is the point of instability which we need to discuss here . from eq . ( [ pot1 ] ) , we notice that prior to the oscillations of the fields , and during the oscillations , the effective mass square for @xmath120 is always positive . however , this is not the case for @xmath0 , and its mass square can be positive as well as negative even during the oscillations of the fields , provided the amplitudes are large enough . if the amplitudes for @xmath120 and @xmath0 are such that they satisfy eq . ( [ traj ] ) , then the effective mass square for the @xmath0 field is in fact always negative for @xmath139 . if the amplitudes are large enough such that after the second order phase transition the initial amplitude for @xmath140 , it is then quite possible that near the critical point the effective mass for the field @xmath0 vanishes completely . as far as the motion of the mean field without including fluctuations is concerned this does not provide any new insight . however , if the fields are quantized then the perturbations in the field , especially for @xmath141 , grow exponentially because @xmath142 in eq . ( [ zerofreq ] ) becomes negative for sufficiently small momentum @xmath143 . this shows that the vacuum is unstable near the critical point @xmath127 . another intuitive way to appreciate this point is to consider the adiabatic condition for the vacuum . the adiabatic evolution for the zero mode evolution for @xmath0 field is given by @xmath144 . this condition is maximally violated at the point where the effective mass square for @xmath0 becomes zero , and , violation in adiabatic evolution of the zero mode for @xmath0 suggests that many fluctuations of @xmath141 are produced during the finite period when the adiabaticity is broken @xcite . this explanation is quite naive because the overall production of particles and fluctuations depends also upon the global behavior of the zero mode fields . the effect of corrections due to fluctuations might affect the production of particles and this is the point we are going to emphasize in our numerical simulations . in some sense the hybrid model is quite different from chaotic inflationary models . in chaotic models , the inflaton field rolls down with an amplitude @xmath145 , where @xmath146 is the mass of the oscillating field . however , in the hybrid model the amplitude of the oscillations die down very slowly , allowing many oscillations of the @xmath120 and @xmath0 fields in one hubble time . thus , one could expect large amplitude oscillations of the fields for a long time . this crucially depends on the parameter @xmath127 . if @xmath147 , then we notice that effective masses for @xmath120 and @xmath0 fields during oscillations are much larger than the hubble parameter . the hubble parameter is given by @xmath148 during inflation , so , @xmath149 provided the scale of @xmath127 is quite small compared to the planck mass , we can effectively neglect the expansion of the universe . in the supersymmetric hybrid model there are two regimes of interest . just after the mass square of the @xmath0 field becomes negative , the fields begin to oscillate with an amplitude which decreases as @xmath150 . when the field amplitude drops below @xmath151 , the amplitude of the oscillations decreases as @xmath152 . in this regime , when the expansion of the universe is neglected , the amplitude of the oscillations remains constant and the oscillations are harmonic : @xmath153 the corresponding evolution equation for the @xmath120 field can be found from eq . ( [ traj ] ) . in this paper we are neglecting the expansion of the universe . we will to concentrate upon two regimes : one with large amplitude oscillations which leads to the following parameters @xmath154 and , the other with small amplitude oscillations with the parameters @xmath155 the coupling constants are dimensionless while the other dimensionful parameters denoted in planck units . we find below a marked difference in the zero mode behavior of the fields @xmath120 and @xmath0 in these two cases depending on whether the fluctuations are taken into account or neglected . in parameter set ( [ par3 ] ) , we study the features of the fields with a large amplitude . this can happen when the @xmath120 and @xmath0 fields begin their oscillations just after the end of inflation . as mentioned earlier , after the end of inflation the maximum amplitude attained by the mean fields can be quite large @xmath138 , and @xmath156 . this is precisely the initial condition we have chosen for the mean fields for our numerics , as shown in fig . the values for @xmath157 and @xmath127 can be evaluated from eq . ( [ relation ] ) , which yields @xmath158 we notice that the evolution for @xmath120 and @xmath0 fields without taking into account the fluctuations are anharmonic , see fig . 7 , and , their trajectories in the @xmath159 plane is a straight line , as shown in fig . however , switching on the fluctuations leads to a completely chaotic trajectory as shown in fig . the departure from the straight line trajectory is quite significant and it tells us that the renormalized zero mode equations have different contributions to the parameter @xmath160 and to the effective mass of the @xmath0 field . this mismatch in the frequencies of the zero mode equations for @xmath161 and @xmath0 leads to an irregular trajectory . the other way to interpret this behavior is to think in terms of different effective mass corrections to @xmath120 and @xmath0 fields , such that the effective frequencies of the oscillations for @xmath120 and @xmath0 do not match each other at the bottom of the potential . this is certainly a nontrivial result . nonetheless , the result is quite expected from the fact that the amplitude of the oscillations are quite large and the effective mass for the @xmath0 field is zero at each and every oscillation when @xmath122 . as we mentioned earlier , the frequencies of the oscillations of the zero modes are different , as can be noticed in figs . 10 and 11 . the zero mode of @xmath120 influenced by the fluctuations oscillates around its minimum @xmath162 with a more rapid frequency than when fluctuations are neglected . this suggests that the effective mass correction to the zero mode for @xmath120 is coming solely from the finite coupling contribution from the @xmath0 field . ( note , that we have already set the bare mass for @xmath163 . ) the oscillations maintain the regularity with increasing and decreasing amplitude . however , the story is not the same for the zero mode behavior for the @xmath0 field . the amplitude of @xmath0 increases gradually and the frequency of the oscillations varies . we mention here that the effective mass for the @xmath0 field can vanish at a critical point . as a result , the adiabatic condition for the @xmath0 field is violated at those instants and this is the reason why the amplitude of the @xmath0 field is enhanced rather than suppressed . the evolution of the energy density is shown in fig . 12 . at first instance it seems quite odd that the energy density of the fluctuations does not increase further . one would naively expect a larger contribution of the energy density of @xmath164 and @xmath141 . this is not the case here . the energy density for the mean fields and the fluctuations are equally shared . the reason is the correction due to the fluctuations . these corrections modify the effective mass of the @xmath0 field and induce corrections to the coupling constants , namely @xmath165 and @xmath107 . the coupling constants are modified in such a way that the trajectory of zero mode fields become irregular . usually the production of fluctuations is not efficient in this case . this is quite similar to the situation of preheating in non - supersymmetric hybrid models @xcite . even though we started with a supersymmetric hybrid model where at the bottom of the potential there is a single effective frequency , the situation changes completely if the fluctuations are taken into account . essentially the coupling constants get a large correction which does not preserve an effective single coupling constant for the evolution of the zero mode fields . this is precisely the reason why the zero mode trajectory becomes irregular and also why the production of @xmath166 and @xmath141 is so low . as a next example of the supersymmetric hybrid model we choose parameter set ( [ par4 ] ) , with a small coupling @xmath124 and small @xmath127 @xmath167 in this example , the coupling between the fields is quite small ; @xmath168 , and also the initial conditions for @xmath120 and @xmath0 have been chosen such that the fields oscillate around their respective minima . the maximum amplitude for @xmath169 and @xmath170 is much below the critical point @xmath127 . we remind the readers that the chosen initial conditions for the oscillations do not come naturally just after the end of inflation , therefore this example does not represent a real situation . in spite of this we study the particular situation in order to notice the contrast in the behavior of the zero modes and the energy densities in the fluctuations . this particular set of initial conditions for @xmath171 offers an alternative example where spinodal instability in the @xmath0 field does not take place . as a result the effective mass square for the @xmath0 field never crosses zero and the adiabatic condition for the @xmath0 field is not strongly broken . the oscillations of the mean fields @xmath120 and @xmath0 are harmonic in nature , as shown in figs . 13 and 14 by the dotted lines . the amplitudes are constant with a frequency given by eq . ( [ frequency ] ) . the oscillations of the mean fields is governed by eqs . ( [ traj ] ) and ( [ evolv ] ) . the trajectory in the @xmath172 plane is a straight line whose slope is governed by eq . ( [ traj ] ) . the effect of the fluctuations is also quite expected in this case . the amplitudes of the zero mode for @xmath120 and @xmath0 fields decreases after a while and , in contrast to the preceding example , the frequency of the oscillations do not change very dramatically ; see the behavior of zero mode in solid lines in figs . 13 and 14 . the trajectories for the zero mode evolution remain a straight line in this case as shown in fig . this is quite reasonable for the parameters we have chosen , but an important observation is that the effect of fluctuations does not alter the straight line trajectory for the zero mode fields . this suggests that for small amplitude oscillations the corrections to the coupling constants ; @xmath173 and @xmath174 are such that the zero mode equations still have a similar oscillating frequency . this can be seen in figs . 13 and 14 . the production of @xmath164 and @xmath141 is not very significant because the energy density stored in @xmath164 and @xmath141 does not grow rapidly . thus the energy transfer from the zero modes to the fluctuation modes is not favorable for such a small amplitude oscillations as can be seen in fig . we conclude this section by mentioning that preheating in this supersymmetric hybrid model is quite interesting . depending on the amplitude of the oscillations of the fields , the behavior of the zero mode can be quite different . as a new feature we noticed that if the amplitude of the oscillations is close to the critical value @xmath127 , the effective mass square for the @xmath0 field becomes negative and as a result the fluctuations of the field grows exponentially . however , the effect of fluctuations alters the coupling constants in such a way that the trajectory of the zero modes become irregular . even though the adiabatic conditions seem to be broken for the @xmath0 field near the critical value , the energy density transferred from the zero mode to the fluctuations is not sufficient . our study reveals some interesting messages which we briefly mention here . we emphasize the point that the departure from the straight line trajectory of the zero mode is an essential feature of a supersymmetric hybrid model if the fluctuations are taken into account . even though , we have not included the hubble expansion , the results we have obtained are quite robust because supersymmetric hybrid inflationary models have a unique behavior of the fields which allows a smaller inflationary scale compared to the effective masses of the fields around their global minima . this suggests that during the oscillations , the expansion is felt much later , on a time scale determined by the parameters . this behavior is not shared by models where inflation is governed by a single field as in chaotic inflationary models . this undermines the production of quanta from the vacuum fluctuations . in several ways this affects the post inflationary radiation era of the universe . supersymmetric , weakly interacting dark matter formation and generation of baryonic asymmetry in the universe during preheating are the two most important frontiers which due to our results may warrant a careful revaluation . in order to substantiate our claim that a due consideration of fluctuations after the end of inflation is an important feature of any supersymmetric hybrid model , we have chosen an unphysical example which serves the purpose of making a vivid distinction . we stress here that the spinodal instability which is actually responsible for producing an irregular trajectory of the zero mode of the fields in a phase space is completely lacking if the amplitudes of the oscillations for @xmath175 are small compared to the critical value @xmath127 . this acts as a comparative study and shows that after the end of inflation , in a supersymmetric hybrid inflationary model , due to the spinodal instability in a field , a proper renormalization of the masses and the coupling constant have to be taken into account . we have introduced a formalism to address the dynamics of @xmath0 nonequilibrium , coupled , time varying scalar fields . we have shown that the one - loop corrections to the mean field evolution can be renormalized by dimensional regularization . for the sake of clarity and simplicity we restricted ourselves to minkowski spacetime while deriving the renormalized equations of motion and the energy density of the system . we applied our formalism to a two field case where we study the behavior of the quantized mode functions and the effect of fluctuations on the zero mode equations of motion for various parameters including small and large amplitude oscillations and large and weak coupling between two scalar fields . the varied couplings and amplitudes illustrate various facets of the intertwined dynamics of the two fields which lead to a deeper understanding of the production of self quanta and transfer of energy density between the fields in a cosmological context . as a special example we have chosen a two field inflationary model which is genuinely motivated by supersymmetry and thus preserves the effective masses of the fields to be the same in their local minima . the model , as a paradigm , predicts inflation which comes to an end via a smooth phase transition , and robustness of the model is confirmed by a slightly tilted spectrum of scalar density fluctuations within the cobe limit . the model parameters can be adjusted to give an inflationary scale covering a wide range of energy scales from tev to @xmath176 gev . the phase transition leads to a spinodal instability in one of the fields which leads to a coherent oscillations of the fields around their global minima . the instability occurs in one of the fields which demands careful study of the back reaction to an otherwise growing mean field in an intertwined coupled bosonic system . an account of influence of the fluctuations gives rise to uneven contribution to the renormalized masses of the fields . this results in an irregular trajectory of the zero mode in a phase space , which breaks the coherent oscillations of the two fields . this prohibits an excessive production of particles from the vacuum fluctuations . this requires a careful revaluation of the successes of the production of weakly interacting massive particles and baryogenesis via out of equilibrium decay in supersymmetric hybrid inflationary models . our study implies that exciting higher spin particles from the vacuum fluctuations of the coherent oscillations of the fields in a supersymmetric hybrid inflationary model demands careful reconsideration . even though , we have neglected the effect of expansion in our calculation , our results are robust enough to claim that the fluctuations in a supersymmetric hybrid model do not grow if the back reaction of the fluctuations are taken into account in the mean field evolution . an extension of our formalism to an expanding universe deserves separate attention . the authors are thankful to mar bastero - gil and michael g. schmidt for helpful discussion . we thank salman habib for helpful comments on the manuscript . a.m. is partially supported by * the early universe network * hprn - ct-2000 - 00152 . a. albrecht , p. j. steinhardt , m. s. turner and f. wilczek , phys . lett . * 48 * , 1437 ( 1982 ) ; a. d. dolgov and a. d. linde , phys . b*116 * , 329 ( 1982 ) ; l. f. abbott , e. farhi and m. wise , phys . b*117 * , 29 ( 1982 ) ; l. kofman , a. linde and a. starobinsky , phys . rev . lett . * 73 * , 3195 ( 1994 ) ; phys . d * 56 * 3258 ( 1997 ) ; d. boyanovsky , m. dattanasio , h.j . de vega , r. holman , d .- s . lee , phys . d*52 * 6805 , ( 1995 ) ; d. boyanovsky , d. cormier , h. j. de vega , r. holman , a. singh , m. srednicki , phys . rev . d*56 * , 1939 ( 1997 ) . examples of reheating involving higher spin fields are included in j. baacke , k. heitmann and c. ptzold , phys . d * 58 * , 125013 ( 1998 ) ; a. l. maroto and a. mazumdar , phys . lett . * 84 * , 1655 ( 2000 ) . d. boyanovsky , d. cormier , h. j. de vega , r. holman and p. kumar phys . d*57 * , 2166 ( 1998 ) ; d. cormier and r. holman , phys . rev.d*60 * , 41301 ( 1999 ) ; phys . rev . d*62 * , 23520 ( 2000 ) ; phys . lett . * 84 * , 5936 ( 2000 ) . f. cooper , s. habib , y. kluger , and e. mottola , phys . rev . * d55 * , 6471 ( 1997 ) ; d. boyanovsky , h. j. de vega , r. holman , and j. salgado , phys . rev . * d59 * , 125009 ( 1999 ) ; j. baacke and k. heitmann , phys . rev . * d62 * , 105022 ( 2000 ) . e. halyo , phys . lett . b * 387 * , 43 ( 1996 ) ; p. bintruy and g. dvali , phys . b * 388 * , 241 ( 1996 ) . for earlier work on this subject see : j. a. casas and c. muoz , phys . b*216 * , 37 ( 1989 ) ; j. a. casas , j. moreno , c. muoz and m. quiros , nucl . b*328 * , 272 ( 1989 ) .

coupled , multi - field models of inflation can provide several attractive features unavailable in the case of a single inflaton field . these models have a rich dynamical structure resulting from the interaction of the fields and their associated fluctuations . we present a formalism to study the nonequilibrium dynamics of coupled scalar fields . this formalism solves the problem of renormalizing interacting models in a transparent way using dimensional regularization . the evolution is generated by a renormalized effective lagrangian which incorporates the dynamics of the mean fields and their associated fluctuations at one - loop order . we apply our method to two problems of physical interest : ( i ) a simple two - field model which exemplifies applications to reheating in inflation , and ( ii ) a supersymmetric hybrid inflation model . this second case is interesting because inflation terminates via a smooth phase transition which gives rise to a spinodal instability in one of the fields . we study the evolution of the zero mode of the fields and the energy density transfer to the fluctuations from the mean fields . we conclude that back reaction effects can be significant over a wide parameter range . in particular for the supersymmetric hybrid model we find that particle production can be suppressed due to these effects .

arxiv

existence of dark matter ( dm ) has been established , and its thermal relic abundance has been determined by the wmap experiment @xcite . if the essence of dm is an elementary particle , the weakly interacting massive particle ( wimp ) would be a promising candidate . it is desired to have a viable candidate for the dark matter in models beyond the standard model ( sm ) . the wimp dark matter candidate can be accommodated economically by introducing only an inert scalar field @xcite , where we use `` inert '' for the @xmath1-odd property . the imposed @xmath1 parity ensures the stability of the dm candidate . phenomenology in such models have been studied in , e.g. , refs . @xcite . on the other hand , it has been confirmed by neutrino oscillation measurements that neutrinos have nonzero but tiny masses as compared to the electroweak scale @xcite . the different flavor structure of neutrinos from that of quarks and leptons may indicate that neutrino masses are of majorana type . in order to explain tiny neutrino masses , many models have been proposed . the seesaw mechanism is the simplest way to explain tiny neutrino masses , in which right - handed neutrinos are introduced with large majorana masses @xcite . another simple model for generating neutrino masses is the higgs triplet model ( htm ) @xcite . however , these scenarios do not contain dark matter candidate in themselves . in a class of models where tiny neutrino masses are generated by higher orders of perturbation , the dm candidate can be naturally contained @xcite . in models in refs . @xcite , the yukawa couplings of neutrinos with the sm higgs boson are forbidden at the tree level by imposing a @xmath1 parity . the same @xmath1 parity also guarantees the stability of the lightest @xmath1-odd particle in the model which can be the candidate of the dm as long as it is electrically neutral . in this paper , we consider an extension of the htm in which by introducing the @xmath1 parity @xmath6 is generated at the one - loop level and the dm candidate appears . in the htm , majorana masses for neutrinos are generated via the yukawa interaction @xmath7 with a nonzero vacuum expectation value ( vev ) of an @xmath2 triplet scalar field @xmath8 with the hypercharge of @xmath9 . the vev of @xmath8 is described by @xmath10 , where @xmath11 is the vev of the higgs doublet field @xmath12 and @xmath13 is the typical mass scale of the triplet field ; the dimensionful parameter @xmath14 breaks lepton number conservation at the trilinear term @xmath15 which we refer to as the @xmath14-term . as the simplest explanation for the smallness of neutrino masses , the mass of the triplet field is assumed to be much larger than the electroweak scale . on the other hand a characteristic feature of the htm is the fact that the structure of the neutrino mass matrix @xmath16 is given by that of the yukawa matrix , @xmath17 . the direct information on @xmath16 would be extracted from the decay @xmath18 @xcite if @xmath19 is light enough to be produced at collider experiments , where @xmath19 is the doubly charged component of the triplet field @xmath8 . at hadron colliders , the @xmath20 can be produced via @xmath21 @xcite and @xmath22 @xcite . the @xmath20 searches at the lhc put lower bound on its mass as @xmath23 @xcite , assuming that the main decay mode is @xmath24 . phenomenological analyses for @xmath20 in the htm at the lhc have also been performed in ref . triplet scalars can contribute to lepton flavor violation ( lfv ) in decays of charged leptons , e.g. , @xmath25 and @xmath26 at the tree level and @xmath27 at the one - loop level . relation between these lfv decays and neutrino mass matrix constrained by oscillation data was discussed in refs . @xcite . in order to explain the small @xmath0 with such a detectable light @xmath19 , the @xmath14 parameter has to be taken to be unnaturally much lower than the electroweak scale . therefore , it would be interesting to extend the htm in order to include a natural suppression mechanism of the @xmath14 parameter ( therefore @xmath0 ) in addition to the dm candidate . in our model , lepton number conservation is imposed to the lagrangian in order to forbid the @xmath14-term in the htm at the tree level while the triplet yukawa term @xmath28 exists . the vev of a @xmath1-even complex singlet scalar @xmath5 breaks the lepton number conservation by a unit . an @xmath2 doublet @xmath3 and a real singlet @xmath4 are also introduced as @xmath1-odd scalars in order to accommodate the dm candidate . then , the @xmath14-term is generated at the one - loop level by the diagram in which the @xmath1-odd scalars are in the loop . by this mechanism , the smallness of @xmath29 is realized , and the tiny neutrino masses are naturally explained without assuming the triplet fields to be heavy . the yukawa sector is then the same as the one in the htm , so that its predictions for the lfv processes are not changed . see refs . @xcite for some discussions about two - loop realization of the @xmath14-term - term in ref . @xcite is given with softly - broken @xmath30 symmetry , but the tree level @xmath14-term would be also accepted as a soft breaking term . the two - loop @xmath14-term in ref . @xcite is given with @xmath31 symmetry which is broken by a vev of a scalar @xmath32 , but the tree level @xmath33 seems allowed by the @xmath31 . ] . this paper is organized as follows . in sec . [ sec : htm ] , we give a quick review for the htm to define notation . in sec . [ sec:1-loop ] , the model for radiatively generating the @xmath14 parameter with the dark matter candidate is presented . some phenomenological implications are discussed in sec . [ sec : pheno ] , and the conclusion is given in sec . [ sec : concl ] . the full expressions of the higgs potential and mass formulae for scalar bosons in our model are given in appendix . in the htm , an @xmath34 triplet of complex scalar fields with hyperchage @xmath9 is introduced to the sm . the triplet @xmath8 can be expressed as @xmath35 where @xmath36 . the triplet has a new yukawa interaction term with leptons as @xmath37 where @xmath38 ( @xmath39 ) are the new yukawa coupling constants , @xmath40 [ @xmath41 are lepton doublet fields , a superscript @xmath42 means the charge conjugation , and @xmath43 ( @xmath44 ) denote the pauli matrices . lepton number ( @xmath45 ) of @xmath8 is assigned to be @xmath46 as a convention such that the yukawa term does not break the conservation . a vacuum expectation value @xmath47 [ @xmath48 breaks lepton number conservation by two units . the new yukawa interaction then yields the majorana neutrino mass term @xmath49 where @xmath50 . the scalar potential in the htm can be written as @xmath51 ^ 2 + \lambda_3\ , { { \text{tr}}}[(\delta^\dagger \delta)^2 ] \nonumber\\ & & { } + \lambda_4\ , ( \phi^\dagger \phi ) { { \text{tr}}}(\delta^\dagger \delta ) + \lambda_5\ , \phi^\dagger \delta \delta^\dagger \phi , \end{aligned}\ ] ] where @xmath52 [ @xmath53 is the higgs doublet field in the sm . the @xmath14 parameter can be real by using rephasing of @xmath8 . because we take @xmath54 , there is no nambu - goldstone boson for spontaneous breaking of lepton number conservation . the small triplet vev @xmath47 is generated by an explicit breaking parameter @xmath14 of the lepton number conservation as @xmath55 where @xmath11 ( @xmath56 ) is the doublet vev defined by @xmath57 . in order to obtain small neutrino masses in the htm , at least one of @xmath58 , @xmath38 , @xmath59 should be tiny . a small @xmath14 is an attractive option because @xmath60 can be small ( @xmath61 ) so that triplet scalars can be produced at the lhc . furthermore , large @xmath38 can be taken , which have direct information on the flavor structure of @xmath16 . there is , however , no reason why the @xmath14 parameter is tiny in the htm . in our model presented below , the @xmath14 parameter is naturally small because it arises at the one - loop level . since we try to generate the @xmath14-term in the htm radiatively , the term must be forbidden at the tree level . the simplest way would be to impose lepton number conservation to the lagrangian . the conservation is assumed to be broken by the vev of a new scalar field @xmath5 which is singlet under the sm gauge symmetry . notice that @xmath5 [ @xmath62 is a complex ( `` charged '' ) field with non - zero lepton number although it is electrically neutral . one might think that the vev of @xmath5 could be generated by using soft breaking terms of @xmath45 . however , the @xmath14-term is also a soft breaking term . therefore lepton number must be broken spontaneously in our scenario . one may worry about nambu - goldstone boson corresponds to the spontaneous breaking of the lepton number conservation ( the so - called majoron , @xmath63 ) . however the majoron which comes from gauge singlet field can evade experimental searches ( constraints ) because it interacts very weakly with matter fields @xcite . it is also possible to make it absorbed by a gauge boson by introducing the @xmath64 gauge symmetry to the model ( see , e.g. , ref . @xcite ) . in this paper we just accept the majoron without assuming the @xmath64 gauge symmetry for simplicity . if @xmath5 has @xmath65 , we can have a dimension-4 operator @xmath66 . this gives a trivial result @xmath67 at the tree level . although the dim.-4 operator could be forbidden by some extra global symmetries with extra scalars to break them , we do not take such a possibility in this paper . we just assume the @xmath5 has @xmath68 . then the lepton number conserving operator which results in the @xmath14-term is of dimension-5 as @xmath69 we consider below how to obtain the dim.-5 operator at the loop level by using renormalizable interactions of @xmath70 . ] . we restrict ourselves to extend only the @xmath71-singlet scalar sector in the htm because it seems a kind of beauty that the htm does not extend the fermion sector and colored sector in the sm . an unbroken @xmath1 symmetry is introduced in order to obtain dark matter candidates , and new scalars which appear in the loop diagram for the @xmath14-term are aligned to be @xmath1-odd particles . we emphasize that the unbroken @xmath72 symmetry is not for a single purpose to introduce dark matter candidates but utilized also for our radiative mechanism for the @xmath14-term . . list of particle contents of our one - loop model . [ cols="^,^,^,^,^,^,^",options="header " , ] we present the minimal model where the dim.-5 operator in eq . ( [ eq : dim5op ] ) is generated by a one - loop diagram with dark matter candidates . table [ tab:1-loop ] shows the particle contents . a real singlet scalar field @xmath4 and the second doublet scalar field @xmath3 [ @xmath73 are introduced to the htm in addition to @xmath5 . lepton numbers of @xmath4 and @xmath3 are 0 and @xmath74 , respectively . then @xmath75 conserves lepton number . in order to forbid the vev of @xmath3 , we introduce an unbroken @xmath1 symmetry for which @xmath4 and @xmath3 are odd . other fields are even under the @xmath1 . the yukawa interactions are the same as those in the htm . the higgs potential is given as @xmath76 here we show only relevant parts for radiative generation of the @xmath14-term . see appendix for the other terms . vacuum expectation values @xmath11 and @xmath77 [ @xmath78 are given by @xmath79 the @xmath1-odd scalars in this model are two cp - even neutral ones ( @xmath80 and @xmath81 ) , a cp - odd neutral one ( @xmath82 ) , and a charged pair ( @xmath83 ) . the cp - even scalars are defined as @xmath84 where @xmath85 and @xmath86 . squared masses of these scalars are given by @xmath87 notice that @xmath88 . we assume @xmath89 and then @xmath80 becomes the dark matter candidate . hereafter it is assumed that the mixing @xmath90 is small . the @xmath14-term is generated by the one - loop diagram . figure [ fig:1-loop ] is the dominant one in the case of small @xmath90 . then , the parameter @xmath14 is calculated as @xmath91 the one - loop induced @xmath14 parameter can be expected to be much smaller than @xmath92 . the suppression factor @xmath93 is estimated in sec . [ subsec : dm ] . we call it `` a. oryzae diagram '' @xcite . , title="fig : " ] -term . we call it `` a. oryzae diagram '' @xcite . , title="fig : " ] if @xmath94 , the dark matter candidate @xmath80 is given by @xmath95 approximately because we assume small mixing . see , e.g. , ref . @xcite for studies about the inert doublet scalar . let us assume @xmath96 and @xmath97 . as shown in ref . @xcite , these values satisfy constraints from the lep experiments @xcite and the wmap experiment @xcite . the mass splitting ( @xmath98 ) suppresses quasi - elastic scattering on nuclei ( @xmath99 mediated by the @xmath100 boson ) enough to satisfy constraints from direct search experiments of the dm @xcite . by using eqs . and , we obtain @xmath101 in order to be consistent with our assumption of small @xmath90 ( e.g. , @xmath102 ) , @xmath103 is required . the value in eq . results in @xmath104 for the greater value of @xmath105 , the larger @xmath106 is predicted . in particular , by taking @xmath105 to be the tev scale , we obtain @xmath107 , which yields @xmath108 for @xmath109 and @xmath60 to be at the electroweak scale . such a value for @xmath47 is suggested in the recent study of radiative corrections to the electroweak parameters @xcite . on the contrary , if we take @xmath110 which is allowed in a tiny region @xcite , values in eqs . and become 10 times smaller . we mention that the wmap constraint might be changed by a characteristic annihilation process @xmath111 where @xmath112 interaction is governed by @xmath109 ( not by a tiny @xmath14 ) . this additional process could sift allowed value of @xmath113 to lower one while @xmath114 due to the lep constraint . then , @xmath106 might become larger than the value in eq . because of larger @xmath115 . this undesired effect would be easily avoided if @xmath116 is away enough from @xmath117 . on the other hand , @xmath80 comes dominantly from @xmath4 if @xmath118 . see , e.g. , ref . @xcite for studies about the real inert singlet scalar . coupling @xmath119 of the @xmath120 interaction ( @xmath121 is the sm higgs boson ) determines annihilation cross section of @xmath80 and scattering cross section on nuclei . if we introduce the @xmath64 gauge symmetry , the scattering of @xmath4 on nuclei can be mediated also by the gauge boson @xmath122 . notice that the parameter @xmath123 ( and also the @xmath64 gauge coupling constant ) does not affect on @xmath14 parameter in eq . . let us estimate the magnitude of @xmath106 . in the usual htm , @xmath38 is expected to be @xmath124 for @xmath125 in order to suppress lfv processes . thus , we may accept @xmath126 as a value which is not too small . assuming @xmath127 for example gauge symmetry in order to eliminate the majoron , @xmath77 should be a little bit larger ( e.g. , @xmath128 ) due to constraint on the mass of @xmath122 . ] , we have a suppression factor as @xmath129 thus , even if the value of @xmath92 is in the tev scale , we can obtain @xmath130 although we need further suppression with @xmath131 to have @xmath132 . if we use @xmath133 , we obtain @xmath134 which can connect the tev scale @xmath92 to the ev scale @xmath6 . . the bosonic decay of @xmath135 contains information of @xmath92 indicated by a red blob . ] the characteristic feature of our model is that @xmath92 is much larger than @xmath14 . let us consider possibility to probe the large @xmath92 in collider experiments . a favorable process is shown in fig . [ fig : lhc ] for @xmath136 . for simplicity , we take @xmath137 which results in @xmath138 . recently , it was found in ref . @xcite that the electroweak precision test prefers @xmath139 in the htm where the electroweak sector is described by four input parameters . however , results in ref . @xcite might not be applied directly to our model is generated at the 1-loop level . ] because the scalar sector is extended . since @xmath140 is the most interesting decay in the htm , we assume @xmath141 in order to forbid @xmath142 . even in this case , the dm @xmath80 can be light enough ( @xmath143 ) so that @xmath1-even charged scalar @xmath144 ( @xmath145 ) can decay into @xmath146 via @xmath92-term which is indicated by a red blob in fig . [ fig : lhc ] . the partial decay width of @xmath147 is determined by @xmath148 while the width of @xmath149 is proportional to @xmath150 . taking @xmath151 , @xmath152 , @xmath153 , and @xmath154 for example , we have @xmath155 and @xmath156 . then , @xmath144 dominantly decays into @xmath146 . finally , @xmath157 decays into @xmath158 . therefore , from a production mechanism @xmath159 , we would have @xmath160 as a final state in fig . [ fig : lhc ] can be replaced with @xmath161 which decays into @xmath162 for @xmath136 . ] for which @xmath163 has the invariant mass @xmath164 at @xmath165 assuming that the value of @xmath165 has been known already . if @xmath166 , then @xmath144 decays via @xmath92-term into @xmath167 or @xmath168 followed by @xmath169 where a sizable @xmath170 is assumed is small , @xmath81 ( @xmath171 ) decays into @xmath172 . ] . because of @xmath173 through @xmath174 , we have again @xmath160 with @xmath175 from @xmath159 . in the usual htm in contrast , the final state with such @xmath163 is likely to include additional charged leptons ( @xmath176 from @xmath177 , @xmath178 from @xmath179 , etc . ) if @xmath20 decay dominantly into @xmath180 . therefore , our model would be supported if experiments observe final states which include jets and only two @xmath181 whose invariant mass gives @xmath175 . this potential signature might be disturbed by hadronic decays of @xmath182 because @xmath183 can result in @xmath160 with @xmath175 . realistic simulation is necessary to see the feasibility . we have presented the simple extension of the htm by introducing a @xmath1-even neutral scalar @xmath5 of @xmath68 , a @xmath1-odd neutral real scalar @xmath4 of @xmath184 , and a @xmath1-odd doublet scalar field @xmath3 of @xmath68 . the dm candidate @xmath185 in our model is made from @xmath4 and @xmath186 . the @xmath187 interaction which is the origin of @xmath0 ( and neutrino masses ) is induced at the one - loop level while the @xmath188 interaction exists at the tree level . because of the loop suppression for @xmath14 parameter , the model gives small neutrino masses naturally without using very heavy particles . for @xmath189 , the suppression factor @xmath190 is constrained by the dm relic abundance measured by the wmap experiment . we have shown that @xmath191 is possible . on the other hand , for @xmath192 , the suppression factor is somewhat free from experimental constraints on the dm . in our estimate , @xmath193 can be obtained as an example with @xmath126 . the characteristic feature of the model is that @xmath92 is not small while @xmath14 can be small . a possible collider signature which depends on @xmath92 would be @xmath160 with the invariant mass @xmath175 because more charged leptons are likely to exist in such final states in the usual htm . the work of s.k . was supported by grant - in - aid for scientific research nos . 22244031 and 23104006 . the work of h.s . was supported by the sasakawa scientific research grant from the japan science society and grant - in - aid for young scientists ( b ) no . the higgs potential of our model is given by @xmath194 where @xmath195 @xmath196 @xmath197 ^ 2 + \lambda_3\ , { { \text{tr}}}[(\delta^\dagger \delta)^2 ] \nonumber\\ & & { } + \lambda_{4\phi}\ , ( \phi^\dagger \phi)\ , { { \text{tr}}}(\delta^\dagger \delta ) + \lambda_{4\eta}\ , ( \eta^\dagger \eta)\ , { { \text{tr}}}(\delta^\dagger \delta ) \nonumber\\ & & { } + \lambda_{5\phi}\ , ( \phi^\dagger \delta \delta^\dagger \phi ) + \lambda_{5\eta}\ , ( \eta^\dagger \delta \delta^\dagger \eta ) \nonumber\\ & & { } + \lambda_{s1}\ , |s_1 ^ 0|^4 + \lambda_{s2}\ , ( s_2 ^ 0)^4 + \lambda_{s3}\ , |s_1 ^ 0|^2 ( s_2 ^ 0)^2 \nonumber\\ & & { } + \lambda_{s\phi 1}\ , |s_1 ^ 0|^2\ , ( \phi^\dagger \phi ) + \lambda_{s\phi 2}\ , ( s_2 ^ 0)^2\ , ( \phi^\dagger \phi ) \nonumber\\ & & { } + \lambda_{s\eta 1}\ , |s_1 ^ 0|^2\ , ( \eta^\dagger \eta ) + \lambda_{s\eta 2}\ , ( s_2 ^ 0)^2\ , ( \eta^\dagger \eta ) + \left\ { \lambda_{s\phi\eta}\ , s_1 ^ 0\ , s_2 ^ 0\ , ( \eta^\dagger \phi ) + \text{h.c . } \right\ } \nonumber\\ & & { } + \lambda_{s\delta 1}\ , |s_1 ^ 0|^2 { { \text{tr}}}(\delta^\dagger \delta ) + \lambda_{s\delta 2}\ , ( s_2 ^ 0)^2 { { \text{tr}}}(\delta^\dagger \delta ) .\end{aligned}\ ] ] all coupling constants are real because the phases of @xmath92 and @xmath174 can be absorbed by @xmath8 and @xmath5 , respectively . mass eigenstates of two @xmath1-even cp - even neutral scalars which are composed of @xmath198 and @xmath199 are obtained as @xmath200 their masses eigenvalues are given by @xmath201 where small contributions from @xmath47 are neglected . two @xmath1-even cp - odd neutral bosons ( @xmath202 and @xmath203 ) are nambu - goldstone bosons ; 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we extend the higgs triplet model so as to include dark matter candidates and a simple suppression mechanism for the vacuum expectation value ( @xmath0 ) of the triplet scalar field . the smallness of neutrino masses can be naturally explained with the suppressed value of @xmath0 even when the triplet fields are at the tev scale . the higgs sector is extended by introducing @xmath1-odd scalars ( an @xmath2 doublet @xmath3 and a real singlet @xmath4 ) in addition to a @xmath1-even complex singlet scalar @xmath5 whose vacuum expectation value violates the lepton number conservation by a unit . in our model , @xmath0 is generated by the one - loop diagram to which @xmath1-odd particles contribute . the lightest @xmath1-odd scalar boson can be a candidate for the dark matter . we briefly discuss a characteristic signal of our model at the lhc .

arxiv

the introduction of new quantum mechanical technologies promises to fundamentally alter the way we communicate . quantum key distribution ( qkd ) , for instance , will allow us to communicate in an intrinsically secure way @xcite . but new quantum communication technologies will require a new telecommunications infrastructure , one which is quantum - enabled . that is , this network must be able to properly accommodate the quantum properties that quantum communications inherently rely on . such a quantum network will contain many novel components , such as quantum memories @xcite , quantum repeaters @xcite , or , most generally , quantum channels . these components must each operate in a strictly quantum way . of course , no technology is perfect , and quantum technologies offer a new set of practical challenges . however , as we have learned from qkd , perfectly ideal devices are not a necessity . by shifting our efforts into classical post - processing of data , we can deal with imperfections in quantum technologies . the question then becomes , how much imperfection can be tolerated before a device is no longer operating in a sufficiently quantum way ? we can enforce a minimal quantum requirement on devices by insisting that they do not act as _ measure and prepare _ channels @xcite ( or , in the parlance of qkd , _ intercept and resend _ channels ) , since communication through such channels is equivalent to classical communication . indeed , this type of channel destroys any quantum correlations in bipartite states when one subsystem is sent through it . of course , this is just the minimum requirement . it is also important to quantify the quantum behaviour , as is done in the field of entanglement measures , or in qkd through the secret key rate . for quantum channels , we can ask , _ how well does the channel preserve quantum correlations in bipartite systems , when only one subsystem passes through it ? _ to study this question , we take a state with well - quantified quantum correlations , send one subsystem through the channel , and examine the output . we then compare the quantum correlations detectable in the output with the input correlations . in fact , as we shall see , we can test for these correlations in a so - called ` prepare and measure ' picture , bypassing the need to use actual bipartite states . a strong quantum channel is one which preserves all or nearly all of the quantum correlations . this idea corresponds to what we shall call the _ quantum throughput_. such a measure would allow us to characterize the suitability of devices for quantum communication tasks . the goal of this work is to illustrate that these ideas about device characterization via quantum throughput can be implemented in a meaningful way . although we will make specific choices regarding device types or quantification measures , the basic idea remains quite general , and our scheme can be extended and adapted to other methods as well . finally , if we picture a future quantum communications network consisting of many components , it should be evident that any device - testing procedure should be as experimentally practical as possible . ideally , we seek a testing scenario where a finite number of test states and a limited set of measurements are sufficient to understand the quantum throughput . the latter requirement is especially important for optical systems , which are perhaps the most natural choice of carrier for quantum information . in these systems , full tomography is not really a practical option because of the dimension of the hilbert space . we have previously examined quantum correlations in optical devices in a qualitative way @xcite ; in the present contribution , we will extend those results to provide a quantitative picture of optical devices . the rest of this paper is organized as follows . in sec . [ sec : quant ] we outline our quantitative device - testing scheme , focusing mainly on optical systems . we show how to estimate important parameters from homodyne measurements on the output , and how to use these estimates to make quantitative statements about the optical device . in sec . [ sec : results ] , we give the results of this quantification procedure for a wide class of optical channels , and examine the strength of our method . sec . [ sec : conclusion ] summarizes the paper , while appendices [ app : overlapbounds]-[app : offdiagbounds ] provide technical details and derivations . the quantum device testing procedure we employ is the same as the one found in @xcite . this protocol is based on the idea that a truly quantum channel should be distinguishable from those channels where the input quantum state is temporarily converted to classical data before a new quantum state is output , a so - called _ measure and prepare _ channel . measure and prepare channels are also called _ entanglement - breaking _ channels , as the two notions are equivalent @xcite . this provides a hint on how to quantify a channel s quantum throughput , namely by sending part of an entangled state through the channel and determining the amount of entanglement that still remains afterwards . to this end , imagine we have an entangled state of the form @xmath0\ ] ] where system @xmath1 is a qubit and system @xmath2 is an optical mode . we can assume , without loss of generality , that @xmath3 , so that @xmath4 and @xmath5 denote coherent states of opposite phase . this is an entangled state for all values @xmath6 , as can be seen by calculating the entropy of entanglement . keeping subsystem a isolated , an optical channel can be probed using subsystem b of this state , followed by local projective measurements @xmath7 by alice and homodyne measurements @xmath8 by bob . these expectation values , along with the knowledge of alice s reduced density matrix @xmath9 , can be used to determine just how much of the initial state s entanglement is remaining . of course , states like eq . ( [ eq : initialstate ] ) may be difficult to create and therefore not suited for practical device testing . however , notice that alice s reduced density matrix does not depend on what happens in the optical channel , nor on any of bob s measurement results . her expectation values can be completely determined from the initial state @xmath10 . indeed , alice s measurement results can be thought of as classical registers which merely record which mode state was sent through the device . this observation allows us to move from an entanglement - based ( eb ) picture to an equivalent ` prepare and measure ' ( pm ) scenario @xcite , in which alice s measurements are absorbed into the initial state preparation . in a pm scenario , we retain full knowledge of @xmath9 , in particular the off - diagonal coherence term @xmath11_{01}={\left\langle{\alpha}\right\vert { { \hspace{-0.1 em}}}{{\hspace{-0.1 em}}}\left . { -\alpha}\right\rangle}$ ] . we must insert this additional information _ by hand _ into the set of expectation values for @xmath12 . this distinguishes the expectation values from data which would come from using just a classical mixture of test states @xmath13 . other than this , the procedure is the same as the eb scenario described above . quantum correlations introduced in this way are referred to as ` effective entanglement . ' using this convenient theoretical trick , the testing protocol can be accomplished simply by probing the channel using a source which prepares one of the two conditional states @xmath14 with equal probability . if the measured expectation values , along with the inserted knowledge of @xmath9 , are not compatible with any separable qubit - mode state , then there is ( effective ) entanglement and the channel is certifiably quantum . exploiting the duality between the pm picture and the eb picture , we can quantify the quantum correlations remaining in the output state through a suitable entanglement measure . in turn , this can be compared to the entanglement of the state in eq . ( [ eq : initialstate ] ) to determine the quantum throughput . our main goal in this work is to give an estimate of the amount of effective entanglement observable in an optical system after transmission through an optical channel . our method is based on the following observation : when the two , initially pure , conditional states @xmath15 pass through the channel , they are subject to loss and noise , and evolve in general to mixed states @xmath16 on the infinite - dimensional mode hilbert space ; however , since we work with coherent states , this change in purity comes only from the noise . thus , for any loss value , if the noise introduced by the channel is not too high , then the output states @xmath17 and @xmath18 will still be nearly pure . in this case , most of the information about the state is still contained in a very small subspace of the full infinite dimensional hilbert space . estimating the ` most significant ' subspace on the mode system can therefore be quite useful . this subspace should contain as much information as possible about both conditional states . additionally , we will concentrate on the simplest non - trivial mode subspace , namely one of dimension 2 . writing the conditional output states @xmath17 and @xmath18 in terms of their eigenvectors , in order of descending eigenvalues , we have @xmath19 the most significant subspace is then the one formed using @xmath20 and @xmath21 as basis vectors . three parameters will be important to identify this subspace : @xmath22 , @xmath23 , and @xmath24 . we will estimate these parameters using homodyne detection . specifically , if @xmath25 is the annihilation operator for the mode at bob s detector , then a balanced homodyne detection scheme allows us to measure the field quadratures , here defined as @xmath26 we will use the mean values @xmath27 and the variances @xmath28 of the quadratures from both conditional states to estimate the three subspace parameters . exactly how this is done will be shown in the next part . with these parameters , we can build a @xmath29 density matrix @xmath30 , which corresponds to the projection of the full qubit - mode state @xmath12 onto the two - qubit subspace spanned by the basis @xmath31 . the idea is now to bound the entanglement of the full state @xmath12 using the entanglement of the projection @xmath30 . for this , we need to exploit the strong monotonicity under local operations and classical communication ( locc ) property found in many entanglement measures . specifically , if we perform a complete set of local measurements on a bipartite state @xmath32 , which yields ( perhaps using some classical communication ) the state @xmath33 with probability @xmath34 , then the strong monotonicity property captures the idea that the entanglement should not increase , on average , under this process . in other words , for a given measure @xmath35 , @xmath36 as the name implies , this is a stronger condition than just monotonicity under locc alone . in our case , the measurement consists of projecting the mode system onto the most significant subspace or onto the orthogonal complement . we denote the former projection by @xmath37 and the latter by @xmath38 . then , if we choose an entanglement measure @xmath35 with the strong monotonicity property , we have @xmath39 where @xmath40 and @xmath41 . for later practicality purposes , we would like to factor the probabilities through the entanglement measure , so that we work directly with unnormalized states . the unnormalized projected state is thus given by @xmath42 . we must be careful to choose an entanglement measure which , in addition to being a strong monotone , can be defined for unnormalized states and which permits a positive prefactor to be absorbed into the state . we will focus on the negativity @xcite in this work , for which this choice is justified . we will not attempt to estimate the second term in eq . ( [ eq : strongmono ] ) coming from the orthogonal projection ; we only note that it is non - negative , so that we have the bound @xmath43 in practice , the projected matrix @xmath30 will not be fully characterized and will contain open parameters . on the other hand , some constraints can be imposed on @xmath30 from our knowledge of the initial conditional states and the homodyne measurement results , as well as natural positivity ( @xmath44 ) and trace constraints @xmath45 for unnormalized @xmath30 ) . as a final step , we must determine the minimal entanglement of @xmath30 compatible with all allowed values of these open parameters , subject to the known constraints . we are left with the final relation @xmath46 which will be used as the basis for calculating bounds on @xmath47 . the next two subsections will cover how to estimate @xmath30 and how we minimize the entanglement over all compatible forms of @xmath30 . the first step in our method requires determining the projection @xmath30 of the full state @xmath12 onto the most - significant subspace . for this , we need to estimate the three parameters @xmath22 , @xmath23 and @xmath24 from the decomposition in eq . ( [ eq : eigendecomp ] ) . in ref . @xcite , which considers the related problem of effective entanglement verification using heterodyne measurements ( i.e. full knowledge of the @xmath48 function ) , several useful formulas for estimating these maximal eigenvalues and overlaps are given . these bounds are later refined in @xcite , where they are used to derive secret key rates for continuous variable quantum key distribution . here , we use these bounds as a starting point to build up a good estimate of the projected state for our quantification scheme . we will roughly follow the notation of @xcite in the following . first , since the conditional output states @xmath49 have unit trace , their maximal eigenvalues can be parameterized by @xmath50 , with @xmath51 $ ] . then eqs . ( 66 ) and ( 69 ) from @xcite give directly the following bound : @xmath52 = : u_j.\ ] ] this bound comes up several times , so it is denoted @xmath53 ( @xmath54 ) to make later equations more readable . importantly , the bound can be calculated using only the measured variances of the conditional states . estimating the overlap @xmath24 is more involved . we need to derive bounds on its magnitude based on our available information . again , we begin with bounds provided in refs . @xcite . with suitable relaxations , their bounds can be put into a specific form which will be more desirable for us later , as we would ultimately like to do a convex optimization . the specific details of this relaxation are straightforward , and are outlined in appendix [ app : overlapbounds ] . we will need an additional parameter , @xmath55 , which can be calculated directly using the measured first moments @xmath56 . defining two coherent states with the same means as the conditional states , @xmath57 the new parameter is given through the overlap of these coherent states , @xmath58 with this definition in place , we can give the relaxed bounds @xmath59 where @xmath60 and @xmath61 having these bounds , obtained purely through homodyne measurements , we can now move on to estimating the elements of the projected density matrix @xmath30 . we can already estimate matrix elements of the form @xmath62 using eq . ( [ eq : noisebound ] ) , but to build @xmath30 we also require bounds on the supplementary elements @xmath63 for @xmath64 . to get these , we first expand @xmath17 into its eigenbasis , eq . ( [ eq : eigendecomp ] ) . then , using the fact that @xmath65 $ ] for any normalized vector @xmath66 , we can easily derive the following bounds on the desired matrix element ( see appendix [ app : suppdiagbounds ] for full details ) : @xmath67 analogous bounds can be given for @xmath68 . finally , we need to estimate some elements of the off - diagonal blocks of @xmath30 , or else there would be no way to differentiate an entangeld state from a classical mixture of the conditional states . to this end , we label the off - diagonal block of the full density matrix @xmath12 by @xmath69 , so that it is naturally split into the form @xmath70 where the diagonal blocks correspond to the two conditional states . in the pm picture , we hold full knowledge of the alice s reduced density matrix @xmath71 where @xmath72 . each element in eq . ( [ eq : rhoa ] ) is the trace of the corresponding element in eq . ( [ eq : rhoblocks ] ) , so we can enforce the condition @xmath73 . using this as our starting point , and with an appropriate basis choice for system b , we can determine the following off - diagonal bounds which can be incorporated into @xmath30 : @xmath74 details on how to arrive at these inequalities can be found in appendix [ app : offdiagbounds ] . we now have sufficient information to construct a useful estimate of the projected state . to summarize , we have the quantities @xmath55 and @xmath75 , which can be calculated from measurements of the first moments and second moments , respectively . we want to determine @xmath30 , which is the projection of @xmath12 from eq . ( [ eq : rhoblocks ] ) onto the subspace spanned by @xmath31 . we have estimated some of the overlaps of @xmath12 with these basis vectors in eqs . ( [ eq : suppdiagbounds1]-[eq : suppdiagbounds2 ] ) and ( [ eq : offdiagbound1]-[eq : offdiagbound2 ] ) . these estimates depend only on the input parameter @xmath72 and on the output state quantities @xmath76 , @xmath77 , and @xmath78 . this last overlap quantity is itself bounded to a region defined by eqs . ( [ eq : overlapbounds]-[eq : bupper ] ) , which depends only on @xmath76 , @xmath77 and @xmath55 . hence , for a fixed input overlap @xmath79 and a fixed set of homodyne measurement results , we have a parameter region which forms a set of constraints on @xmath30 . this region must be searched to find the minimal entanglement compatible with @xmath30 . we will now move on to address the question of how to find the minimal entanglement compatible with our constraints . as mentioned earlier , we will choose the negativity as the entanglement measure for demonstrating our method . in principle , we would like to find the minimal entanglement using the methods of semidefinite programming . but we must make some simplifications and relaxations which will allow us to do so . first , we exploit the fact that local unitary operations can not change the quantity of entanglement . therefore , without loss of generality , we can assume that the overlap of the maximal eigenstates is real and positive ( since this can be accomplished by a relative change of phase on subsystem b ) . @xmath80 as well , we can perform local phase changes on subsystem a , which allows us to also make the restriction @xmath81 the other off - diagonal element of interest , @xmath82 , is in general still a complex number . the main problem is that eq . ( [ eq : offdiagbound2 ] ) is a _ non - convex _ constraint on @xmath83 . to use this constraint in a semidefinite program , we have to replace it with a set of convex constraints . we accomplish this by denoting the right - hand side of eq . ( [ eq : offdiagbound2 ] ) as @xmath84 and expanding our constraints to the region @xmath85 this new constraint still non - convex , but we can search for the minimum entanglement independently in each of the four quadrants , where the constraints are convex ( see fig . [ fig : convexregions ] ) , and take the minimum over these four searches . the final result will be a lower bound to the minimum entanglement in the region constrained by eq . ( [ eq : offdiagbound2 ] ) . we can extend this idea further , replacing the inscribed square from fig . [ fig : convexregions ] with any other inscribed polygon . with more sides , we can better approximate the non - convex constraint eq . ( [ eq : offdiagbound2 ] ) , but this will also increase the number of convex subregions which must be searched to find the overall minimum . numerical evidence indicates that the minimum entanglement is often , though not always , found at a point outside the circle . we tested with an inscribed octagon and it was not found to alter the final results significantly . the final hurdle comes from the overlap @xmath24 . since the maximal eigenstates will in general have a non - zero overlap ( indeed , for zero overlap , we will not find any entanglement in @xmath30 ) , we must construct an orthogonal basis in order to explicitly write down a matrix representing @xmath30 . doing so introduces matrix elements that are both linear and quadratic in the overlap @xmath24 . if the overlap is used as a parameter in the semidefinite programming , this non - linear dependence becomes problematic . fortunately , it turns out that to find the minimal entanglement we only need to consider the case where the overlap takes the largest allowed value , i.e. @xmath86 . the reason for this is that , for fixed values of @xmath22 , @xmath23 , and @xmath24 , there always exists a cptp map on the b subsystem which preserves the maximal eigenvalues while making the corresponding overlap larger . such a local map can not increase the entanglement , so indeed the minimal entanglement will be found at @xmath86 . this useful result will be shown in detail elsewhere @xcite . in the previous section , we outlined a method for calculating the effective entanglement in optical systems . this began with the observation that we can get bounds just by looking at the most significant two - qubit subsystem . the remainder of sec . [ sec : quant ] provided the necessary tools to allow us to calculate these bounds efficiently as a semidefinite program . now that all the pieces are in place , we can turn to applying our scheme . to illustrate our quantification method , we use data corresponding to the action of the optical channel on the field quadratures , which we assume to be symmetric for both signal states and for both quadratures . these symmetry assumptions are made solely to aid the graphical representation of our results , and our method does not rely on them . it is also important to note that , beyond the symmetry , we do not make any assumptions about how the channel works . in the absence of experimental data , we merely parameterize the channel s effect on the first quadrature moments by a loss parameter and on the second moments by the excess noise . specifically , if the means of the two conditional output states are denoted by @xmath87 from eq . ( [ eq : cohmean ] ) , then the loss is parameterized through the transmittivity @xmath88 and the symmetric excess noise ( expressed in shot noise units ) by @xmath89 the input states are characterized entirely by the overlap parameter @xmath90 . the quantification program was carried out using the negativity @xcite , @xmath91 this measure has all the properties demanded by our quantification method , but more importantly , the trace norm @xmath92 of a matrix can be computed efficiently as a semidefinite program @xcite . we have normalized the negativity so that a maximally entangled two - qubit state has @xmath93 . our calculations were done in matlab using the yalmip interface @xcite along with the solver sdpt3 @xcite . our main results are shown in fig . [ fig : mainresults ] , where the minimal negativity of @xmath30 compatible with the initial overlap @xmath90 and excess noise @xmath94 is given , for various values of the transmittivity @xmath95 . this quantity gives a lower bound on the negativity of the full state @xmath12 . the entanglement of the initial state , eq . ( [ eq : initialstate ] ) , is also shown as a function of the initial overlap in fig . ( [ fig : test1 ] ) . for figs . ( [ fig : test2]-[fig : test3 ] ) , the modification @xmath96 is made to eq . ( [ eq : initialstate ] ) for these comparisons . the initial entanglement can be compared with the calculated bounds to help understand the quantum throughput of a device . in the limit of zero excess noise and zero loss , our entanglement bound is tight with the initial entanglement . our bounds are quite high for very low noise , but they become lower as the measurement results get more noisy . at some point , a non - trivial entanglement bound can no longer be given , despite the fact that quantum correlations can still be proven for higher noise values ( cf . @xcite ) . as well , for larger loss values , the tolerance for excess noise is lower , and the region where non - trivial bounds can be given becomes smaller . the exact noise value where our bounds become trivial depends on the initial overlap and on the measured loss , but the highest tolerable excess noise is around 5% of the vacuum for @xmath97 . this shrinks to about 3% for a transmittivity of @xmath98 . though the quantification region is small , it is within the limits of current experimental technology @xcite . some entanglement degradation should be expected as the noise is increased , but , as mentioned earlier , entanglement can still be verified ( though not previously quantified ) under the same testing scenario up to much higher noise values than seen here @xcite . thus , our bounds do not provide the full picture . the weakening of the bounds with higher noise is mainly due to the estimation procedure . certain approximations become cruder ( though still valid ) as the noise increases . first , for higher noise , the conditional states become more mixed , spreading out into more of the infinite - dimensional mode hilbert space . this leads to additional information being lost when we truncate down from @xmath12 to @xmath30 . another problem stems from the bounds we use to estimate @xmath30 . higher noise leads to weaker bounds on the maximal eigenvalues from eq . ( [ eq : noisebound ] ) , which weakens all other inequalities . to examine the effects of these two approximations , we briefly consider a simple channel where the test state , eq . ( [ eq : initialstate ] ) , is mixed at a @xmath99 beam - splitter with a thermalized vacuum . the first moments reduce by a factor of @xmath100 , and the increased variances of the output optical states can be determined from the mean photon number @xmath101 of the thermal state . for @xmath102 , the conditional output states are displaced thermal states . the reason for studying this channel is that we can _ exactly _ determine the maximal eigenvalues @xmath22 , @xmath23 , and the overlap @xmath24 . this allows us to study our approximations independently , since we decouple the effects of the two - qubit projection from the homodyne parameter estimation ( in practice , of course , our quantification scheme must use both ) . in fig . ( [ fig : comparison ] ) we show the result of the quantification scheme , when this extra information is included . we see that the tolerable excess noise is @xmath103 of the vacuum , more than three times what it would be if we had to estimate the eigenvalues and overlap using homodyne results ( cf ( [ fig : test3 ] ) ) . also included in fig . ( [ fig : comparison ] ) is an entanglement verification curve , obtained using the methods of @xcite . any points with lower noise than this verification curve must come from entangled states . the two - qubit projection is tight to the entanglement verification curve for low overlaps . for higher values , the projection becomes weaker , only working to about half the noise value that the entanglement verification curve reaches . ideally , we want to be able to calculate non - trivial values for the entanglement wherever it be verified . this would give us a true quantitative complement to existing entanglement verification methods . one obvious extension to our method would be to truncate the mode subspace using the two largest eigenstates from each conditional state , or even more . in theory , this would strictly improve the estimates . however , in practice , this will increase the complexity of the quantification calculation , since some simplifying assumptions ( i.e. certain overlaps are real ) may no longer be valid . as well , the number of additional minimizations we have to do , as in our non - convex relaxation of eq . ( [ eq : offdiagbound2 ] ) , increases fourfold with each added dimension . another approach might therefore be necessary to overcome this problem . nevertheless , the quantification scheme outlined here is a useful method for characterizing the degree of quantumness of optical channels , especially when these channels introduce low noise . we have outlined a method for quantifying the effective entanglement in qubit - mode systems using only homodyne measurement results and knowledge of the initial preparation . this quantification method works particularly well if the mode subsystem exhibits low noise . by combining this quantification scheme with a device testing scenario which uses two nonorthogonal test states , one can examine how strongly an optical device or experiment is operating in the quantum domain . our scheme provides a useful tool for understanding the quantum nature of optical devices , especially the question of how well they preserve quantum correlations . in this appendix , we derive the bounds from eqs . ( [ eq : blower]-[eq : bupper ] ) for the absolute value of the overlap of the maximal eigenstates , @xmath104 . from @xcite , we have the following : _ overlap bounds . _ let the largest eigenvalue of @xmath105 be parameterized by @xmath106 and let the fidelity between the conditional states and the coherent states @xmath107 from eq . ( [ eq : cohmean ] ) be given by @xmath108 and let @xmath58 then the following holds : @xmath109 with @xmath110 and @xmath111 since we can not calculate @xmath112 or @xmath113 in practice , we now modify these bounds from the above form found in @xcite to one involving only the parameters @xmath55 ( calculated from first moments ) and the @xmath53 ( calculated from second moments ) . to do this , we make use only of the obvious inequality @xmath114 from this , we can easily derive the following auxiliary inequalities : @xmath115 it is important to note that the second and third inequalities only hold so long as @xmath116 . for symmetric noise , the value @xmath117 corresponds to @xmath118 , almost twice the vacuum variance . this value is far outside the region where our method gives non - trivial bounds , so it is not an issue . substituting the inequalities ( [ eq : aux1]-[eq : aux3 ] ) into eqs . ( [ eq : oldoverlaplowerbound ] ) and ( [ eq : oldoverlapupperbound ] ) , we arrive at the bounds given in eqs . ( [ eq : blower]-[eq : bupper ] ) . here we aim to bound the quantities @xmath63 for @xmath64 , as found in eqs . ( [ eq : suppdiagbounds1]-[eq : suppdiagbounds2 ] ) . an eigenbasis expansion of @xmath17 leads to @xmath119 a lower bound can be derived in a similar way : @xmath120 the bounds for @xmath121 follow by interchanging indices . this appendix outlines the derivation of the off - diagonal bounds from eqs . ( [ eq : offdiagbound1]-[eq : offdiagbound2 ] ) . we completely know @xmath9 , which constrains that we must have @xmath122 . first , we consider the full density matrix @xmath12 in the basis defined by @xmath123 for system @xmath1 and the eigenbasis of @xmath17 , @xmath124 , for system b. we can still write this in the block form of eq . ( [ eq : rhoblocks ] ) , where we denote the diagonal elements of the block @xmath18 by @xmath125 and the diagonal elements of the block @xmath69 by @xmath126 ( the diagonal elements of @xmath17 are its eigenvalues ) . using the triangle inequality , we have @xmath127 from positivity of @xmath12 , we find @xmath128 and from the cauchy - schwarz inequality , @xmath129 the first sum is just @xmath130 and the second is @xmath131 . now , using the bounds from appendix [ app : suppdiagbounds ] , we get @xmath132 which we can substitute above to obtain @xmath133 replacing @xmath134 with @xmath135 , we are led to the off - diagonal bound @xmath136 by applying the same arguments using the eigenbasis of @xmath18 , we can arrive at an analogous bound for @xmath137 .

quantum communication relies on optical implementations of channels , memories and repeaters . in the absence of perfect devices , a minimum requirement on real - world devices is that they preserve quantum correlations , meaning that they have some thoughput of a quantum mechanical nature . previous work has verified throughput in optical devices while using minimal resources . we extend this approach to the quantitative regime . our method is illustrated in a setting where the input consists of two coherent states while the output is measured by two homodyne measurement settings .

arxiv

over the past 83 years , the study of dipole moments of elementary particles has provided a wealth of information on subatomic physics . from the pioneering work of stern@xcite through the discovery of the large anomalous magnetic moments of the proton@xcite and neutron@xcite , the ground work was laid for the discovery of spin , of radiative corrections and the renormalizable theory of qed , of the quark structure of baryons and the development of qcd . a charged particle with spin @xmath2 has a magnetic moment @xmath3 where @xmath4 is the gyromagnetic ratio , @xmath5 is the anomaly , and the latter expression is what one finds in the particle data tables.@xcite the dirac equation tells us that for spin one - half point - like particles , @xmath6 for spin angular momentum , and is unity for orbital angular momentum ( the latter having been verified experimentally@xcite ) . for point particles , the anomaly arises from radiative corrections , two examples of which are shown in fig . [ fg : aexpan ] . the lowest - order correction gives the famous schwinger@xcite result , @xmath7 , which was verified experimentally by foley and kusch.@xcite the situation for baryons is quite different , since their internal quark structure gives them large anomalies . in general @xmath5 ( or @xmath8 ) is an expansion in @xmath9 , @xmath10 with 1 diagram for the schwinger ( second - order ) contribution , 5 for the fourth order , 40 for the sixth order , 891 for the eighth order . the qed contributions to electron and muon 2 have now been calculated through eighth order , @xmath11 and the tenth - order contribution has been estimated.@xcite .,scaledwidth=45.0% ] .transformation properties of the magnetic and electric fields and dipole moments . [ cols="^,^,^,^",options="header " , ] the magnetic and electric dipole moments can be represented as the real and imaginary parts of a generalized dipole operator @xmath12 , and the interaction lagrangian becomes @xmath13 \mu f_{\alpha \beta}\ ] ] with @xmath14 and @xmath15 . the electron anomaly is now measured to a relative precision of about four parts in a billion ( ppb),@xcite which is better than the precision on the fine - structure constant @xmath16 , and kinoshita has used the measured electron anomaly to give the best determination of @xmath16.@xcite the electron anomaly will be further improved over the next few years.@xcite the muon anomaly is measured to 0.5 parts per million ( ppm).@xcite the relative contributions of heavier particles to @xmath5 scales as @xmath17 , so the muon has an increased sensitivity to higher mass scale radiative corrections of about 40,000 over the electron . at a precision of @xmath18 ppm , the muon anomaly is sensitive to @xmath19 gev scale physics . the standard model value of @xmath20 has measurable contributions from three types of radiative processes : qed loops containing leptons ( @xmath21 ) and photons;@xcite hadronic loops containing hadrons in vacuum polarization loops;@xcite and weak loops involving the @xmath22 and @xmath23 weak gauge bosons ( the standard model higgs contribution is negligible),@xcite @xmath24 a significant difference between the experimental value and the standard model prediction would signify the presence of new physics . a few examples of such potential contributions are lepton substructure , anomalous @xmath25 couplings , and supersymmetry.@xcite the cern experiment@xcite observed the contribution of hadronic vacuum polarization shown in fig . [ fg : had](a ) at the 8 standard deviation level . unfortunately , the hadronic contribution can not be calculated directly from qcd , since the energy scale is very low ( @xmath26 ) , although blum@xcite has performed a proof of principle calculation on the lattice . fortunately dispersion theory gives a relationship between the vacuum polarization loop and the cross section for @xmath27 , @xmath28 where @xmath29 and experimental data are used as input . the factor @xmath30 in the dispersion relation , means that values of @xmath31 at low energies ( the @xmath32 resonance ) dominate the determination of @xmath33 . in principle , this information could be obtained from hadronic @xmath34 decays such as @xmath35 , which can be related to @xmath36 annihilation through the cvc hypothesis and isospin conservation.@xcite however , inconsistencies between information obtained from @xmath36 annihilation and hadronic tau decays , plus an independent confirmation of the cmd2 high - precision @xmath36 cross - section measurements by the kloe collaboration,@xcite have prompted davier , hcker , et al , to state that until these inconsistencies can be understood only the @xmath36 data should be used to determine @xmath33.@xcite conversion , showing the relevant slepton mixing matrix elements . the mdm and edm give the real and imaginary parts of the matrix element , respectively . ] the hadronic light - by - light contribution ( see fig . [ fg : had](e ) ) has been the topic of much theoretical investigation.@xcite unlike the lowest - order contribution , it can only be calculated from a model , and this contribution is likely to provide the ultimate limit to the precision of the standard - model value of @xmath20 . one of the very useful roles the measurements of @xmath20 have played in the past is placing serious restrictions on physics beyond the standard model . with the development of supersymmetric theories as a favored scheme of physics beyond the standard model , interest in the experimental and theoretical value of @xmath20 has grown substantially . susy contributions to @xmath20 could be at a measurable level in a broad range of models . furthermore , there is a complementarity between the susy contributions to the mdm , edm and transition moment for the lepton - flavor violating ( lfv ) process @xmath37 in the field of a nucleus . the mdm and edm are related to the real and imaginary parts of the diagonal element of the slepton mixing matrix , and the transition moment is related to the off diagonal one , as shown in fig . [ fg : susy ] . this reaction , along with the companion lfv decay @xmath38 , will be searched for in `` next generation '' experiments now under construction.@xcite from neutrino oscillations we already know that lepton flavor is violated , and this violation will be enhanced if there is new dynamics at the tev scale . this same new physics could also generate measurable effects in the magnetic and electric dipole moments of the muon as well.@xcite the method used in the third cern experiment and the bnl experiment are very similar , save the use of direct muon injection@xcite into the storage ring,@xcite which was developed by the e821 collaboration . these experiments are based on the fact that for @xmath39 the spin precesses faster than the momentum vector when a muon travels transversely to a magnetic field . the spin precession frequency @xmath40 consists of the larmor and thomas spin - precession terms . the spin frequency @xmath40 , the momentum precession ( cyclotron ) frequency @xmath41 , are given by @xmath42 the difference frequency @xmath43 is the frequency with which the spin precesses relative to the momentum , and is proportional to the anomaly , rather than to @xmath8 . a precision measurement of @xmath20 requires precision measurements of the muon spin precession frequency @xmath44 , and the magnetic field , which is expressed as the free - proton precession frequency @xmath45 in the storage ring magnetic field . the muon frequency can be measured as accurately as the counting statistics and detector apparatus permit . the design goal for the nmr magnetometer and calibration system was a field accuracy of 0.1 ppm . the @xmath46 which enters in eq . [ eq : omeganoe ] is the average field seen by the ensemble of muons in the storage ring . in e821 we reached a precision of 0.17 ppm in the magnetic field measurement . an electric quadrupole@xcite is used for vertical focusing , taking advantage of the `` magic '' @xmath47 at which an electric field does not contribute to the spin motion relative to the momentum . with both an electric and a magnetic field , the spin difference frequency is given by @xmath48 , \label{eq : tbmt}\ ] ] which reduces to eq . [ eq : omeganoe ] in the absence of an electric field . for muons with @xmath49 in an electric field alone , the spin would follow the momentum vector . the experimental signal is the @xmath50 from @xmath51 decay , which were detected by lead - scintillating fiber calorimeters.@xcite the time and energy of each event was stored for analysis offline . muon decay is a three - body decay , so the 3.1 gev muons produce a continuum of positrons ( electrons ) from the end - point energy down . since the highest energy @xmath50 are correlated with the muon spin , if one counts high - energy @xmath50 as a function of time , one gets an exponential from muon decay modulated by the @xmath52 precession . the expected form for the positron time spectrum is @xmath53 $ ] , however in analyzing the data it is necessary to take a number of small effects into account in order to obtain a satisfactory @xmath54 for the fit.@xcite the data from our 2000 running period is shown in fig . [ fg : wig00 ] the experimental results from e821 are shown in fig . [ fg : amu ] , with the average @xmath55 which determines the `` world average '' . the theory value@xcite @xmath56 is taken from hcker et al.,@xcite , which updates their earlier analysis@xcite with the kloe data;@xcite and from hagiwara , et al.,@xcite who use a different weighting scheme for the experimental data when evaluating the dispersion integral but do not include the kloe data . when this theory value is compared to the standard model value using either of these two analyses@xcite for the lowest - order hadronic contribution , one finds @xmath57 or a discrepancy of 2.7 standard deviations . . the strong interaction contribution is taken from references @xcite and @xcite . ] to show the sensitivity of our measurement of @xmath20 to the presence of virtual electroweak gauge bosons , we subtract off the electroweak contribution of @xmath58 from the standard model value , compare with experiment and obtain @xmath59 a 4.7 standard deviation discrepancy . this difference shows conclusively that e821 was sensitive to physics at the 100 gev scale . at present , it is inconclusive whether we see evidence for contributions from physics beyond the standard - model gauge bosons . with each data set , the systematic error was reduced , and for the final data set taken in 2001 the systematic error on @xmath60 was 0.27 ppm with a statistical error of 0.66 ppm . given the tantalizing discrepancy between our result and the latest standard - model value , and the fact that the hadronic error could be reduced by about a factor of two over the next few years,@xcite we submitted a new proposal to brookhaven to further improve the experimental measurement . the goal of this new experiment is @xmath61 ppm total error , with the goal of controlling the total systematic errors on the magnetic field and on the muon frequency measurement to 0.1 ppm each . our proposal@xcite was given enthusiastic scientific approval in september 2004 by the laboratory , and has been given the new number , e969 . negotiations are underway between the laboratory and the funding agencies to secure funding . a letter of intent ( loi ) for an even more precise 2 experiment was also submitted to j - parc.@xcite in that loi we proposed to reach a precision below 0.1 ppm . since it is not clear how well the hadronic contribution can be calculated , and whether the new brookhaven experiment e969 will go ahead , we will evaluate whether to press forward with this experiment at a later time . our loi at j - parc@xcite was predicated on pushing as far as possible at brookhaven before moving to japan . while the mdm has a substantial standard model value , the predicted edms for the leptons are unmeasurably small and lie orders of magnitude below the present experimental limits given in table [ tb : edm ] . an edm at a measurable level would signify physics beyond the standard model . susy models , and other dynamics at the tev scale do predict edms at measurable levels.@xcite a new experiment to search for a permanent edm of the muon with a design sensitivity of @xmath62 @xmath63-cm is being planned for j - parc.@xcite this sensitivity lies well within values predicted by some susy models.@xcite feng , et al.,@xcite have calculated the range of @xmath64 available to such an experiment , assuming a new physics contribution to @xmath20 of @xmath65 , @xmath66 where @xmath67 is a _ cp _ violating phase . this range is shown in fig . [ fg : phicp ] . available to a dedicated muon edm experiment.@xcite the two bands show the one and two standard - deviation ranges if @xmath20 differs from the standard model value by @xmath68 . ] of course one wishes to measure as many edms as possible to understand the nature of the interaction . while naively the muon and electron edms scale linearly with mass , in some theories the muon edm is greatly enhanced relative to linear scaling relative to the electron edm when the heavy neutrinos of the theory are non - degenerate.@xcite and @xmath69 . ] with an edm present , the spin precession relative to the momentum is given by @xmath70 \nonumber \\ & \qquad \ \ + & { e \over m}\left [ { \eta \over 2 } \left ( { \vec e \over c } + \vec \beta \times \vec b \right ) \right ] \label{eq : omegawedm}\end{aligned}\ ] ] where @xmath71 and @xmath72 . for reasonable values of @xmath73 , the motional electric field @xmath74 is much larger than electric fields that can be obtained in the laboratory , and the two vector frequencies are orthogonal to each other . the edm has two effects on the precession : the magnitude of the observed frequency is increased , and the precession plane is tipped relative to the magnetic field , as illustrated in fig . [ fg : omegaeta ] . e821 was operated at the magic @xmath75 so that the focusing electric field did not cause a spin precession . in e821 the tipping of the precession plane is very small , ( @xmath76 mrad ) if one uses the cern limit@xcite given in table [ tb : edm ] . this small tipping angle makes it very difficult to observe an edm effect in e821 , since the 2 precession ( @xmath44 ) is such a large effect . we have recently introduced a new idea which optimizes the edm signal , and which uses the motional electric field in the rest frame of the muon interacting with the edm to cause spin motion.@xcite the dedicated experiment will be operated off of the magic @xmath75 , for example at @xmath77 mev / c , and will use a radial electric field to stop the @xmath52 precession.@xcite then the spin will follow the momentum as the muons go around the ring , except for any movement arising from an edm . thus the edm would cause a steady build - up of the spin out of the plane with time . detectors would be placed above and below the storage region , and a time - dependent up - down asymmetry @xmath78 would be the signal of an edm , @xmath79 a simulation for @xmath80 cm is given in fig . [ fg : edmsig ] . or @xmath81 ] the figure of merit for statistics in the edm experiment is the number of muons times the polarization . in order to reach @xmath82 cm , the muon edm experiment would need @xmath83 , a flux only available at a future facility . while progress can still be made at brookhaven on @xmath20 , a dedicated muon edm experiment must be done elsewhere . measurements of the muon and electron anomalies played an important role in our understanding of sub - atomic physics in the 20th century . the electron anomaly was tied closely to the development of qed . the subsequent measurement of the muon anomaly showed that the muon was indeed a `` heavy electron '' which obeyed qed.@xcite with the sub - ppm accuracy now available for the muon anomaly,@xcite there may be indications that new physics is beginning to appear in loop processes.@xcite the non - observation of an electron edm is becoming an issue for supersymmetry , just as the non - observation of a neutron edm implies such a mysteriously ( some would say un - naturally ) small @xmath84-parameter for qcd . the search for edms will continue , and if one is observed , the motivation for further searches in other systems will be even stronger . the muon presents a unique opportunity to observe an edm in a second - generation particle , where the _ cp _ phase might be different from the first generation , or the scaling with mass might be quadratic rather than linear . if susy turns out to be _ the _ extension to the standard model , then there will be susy enhancements to @xmath20 to the muon edm and also to the amplitudes for lepton flavor violating muon decays . once the susy mass spectrum is measured , @xmath20 will provide a very clean measurement of @xmath85.@xcite if susy or other new dynamics at the tev scale are not found at lhc , then precision experiments , which are sensitive through virtual loops to much higher mass scales than direct searches for new particles , become even more important . experiments such as edm searches , 2 and searches for lepton flavor violation , all carried out at high intensity facilities , may provide the only way to probe these higher energy scales . opportunities at future high intensity facilities are actively being pursued , and both the theoretical and experimental situations are evolving . it is clear that the study of lepton moments and lepton flavor violation , along with neutron edm searches , will continue to be a topic of great importance in the first part of the 21st century . _ acknowledgments : i wish to thank my colleagues on the muon 2 experiment , as well as m. davier , j. ellis , e. de rafael , w. marciano and t. teubner for helpful discussions . special thanks to y. semertzidis for critically reading this manuscript . _ j.l . feng , k.t . matchev .y , shadmi , nucl * b 613 * , 366 ( 2001 ) , and phys . lett . * b555 * , 89 ( 2003 ) . r.s . van dyck et al . , phys . lett . , * 59 * , 26(1987 ) and in _ quantum electrodynamics _ , ( directions in high energy physics vol . 7 ) t. kinoshita ed . , world scientific , 1990 , p.322 . a. hcker , ichep04 , which can be found at http://ichep04.ihep.ac.cn/program.htm , also m. davier 8th international workshop on tau - lepton physics , september 2004,http://www.hepl.phys.nagoya - u.ac.jp / public / tau04/ r.m . carey , a. gafarov , i. logashenko , k.r . lynch , j.p . miller , b.l . roberts ( co - spokesperson ) , g. bunce , w. meng , w.m . morse ( resident spokesperson ) , y.k . semertzidis , d. grigoriev , b.i . khazin , s.i . redin , yuri m. shatunov , e. solodov , y. orlov , p. debevec , d.w . hertzog ( co - spokesperson ) , p. kammel , r. mcnabb , f. mlhauser , k.l . giovanetti , k.p . jungmann , c.j.g . onderwater , s. dhamija , t.p . gorringe , w. korsch , f.e . gray , b. lauss , e.p . sichtermann , p. cushman , t. qian , p. shagin , s. dhawan and f.j.m . farley , which can be found at : http://g2pc1.bu.edu/@xmath88roberts/

from the famous experiments of stern and gerlach to the present , measurements of magnetic dipole moments , and searches for electric dipole moments of `` elementary '' particles have played a major role in our understanding of sub - atomic physics . in this talk i discuss the progress on measurements and theory of the magnetic dipole moment of the muon . i also discuss a new proposal to search for a permanent electric dipole moment ( edm ) of the muon and put it into the more general context of other edm searches . these experiments , along with searches for the lepton flavor violating decays @xmath0 and @xmath1 , provide a path to the high - energy frontier through precision measurements .

arxiv

in a recent paper li _ et al . _ @xcite presented a new design for an optomechanical system that consists of a microdisk resonator coupled to a waveguide . this design has several attractive features . besides its universality , it enables one to study the reactive effects @xcite in optomechanical coupling . the origin of the reactive coupling is well explained in ref . its origin lies in the mechanical motion dependence of the extrinsic losses of the disk resonator . further phase - dependent gradient forces lead to reactive coupling . have also argued that this design is more effective in achieving cooling of the system to its ground state . while cooling is desirable for studying quantum effects at the macroscopic scale @xcite , we examine other possibilities , which do not depend on the cooling of the system , to investigate the effects arising from strong reactive coupling . since optomechanical coupling effects are intrinsically nonlinear , we examine the nonlinear response of the microdisk resonator to pump probe fields . we report reactive - coupling - induced normal mode splitting . note that in previous works @xcite on normal mode splitting in optomechanical devices , only dispersive coupling was used . in this paper , we report on normal mode splitting due to reactive effects . the paper is organized as follows . in sec . ii , the physical system is introduced and the time evolutions of the expectation values of the system operators are given and solved . in sec . iii , the expectation value of the output fields is calculated , and the nonlinear susceptibilities for stokes and anti - stokes processes are obtained . in sec . iv , we discuss normal mode splitting in output fields with or without reactive coupling . we find that there is no normal mode splitting in output fields in the absence of reactive coupling . however , normal mode splitting occurs in output fields in the presence of reactive coupling . we consider the system shown in fig . [ fig1 ] , in which a microdisk cavity is coupled to a freestanding waveguide . a strong pump field with frequency @xmath0 and a weak stokes field with frequency @xmath1 enter the system through the waveguide . the waveguide will move along the @xmath2 direction under the action of the optical force exerted by the photons from the cavity . further , considering the dispersive coupling and reactive coupling between the waveguide and the cavity , displacement @xmath3 of the waveguide from its equilibrium position will change the resonant frequency of the cavity field and the cavity decay rate , represented by @xmath4 and @xmath5 , respectively . in a rotating frame at pump frequency @xmath0 , the hamiltonian of the system is given by @xcite @xmath6c^{\dag}c+\frac{p^2}{2m}+\frac{1}{2}m\omega_{m}^2q^2\vspace{0.1in}\\\hspace{0.3in}+\hbar\frac { l}{c}\tilde{n}_{g}(\omega_{l}\varepsilon_{l}^2+\omega_{s}|\varepsilon_{s}|^2)+i\hbar\sqrt{2\kappa_{e}(q)}\varepsilon_{l}(c^{\dag}-c)\vspace{0.1in}\\\hspace{0.3 in } + i\hbar\sqrt{2\kappa_{e}(q)}(\varepsilon_{s}e^{-i\delta t}c^{\dag}-\varepsilon^{*}_{s}e^{i\delta t}c ) . \end{array}\ ] ] the first term is the energy of the cavity field , whose annihilation ( creation ) operators are denoted @xmath7 . the second and third terms are the energy of the waveguide with mass @xmath8 , frequency @xmath9 , and momentum operator @xmath10 . the fourth term gives the interactions between the waveguide and the incident fields ( the pump field and the stokes field ) , @xmath11 is the length of the waveguide , @xmath12 is the speed of light in vacuum , @xmath13 is the group index of the waveguide optical mode @xcite , @xmath14 and @xmath15 are the amplitudes of the pump field and the stokes field , respectively , and they are related to their corresponding power @xmath16 and @xmath17 by @xmath18 and @xmath19 . the latter two terms describe the coupling of the cavity field to the pump field and the stokes field , respectively . and @xmath20 is the detuning between the stokes field and the pump field . we would study the physical effects by scanning the stokes laser . for a small displacement @xmath3 , @xmath4 and @xmath5 can be expanded to the first order of @xmath3 , @xmath21 thus the quantities @xmath22 and @xmath23 describe the cavity - waveguide dispersive and reactive coupling strength , respectively . further , note that the photons in the cavity can leak out of the cavity by an intrinsic damping rate @xmath24 of the cavity and by a rate of @xmath5 due to the reactive coupling between the waveguide and the cavity . in addition , the velocity of the waveguide is damped at a rate of @xmath25 . applying the heisenberg equation of motion and adding the damping terms , the time evolutions of the expectation values ( @xmath26 , and @xmath27 ) for the system can be expressed as @xmath28-\gamma_{m}\langle p\rangle,\vspace{0.2in}\\ \displaystyle\langle\dot{c}\rangle=-[\kappa+\langle q\rangle\kappa_{om}+i(\omega_{c}-\omega_{l}+\langle q\rangle\chi)]\langle c\rangle\vspace{0.2in}\\\hspace{0.4in}\displaystyle+\sqrt{\kappa}[1+\langle q\rangle\frac{\kappa_{om}}{\kappa}](\varepsilon_{l}+\varepsilon_{s}e^{-i\delta t } ) , \end{array}\ ] ] where we have used the mean field assumption @xmath29 , expanded @xmath5 to the first order of @xmath3 , and assumed @xmath30 , where @xmath31 is the half - linewidth of the cavity field . it should be noted that the steady - state solution of eq . ( [ 3 ] ) contains an infinite number of frequencies . since the stokes field @xmath32 is much weaker than the pump field @xmath14 , the steady - state solution of eq . ( [ 3 ] ) can be simplified to first order in @xmath32 only . we find that in the limit @xmath33 , each @xmath34,@xmath35 , and @xmath27 has the form @xmath36 where @xmath37 stands for any of the three quantities @xmath3 , @xmath10 , and @xmath12 . thus the expectation values @xmath38 , and @xmath27 ) oscillate at three frequencies ( @xmath0 , @xmath1 , and 2@xmath39 ) . substituting eq . ( [ 4 ] ) into eq . ( [ 3 ] ) , ignoring those terms containing the small quantities @xmath40 , @xmath41 , @xmath42 , and equating coefficients of terms with the same frequency , respectively , we obtain the following results @xmath43,\vspace{.1in}\\ c_{+}=\displaystyle \frac{1}{d(\delta)}[a(be+fj)-i\hbar\frac{\kappa_{om}}{\sqrt{\kappa}}c_{0}^{*}bf^{*}],\vspace{.1in}\\ c_{-}=\displaystyle \frac{f^{*}}{d^{*}(\delta)}(-aj+i\hbar\frac{\kappa_{om}}{\sqrt{\kappa}}c_{0}v),\vspace{0.1in}\\ q_{+}=\displaystyle \frac{b}{d(\delta)}(-aj^{*}-i\hbar\frac{\kappa_{om}}{\sqrt{\kappa}}c_{0}^{*}v^{*}),\vspace{0.1in}\\ q_{-}=(q_{+})^ { * } , \end{array}\ ] ] where @xmath44 @xmath45 and @xmath46 , @xmath47 , @xmath48 , @xmath49 , @xmath50 , @xmath51 . the approach used in this paper is similar to our earlier work @xcite which dealt with optomechanical systems with dispersive coupling only . to investigate the normal mode splitting of the output fields , we need to calculate their expectation value . it can be obtained by using the input - output relation @xcite @xmath52 . if we write @xmath53 as @xmath54 where @xmath55 is the response at the pump frequency @xmath0 , @xmath56 is the response at the stokes frequency @xmath1 , and @xmath57 is the field generated at the new anti - stokes frequency @xmath58 . then we have @xmath59 furthermore , whether there is normal mode splitting in the output fields is determined by the roots of the denominator @xmath60 of @xmath56 . here we examine the roots of @xmath60 given by eq . ( [ 7 ] ) numerically . the response of the system is expected to be especially significant if we choose @xmath1 corresponding to a sideband @xmath61 or @xmath62 , so we consider the case @xmath63 . the other parameters are chosen from a recent experiment focusing on the effect of the reactive force on the waveguide @xcite : the wavelength of the laser @xmath64 nm , @xmath65 mhz / nm , @xmath66 pg ( density of the silicon waveguide , 2.33 g/@xmath67 ; length , 10 @xmath68 m ; width , 300 nm ; height , 300 nm ) , @xmath69 , @xmath70 mhz , and the mechanical quality factor @xmath71 . in the following , we work in the stable regime of the system . figure [ fig2 ] shows the variation of the real parts of the roots of @xmath60 in the domain re@xmath72 with increasing pump power for no reactive coupling , @xmath73 , and for @xmath74 mhz / nm . for @xmath73 , the interaction of the waveguide with the cavity is purely dispersive ; the cavity decay rate does not depend on the displacement of the waveguide . in this case , the real parts of the roots of @xmath60 always have two equal values with increasing pump power . thus there is no splitting because the dispersive coupling is not strong enough . however , for @xmath74 mhz / nm , the system has both dispersive and reactive couplings , the cavity decay rate depends on the displacement of the waveguide , and the real parts of the roots of @xmath60 will change from two equal values to two different values with increasing pump power . and the difference between two real parts of the roots of @xmath60 in the domain re@xmath72 is increased with increasing pump power . therefore , the reactive coupling between the waveguide and the cavity can result in normal mode splitting of the output fields , and the peak separation becomes larger with increasing pump power . figure [ fig3 ] shows the variation of the imaginary parts of the roots of @xmath60 with increasing pump power for zero reactive coupling @xmath73 and nonzero reactive coupling @xmath74 mhz / nm . for @xmath73 , the imaginary parts of the roots of @xmath60 do not change with increasing pump power . however , for @xmath74 mhz , the imaginary parts of the roots of @xmath60 change with increasing pump power . we thus conclude that for the present microdisk resonator coupled to a waveguide the normal mode splitting is solely due to the reactive coupling . we now discuss how the output fields depend on the behavior of the roots of @xmath60 . for convenience , we normalize all quantities to the input stokes power @xmath17 . assuming that @xmath32 is real , we express the output power at the stokes frequency @xmath1 in terms of the input stokes power @xmath75 further , we introduce the two quadratures of the stokes component of the output fields by @xmath76 and @xmath77 . one can measure either the quadratures of the output by homodyne techniques or the intensity of the output . for brevity , we only show @xmath78 and @xmath79 as a function of the normalized detuning between the stokes field and the pump field @xmath80 for this model , without reactive coupling ( @xmath23=0 ) and with it ( @xmath74 mhz / nm ) , for different pump powers in figs . [ fig4][fig5 ] . for @xmath23=0 , it is found that @xmath78 has a lorentzian lineshape corresponding to the absorptive behavior . note that @xmath78 and @xmath79 exhibit no splitting when @xmath23=0 . however , for @xmath74 mhz / nm , it is clearly seen that normal mode splitting appears in @xmath78 and @xmath79 . therefore reactive coupling can lead to the appearance of normal mode splitting in the output stokes field . and the peak separation increases with increasing pump power @xcite . the dip at the line center exhibits power broadening . we also find that the stokes field can be amplified by the stimulated process . obviously the maximum gain @xmath79 for the stokes field depends on the system parameters . for a pump power @xmath81 @xmath68w , the maximum gain for the stokes field is about 1.3 . note that the nonlinear nature of the reactive coupling generates anti - stokes radiation . in a similar way , we define a normalized output power at the anti - stokes frequency @xmath58 as @xmath82 the plots of @xmath83 versus the normalized detuning between the stokes field and the pump field @xmath80 for this model , without reactive coupling ( @xmath23=0 ) and with it ( @xmath74 mhz / nm ) , for different pump powers are presented in fig . we can see that @xmath84 for @xmath23=0 . the reason is that the dispersive coupling constant @xmath22 is too small . however , for @xmath74 mhz / nm , @xmath83 is not equal to zero . this shows that the optomechanical system can generate an anti - stokes field with frequency @xmath85 due to the reactive coupling . for pump power @xmath81 @xmath68w , the maximum gain defined with reference to the input stokes power for the anti - stokes field is about 0.1 . in conclusion , we have observed normal mode splitting of output fields due to reactive coupling between the waveguide and the cavity . meanwhile , the separation of the peaks increases for larger pump powers . further , the reactive coupling can also cause four - wave mixing , which creates an anti - stokes component generated by the optomechanical system . 99 m. li , w. h. p. pernice , and h. x. tang , phys . lett . * 103 * , 223901 ( 2009 ) . f. elste , s. m. girvin , and a. a. clerk , phys . . lett . * 102 * , 207209 ( 2009 ) . m. li , w. h. p. pernice , and h. x. tang , nat . photon . * 3 * , 464 ( 2009 ) . m. j. hartmann and m. b. plenio , phys * 101 * , 200503 ( 2008 ) . m. bhattacharya and p. meystre , phys . rev . lett . * 99 * , 073601 ( 2007 ) . s. bose , k. jacobs , and p. l. knight , phys . a * 56 * , 4175 ( 1997 ) . s. huang and g. s. agarwal , new j. phys . * 11 * , 103044 ( 2009 ) . m. paternostro , d. vitali , s. gigan , m. s. kim , c. brukner , j. eisert , and m. aspelmeyer , phys . * 99 * , 250401 ( 2007 ) . d. vitali , s. gigan , a. ferreira , h. r. bhm , p. tombesi , a. guerreiro , v. vedral , a. zeilinger , and m. aspelmeyer , phys . * 98 * , 030405 ( 2007 ) . f. marquardt , j. p. chen , a. a. clerk , and s. m. girvin , phys . lett . * 99 * , 093902 ( 2007 ) . j. m. dobrindt , i. wilson - rae , and t. j. kippenberg , phys . * 101 * , 263602 ( 2008 ) . s. grblacher , k. hammerer , m. vanner , and m. aspelmeyer , nature ( london ) * 460 * , 724 ( 2009 ) . s. huang and g. s. agarwal , phys . a * 81 * , 033830 ( 2010 ) . [ this paper deals exclusively with designs where only dispersive optomechanical coupling occurs . ] w. h. p. pernice , m. li , and h. x. tang , opt . express * 17 * , 1806 ( 2009 ) . d. f. walls and g. j. milburn , _ quantum optics _ ( springer - verlag , berlin , 1994 ) . g. s. agarwal and s. huang , phys . rev . a * 81 * , 041803(r ) ( 2010 ) ; p. anisimov and o. kocharovskaya , j. mod . opt . * 55 * , 3159 ( 2008 ) .

we study the optomechanical design introduced by m. li _ et al . _ [ phys . rev . lett . * 103 * , 223901 ( 2009 ) ] , which is very effective for investigations of the effects of reactive coupling . we show the normal mode splitting that is due solely to reactive coupling rather than due to dispersive coupling . we suggest feeding the waveguide with a pump field along with a probe field and scanning the output probe for evidence of reactive - coupling - induced normal mode splitting .

arxiv

quantum many - body system out of equilibrium has been subjected to upsurging interest in recent times . extensive studies have been done over decades giving rise to many new findings in this field ( see @xcite for a review ) . lots of questions still remain unresolved too . from the results obtained so far , it is impossible to draw a generic picture of the dynamics out of equilibrium . emergence of current of a physical quantity generally renders a system to a nonequilibrium one . transport of that quantity due to presence of current thereby gets attention as an important aspect of nonequilibrium dynamics . transport properties , especially magnetization transport in low dimensional spin systems have received attention of the experimentalists too gaining importance in the area of spintronics , nano - device applications @xcite . earlier theoretical studies @xcite have confirmed the fact that the properties of transportation in closed quantum system differ significantly from its classical counterpart . a number of works , both analytical and numerical , have been done on introducing current of a quantity in the integrable hamiltonian to observe its relaxation dynamics @xcite . in these papers , current - carrying states have been generated in the system in two ways , either by adding current to the hamiltonian or by preparing an inhomogeneous initial state . introduction of magnetization current results in oscillatory nature and power - law decay of correlations in space . a typical initial state with a steplike inhomogeneity in magnetization even yields a scaling form of the same in the large - time limit . analytical as well as simulative studies have also been made on nonequilibrium transport of magnetization in open quantum spin chain @xcite . these studies have been able to shed some light upon the behavior of physical quantities in an integrable system while trying to equilibrate . + integrability has become a very crucial perspective in these explorations of nonequilibrium phenomena . integrable systems with many local conserved quantities have restricted dynamics and unlike non - integrable systems , their relaxation process is dependent on the initial state . another behavior believed to be the line of division between integrable and non - integrable system is thermalization . it was claimed to be absent in integrable systems @xcite . numerical studies on non - integrable systems has been done recently in this regard @xcite . evidences are however present where non - integrable systems are shown to have weak thermalization or even no thermalization @xcite . owing to some more elaborate studies the role of integrability on the behavior of response functions have been put under question . it has been shown @xcite that in a completely integrable system , apart from the usual non - thermal behavior of some quantities , there do exist some other quantities that exhibit thermal behavior which is counterintuitive to the preceding ideas on thermalization . another important question is whether their lies any generic relation between integrability and transport properties . efforts have been given in that part too by some preceding works @xcite . + a scattered context like this has been the motivation of more investigation in relaxation dynamics of integrable quantum system generally followed by a quench @xcite ( see @xcite for review ) . in this article , we study the dynamics due to transport of transverse magnetization in a quantum ising chain . transportation of magnetization is induced by preparing an initial state which produces inhomogeneity in transverse magnetization and thus puts the system into nonequilibrium . we let it evolve in presence of a homogeneous and constant ( in time ) transverse field with a view to studying whether the transport properties finally make the magnetization homogeneous or not and how the magnetization relaxes with time . somewhat similar study has been made on xxz spin chain showing oscillatory behavior and power - law decay @xcite . power - law relaxation of local magnetization has been observed in heisenberg spin chain too @xcite . study of temporal behavior of transverse magnetization from a thermally inhomogeneous state has also been done in quantum xx chain @xcite . in our case , to prepare such an initial configuration , a new protocol is introduced : we include two odd - occupation basis states in fermionic momentum space which has zero eigenvalue . a surprising observation is that the later dynamics driven by the external transverse field can not make the magnetization homogeneous even after infinite time which is counterintuitive to our ideas in similar context of classical counterpart . this immediately renders the strength of the external field less important than the presence of odd - occupation states . although the system relaxes with time , the local magnetization at different sites do not attain the same value . the underlying dynamics at each site are shown to possess two different timescales of oscillation . the smaller timescale contributes insignificant undulations whereas the oscillations of larger timescale exhibit generic power - law decay . the exponent of the decay is proved to be independent of the strength of the external field . this type of decay also indicates absence of thermalization , in agreement with previous works @xcite , whereas in contrast with that work , no scaling form is observed in the large - time limit for this type of initial configuration . a very recent work on thermalization also supports the fact that such type of decay indicates lack of thermalization @xcite . + another important feature of our work is that the phenomena of quantum phase transition is manifested in the dynamics in two ways as can be established analytically , ( i ) the frequency of the characteristic oscillation is a monotonic function of the external field as long as the system is in the ordered phase and becomes independent of the field when it crosses the critical value to the disordered phase , ( ii ) the transverse magnetization after infinite time at each site shows different functional behavior in the ordered and disordered phase and thus produces nonanalyticity at the critical point . + lastly , our initial configuration contains several parameters and all the features of the dynamics we observe , is true for arbitrary values of those parameters . thus , inspite of the fact that the hamiltonian is integrable , some characteristics of the dynamics is valid for a class of initial configurations . + in the next section we give detailed description of the model and of the initial state . section iii contains the exact analytical treatment of transverse magnetization and its dynamics . in the last section , conclusions are drawn from the results obtained along with general discussions . one dimensional spin-@xmath1 quantum ising chain of @xmath2 sites is described by the hamiltonian = -_i=1^n s^x_i s^x_i+1 - _ i=1^n s^z_i where @xmath0 , scaled by the coupling constant , is the external transverse field and @xmath3 are two components of pauli spin matrices . + the well - known jordan - wigner transformation enables us to transform the hamiltonian into a direct sum of hamiltonians ( @xmath4 ) of nonlocal free fermions of momenta @xmath5 @xcite _ k = ( -2 i k ) - 2(+ k ) [ hk_def ] where @xmath6 and @xmath7 are fermionic creation and annihilation operator in momentum space and @xmath8 , with @xmath9 . the operator @xmath4 has four eigenstates . the even - occupation eigenstates of @xmath4 are spanned by two basis states namely , @xmath10 and @xmath11 where the numbers signify the occupation status of the fermions having momenta @xmath12 and @xmath13 respectively . we denote the eigenstates of @xmath4 within these subspaces as @xmath14 with eigenvalues @xmath15 , where @xmath16 $ ] . the other basis states @xmath17 and @xmath18 have zero eigenvalues . for any wave function @xmath19 , the transverse magnetization at the @xmath20-th site is given by @xmath21 this magnetization is homogeneous if @xmath19 is the ground state or any other state comprising of @xmath22 and @xmath23 . + we construct an initial configuration incorporating the odd - occupation basis ( @xmath24 and @xmath25 ) in momentum space alongwith the states @xmath22 and @xmath23 . we choose |(0)= |_k |_k= _ k |11_k + _ k |00_k + _ k |10_k + _ k |01_k [ def_psi ] the normalization condition requires note that this state is not an eigenstate of the fermionic hamiltonian @xmath4 . we shall now show that one can calculate analytically the magnetization at any given site for this initial state under certain conditions . the expression for magnetization eq.([mnz ] ) involves jump of fermions from any site @xmath26 to any site @xmath27 . let us first assume that @xmath27 and @xmath26 are both positive and @xmath28 . for such a jump , the sign of the resulting term is then determined by whether the fermion crosses an even or odd number of fermions during its flight . @xmath29 \nonumber \\ & & \left(\alpha_{k_2 } |01\rangle_{k_2 } + \gamma_{k_2 } |00\rangle_{k_2 } \right ) \end{aligned}\ ] ] the sign will be plus ( minus ) if the number of fermions in the range @xmath30 for a given term in the expanded form of the portion within square brackets is even ( odd ) . however , one observes that , the correct sign is obtained from the equality , @xmath31 \nonumber \\ & & \left(\alpha_{k_2 } |01\rangle_{k_2 } + \gamma_{k_2 } |00\rangle_{k_2 } \right ) \label{ak } \end{aligned}\ ] ] in order to calculate the magnetization @xmath32 at site @xmath20 , we need to operate @xmath33 on eq.([ak ] ) . however , a simplification occurs if we note that ( _ k 11| + _ k 00|_k + _ k 10|_k + _ k 01|_k ) ( _ k |11_k + _ k |00_k - _ k |10_k - _ k |01_k ) = |_k|^2 + |_k|^2 - 2|_k|^2 . hence , if we ensure that |_k|^2 + |_k|^2 = 2|_k|^2 [ zero ] for all @xmath34 , the quantity @xmath35 will be non - zero _ only when _ either @xmath36 or @xmath27 and @xmath26 are two successive points so that @xmath37 ( say ) . note that the constraints ( [ normalization ] ) and ( [ zero ] ) need @xmath38 and @xmath39 for all @xmath34 . we assume additionally , that @xmath40 for all @xmath34 . one can now obtain the expression for magnetization @xmath41 as , @xmath42 here the sum runs over @xmath43 values between @xmath44 and @xmath45 and @xmath46 stands for complex conjugate . in order to study the dynamics we first note that , from eq.([def_psi ] ) the wave function at time @xmath47 may be obtained as @xmath48 + \beta_k[-sin\theta_k e^{i\lambda_k t}|(\gamma , k)_{-}\rangle + \cos\theta_k e^{-i\lambda_k t}|(\gamma , k)_{+}\rangle ] \nonumber \\ % & & + \ ; \gamma_k|10\rangle + \ ; \gamma_k|01\rangle \\ & = & \alpha_k^{\prime } |11\rangle + \beta_k^{\prime } |00\rangle + \gamma |10\rangle + \gamma |01\rangle \end{aligned}\ ] ] where @xmath49 @xmath50 with @xmath47 is scaled by @xmath51 . a very crucial point is that the coefficients @xmath52 , @xmath53 and @xmath54 also satisfy the condition eq.([zero ] ) . hence , the magnetization at time @xmath47 can also be calculated following the previous procedure : m_z(n , t ) = -1 + _ k=0^ m_k [ m1 ] where @xmath55 is given by the expression in eq.([mn0 ] ) with @xmath56 , @xmath57 replaced by @xmath58 , @xmath59 . after this replacement one can express @xmath55 as m_k = a_k + b_k ( 2_k t + b_k ) + c_k(n ) 2(_k+u+_k)t + c_k(n ) + d_k(n)2(_k+u-_k)t + d_k(n ) [ mk ] the precise expressions for the amplitudes @xmath60 and the phases @xmath61 , @xmath62 , @xmath63 in terms of @xmath34 , @xmath20 , @xmath56 , @xmath57 and @xmath64 can be obtained but is not required for our subsequent calculations . + from now on we shall be restricted to the case where @xmath56 and @xmath57 are real and equal for all @xmath34 , i.e. @xmath65 and @xmath66 . then , in the thermodynamic limit , @xmath41 can be expressed as @xmath67 \ ; dk % m_k & = & 2\gamma^2(\sin^4\theta_k + \cos^4\theta_k)\left[2(\beta^2-\alpha^2)\cos un + 2(\beta^2+\alpha^2)\cos 2kn \right]\cos u\lambda'_k t \nonumber \\ % & & - 8\gamma^2\alpha\beta\cos 2\theta_k \sin 2kn \sin u\lambda'_k t + 2\gamma^2 \sin^2 2\theta_k \left[(\beta^2-\alpha^2)\cos un \right](\cos^2\lambda_k t-3\sin^2\lambda_k t ) \nonumber \\ % & & - 8\gamma^2\alpha\beta\sin 2\theta_k \cos un \sin 2\lambda_k t + 4\gamma^2(\beta^2+\alpha^2)\sin 4\theta_k\sin un \sin^2\lambda_k t \nonumber \\ % & & + 2\gamma^2\left[(\beta^2+\alpha^2)\sin^2 2\theta_k + 1\right]\cos 2kn \nonumber \\ % & & + \alpha^2 \sin^2 2\theta_k\cos 2\lambda_k t + 2\alpha\beta \sin 2\theta_k \sin 2\lambda_k t + 2\beta^2 \sin^2 2\theta_k \sin^2\lambda_k t + 2\alpha^2 ( \cos^4\theta_k + \sin^4\theta_k ) \label{m_int}\ ] ] where @xmath68\cos 2\lambda_k t + [ 2\alpha\beta\sin 2\theta_k(1-\cos un ) ] \sin 2\lambda_k t \\ m_k(t / n)&= & \frac{1}{2}(\cos^4\theta_k + \sin^4\theta_k)[2(\beta^2-\alpha^2)\cos un + \cos 2kn ] \cos 2\pi\lambda'_k\frac{t}{n } \;-\ ; 2\alpha\beta\cos 2\theta_k \sin 2kn \sin 2\pi\lambda'_k\frac{t}{n}\end{aligned}\ ] ] we have written @xmath69 as in the thermodynamic limit the difference between consecutive @xmath34-points become infinitesimally small . + the dynamics is characterized by the behavior of transverse magnetization @xmath41 at a time @xmath47 at site @xmath20 , as given by eqs ( [ m_int ] ) . at any given time ( including @xmath70 ) magnetization is a continuously varying function of @xmath20 . , starting from initial state ( [ def_psi ] ) with @xmath71 for all @xmath34 and @xmath72.,title="fig:",width=377 ] initial magnetization as derived from eq.([m_int ] ) is given by m_z(n , t=0)=-+2 ^ 2+(^2-^2)un [ mz - init ] eqs ( [ m_int ] ) and ( [ mz - init ] ) give clear indication that the magnetization is inhomogeneous at all time and is shown in fig [ mzvsn ] . + the temporal behavior of transverse magnetization at a given site is shown in fig.[mzvst ] for different external fields . it exhibits oscillatory behavior with the envelope decaying algebraically in the large time limit . it is evident that eq.([m_int ] ) has two time - dependent parts , @xmath73 and @xmath74 of two separate timescales . the former part gives tiny undulations ( fig . [ mzvst](a ) ) and the other is responsible to produce larger oscillation with algebraic decay of envelope and dominates in the large time limit . hence , for @xmath75 and @xmath76 , we have @xmath77 \cos 2\pi\lambda'_k\frac{t}{n } \nonumber \\ & & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\ ; 2\alpha\beta\cos 2\theta_k \sin 2kn \sin 2\pi\lambda'_k\frac{t}{n } \;]\;dk \nonumber \\ & \approx & m_{\infty } + \frac{1}{\pi}\int_0^\pi \mathit{g}_k(n)\cos(\omega_k\tau + \phi_k(n))\;dk \label{mz - tau } \end{aligned}\ ] ] where @xmath78 and @xmath79 . the quantities @xmath80 , @xmath81 can be written in terms of @xmath82 and @xmath34 from the last equation , but the explicit form is not necessary for our calculations below . we call the @xmath83-independent term @xmath84 because it will give us the magnetization at @xmath85 which we shall discuss later . and @xmath86 at @xmath87 against @xmath88 , starting from initial state ( [ def_psi ] ) with @xmath89 for all @xmath34 . ( a ) @xmath32 vs @xmath83 alongwith small undulations due to time scale @xmath47 . variation of @xmath32 for very small change of @xmath83 given as inset shows oscillations in small timescale . ( b ) oscillations in large timescale are shown in @xmath32 vs @xmath83 plot for @xmath72 and @xmath90 . the envelopes of oscillations decay as @xmath91 both for @xmath92 and @xmath93 . ( c ) & ( d ) frequency of large timescale oscillation is shown to increase with @xmath0 for @xmath94 in @xmath32 vs @xmath83 plot whereas for @xmath95 , it becomes constant.,title="fig:",width=604 ] we first observe that for sufficiently large @xmath83 , the quantity @xmath96 will be large ( so that its cosine will fluctuate very rapidly and vanish on integration ) unless @xmath97 is very small . hence the region where @xmath97 is minimum with respect to @xmath34 will only contribute to the integral . this minima is found to occur at k = k_0 = \ { rl ^-1(- ) & + ^-1 ( - ) & . [ k0 ] it is hence sufficient to integrate the oscillatory term over a small region @xmath98 ( @xmath99 becoming smaller and smaller with increasing @xmath83 ) where @xmath100 and @xmath81 do not vary appreciably . expanding @xmath97 about @xmath101 , _ k = _ k_0 + f(k - k_0)^2 [ omega - taylor ] ( where @xmath102 ) we get the expression for magnetization at large @xmath83 as , @xmath103 where @xmath104 . thus it is evident that for the chosen initial configuration , the transverse magnetization always exhibits @xmath91 decay in the envelope of oscillation irrespective of the magnitude of the transverse field . + as shown in fig.[mzvst](c ) and ( d ) , the frequency of oscillation varies monotonically with @xmath0 in the ferromagnetic phase and becomes independent of @xmath0 in paramagnetic region . it can be explained analytically . we note from eq.([k0 ] ) that @xmath105 changes from @xmath106 to @xmath107 as @xmath0 crosses the critical point . this means that the frequency in the large @xmath83 limit behaves as |_k| = 2|| = \ { rl 4 & + 4 & . [ omegak0 ] the significance of this nonanalytic behavior of frequency is that it coincides with the occurrence of order - disorder quantum phase transition of the system . + behavior at @xmath85 ( and hence @xmath108 as well ) can also be obtained form eq.([m_int ] ) by performing the integrations involved : m _ ( n ) = - + 2 ^ 2 - + ( 2 - un ) ^p - ( - ) ^2 ^p(2n-1 ) [ mz - inf ] where the index @xmath109 is @xmath110 for @xmath111 and @xmath112 for @xmath113 . the persistence of inhomogeneity of magnetization at @xmath85 is evident . another important aspect of the long - time behaviour is that the quantity @xmath114 shows different behavior as a function of @xmath0 in ferromagnetic and paramagnetic region and is non - analytic at the critical point . such behaviour originates from two integrals involved in the calculation of @xmath115 . they can be written as a contour integral over the unit circle and the integrand has poles at @xmath116 and @xmath117 . for @xmath111 , the former pole is inside the unit circle and for @xmath113 , the latter is inside . this change of pole structure leads to a change in functional behavior and to non - analyticity , as has been observed elsewhere @xcite . vs @xmath0 plot & ( b ) @xmath118 vs @xmath0 plot , for @xmath119 with the initial state ( [ def_psi ] ) taking @xmath71 for all @xmath34.,title="fig:",width=529 ] we have shown analytically that in a transverse ising chain at zero temperature , a state can be manipulated in the fermioninc momentum space so that it produces spatial variation in transverse magnetization . such inhomogeneity gives rise to transport of transverse magnetization from one site to another as a result of which both spatial and temporal variations take place in presence of homogeneous and constant ( in time ) external transverse field . at each site the magnetization evolves in an oscillatory manner with the envelope decaying algebraically . the oscillation and decay however do not lead to homogeneous magnetization because the odd - occupation states have zero energy and their coefficients , which are the prime factor for inhomogeneity , remain nonzero forever . the exponent of decay is independent of the field i.e. , it remains unaltered in ferromagnetic and paramagnetic phases . the characteristic change is shown by the frequency of oscillation at long time and the final magnetization at each site . they show different behavior in ferromagnetic and paramagnetic phases . thus we find no signature of criticality in the exponent of decay whereas two new quantities are found to bear it . moreover , starting with such configuration , the inhomogeneity in transverse magnetization can not be removed by the external field . both the phenomenon are counterintuitive to the prevailing ideas . the scenario may be thought of as a quench from an initial state and the preceding works on quench dynamics report the change of the exponent of decay in different phases @xcite although in our case , the presence of odd - occupation states becomes an important factor in the dynamics . although the initial configuration we have worked with contains real and @xmath34-independent values of the coefficients , it is important to note that , all our observations are valid for arbitrary phases of these quantities and arbitrary modulus of @xmath56 or @xmath57 . thus , although the hamiltonian is integrable , we observe a generic behaviour for a wide range of initial states . + a question arises regarding other observables . integrable system has been reported to have some non - local observables which show typically different behaviors including even thermalization in course of quench dynamics @xcite . now in our system , with this initial configuration , how do non - local observables behave ? can thermalization or field - dependent decay exponent be found out in any of them ? how do the non - integrable systems behave starting from such inhomogeneity ? search for answers of many such open questions may thus be quite interesting to investigate in future .

we study the dynamics caused by transport of transverse magnetization in one dimensional transverse ising chain at zero temperature . we observe that a class of initial states in fermionic momentum - space produce spatial variation in transverse magnetization . starting from such a state , we obtain the transverse magnetization analytically and then observe its dynamics in presence of a homogeneous constant field @xmath0 . in contradiction with general expectation , whatever be the strength of the field , the magnetization of the system does not become homogeneous even after infinite time . in each site , the dynamics is associated with oscillations having two different timescales . the envelope of the larger timescale oscillation decays algebraically having an exponent which is universal for the entire class of such initial states . signature of quantum criticality is also embedded in the dynamics in two aspects : behavioral change shown by the frequency of large timescale oscillation at any site and the corresponding magnetization at infinity when they cross the quantum critical point .

arxiv

studying qcd at finite baryon density is the traditional subject of nuclear physics . the behaviour of qcd at finite baryon density and low temperature is central for astrophysics to understand the structure of compact stars , and conditions near the core of collapsing stars ( supernovae , hypernovae ) . it is known that sufficiently cold and dense baryonic matter is in the color superconducting phase . this was proposed several decades ago by frautschi @xcite and barrois @xcite by noticing that one - gluon exchange between two quarks is attractive in the color antitriplet channel . from bcs theory @xcite , we know that if there is a weak attractive interaction in a cold fermi sea , the system is unstable with respect to the formation of particle - particle cooper - pair condensate in the momentum space . studies on color superconducting phase in 1980 s can be found in ref . the topic of color superconductivity stirred a lot of interest in recent years @xcite . for reviews on recent progress of color superconductivity see , for example , ref . @xcite . the color superconducting phase may exist in the central region of compact stars . to form bulk matter inside compact stars , the charge neutrality condition as well as @xmath0 equilibrium are required @xcite . this induces mismatch between the fermi surfaces of the pairing quarks . it is clear that the cooper pairing will be eventually destroyed with the increase of mismatch . without the constraint from the charge neutrality condition , the system may exhibit a first order phase transition from the color superconducting phase to the normal phase when the mismatch increases @xcite . it was also found that the system can experience a spatial non - uniform loff ( larkin - ovchinnikov - fudde - ferrell ) state @xcite in a certain window of moderate mismatch . it is still not fully understood how the cooper pairing will be eventually destroyed by increasing mismatch in a charge neutral system . the charge neutrality condition plays an essential role in determining the ground state of the neutral system . if the charge neutrality condition is satisfied globally , and also if the surface tension is small , the mixed phase will be favored @xcite . it is difficult to precisely calculate the surface tension in the mixed phase , thus in the following , we would like to focus on the homogeneous phase when the charge neutrality condition is required locally . it was found that homogeneous neutral cold - dense quark matter can be in the gapless 2sc ( g2sc ) phase @xcite or gapless cfl ( gcfl ) phase @xcite , depending on the flavor structure of the system . the gapless state resembles the unstable sarma state @xcite . however , under a natural charge neutrality condition , i.e. , only neutral matter can exist , the gapless phase is indeed a thermal stable state as shown in @xcite . the existence of thermal stable gapless color superconducting phases was confirmed in refs . @xcite and generalized to finite temperatures in refs . recent results based on more careful numerical calculations show that the g2sc and gcfl phases can exist at moderate baryon density in the color superconducting phase diagram @xcite . one of the most important properties of an ordinary superconductor is the meissner effect , i.e. , the superconductor expels the magnetic field @xcite . in ideal color superconducting phases , e.g. , in the 2sc and cfl phases , the gauge bosons connected with the broken generators obtain masses , which indicates the meissner screening effect @xcite . the meissner effect can be understood using the standard anderson - higgs mechanism . unexpectedly , it was found that in the g2sc phase , the meissner screening masses for five gluons corresponding to broken generators of @xmath1 become imaginary , which indicates a type of chromomagnetic instability in the g2sc phase @xcite . the calculations in the gcfl phase show the same type of chromomagnetic instability @xcite . remembering the discovery of superfluidity density instability @xcite in the gapless interior - gap state @xcite , it seems that the instability is a inherent property of gapless phases . ( there are several exceptions : 1 ) it is shown that there is no chromomangetic instability near the critical temperature @xcite ; 2 ) it is also found that the gapless phase in strong coupling region is free of any instabilities @xcite . ) the chromomagnetic instability in the gapless phase still remains as a puzzle . by observing that , the 8-th gluon s chromomagnetic instability is related to the instability with respect to a virtual net momentum of diquark pair , giannakis and ren suggested that a loff state might be the true ground state @xcite . their further calculations show that there is no chromomagnetic instability in a narrow loff window when the local stability condition is satisfied @xcite . latter on , it was found in ref . @xcite that a charge neutral loff state can not cure the instability of off - diagonal 4 - 7th gluons , while a gluon condensate state @xcite can do the job . in ref . @xcite we further pointed out that , when charge neutrality condition is required , there exists another narrow unstable loff window , not only off - diagonal gluons but the diagonal 8-th gluon can not avoid the magnetic instability . in a minimal model of gapless color supercondutor , hong showed in ref . @xcite that the mismatch can induce a spontaneous nambu - goldstone current generation . the nambu - goldstone current generation state in u(1 ) case resembles the one - plane wave loff state or diagonal gauge boson s condensate . we extended the nambu - goldstone current generation picture to the 2sc case in the nonlinear realization framework in ref . we show that five pseudo nambu - goldstone currents can be spontaneously generated by increasing the mismatch between the fermi surfaces of the pairing quarks . the nambu - goldstone currents generation state covers the gluon phase as well as the one - plane wave loff state . this article is organized as follows . in sec . [ sec - gnjl ] , we describe the framework of the gauged su(2 ) nambu jona - lasinio ( gnjl ) model in @xmath0-equilibrium . we review chromomagnetic instabilities in sec . [ sec - mag - ins ] . we discuss neutral baryon current instability and the loff state in sec . [ sec - bc ] . then sec . [ sec - ng - current ] gives a general nambu - goldstone currents generation description in the non - linearization framework . at the end , we give the discussion and summary in sec . [ sec - sum ] . we take the gauged form of the extended nambu jona - lasinio model @xcite , the lagrangian density has the form of @xmath2 \nonumber \\ & + & g_d[(i { \bar q}^c \varepsilon \epsilon^{b } \gamma_5 q ) ( i { \bar q } \varepsilon \epsilon^{b } \gamma_5 q^c ) ] , \label{lg}\end{aligned}\ ] ] with @xmath3 . here @xmath4 are gluon fields and @xmath5 with @xmath6 are generators of @xmath7 gauge groups . please note that we regard all the gauge fields as external fields , which are weakly interacting with the system . the property of the color superconducting phase characterized by the diquark gap parameter is determined by the unkown nonperturbative gluon fields , which has been simply replaced by the four - fermion interaction in the njl model . however , the external gluon fields do not contribute to the properties of the system . therefore , we do not have the contribution to the lagrangian density from gauge field part @xmath8 as introduced in ref . @xcite . . [ sec - ng - current ] , by using the non - linear realization in the gnjl model , we will derive one nambu - goldstone currents state , which is equivalent to the so - called gluon - condensate state . ) in the lagrangian density eq . ( [ lg ] ) , @xmath9 , @xmath10 are charge - conjugate spinors , @xmath11 is the charge conjugation matrix ( the superscript @xmath12 denotes the transposition operation ) . the quark field @xmath13 with @xmath14 and @xmath15 is a flavor doublet and color triplet , as well as a four - component dirac spinor , @xmath16 are pauli matrices in the flavor space , where @xmath17 is antisymmetric , and @xmath18 , @xmath19 are totally antisymmetric tensors in the flavor and color spaces . @xmath20 is the matrix of chemical potentials in the color and flavor space . in @xmath0-equilibrium , the matrix of chemical potentials in the color - flavor space @xmath21 is given in terms of the quark chemical potential @xmath22 , the chemical potential for the electrical charge @xmath23 and the color chemical potential @xmath24 , @xmath25 @xmath26 and @xmath27 are the quark - antiquark coupling constant and the diquark coupling constant , respectively . in the following , we only focus on the color superconducting phase , where @xmath28 and @xmath29 . after bosonization , one obtains the linearized version of the model for the 2-flavor superconducting phase , @xmath30 with the bosonic fields @xmath31 in the nambu - gorkov space , @xmath32 the inverse of the quark propagator is defined as @xmath33^{-1 } = \left(\begin{array}{cc } \left[g_0^{+}(p)\right]^{-1 } & \delta^- \\ \delta^+ & \left[g_0^{-}(p)\right]^{-1 } \end{array}\right ) , \label{prop}\ ] ] with the off - diagonal elements @xmath34 and the free quark propagators @xmath35 taking the form of @xmath36^{-1 } = \gamma^0 ( p_0 \pm \hat{\mu } ) - \vec{\gamma } \cdot \vec{p}. \label{freep}\ ] ] the 4-momenta are denoted by capital letters , e.g. , @xmath37 . we have assumed the quarks are massless in dense quark matter , and the external gluon fields do not contribute to the quark self - energy . the explicit form of the functions @xmath38 and @xmath39 reads @xmath40 [ g_i ] with @xmath41 and @xmath42 [ xi_i ] where @xmath43 is an alternative set of energy projectors , and the following notation was used : @xmath44 from the dispersion relation of the quasiparticles eq . ( [ g - disp ] ) , we can read that , when @xmath45 , there will be excitations of gapless modes in the system . the thermodynamic potential corresponding to the solution of the gapless state @xmath45 is a local maximum . however , under certain constraint , e.g. , the charge neutrality condition , the gapless 2sc phase can be a thermal stable state @xcite . in this section , we review the chromomagnetic instabilities driven by mismatch in 2sc and g2sc phases as shown in ref . the polarization tensor in momentum space has the following general structure : @xmath46 . \label{pipscfng}\ ] ] the trace here runs over the dirac indices and the vertices @xmath47 with @xmath48 . gluons @xmath52 of the unbroken @xmath53 subgroup couple only to the red and green quarks . the general expression for the polarization tensor @xmath54 with @xmath55 is diagonal . after performing the traces over the color , flavor and nambu - gorkov indices , the expression has the form of @xmath56 , \label{pi11}\end{aligned}\ ] ] by making use of the definition in eq . ( [ def - debye ] ) and eq . ( [ def - meissner ] ) . we arrive at the following result for the threefold degenerate debye mass square : @xmath57 with @xmath58 . the meissner mass square reads @xmath59 the debye screening mass in eq . ( [ m_d_1 ] ) vanishes in the gapped phase ( i.e. , @xmath60 ) . as in the case of the ideal 2sc phase , this reflects the fact that there are no gapless quasiparticles charged with respect to the unbroken su(2)@xmath61 gauge group . in the gapless 2sc phase , such quasiparticles exist and the value of the debye screening mass is proportional to the density of states at the corresponding `` effective '' fermi surfaces . the 8-th gluon can probe the cooper - paired red and green quarks , as well as the unpaired blue quarks . after the traces over the color , the flavor and the nambu - gorkov indices are performed , the polarization tensor for the 8th gluon can be expressed as @xmath63 , \label{pi88 } \\ \pi_{88,b}^{\mu\nu}(p ) & = & \frac{g^2t}{4}\sum_n\int \frac{d^3 { \mathbf k}}{(2\pi)^3 } \mbox{tr}_{\rm d } \left [ \gamma^{\mu } g_{3}^+(k ) \gamma^{\nu } g_{3}^+(k ' ) + \gamma^{\mu } g_{3}^-(k ) \gamma^{\nu}g_{3}^-(k ' ) \right.\nonumber\\ & + & \left . \gamma^{\mu } g_{4}^+(k ) \gamma^{\nu}g_{4}^+(k ' ) + \gamma^{\mu } g_{4}^-(k ) \gamma^{\nu}g_{4}^-(k ' ) \right ] . \label{pi88b}\end{aligned}\ ] ] after performing the traces over the color , the flavor and the nambu - gorkov indices , the diagonal components of the polarization tensor @xmath68 with @xmath69 have the form @xmath70.\end{aligned}\ ] ] note that @xmath71 . apart from the diagonal elements , there are also nonzero off - diagonal elements , @xmath72 with @xmath73.\end{aligned}\ ] ] the physical gluon fields in the 2sc / g2sc phase are the following linear combinations : @xmath74 and @xmath75 . these new fields , @xmath76 and @xmath77 describe two pairs of massive vector particles with well defined electomagnetic charges , @xmath78 . the components of the polarization tensor in the new basis read in the static limit , all four eigenvalues of the polarization tensor are degenerate . by making use of the definition in eq . ( [ def - debye ] ) , we derive the following result for the corresponding debye masses : @xmath81 \!\ ! . \label{m_d_4}\ ] ] here we assumed that @xmath24 is vanishing which is a good approximation in neutral two - flavor quark matter . the fourfold degenerate meissner screening mass of the gluons with @xmath67 reads @xmath82 \!\ ! . \label{m_m_4}\ ] ] both results in eqs . ( [ m_d_4 ] ) and ( [ m_m_4 ] ) interpolate between the known results in the normal phase ( i.e. , @xmath83 ) and in the ideal 2sc phase ( i.e. , @xmath84 ) . the instability of off - diagonal gluons appears in the whole region of gapless 2sc phases ( with @xmath66 ) and even in some gapped 2sc phases ( with @xmath85 ) . it is noticed that the meissner screening mass square for the off - diagonal gluons decreases monotonously to zero when the mismatch increases from zero to @xmath86 , then goes to negative value with further increase of the mismatch in the gapped 2sc phase . however , the behavior of diagonal 8-th gluon s meissner mass square is quite different . it keeps as a constant in the gapped 2sc phase . in the gapless 2sc phase , all these five gluons meissner mass square are negative . it is not understood why gapless color superconducting phases exhibit chromomagnetic instability . it sounds quite strange especially in the g2sc phase , where it is the electrical neutrality not the color neutrality playing the essential role . it is a puzzle why the gluons can feel the instability by requiring the electrical neutrality on the system . in order to understand what is really going ` wrong ' with the homogeneous g2sc phase , we want to know whether there exists other instabilities except the chromomagnetic instability . for that purpose , we probe the g2sc phase using different external sources , e.g. , scalar and vector diquarks , mesons , vector current , and so on . here we report the most interesting result regarding the response of the g2sc phase to an external vector current @xmath87 , the time - component and spatial - components of this current correspond to the baryon number density and baryon current , respectively . from the linear response theory , the induced current and the external vector current is related by the response function @xmath88 , @xmath89 .\ ] ] the trace here runs over the nambu - gorkov , flavor , color and dirac indices . the explicit form of vertices is @xmath90 . the explicit expression of the vector current response function takes the form of @xmath91 with @xmath92 , \\ % \label{piv } \pi_{v , b}^{\mu\nu}(p ) & = & \frac{t}{2}\sum_n\int \frac{d^3 { \mathbf k}}{(2\pi)^3 } \mbox{tr}_{\rm d } \left [ \right . \nonumber\\ & & \left . \gamma^{\mu } g_{3}^+(k ) \gamma^{\nu } g_{3}^+(k ' ) + \gamma^{\mu } g_{3}^-(k ) \gamma^{\nu}g_{3}^-(k ' ) \right.\nonumber\\ & + & \left . \gamma^{\mu } g_{4}^+(k ) \gamma^{\nu}g_{4}^+(k ' ) + \gamma^{\mu } g_{4}^-(k ) \gamma^{\nu}g_{4}^-(k ' ) \right ] , % \label{pivb}\end{aligned}\ ] ] here the trace is over the dirac space . comparing the explicit expression of @xmath93 with that of the 8-th gluon s self - energy @xmath94 , i.e. , eq . ( [ pi88 ] ) , it can be clearly seen that , @xmath93 and @xmath94 almost share the same expression , except the coefficients . this can be easily understood , because the color charge and color current carried by the 8-th gluon is proportional to the baryon number and baryon current , respectively . in the static long - wavelength ( @xmath95 and @xmath96 ) limit , the time - component and spatial component of @xmath93 give the baryon number susceptibility @xmath97 and baryon current susceptibility @xmath98 , respectively , @xmath99 in the g2sc phase , @xmath100 as well as @xmath98 become negative . this means that , except the chromomagnetic instability corresponding to broken generators of @xmath1 , and the instability of a net momentum for diquark pair , the g2sc phase is also unstable with respect to an external color neutral baryon current @xmath101 . the 8-th gluon s magnetic instability , the diquark momentum instability and the color neutral baryon current in the g2sc phase can be understood in one common physical picture . the g2sc phase exhibits a paramagnetic response to an external baryon current . naturally , the color current carried by the 8-th gluon , which differs from the baryon current by a color charge , also experiences the instability in the g2sc phase . the paramagnetic instability of the baryon current indicates that the quark can spontaneously obtain a momentum , because diquark carries twice of the quark momentum , it is not hard to understand why the g2sc phase is also unstable with respect to the response of a net diquark momentum . it is noticed that , the instability of @xmath101 will be induced by mismatch in all the asymmetric fermi pairing systems , including superfluid systems , where @xmath101 can be interpreted as particle current . the paramagnetic response to an external vector current naturally suggests that a vector current can be spontaneously generated in the system . the generated vector current behaves as a vector potential , which modifies the quark self - energy with a spatial vector condensate @xmath102 , and breaks the rotational symmetry of the system . it can also be understood that the quasiparticles in the gapless phase spontaneously obtain a superfluid velocity , and the ground state is in an anisotropic state . the quark propagator @xmath35 in eq . ( [ freep ] ) is modified as @xmath103^{-1 } = \gamma^0 ( p_0 \pm \hat{\mu } ) - \vec{\gamma } \cdot \vec{p } \mp \vec{\gamma } \cdot \vec{\sigma}_v , \label{freep - m}\ ] ] with a subscript @xmath104 indicating the modified quark propagator . correspondingly , the inverse of the quark propagator @xmath105^{-1}$ ] in eq . ( [ prop ] ) is modified as @xmath106^{-1 } = \left(\begin{array}{cc } \left[g_{0,v}^{+}(p)\right]^{-1 } & \delta^- \\ \delta^+ & \left[g_{0,v}^{-}(p)\right]^{-1 } \end{array}\right ) . \label{prop - m}\ ] ] it is noticed that the expression of the modified inverse quark propagator @xmath107^{-1}$ ] takes the same form as the inverse quark propagator in the one - plane wave loff state shown in ref . the net momentum @xmath108 of the diquark pair in the loff state @xcite is replaced here by a spatial vector condensate @xmath109 . the spatial vector condensate @xmath102 breaks rotational symmetry of the system . this means that the fermi surfaces of the pairing quarks are not spherical any more . it has to be pointed out , the baryon current offers one doppler - shift superfluid velocity for the quarks . a spontaneously generated nambu - goldstone current in the minimal gapless model @xcite or a condensate of 8-th gluon s spatial component can do the same job . all these states mimic the one - plane wave loff state . in the following , we just call all these states as the single - plane wave loff state . in order to determine the deformed structure of the fermi surfaces , one should self - consistently minimize the free energy @xmath110 . the explicit form of the free energy can be evaluated directly using the standard method , in the framework of nambu jona - lasinio model @xcite , it takes the form of @xmath111^{-1 } ) + \frac{\delta^2}{4 g_d},\end{aligned}\ ] ] where @xmath12 is the temperature , and @xmath27 is the coupling constant in the diquark channel . when there is no charge neutrality condition , the ground state is determined by the thermal stability condition , i.e. , the local stability condition . the ground state is in the 2sc phase when @xmath112 with @xmath113 , in the loff phase when @xmath114 correspondingly @xmath115 , and then in the normal phase with @xmath116 when the mismatch is larger than @xmath117 . here @xmath118 indicate the diquark gap in the case of @xmath119 and @xmath120 , respectively . when charge neutrality condition is required , the ground state of charge neutral quark matter should be determined by solving the gap equations as well as the charge neutrality condition , i.e. , @xmath121 by changing @xmath122 or coupling strength @xmath27 , the solution of the charge neutral loff state can stay everywhere in the full loff window , including the window not protected by the local stability condition , as shown explicitly in ref . @xcite . from the lesson of charge neutral g2sc phase , we learn that even though the neutral state is a thermal stable state , i.e. , the thermodynamic potential is a global minimum along the neutrality line , it can not guarantee the dynamical stability of the system . the stability of the neutral system should be further determined by the dynamical stability condition , i.e. , the positivity of the meissner mass square . the polarization tensor for the gluons with color @xmath123 should be evaluated using the modified quark propagator @xmath124 in eq . ( [ prop - m ] ) , i.e. , @xmath125 , \label{piab}\ ] ] with @xmath126 and the explicit form of the vertices @xmath127 has the form @xmath128 . in the loff state , the meissner tensor can be decomposed into transverse and longitudinal component . the transverse and longitudinal meissner mass square for the off - diagonal 4 - 7 gluons and the diagonal 8-th gluon have been performed explicitly in the one - plane wave loff state in ref . @xcite . \1 ) the stable loff ( s - loff ) window in the region of @xmath129 , which is free of any magnetic instability . please note that this s - loff window is a little bit wider than the window @xmath130 protected by the local stability condition . \2 ) the stable window for diagonal gluon characterized by ds - loff window in the region of @xmath131 , which is free of the diagonal 8-th gluon s magnetic instability but not free of the off - diagonal gluons magnetic instability ; \3 ) the unstable loff ( us - loff ) window in the region of @xmath132 , with @xmath133 . in this us - loff window , all the magnetic instabilities exist . please note that , it is the longitudinal meissner mass square for the 8-th gluon is negative in this us - loff window , the transverse meissner mass square of 8-th gluon is always zero in the full loff window , which is guaranteed by the momentum equation . us - loff is a very interesting window , it indicates that the loff state even can not cure the 8-th gluon s magnetic instability . in the charge neutral 2-flavor system , it seems that the diagonal gluon s magnetic instability can not be cured in the gluon phase , because there is no direct relation between the diagonal gluon s instability and the off - diagonal gluons instability . ( of course , it has to be carefully checked , whether all the instabilities in this us - loff window can be cured by off - diagonal gluons condensate in the charge neutral 2-flavor system . ) it is also noticed that in this us - loff window , the mismatch is close to the diquark gap , i.e. , @xmath134 . therefore it is interesting to check whether this us - loff window can be stabilized by a spin-1 condensate @xcite as proposed in ref . @xcite . in the charge neutral 2sc phase , though it is unlikely , we might have a lucky chance to cure the diagonal instability by the condensation of off - diagonal gluons . it is expected that this instability will show up in some constrained abelian asymmetric superfluidity system , e.g. , in the fixed number density case @xcite . it will be a new challenge for us to really solve this problem . we have seen that , except chromomagnetic instability corresponding to broken generators of @xmath1 , the g2sc phase is also unstable with respect to the external neutral baryon current . it is noticed that all the instabilities are induced by increasing the mismatch between the fermi surfaces of the cooper pairing . in order to understand the instability driven by mismatch , in the following , we give some general analysis . a superconductor will be eventually destroyed and goes to the normal fermi liquid state , so one natural question is : how an ideal bcs superconductor will be destroyed by increasing mismatch ? to answer how a superconductor will be destroyed , one has to firstly understand what is a superconductor . the superconducting phase is characterized by the order parameter @xmath135 , which is a complex scalar field and has the form of e.g. , for electrical superconductor , @xmath136 , with @xmath137 the amplitude and @xmath138 the phase of the gap order parameter or the pseudo nambu - goldstone boson . there are two ways to destroy a superconductor . one way is by driving the amplitude of the order parameter to zero . this way is bcs - like , because it mimics the behavior of a conventional superconductor at finite temperature , the gap amplitude monotonously drops to zero with the increase of temperature ; another way is non - bcs like , but berezinskii - kosterlitz - thouless ( bkt)-like @xcite , even if the amplitude of the order parameter is large and finite , superconductivity will be lost with the destruction of phase coherence , e.g. the phase transition from the @xmath144wave superconductor to the pseudogap state in high temperature superconductors @xcite . stimulating by the role of the phase fluctuation in the unconventional superconducting phase in condensed matter , we follow ref . @xcite to formulate the 2sc phase in the nonlinear realization framework in order to naturally take into account the contribution from the phase fluctuation or pseudo nambu - goldstone current . in the 2sc phase , the color symmetry @xmath145 breaks to @xmath146 . the generators of the residual @xmath53 symmetry h are @xmath147 with @xmath148 and the broken generators @xmath149 with @xmath150 . more precisely , the last broken generator is a combination of @xmath151 and the generator @xmath152 of the global @xmath153 symmetry of baryon number conservation , @xmath154 of generators of the global @xmath155 and local @xmath1 symmetry . the coset space @xmath156 is parameterized by the group elements @xmath157\,\,,\ ] ] here @xmath158 and @xmath159 are five nambu - goldstone diquarks , and we have neglected the singular phase , which should include the information of the topological defects @xcite . operator @xmath160 is unitary , @xmath161 . introducing a new quark field @xmath162 , which is connected with the original quark field @xmath163 in eq . ( [ lagr-2sc ] ) in a nonlinear transformation form , @xmath164 and the charge - conjugate fields transform as @xmath165 in high-@xmath166 superconductor , this technique is called charge - spin separation , see ref . the advantage of transforming the quark fields is that this preserves the simple structure of the terms coupling the quark fields to the diquark sources , @xmath167 in mean - field approximation , the diquark source terms are proportional to @xmath168 introducing the new nambu - gorkov spinors @xmath169 the nonlinear realization of the original lagrangian density eq.([lagr-2sc ] ) takes the form of @xmath170 where @xmath171^{-1 } & \phi^- \\ \phi^+ & [ g^-_{0,nl}]^{-1 } \end{array } \right)\,\ , .\ ] ] here the explicit form of the free propagator for the new quark field is @xmath172^{-1 } & = & i\ , \fsl{d } + { \hat \mu } \ , \gamma_0 + \gamma_\mu \ , v^\mu , % [ g^+_{0,nl}]^{-1 } & = & i\ , \gamma^\mu \partial_\mu + { \hat \mu } \ , \gamma_0 + \gamma_\mu \ , v^\mu , \end{aligned}\ ] ] and @xmath173^{-1 } & = & i\ , \fsl{d}^t - { \hat \mu } \ , \gamma_0 + \gamma_{\mu } \ , v_c^\mu .\end{aligned}\ ] ] comparing with the free propagator in the original lagrangian density , the free propagator in the non - linear realization framework naturally takes into account the contribution from the nambu - goldstone currents or phase fluctuations , i.e. , @xmath174 which is the @xmath175-dimensional maurer - cartan one - form introduced in ref . the linear order of the nambu - goldstone currents @xmath176 and @xmath177 has the explicit form of @xmath178 the lagrangian density eq . ( [ lagr - nl ] ) for the new quark fields looks like an extension of the theory in ref . @xcite for high-@xmath166 superconductor to non - abelian system , except that here we neglected the singular phase contribution from the topologic defects . the advantage of the non - linear realization framework eq . ( [ lagr - nl ] ) is that it can naturally take into account the contribution from the phase fluctuations or nambu - goldstone currents . the task left is to correctly solve the ground state by considering the phase fluctuations . the free energy @xmath179 can be evaluated directly and it takes the form of @xmath180^{-1 } ) + \frac{\phi^2}{4 g_d}. \label{free - energy - ng}\end{aligned}\ ] ] to evaluate the ground state of @xmath179 as a function of mismatch is tedious and still under progress . in the following we just give a brief discussion on the nambu - goldstone current generation state @xcite , one - plane wave loff state @xcite , as well as the gluon phase @xcite . if we expand the thermodynamic potential @xmath179 of the non - linear realization form in terms of the nambu - goldstone currents , we will naturally have the nambu - goldstone currents generation in the system with the increase of mismatch , i.e. , @xmath181 and/or @xmath182 at large @xmath183 . this is an extended version of the nambu - goldstone current generation state proposed in a minimal gapless model in ref . @xcite . from eq . ( [ lagr - nl ] ) , we can see that @xmath184 contributes to the baryon current . @xmath182 indicates a baryon current generation or 8-th gluon condensate in the system , it is just the one - plane wave loff state . this has been discussed in sec . [ sec - usloff ] . the other four nambu - goldstone currents generation @xmath181 indicates other color current generation in the system , and is equivalent to the gluon phase described in ref . @xcite . we do not argue whether the system will exprience a gluon condensate phase or nambu - goldstone currents generation state . we simply think they are equivalent . in fact , the gauge fields and the nambu - goldstone currents share a gauge covariant form as shown in the free propogator . however , we prefer to using nambu - goldstone currents generation than the gluon condensate in the gnjl model . as mentioned in sec . [ sec - gnjl ] , in the gnjl model , all the information from unkown nonperturbative gluons are hidden in the diquark gap parameter @xmath185 . the gauge fields in the lagrangian density are just external fields , they only play the role of probing the system , but do not contribute to the property of the color superconducting phase . therefore , there is no gluon free - energy in the gnjl model , it is not clear how to derive the gluon condensate in this model . in order to investigate the problem in a fully self - consistent way , one has to use the ambitious framework by using the dyson - schwinger equations ( dse ) @xcite including diquark degree of freedom @xcite or in the framework of effective theory of high - density quark matter as in ref . @xcite . except the chromomagnetic instability , the g2sc phase also exhibits a paramagnetic response to the perturbation of an external baryon current . this suggests a baryon current can be spontaneously generated in the g2sc phase , and the quasiparticles spontaneously obtain a superfluid velocity . the spontaneously generated baryon current breaks the rotational symmetry of the system , and it resembles the one - plane wave loff state . we further describe the 2sc phase in the nonlinear realization framework , and show that each instability indicates the spontaneous generation of the corresponding pseudo nambu - goldstone current . we show this nambu - goldstone currents generation state can naturally cover the gluon phase as well as the one - plane wave loff state . we also point out that , when charge neutrality condition is required , there exists a narrow unstable loff ( us - loff ) window , where not only off - diagonal gluons but the diagonal 8-th gluon can not avoid the magnetic instability . the diagonal gluon s magnetic instability in this us - loff window can not be cured by off - diagonal gluon condensate in color superconducting phase . more interestingly , this us - loff window will also show up in some constrained abelian asymmetric superfluid system . the us - loff window brings us a new challenge . we need new thoughts on understanding how a bcs supercondutor will be eventually destroyed by increasing the mismatch , we also need to develop new methods to really resolve the instability problem . some methods developed in unconventional superconductor field , e.g. , high-@xmath166 superconductor , might be helpful . till now , the results on instabilities are based on mean - field ( mf ) approximation . the bcs theory at mf can describe strongly coherent or rigid superconducting state very well . however , as we pointed out in @xcite , with the increase of mismatch , the low degrees of freedom in the system have been changed . for example , the gapless quasi - particle excitations in the gapless phase , and the small meissner mass square of the off - diagonal gluons around @xmath186 . this indicates that these quasi - quarks and gluons become low degrees of freedom in the system , their fluctuations become more important . in order to correctly describe the system , the low degrees of freedom should be taken into account properly . the work toward this direction is still in progress . the author thanks m. alford , f.a . bais , k. fukushima , e. gubankova , m. hashimoto , t. hatsuda , l.y . hong , w.v . liu , m. mannarelli , t. matsuura , y. nambu , k. rajagopal , h.c . ren , d. rischke , t. schafer , a. schmitt , i. shovkovy , d. t. son , m. tachibana , z.tesanovic , x. g. wen , z. y. weng , f. wilczek and k. yang for valuable discussions . the work is supported by the japan society for the promotion of science fellowship program . r. rapp , t. schfer , e. v. shuryak and m. velkovsky , phys . lett . * 81 * , 53 ( 1998 ) ; 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we review recent progress of understanding and resolving instabilities driven by mismatch between the fermi surfaces of the pairing quarks in 2-flavor color superconductor . with the increase of mismatch , the 2sc phase exhibits chromomagnetic instability as well as color neutral baryon current instability . we describe the 2sc phase in the nonlinear realization framework , and show that each instability indicates the spontaneous generation of the corresponding pseudo nambu - golstone current . the nambu - goldstone currents generation state covers the gluon phase as well as the one - plane wave loff state . we further point out that , when charge neutrality condition is required , there exists a narrow unstable loff ( us - loff ) window , where not only off - diagonal gluons but the diagonal 8-th gluon can not avoid the magnetic instability . in this us - loff window , the diagonal magnetic instability can not be cured by off - diagonal gluon condensate in the color superconducting phase .

arxiv

the debate about the properties of the melting phase transition of two - dimensional ( 2d ) systems did not lose its intensity over the past several decades . recent developments in the fabrication of 2d materials @xcite simultaneously seek for , and may provide clarification of the details of the transition . a milestone , and still the most widely accepted theory available , is the kosterlitz - thouless - halperin - nelson - young ( kthny ) picture @xcite . in the underlying physical process two separate , continuous transitions can be distinguished , as the solid transforms into a liquid in quasi - equilibrium steps by slow heating . during the first stage the translational ( positional ) order vanishes , while in the second stage the orientational order decays . all this is mediated by the unbinding of ( i ) dislocation pairs into individual dislocations , and ( ii ) dislocations into point defects @xcite . the strength of this theory consists in its compatibility with the mermin - wagner theorem that forbids the existence of exact long range positional order in 2d for a wide range of pair potentials , at finite temperatures @xcite . the most criticized weakness of it , however , is that it assumes a dilute , unstructured distribution of the lattice defects , which is in contradiction with observations , where the alignment and accumulation of dislocations into small angle domain walls was found @xcite . since the birth of the kthny theory , the examination of its validity for systems with different pair interactions has been in focus . investigations started with hard - sphere ( disk ) , lennard - jones , and coulomb systems . more recently , systems characterized by dipole - dipole and debye - hckel ( screened coulomb or yukawa ) inter - particle interactions became important due to the significant advances achieved in the field of colloid suspensions @xcite and dusty plasmas @xcite . to illustrate the incongruity of both experimental and simulation results that had accumulated over the last three decades on investigations of classical single - layer ( 2d ) many - body systems , we list a few examples : * first order phase transition to exist was reported for lennard - jones systems @xcite and hard - disk systems @xcite , for the phase - field - crystal ( pfc ) model @xcite , as well as for coulomb and dipole systems @xcite in simulations , and in experiments with halomethanes and haloethanes physisorbed on exfoliated graphite @xcite , as well as in experiments on a quasi - two - dimensional suspension of uncharged silica spheres @xcite . * second order ( or single step continuous ) transition was found in dusty plasma experiments @xcite , for a hard - disk system @xcite , electro - hydrodynamicly excited colloidal suspensions @xcite , as well as for coulomb @xcite and yukawa @xcite systems . * kthny - like transition was reported in a dusty plasma experiment @xcite and related numerical simulations @xcite , for the harmonic lattice model @xcite , in experiments and simulations of colloidal suspensions @xcite , for lennard - jones @xcite , yukawa @xcite , hard disk @xcite , dipole - dipole @xcite , gaussian - core @xcite , and electron systems @xcite , as well as for a system with @xmath1 repulsive pair potential @xcite , weakly softened core @xcite , and for vortices in a w - based superconducting thin film @xcite . the effect of the range of the potential on two - dimensional melting was studied in @xcite for a wide range of morse potentials . it has been shown , that extended - ranged interatomic potentials are important for the formation of a `` stable '' hexatic phase . similar conclusion was drawn in @xcite for modified hard - disk potentials . the effect of the dimensionality ( deviation from the mathematically perfect 2d plane ) on the hexatic phase was discussed for lennard - jones systems in @xcite . it was found , that an intermediate hexatic phase could only be observed in a monolayer of particles confined such that the fluctuations in the positions perpendicular to the particle layer was less than 0.15 particle diameters . the timeline of the results listed above shows a general trend : in earlier studies , first or second order phase transitions were identified in particle simulations , but subsequently , as the computational power increased with time , since approximately the year of 2000 , particle based numerical studies became in favor of the kthny theory . a possible resolution of the ongoing debate is given in @xcite , where extensive monte carlo simulations of 2d lennard - jones systems have revealed the metastable nature of the hexatic phase . this seems to support pfc simulations @xcite operating on the diffusive time - scale ( averaging out single particle oscillations ) , which is significantly longer , than what monte carlo ( mc ) or molecular dynamics ( md ) methods can cover . in this paper we will show that the observation of the hexatic phase is strongly linked with the thermodynamic equilibration of the systems . the necessary equilibration time , in turn , strongly depends on the measured quantity of interest . local , or single particle properties can equilibrate very rapidly , while long - range , or collective relaxations usually take significantly longer . we find , consequently , that monitoring the velocity distribution function alone to verify the equilibration of the system is insufficient . the idea , that numerical simulations may have related equilibration issues ( called as kinetic bottlenecks ) was raised already in 1993 in @xcite . we have performed extensive microcanonical md simulations @xcite in the close vicinity of the expected solid - liquid phase transition temperature , @xmath2 , for repulsive screened coulomb ( also called yukawa or debye - hckel ) pair - potential with the potential energy in form of @xmath3 where @xmath4 is the debye screening length , @xmath5 is the electric charge of the particles , and @xmath6 is the vacuum permittivity . to characterize the screening we use the dimensionless screening parameter @xmath7 , where @xmath8 is the wigner - seitz radius , and @xmath9 is the particle density this model potential was chosen because of its relevance to several experimental systems consisting of electrically charged particles , like dusty plasmas , charged colloidal suspensions , and electrolytes . here we show results obtained for @xmath10 . our earlier studies @xcite identified the melting transition ( without clarifying its nature ) to take place around the coulomb coupling parameter @xmath11 for this strength of screening . time is measured in units of the nominal 2d plasma oscillation period with @xmath12 where @xmath13 is the mass of a particle . our simulations are initialized by placing @xmath14 particles ( in the range of 1,920 to 740,000 ) into a rectangular simulation cell that has periodic boundary conditions . the particles are released from hexagonal lattice positions , with initial velocities randomly sampled from a predefined distribution . at the initial stage , which has a duration @xmath15 ( thermalization time ) , the system is thermostated by applying the velocity back - scaling method ( to follow the usual approach used in many previous studies ) to reach near - equilibrium state at the desired ( kinetic ) temperature . data collection starts only after this initial stage and runs for a time period @xmath16 ( measurement time ) without any additional thermostation . to characterize the level of equilibration we study the time and system size dependence of the following quantities : * momenta of the velocity distribution function , @xmath17 , * the configurational temperature , @xmath18 @xcite , and * the long - range decay of the @xmath19 pair - correlation and @xmath20 bond - angle correlation functions @xcite . while in the case of the first two quantities @xmath21 , in the simulations targeting the correlation functions , @xmath15 is varied over a wide range and the measurement time is chosen to be @xmath22 to avoid significant changes ( due to ongoing equilibration ) during the measurement . using the maxwell - boltzmann assumption for the velocity distribution in thermal equilibrium in the form @xmath23 where @xmath24 , in two - dimensions the first four velocity moments are : @xmath25 to measure the relaxation time of the velocity distribution function we have performed md simulations with particle numbers @xmath26 and @xmath27 , with initial velocity components ( @xmath28 and @xmath29 ) sampled from a uniform distribution between @xmath30 and @xmath31 , in order to start with the desired average kinetic energy , but being far from equilibrium . figure [ fig : vmom ] shows the time evolution of the first eight velocity moments normalized with their theoretical equilibrium values . as already mentioned , the initial conditions are far from the equilibrium configuration ( perfect lattice position and non - thermal velocity distribution ) . ( color online ) moments of the computed velocity distribution functions relative to the theoretical equilibrium values vs. simulation time at temperatures slightly above ( full lines ) and below ( dashed lines ) the melting point , @xmath2 . the dashed lines are mostly hidden behind the full lines , indicating a low sensitivity on the temperature . the dark red curve shows functional fit in the form @xmath32 to @xmath33 . @xmath26.,width=302 ] we can observe , that the velocity momenta have initial values very different from the expected maxwell - boltzmann equilibrium distribution . the values approach the equilibrium value asymptotically with regular oscillations . these oscillations ( or fluctuations ) are typical for microcanonical md simulations , where the total energy of the system is constant , while there is a permanent exchange of potential and kinetic energies . the relaxation time can be found by fitting the curves with an exponential asymptotic formula in the form @xmath32 . we find , that the relaxation of the velocity distribution can be characterized by a short relaxation time of @xmath34 , and this is independent of system size and temperature in the vicinity of the melting point . in 1997 , rugh @xcite pointed out that the temperature can also be expressed as ensemble average over geometrical and dynamical quantities and derived the formula for the configurational temperature : @xmath35 where @xmath36 . as the central quantity in this expression is the inter - particle force acting on each particle , in case of finite range interactions ( like the yukawa potential ) , the configurational temperature is sensitive on the local environment within this range . simulations were performed for a series of particle numbers between @xmath37 and @xmath38 with initial velocities sampled from maxwellian distribution . figure [ fig : tconf](a ) shows examples from runs with @xmath26 for the time evolutions , while fig . [ fig : tconf](b ) presents relaxation time data computed ( similarly as above ) for different kinetic temperatures . ( color online ) ( a ) time evolution of the configurational temperature @xmath18 for different kinetic temperatures @xmath39 . ( b ) relaxation time vs. kinetic temperature . ( @xmath26).,width=302 ] we observe relaxation times about an order of magnitude longer ( @xmath40 ) compared to the velocity distribution , and a strong temperature dependence in the vicinity of the melting point . no significant system size dependence was found . the central property used to identify the hexatic phase is traditionally the long - range behavior of the pair- , and bond - order correlation functions , @xmath19 and @xmath20 , respectively @xcite . to be able to compute correlations at large distances , one naturally has to use large particle numbers , otherwise the periodic boundary conditions introduce artificial correlation peaks . this trivial constraint led to investigations of larger and larger systems by different groups . figures [ fig : corr ] and [ fig : corr2 ] show correlation functions for systems consisting of @xmath41 particles , for a set of increasing equilibration times provided to the systems before performing the data collection . ( color online ) log log plots of ( a ) an example of @xmath42 pair correlation function with its upper envelope , ( b ) a series of envelope curves of pair correlation functions , ( c ) @xmath20 bond - order correlation functions measured after letting the systems equilibrate for various times indicated , @xmath15 , at a temperature 1 percent above the melting point . the systems consisted of @xmath41 particles , the data acquisition took @xmath43 and started after @xmath15 has elapsed.,width=302 ] ( color online ) same as fig [ fig : corr ] with semi - logarithmic scales.,width=302 ] we can observe a clear long - time evolution of the correlation functions . on the double - logarithmic plot the @xmath19 pair - correlation functions show already at early times a long - range decay , which is faster than power - law [ fig . [ fig : corr](a , b ) ] , while the @xmath20 orientational correlations smooth out to near perfect straight lines [ fig . [ fig : corr](c ) ] , representing power - law type decay for relatively long times . on the semi - logarithmic graphs all the @xmath19 functions have almost straight upper envelopes [ fig . [ fig : corr2](a , b ) ] in the intermediate distance range @xmath44 , where the statistical noise is still negligible . this indicates almost pure exponential decay , although the characteristic decay distance does decrease with increasing simulation time . on the other hand , it is only the last @xmath20 orientational correlation function , belonging to the longest simulation , that shows linear apparent asymptote on the semi - logarithmic scale [ fig . [ fig : corr2](c ) ] , representing a clear exponential decay , meaning the lack of long range order . to conclude these observations : in short simulations we observe short - range positional and quasi - long - range orientational order , signatures of the hexatic phase , which , however vanish if we provide the system longer time for equilibration . as a consequence , in the case we would stop the simulation at , e.g. , @xmath45 ( which already means simulation time - steps in the order of @xmath46 , as plasma oscillations have to be resolved smoothly ) we may identify the system to be in the hexatic phase , exactly as shown in @xcite , which , however is not the true equilibrium configuration . in addition , as the accessible length scale strongly depends on the system size ( typically less than 1/3 of the side length of the simulation box ) , smaller systems apparently equilibrate faster . we have found @xmath47 to be sufficient to reach equilibrium for a system of @xmath37 particles , while @xmath48 was needed for @xmath41 . to verify , that the observed slowdown of relaxation is not an artifact of the applied microcanonical ( constant @xmath49 ) simulation , we have implemented the computationally much more demanding , but in principle for phase transition studies better suited isothermal - isobaric ( constant @xmath50 ) molecular dynamics scheme @xcite . although the @xmath50 simulations were performed for much smaller systems ( @xmath51 ) , limiting the calculation of the correlation functions to a shorter range and resulting in higher noise levels , the same long - time tendency of decaying long - range correlations could be identified as already shown with the computationally much more efficient microcanonical simulations . during the equilibration of an interacting charged many - particle system we have identified three different stages of relaxation : * the velocity distribution does approach the maxwellian distribution within a few plasma oscillation cycles . in the close vicinity of the melting transition the speed of this process is found to be independent of temperature and system size . * compared to the velocity distribution function , the configurational temperature ( determined by the local neighborhood within the range of the inter - particle interaction potential ) relaxes at time scales about an order of magnitude longer for our systems . the relaxation time scale is not sensitive to the system size , but has a strong dependence on the temperature . * the equilibration of the long range correlations is significantly slower compared to the above quantities , and depends strongly on the systems size ( larger systems need longer time to equilibrate ) . from this study we can conclude , that increasing the system size in particle simulations alone can be insufficient and can result in misleading conclusions , as the length of the equilibration period also plays a crucial role in building up or destroying correlations . in the vast majority of the earlier numerical studies on charged particle ensembles ( as listed in the introduction ) no simulation time is specified , given to the system to equilibrate before the actual measurement were performed , neither is the method of characterizing the quality of the equilibrium described . based on these results , we suspect , that the rapidly increasing computational resources in the first decade of the 21st century beguiled increasing the system sizes in particle simulations without increasing the length of the simulated time intervals . in the majority of these studies the systems may got stuck in the metastable hexatic phase , instead of settling in the true equilibrium configuration . 67ifxundefined [ 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 @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' ( , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , ed . 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for two - dimensional many - particle systems first - order , second - order , single step continuous , as well as two - step continuous ( kthny - like ) melting transitions have been found in previous studies . recent computer simulations , using particle numbers in the @xmath0 range , as well as a few experimental studies , tend to support the two - step scenario , where the solid and liquid phases are separated by a third , so called hexatic phase . we have performed molecular dynamics simulations on yukawa ( debye - hckel ) systems at conditions earlier predicted to belong to the hexatic phase . our simulation studies on the time needed for the equilibration of the systems conclude that the hexatic phase is metastable and disappears in the limit of long times . we also show that simply increasing the particle number in particle simulations does not necessarily result in more accurate conclusions regarding the existence of the hexatic phase . the increase of the system size has to be accompanied with the increase of the simulation time to ensure properly thermalized conditions .

arxiv

the kepler mission @xcite , with the discovery of over 4100 planetary candidates in 3200 systems , has spawned a revolution in our understanding of planet occurrence rates around stars of all types . one of kepler s profound discoveries is that small planets ( @xmath8 ) are nearly ubiquitous ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and , in particular , some of the most common planets have sizes between earth - sized and neptune - sized a planet type not found in our own solar system . indeed , it is within this group of super - earths to mini - neptunes that there is a transition from `` rocky '' planets to `` non - rocky planets '' ; the transition is near a planet radius of @xmath9 and is very sharp occurring within @xmath10 of this transition radius @xcite . unless an intra - system comparison of planetary radii is performed where only the relative planetary sizes are important @xcite , having accurate ( as well as precise ) planetary radii is crucial to our comprehension of the distribution of planetary structures . in particular , understanding the radii of the planets to within @xmath11 is necessary if we are to understand the relative occurrence rates of `` rocky '' to `` non - rocky '' planets , and the relationship between radius , mass , and bulk density .. while there has been a systematic follow - up observation program to obtain spectroscopy and high resolution imaging , only approximately half of the kepler candidate stars have been observed ( mostly as a result of the brightness distribution of the candidate stars ) . those stars that have been observed have been done mostly to eliminate false positives , to determine the stellar parameters of host stars , and to search for nearby stars that may be blended in the kepler photometric apertures . stars that are identified as possible binary or triple stars are noted on the kepler community follow - up observation program website , and are often handled in individual papers ( e.g. , * ? ? ? * ; * ? ? ? * ) . the false positive assessment of an koi ( or all of the kois ) can take into account the likelihood of stellar companions ( e.g. , * ? ? ? * ; * ? ? ? * ) , and a false positive probability will likely be included in future koi lists . but presently , the current production of the planetary candidate koi list and the associated parameters are derived assuming that _ all _ of the koi host stars are single . that is , the kepler pipeline treats each kepler candidate host star as a single star ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? thus , statistical studies based upon the kepler candidate lists are also assuming that all the stars in the sample set are single stars . the exact fraction of multiple stars in the kepler candidate list is not yet determined , but it is certainly not zero . recent work suggests that a non - negligible fraction ( @xmath12 ) of the kepler host stars may be multiple stars @xcite , although other work may indicate that ( giant ) planet formation may be suppressed in multiple star systems @xcite . the presence of a stellar companion does not necessarily invalidate a planetary candidate , but it does change the observed transit depths and , as a result , the planetary radii . thus , assuming all of the stars in the kepler candidate list are single can introduce a systematic uncertainty into the planetary radii and occurrence rate distributions . this has already been discussed for the occurrence rate of hot jupiters in the kepler sample where it was found that @xmath13 of hot jupiters were classified as smaller planets because of the unaccounted effects of transit dilution from stellar companions @xcite . in this paper , we explore the effects of undetected stellar gravitationally bound companions on the observed transit depths and the resulting derived planetary radii for the entire kepler candidate sample . we do not consider the dilution effects of line - of - sight background stars , rather only potential bound companions , as companions within @xmath14 are most likely bound companions ( e.g. , * ? ? ? * ; * ? ? ? * ) , and most stars beyond @xmath14 are either in the kepler input catalog @xcite or in the ukirt survey of the kepler field and , thus , are already accounted for with regards to flux dilution in the kepler project transit fitting pipeline . within 1 , the density of blended background stars is fairly low , ranging between @xmath15 stars/@xmath16 @xcite . thus , within a radius of 1 , we expect to find a blended background ( line - of - sight ) star only @xmath17% of the time . therefore , the primary contaminant within 1 of the host stars are bound companions . we present here probabilistic uncertainties of the planetary radii based upon expected stellar multiplicity rates and stellar companion sizes . we show that , in the absence of any spectroscopic or high resolution imaging observations to vet companions , the observed planetary radii will be systematically too small . however , if a candidate host star is observed with high resolution imaging ( hri ) or with radial velocity ( rv ) spectroscopy to screen the star for companions , the underestimate of the true planet radius is significantly reduced . while imaging and radial velocity vetting is effective for the kepler candidate host stars , it will be even more effective for the k2 and tess candidates which will be , on average , 10 times closer than the kepler candidate host stars . the planetary radii are not directly observed ; rather , the transit depth is the observable which is then related to the planet size . the observed depth ( @xmath18 ) of a planetary transit is defined as the fractional difference in the measured out - of - transit flux ( @xmath19 ) and the measured in - transit flux @xmath20 : @xmath21 if there are @xmath22 stars within a system , then the total out - of - transit flux in the system is given by @xmath23 and if the planet transits the @xmath24 star in the system , then the in transit flux can be defined as @xmath25 where @xmath26 is the flux of the star with the transiting planet , @xmath27 is the radius of the planet , and @xmath28 is the radius of the star being transited . substituting into equation ( [ eq - single - flux ] ) , the generalized transit depth equation ( in the absence of limb darkening or star spots ) becomes @xmath29 for a single star , @xmath30 and the transit depth expression simplifies to just the square of the size ratio between the planet and the star . however , for a multiple star system , the relationship between the observed transit depth and the true planetary radius depends upon the brightness ratio of the transited star to the total brightness of the system _ and _ on the stellar radius which changes depending on which star the planet is transiting : @xmath31 the kepler planetary candidates parameters are estimated assuming the star is a single star @xcite , and , therefore , may incorrectly report the planet radius if the stellar host is really a multiple star system . the extra flux contributed by the companion stars will dilute the observed transit depth , and the derived planet radius depends on the size of star presumed to be transited . the ratio of the true planet radius , @xmath32 , to the observed planet radius assuming a single star with no companions , @xmath33 , can be described as : @xmath34 where @xmath35 is the radius of the ( assumed single ) primary star , and @xmath26 and @xmath28 are the brightness and the radius , respectively , of the star being transited by the planet . this ratio reduces to unity in the case of a single star ( @xmath36 and @xmath37 ) . for a multiple star system where the planet orbits the primary star ( @xmath38 ) , the planet size is underestimated only by the flux dilution factor : @xmath39 however , if the planet orbits one of the companion stars and not the primary star , then the ratio of the primary star radius ( @xmath35 ) to the radius of the companion star being transited ( @xmath28 ) affects the observed planetary radius , in addition to the flux dilution factor . ) are plotted as a function of companion - to - primary brightness ratios ( _ bottom axis _ ) and mass ratios ( _ top axis _ ) for possible binary systems ( _ top plot _ ) or triple ( _ bottom plot _ ) systems . this figure is an example for the g - dwarf koi-299 ; similar calculations have been made for every koi . in each plot , the dark blue stars represent the correction factors if the planet orbits the primary star ( equation [ eq - primary - ratio ] ) ; the red circles represent the correction factors if the planet orbits the secondary star , and the light blue triangles represent the correction factors if the planet orbits the tertiary star ( equation [ eq - full - ratio ] ) . the lines are third order polynomials fit to the distributions . unity is marked with a horizontal dashed line . ] , but for the m - dwarf koi-1085 , demonstrating that the details of the derived correction factors are dependent upon the koi properties . ] to explore the possible effects of the undetected stellar companions on the derived planetary parameters , we first assess what companions are possible for each koi . for this work , we have downloaded the cumulative kepler candidate list and stellar parameters table from the nasa exoplanet archive . the cumulative list is updated with each new release of the koi lists ; as a result , the details of any one star and planet may have changed since the analysis for this paper was done . however , the overall results of the paper presented here should remain largely unchanged . for the koi lists , the stellar parameters for each koi were determined by fitting photometric colors and spectroscopically derived parameters ( where available ) to the dartmouth stellar evolution database @xcite . the planet parameters were then derived based upon the transit curve fitting and the associated stellar parameters . other stars listed in the kepler input catalog or ukirt imaging that may be blended with the koi host stars were accounted for in the transit fitting , but , in general , as mentioned above , each planetary host star was assumed to be a single star . we have restricted the range of possible bound stellar companions to each koi host star by utilizing the same dartmouth isochrones used to determine the stellar parameters . possible gravitationally bound companions are assumed to lie along on the same isochrone as the primary star . for each koi host star , we found the single best fit isochrone ( characterized by mass , metallicity , and age ) by minimizing the chi - square fit to the stellar parameters ( effective temperature , surface gravity , radius , and metallicity ) listed in the koi table . we did not try to re - derive stellar parameters or independently find the best isochrone fit for the star ; we simply identified the appropriate dartmouth isochrone as used in the determination of the stellar parameters @xcite . we note that there exists an additional uncertainty based upon the isochrone finding . in this work , we did not try to re - derive the stellar parameters of the host stars , but rather , we simply find the appropriate isochrone that matches the koi stellar parameters . thus , any errors in the stellar parameters derivations in the koi list are propagated here . this is likely only a significant source of uncertainty for nearly equal brightness companions . once an isochrone was identified for a given star , all stars along an isochrone with ( absolute ) kepler magnitudes fainter than the ( absolute ) kepler magnitude of the host star were considered to be viable companions ; i.e. , the primary host star was assumed to be the brightest star in the system . the fainter companions listed within that particular isochrone were then used to establish the range of possible planetary radii corrections ( equation [ eq - full - ratio ] ) assuming the host star is actually a binary or triple star . higher order ( e.g. , quadruple ) stellar multiples are not considered here as they represent only @xmath40% of the stellar population @xcite . we have considered six specific multiplicity scenarios : 1 . single star ( @xmath41 ) 2 . binary star planet orbits primary star 3 . binary star planet orbits secondary star 4 . triple star planet orbits primary star 5 . triple star planet orbits secondary star 6 . triple star planet orbits tertiary star . based upon the brightness and size differences between the primary star and the putative secondary or tertiary companions , we have calculated for each koi the possible factor by which the planetary radii are underestimated ( @xmath0 ) . if the star is single , the correction factor is unity , and if , in a multiple star system , the planet orbits the primary star , only flux dilution affects the observed transit depth and the derived planetary radius ( eq . [ eq - primary - ratio ] ) . for the scenarios where the planets orbits the secondary or tertiary star , the planet size correction factors ( eq . [ eq - full - ratio ] ) were determined only for stellar companions where the stellar companion could physically account for the observed transit depth . if more than 100% of the stellar companion light had to be eclipsed in order to produce the observed transit in the presence of the flux dilution , then that star ( and all subsequent stars on the isochrone with lower mass ) was not considered viable as a potential source of the transit . for example , for an observed 1% transit , no binary companions can be fainter than the primary star by 5 magnitudes or more ; an eclipse of such a secondary star would need to be more than 100% deep . the stellar brightness limits were calculated independently for each planet within a koi system so as to not assume that all planets within a system necessarily orbited the same star . figures [ fig - koi299 ] and [ fig - koi1085 ] show representative correction factors ( @xmath0 ) for koi-299 ( a g - dwarf with a super - earth sized @xmath42 planet ) and for koi-1085 ( an m - dwarf with an earth - sized @xmath43 planet ) . the planet radius correction factors ( @xmath0 ) are shown as a function of the companion to primary brightness ratio ( bottom x - axis of plots ) and the companion to primary mass ratio ( top x - axis of plots ) and are determined for the koi assuming it is a binary - star system ( top plot ) or a triple - star system ( bottom plot ) . the amplitude of the correction factor ( @xmath0 ) varies strongly depending on the particular system and which star the planet may orbit . if the planet orbits the primary star , then the largest the correction factors are for equal brightness companions ( @xmath44 for a binary system and @xmath45 for a triple system ) with an asymptotic approach to unity as the companion stars become fainter and fainter . if the planet orbits the secondary or tertiary star , the planet radius correction factor can be significantly larger ranging from @xmath46 for binary systems and @xmath47 or more for triple systems depending on the size and brightness of the secondary or tertiary star . it is important to recognize the full range of the possible correction factors , but in order to have a better understanding of the statistical correction any given koi ( or the koi list as a whole ) may need , we must understand the mean correction for any one multiplicity scenario and convert these into a single mean correction factor for each star . to do this , we must take into account the probability the star may be a multiple star , the distribution of mass ratios if the star is a multiple , the probability that the planet orbits any one star if the stellar system has multiple stars , and whether or not the star has been vetted ( and how well it is has been vetted ) for stellar companions . in order to calculate an average correction factor for each multiplicity scenario , we have fitted the individual scenario correction factors as a function of mass ratio with a 3@xmath48-order polynomial ( see fig . [ fig - koi299 ] and [ fig - koi1085 ] ) . because the isochrones are not evenly sampled in mass , taking a mean straight from the isochrone points would skew the results ; the polynomial parameterization of the correction factor as a function of the mass ratio enables a more robust determination of the mean correction factor for each multiplicity scenario . if the companion to primary mass ratio distribution was uniform across all mass ratios , then a straight mean of the correction values determined from each polynomial curve would yield the average correction factor for each multiplicity scenario . however , the mass ratio distribution is likely not uniform , and we have adopted the form displayed in figure 16 of @xcite . that distribution is a nearly - flat frequency distribution across all mass ratios with a @xmath49 enhancement for nearly equal mass companion stars ( @xmath50 ) . this distribution is in contrast to the gaussian distribution shown in @xcite ; however , the more recent results of @xcite incorporate more stars , a broader breadth of stellar properties , and multiple companion detection techniques . the mass ratio distribution is convolved with the polynomial curves fitted for each multiplicity scenario , and a weighted mean for each multiplicity scenario was calculated for every koi . for example , in the case of koi-299 ( fig . [ fig - koi299 ] ) , the single star mean correction factor is 1.0 ( by definition ) . for the binary star cases , the average scenario correction factors are 1.14 ( planet orbits primary ) and 2.28 ( planet orbits secondary ) ; for the triple stars cases , the correction factors are 1.16 ( planet orbits primary ) , 2.75 ( planet orbits secondary ) , and 4.61 ( planet orbits tertiary ) . for koi-1085 ( fig . [ fig - koi1085 ] ) , the weighted mean correction factors are 1.18 , 1.56 , 1.24 , 1.61 , and 2.29 , respectively . to turn these individual scenario correction factors into an overall single mean correction factor @xmath3 per koi , the six scenario corrections are convolved with the probability that a koi will be a single star , a binary star , or a triple star . the multiplicity rate of the kepler stars is still unclear @xcite , and , indeed , there may be some contradictory evidence for the the exact value for the multiplicity rates of the koi host stars ( e.g. , * ? ? ? * ; * ? ? ? * ) , but the multiplicity rates appear to be near @xmath51 , similar to the general field population . in the absence of a more definitive estimate , we have chosen to utilize the multiplicity fractions from @xcite : a 54% single star fraction , a 34% binary star fraction , and a 12% triple star fraction @xcite . we have grouped all higher order multiples ( @xmath52 ) into the single category of `` triples '' , given the relatively rarity of the quadruple and higher order stellar systems . for the scenarios where there are multiple stars in a system , we have assumed that the planets are equally likely to orbit any one of the stars ( 50% for binaries , 33.3% for triples ) . the final mean correction factors @xmath3 per koi are displayed in figure [ fig - mean - factor ] ; the median value of the correction factor and the dispersion around that median is @xmath53 . this median correction factor implies that assuming a star in the koi list is single , in the absence of any ( observational ) companion vetting , yields a statistical bias on the derived planetary radii where the radii are underestimated , _ on average _ , by a factor @xmath54 , and the mass density of the planets are overestimated by a factor of @xmath55 . from figure [ fig - mean - factor ] , it is clear that the mean correction factor @xmath3 depends upon the stellar temperature of the host star . as most of the stars in the koi list are dwarfs , the lower temperature stars are typically lower mass stars and , thus , have a smaller range of possible stellar companions . thus , an average value for the correction factor 1.5 represents the sample as a whole , but a more accurate value for the correction factor can be derived for a given star , with a temperature between @xmath56k , using the fitted @xmath57-order polynomial : @xmath58 where @xmath59 . in the absence of any specific knowledge of the stellar properties ( other than the effective temperature ) and in the absence of any radial velocity or high resolution imaging to assess the specific companion properties of a given koi , ( see section [ sub - vetting ] ) , the above parameterization ( equation [ eq - factor - unvetted ] ) can be used to derive a mean radii correction factor @xmath3 for a given star . for g - dwarfs and hotter stars , the correction factor is near @xmath60 . as the stellar temperature ( mass ) of the primary decreases to the range of m - dwarfs , the correction factor can be as low as @xmath61 . ) to the quoted radii uncertainties ( @xmath62)from the cumulative koi list ( see equation [ eq - total - unc ] ) . for the red histogram , it is assumed that the kois are single as is the case in the published koi list ; for the blue histogram , it is assumed that each koi has been vetted with radial velocity ( rv ) and high resolution imaging ( see section [ sub - vetting ] ) . the vertical dashed lines represent the median values of the distributions : @xmath63 for the unvetted kois and @xmath64 for the vetted kois ( see section [ sub - vetting ] ) . ] the mean correction factor is useful for understanding how strongly the planetary radii may be underestimated , but an additional uncertainty term derived from the mean radius correction factor is potentially more useful as it can be added in quadrature to the formal planetary radii uncertainties . the formal uncertainties , presented in the koi list , are derived from the uncertainties in the transit fitting and the uncertainty in the knowledge of the stellar radius , and they are calculated assuming the kois are single stars . we can estimate an additional planet radius uncertainty term based upon the mean radii correction factor as @xmath66 where @xmath27 is the observed radius of the planet . adding in quadrature to the reported uncertainty , a more complete uncertainty on the planetary radius can be reported as @xmath67 where @xmath68 is the uncertainty of the planetary radius as presented in the koi list . the distribution of the ratio of the more complete koi radius uncertainties ( @xmath69 ) to the reported koi radius uncertainties ( @xmath68 ) is shown in figure [ fig - ratio - unc ] . including the possibility that a koi may be a multiple star increases the planetary radii uncertainties . while the distribution has a long tail dependent upon the specific system , the planetary radii uncertainties are underestimated as reported in the koi list , _ on average _ , by a factor of 1.7 . the above analysis has assumed that the kois have undergone no companion vetting , as is the assumption in the current koi list . in reality , the kepler project has funded a substantial ground - based follow - up observation program which includes radial velocity vetting and high resolution imaging . in this section , we explore the effectiveness of the observational vetting . the observational vetting reduces the fraction of undetected companions . if there is no vetting or all stars are assumed to be single , as is the case for the published koi list , then the fraction of undetected companions is 100% and the mean correction factors @xmath3 are as presented above . if every stellar companion is detected and accounted for in the planetary parameter derivations , then the fraction of undetected companions is 0% , and the mean correction factors are unity . reality is somewhere in between these two extremes . to explore the effectiveness of the observational vetting on reducing the radii corrections factors ( and the associated radii uncertainties ) , we have assumed that every koi has been vetted equally , and all companions within the reach of the observations have been detected and accounted . thus , the corrections factors depend only on the fraction of companions stars that remain out of the reach of vetting and undetected . in this simulation , we have assumed that all companions with orbital periods of 2 years or less and all companions with angular separations of @xmath70 or greater have been detected . this , of course , will not quite be true as random orbital phase effects , inclination effects , companion mass distribution , stellar rotation effects , etc . will diminish the efficiency of the observations to detect companions . we recognize the simplicity of these assumptions ; however , the purpose of this section is to assess the usefulness of observational vetting on reducing the uncertainties of the planetary radii estimates , not to explore fully the sensitivities and completeness of the vetting . typical follow - up observations include stellar spectroscopy , a few radial velocity measurements , and high resolution imaging . the radial velocity observations usually include @xmath71 measurements over the span of @xmath72 months and are typically sufficient to identify potential stellar companions with orbital periods of @xmath73 years or less . while determining full orbits and stellar masses for any stellar companions detected typically requires more intensive observing , we have estimated that 3 measurements spanning @xmath74 months is sufficient to enable the detection of an rv trend for orbital periods of @xmath75 years or less and mark the star as needing more detailed observations . the amplitude of the rv signature , and hence the ability to detect companions , does depend upon the masses of the primary and companion stars ; massive stars with low mass companions will display relatively low rv signatures . however , rv vetting for the kepler program has been done at a level of @xmath76 m / s , which is sufficient to detect ( at @xmath77 ) a late - type m - dwarf companion in a two - year orbit around a mid b - dwarf primary . indeed , the rv vetting is made even more effective by searching for companions via spectral signatures @xcite . the high resolution imaging via adaptive optics , `` lucky imaging '' , and/or speckle observations typically has resolutions of @xmath78 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? based upon monte carlo simulations in which we have averaged over random orbital inclinations and eccentricities , we have calculated the fraction of time within its orbit a companion will be detectable via high resolution imaging . with typical high resolution imaging of 0.05 , we have estimated that @xmath79 of the stellar companions will be detected at one full - width half - maximum ( fwhm=0.05 ) of the image resolution and beyond and @xmath80% at @xmath81 fwhm ( 0.1 ) of the image resolution and beyond . to determine what fraction of possible stellar companions would be detected in such a scenario , we have used the nearly log - normal orbital period distribution from @xcite . to convert the high resolution imaging limits into period - limits , we have estimated the distance to each koi by determining a distance modulus from the observed kepler magnitude and the absolute kepler magnitude associated with the fitted isochrone . the median distance to the kois was found to be @xmath82 pc , corresponding to @xmath83 au for 0.1 imaging . using the isochrone stellar mass , the semi - major axis detection limits were converted to orbital period limits ( assuming circular orbits ) . combining the 2-year radial velocity limit and the @xmath70 imaging limit , we were able to estimate the fraction of undetected companions for each individual koi ( see figure [ fig - logp ] ) . the distribution of the fraction of undetected companions ranges from @xmath84% and , on average , the ground - based observations leave @xmath85 of the possible companions undetected for the kois ( see figure [ fig - logp ] ) . the mean correction factors @xmath3 are only applicable to the undetected companions . for the stars that are vetted with radial velocity and/or high resolution imaging , the intrinsic stellar companion rate for the kois of 46% @xcite is reduced by the unvetted companion fraction for each koi . that is , we assume that companion stars detected in the vetting have been accounted for in the planetary radii determinations , and the unvetted companion fraction is the relevant companion rate for determining the correction factors . in the koi-299 example ( fig . [ fig - logp ] ) , the undetected companion rate used to calculate the mean radii correction factor is @xmath86 . this lower fraction of undetected companions in turn reduces the mean correction factors for the vetted stars which are displayed in figure [ fig - mean - factor ] ( blue points ) . instead of a mean correction factor of @xmath87 , the average correction factor is @xmath88 if the stars are vetted with radial velocity and high resolution imaging . the mean correction factor still changes as a function of the primary star effective temperature but the dependence is much more shallow with coefficients for equation [ eq - factor - unvetted ] of @xmath89 ( see figure [ fig - mean - factor ] ) . the above analysis has concentrated on the kepler mission and the associated koi list , but the same effects will apply to all transit surveys including k2 @xcite and tess @xcite . if the planetary host stars from k2 and tess are also assumed to be single with no observational vetting , the planetary radii will be underestimated by the same amount as the kepler kois ( fig . [ fig - mean - factor ] and eq . [ eq - factor - unvetted ] ) . many k2 targets and nearly all of tess targets will be stars that are typically @xmath90 magnitudes brighter than the stars observed by kepler , and therefore , k2 and tess targets will be @xmath91 times closer than the kepler targets . the effectiveness of the radial velocity vetting will remain mostly unaffected by the brighter and closer stars , but the effectiveness of the high resolution imaging will be significantly enhanced . instead of probing the stars to within @xmath92au , the imaging will be able to detect companion stars within @xmath93 au of the stars . as a result , the fraction of undetected companions will decrease significantly . even for the kepler stars that undergo vetting via radial velocity and high resolution imaging , @xmath94% of the companions remain undetected . but for the stars that are 10 times closer that fraction decreases to @xmath95% ( see figure [ fig - logp ] ) . this has the strong benefit of greatly reducing the mean correction factors for the stars that are observed by k2 and tess and are vetted for companions with radial velocity and high resolution imaging . the mean correction factor for vetted k2/tess - like stars is only @xmath96 . the correction factor has a much flatter dependence on the primary star effective temperature , because the majority of the possible stellar companions are detected by the vetting . the coefficients for equation [ eq - factor - unvetted ] become @xmath97 . the mean radii correction factors for vetted k2/tess planetary host stars correspond to a correction to the planetary radii uncertainties of only @xmath75% , in comparison to a correction of @xmath98% if the k2/tess stars remain unvetted . for k2 and tess , where the number of candidate planetary systems may outnumber the kois by an order of magnitude ( or more ) , single epoch high resolution imaging may prove to be the most important observational vetting performed . while the imaging will not reach the innermost stellar companions , radial velocity observations require multiple visits over a baseline comparable to the orbital periods an observer is trying to sample . in contrast , the high resolution imaging requires a single visit ( or perhaps one per filter on a single night ) and will sample the majority of the expected stellar companion period distribution . understanding the occurrence rates of the earth - sized planets is one of the primary goals of the kepler mission and one of the uses of the koi list @xcite . it has been shown that the transition from rocky to non - rocky planets occurs near a radius of @xmath99 and the transition is very sharp @xcite . however , the amplitude of the uncertainties resulting from undetected companions may be large enough to push planets across this boundary and affect our knowledge of the fraction of earth - sized planets . we have explored the possible effects of undetected companions on the derived occurrence rates . the planetary radii can not simply be multiplied by a mean correction factor @xmath0 , as that factor is only a measure of the statistical uncertainty of the planetary radius resulting from assuming the stars are single and only a fraction of the stars are truly multiples . instead a monte carlo simulation has been performed to assign randomly the effect of unseen companions on the kois . the simulation was performed 10,000 times for each koi . for each realization of the simulation , we have randomly assigned the star to be single , binary , or triple star via the 54% , the 34% and the 12% fractions @xcite . if the koi is assigned to be a single star , the mean correction factors for the planets in that system are unity : @xmath100 . if the koi star is a multiple star system , we have randomly assigned the stellar companion masses according to the masses available from the fitted isochrones and using the mass ratio distribution of @xcite . finally , the planets are randomly assigned to the primary or to the companion stars ( i.e. , 50% fractions for binary stars and 33.3% fractions for triple stars ) . once the details for the system are set for a particular realization , the final correction factor for the planets are determined from the polynomial fits for the individual multiplicity scenarios ( e.g. , fig . [ fig - koi299 ] and [ fig - koi1085 ] ) . for each set of the simulations , we compiled the fraction of planets within the following planet - radii bins : @xmath101 ; @xmath102 ; @xmath103 corresponding to earth - sized , super - earth / mini - neptune - sized , and neptune - to - jupiter - sized planets . the raw fractions directly from the koi - list , for these three categories of planets , are 33.3% , 46.0% , and 20.7% . note that these are the raw fractions and are not corrected for completeness or detectability as must be done for a true occurrence rate calculation ; these fractions are necessary for comparing how unseen companions affect the determination of fractions . finally , we repeated the simulations , but using the undetected multiple star fractions after vetting with radial velocity and high resolution imaging had been performed , thus , effectively increasing the fraction of stars with correction factors of unity . the distributions of the change in the fractions of planets in each planet category , compared to the raw koi fractions , are shown in figure [ fig - occurrence - rates ] . if the occurrence rates utilize the assumed - single koi list ( i.e. , unvetted ) , then the earth - sized planet fraction may be overestimated by as much as @xmath7% and the giant - planet fraction may be underestimated by as much as 30% . interestingly , the fraction of super - earth / mini - neptune planets does not change substantially ; this is a result of smaller planets moving into this bin , and larger planets moving out of the bin . in contrast , if all of the kois undergo vetting via radial velocity and high resolution imaging , the fractional changes to these bin fractions are much smaller : @xmath104% for the earth - sized planets and @xmath105% for the neptune / jupiter - sizes planets . we present an exploration of the effect of undetected companions on the measured radii of planets in the kepler sample . we find that if stars are assumed to be single ( as they are in the current kepler objects of interest list ) and no companion vetting with radial velocity and/or high resolution imaging is performed , the planetary radii are underestimated , _ on average _ , by a factor of @xmath106 , corresponding to an overestimation of the planet bulk density by a factor of @xmath107 . because lower mass stars will have a smaller range of stellar companion masses than higher mass stars , the planet radius mean correction factor has been quantified as a function of stellar effective temperature . if the kois are vetted with radial velocity observations and high resolution imaging , the planetary radius mean correction necessary to account for undetected companions is reduced significantly to a factor of @xmath108 . the benefit of radial velocity and imaging vetting is even more powerful for missions like k2 and tess , where the targets are , on average , ten times closer than the kepler objects of interest . with vetting , the planetary radii for k2 and tess targets will only be underestimated , on average , by 10% . given the large number of candidates expected to be produced by k2 and tess , single epoch high resolution imaging may be the most effective and efficient means of reducing the mean planetary radius correction factor . finally , we explored the effects of undetected companions on the occurrence rate calculations for earth - sized , super - earth / mini - neptune - sized , and neptune - sized and larger planets . we find that if the kepler objects of interest are all assumed to be single ( as they currently are in the koi list ) , then the fraction of earth - sized planets may be overestimated by as much as 15 - 20% and the fraction of large planets may be underestimated by as much as 30% the particular radial velocity observations or high resolution imaging vetting that any one koi may ( or may not ) have undergone differs from star to star . companion vetting simulations presented here show that a full understanding and characterization of the planetary companions is dependent upon also understanding the presence of stellar companions , but is also dependent upon understanding the limits of those observations . for a final occurrence rate determination of earth - sized planets and , more importantly , an uncertainty on that occurrence rate , the stellar companion detections ( or lack thereof ) must be taken into account . the authors would like to thank ji wang , tim morton , and gerard van belle for useful discussions during the writing of this paper . this research has made use of the nasa exoplanet archive , which is operated by the california institute of technology , under contract with the national aeronautics and space administration under the exoplanet exploration program . portions of this work were performed at the california institute of technology under contract with the national aeronautics and space administration .

we present a study on the effect of undetected stellar companions on the derived planetary radii for the kepler objects of interest ( kois ) . the current production of the koi list assumes that the each koi is a single star . not accounting for stellar multiplicity statistically biases the planets towards smaller radii . the bias towards smaller radii depends on the properties of the companion stars and whether the planets orbit the primary or the companion stars . defining a planetary radius correction factor @xmath0 , we find that if the kois are assumed to be single , then , _ on average _ , the planetary radii may be underestimated by a factor of @xmath1 . if typical radial velocity and high resolution imaging observations are performed and no companions are detected , this factor reduces to @xmath2 . the correction factor @xmath3 is dependent upon the primary star properties and ranges from @xmath4 for a and f stars to @xmath5 for k and m stars . for missions like k2 and tess where the stars may be closer than the stars in the kepler target sample , observational vetting ( primary imaging ) reduces the radius correction factor to @xmath6 . finally , we show that if the stellar multiplicity rates are not accounted for correctly , occurrence rate calculations for earth - sized planets may overestimate the frequency of small planets by as much as @xmath7% .

arxiv

the range of hi column densities typically seen in routine 21-cm emission line observations of the neutral gas disks in nearby galaxies is very similar to those that characterise the damped lyman-@xmath6 systems or dlas with @xmath7 . an attractive experiment would therefore be to map the hi gas of dla absorbing systems in 21-cm emission , and measure the dlas total gas mass , the extent of the gas disks and their dynamics . this would provide a direct observational link between dlas and local galaxies , but unfortunately such studies are impossible with present technology ( see e.g. , kanekar et al . the transition probability of the hyperfine splitting that causes the 21-cm line is extremely small , resulting in a weak line that can only be observed in emission in the very local ( @xmath8 ) universe , with present technology . on the other hand , the identification of dlas as absorbers in background qso spectra is , to first order , not distance dependent because the detection efficiency depends mostly on the brightness of the background source , not on the redshift of the absorber itself . in fact , the lowest redshift ( @xmath9 ) lyman-@xmath6 absorbers can not be observed from the ground because the earth s atmosphere is opaque to the uv wavelength range in which these are to be found . furthermore , due to the expansion of the universe the redshift number density of dlas decreases rapidly toward lower redshifts . consequently , there are not many dlas known whose 21-cm emission would be within the reach of present - day radio telescopes . so , we are left with a wealth of information on the cold gas properties in local galaxies , which has been collected over the last half century , and several hundreds dla absorption profiles at intermediate and high redshift , but little possibility to bridge these two sets of information . obviously , most observers resort to the optical wavelengths to study dlas but attempts to directly image their host galaxies have been notably unsuccessful ( see e.g. , warren et al . 2001 and mller et al . 2002 for reviews ) . a few positive identifications do exist , mostly the result of hst imaging . although the absolute number of dlas at low @xmath3 is small , the success rate for finding low-@xmath3 host galaxies is better for obvious reasons : the host galaxies are expected to be brighter and the separation on the sky between the bright qso and the dla galaxy is likely larger . early surveys for low-@xmath3 dla host galaxies consisted of broad band imaging and lacked spectroscopic follow - up ( e.g. , le brun et al.1997 ) . later studies aimed at measuring redshifts to determine the association of optically identified galaxies with dlas , either spectroscopically ( e.g. , rao et al . 2003 ) , or using photometric redshifts ( chen & lanzetta 2003 ) . all together , there are now @xmath10 dla galaxies known at @xmath11 . the galaxies span a wide range in galaxy properties , ranging from inconspicuous lsb dwarfs to giant spirals and even early type galaxies . obviously , it is not just the luminous , high surface brightness spiral galaxies that contribute to the hi cross section above the dla threshold . as explained above , we can not study these galaxies in the 21-cm line on a case - by - case basis , but we can do a study of a statistical nature to see if the properties of dlas and dla galaxies agree with our knowledge of hi in the local universe . blind 21-cm emission line surveys in the local universe with single dish radio telescopes such as parkes or arecibo have resulted in an accurate measurement of @xmath12 , which can be used as a reference point for higher redshift dla studies . @xmath13 is simply calculated by integrating over the hi mass function of galaxies , which is measured with surveys such as hipass ( zwaan et al . however , due to the large beam widths of the singe dish instruments , these surveys at best only barely resolve the detected galaxies and are therefore not very useful in constraining the column density distribution function of @xmath0 hi . hence , for this purpose we use the high resolution 21-cm maps of a large sample of local galaxies that have been observed with the westerbork synthesis radio telescope . this sample is known as whisp ( van der hulst et al . 2001 ) and consists of 355 galaxies spanning a large range in hi mass and optical luminosity . the total number of independent column density measurements above the dla limit is @xmath14 , which implies that the data volume of our present study is the equivalent of @xmath14 dlas at @xmath1 ! each galaxy in the sample is weighted according to the hi mass function of galaxies . we can now calculate the column density distribution function , @xmath15 where @xmath16 is the area function that describes for galaxies with hi mass the area in @xmath17 corresponding to a column density in the range to @xmath18 , and @xmath19 is the hi mass function . @xmath20 converts the number of systems per mpc to that per unit redshift . figure [ whispfn2.fig ] shows the resulting on the left , and the derived hi mass density per decade of on the right . for comparison with higher redshift observations , we also plot the results from two other studies . the proux ( 2005 ) measurements of below the dla limit are the result of their new uves survey for `` sub - dlas '' . the intermediate redshift points from rao et al . ( 2005 ) are based on mgii - selected dla systems . the surprising result from this figure is that there appears to be only very mild evolution in the intersection cross section of hi from redshift @xmath21 to the present . from this figure we can determine the redshift number density of @xmath22 gas and find that @xmath23 , in good agreement with earlier measurements at @xmath1 . compared to the most recent measurements of @xmath24 at intermediate and high @xmath3 , this implies that the comoving number density ( or the `` space density times cross section '' ) of dlas does not evolve after @xmath4 . in other words , the local galaxy population explains the incidence rate of low and intermediate @xmath3 dlas and there is no need for a population of hidden very low surface brightness ( lsb ) galaxies or isolated hi clouds ( dark galaxies ) . the right hand panel shows that at @xmath1 most of the hi atoms are in column densities around @xmath25 . this also seems to be the case at higher redshifts , although the distribution might flatten somewhat . the one point that clearly deviates is the highest point from rao et al . ( 2005 ) at @xmath26 . the figure very clearly demonstrates that this point dominates the measurement at intermediate redshifts . it is therefore important to understand whether the mgii - based results really indicate that high column densities ( @xmath27 ) are rare at high and low redshift , but much more ubiquitous at intermediate redshifts , or whether the mgii selection introduces currently unidentified biases . now that we have accurate cross section measurement of all galaxies in our sample , and know what the space density of our galaxies is , we can calculate the cross - section weighted probability distribution functions of various galaxy parameters . figure [ pdfs.fig ] shows two examples . the left panel shows the @xmath28-band absolute magnitude distribution of cross - section selected galaxies above four different hi column density cut - offs . 87% of the dla cross - section appears to be in galaxies that are fainter than @xmath29 , and 45% is in galaxies with @xmath30 . these numbers agree very well with the luminosity distribution of @xmath11 dla host galaxies . taking into account the non - detections of dla host galaxies and assuming that these are @xmath31 , we find that 80% of the @xmath11 dla galaxies are sub-@xmath29 . the median absolute magnitude of a @xmath1 dla galaxy is expected to be @xmath32 ( @xmath33 ) . the conclusion to draw from this is that we should not be surprised to find that identifying dla host galaxies is difficult . most of them ( some 87% ) are expected to be sub-@xmath29 and many are dwarfs . using similar techniques , we find that the expected median impact parameter of @xmath34 systems is 7.8 kpc , whereas the median impact parameter of identified @xmath11 dla galaxies is 8.3 kpc . assuming no evolution in the properties of galaxies gas disks , these numbers imply that 37% of the impact parameters are expected to be less than @xmath35 for systems at @xmath36 . this illustrates that very high spatial resolution imaging programs are required to successfully identify a typical dla galaxy at intermediate redshifts . the right panel in figure [ pdfs.fig ] shows the probability distribution of oxygen abundance in @xmath1 dlas . we constructed this diagram by assigning to every hi pixel in our 21-cm maps an oxygen abundance , based on the assumption that the galaxies in our sample follow the local metallicity luminosity ( @xmath37 ) relation ( e.g. garnett 2002 ) , and that each disk shows an abundance gradient of [ o / h ] of @xmath38 ( e.g. , ferguson et al . 1998 ) along the major axis . the solid lines correspond to these assumptions , the dotted lines are for varying metallicity gradients in disks of different absolute brightness . the main conclusion is that the metallicity distribution for hi column densities @xmath34 peaks around [ o / h]=@xmath39 to @xmath40 , much lower than the mean value of an @xmath29 galaxy of [ o / h]@xmath41 . the reason for this being that _ 1 ) _ much of the dla cross section is in sub-@xmath29 galaxies , which mostly have sub - solar metallicities , and _ 2 ) _ for the more luminous , larger galaxies , the highest interception probability is at larger impact parameters from the centre , where the metallicity is lower . interestingly , this number is very close to the @xmath1 extrapolations of metallicity measurements in dlas at higher @xmath3 from prochaska et al . ( 2003 ) and kulkarni et al . for the mean mass - weighted metallicity of hi gas with @xmath34 at @xmath1 we find the value of @xmath42 , also consistent with the @xmath1 extrapolation of -weighted metallicities in dlas , although we note this extrapolation has large uncertainties given the poor statistics from dlas at @xmath43 . these results are in good agreement with the hypothesis that dlas arise in the hi disks of galaxies . the local galaxy population can explain the incidence rate and metallicities of dlas , the luminosities of their host galaxies , and the impact parameters between centres of host galaxies and the background qsos . 1 chen , h. & lanzetta , k. m. 2003 , apj , 597 , 706 ferguson a. m. n. , gallagher j. s. , wyse r. f. g. , 1998 , aj , 116 , 673 garnett d. r. , 2002 , apj , 581 , 1019 kanekar n. , chengalur j. n. , subrahmanyan r. , petitjean p. , 2001 , a&a , 367 , 46 kulkarni v. p. , fall s. m. , lauroesch j. t. , york d. g. , welty d. e. , khare p. , truran j. w. , 2005 , apj , 618 , 68 le brun , v. , bergeron , j. , boisse , p. , & deharveng , j. m. 1997 , a&a , 321 , 733 mller , p. , warren , s. j. , fall , s. m. , fynbo , j. u. , & jakobsen , p. 2002 , apj , 574 , 51 mller p. , fynbo j. p. u. , fall s. m. , 2004 , a&a , 422 , l33 p ' eroux , c. , dessauges - zavadsky , m. , dodorico , s. , kim , t. , & mcmahon , r. g. 2005 , mnras , _ in press _ prochaska j. x. , gawiser e. , wolfe a. m. , castro s. , djorgovski s. g. , 2003 , apj , 595 , l9 rao , s. m. , nestor , d. b. , turnshek , d. a. , lane , w. m. , monier , e. m. , & bergeron , j. 2003 , apj , 595 , 94 rao , s. m. 2005 , astro - ph/0505479 van der hulst j. m. , van albada t. s. , sancisi r. , 2001 , asp conf . ser . 240 : gas and galaxy evolution , 240 , 451 warren s. j. , mller p. , fall s. m. , jakobsen p. , 2001 , mnras , 326 , 759 zwaan , m. a. , et al . 2005a , mnras , 359 , l30 zwaan , m. a. , van der hulst , j. m. , briggs , f. h. , verheijen , m. a. w. , ryan - weber , e. v. , 2005b , mnras , _ submitted _

we calculate in detail the expected properties of low redshift dlas under the assumption that they arise in the gaseous disks of galaxies like those in the @xmath0 population . a sample of 355 nearby galaxies is analysed , for which high quality hi 21-cm emission line maps are available as part of an extensive survey with the westerbork telescope ( whisp ) . we find that expected luminosities , impact parameters between quasars and dla host galaxies , and metal abundances are in good agreement with the observed properties of dlas and dla galaxies . the measured redshift number density of @xmath1 gas above the dla limit is @xmath2 , which compared to higher @xmath3 measurements implies that there is no evolution in the comoving density of dlas along a line of sight between @xmath4 and @xmath1 , and a decrease of only a factor of two from @xmath5 to the present time . we conclude that the local galaxy population can explain all properties of low redshift dlas . galaxies : ism ; ( galaxies : ) quasars : absorption lines ; galaxies : evolution

arxiv

it is thought that the vast majority of stars are formed in star clusters ( lada & lada 2003 ) . during the collapse and fragmentation of a giant molecular cloud into a star cluster , only a modest percentage ( @xmath2 % ) of the gas is turned into stars ( e.g. lada & lada 2003 ) . thus , during the initial phases of its lifetime , a star cluster will be made up of a combination of gas and stars . however , at the onset of stellar winds and after the first supernovae explosions , enough energy is injected into the gas within the embedded cluster to remove the gas on timescales shorter than a crossing time ( e.g. hills 1980 ; lada et al . 1984 ; goodwin 1997a ) . the resulting cluster , now devoid of gas , is far out of equilibrium , due to the rapid change in gravitational potential energy caused by the loss of a significant fraction of its mass . while this process is fairly well understood theoretically ( e.g. hills 1980 ; mathieu 1983 ; goodwin 1997a , b ; boily & kroupa 2003a , b ) , its effects have received little consideration in observational studies of young massive star clusters . in particular , many studies have recently attempted to constrain the initial stellar mass function ( imf ) in clusters by studying the internal dynamics of young clusters . by measuring the velocity dispersion and half - mass radius of a cluster , and assuming that the cluster is in virial equilibrium , an estimate of the dynamical mass can be made . by then comparing the ratio of dynamical mass to observed light of a cluster to simple stellar population models ( which require an input imf ) one can constrain the slope or lower / upper mass cuts of the imf required to reproduce the observations . studies which have done such analyses have found discrepant results , with some reporting non - standard imfs ( e.g. smith & gallagher 2001 , mengel et al . 2002 ) and others reporting standard kroupa ( 2002 ) or salpeter ( 1955 ) type imfs ( e.g. maraston et al . 2004 ; larsen & richtler 2004 ) . however , bastian et al . ( 2006 ) noted an age - dependence in how well clusters fit standard imfs , in the sense that all clusters @xmath1100 myr were well fit by kroupa or salpeter imfs , while the youngest clusters showed a significant scatter . they suggest that this is due to the youngest ( tens of myr ) clusters being out of equilibrium , hence undercutting the underlying assumption of virial equilibrium needed for such studies . in order to test this scenario , in the present work we shall look at the detailed luminosity profiles of three young massive clusters , namely m82-f , ngc 1569-a , & ngc 1705 - 1 , all of which reside in nearby starburst galaxies . m82-f and ngc 1705 - 1 have been reported to have non - standard stellar imfs ( smith & gallagher 2001 , mccrady et al . 2005 , sternberg 1998 ) . here we provide evidence that they are likely not in dynamical equilibrium due to rapid gas loss , thus calling into question claims of a varying stellar imf . ngc 1569-a appears to have a standard imf ( smith & gallagher 2001 ) based on dynamical measurements , however we show that this cluster is likely also out of equilibrium . throughout this work we adopt ages of m82-f , ngc 1569-a , and ngc 1705 to be @xmath3 myr ( gallagher & smith 1999 ) , @xmath4 myr ( anders et al . 2004 ) and 1020 myr ( heckman & leitherer 1997 ) respectively . studies of star clusters in the galaxy ( e.g. lada & lada 2003 ) as well as extragalactic clusters ( bastian et al . 2005a , fall et al . 2005 ) have shown the existence of a large population of young ( @xmath5 10 - 20 myr ) short - lived clusters . the relative numbers of young and old clusters can only be reconciled if many young clusters are destroyed in what has been dubbed `` infant - mortality '' . it has been suggested that rapid gas expulsion from young cluster which leaves the cluster severely out of equilibrium would cause such an effect ( bastian et al . we provide additional evidence for this hypothesis in the present work . the paper is structured in the following way . in [ data ] and [ models ] we present the observations ( i.e. luminosity profiles ) and models of early cluster evolution , respectively . in [ disc ] we compare the observed profiles with our @xmath0-body simulations and in [ conclusions ] we discuss the implications with respect to the dynamical state and the longevity of young clusters . for the present work , we concentrate on _ f555w _ ( v ) band observations of m82-f , ngc 1569-a , and ngc 1705 - 1 taken with the _ high - resolution channel _ ( hrc ) of the _ advanced camera for surveys _ ( acs ) on - board the _ hubble space telescope _ ( hst ) . the acs - hrc has a plate scale of 0.027 arcseconds per pixel . all observations were taken from the hst archive fully reduced by the standard automatic pipeline ( bias correction , flat - field , and dark subtracted ) and drizzled ( using the multidrizzle package - koekemoer et al . 2002 ) to correct for geometric distortions , remove cosmic rays , and mask bad pixels . the observations of m82-f are presented in more detail in mccrady et al . total exposures were 400s , 130s , and 140s for m82-f , ngc 1569-a , and ngc 1705 - 1 respectively . due to the high signal - to - noise of the data , we were able to produce surface brightness profiles for each of the three clusters on a per - pixel basis . the flux per pixel was background subtracted and transformed to surface brightness . the inherent benefit of using this technique , rather than circular apertures , is that it does not assume that the cluster is circularly symmetric . this is particularly important for m82-f , which is highly elliptical ( e.g. mccrady et al . 2005 ) . for m82-f we took a cut through the major axis of the cluster . the results are shown in the top panel of fig . [ fig : obs ] . we note that a cut along the minor - axis of this cluster as well as using different filters ( u , b , and i - also from _ hst - acs / hrc _ imaging ) would not change the conclusions presented in [ disc ] & [ conclusions ] . for ngc 1569-a and ngc 1705 - 1 we were able to assume circular symmetry ( after checking the validity of this assumption ) and hence we binned the data as a function of radius from the centre . the results for these clusters are shown in the centre and bottom panels of fig . [ fig : obs ] , where the circular data points represent mean binning in flux and the triangles represent median binning . the standard deviation of the binned ( mean ) data points is shown . we also note that our conclusions would remain unchanged ( [ disc ] & [ conclusions ] ) if we used the _ f814w _ ( i ) _ hst - acs / hrc _ observations . we did not correct the surface brightness profiles for the psf as the effects that we are interested in happen far from the centre of the clusters and therefore should not be influenced by the psf . in all panels of fig . [ fig : obs ] we show the psf as a solid green line ( taken from an _ acs - hrc _ observation of a star in a non - crowded region ) . the background of the area surrounding each cluster is shown by a horizontal dashed line . in order to quantify our results , we fit two analytical profiles to the observed lps . the first is a king ( 1962 ) function , which fits well the galactic globular clusters and is characterised by centrally concentrated profiles with distinct tidal cut - offs in their outer regions . the second analytical profile used is an elson , fall , & freeman ( eff - 1987 ) profile , which is also centrally concentrated with a non - truncated power - law envelope . the eff profile has been shown to fit young clusters in the lmc better ( eff ) as well as young massive clusters in galaxies outside the local group ( e.g. larsen 2004 ; schweizer 2004 ) . the best fitting king and eff profiles are shown as blue / dashed and red / solid lines respectively . the fits were carried out on all points within 0.5 of the centre of the clusters , i.e. the point at which , from visual inspection , the profile deviates from a smoothly decreasing function . as is evident in fig . [ fig : obs ] all cluster profiles are well fit by both king and eff profiles in their inner regions . _ however , none of the clusters appear tidally truncated , in fact all three clusters display an excess of light at large radii with respect to the best fitting power - law profile_. the points of deviation from the best fitting eff profiles are marked with arrows . this result will be further discussed in [ disc ] . due to the rather large distance of the galaxies as well as the non - uniform background around the clusters presented here , background subtraction is non - trivial . however , we have checked the effect of selecting different regions surrounding the clusters and note that our conclusions remain unchanged . we also note that in the lmc , where the background can be much more reliably determined , many clusters show excess light at large radii ( e.g. eff ; elson 1991 ; & mackey & gilmore 2003 ) . we model star clusters using @xmath0-body simulations . star clusters are constructed as plummer ( 1911 ) spheres using the prescription of aarseth et al . ( 1974 ) which require the plummer radius @xmath6 and total mass @xmath7 to be specified . clusters initially contain 30000 equal - mass stars , or the gravitational softening . this is because the dynamics we model are those of violent relaxation to a new potential and so 2-body encounters are unimportant . ] . simulations were conducted on a grape-5a special purpose computer at the university of cardiff using a basic @xmath0-body integrator ( the speed of the grape hardware means that sophisticated codes are not required for a simple problem such as this ) . the expulsion of residual gas from star clusters has been modelled by several authors ( see in particular lada et al . 1984 ; goodwin 1997a , b ; geyer & burkert 2001 ; kroupa et al . 2001 ; boily & kroupa 2003a , b ) . the typical method is to represent the gas as an external potential which is removed on a certain timescale . gas removal is expected to be effectively instantaneous , i.e. to occur in less than a crossing time ( e.g. goodwin 1997a ; melioli & de gouveia dal pino 2006 ) . as such we require no gas potential , and can model the cluster as a system that is initially out of virial equilibrium ( equivalent to starting the simulations at the end of the gas expulsion ) . the subsequent evolution is the violent relaxation ( lynden - bell 1967 ) of the cluster as it attempts to return to virial equilibrium . we define an _ effective _ star formation efficiency @xmath8 which parameterises how far out of virial equilibrium the cluster is after gas expulsion . a cluster which initially contains @xmath9 % stars and @xmath9 % gas ( i.e. a @xmath9 % star formation efficiency ) which is initially in virial equilibrium will have a stellar velocity dispersion that is a factor of @xmath10 - more generally @xmath11 - too large to be virialised after the gas is ( instantaneously ) lost . we define the efficiency as effective , as it assumes that the gas and stars are initially in virial equilibrium , which may not be true . we choose as initial conditions , @xmath12 pc ( corresponding to a half mass radius of @xmath13 pc ) and @xmath14 or @xmath15 ( i.e. the total initial stellar plus gas mass was @xmath16 or @xmath15 ) as representative of young massive star clusters . in order to compare the simulations with our observations we place the simulations at our assumed distance of m82 , namely 3.6 mpc ( assuming that it is at the same distance as m81 - freedman et al . previous simulations have shown that for @xmath17 clusters are totally destroyed by gas expulsion , but for higher @xmath8 significant ( stellar ) mass loss occurs , but a bound core remains ( goodwin 1997a , b ; boily & kroupa 2003a , b ) . for @xmath18 and @xmath19 respectively , @xmath20 and @xmath21 % of the initial stellar mass is lost within @xmath22 myr . we confirm those results . the escaping stars are not lost instantaneously , however . stars escape with a velocity of order of the initial velocity dispersion of the cluster , typically a few km s@xmath23 . therefore , escaping stars will still be physically associated with the cluster for @xmath24 @xmath25 myr _ after _ gas expulsion . these stars produce a ` tail ' in the surface brightness profile and produce the observed excess light at large radii . we assume a constant mass - to - light ratio for the simulation and convert the projected mass density into a luminosity and hence surface brightness profile . the normalisation of the surface brightness is arbitrary and scaled so that the central surface brightness is similar to that of the observed clusters . two of the simulations are shown in fig . [ fig : model ] . the filled circles are the surface brightness of the simulated cluster , with the specific parameters ( total initial mass , @xmath8 , and time since gas expulsion ) of the simulations shown . we follow the same fitting technique as with the observations , namely fitting king and eff profiles ( dashed blue and solid red lines respectively ) to the profile . as was seen in the observations , the simulations display excess light at large radii . the detailed correspondence between the observations and simulations presented here lead us to conclude that m82-f , ngc 1569-a and ngc 1705 - 1 display the signature of rapid gas removal and hence are _ not in dynamical equilibrium_. in future works we will provide a large sample of luminosity profiles of young massive extragalactic star clusters , as well as a detailed set of models which can be used to constrain the star formation efficiency of the clusters . here we simply note that models with a sfe between 40 - 50% best reproduce the observations . similar surface brightness profiles with an excess of light at large radii are seen in young lmc clusters : see eff and elson ( 1991 ) in which many clusters clearly show these unusual profiles , and also mackey & gilmore ( 2003 ) - in particular for r136 . these profiles are also well matched by our simulations . mclaughlin & van der marel ( 2005 ) have compiled a data base of structural parameters for young lmc / smc clusters and compare the m / l ratio from dynamical estimates to that predicted by simple stellar population models ( i.e. to check the dynamical state of the young clusters ) . however , the study was limited as the young clusters tend to be of relatively low - mass , making it difficult to measure accurate velocity dispersions . here we simply note that the five clusters in their sample younger than 100 myr all show significant deviations in the m / l ratio , but also note that this may simply be due to stochastic measurement errors . it appears likely that the excess light at large radii seen in many massive young star clusters is a signature of violent relaxation after gas expulsion . this suggests that these clusters have effective star formation efficiencies of around @xmath25 @xmath9 % , such that they show a significant effect , but do not destroy themselves rapidly . it should also be noted that the escaping stars are not just physically associated with a cluster in the surface brightness profiles . measurements of the velocity dispersion of the cluster will also include the escaping stars . this will result in an artificially high velocity dispersion that reflects the initial total stellar _ and _ gaseous mass . thus , mass estimates based on the assumption of stellar virial equilibrium may be wrong by a factor of up to three for 1020 myr after gas expulsion as is shown in fig . [ fig : virial ] for @xmath26 and @xmath27 % clusters ( i.e. at the ages of ngc 1569-a and ngc 1705 - 1 ) . clusters with @xmath28 @xmath27 % rapidly readjust to their new potential and the virial mass estimates become fairly accurate @xmath24 @xmath21 myr after gas expulsion ( i.e. for a cluster age of @xmath21 @xmath29 myr ) . however , for @xmath30 % , the virial mass is significantly greater than the actual mass for @xmath31 myr and clusters do not settle into virial equilibrium for @xmath32 myr indeed , between @xmath33 and @xmath25 myr after gas expulsion the virial mass estimate _ underestimates _ the total mass by up - to @xmath33 % as the cluster has over - expanded . a few recent studies have reported non - kroupa ( 2002 ) or non - salpeter ( 1955 ) type initial stellar mass functions ( imf ) in young star clusters ( e.g. smith & gallagher 2001 ; mengel et al . these results were based on comparing dynamical mass estimates ( found by measuring the velocity dispersion and half - mass radius of a cluster and assuming virial equilibrium ) and the light observed from the cluster with simple stellar population models ( which assume an input stellar imf ) . other studies based on the same technique ( e.g. larsen & ritchler 2004 ; maraston et al . 2004 ) have reported standard kroupa- or salpeter - type imfs . recently , bastian et al . ( 2006 ) noted a strong age dependence on how well young clusters fit ssp models with standard imfs , with all clusters older than @xmath34 myr being will fit by a kroupa imf . based on this age dependence , they suggested that the youngest star clusters ( @xmath35 myr ) may not be in virial equilibrium . the observations presented here strongly support this interpretation as m82-f and ngc 1705 - 1 both seem to have been strongly affected by rapid gas loss . while ngc 1569-a has been reported to have a salpeter - type imf ( smith & gallagher 2001 ) , the excess light at large radii suggests that this cluster has also undergone a period of violent relaxation and stars lost during this are still associated with the cluster even though its velocity dispersion correctly measures its mass .. it should be noted that the obvious signature of violent relaxation in the profile of m82-f suggests that it is at the lower end of its age estimate of @xmath36 myr ( gallagher & smith 1999 ) , as by @xmath25 @xmath9 myr the tail of stars becomes disassociated from the cluster . another possibility is that m82-f has been tidally shocked and has had a significant amount of energy input into the cluster , thus mimicking the effects of gas expulsion . whichever is the case , the tail of stars from m82-f - whatever its age - is a signature of violent relaxation and strongly suggests that it is out of virial equilibrium . if a young star cluster has a low enough effective star formation efficiency ( @xmath37 % ) it can become completely unbound and dissolve over the course of a few tens of myr . this mechanism has been invoked to explain the expanding ob associations in the galaxy ( hills 1980 ) . recent studies of large extragalactic cluster populations in m 51 ( bastian et al . 2005a ) and ngc 4038/39 ( fall et al . 2005 ) have shown a large excess of young ( @xmath510 myr ) clusters relative to what would be expected for a continuous cluster formation history . both of these studies suggest that the excess of extremely young clusters is due to a population of short - lived unbound clusters . the rapid dissolution of these clusters has been dubbed `` infant mortality '' . the observations and simulations presented here support such a scenario . if the star formation efficiency is less than 30% - no matter what the mass - the rapid removal of gas completely disrupts a cluster ( although see fellhauer & kroupa 2005 for a mechanism which can produce a bound cluster with @xmath38% ) . even if @xmath8 is large enough to leave a bound cluster , the cluster may be out of equilibrium enough for external effects to completely dissolve it , such as the passage of giant molecular clouds ( gieles et al . 2006 ) or in the case of large cluster complexes , other young star clusters . interestingly , gas expulsion often significantly lowers the _ stellar _ mass of the cluster even if a bound core remains ( see [ models ] ) . thus , relating the observed mass function of clusters to the birth mass function needs to account not only for infant mortality , but also for ` infant weight - loss ' in which a cluster could lose @xmath39 % of its initial _ stellar _ mass in @xmath40 myr . the current simulations do not include either a stellar imf , nor the evolution of stars . the inclusion of these effects do not significantly effect the results as the mass - loss due to stellar evolution is low compared to that due to gas expulsion ( see goodwin 1997a , b ) . in particular , we do not expect the preferential loss of low - mass stars as these clusters are too young for equipartition to have occured , thus stars of all masses are expected to have similar velocities . one caveat to this is the effect of primordial mass ( hence velocity ) segregation which may mean that the most massive stars are very unlikely to be lost as they have the lowest velocity dispersion . we will consider such points in more detail in a future paper . observations of the surface brightness profiles of the massive young clusters m82-f , ngc 1569-a , and ngc 1705 - 1 show a significant excess of light at large radii compared to king or eff profiles . simulations of the effects of gas expulsion on massive young clusters produce exactly the same excess due to stars escaping during a period of violent relaxation . gas expulsion can also cause virial mass estimates to be significantly wrong for several 10s of myr . these signatures are also seen in many other young star clusters ( e.g. elson 1991 ; mackey & gilmore 2003 ) and suggest that gas expulsion is an important phase in the evolution of young clusters that can not be ignored . in particular , this shows that claims of unusual imfs for young star clusters are probably in error as these clusters are _ not _ in virial equilibrium as is assumed . in future work we will further explore the dynamical state of young clusters in order to constrain the star - formation efficiency within the clusters . we would like to thank mark gieles and francois schweizer for interesting and useful discussions , as well as markus kissler - patig and linda smith for critical readings of earlier drafts of the manuscript . the anonymous referee is thanked for useful suggestions and comments . this paper is based on observations with the nasa / esa _ hubble space telescope _ which is operated by the association of universities for research in astronomy , inc . under nasa contract nas5 - 26555 . spg is supported by a uk astrophysical fluids facility ( ukaff ) fellowship . the grape-5a used for the simulations was purchased on pparc grant ppa / g / s/1998/00642 . 99 aarseth , s.j . , hnon , m. , & wielen , r. 1974 , a&a , 37 , 183 anders , p. , de grijs , r. , fritze - v . alvensleben , u. , & bissantz , n. 2004 , mnras , 347 , 17 bastian , n. , gieles , m. , lamers , h.j.g.l.m . , scheepmaker , r. a. , & de grijs , r. 2005a , a&a 431 , 905 bastian , n. , saglia , r.p . , goudfrooij , p. , kissler - patig , m. , maraston , c. , schweizer , f. , zoccali , m .. 2006 , a&a , 448 , 881 boily , c.m . & kroupa , p. 2003a , mnras , 338 , 665 boily , c.m . & kroupa , p. 2003b , mnras , 338 , 673 elson , r.a.w . 1991 , apjs , 76 , 185 elson , r.a.w . , fall , m.s . , & freeman , k.c . 1987 , apj 323 , 54 ( eff ) fall , s.m . , chandar , r. , whitmore , b.c . 2005 , apj , 631 , 133 fellhauer , m. & kroupa , p. 2005 , mnras , 359 , 223 freedman , w. , hughes , s.m . , madore , b.f . 1994 , apj , 427 , 628 gallagher , j.s . , iii & smith , l.j . 1999 , mnras , 304 , 540 geyer , m.p . & burkert , a. 2001 , mnras , 323 , 988 gieles , m. , portegies zwart , s. f. , sipior , m. , baumgardt , h. , lamers , h.j.g.l.m . , leenaarts , j. , 2006 , mnras in prep goodwin , s.p . 1997a , mnras , 284 , 785 goodwin , s.p . 1997b , mnras , 286 , 669 heckman , t.m . & leitherer , c. 1997 , aj , 114 , 69 hills , j.g . 1980 , apj , 235 , 986 king , i. 1962 , aj 67 , 471 koekemoer , a. m. , fruchter , a. s. , hook , r. n. , & hack , w. 2002 , in the 2002 hst calibration workshop , ed . s. arribas , a. koekemoer , & b. whitmore ( baltimore : stsci ) , 339 2001 , mnras , 321 , 699 kroupa , p. 2002 , science , 295 , 82 lada , c.j . , margulis , m. , & dearborn , d. 1984 , apj , 285 , 141 lada , c.j . & lada , e.a . 2003 , ara&a , 41 , 57 larsen , s.s . 2004 , a&a , 416 , 537 larsen , s.s . & richtler , t. 2004 , a&a , 427 , 495 lynden - bell , d. 1967 , mnras , 136 , 101 mackey , a.d . & gilmore , g.f . 2003 , mnras , 338 , 85 maraston , c. , bastian n. , saglia r. p. , kissler - patig m. , schweizer f. , & goudfrooij p. 2004 , a&a , 416 , 467 mathieu , r.d . 1983 , apj , 267 , l97 mccrady , n. , graham , j.r . , & vacca , w.d . 2005 , apj , 621 , 278 mclaughlin , d.e . & van der marel , r.p . 2005 , apjs , 161 , 304 melioli , c. & de gouveia dal pino , e. m. 2006 , a&a , 445 , l23 mengel , s. , lehnert , m.d . , thatte , n. , & genzel , r. 2002 , a&a , 383 , 137 meylan g. 1993 , in asp conf . 48 , the globular cluste - galaxy connection . smith , j.p . brodie , astron . , san fransisco , p. 588 plummer , h.c . 1911 , mnras , 71 , 460 salpeter , e.e . 1955 , apj , 121 , 161 schweizer , f. 2004 in asp conf . 322 , `` the formation and evolution of massive star clusters '' , eds . lamers , l.j . smith , a. nota , p. 111 smith , l.j . , & gallagher , j.s . 2001 , mnras , 326 , 1027 sternberg , a. 1998 , apj , 506 , 721

we present detailed luminosity profiles of the young massive clusters m82-f , ngc 1569-a , and ngc 1705 - 1 which show significant departures from equilibrium ( king and eff ) profiles . we compare these profiles with those from @xmath0-body simulations of clusters which have undergone the rapid removal of a significant fraction of their mass due to gas expulsion . we show that the observations and simulations agree very well with each other suggesting that these young clusters are undergoing violent relaxation and are also losing a significant fraction of their _ stellar _ mass . that these clusters are not in equilibrium can explain the discrepant mass - to - light ratios observed in many young clusters with respect to simple stellar population models without resorting to non - standard initial stellar mass functions as claimed for m82-f and ngc 1705 - 1 . we also discuss the effect of rapid gas removal on the complete disruption of a large fraction of young massive clusters ( `` infant mortality '' ) . finally we note that even bound clusters may lose @xmath1 50% of their initial _ stellar _ mass due to rapid gas loss ( `` infant weight - loss '' ) . [ firstpage ] galaxies : star clusters stellar dynamics methods : @xmath0-body simulations

arxiv

quantum fluctuation can suppress chaotic motion of wave packet in the phase space due to the quantum interference , as seen in kicked rotor @xcite . on the contrary , the quantum fluctuation can enhance the chaotic motion of wave packet due to tunneling effect as seen in kicked double - well model @xcite . the relation between chaotic behavior and tunneling phenomenon in classically chaotic systems is interesting and important subject in study of quantum physics @xcite . recently , the semiclassical description for the tunneling phenomena in a classically chaotic system have been developed by several groups @xcite . lin and ballentine studied interplay between the tunneling and classical chaos for a particle in a double - well potential with oscillatory driving force @xcite . they found that coherent tunneling takes place between small isolated classical stable regions of phase space bounded by kolmogorov - arnold - moser ( kam ) surfaces , which are much smaller than the volume of a single potential well . hnggi and the coworkers studied the chaos - suppressed tunneling in the driven double - well model in terms of the floquet formalism @xcite . they found a one - dimensional manifold in the parameter space , where the tunneling completely suppressed by the coherent driving . the time - scale for the tunneling between the wells diverges because of intersection of the ground state doublet of the quasienergies . while the mutual influence of quantum coherence and classical chaos has been under investigation since many years ago , the additional effects caused by coupling the chaotic system to the other degrees of freedom ( dof ) or an environment , namely _ decoherence and dissipation _ , have been studied only rarely @xcite as well as the tunneling phenomena in the chaotic system . since mid - eighties there are some studies on environment - induced quantum decoherence by coupling the quantum system to a reservoir @xcite . recently quantum dissipation due to the interaction with chaotic dof has been also studied@xcite . in this paper we numerically investigate the relation _ quantum fluctuation , tunneling and decoherence _ combined to the delocalization in wave packet dynamics in one - dimensional double - well system driven by polychromatic external field . before closing this section , we refer to a study on a delocalization phenomenon by a perturbation with some frequency components in the other model . _ have reported that the kicked rotator model with a frequency modulation amplitude of kick can be mapped to the tight - binding form ( loyld model ) on higher - dimensional lattice in solid - state physics under very specific condition @xcite . then the number @xmath0 of the incommensurate frequencies corresponds the dimensionality of the tight - binding system . the problem can be efficiently reduced to a localization problem in @xmath1 dimension . as seen in the case of kicked rotators , we can also expect that in the double - well system the coupling with oscillatory perturbation is roughly equivalent to an increase in effective degrees of freedom and a transition from a localized wave packet to delocalized one is enhanced by the polychromatic perturbation . the concrete confirmation of the naive expectation is one of aims of this numerical work . we present the model in the next section . in sect.3 , we show the details of the numerical results of the time - dependence of the transition probability between the wells based on the quantum dynamics . section 4 contains the summary and discussion . furthermore , in appendix a , we gave details of the classical phase space portraits in the polychromatically perturbed double - well system and some considerations to the effect of polychromatic perturbation . in appendix b , a simple explanation for the perturbed instanton tunneling picture is given . we consider a system described by the following hamiltonian , @xmath2 for the sake of simplicity , @xmath3 and @xmath4 are taken as @xmath5 , @xmath6 , @xmath7 in the present paper . then @xmath0 is the number of frequency components of the external field and @xmath8 is the perturbation strength respectively . \{@xmath9 } are order of unity and mutually incommensurate frequencies . we choose off - resonant frequencies which are far from both classical and quantum resonance in the corresponding unperturbed problem . the parameter @xmath10 adjusts the distance between the wells and we set @xmath11 to make some energy doublets below the potential barrier . note that lin _ dealt with a double - well system driven by forced oscillator ( duffing - like model ) , therefore , the asymmetry of the potential plays an important role in the chaotic behavior and tunneling transition between the symmetry - related kam tori @xcite . however , in our model the potential is remained symmetric during the time evolution process , and different mechanism from the forced oscillation makes the classical chaotic behavior @xcite . in the previous paper @xcite we presented numerical results concerning a classical and quantum description of the field - induced barrier tunneling under the monochromatic perturbation ( @xmath12 ) . in the unperturbed double - well system ( @xmath13 ) the instanton describes the coherent tunneling motion of the initially localized wave packet . it is also shown that the monochromatic perturbation can breaks the coherent motion as the perturbation strength increases near the resonant frequency in the previous paper . in the classical dynamics of our model , outstanding feature different from previous studies is parametric instability caused by the polychromatic perturbation . based on our criterion given below , we roughly estimate the type of the motion , i.e. the coherent and irregular motions , in a regime of the parameter space spanned by the amplitude and the number of frequency components of the oscillatory driving force . it is suggested that the occurrence of the irregular motion is related to dissipative property which is organized in the quantum physics @xcite . the classical phase space portraits and simple explanation of relation to the dissipative property are given in appendix a. we use gaussian wavepacket with zero momentum as the initial state , which is localized in the right well of the potential . @xmath14 where @xmath15 means a bottom of the right well . the gaussian wavepacket can be approximately generated by the linear combination of the ground state doublet as @xmath16 , where @xmath17 and @xmath18 denote the ground state doublet . the recurrence time for the wavepacket is @xmath19 in the unperturbed case ( @xmath13 ) , where @xmath20 is the energy difference between the tunneling doublet of the ground state . we set the spread of the initial packet @xmath21 and @xmath22 for simplicity throughout this paper . indeed , the ammonia molecule is well described by two doublets below the barrier heigth in unperturbed case . we numerically calculate the solution @xmath23 of time - dependent schrdinger equation by using second order unitary integration with time step @xmath24 . we define _ transition probability _ of finding the wave packet in the left well , @xmath25 in the cases that the perturbation strength is relatively small , @xmath26 can be interpreted as the tunneling probability that the initially localized wave packet goes through the central energy barrier and reaches the left well . we can expect that the transition probability @xmath26 is enhanced as the number @xmath0 of the frequency components increases up to some extent because of the increasing of the stochasticity in the total system . as a function of time @xmath27 for various @xmath0 s . ( a)@xmath28 . ( b)@xmath29 . the calculation time is same order to heisenberg time in the unperturbed case . ] figure 1 shows the time - dependence of @xmath30 for various combinations of @xmath8 and @xmath0 . apparently we can observe the coherent and irregular motions . the coherent motion of the wave packet can be well - described by the semiquantal picture in a sense that the wave packet does not delocalize to the fully delocalized state . the semiquantal picture decomposes the motion of the wave packet into _ evolution of the centroid motion _ and _ the spreading and squeezing _ of the packet @xcite . ( see subsect.3.5 . ) for example , in cases of relatively small perturbation strength ( @xmath28 ) , coherent motion remains still up to relatively large @xmath31 . it is important to emphasize that the tunneling contribution to the transition probability @xmath32 is not so significant for large @xmath8 and/or @xmath0 . then @xmath32 may be interpreted as a barrier crossing probability due to the activation - transition because the energy of wave packet increases over the barrier height in the parameter range . especially , in the relatively large perturbation regime we can interpret the delocalized states as chaos - induced delocalization in a sense that the classical chaos enhances the quantum barrier crossing rate quite significantly . the chaotic behavior in the classical dynamics is given in appendix a , based on the classical poincar section and so on @xcite . in the present section , we mainly focus on the transition of the quantum state from the localized wavepacket to delocalized state based on the data of numerical calculation . once the wave packet incoherently spreads into the space as the @xmath0 and/or @xmath8 increase , the wavepacket is delocalized and never return to gaussian shape again within the numerically accessible time . apparently , we regard the delocalized quantum state as a decoherent state in a sense that the behavior of the wave packet is similar to that of the stochastically perturbed case . ( see fig.5(a ) . ) in case of relatively small perturbation strength ( @xmath28 ) , the decoherence of quantum dynamics appears at around @xmath33 , and @xmath26 fluctuates irregularly in case of large @xmath34 . in short , the irreversible delocalization of a gaussian wave packet generates a transition from coherent oscillation to irregular fluctuation of @xmath26 . we have confirmed that the similar behavior is also observed for other sets of values of the frequencies and the different initial phases @xmath35 of the perturbation . here , we define a _ degree of coherence _ @xmath36 of the time - dependence of @xmath26 , based on the fluctuation of the transition probability in order to estimate quantitatively the difference between coherent and incoherent motions . @xmath37 where @xmath38 represents time average value for a period @xmath39 . note that we used @xmath36 in order to express the decoherence of the tunneling osccilation of the transition probability in the parametrically perturbed double - well system . on the other hand , the other quantities such as _ purity _ , _ linear entropy _ and _ fidelity _ , are sometimes used to characterize the decoherence of the quantum system @xcite . the transition of the dynamical behavior based on the fidelity for description of the decoherence in the double - well system will be given elsewhere @xcite . dependence of the degree of coherence @xmath36 of tunneling probability for various @xmath0 s . the @xmath40 is numerically estimated by @xmath26 . ] figure 2 shows the perturbation strength dependence of @xmath36 for various @xmath0 s . we roughly divide the type of motion of wave packet into three ones as follows . in the _ coherent motions _ , the value of @xmath36 s is almost same to the unperturbed case , i.e. @xmath41 , in which cases the instanton - like picture is valid @xcite . a simple explanation of the perturbed instanton is given in appendix b. in the _ irregular motions _ which are similar to the stochastically perturbed case , the value of @xmath36 s becomes much smaller , i.e. @xmath42 . as a matter of course , there are the intermediate cases between the coherent and the irregular motions , @xmath43 . note that the exact criterion of the intermediate motion is not important in the present paper because we can expect that the transitional cases approach to the irregular case in the long - time behavior . it should be stressed that the critical value @xmath44 exists , which divides the behavior of @xmath26 into regular and irregular motions . . circles(@xmath45 ) , crosses(@xmath46 ) , and triangles(@xmath47 ) denote coherent motions ( @xmath48 ) , irregular motions ( @xmath49 ) , and the transitional cases ( @xmath50 ) respectively . ] figure 3 shows a classification of the motion in the parameter space which is estimated by the value of the degree of coherence @xmath51 . it seems that two kinds of the motion , i.e. coherent and irregular motions , are divided by the thin layer corresponding to the `` transitional case '' . as @xmath0 increases , decoherence of the motion appears even for small @xmath8 . the numerical estimation suggests that there are the critical values @xmath52 of the perturbation strength depending on @xmath0 . when the perturbation strength @xmath8 exceeds the critical value @xmath53 for some @xmath0 , the tunneling oscillation loses the coherence . the approximated phase diagram roughly same as the diagram generated by maximal lyapunov exponent of the classical dynamics . ( see appendix a. ) in this subsection we give a consideration to the reduction of the tunneling period in the regular motion regime @xmath54 . for @xmath12 and @xmath55 in the coherent motion regime @xmath54 . ] figure 4 shows the @xmath56dependence of the the period @xmath57 of the tunneling oscillation estimated by the numerical data @xmath26 in the coherent motion regime @xmath54 . we can observe the monotonically decreasing of the tunneling period as the perturbation strength increases . in the monochromatically perturbed case , the reduction of the tunneling period can be interpreted by applying the floquet theorem to the quasi - energy states and the quasi - energy as the hamiltonian is time - periodic @xmath58 . when the wave packet does not effecively absorb the energy from the external perturbation the time - dependence of the quantum state can be described by the linear combination of a doublet of quasi - degenerate ground states with opposite parity because we prepare the initial state in @xmath16 and the evolution is adiabatic . in the two - state approximation that the avoided crossing of the eigenvalues dynamics does not appear during the time evolution , it is expected that the state evolves as , @xmath59 where @xmath60 and @xmath61 denote the quasi - energies and floquet states of the time - periodic hamiltonian @xcite . under the approximation we expect the following relation , @xmath62 where @xmath63 means quasi - energy splitting of the ground state doublet due to the tunneling between the wells . in the monochromatically perturbed case ( @xmath12 ) . ] let us confirm the relation in eq.(7 ) numerically . in fig . 5 we show the @xmath56dependence of @xmath64 . the behavior is analogus to the @xmath56dependence of the tunneling period of the oscillation @xmath26 in fig . 4 , in the weak perturbation regime . the similar correspondence between the tunneling period and the change of the quasi - energy splitting have been reported for the other double - well system by tomsovic _ it is well - known that the chaos around the separatrix contributes to the enhancement of the tunneling split between the doublet , i.e. chaos - assisted tunneling . the reduction of the tunneling period can be approximately explained by the chaos - assisted instanton picture in the coherent oscilation regime @xmath65 . the simple explanation for the perturbed instanton picture based on the width of the chaotic layer in the classical dynamics is given in appendix b. ( see also appendix a. ) generally speaking , as the number of frequencies @xmath0 increases the tunneling period is more reduced as seen in fig.4 although we do not have analytic representation in the polychromatically perturbed cases . we conjecture that as seen in appendix a the increasing of the width of the stochastic layer contributes the reduction of the tunneling period even in the polychromatically perturbed cases . as a function of time for some combinations of the parameters . ( a ) @xmath55 , @xmath66 . ( b ) @xmath55 , @xmath67 . ( c ) @xmath68 , @xmath69 . ( d ) @xmath68 , @xmath67 . ] as a function of time @xmath27 under stochastic perturbation with @xmath28 . ( b)plots of the uncertainty product @xmath70 versus time for various @xmath71s with the stochastic perturbation . the stochastic perturbation strength @xmath8 is normalized to be equivalent to one of the polychromatic perturbation . ] here let us investigate the spread of the wave packet in the phase space ( @xmath72 ) . hitherto we mainly investigated the dynamics in @xmath73space by @xmath26 . the phase space volume gives a part of the compensating information for the phase space dynamics of the wave packet . figure 6 presents the uncertainly product , i.e. phase space volume , as a function of time for various cases , which is defined by , @xmath74 where @xmath75 denotes quantum mechanical average . the uncertainty product can be used as a measure of quantum fluctuation @xcite . the initial value is @xmath76 for the gaussian wave packet . it is found that in the case @xmath55 the increase of the perturbation strength does not break the coherent oscillation and enhances the frequency of the time - dependence of the uncertainty product . for the relatively large @xmath8 in @xmath68 , @xmath77 increases until the wave packet is relaxed in the space , and it can not return to gaussian wave packet anymore . for the larger time scale , it fluctuates around the corresponding certain level . we can expect that the structure of the time dependence well corresponds to the behavior of the transition probability @xmath26 in fig . it will be instructive to compare the above irregular motion under the polychromatic perturbation with the stochastically perturbed one . we recall that the stochastic perturbation , composed of the infinite number of the frequency components ( @xmath78 ) with absolute continuous spectrum , can break the coherent dynamics . indeed , if the time dependence of the potential comes up with the stochastic fluctuation as @xmath79 , where @xmath80 and @xmath81 denote ensemble average and the temperature respectively , the stochastic perturbation partially models a heat bath coupled with the system @xcite . then the number of the frequency component corresponds to the number of degrees of freedom coupled with the double - well system . the @xmath26 for the stochastic perturbation is shown in fig . the stochastic perturbation can be achieved numerically by replacing @xmath82 in the eq.(2 ) by random number and we use uniform random number which is normalized so that the power of the perturbation is the same order to one of the polychromatic case . in the limit of large @xmath0 the motion under the polychromatic perturbation tends to approach the one driven by the stochastic perturbation provided with the same perturbation strengths @xmath8 . figure 7(b ) shows the uncertainty product @xmath77 for the stochastically perturbed cases . it is found that the time - dependence of the uncertainty product in the stochastically perturbed case behaves similarly to the polychromatically perturbed ones for the relatively small @xmath83 . on the other hand , for the relatively larger @xmath8(=0.4 ) the time - dependence shows quite different behavior . while @xmath77 grows linearly with time in the stochastically perturbed case , in the polychromatically perturbed cases the growth of @xmath77 saturates at a certain level . the linear growth of @xmath77 shows that the external stochasticity breaks the quantum interference in the internal dynamics . the growth of @xmath77 is strongly related to the growth of the energy of the packet @xcite . in the polychromatically perturbed cases the energy growth saturates at certain level due to quantum interference . on the other hand , in the case the energy grows unboundedly , the activation transition becomes much more dominant than the tunneling transition when the wave packet transfers the opposite well . the details concerning relation between the stochastic resonance @xcite and suppression of the energy growth will be given elsewhere @xcite . note that the polychromatic perturbation can be identified with a white noise ( or a colored noise if the frequencies are distributed over a finite band width ) only in the limit of @xmath78 , while the stochastic perturbation can model a heat bath that breaks the quantum interference of the system . a similar phenomenon by the different property of the perturbation has been observed as `` dynamical localization '' and the `` noise - assisted mixing '' of the quantum state in the momentum space in the quantum kicked rotor model @xcite . and the same parameters at @xmath84 , @xmath85 . the selection of initial condition of the fluctuation follows the minimum uncertainty . contour plots ( ( e ) , ( f ) ) of the hushimi functions for the corresponding the quantum state at @xmath86 . contour lines in the panel ( e ) at the values 0.01 , 0.02 , 0.05 , 0.08 and 0.1 , in the panel ( f ) at 0.01 , 0.02 , 0.03 , 0.04 and 0.05 . @xmath87 and @xmath28 for ( a ) , ( c ) and ( e ) . @xmath87 and @xmath29 for ( b ) , ( d ) and ( f ) . ] finally , in order to see effect of quantum fluctuation , we compare the quantum states with the classical and semiquantal motions in the phase space for some cases . the semiquantal equation of motion is given by generalized hamilton - like equations as , @xmath88 where the canonical conjugate pair @xmath89 is defined by the quantum fluctuation @xmath90 and @xmath91 as , @xmath92 . for more details consult @xcite . it is directly observed that the quantum tunneling phenomenon enhances chaotic motion in comparing to the classical and semiquantal trajectories . in fig.8(a ) and ( b ) poincar surface of section of the classical trajectories in the phase plane at @xmath87 are shown . the stroboscopic plots are taken at @xmath93 , due to non time - periodic structure of the hamiltonian . in the relatively small perturbation strength @xmath28 , the trajectories stay the single well , and are stable even for the long - time evolution . figure 7(c ) and ( d ) show the poincar section of the semiquantal trajectories for a polychromatically perturbed double - well system with @xmath87 ( the stroboscopic plots are taken at @xmath94 again ) . the semiquantal trajectories for the squeezed quantum coherent state can be obtained by an effective action which includes partial quantum fluctuation to all order in @xmath95 @xcite . it can be seen that in comparing with ones of classical dynamics the trajectories in the semiquantal dynamics spreads into the opposite well even for the small @xmath8 . this corresponds to the quantum tunneling phenomenon through the semiquantal dynamics . apparently , the partial quantum fluctuation in the semiquantal approximation enhances the the chaotic behavior . notice that the semiquantal picture breaks down for the irregular quantum states because the centroid motion becomes irrelevant . in fig.8(e ) and ( f ) the corresponding coherent state representation for the quantum states are shown . it is directly seen that the wave packet spreads over the two - wells and the shape is not symmetric . once the wave packet incoherently spreads over the space , it can not return to the initial state anymore . we have confirmed that in a case without separatrix ( single - well ) , namely the case that @xmath10 in eq . ( 2 ) is replaced by @xmath96 , in the classical phase space the coherent oscillations have remained against the relatively large @xmath0 and/or @xmath8 . it follows that the full quantum interference suppresses the chaotic behavior as seen in the semiquantual trajectories . we numerically investigated influence of a polychromatic perturbation on wave packet dynamics in one - dimensional double - well potential . the calculated physical quantities are the transition rate @xmath26 , the time - fluctuation @xmath40 , uncertainty product @xmath77 and phase space portrait . the results we obtained in the present investigation are summarized as follows . \(1 ) we classified the motions in the parameter space spanned by the amplitude and the number of frequency components of the oscillatory driving force , i.e. _ coherent motions _ and _ irregular motions_. the critical value @xmath52 which divides the behavior of @xmath26 into regular and irregular motions depends on the number of the frequency component @xmath0 . \(2 ) within the regular motion range , the period of the tunneling oscillation is reduced with increase of the number of colors and/or strength of the perturbation . it could be explained by the increase of the instanton tunneling rate due to appearance of the stochastic layer near separatrix @xcite . in this parameter regime the perturbed instanton picture is one of expression for chaos - assisted tunneling @xcite and chaos - assisted ionization picture reported for some quantum chaos systems @xcite . \(3 ) in the irregular motion in the polychromatically perturbed cases , the growth of @xmath77 initially increases and saturates at certain level due to quantum interference . on the other hand , in the stochastically perturbed case the uncertainty product grows unboundedly because the external stochasticity breaks the quantum interference in the internal dynamics . the growth of @xmath77 is strongly related to the growth of the energy of the wave packet . \(4 ) it is expected that the quantum fluctuation are always large for the classically chaotic trajectories compared to the regular ones . this implies that the quantum corrections to the evolution of the phase space fluctuation become more dominant for classically chaotic trajectories . \(5 ) in the semiquantal approximation the partial quantum fluctuation enhances the chaotic behavior , and simultaneously the chaos enhances the tunneling and decoherence of the wave packet . the quantum fluctuation observed in the semiquantal picture is suppressed by interference effect in the fully quantum motion . the semiquantal picture can not apply to the chaos - induced delocalized states . furthermore , in the appendices , we gave classical phase space portraits in the polychromatically perturbed double - well system and a simple explanation for the perturbed instanton tunneling picture for the reduction of the tunneling period in the coherent motion regime . although we have dealt with quantum dynamics of wave packet with paying attention to existence of the energetic barrier , we can expect that the similar phenomena would appear by dynamical barrier in the system . the details will be given elsewhere @xcite . we show classical stroboscopic phase space portrait in this appendix with paying an attention to the effect of polychromatic perturbation on the chaotic behavior . in the classical dynamics , such a system shows chaotic behavior by the oscillatory force @xmath97 @xcite . the newton s equation of the motion is @xmath98 note that in the monochromatically perturbed case ( @xmath12 , @xmath99 , the equation is known as nonlinear mathieu equation which can be derived from surface acoustic wave in piezoelectric solid @xcite and nanomechanical amplifier in micronscale devices @xcite . in fig . a.1 , we show the change of the classical stroboscopic phase space portrait changing the perturbation parameters . increasing the perturbation strength @xmath8 destroys the separatrix and forms a chaotic layer in the vicinity of the separatrix . needless to say , the phenomena have been observed even in the monochromatically perturbed cases @xcite . in the polychromatically perturbed cases ( @xmath100 ) the smaller the strength @xmath8 can generate chaotic behavior of the classical trajectories the larger @xmath0 is @xcite . it should be emphasized that in the polychromatically perturbed cases the width of the chaotic layer grows faster than the monochromatically perturbed case as the perturbation strength increases . as a result , the increase of the color contributes the increase of the width of the stochastic layer in the polychromatically perturbed cases . ) space are plotted at @xmath101 . the @xmath56dependence for @xmath102 is shown in ( a)@xmath66 , ( b)@xmath69 , ( c)@xmath28 and ( d)@xmath29 . the @xmath103dependence for @xmath104 is shown in ( e)@xmath12 , ( f)@xmath55 , ( g)@xmath102 and ( h)@xmath105 . ] here , we use the increasing rate of infinitesimal displacement along the classical trajectory for the extent of chaotic behavior as a finite - time lyapunov exponent @xmath106 . we prepare various initial points in the phase space , and conveniently adapt a trajectory with maximal increasing rate among the ensemble within the finite - time interval as the finite - time lyapunov exponent . note that an exact lyapunov exponent should be defined for the long - time limit . however , the roughly estimated lyapunov exponent is also useful to observe the classical - quantum correspondence . figure a.2 shows the @xmath56dependence of classical lyapunov exponents for various cases estimated by the numerical data of the classical trajectories . estimated by some classical trajectories within the finite - time @xmath107 $ ] , where @xmath108 is tunneling time given insubsect.3.1 . ] we can roughly observe a transition from motion of kam system ( @xmath109 ) to chaotic motion ( @xmath110 ) as the perturbation strength increases . the increasing of the number of color @xmath0 reduces the value of the critical perturbation strength @xmath111 of a transition from a motion of kam system to fully chaotic motion . roughly speaking , a transition of the classical dynamics corresponds to the transition from coherent motion to irregular one in the quantum dynamics . we expect that the transition observed in sect.3 will corresponds to quantum signatures of the kam transition from the regular to chaotic dynamics . in this subsection , we conceptually consider a role of the polychromatic perturbation different from monochromatic one . note that the change of the number @xmath0 of colors also changes the qualitative nature of the underlying dynamics because @xmath0 corresponds to the effective number of dof under some conditions @xcite . in the our model , when the number of dof of the total system is more than four , i.e. @xmath112 , the classical trajectories can diffuse along the stochastic layers of many resonances that cover the whole phase space if the trajectory starts in the vicinity of a nonlinear resonance . the number of resonances increases rapidly with dof , changing the characteristic of population transfer from bounded to diffusive . such a global instability is known as arnold diffusion in nonlinear hamiltonian system with many dof @xcite . the effect of arnold diffusion in quantum system is not trivial and the study just has started recently @xcite . the more detail is out of scope of this paper . moreover , we can regard the time - dependent model of eq.(1 ) as nonautonomous approximation for an autonomous model , consisting of the double - well system coupled finite number @xmath0 of harmonic oscillators with the incommensurate frequencies @xmath113 . it is worth noting that the linear oscillators can be identified with a highly excited quantum harmonic oscillators , which all phonon modes are excited around fock states with large quantum numbers . then the above model can be regarded as a double - well system coupled with @xmath0 phonon modes . without the interaction with the phonon modes the gaussian wavepacket remains the coherent motion . then the number of dof of total system is @xmath114 and the number of the frequency components @xmath0 corresponds to the that of the highly excited quantum harmonic oscillators . the detail of the correspondence is given in ref.@xcite . in quantum chaotic system with finite and many dof we expect occurrence of a dissipative behavior . for example , we consider simulated light absorption by coupling a system in the ground state with radiation field . then stationary one - way energy transport from photon source to the system can be interpreted as occurrence of quantum irreversibility in the total system . such a irreversibility is called chaos - induced dissipation in quantum system with more than two dof @xcite . in this sense , we can expect occurrence of the one - way transport phenomenon in the delocalized state in the irregular motion phase if it couples with the other dof in the ground state as seen in ref.@xcite . in this appendix , we consider the reduction of the tunneling period as the perturbation strength increases in the coherent oscilation regime @xmath65 , based on a perturbed instanton tunneling . in a double - well system with dipole - type interaction , @xmath115 , the energitical barrier tunneling between the symmetric double - well can be explained by a three - state model or chaos - assisted tunneling ( cat ) @xcite . the three states that take part in the tunneling are a doublet of quasi - degenerate states with opposite parity , localized in the each well , and a third state localized in the chaotic layer around the separatrix . however , note that less attention has been paid to tunneling in kam system while chaotic dynamics has been modeled by multi - level hamiltonian and random matrix model to describe the chaos - assisted tunneling . we give an expresion of the tunneling amplitude in `` chaos - assisted instanton tunneling '' firstly proposed by kuvshinov _ et al _ for a hamiltonian system with time - periodic perturbation @xcite . let us consider only monochromatically perturbed case ( @xmath12 in eq.(2 ) , @xmath116 ) because the separatrix destruction mechanism by the time - periodic perturbation has an universality although our system is different from their one . . indeed , trajectories in the neighborhood of the separatrix of the system are well reproduced by the whisker map of the system . whisker map is a map of the energy change @xmath117 and phase change @xmath118 of a trajectory in the neighborhood of the separatrix for each of its motion during one period of the perturbation , i.e. action - angle variable . moreover , if we linearize the whisker map which describes the behaviors of the trajectories in the neighborhood of the fixed point , we can obtain the following standard map , @xmath119 , where @xmath120 is a nonlinear parameter of local instability that the exact function form which is not essential for our purpose . @xmath121 increases with @xmath8 , and @xmath122 means that the dynamics of the system is locally unstable . a comparison has done between the whisker map and the strobe plots in the time continuous version by yamaguchi @xcite . the form of the mapping is convenient for the estimate of the width of the stochastic layer . the perturbation destroys separatrix of the unperturbed system and the stochastic layer appears . in the regular motion denoted by the circles in fig.3 , classical chaos can increase the rate of instanton tunneling due to appearance of the stochastic layer near separatrix of the unperturbed system . as a result the frequency of time - dependence @xmath26 increases as the classical chaos becomes remarkable in the parameter regime . note that the perturbed instanton tunneling picture disappears in the strongly perturbed regime due to the delocalization of wavepacket . here we give only relation between the width of the stochastic layer and the tunneling amplitude in terms of path integral in imaginary time @xmath123 , found by kuvshivov _ et al_. tunneling amplitude between the two wells in the perturbed system can be given by integration over energy of tunneling amplitude @xmath124 in unperturbed system as , @xmath125 \exp\ { -s[q(\tau , e ) ] \ } , \end{aligned}\ ] ] where @xmath126 $ ] denotes the euclidian action . @xmath127 denotes the width of stochastic layer , where @xmath128 and @xmath129 are the energy of the unperturbed system on the separatrix and on the bound of stochastic layer , respectively . @xmath130 is classical solution of euclidian equation of motion . the contribution of the chaotic instanton solution are taken into account by means of integration over @xmath131 which is energy of the instanton . the perturbed instanton solutions correspond to the motions in vicinity of the separatrix inside the layer . the only manifestation of the perturbation in this approximation is the appearance of a number of additional solutions of the euclidian equation of motion with energy close to the energy of the unperturbed one - instanton solution inside the stochastic layer . accordingly , we can expect that the appearance of the stochastic layer enhances the tunneling rate as reported in the other systems @xcite . however , we have to have in mind that the result is obtained in the first order on coupling constant @xmath8 of the time - periodic perturbation and does not take into account the structure of stochastic layer . the approximation is valid if the layer is narrow by neglecting the higher order resonances in the phase space . for the more details of the perturbed instanton see ref.@xcite . the increasing of the tunneling amplitude is directly related to energy splitting @xmath63 between the quasi - degenerate ground floquet states . as seen in fig.a.1 , the increasing of number of color @xmath0 can enhance the width of the stochastic layer with the perturbation strength being kept at a constant value . the theoretical explanation for the reduction of the tunneling period with the number of color is open for further study . we expect that in the double - well system under the polychromatic perturbation this numerical study will be useful for the analytical derivation of `` reduction of tunneling period '' and `` critical strength of a transition from localized to delocalized behavior of wavepacket '' by extension of the monochromatically perturbed case . the chaos assisted instanton theory might be applicable if we will exactly estimate the width of the stochastic layer in the system under the polychromatic perturbation . it should be noted that the uncertainty product is not always good measure of quantum fluctuation because it does not correspond to the real area ( or volume ) of phase space . however , we would like to pay attention to the initial growth and the mean value during the time - evolution process instead of the detail of the definition of the exact quantum fluctuation in the dynamics . practically , the @xmath132 remains small thorough the regular instanton - like motion , on the other hand , it shows sharp increase and the large value remains for the strongly chaotic cases . see papers , s. chaudhuri , g. gangopadhyay and d.s.ray , phys . a * 216 * , 53(1996 ) ; p. k. chattaraj , b. maiti , and s. sengupta , int . j. quant . chem . * 100 * , 254(2004 ) . as a result the classical chaos enhances quantum fluctuation in the restricted sense .

we numerically study influence of a polychromatic perturbation on wave packet dynamics in one - dimensional double - well potential . it is found that time - dependence of the transition probability between the wells shows two kinds of the motion typically , coherent oscillation and irregular fluctuation combined to the delocalization of the wave packet , depending on the perturbation parameters . the coherent motion changes the irregular one as the strength and/or the number of frequency components of the perturbation increases . we discuss a relation between our model and decoherence in comparing with the result under stochastic perturbation . furthermore we compare the quantum fluctuation , tunneling in the quantum dynamics with ones in the semiquantal dynamics .

arxiv

understanding dissipative quantum dynamics of a system embedded in a complex environment is an important topic across various sub - disciplines of physics and chemistry . significant progress in the understanding of condensed phase dynamics have been achieved within the context of a few prototypical models@xcite such as caldeira - leggett model and spin - boson model . in most cases the environment is modeled as a bosonic bath , a set of non - interacting harmonic oscillators whose influences on the system is concisely encoded in a spectral density . the prevalent adoption of bosonic bath models is based on the arguments that knowing the linear response of an environment near equilibrium should be sufficient to predict the dissipative quantum dynamics of the system . despite many important advancements in the quantum dissipation theory have been made with the standard bosonic bath models in the past decades , more and more physical and chemical studies have suggested the essential roles that other bath models assume . we briefly summarize three scenarios below . 1 . a standard bosonic bath model fails to predict the correct electron transfer rate in donor - acceptor complex strongly coupled to some low - frequency intramolecular modes . some past attempts to model such an anharmonic , condensed phase environment include ( a ) using a bath of non - interacting morse@xcite or quartic oscillatorsand ( b ) mapping anharmonic environment onto effective harmonic modes@xcite with a temperature - dependent spectral density . another prominent example is the fermonic bath model . electronic transports through nanostructures , such as quantum dots or molecular junctions , involves particle exchange occurs across the system - bath boundary . recent developments of several many - body physics and chemistry methods , such as the dynamical mean - field theory@xcite and the density matrix embedding theory@xcite , reformulate the original problem in such a way that a crucial part of the methods is to solve an open quantum impurity model embedded in a fermionic environment . the spin ( two - level system ) bath models have also received increased attention over the years due to ongoing interests in developing various solid - state quantum technologies@xcite under the ultralow temperature when the phonon or vibrational modes are frozen and coupling to other physical spins ( such as nuclear spins carried by the lattice atoms ) , impurities or defects in the host material emerge as the dominant channels of decoherence . both bosonic and fermionic environments are gaussian baths , which can be exactly treated by the linear response@xcite in the path integral formalism . for the non - gaussian baths , attaining numerically exact open quantum dynamics would require either access to higher order response function of the bath in terms of its multi - time correlation functions or explicit dynamical treatments of the bath degrees of freedom ( dofs ) . in this work , we extend a stochastic formulation@xcite of quantum dissipation by incorporating all three fundamental bath models : non - interacting bosons , fermions and spins . the stochastic liouville equation ( sle ) , eq . ( [ eq : sleq ] ) , prescribes a simple yet general form of quantum dissipative dynamics when the bath effects are modelled as colored noises @xmath0 and @xmath1 . different bath models and bath properties are distinguished in the present framework by assigning distinct noise variables and associated statistics . for instance , in dealing with bosonic and fermionic baths , the noises are complex - valued and grassmann - valued gaussian processes , respectively , and characterized by the two - time correlation functions such as eq . ( [ eq : xi_corr ] ) . the grassmann - valued noises are adopted whenever the environment is composed of fermionic modes as these algebraic entities would bring out the gaussian characteristics of fermionic modes . for anharmonic environments , such as a spin bath , the required noises are generally non - gaussian . two - time statistics can not fully distinguish these processes and higher order statistics furnished with bath multi - time correlation functions are needed . despite the conceptual simplicity of the sle , achieving stable convergences in stochastic simulations has proven to be challenging in the long - time limit . even for the most well - studied bosonic bath models , it is still an active research topic to develop efficient stochastic simulation schems@xcite today . our group has successfully applied stochastic path integral simulations to calculate ( imaginary - time ) thermal distributions@xcite , absorption / emission spectra@xcite and energy transfer@xcite ; however , a direct stochastic simulation of real - time dynamics remains formidable . in this study , we consider generic quantum environments that either exhibit non - gaussian characteristics or invovles ferimonic degrees of freedoms ( and associated grassmann noise in the stochastic formalism ) . both scenarios present new challenges to developing efficient stochastic simulations . hence , in subsequent discussions , all numerical methods developed are strictly deterministic . we note that it is common to derive exact master equation@xcite , hierarchical equation of motions@xcite , and hybrid stochastic - deterministic numerical methods@xcite from a stochastic formulation of open quantum theory . in sec . [ sec : spectral ] , we further illustrate the usefulness of our stochastic frmulation by presenting a numerical scheme that would be difficult to obtain within a strictly deterministic framework of open quantum theory . furthermore , the stochastic formalism gives straightforward prescriptions to compute dynamical quantities such as @xmath2 , which represents system - bath joint observables , as done in a recently proposed theory@xcite . staring from the sle , we derive numerical schemes to deterministically simulate quantum dynamics for all three fundamental bath models . the key step is to formally average out the noise variables in the sle . a common approach is to introduce auxiliary density matrices ( adms ) , in close parallel to the hierarchical equation of motion ( heom ) formalism@xcite , that fold noise - induced fluctuations on reduced density matrix in these auxiliary constructs with their own equations of motions . to facilitate the formal derivation with the noise averaging , we consider two distinct ways to expand the adms with respect to a complete set of orthonormal functions in the time domain . in the first case , the basis set corresponds to the eigenfunctions of the bath s two - time correlation functions . this approach provides an efficient description of open quantum dynamics for bosonic bath models . unfortunately , it is not convenient to extend this approach to study non - gaussian bath models . we then investigate another approach inspired by a recent work on the extended heom ( eheom)@xcite . in this case , we expand the bath s multi - time correlation functions in an arbitrary set of orthonormal functions . this approach generalizes eheom to the study of non - gaussian and fermionic bath models with arbitrary spectral densities and temperature regimes . despite having a slightly more complex form , our fermionic heom can be easily related to the existing formalism@xcite . in this work , we refer to the family of numerical schemes discussed in this work collectively as the generalized hierarchical equations ( ghe ) . among the three fundamental bath models , spin baths deserve more attentions . a spin bath can feature very different physical properties@xcite from the standard heat bath composed of non - interacting bosons ; especially , when the bath is composed of localized nuclear / electron spins@xcite , defects and impurities . this kind of spin environment is often of a finite - size and has an extremely narrow bandwidth of frequencies . to more efficiently handle this situation , we consider a dual - fermion mapping that transforms each spin into a pair of coupled fermions . at the expense of introducing an extra set of fermionic dofs , it becomes possible to recast the non - gaussian properties of the original spin bath in terms of gaussian processes in the extended space . in a subsequent work , the paper ii@xcite , we should further investigate physical properties of spin bath models . the paper is organized as follows . in sec . [ sec:2 ] , we introduce thoroughly the stochastic formalism for open quantum systems embedeed in a generic quantum environment . the sle is the starting point that we build upon to construct generalized hierarchical equations ( ghe ) , a family of deterministic simulation methods after formally averaging out the noise variables . in sec . [ sec : spectral ] , we study bosonic baths by expanding the noise processes in terms of the spectral eigenfunctions of the bath s two - time correlation function . in sec . [ sec : heom ] , we discuss the alternative derivation that generalizes the recently introduced extended heom to study non - gaussian bath models and fermionic bath models . in sec . [ sec : spin ] , we introduce the dual - fermion representation and derive an alternative ghe more suitable for spin bath models composed of nuclear / electron spins . a brief summary is given in sec . [ sec : last ] . in app.[app : stochprocess]-[app : grassmann ] , we provide additional materials on the stochastic calculus and grassmann number to clarify some details of the present work . appendix [ app : inf - func ] shows how to recover the influence functional theory from the present stochastic formalism . in the last appendix , we present numerical examples to illustrate the methods discussed in the main text . in this study , we consider the following joint system and bath hamiltonian , @xmath3 where the interacting hamiltonian can be decomposed into factorized forms with @xmath4 and @xmath5 acting on the system and bath , respectively . more specifically , we assume @xmath6 and @xmath7 where the operator @xmath8 can be taken as the bosonic creation operator @xmath9 , the fermionic creation operator @xmath10 or the spin raising operator @xmath11 depending on the specific bath model considered . throughout the rest of this paper , the system hamiltonian reads @xmath12 where we simply take the system as a spin and the index 0 always refers to the system . although we adopt a specific system hamiltonian in eq . ( [ eq : sysh ] ) , it can be more general . we consider the factorized initial conditions , @xmath13 where two parts are initially uncorrelated and the bath density matrix is commonly taken as a tensor product of thermal states for each individual mode . with eqs.([eq : genh])-([eq : genrhoi ] ) , the dynamics of the composite system ( system and bath ) is obtained by solving the von neumann equation @xmath14.\end{aligned}\ ] ] if only the system is of interest , then one can trace over the bath dofs , i.e. @xmath15 . this straightforward computation ( full many - body dynamics then partial trace ) soon becomes intractable as the dimension of the hilbert space scales exponentially with respect to a possibly large number of bath modes . many open quantum system techniques have been proposed to avoid a direct computation of eq . ( [ eq : vneq ] ) . our starting point is to replace eq . ( [ eq : vneq ] ) with a set of coupled stochastic differential equations , @xmath16 -\frac{i}{\sqrt 2 } \sum_{\alpha=\pm } a^\alpha\tilde\rho_s dw_\alpha + \frac{i}{\sqrt 2}\sum_{\alpha=\pm } \tilde\rho_s a^\alpha dv_\alpha , \nonumber \\ \label{eq : ito_unnorm2 } & & d\tilde\rho_b = \\ & & -i dt \left[\hat{h}_b , \tilde\rho_b\right ] + \frac{1}{\sqrt 2 } \sum_{\alpha=\pm } dw^*_\alpha b^{\bar \alpha } \tilde \rho_b + \frac{1}{\sqrt 2}\sum_{\alpha=\pm } dv^*_\alpha \tilde \rho_b b^{\bar \alpha } , \nonumber\end{aligned}\ ] ] where the stochastic noises appear as differential wiener increments , @xmath17 and @xmath18 ( the white noises @xmath19 and @xmath20 will be explicitly defined later ) . noting the stochastic relation@xcite between density matrix and wave function , @xmath21 , the two forward / backward wave functions evolves under the time - dependent hamiltonian , @xmath22 and @xmath23 , respectively . this connection implies an equivalent formulation of stochastic wave function - based theory of open quantum systems with better scaling@xcite for large system simulations . in this work , we focus on the density matrix presentation and attach the tilda symbol on top of all stochastically - evolved density matrices as in eqs . ( [ eq : ito_unnorm])-([eq : ito_unnorm2 ] ) . equation ( [ eq : ito_unnorm2 ] ) can be further decomposed into corresponding equations for individual modes , @xmath24 \nonumber \\ & & + \frac{1}{\sqrt 2 } \sum_{\alpha=\pm } g_k dw^*_\alpha b^{\bar\alpha}_k \tilde \rho_k + \frac{1}{\sqrt 2}\sum_{\alpha=\pm } g_k dv^*_\alpha \tilde \rho_k b^{\bar\alpha}_k,\end{aligned}\ ] ] where @xmath25 , @xmath26 and @xmath27 . the system and bath are decoupled from each other but subjected to the same set of random fields . the stochastic processes ( @xmath28 and @xmath29 ) in this work can be either complex - valued or grassmann - valued depending on the bath model under study . to manifest the gaussian properties of fermionic baths , it is essential to adopt the grassmann - valued noises . in these cases , it is crucial to maintain the order between fermionic operators and grassmann - valued noise variables presented in eqs . ( [ eq : ito_unnorm2])-([eq : ito_unnorm3 ] ) as negative signs arise when the order of variables and operators are switched . the white noises satisfy the standard relations @xmath30 where the overlines denote averages over noise realizations . any other unspecified two - time correlation functions vanish exactly . the order of variables in eq . ( [ eq : noise ] ) also matters for grassmann noises as explained earlier . with these basic properties laid out , we elucidate how to recover eq . ( [ eq : vneq ] ) from the stochastic formalism . first , the equation of motion for the joint density matrix @xmath31 is given by @xmath32 \tilde \rho_b(t ) + \tilde \rho_s(t ) [ d\tilde \rho_b(t ) ] + d \tilde \rho_s(t ) d \tilde \rho_b(t),\end{aligned}\ ] ] where the last term is needed to account for all differentials up to @xmath33 as the product of the conjugate pairs of differential wiener increments such as @xmath34 , contributes a term proportional to @xmath35 on average . taking the noise averages of eq . ( [ eq : itodiff ] ) , the first two terms together yield @xmath36 $ ] and the last term gives the system - bath interaction , @xmath37 $ ] . due to the linearity of the von neumann equation and the factorized initial condition , the composite system dynamics is given by @xmath38 . to extract the reduced density matrix , we trace out the bath dofs before taking the noise average , @xmath39 in this formulation , it is clear that all the bath - induced dissipative effects are encoded in the trace of the bath s density matrix . because of the non - unitary dynamics implied in the eqs . ( [ eq : ito_unnorm])-([eq : ito_unnorm2 ] ) , the norm of the stochastically evolved bath density matrices are not conserved along each path of noise realization . the norm conservations only emerge after the noise averaging . this is a common source of numerical instabilities one encounters when directly simulating the simple stochastic dynamics presented so far . the norm fluctuations of @xmath40 can be suppressed by modifying the stochastic differential equations above to read , @xmath41 \mp i dt \sum_{\alpha } \left [ a^{\alpha } , \tilde\rho_s \right]_{\mp } \mathcal{b}^{\bar \alpha}(t ) \\ & & \ , \ , -\frac{i}{\sqrt 2 } \sum_{\alpha } a^{\alpha}\tilde\rho_s dw_{\alpha } + \frac{i}{\sqrt 2}\sum_{\alpha } \tilde\rho_s a^{\alpha } dv_{\alpha } , \nonumber \\ \label{eq : ito2 } & & d\tilde\rho_b = -i dt \left[\hat{h}_b , \tilde\rho_b\right ] + \frac{1}{\sqrt 2 } \sum_{\alpha } dw^*_{\bar \alpha } \left ( b^{\alpha } \mp \mathcal b^\alpha(t ) \right ) \tilde \rho_b \nonumber \\ & & \ , \ , + \frac{1}{\sqrt 2}\sum_{\alpha } dv^*_{\bar \alpha } \tilde \rho_b \left ( b^\alpha - \mathcal b^\alpha ( t ) \right).\end{aligned}\ ] ] where additional stochastic fields @xmath42 is introduced to ensure @xmath43 is conserved along each noise path . in eqs . ( [ eq : ito])-([eq : ito2 ] ) , the top / bottom sign in the symbols @xmath44 ( @xmath45 ) refers to complex - valued / grassmann - valued noises , respectively . similarly , @xmath46_{\mp}$ ] refers to commutator ( complex - valued noise ) and anti - commutator ( grassmann - valued noise ) in eq . ( [ eq : ito ] ) . after these modifications , the exact reduced density matrix of the system is given by @xmath47 . by introducting @xmath48 , we directly incorporate the bath s response to the random noise in system s dynamical equation . equation ( [ eq : ito ] ) and the determination of @xmath48 constitute the foundation of open system dynamics in the stochastic framework . in addition to being a methodology of open quantum systems , it should be clear that the present framework allows one to calculate explicitly bath operator involved quantities of interest@xcite . from eq . ( [ eq : bfield ] ) , it is clear that @xmath48 can be obtained by formally integrating eq . ( [ eq : ito2 ] ) . for simplicity , we take @xmath49 and similarly for @xmath50 . this simplification does not compromise the generality of the results presented below and applies to the common case of the spin boson like model when the system - bath interacting hamiltonian is given by @xmath51 with @xmath52 . we first consider the gaussian baths composed of non - interacting bosons or fermions . the equations of motion for the creation and annihilation operators for individual modes read @xmath53 where the top ( bottom ) sign of @xmath44 should be used when complex - valued ( grassmann - valued ) noises are adopted . the expectation values in eqs . ( [ eq : boson_mode1])-([eq : boson_mode2 ] ) are taken with respect to @xmath54 . the generalized cumulants are defined as @xmath55 for bosonic and fermionic bath models , it is straightforward to show that the time derivatives of @xmath56 vanish exactly . hence , the second order cumulants are determined by the thermal equilibrium conditions of the initial states . immediately , one can identify the relevant quantity @xmath57 representing either the bose - einstein or fermi - dirac distribution depending on whether it is a bosonic or fermionic mode . replacing the second order cumulants in eqs . ( [ eq : boson_mode1])-([eq : boson_mode2 ] ) with an appropriate thermal distribution , one can derive a closed form expression @xmath58 where @xmath59 stands for the corresponding two - time bath correlation functions . for the gaussian bath models , eqs . ( [ eq : ito ] ) and ( [ eq : gaussbfield ] ) together provide an exact account of the reduced system dynamics . next we illustrate the treatment of non - gaussian bath models within the present framework with the spin bath as an example . in the rest of this section , the analysis only applies to the complex - valued noises . the determination of @xmath60 still follows the same procedure described at the beginning of this section up to eqs . ( [ eq : boson_mode1])-([eq : boson_mode2 ] ) . the deviations appear when one tries to compute the time derivatives of the second cumulants . common to all non - gaussian bath models , the second cumulants are not time invariant . instead , by iteratively using eq . ( [ eq : ito2 ] ) , the second and higher order cumulants can be shown to obey the following general equation , @xmath61}_{k}+\mathcal{g}^{[\pmb{\alpha},-]}_{k}\right ) \nonumber \\ & & + \frac{1}{\sqrt{2}}dv^*_s \left(\mathcal{g}^{[+,\pmb{\alpha}]}_{k}+\mathcal{g}^{[-,\pmb{\alpha}]}_{k}\right),\end{aligned}\ ] ] where @xmath62 specifies a sequence of raising and lowering spin operators that constitute this particular @xmath63-th order cumulant and @xmath64 with @xmath65 depending on whether it refers to a raising ( + ) or lowering ( - ) operator , respectively . the time evolution of these bath cumulants form a simple hierarchical structure with an @xmath63-th order cumulant influenced directly by the @xmath66-th order cumulants according to the equation above where we use @xmath67 \equiv ( \alpha_1 , \dots \alpha_n , \pm)$ ] to denote an @xmath66-th cumulant obtained by appending a spin operator to @xmath68 . a similar definition is implied for @xmath69 $ ] . more specifically , these cumulants are defined via an inductive relation that we explicitly demonstrate with an example to obtain a third - order cumulant starting from a second - order one given in eq . ( [ eq:2ndcum ] ) , @xmath70}_k = \langle b^{\alpha_1 } b^{\alpha_2 } ( b^\pm - \langle b^\pm \rangle ) \rangle + \langle b^{\alpha_1}(b^\pm - \langle b^\pm \rangle ) \rangle \langle b^{\alpha_2}\rangle + \langle b^{\alpha_1 } \rangle \langle b^{\alpha_2 } ( b^\pm - \langle b^\pm \rangle ) \rangle.\end{aligned}\ ] ] the key step in this inductive procedure is to insert an operator identity @xmath71 at the end of each expectation bracket defining the @xmath63-th cumulant . if a term is composed of m expectation brackets , then this insertion should apply to one bracket at a time and generate m terms for the @xmath66-th cumulant . similarly , we get @xmath72}$ ] by inserting the same operator identity to the beginning of each expectation bracket of @xmath73 . for the spin bath , these higher order cumulants do not vanish and persist up to all orders . in any calculations , one should certainly truncate the cumulants at a specific order by imposing the time invariance , @xmath74 and evaluate the lower order cumulants by recursively integrating eq . ( [ eq : gencum ] ) . through this simple prescription , one derives @xmath75 where @xmath76 stands for an @xmath63-time bath correlation functions with the subscript @xmath77 used to distinguish the @xmath78 @xmath63-time correlation functions appearing in the stochastic integrations ( each involves a unique sequence of noise variables ) in eq . ( [ eq : generalfield ] ) . comparing to eq.([eq : gaussbfield]),one can identify @xmath79 and @xmath80 with @xmath81 and @xmath82 , respectively . note this derivation assumes the odd - time correlation functions vanish with respect to the initial thermal equilibrium state . at this point , we briefly summarize the unified stochastic formalism . once @xmath60 is fully determined , eq . ( [ eq : ito ] ) can be presented in a simple form , the stochastic liouville equaiton ( sle ) , @xmath83 \mp i a \tilde\rho_s(t)\xi(t ) + i \tilde\rho_s(t)a\eta(t),\nonumber \\\end{aligned}\ ] ] where the newly defined color noises are @xmath84 in these equations above , the top / bottom signs are associated with complex - valued / grassmann - valued noises , respectively . in the cases of bosonic baths , @xmath60 is given by eq . ( [ eq : gaussbfield ] ) and driven by the complex - valued noises . the color noises then are fully characterized by the statistical properties , @xmath85 several stochastic simulation algorithms have been proposed to solve the sle with the gaussian noises . on the other hand , all previous efforts in the stochastic formulations of fermionic bath end up with derivations of either master equations@xcite or hierarchical@xcite type of coupled equations . the stochastic framework has simply served as a mean to derive deterministic equations for numerical simulations . the lack of direct stochastic algorithm is due to the numerical difficulty to model grassmann numbers . in this study , we support the view that grassmann numbers are simply formal bookkeeping devices " to help formulate the fermionic path integrals and the formal stochastic equations of motion with gaussian properties . hence , it is critical to formally eliminate the grassmann number and the associated stochastic processes , which will be demonstrated in sec . [ sec : spin ] with numerical illustrations in app . [ app : num ] . going beyond the gaussian baths , @xmath60 is given by eq . ( [ eq : generalfield ] ) which involves multiple - time stochastic integrals . formally , one can still use the same definition of noises , eq . ( [ eq : defcolornoise ] ) , and the sle still prescribes the exact dynamics for the reduced density matrix . the primary factor distinguishing from the gaussian baths is the statistical characterization of the noises . higher order statistics are no longer trivial for non - gaussian processes , and they are determined by the multi - time correlation functions @xmath86 in eq . ( [ eq : generalfield ] ) . for instance , when the fourth order correlation functions are included in the definition of @xmath60 , additional statistical conditions such as @xmath87 would have to be imposed and related to @xmath88 to fully specify this noise . since constructing a purely stochastic method to simulate gaussian processes is already a non - trivial task , simulating non - gaussian random processes is an even tougher goal . in the subsequent discussions , we should devise deterministic numerical methods based on the sle , eq . ( [ eq : sleq ] ) , by formally averaging out the noises . we name the proposed methods in sec . [ sec : spectral ] - sec . [ sec : heom ] collectively as the generalized hierarchical equations ( ghe ) in this work . besides the ghe to be presented , we note sophisticated hybrid algorithms@xcite could also be constructed to combine advantages of both stochastic and deterministic approaches . we shall leave these potential extensions in a future study . in this section , we consider a bosonic gaussian bath . we note eq . ( [ eq : xi_corr ] ) encodes the full dissipative effects induced by a bath in the stochastic formalism . the microscopic details , such as eq . ( [ eq : defcolornoise ] ) , of the noise variables become secondary concerns . this observation allows us to substitute any pairs of correlated color noises that satisfy eq . ( [ eq : xi_corr ] ) as these statistical conditions alone do not fully specify the noises present in eq . ( [ eq : sleq ] ) . in other words , more than one set of noises can generate identical quantum dissipative dynamics as long as they all satisfy eq . ( [ eq : xi_corr ] ) but may differ in other unspecified statistics such as @xmath89 and @xmath90 etc . this flexibility with the choice@xcite of stochastic processes provides opportunities to fine - tune performances of nuemrical algorithms . we propose the following decomposition@xcite of the noise variables @xmath91 where ( @xmath92,@xmath93 ) and ( @xmath94 , @xmath95 ) are independent and real - valued normal variables with mean @xmath96 and variance @xmath97 while @xmath98 are similarly defined but complex - valued . the other unspecified functions are obtained from the spectral expansion of the correlation functions , @xmath99 where @xmath100 , the functions @xmath101 ( complex - valued in general ) , @xmath102 ( real - valued ) and @xmath103 ( real - valued ) form independent sets of orthonormal basis function over time domain @xmath104 $ ] for the simulation . the spectral components can be determined explicitly by solving @xmath105 and similar integral equations will yield other sets of basis functions and the associated eigenvalues . the newly defined noises in eq . ( [ eq : corr - stoch ] ) can be shown to reproduce the two - time statistics given in eq . ( [ eq : xi_corr ] ) . a crucial assumption of the spectral expansion above is that correlation functions should be positive semi - definite . this could be a concern with the quantum correlation functions in the low temperature regime . however , this problem can be addressed by modifying the hamiltonian and re - define the correlation functions in order to shift the spectral values by a large constant to avoid negative eigenvalues . we re - label the newly introduced random variables @xmath106 and expand the reduced density matrix by@xcite @xmath107 where the function @xmath108 with @xmath109 is explicitly defined in the second line . the @xmath77-th hermite polynomial @xmath110 takes argument of random variables @xmath111 . the total number of random variables @xmath112 is given by @xmath113 . every auxiliary density matrices @xmath114 directly contributes to the determination of @xmath115 . substituting eqs . ( [ eq : corr - stoch ] ) and ( [ eq : rho - stochexp ] ) into eq . ( [ eq : sleq ] ) and average over all random variables @xmath112 , one obtains a set of coupled equations for the density matrices , @xmath116 + i \sum_{k,\mathbf n } a \sigma_{\mathbf n } \theta_k(t ) g^{k}_{\mathbf m \mathbf n } - i \sum_{k,\mathbf n } \sigma_{\mathbf n}a \theta^\prime_k(t ) g^{k}_{\mathbf m \mathbf n},\end{aligned}\ ] ] where @xmath117 , the components of @xmath0 in eq . ( [ eq : corr - stoch ] ) , and similarly @xmath118 correspond to the components of @xmath1 , respectively . in the above equation , @xmath119 is defined by @xmath120 where these averages can be done analytically by exploiting the properties of the hermite polynomials and gaussian integrals@xcite . finally , the exact reduced density matrix is obtained after averaging out random variables in eq . ( [ eq : rho - stochexp ] ) , which can be done by invoking the gaussian integral identities . the present approach introduces an efficient decomposition of the noise variables and provides an alternative coupling structure for a system of differential equations than the standard heom in solving open quantum dynamics . unfortunately , the present method is not easily generalizable to accommodate the non - gaussian processes . we next present another approach , starting from eq . ( [ eq : sleq ] ) again , that yields deterministic equations and more easily to accommodate non - gaussian bath models . this time we utilize eq . ( [ eq : defcolornoise ] ) as the definitions for the noises , @xmath0 and @xmath1 . following the basic procedure of ref . , we average over the noises in eq . ( [ eq : sleq ] ) to get @xmath121 - i \left [ a , \overline{\tilde \rho_s \mathcal{b } } \right ] , \end{aligned}\ ] ] the noise averages yield an auxiliary density matrix ( adm ) , @xmath122 , by eq . ( [ eq : defcolornoise ] ) . working out the equation of motion for the adm , one is then required to define additional adms and solve their dynamics too . in this way , a hierarchy of equation of motions for adms develops with the general structure @xmath123 = \overline { d_t [ \tilde\rho_s ] \mathcal{b}^m } + \overline{\tilde \rho_s d_t \mathcal{b}^m } + \overline{d_t \tilde \rho_s d_t \mathcal b^m}.\end{aligned}\ ] ] the time derivatives of @xmath124 and @xmath125 are given by eqs . ( [ eq : ito ] ) and ( [ eq : generalfield ] ) , respectively . if we group the adm s into a hierarchical tier structure according to the exponent @xmath77 of @xmath126 , then it would be clear soon that the first term of the rhs of eq . ( [ eq : genheom ] ) couples the present adm to ones in the ( @xmath127)-th tier , and the last term couples the present adm to others in the ( @xmath128)-th tier . in the rest of this section , we should materialize these ideas by formulating generalized heoms in detail . we separately consider the cases of complex - valued noises ( for bosonic and non - gaussian bath models ) and grassmann - valued noises ( for fermionic bath models ) . following a recently proposed scheme , we introduce a complete set of orthonormal functions @xmath129 and express all the multi - time correlation functions in eq . ( [ eq : generalfield ] ) as @xmath130 where @xmath131 . due to the completeness , one can also cast the derivatives of the basis functions in the form , @xmath132 next we define cumulant matrices @xmath133,\end{aligned}\ ] ] where each composed of @xmath134 row vectors with indefinite size . for instance , @xmath135 has two row vectors while @xmath136 has four row vectors etc . the @xmath77-th row vector of matrix @xmath137 contains matrix elements denoted by @xmath138 . each of this matrix element can be further interpreted by @xmath139 where @xmath140 can be either a @xmath141 or @xmath142 stochastic variable depending on index @xmath77 . with these new notations , the multi - time correlation functions in eq . ( [ eq : generalfield ] ) can be concisely encoded by @xmath143 now we introduce a set of adm s @xmath144 \left [ \mathbf a_2 \right ] \left [ \mathbf a_3 \right ] } \dots \equiv \overline { \tilde\rho_s(t ) \prod_{n , m , k } a^n_{m \mathbf{j}_k}(t)},\end{aligned}\ ] ] which implies the noise average over a product of all non - zero elements of each matrix @xmath145 with the stochastically evolved reduced density matrix of the central spin . the desired reduced density matrix would correspond to the adm at the zero - th tier with all @xmath146 being null . furthermore , the very first adm we discuss in eq . ( [ eq : first - heom ] ) can be cast as @xmath147 \dots } ( t),\end{aligned}\ ] ] where each adm on the rhs of the equation carries only one non - trivial matrix element @xmath148 in @xmath149 . finally , the hierarchical equations of motion for all adms can now be put in the following form , @xmath150 \left [ \mathbf a_2 \right ] \left [ \mathbf a_3 \right ] \cdots } = -i \left [ h_s , \rho^{\left [ \mathbf a_1 \right ] \left [ \mathbf a_2 \right ] \left [ \mathbf a_3 \right ] \cdots } \right ] -i \sum_{n , m,\pmb j } \chi^{n+1,m}_{\pmb j } \left [ a , \rho^{\cdots \left [ \mathbf{a}_n + ( m,\pmb j ) \right ] \cdots } \right ] \nonumber \\ & & -i \sum_{n , m,\mathbf j } \phi_{j_1}(0 ) a \rho^{\cdots \left [ \mathbf{a}_{n-1 } + ( m',\pmb{j}_1 ) \right]\left [ \mathbf{a}_n - ( m,\pmb j ) \right ] \cdots } -i \sum_{n , m,\mathbf j } \phi_{j_1}(0 ) \rho^{\cdots \left [ \mathbf{a}_{n-1 } + ( m',\pmb{j}_1 ) \right]\left [ \mathbf{a}_n - ( m,\pmb j ) \right ] \cdots } a \nonumber \\ & & + \sum_{n , m,\mathbf{j}\mathbf{j ' } } \eta_{\mathbf{j } \mathbf{j ' } } \rho^{\cdots \left [ a^{n}_{m \mathbf{j } } \rightarrow a^{n}_{m \mathbf{j ' } } \right ] \cdots}\end{aligned}\ ] ] this equation involves a few compact notations that we now explain . we use @xmath151 $ ] to mean adding or removing an element @xmath152 to the @xmath77-th row . we also use @xmath153 $ ] to denote a replacement of an element in the @xmath77-th row of @xmath149 . on the second line , we specify an element in a lower matrix given by @xmath154 . the variable @xmath155 implies removing the first element of the @xmath156 array and the associated index @xmath157 is determined by removing the first stochastic integral in eq . ( [ eq : adef ] ) . we caution that there is no @xmath158 matrix and such a term whenever arises should simply be ignored when interpreting the above equation . after the first term on the rhs of eq . ( [ eq : gheom ] ) , we only explicitly show the matrices @xmath149 affected in each term of the equation . this generalized heom structure reduces exactly to the recently proposed eheom@xcite when only @xmath159 cumulant matrix carries non - zero elements , i.e. only the second cumulant expansion of an influence functional is taken into account . it is clear that the higher - order non - linear effects induced by the bath s @xmath63-time correlation functions will only appear earliest at the ( @xmath160)-th tier expansion . next we consider the fermionic bath models with grassmann - valued noises . as discussed earlier , the grassmann numbers are essential to manifest the gaussian properties of fermionic baths . in this case , one can significantly simplify the generalized heom in the previous section . first , the bath - induced stochastic field can still be decomposed in the form , @xmath161 where the indices @xmath77 and @xmath63 are suppressed when compared to eq . ( [ eq : bosonb ] ) . this is because the hamiltonian we consider in this study only allows fermion gaussian bath model . each element @xmath162 are similarly defined @xmath163 where @xmath164 ( grassmann - valued ) when @xmath165 ( or @xmath166 ) . similar to the algebraic properties of fermionic operators , there are no higher powers of grassmann numbers and each element @xmath167 can only appear once . this pauli exclusion constraint allows us to simplify the representation of fermionic adms . we may specify an m - th tier adm by @xmath168 where @xmath169 with @xmath170 or @xmath97 . in this simplified representation , instead of specifying the non - zero elements as in eq . ( [ eq : boson - adm ] ) , we layout all elements @xmath171 in an ordered fashion and employ the binary index @xmath172 to denote which basis functions contribute to a particular adm . the tier level of an adm is determined by the number of basis participating functions , i.e. @xmath173 . following the general procedure outlined in eq . ( [ eq : genheom ] ) , the m - th tier heom reads , @xmath174 + i \sum_{j } \left ( \chi_j a \bar\rho^{m+1}_{\mathbf n + \mathbf 1_{j } } ( -1)^{\vert \mathbf n \vert_j } + \chi_j \bar\rho^{m+1}_{\mathbf n + \mathbf 1_{j } } a ( -1)^{\vert \mathbf n \vert_j } \right)(1-n_j ) \nonumber \\ & & + i\sum_{j } \phi_{j}(0 ) a \rho^{m-1}_{\mathbf n - \mathbf 1_j}(-1)^{\vert \mathbf n \vert_j}n_j \nonumber \\ & & + i\sum_{j } \phi_j(0 ) \rho^{m-1}_{\mathbf n - \mathbf 1_j } a ( -1)^{\vert \mathbf n \vert_j}n_j \nonumber \\ & & + \sum_{j , j ' } \eta_{jj ' } \rho^{m}_{\mathbf n_{j , j ' } } ( -1)^{\vert \mathbf n \vert_j+\vert \mathbf n \vert_{j ' } } n_j ( 1-n_{j ' } ) , \ ] ] where @xmath175 and @xmath176 implies setting @xmath177 and @xmath178 . in the above equation , @xmath179 is a vector of zero s except an one at the @xmath180-th component . the factor such as @xmath181 and @xmath182 are present to enforce the pauli exclusion principle associated with the fermions . the structure of fermionic heom certainly resembles that of the bosonic case . however , a few distinctions are worth emphasized . first , it is just the gaussian bath result including only @xmath159 block matrix when compared to the results in sec . [ sec : complex ] . secondly , the fermionic heom truncates exactly at some finite number of tiers due to the constraint on the array of binary indices , @xmath172 . extra negative signs arise from the permutations to shift the underlying grassmann - valued stochastic variables from their respective positions in eq . ( [ eq : fm - tier - defn ] ) to the left end of the sequence . we dedicate an entire section to discuss the spin bath from two distinct perspectives . if one formulates the sle for a spin bath model in terms of complex - valued noises , then the bath induced stochastic field @xmath60 is given by eq . ( [ eq : generalfield ] ) . in this way , the spin bath is a specific example of non - gaussian bath models . the generalized heom formulated earlier can be directly applied in this case . however , in any realistic computations , it is necessary to truncate the statistical characterization of @xmath60 up to a finite order of multi - time correlation functions in eq . ( [ eq : generalfield ] ) . while the method is numerically exact , it is computationally prohibitive to calculate beyond the first few higher - order corrections . when the spin bath is large and can be considered as a finite - size approximation to a heat bath , one can show that the linear response approximation@xcite often yields accurate results and a leading order correction should be sufficient whenever needed . the relevance of this leading order correction for spin bath models will be investigated in the paper ii@xcite . on the other hand , a spin bath composed of nuclear / electrons spins , as commonly studied in artificial nanostructures at ultralow temperature regime , can beahve very differently from a heat bath composed of non - interacting bosons . there is no particular reason that the linear response and the first leading order correction should sufficiently account for quantum dissipations under all circumstances . in this scenario , it could be useful to map each spin mode onto a pair of coupled fermions . the non - linear mapping allows us to efficiently capture the exact dynamics in an extended gaussian bath model . we consider the following transformation that maps each spin mode into two fermions via , @xmath183 where the fermion operators satisfy the canonical anti - commutation relations . one can verify the above mapping reproduces the correct quantum angular momentum commutation relations with the @xmath184 fermion operators for each spin , while the presence of additional @xmath185 fermions makes the spin operators associated with different modes commute with each other . we now re - write the hamiltonian as @xmath186 the initial density matrix still maintains a factorized form in the dual - fermion representation , @xmath187 where @xmath188 is the thermal equilibrium state of the @xmath184 fermions at the original inverse temperature @xmath189 of the spin bath and the @xmath185 fermions are in the maximally mixed state which is denotes by the identity matrix with dimension @xmath190 where @xmath63 is the number of bath modes . a normalization constant is implied to associate with the @xmath191 matrix . according to the transformed hamiltonian in eq . ( [ eq : dfh ] ) , the system bath coupling now involves three - body interactions , @xmath192 . furthermore , the two femrionic baths portrait a non - equilibrium setting with @xmath184 fermionic bath inherits all physical properties of the original spin bath while @xmath185 fermionic bath is always initialized in the infinite - temperature limit regardless of the actual state of the spin bath . we first take the system and the d fermions together as an enlarged system and treat the c fermions collectively as a fermionic bath . we introduce the grassmann noises to stochastically decouple the two subsystems , @xmath193 - i \sum_{k } \sigma^z_0 a_k \tilde \rho_{sd } d\mathcal{w}_k , \nonumber \\ & & \ , + i \sum_k \tilde\rho_{sd } \sigma^z_0 a_k d\mathcal{v}_k \nonumber \\ \label{eq : sd - ito2 } d\tilde \rho_{k } & = & -i dt \left [ \hat h_{b , k } , \tilde \rho_k \right ] + \frac{dw^*_k}{\sqrt 2 } \left(b_k+\mathcal{b}_k\right ) \tilde \rho_k \nonumber \\ & & \ , + \frac{dv^*_k}{\sqrt 2 } \tilde \rho_k \left(b_k-\mathcal{b}_k\right),\end{aligned}\ ] ] where @xmath194 , @xmath195 , @xmath196 and @xmath197 . the density matrices @xmath198 denotes the extended system including system spin and all @xmath185 fermions and @xmath199 denotes the individual c fermions with @xmath200 . in eq . ( [ eq : sd - ito ] ) , the noises are defined by @xmath201 where @xmath202 and @xmath203 are the standard grassmann noises defined earlier . ( [ eq : sd - ito2 ] ) clearly conserve the norm of @xmath54 along each noise path , and we will focus on eq . ( [ eq : sd - ito ] ) and the stochastic fields , @xmath204 . our main interest is just the system spin . hence , we trace out the d fermions in eq . ( [ eq : sd - ito ] ) and get @xmath205 + i \sum_k \left [ \sigma^z_0 , \psi^{1}_k\right ] \frac{dx_k}{2 } \nonumber \\ & & + i \sum_{k } \left\{\sigma^z_0 , \psi^{1}_k \right\}\left(\frac{dy_k}{2 } - \mathcal{b}_k \right).\end{aligned}\ ] ] the auxiliary objects , @xmath206 , appearing in eq . ( [ eq : dual - stoch1 ] ) are defined via @xmath207 where @xmath208 , ( s)tr either implies standard trace ( n is even ) or super - trace ( n is odd ) , and the new noises @xmath209 a hierarchical structure is implied in eq . ( [ eq : dual - stoch1 ] ) , so we derive the equations of motions for the auxiliary objects , @xmath210 + i \sum_{j \notin \mathbf{k } } \left\{\sigma^z_0 , \psi^{n+1}_{\mathbf{k}+j}\right\ } \frac{dx_k}{2}(-1)^{\vert \mathbf k \vert _ { > j } } \ , + i \sum_{j \in \mathbf{k } } \left[\sigma^z_0 , \psi^{n-1}_{\mathbf{k}-j}\right]\frac{dx_k}{2 } ( -1)^{\vert \mathbf k \vert_{\geq j}}\nonumber \\ & & + i \sum_{j \notin \mathbf{k } } \left[\sigma^z_0 , \psi^{n+1}_{\mathbf{k}+j}\right ] \left(\frac{dy_k}{2 } - \mathcal{b}_k \right ) ( -1)^{\vert \mathbf k \vert _ { > j } } + i \sum_{j \in \mathbf{k } } \left\{\sigma^z_0 , \psi^{n-1}_{\mathbf{k}-j}\right\}\left(\frac{dy_k}{2 } - \mathcal{b}_k \right ) ( -1)^{\vert \mathbf k \vert_{\geq j}},\end{aligned}\ ] ] where @xmath211 . since the spin bath model is mapped onto an effective fermionic problem , the uses of grassmann noises , eq . ( [ eq : dual - stoch2 ] ) , will serve as a starting point to develop deterministic numerical methods once the grassmann noises are integrated out . to solve eq . ( [ eq : dual - stoch2 ] ) , we first define the generalized adms @xmath212 where @xmath213 . the 2 index vectors @xmath214 and @xmath215 label pairs of coupled @xmath184 and @xmath185 fermionic modes ; furthermore , the two index vectors are mutually exclusive in the sense a bath mode can appear in just one of the two vectors each time . in this case , the tier - structure of the adms are determined by @xmath216 . same as the bosonic and fermionic bath results , the desired reduced density matrix is exactly given by the zero - th tier of adms . further notational details of eq . ( [ eq : adm_sbath ] ) are explained now . the vector @xmath217 is to be paired with the vector @xmath214 to characterize the first set of stochastic fields @xmath218 in eq . ( [ eq : adm_sbath ] ) . more precisely , each @xmath219 labels one of the two possible stochastic fields , @xmath220 and @xmath221 , associated with @xmath222-th c fermion . in dealing with bosonic , fermionic and non - gaussian baths , we need to explicitly use bath s multi - time correlation functions via @xmath223 when formulating the generalized heom approach . in the present case , the stochastic decoupling we introduced in sec . [ sec : spin - dualf ] dictates that each c fermion acts as a bath and equipped with its own set of stochastic fields as shown in eq . ( [ eq : dual - stoch2 ] ) . there is no need to expand the bath correlation functions in some orthonormal basis , as each mode s correlation functions will be treated explicitly in a fourier decomposition . repeat the same steps of the derivation as before , we obtain the generalized heom for the spin bath , @xmath224 + i \sum_{l \in \mathbf k } \alpha_l \omega_l \rho^{n , m}_{\omega , \mathbf j } + i \sum_{\substack{l \notin \mathbf k , l \notin \mathbf j \\ \gamma } } \left [ a , \rho^{n+1,m}_{\omega+(l,\gamma ) , j}\right ] + i \sum_{l \in \mathbf k } \alpha_l \left\ { a , \rho^{n-1,m+1}_{\omega-(l,\alpha_l),\mathbf j+\mathbf 1_l}\right\ } \nonumber \\ & & -\frac{i}{2 } \sum_{l \in \mathbf j , \gamma } \gamma g_l^2 ( 1 - 2n_f(\omega_l ) ) \left\ { a , \rho^{n+1,m-1}_{\omega+(l,\bar\gamma),\mathbf j - \mathbf 1_l } \right\ } -\frac{i}{2 } \sum_{l \in \mathbf k } g_l^2 ( 1 - 2n_f(\omega_{l } ) ) \left [ a , \rho^{n-1,m}_{\omega-(l,\alpha_l),\mathbf j } \right ] , \nonumber \\ & & + \frac{i}{2 } \sum_{l \in \mathbf j , \gamma } g_l^2 \left [ a , \rho^{n+1,m-1}_{\omega+(l,\gamma),\mathbf j - \mathbf 1_l}\right ] + \frac{i}{2 } \sum_{l \in \mathbf k } \alpha_l g_l^2 \left\{a , \rho^{n-1,m}_{\omega-(l,\alpha_l),\mathbf j}\right\ } , \end{aligned}\ ] ] where @xmath225 means a stochastic field @xmath226 is either added or removed from the vectors @xmath214 and @xmath217 and , similarly , @xmath227 means an index @xmath228 is either added or removed from @xmath215 . the range of the index values @xmath214 and @xmath215 can be extremely large as we explicitly label each microscopic bath modes . due to the hierarchical structure and the way adms are defined , it becomes prohibitively expensive to delve deep down the hierarchical tiers in many realistic calculations . however , the situation might not be as dire as it appears . we already discuss how the present formulation is motivated by the physical spin based environment , such as a collection of nuclear spins in a solid . in such cases , the bath often possess some symmetries allowing simplifications . for instance , most nuclear spins will precess at the same lamour frequency , and the coupling constant is often distance - dependent . hence , one can construct spatial symmetric shells " centered around the system spin in the 3-dimensional real space such that all bath spins inside a shell will more or less share the same frequency and system - bath coupling coefficient . by exploiting this kind of symmetry arguments , one can combine many adms defined in eq . ( [ eq : adm_sbath ] ) together to significantly reduce the complexity of the hierarchical structures . for a perfectly symmetric bath ( i.e. one frequency and one system - bath coupling term ) , one can use the following compressed adm , @xmath229 where the sum takes into account of all possible combinations of @xmath63 modes compatible with the requirement that @xmath230 . on the other hand , if one deals with a large spin bath described by an effective spectral density then treating the spin bath as an anharmonic environment and usage of the generalize heom in sec . [ sec : complex ] will be more appropriate . in fact , in the thermodynamical limit , the spin bath can be accurately approximated as a gaussian bath and one only needs to invoke @xmath159 block matrix in most calculations . in summary , we advocate the present stochastic framework as a unified approach to extend the study of dissipative quantum dynamics beyond the standard bosonic bath models . we exploit the it calculus rule to represent any bilinear interaction between two quantum dof as white noises . starting from eqs . ( [ eq : ito])-([eq : ito2 ] ) , one can derive the sle , eq . ( [ eq : sleq ] ) , with appropriate statistical conditions , such as eq . ( [ eq : xi_corr ] ) , that the noises must satisfy . in the gaussian bath models , the required conditions only involve two - time statistics determined by the bath s correlation functions . in the case of non - gaussian bath models , the noises are further characterized by higher order statistics and the multi - time correlation functions . we devise a family of ghe to solve the sle with deterministic simulations . we consider two separate orthonormal basis expansions : ( 1 ) spectral expansion and ( 2 ) generalized heom . the spectral expansion , in sec . [ sec : spectral ] , allows us to solve bosonic bath models efficiently when bath s two - time correlations assume a simple spectral expansion . this is often the case for correlation functions with a slow decay . the second approach , in sec . [ sec : heom ] , generalizes the eheom method to handle multi - time correlation functions in some arbitary set of orthnormal functions . this generalization can provide numerically exact simulations for non - gaussian ( including spin ) , fermionic and bosonic bath models with arbitray sepctral densities and temperature regimes . among the bath models , we extensively discuss the spin bath . when a spin bath is characterized by a well - behaved spectral density@xcite , the generalized heom in sec . [ sec : heom ] serves as an efficent approach to simulate dissipative quantum dynamics in a non - gaussian bath . for situations requiring more than a few higher - order response functions , such as baths composed of almost identical nuclear / electron spins , an alternative approach is to first map the spin bath onto an enlarged gaussian bath model of fermions via the dual - fermion representation and apply the dual - fermion ghe in sec . [ sec : heom - spin ] . numerical examples are illustrated in app . [ app : num ] . c.h . acknolwedges support from the sutd - mit program . is supported by nsf ( grant no . che-1112825 ) and smart . we focus on the complex - valued stochastic processes in this appendix . additional remarks on grassmann noises will be made in the following section . the basic wiener processes considered in this work is taken to be @xmath231 where the complex - valued noise has a mean @xmath232 and a variance @xmath233 . take a uniform discretization of time domain , in each time interval @xmath234 , each white noise path reduces to a sequence of normal random variables @xmath235 . hence , at each time interval , an identical normal distribution is given , @xmath236 the variance is chosen to reproduce the dirac delta function in the limit @xmath237 . furthermore , the differential wiener increments @xmath238 satisfy @xmath239 as required for brownian motion . the averaging process , implied by the bar on top of stochastic variables , can now be explicitly defined as @xmath240 grassmann numbers are algebraic constructs that anti - commute among themselves and with any fermionic operators . given any two grassmann numbers , @xmath241 and @xmath242 , and a fermionic operator , @xmath243 , they satisfy @xmath244 furthermore , the grassmann numbers commute with the vacuum state @xmath245 and , consequently , anti - commute with @xmath246 . besides the fermionic operators , these numbers commute with everything else such as the bosonic operators and spin pauli matrices . due to the anti - commutativity , there is no higher powers of grassmann numbers , i.e. @xmath247 . for instance , a single - variate grassmann function @xmath248 ( all variables , a , b , and x , are grassmann - valued ) can only assume this finite taylor - expanded form . in general every grassmann function can be decomposed into odd and even parity , @xmath249 . such that @xmath250 where @xmath251 is another arbitrary function with no particular parity assumed . the fermionic thermal equilibrium states are are even - parity grassmann function when represented in terms of the fermonic coherent states . this even - parity is preserved under linear driving with grassmann - valued noises . this means all the grassmann numbers will commute with the fermonic bath density matrices in our study . another relevant algebraic property for our study is @xmath252 where @xmath241 is a grassmann number and str@xmath253 is often termed the super - trace . finally , we discuss grassmann - valued white noises . similar to the discretized complex - valued white noises introduced earlier , we shall take the noise path as a continuum limit of a sequence of grassmann numbers , @xmath254 . we will formally treat them as random numbers with respect to grassmann gaussians as probability distributions . more precisely , the following integrals yield the desired first two moments ( in analogy to the complex - valued normal random variables ) , @xmath255 where the gaussians should be interpreted by the taylor expansion : @xmath256 . in evaluating the integrals above , we recall the standard grassmann calculus rule that integration with respect to @xmath241 is equivalent to differentiation with respect to @xmath241 . with these basic set - ups , one can operationally formulate grassmann noises in close analogy to the complex - valued cases . the connection between the two formalisms is usually investigated by deriving the stochastic equations from the influence functional theory via the hubbard - stratonovich transformation@xcite . nevertheless , to advocate the stochastic view of quantum dynamics as a rigorous foundation , we establish the connections in the reversed order . we should restrict to the standard bosonic bath models , but extension should be obvious . we first re - write eq . ( [ eq : ito_unnorm ] ) as @xmath257 \tilde\rho_s + i \tilde\rho_s \left[h_s -i \frac{\nu^*(t)}{\sqrt{2}}a\right],\end{aligned}\ ] ] where the system - bath interaction is given by eq . ( [ eq : genh ] ) . in this revised form , it is immediately clear that @xmath258 where @xmath259 is the time - ordering ( + ) and anti - time - ordering ( - ) operator . by inserting a complete set of basis @xmath260 at each time slice , eq . ( [ eq : appc - rdm ] ) can be put in the form , @xmath261 e^{is[\alpha_\tau]-is[\alpha^\prime_\tau ] } \nonumber \\ & & \,\,\ , \times e^{-\frac{i}{\sqrt 2 } \int^t_0 d\tau\left(\mu^\prime_\tau\alpha_\tau + i \nu^\prime_\tau \alpha'_\tau\right ) } , \nonumber \end{aligned}\ ] ] where @xmath262 and @xmath263 . on the other hand , the trace of @xmath264 , governed by eq . ( [ eq : ito_unnorm2 ] ) , can be expressed as @xmath265 where @xmath266 is given by eq . ( [ eq : gaussbfield ] ) . the exact reduced density matrix is then obtained after formally averaging out the noises in the following equation , @xmath267e^{is[\alpha_\tau]-is[\alpha^\prime_\tau ] } \nonumber \\ & & \,\,\ , \times \overline { \exp\left(-\frac{i}{\sqrt 2}\int^t_0 ds \left\ { \mu^\prime_s\alpha_s + i \nu^\prime_s \alpha'_s + i \left ( \mu_s + i \nu_s\right ) \mathcal b_s\right\ } \right ) } , \nonumber\end{aligned}\ ] ] where an explicit evaluation of the noise average on the last line should yield the standard bosonic bath influence functional . to get the influence function , it is useful to contemplate the discretized integrals for the noise averaging , @xmath268 = \int \prod_{i } \left [ d\mu_i d\mu^*_i d\nu_i d\nu^*_i \left(\frac{\delta t}{2\pi}\right)^2 e^{-\frac{\delta t}{2 } \left(\vert \mu_i\vert^2 + \vert \nu_i\vert^2 \right ) } \right ] \nonumber \\ & & \,\ , \times \exp\left ( -\frac{i}{\sqrt 2 } \sum_i \left\ { \mu^*_i \alpha_i + i \nu^*_i \alpha^\prime_i \right\ } \right ) \nonumber \\ & & \,\ , \times\exp\left(\frac{1}{\sqrt 2 } \sum_{i\geq j}\left\ { ( \mu_i + i\nu_i ) ( c_{i - j}\mu_j - i c^*_{i - j}\nu_j ) \right\ } \right ) , \nonumber \end{aligned}\ ] ] where the bath correlation function @xmath269 is given by @xmath270 by using the complex - valued gaussian integral identity , @xmath271 the standard feymann - vermon influence functional is recovered . the present result is easily generalized when dealing with non - gaussian baths and @xmath60 is potentially characterized by an infinite number of multi - time correlation functions . the noise averaging in this general case will give the cumulant expansion of an influence functional for any bath . we present a few numerical results to illustrate the dual - fermion ghe method introduced in this work . numerical examples with generalized heom approach will be further studied in a separate work , the paper ii@xcite . we will consider various cases of a pure dephasing model , @xmath272 an analytical expression for the off - diagonal matrix element of the reduced density matrix reads , @xmath273 with @xmath274,\end{aligned}\ ] ] where @xmath275 . first , we consider a 50-spin bath with the parameters @xmath276 sampled from the discretization of an ohmic bath . we use the general dual - fermion ghe scheme , eq . ( [ eq : dualfheom ] ) , to simulate the off - diagonal matrix element for the density matrix . figure [ fig : ohmic ] shows the results in the weak coupling ( panel a ) and the strong coupling ( panel b ) cases . due to each spin is modelled as a bath , it becomes prohibitive to delve into further tiers . nevertheless , with a shallow 2-tier hierarchy , the results seem to do reasonably well in the short - time limit . in the second case , we consider the spin star model@xcite where all the bath spins look identical , i.e. @xmath277 and @xmath278 . this is an often used model to analyze spin bath models . as shown by the results in fig . [ fig : nuclear ] , it is critical to go deep down the hierarchical tiers in order to recover the correct quantum dissipations . one can only generate this many tiers through compressing the auxiliary density matrices as in eq . ( [ eq : dualfc ] ) . this second example illustrates the kind of scenarios where dual - fermion ghe could provide an accurate account of quantum dynamics induced by a spin bath . @xmath279 , and @xmath280 ( kondo parameter ) assumes the value 0.1 ( a ) and 0.8 ( b ) . 2 hierarchical tiers are used in both cases . red curves are the numerical results and black curves are the exact results . ]

we extend a standard stochastic theory to study open quantum systems coupled to generic quantum environments including the three fundamental classes of non - interacting particles : bosons , fermions and spins . in this unified stochastic approach , the generalized stochastic liouville equation ( sle ) formally captures the exact quantum dissipations when noise variables with appropriate statistics for different bath models are applied . anharmonic effects of a non - gaussian bath are precisely encoded in the bath multi - time correlation functions that noise variables have to satisfy . staring from the sle , we devise a family of generalized hierarchical equations by averaging out the noise variables and expand bath multi - time correlation functions in a complete basis of orthonormal functions . the general hiearchical equations constitute systems of linear equations that provide numerically exact simulations of quantum dynamics . for bosonic bath models , our general hierarchical equation of motion reduces exactly to an extended version of hierarchical equation of motion which allows efficient simulation for arbitrary spectral densities and temperature regimes . similar efficiency and flexibility can be achieved for the fermionic bath models within our formalism . the spin bath models can be simulated with two complementary approaches in the presetn formalism . ( i ) they can be viewed as an example of non - gaussian bath models and be directly handled with the general hierarchical equation approach given their multi - time correlation functions . ( ii ) alterantively , each bath spin can be first mapped onto a pair of fermions and be treated as fermionic environments within the present formalism .

arxiv

the transition from a liquid to an amorphous solid that sometimes occurs upon cooling remains one of the largely unresolved problems of statistical physics @xcite . at the experimental level , the so - called glass transition is generally associated with a sharp increase in the characteristic relaxation times of the system , and a concomitant departure of laboratory measurements from equilibrium . at the theoretical level , it has been proposed that the transition from a liquid to a glassy state is triggered by an underlying thermodynamic ( equilibrium ) transition @xcite ; in that view , an `` ideal '' glass transition is believed to occur at the so - called kauzmann temperature , @xmath5 . at @xmath5 , it is proposed that only one minimum - energy basin of attraction is accessible to the system . one of the first arguments of this type is due to gibbs and dimarzio @xcite , but more recent studies using replica methods have yielded evidence in support of such a transition in lennard - jones glass formers @xcite . these observations have been called into question by experimental data and recent results of simulations of polydisperse hard - core disks , which have failed to detect any evidence of a thermodynamic transition up to extremely high packing fractions @xcite . one of the questions that arises is therefore whether the discrepancies between the reported simulated behavior of hard - disk and soft - sphere systems is due to fundamental differences in the models , or whether they are a consequence of inappropriate sampling at low temperatures and high densities . different , alternative theoretical considerations have attempted to establish a connection between glass transition phenomena and the rapid increase in relaxation times that arises in the vicinity of a theoretical critical temperature ( the so - called `` mode - coupling '' temperature , @xmath6 ) , thereby giving rise to a `` kinetic '' or `` dynamic '' transition @xcite . in recent years , both viewpoints have received some support from molecular simulations . many of these simulations have been conducted in the context of models introduced by stillinger and weber and by kob and andersen @xcite ; such models have been employed in a number of studies that have helped shape our current views about the glass transition @xcite . in its simplest ( `` idealized '' ) version , firstly analyzed in the `` schematic '' approach by bengtzelius et al . @xcite and independently by leutheusser @xcite , the mct predicts a transition from a high temperature liquid ( `` ergodic '' ) state to a low temperature arrested ( `` nonergodic '' ) state at a critical temperature @xmath0 . including transversale currents as additional hydrodynamic variables , the full mct shows no longer a sharp transition at @xmath0 but all structural correlations decay in a final @xmath7-process @xcite . similar effects are expected from inclusion of thermally activated matter transport , that means diffusion in the arrested state @xcite . in the full mct , the remainders of the transition and the value of @xmath0 have to be evaluated , e.g. , from the approach of the undercooled melt towards the idealized arrested state , either by analyzing the time and temperature dependence in the @xmath8-regime of the structural fluctuation dynamics @xcite or by evaluating the temperature dependence of the so - called @xmath3-parameter @xcite . there are further posibilities to estimates @xmath0 , e.g. , from the temperature dependence of the diffusion coefficients or the relaxation time of the final @xmath7-decay in the melt , as these quantities for @xmath9 display a critical behaviour @xmath10 . however , only crude estimates of @xmath0 can be obtained from these quantities , since near @xmath0 the critical behaviour is masked by the effects of transversale currents and thermally activated matter transport , as mentioned above . on the other hand , as emphasized and applied in @xcite , the value of @xmath0 predicted by the idealized mct can be calculated once the partial structure factors of the system and their temperature dependence are sufficiently well known . besides temperature and particle concentration , the partial structure factors are the only significant quantities which enter the equations of the so - called nonergodicity parameters of the system . the latter vanish identically for temperatures above @xmath0 and their calculation thus allows a rather precise determination of the critical temperature predicted by the idealized theory . at this stage it is tempting to consider how well the estimates of @xmath0 from different approaches fit together and whether the @xmath0 estimate from the nonergodicity parameters of the idealized mct compares to the values from the full mct . regarding this , we here investigate a molecular dynamics ( md ) simulation model adapted to the glass - forming ni@xmath1zr@xmath2 transition metal system . the ni@xmath11zr@xmath12-system is well studied by experiments @xcite and by md - simulations @xcite , as it is a rather interesting system whose components are important constituents of a number of multi - component massive metallic glasses . in the present contribution we consider , in particular , the @xmath13 composition and concentrate on the determination of @xmath0 from evaluating and analyzing the nonergodicity parameter , the @xmath14-parameter in the ergodic regime , and the diffusion coefficients . our paper is organized as follows : in section ii , we present the model and give some details of the computations . section iii . gives a brief discussion of some aspects of the mode coupling theory as used here . results of our md - simulations and their analysis are then presented and discussed in section iv . the present simulations are carried out as state - of - the - art isothermal - isobaric ( @xmath15 ) calculations . the newtonian equations of @xmath16 648 atoms ( 518 ni and 130 zr ) are numerically integrated by a fifth order predictor - corrector algorithm with time step @xmath17 = 2.5 10@xmath18s in a cubic volume with periodic boundary conditions and variable box length l. with regard to the electron theoretical description of the interatomic potentials in transition metal alloys by hausleitner and hafner @xcite , we model the interatomic couplings as in @xcite by a volume dependent electron - gas term @xmath19 and pair potentials @xmath20 adapted to the equilibrium distance , depth , width , and zero of the hausleitner - hafner potentials @xcite for ni@xmath1zr@xmath2 @xcite . for this model , simulations were started through heating a starting configuration up to 2000 k which leads to a homogeneous liquid state . the system then is cooled continuously to various annealing temperatures with cooling rate @xmath21 = 1.5 10@xmath22 k / s . afterwards the obtained configurations at various annealing temperatures ( here 1500 - 600 k ) are relaxed by carrying out additional isothermal annealing runs . finally the time evolution of these relaxed configurations is modelled and analyzed . more details of the simulations are given in @xcite . in this section we provide some basic formulae that permit calculation of @xmath0 and the nonergodicity parameters @xmath23 for our system . a more detailed presentation may be found in refs . . the central object of the mct are the partial intermediate scattering functions which are defined for a binary system by @xcite @xmath24)\right\rangle \quad , \label{t.1}\end{aligned}\ ] ] where @xmath25 is a fourier component of the microscopic density of species @xmath26 . the diagonal terms @xmath27 are denoted as the incoherent intermediate scattering function @xmath28)\right\rangle \quad . \label{t.2}\ ] ] the normalized partial- and incoherent intermediate scattering functions are given by @xmath29 where the @xmath30 are the partial static structure factors . the basic equations of the mct are the set of nonlinear matrix integrodifferential equations @xmath31 where @xmath32 is the @xmath33 matrix consisting of the partial intermediate scattering functions @xmath34 , and the frequency matrix @xmath35 is given by @xmath36_{ij}=q^2k_b t ( x_i / m_i)\sum_{k}\delta_{ik } \left[{\bf s}^{-1}(q)\right]_{kj}\quad . \label{t.6}\ ] ] @xmath37 denotes the @xmath33 matrix of the partial structure factors @xmath38 , @xmath39 and @xmath40 means the atomic mass of the species @xmath26 . the mct for the idealized glass transition predicts @xcite that the memory kern @xmath41 can be expressed at long times by @xmath42 where @xmath43 is the particle density and the vertex @xmath44 is given by @xmath45 and the matrix of the direct correlation function is defined by @xmath46_{ij } \quad . \label{t.9}\ ] ] the equation of motion for @xmath47 has a similar form as eq.([t.5 ] ) , but the memory function for the incoherent intermediate scattering function is given by @xmath48 @xmath49 in order to characterize the long time behaviour of the intermediate scattering function , the nonergodicity parameters @xmath50 are introduced as @xmath51 these parameters are the solution of eqs . ( [ t.5])-([t.9 ] ) at long times . the meaning of these parameters is the following : if @xmath52 , then the system is in a liquid state with density fluctuation correlations decaying at long times . if @xmath53 , the system is in an arrested , nonergodic state , where density fluctuation correlations are stable for all times . in order to compute @xmath54 , one can use the following iterative procedure @xcite : @xmath55(q ) \cdot { \bf s } ( q)}{{\bf z } } \nonumber \\ & & + \frac{q^{-2}|{\bf s}(q)| |{\bf n}[{\bf f}^{(l)},{\bf f}^{(l)}](q)| { \bf s}(q)}{\bf z } \quad , \label{t.13}\end{aligned}\ ] ] @xmath56(q ) ) \nonumber \\ & & + q^{-2}| { \bf s}(q)| | { \bf n}[{\bf f}^{(l)},{\bf f}^{(l)}](q)| \nonumber \quad,\end{aligned}\ ] ] where the matrix @xmath57 is given by @xmath58 this iterative procedure , indeed , has two type of solutions , nontrivial ones with @xmath59 and trivial solutions @xmath60 . the incoherent nonergodicity parameter @xmath61 can be evaluated by the following iterative procedure : @xmath62(q ) \quad . \label{t.15}\ ] ] as indicated by eq.([t.15 ] ) , computation of the incoherent nonergodicity parameter @xmath63 demands that the coherent nonergodicity parameters are determined in advance . beyond the details of the mct , equations of motion like ( [ t.5 ] ) can be derived for the correlation functions under rather general assumptions within the lanczos recursion scheme @xcite resp . the mori - zwanzig formalism @xcite . the approach demands that the time dependence of fluctuations a , b , ... is governed by a time evolution operator like the liouvillian and that for two fluctuating quantitites a scalar products ( b , a ) with the meaning of a correlation function can be defined . in case of a tagged particle , this leads for @xmath65 to the exact equation @xmath66 with memory kernel @xmath67 in terms of a continued fraction . within @xmath67 are hidden all the details of the time evolution of @xmath65 . as proposed and applied in @xcite , instead of calculating @xmath67 from the time evolution operator as a continued fraction , it can be evaluated in closed forms once @xmath65 is known , e.g. , from experiments or md - simulations . this can be demonstrated by introduction of @xmath68 with @xmath69 the laplace transform of @xmath70 , and @xmath71 eq.([g.1 ] ) then leads to @xmath72 ^{2}+\left [ \omega \phi _ { c}(\omega ) \right ] ^{2 } } \quad . \label{g.5}\ ] ] on the time axis , @xmath73 is given by @xmath74 adopting some arguments from the schematic mct , eq.([g.1 ] ) allows asymptotically finite correlations @xmath75 , that means an arrested state , if @xmath76 remains finite where the relationship holds @xmath77 in order to characterize the undercooled melt and its transition into the glassy state , we introduced in @xcite the function @xmath78 according to ( [ g.7 ] ) , @xmath79 has the property that @xmath80 in the arrested , nonergodic state . on the other hand , if @xmath81 there is no arrested solution and the correlations @xmath65 decay to zero for @xmath82 , that means , the system is in the liquid state . from that we proposed @xcite to use the value of @xmath3 as a relative measure how much the system has approached the arrested state and to use the temperature dependence of @xmath14 in the liquid state as an indication how the system approaches this state . first we show the results of our simulations concerning the static properties of the system in terms of the partial structure factors @xmath38 and partial correlation functions @xmath83 . to compute the partial structure factors @xmath38 for a binary system we use the following definition @xcite @xmath84 where @xmath85 are the partial pair correlation functions . the md simulations yield a periodic repetition of the atomic distributions with periodicity length @xmath86 . truncation of the fourier integral in eq.([e.5 ] ) leads to an oscillatory behavior of the partial structure factors at small @xmath87 . in order to reduce the effects of this truncation , we compute from eq.([e.5a ] ) the partial pair correlation functions for distance @xmath88 up to @xmath89 . for numerical evaluation of eq.([e.5 ] ) , a gaussian type damping term is included @xmath90 with @xmath91 . fig.[fig1]- fig.[fig2a ] shows the partial structure factors @xmath38 versus @xmath87 for all temperatures investigated . the figure indicates that the shape of @xmath38 depends weakly on temperature only and that , in particular , the positions of the first maximum and the first minimum in @xmath38 are more or less temperature independent . to investigate the dynamical properties of the system , we have calculated the incoherent scattering function @xmath92 and the coherent scattering function @xmath34 as defined in equations ( [ t.1 ] ) and ( [ t.2 ] ) . fig.[fig2b ] and fig.[fig3a ] presents the normalized incoherent intermediate scattering functions @xmath65 of both species evaluated from our md data for wave vector @xmath93=@xmath94 with n = 9 , that means @xmath95 nm @xmath96 . from the figure we see that @xmath65 of both species shows at intermediate temperatures a structural relaxation in three succesive steps as predicted by the idealized schematic mct @xcite . the first step is a fast initial decay on the time scale of the vibrations of atoms ( @xmath97 ps ) . this step is characterized by the mct only globaly . the second step is the @xmath98-relaxation regime . in the early @xmath8-regime the correlator should decrease according to @xmath99 and in the late @xmath8-relaxation regime , which appears only in the melt , according the von schweidler law @xmath100 between them a wide plateau is found near the critical temperature @xmath101 . in the melt , the @xmath7-relaxation takes place as the last decay step after the von schweidler - law . it can be described by the kohlrausch - williams - watts ( kww ) law @xmath102 where the relaxation time @xmath103 near the glass transition shifts drastically to longer times . the inverse power - law decay for the early @xmath8-regime @xmath104 is not seen in our data . this seems to be due to the fact that in our system the power - law decay is dressed by the atomic vibrations ( @xcite and references therein ) . according to our md - results , @xmath65 decays to zero for longer times at all temperatures investigated . this is in agreement with the full mct . including transversal currents as additional hydrodynamic variables , the full mct @xcite comes to the conclusion that all structural correlations decay in the final @xmath7-process , independent of temperature . similar effects are expected from inclusion of thermally activated matter transport , that means diffusion in the arrested state . at @xmath105 900 k - 700 k , the @xmath65 drop rather sharply at large @xmath106 . this reflects aging effects which take place , if a system is in a transient , non - steady state @xcite . such a behaviour indicates relaxations of the system on the time scale of the measuring time of the correlations . the nonergodicity parameters are defined by eq.([t.12 ] ) as a non - vanishing asymptotic solution of the mct eq.([t.5 ] ) . fig . [ fig3b ] presents the estimated @xmath87-dependent nonergodicity parameters from the coherent and incoherent scattering functions of ni and zr at t=1005 k. in order to compute the nonergodicity parameters @xmath23 analytically , we followed for our binary system the self - consistent method as formulated by nauroth and kob @xcite and as sketched in section iii.a . input data for our iterative determination of @xmath107 are the temperature dependent partial structure factors @xmath38 from the previous subsection . the iteration is started by arbitrarily setting @xmath108 , @xmath109 , @xmath110 . for @xmath111 k we always obtain the trivial solution @xmath112 while at t = 1000 k and below we get stable non - vanishing @xmath113 . the stability of the non - vanishing solutions was tested for more than 3000 iteration steps . from this results we expect that @xmath0 for our system lies between 1000 and 1100 k. to estimate @xmath0 more precisely , we interpolated @xmath38 from our md data for temperatures between 1000 and 1100 k by use of the algorithm of press et.al . we observe that at @xmath114 k a non - trivial solution of @xmath23 can be found , but not at @xmath115 k and above . it means that the critical temperature @xmath0 for our system is around 1005 k. the non - trivial solutions @xmath23 for this temperature shall be denoted the critical nonergodicty parameters @xmath116 . they are included in fig . [ fig3b ] . by use of the critical nonergodicity parameters @xmath116 , the computational procedure was run to determine the critical nonergodicity parameters @xmath117 for the incoherent scattering functions at t = 1005 k . [ fig3b ] also presents our results for the so calculated @xmath117 . here we present our results about the @xmath118-function @xcite described in section iii.b . the memory functions @xmath119 are evaluated from the md data for @xmath120 by fourier transformation along the positive time axis . for completeness , also @xmath121 and 800 k data are included where the corresponding @xmath120 are extrapolated to longer times by use of an kww approximation . [ fig4a ] and fig . [ fig4b ] show the thus deduced @xmath119 for @xmath122 nm@xmath96 . regarding their qualitative features , the obtained @xmath119 are in full agreement with the results in @xcite for the ni@xmath123zr@xmath123 system . a particular interesting detail is the fact that there exists a minimum in @xmath119 for both species , ni and zr , at all investigated temperatures around a time of 0.1 ps . below this time , @xmath120 reflects the vibrational dynamics of the atoms . above this value , the escape from the local cages takes place in the melt and the @xmath8-regime dynamics are developed . apparently , the minimum is related to this crossover . in fig . [ fig5 ] and fig . [ fig5a ] we display @xmath124 , that means @xmath119 versus @xmath120 . in this figure we again find the features already described for ni@xmath123zr@xmath123 in @xcite . according to the plot , there exist ( @xmath87-dependent ) limiting values @xmath125 so that @xmath119 for @xmath126 is close to an universal behavior , while for @xmath127 marked deviations are seen . @xmath125 significantly decreases with increasing temperature . it is tempting to identify @xmath119 below @xmath125 with the polynomial form for @xmath119 assumed in the schematic version of the mct @xcite . in fig . [ fig5 ] and fig . [ fig5a ] , the polynomial obtained by fitting the 1000 k data below @xmath125 is included by a dashed line , extrapolating it over the whole @xmath128-range . by use of the calculated memory functions , we can evaluate the @xmath118 , eq.([g.8 ] ) . in fig.[fig6 ] and fig . [ fig7 ] this quantity is presented versus the corresponding value of @xmath120 and denoted as @xmath129 . for all the investigated temperatures , @xmath129 has a maximum @xmath130 at an intermediate value of @xmath128 . in the high temperature regime , the values of @xmath130 move with decreasing temperature towards the limiting value 1 . this is , in particular , visible in fig . [ fig8 ] where we present @xmath130 as function of temperature for both species , ni and zr , and wave - vectors @xmath95 nm@xmath96 . at temperatures above 1000 k , the @xmath3-values increase approximately linear towards 1 with decreasing temperatures . below 1000 k , they remain close below the limiting value of 1 , a behavior denoted in @xcite as a balancing on the borderline between the arrested and the non - arrested state due to thermally induced matter transport by diffusion in the arrested state at the present high temperatures . linear fit of the @xmath3-values for ni above 950 k and for zr above 1000 k predicts a crossover temperature @xmath131 from liquid ( @xmath132 ) to the quasi - arrested ( @xmath133 ) behavior around 970 k from the ni data and around 1020 k from the zr data . we here identify this crossover temperature with the value of @xmath0 as visible in the ergodic , liquid regime and estimate it by the mean value from the ni- and zr - subsystems , that means by @xmath134 k. while in @xcite for the ni@xmath123zr@xmath123 melt a @xmath0-value of 1120 k was estimated from @xmath14 , the value for the present composition is lower by about 120 k. a significant composition dependence of @xmath0 is expected according to the results of md simulation for the closely related co@xmath11zr@xmath12 system @xcite . over the whole @xmath135-range , @xmath0 was found to vary between 1170 and 650 k in co@xmath11zr@xmath12 , with @xmath0(@xmath136 ) @xmath137 800 k. regarding this , the present data for the ni@xmath11zr@xmath12 system reflect a rather weak @xmath0 variation . from the simulated atomic motions in the computer experiments , the diffusion coefficients of the ni and zr species can be determined as the slope of the atomic mean square displacements in the asymptotic long - time limit @xmath138 fig . [ fig9 ] shows the thus calculated diffusion coefficients of our ni@xmath1zr@xmath2 model for the temperature range between 600 and 2000 k. at temperatures above approximately 1000 k , the diffusion coefficients for both species run parallel to each other in the arrhenius plot , indicating a fixed ratio @xmath139 in this temperature regime . at lower temperatures , the ni atoms have a lower mobility than the zr atoms , yielding around 800 k a value of about 10 for @xmath140 . that means , here the zr atoms carry out a rather rapid motion within a relative immobile ni matrix . according to the mct , above @xmath0 the diffusion coefficients follow a critical power law @xmath141 with non - universal exponent @xmath142 @xcite . in order to estimate @xmath0 from this relationship , we have adapted the critical power law by a least mean squares fit to the simulated diffusion data for 1000 k and above . according to this fit , the system has a critical temperature of about 850 - 900 k. similar results for the temperature dependence of the diffusion coefficients have been found in md simulations for other metallic glass forming systems , e.g. , for ni@xmath123zr@xmath123 @xcite , for ni@xmath143zr@xmath12 @xcite , cu@xmath144zr@xmath145 @xcite , or ni@xmath146b@xmath147 @xcite . in all cases , like here , a break is observed in the arrhenius slope . in the mentioned zr - systems , this break is related to a change of the atomic dynamics around @xmath0 whereas for ni@xmath146b@xmath147 system it is ascribed to @xmath148 . as in @xcite @xmath0 and @xmath148 apparently fall together , there is no serious conflict between the obervations . the present contribution reports results from md simulations of a ni@xmath1zr@xmath2 computer model . the model is based on the electron theoretical description of the interatomic potentials for transition metal alloys by hausleitner and hafner @xcite . there are no parameters in the model adapted to the experiments . there is close agreement between the @xmath0 values estimated from the dynamics in the undercooled melt when approaching @xmath0 from the high temperature side . the values are @xmath149 k from the @xmath3-parameters , and @xmath150 k from the diffusion coefficients . as discussed in @xcite , the @xmath0-estimates from the diffusion coefficients seem to depend on the upper limit of the temperature region taken into account in the fit procedure , where an increase in the upper limit increases the estimated @xmath0 . accordingly , there is evidence that the present value of 950 k may underestimate the true @xmath0 by about 10 to 50 k , as it based on an upper limit of 2000 k only . taking this into account , the present estimates from the melt seem to lead to a @xmath0 value around 1000 k. the @xmath0 from the nonergodicity parameters describe the approach of the system towards @xmath0 from the low temperature side . they predict a @xmath0 value of 1005 k. this value is clearly outside the range of our @xmath0 estimates from the high temperature , ergodic melt . we consider this as a significant deviation which , however , is much smaller than the factor of two found in the modelling of a lennard - jones system @xcite . the here observed deviation between the @xmath0 estimates from the ergodic and the so - called nonergodic side reconfirm the finding from the soft spheres model@xcite of an agreement within some 10 @xmath4 between the different @xmath0-estimates .

we use molecular dynamics computer simulations to investigate a critical temperature @xmath0 for a dynamical glass transition as proposed by the mode - coupling theory ( mct ) of dense liquids in a glass forming ni@xmath1zr@xmath2-system . the critical temperature @xmath0 are analyzed from different quantities and checked the consistency of the estimated values , i.e. from ( i ) the non - vanishing nonergodicity parameters as asymptotic solutions of the mct equations in the arrested state , ( ii ) the @xmath3-parameters describing the approach of the melt towards the arrested state on the ergodic side , ( iii ) the diffusion coefficients in the melt . the resulting @xmath0 values are found to agree within about 10 @xmath4 .

arxiv

[ secintro ] dyadic data are common in psychosocial and behavioral studies [ @xcite ] . many social phenomena , such as dating and marital relationships , are interpersonal by definition , and , as a result , related observations do not refer to a single person but rather to both persons involved in the dyadic relationship . members of dyads often influence each other s cognitions , emotions and behaviors , which leads to interdependence in a relationship . for example , a husband s ( or wife s ) drinking behavior may lead to lowered marital satisfaction for the wife ( or husband ) . a consequence of interdependence is that observations of the two individuals are correlated . for example , the marital satisfaction scores of husbands and wives tend to be positively correlated . one of the primary objectives of relationship research is to understand the interdependence of individuals within dyads and how the attributes and behaviors of one dyad member impact the outcome of the other dyad member . in many studies , dyadic outcomes are measured over time , resulting in longitudinal dyadic data . repeatedly measuring dyads brings in two complications . first , in addition to the within - dyad correlation , repeated measures on each subject are also correlated , that is , within - subject correlation . when analyzing longitudinal dyadic data , it is important to account for these two types of correlations simultaneously ; otherwise , the analysis results may be invalid . the second complication is that longitudinal dyadic data are prone to the missing data problem caused by dropout , whereby subjects are lost to follow - up and their responses are not observed thereafter . in psychosocial dyadic studies , the dropouts are often nonignorable or informative in the sense that the dropout depends on missing values . in the presence of the nonignorable dropouts , conventional statistical methods may be invalid and lead to severely biased estimates [ @xcite ] . there is extensive literature on statistical modeling of nonignorable dropouts in longitudinal studies . based on different factorizations of the likelihood of the outcome process and the dropout process , @xcite identified two broad classes of likelihood - based nonignorable models : selection models [ @xcite ; @xcite ; follman and wu ( @xcite ) ; @xcite ] and pattern mixture models [ @xcite ; little ( @xcite , @xcite ) ; hogan and laird ( @xcite ) ; @xcite ; @xcite ] . other likelihood - based approaches that do not directly belong to this classification have also been proposed in the literature , for example , the mixed - effects hybrid model by @xcite and a class of nonignorable models by @xcite . another general approach for dealing with nonignorable dropouts is based on estimation equations and includes @xcite , @xcite , @xcite and @xcite . recent reviews of methods handling nonignorable dropouts in longitudinal data can be found in @xcite , @xcite , little ( @xcite ) , @xcite and @xcite . in spite of the rich body of literature noted above , to the best of our knowledge , the nonignorable dropout problem has not been addressed in the context of longitudinal dyadic data . the interdependence structure within dyads brings new challenges to this missing data problem . for example , within dyads , one member s outcome often depends on his / her covariates , as well as the other member s outcome and covariates . thus , the dropout of the other member in the dyad causes not only a missing ( outcome ) data problem for that member , but also a missing ( covariate ) data problem for the member who remains in the study.=-1 we propose a fully bayesian approach to deal with longitudinal dyadic data with nonignorable dropouts based on a selection model . specifically , we model each subject s longitudinal measurement process using a transition model , which includes both the patient s and spouse s characteristics as covariates in order to capture the interdependence between patients and their spouses . we account for the within - dyad correlation by introducing dyad - specific random effects into the transition model . to accommodate the nonignorable dropouts , we take the selection model approach by directly modeling the relationship between the dropout process and missing outcomes using a discrete time survival model . the remainder of the article is organized as follows . in section [ sec2 ] we describe our motivating data collected from a longitudinal dyadic breast cancer study . in section [ sec3 ] we propose a bayesian selection - model - based approach for longitudinal dyad data with informative nonresponse , and provide estimation procedures using a gibbs sampler in section [ sec4 ] . in section [ sec5 ] we present simulation studies to evaluate the performance of the proposed method . in section [ sec6 ] we illustrate our method by analyzing a breast cancer data set and we provide conclusions in section [ sec7 ] . our research is motivated by a single - arm dyadic study focusing on physiological and psychosocial aspects of pain among patients with breast cancer and their spouses [ @xcite ] . for individuals with breast cancer , spouses are most commonly reported as being the primary sources of support [ @xcite ] , and spousal support is associated with lower emotional distress and depressive symptoms in these patients [ @xcite ] . one specific aim of the study is to characterize the depression experience due to metastatic breast cancer from both patients and spouses perspectives , and examine the dyadic interaction and interdependence of patients and spouses over time regarding their depression . the results will be used to guide the design of an efficient prevention program to decrease depression among patients . for example , conventional prevention programs typically apply interventions to patients directly . however , if we find that the patient s depression depends on both her own and spouse s previous depression history and chronic pain , when designing a prevention program to improve the depression management and pain relief , we may achieve better outcomes by targeting both patients and spouses simultaneously rather than targeting patients only . in this study , female patients who had initiated metastatic breast cancer treatment were approached by the project staff . patients meeting the eligibility criteria ( e.g. , speak english , experience pain due to the breast cancer , having a male spouse or significant other , be able to carry on pre - disease performance , be able to provide informed consent ) were asked to participate the study on a voluntary basis . the participation of the study would not affect their treatment in any way . depression in patients and spouse was measured at three time points ( baseline , 3 months and 6 months ) using the center for epidemiologic studies depression scale ( cesd ) questionnaires . however , a substantial number of dropouts occurred . baseline cesd measurements were collected from 191 couples ; however , at 3 months , 101 couples ( 105 patients and 107 spouses ) completed questionnaires , and at 6 months , 73 couples ( 76 patients and 79 spouses ) completed questionnaires . the missingness of the cesd measurements is likely related to the current depression levels of the patients or spouses , thus an nonignorable missing data mechanism is assumed for this study . consequently , it is important to account for the nonignorable dropouts in this data analysis ; otherwise , the results may be biased , as we will show in section [ sec6 ] . consider a longitudinal dyadic study designed to collect @xmath0 repeated measurements of a response @xmath1 and a vector of covariates @xmath2 for each of @xmath3 dyads . let @xmath4 , @xmath5 and @xmath6 denote the outcome , @xmath7 covariate vector and outcome history , respectively , for the member @xmath8 of dyad @xmath9 at the @xmath10th measurement time with @xmath11 . we assume that @xmath2 is fully observed ( e.g. , is external or fixed by study design ) , but @xmath1 is subject to missingness due to dropout . the random variable @xmath12 , taking values from @xmath13 to @xmath14 , indicates the time the member @xmath8 of the @xmath9th dyad drops out , where @xmath15 if the subject completes the study , and @xmath16 if the subject drops out between the @xmath17th and @xmath10th measurement time , that is , @xmath18 are observed and @xmath19 are missing . we assume at least 1 observation for each subject , as subjects without any observations have no information and are often excluded from the analysis . when modeling longitudinal dyadic data , we need to consider two types of correlations : the within - subject correlation due to repeated measures on a subject , and the within - dyad correlation due to the dyadic structure . we account for the first type of correlation by a transition model , and the second type of correlation by dyad - specific random effects @xmath20 , as follows : @xmath21 regression parameters in this random - effects transition model have intuitive interpretations similar to those of the actor partner interdependence model , a conceptual framework proposed by @xcite to study dyadic relationships in the social sciences and behavior research fields . specifically , @xmath22 and @xmath23 represent the `` actor '' effects of the patient , which indicate how the covariates and the outcome history of the patient ( i.e. , @xmath24 and @xmath25 ) affect her own current outcome , whereas @xmath26 and @xmath27 represent the `` partner '' effects for the patient , which indicate how the covariates and the outcome history of the spouse ( i.e. , @xmath28 and @xmath29 ) affect the outcome of the patient . similarly , @xmath30 and @xmath31 characterize the actor effects and @xmath32 and @xmath33 characterize the partner effects for the spouse of the patient . estimates of the actor and partner effects provide important information about the interdependence within dyads . we assume that residuals @xmath34 and @xmath35 are independent and follow normal distributions @xmath36 and @xmath37 , respectively ; and @xmath34 and @xmath35 are independent of random effects @xmath20 s . the parameters @xmath38 and @xmath39 are intercepts for the patients and spouses , respectively . in many situations , the conditional distribution of @xmath4 given @xmath40 and @xmath5 depends only on the @xmath41 prior outcomes @xmath42 and @xmath5 . if this is the case , we obtain the so - called @xmath41th - order transition model , a type of transition model that is most useful in practice [ @xcite ] . the choice of the model order @xmath41 depends on subject matters . in many applications , it is often reasonable to set @xmath43 when the current outcome depends on only the last observed previous outcome , leading to commonly used markov models . the likelihood ratio test can be used to assess whether a specific value of @xmath41 is appropriate [ @xcite ] . auto - correlation analysis of the outcome history also can provide useful information to determine the value of @xmath41 [ @xcite ; @xcite ] . define @xmath44 and @xmath45 for @xmath46 . given \{@xmath47 and the random effect @xmath20 , the joint log likelihood of @xmath48 for the @xmath9th dyad under the @xmath41th - order ( random - effects ) transition model is given by @xmath49 where @xmath50 is the likelihood corresponding to model ( [ transition ] ) , and @xmath51 is assumed free of @xmath52 , for @xmath46 . an important feature of model ( [ transition ] ) that distinguishes it from the standard transition model is that the current value of the outcome @xmath1 depends on not only the subject s outcome history , but also the spouse s outcome history . such a `` partner '' effect is of particular interest in dyadic studies because it reflects the interdependence between the patients and spouses . this interdependence within dyads also makes the missing data problem more challenging . consider a dyad consisting of subjects @xmath53 and @xmath54 and that @xmath54 drops out prematurely . because the outcome history of @xmath54 is used as a covariate in the transition model of @xmath53 , when @xmath54 drops out , we face not only the missing outcome ( for @xmath54 ) but also missing covariates ( for @xmath53 ) . we address this dual missing data problem using the data augmentation approach , as described in section [ sec4 ] . to account for nonignorable dropouts , we employ the discrete time survival model [ @xcite ] to jointly model the missing data mechanism . specifically , we assume that the distribution of @xmath12 depends on both the past history of the longitudinal process and the current outcome @xmath4 , but not on future observations . define the discrete hazard rate @xmath55 . it follows that the probability of dropout for the member @xmath8 in the @xmath9th dyad is given by @xmath56 we specify the discrete hazard rate @xmath57 using the logistic regression model : @xmath58 where @xmath59 is the random effect accounting for the within - dyadic correlation , and @xmath60 and @xmath61 are unknown parameters . in this dropout model , we assume that , conditioning on the random effects , a subject s covariates , past history and current ( unobserved ) outcome , the dropout probability of this subject is independent of the characteristics and outcomes of the other member in the dyad . the spouse may indirectly affect the dropout rate of the patient through influencing the patient s depression status ; however , when conditional on the patient s depression score , the dropout of the patient does not depend on her spouse s depression score . in practice , we often expect that , given @xmath4 and @xmath62 , the conditional dependence of @xmath12 on @xmath63 will be negligible because , temporally , the patient s ( current ) decision of dropout is mostly driven by his ( or her ) current and the most recent outcome statuses . using the breast cancer study as an example , we do not expect that the early history of depression plays an important role for the patient s current decision of dropout ; instead , the patient drops out typically because she is currently experiencing or most recently experienced high depression . the early history may influence the dropout but mainly through its effects on the current depression status . once conditioning on the current and the most recent depression statuses , the influence from the early history is essentially negligible . thus , we use a simpler form of the discrete hazard model @xmath64 [ secestimation ] under the bayesian paradigm , we assign the following vague priors to the unknown parameters and fit the proposed model using a gibbs sampler : @xmath65 where @xmath66 denote an inverse gamma distribution with a shape parameter @xmath67 and a scale parameter @xmath68 . we set @xmath67 and @xmath68 at smaller values , such as 0.1 , so that the data dominate the prior information . let @xmath69 and @xmath70 denote the observed and missing part of the data , respectively . considering the @xmath8th iteration of the gibbs sampler , the first step of the iteration is `` data augmentation '' [ @xcite ] , in which the missing data @xmath70 are generated from their full conditional distributions . without loss of generality , suppose for the @xmath9th dyad , member 2 drops out of the study no later than member 1 , that is , @xmath71 , and let @xmath72 . assuming a first - order ( @xmath73 ) transition model ( or markov model ) and letting @xmath74 denote a generic symbol that represents the values of all other model parameters , the data augmentation consists of the following 3 steps : for @xmath75 , draw @xmath76 from the conditional distribution @xmath77 where @xmath78 draw @xmath79 from the conditional distribution @xmath80 draw @xmath81 from the conditional distribution @xmath82 now , with the augmented complete data @xmath83 , the parameters are drawn alternatively as follows : for @xmath84 , draw random effects @xmath20 from the conditional distribution @xmath85 where @xmath86 draw @xmath87 from the conditional distribution @xmath88 where @xmath89 draw @xmath90 from the conditional distribution @xmath91 draw @xmath92 from the normal distribution @xmath93 where @xmath94 and @xmath95 similarly , draw @xmath96 from the conditional distribution @xmath97 where @xmath98 and @xmath99 are defined in a similar way to @xmath100 and @xmath101 draw @xmath102 and @xmath103 from the conditional distributions @xmath104 draw random effects @xmath59 from the conditional distribution @xmath105 draw @xmath106 from the conditional distribution @xmath107 [ secsimu ] we conducted two simulation studies ( a and b ) . simulation a consists of 500 data sets , each with 200 dyads and three repeated measures . for the @xmath9th dyad , we generated the first measurements , @xmath108 and @xmath109 , from normal distributions @xmath110 and @xmath111 , respectively , and generated the second and third measurements based on the following random - effects transition model : @xmath112 where @xmath113 , @xmath114 , @xmath115 , and covariates @xmath116 and @xmath117 were generated independently from @xmath118 . we assumed that the baseline ( first ) measurements @xmath108 and @xmath109 were observed for all subjects , and the hazard of dropout at the second and third measurement times depended on the current and last observed values of @xmath1 , that is , @xmath119 under this dropout model , on average , 24% ( 12% of member 1 and 13% of member 2 ) of the dyads dropped out at the second time point and 45% ( 26% of member 1 and 30% of member 2 ) dropped out at the third measurement time . we applied the proposed method to the simulated data sets . we used 1,000 iterations to burn in and made inference based on 10,000 posterior draws . for comparison purposes , we also conducted complete - case and available - case analyses . the complete - case analysis was based on the data from dyads who completed the follow - up , and the available - case analysis was based on all observed data ( without considering the missing data mechanism ) . [ tab1 ] @ld2.2ccd2.2ccd2.2cc@ & & & + & & & + * parameter * & & * se * & * coverage * & & * se * & * coverage * & & * se * & * coverage * + @xmath120 & -0.03 & 0.06 & 0.93 & -0.01 & 0.05 & 0.94 & -0.01 & 0.05 & 0.95 + @xmath121 & -0.06 & 0.05 & 0.81 & -0.03 & 0.04 & 0.88 & 0.07 & 0.04 & 0.96 + @xmath122 & -0.16 & 0.12 & 0.72 & -0.10 & 0.10 & 0.81 & 0.05 & 0.08 & 0.94 + @xmath123 & -0.17 & 0.12 & 0.75 & -0.10 & 0.10 & 0.78 & 0.02 & 0.09 & 0.97 + @xmath124 & -0.06 & 0.06 & 0.89 & -0.06 & 0.05 & 0.84 & 0.08 & 0.05 & 0.97 + @xmath125 & -0.04 & 0.05 & 0.87 & -0.00 & 0.04 & 0.95 & -0.04 & 0.06 & 0.96 + @xmath126 & -0.17 & 0.12 & 0.73 & -0.10 & 0.10 & 0.84&-0.01 & 0.12 & 0.95 + @xmath127 & -0.17 & 0.12 & 0.72 & -0.10 & 0.10 & 0.81 & 0.01 & 0.09 & 0.97 + table [ tbtransmodel1 ] shows the bias , standard error ( se ) and coverage rate of the 95% credible interval ( ci ) under different approaches . we can see that the proposed method substantially outperformed the complete - case and available - case analyses . our method yielded estimates with smaller bias and coverage rates close to the 95% nominal level . in contrast , the complete - case and available - case analyses often led to larger bias and poor coverage rates . for example , the bias of the estimate of @xmath122 under the complete - case and available - case analyses were @xmath128 and @xmath129 , respectively , substantially larger than that under the proposed method ( i.e. , 0.05 ) ; the coverage rate using the proposed method was about 94% , whereas those using the complete - case and available - case analyses were under 82% . the second simulation study ( simulation b ) was designed to evaluate the performance of the proposed method when the nonignorable missing data mechanism is misspecified , for example , data actually are missing at random ( mar ) . we generated the first measurements , @xmath108 and @xmath109 , from normal distribution @xmath130 independently , and generated the second and third measurements based on the same transition model as in simulation a. we assumed the hazard of dropout at the second and third measurement times depended on the previous ( observed ) value of @xmath1 quadratically , but not on the current ( missing ) value of @xmath1 , that is , @xmath131 under this mar dropout model , on average , 37% ( 21% of member 1 and 21% of member 2 ) of the dyads dropped out at the second time point and 27% ( 24% of member 1 and 33% of member 2 ) dropped out at the third measurement time . to fit the simulated data , we considered two nonignorable models with different specifications of the dropout ( or selection ) model . the first nonignorable model assumed a flexible dropout model @xmath132 which included the true dropout process ( [ simu2 ] ) as a specific case with @xmath133 ; and the second nonignorable model took a misspecified dropout model of the form @xmath134 table [ simulationc ] shows the bias , standard error and coverage rate of the 95% ci under different approaches . when the missing data were mar , the complete - case analysis was invalid and led to biased estimates and poor coverage rates because the complete cases are not random samples from the original population . in contrast , the available - case analysis yielded unbiased estimates and coverage rates close to the 95% nominal level . for the nonignorable models , the one with the flexible dropout model yielded unbiased estimates and reasonable coverage rates , whereas the model with the misspecified dropout model led to biased estimates ( e.g. , @xmath135 and @xmath136 ) and poor coverage rates . this result is not surprising because it is well known that selection models are sensitive to the misspecification of the dropout model [ @xcite ; @xcite ] . for nonignorable missing data , the difficulty is that we can not judge whether a specific dropout model is misspecified or not based solely on observed data because the observed data contain no information about the ( nonignorable ) missing data mechanism . to address this difficulty , one possible approach is to specify a flexible dropout model to decrease the chance of model misspecification . alternatively , maybe a better approach is to conduct sensitivity analysis to evaluate how the results vary when the dropout model varies . we will illustrate the latter approach in the next section . = [ tab2 ] @ld2.2cccccd2.2ccd2.2cc@ & & & & & & & & + & & & & + & & & & + * parameter * & & * se * & * coverage * & * bias * & * se * & * coverage * & & * se * & * coverage * & & * se * & * coverage * + @xmath120 & -0.06 & 0.08 & 0.86 & 0.00 & 0.06 & 0.95 & -0.01 & 0.06 & 0.95 & 0.14 & 0.06 & 0.78 + @xmath121 & -0.09 & 0.08 & 0.82 & 0.00 & 0.05 & 0.96 & 0.07 & 0.05 & 0.97 & -0.01 & 0.05 & 0.95 + @xmath122 & -0.11 & 0.14 & 0.84 & 0.00 & 0.10 & 0.95 & 0.04 & 0.08 & 0.96 & 0.03 & 0.08 & 0.94 + @xmath123 & -0.13 & 0.14 & 0.84 & 0.00 & 0.10 & 0.96 & 0.02 & 0.09 & 0.97 & 0.02 & 0.09 & 0.98 + @xmath124 & -0.07 & 0.08 & 0.87 & 0.00 & 0.06 & 0.96 & 0.02 & 0.06 & 0.97 & 0.12 & 0.06 & 0.79 + @xmath125 & -0.10 & 0.08 & 0.78 & 0.00 & 0.07 & 0.96 & 0.00 & 0.06 & 0.96 & -0.08 & 0.06 & 0.93 + @xmath126 & -0.14 & 0.14 & 0.82 & 0.00 & 0.10 & 0.96&0.01 & 0.12 & 0.94 & 0.01 & 0.12 & 0.95 + @xmath127 & -0.14 & 0.13 & 0.83 & 0.01 & 0.10 & 0.96 & 0.01 & 0.09 & 0.97 & 0.01 & 0.09 & 0.98 + [ secapplication ] we applied our method to the longitudinal metastatic breast cancer data . we used the first - order random - effects transition model for the longitudinal measurement process . in the model , we included 5 covariates : chronic pain measured by the multidimensional pain inventory ( mpi ) and previous cesd scores from both the patients and spouses , and the patient s stage of cancer . in the discrete - time dropout model , we included the subject s current and previous cesd scores , mpi measurements and the patient s stage of cancer as covariates . age was excluded from the models because its estimate was very close to 0 and not significant . we used 5,000 iterations to burn in and made inference based on 5,000 posterior draws . we also conducted the complete - case and available - case analyses for the purpose of comparison . [ tab3 ] @lcd2.14d2.14d2.14@ & & & & + & intercept & 2.53 ( -1.71 , 6.77 ) & 0.99 ( -2.55 , 4.52 ) & 5.10 ( 3.31 , 6.59 ) + & patient cesd & 0.43 ( 0.29 , 0.58 ) & 0.56 ( 0.44 , 0.68)&0.87 ( 0.80 , 0.93 ) + & spouse cesd & 0.07 ( -0.06 , 0.20 ) & 0.06 ( -0.06 , 0.17 ) & 0.14 ( 0.09 , 0.19 ) + & patient mpi & 0.94 ( 0.22 , 1.67 ) & 0.82 ( 0.21 , 1.43 ) & 1.24 ( 0.83 , 1.64 ) + & spouse mpi & 1.06 ( 0.29 , 1.82 ) & 0.90 ( 0.31 , 1.48 ) & 0.62 ( 0.40 , 0.84 ) + & cancer stage & 0.39 ( -0.81 , 1.60 ) & 0.59 ( -0.43 , 1.60 ) & 0.10 ( -0.47 , 0.66 ) + [ 4pt ] spouses & intercept & 3.68 ( -0.55 , 7.92 ) & 2.00 ( -1.63 , 5.64 ) & 8.16 ( 4.26 , 11.9 ) + & patient cesd & -0.05 ( -0.19 , 0.09 ) & 0.01 ( -0.11 , 0.13 ) & 0.68 ( 0.63 , 0.74 ) + & spouse cesd & 0.77 ( 0.64 , 0.90 ) & 0.78 ( 0.66 , 0.89 ) & 0.76 ( 0.71 , 0.81 ) + & patient mpi & 0.43 ( -0.29 , 1.15 ) & 0.27 ( -0.27 , 0.81 ) & 0.53 ( 0.33 , 0.73 ) + & spouse mpi & 0.55 ( -0.22 , 1.31 ) & 0.58 ( -0.04 , 1.20)&0.36 ( -0.64 , 1.15 ) + & cancer stage & -0.42 ( -1.63 , 0.79)&-0.21 ( -1.23 , 0.80)&-0.50 ( -0.92 , 0.09 ) + as shown in table [ tab3 ] , the proposed method suggests significant `` partner '' effects for the patients . specifically , the patient s depression increases with her spouse s mpi [ @xmath137 and 95% @xmath138 and previous cesd [ @xmath139 and 95% @xmath140 . in addition , there are also significant `` actor '' effects for the patients , that is , the patient s depression is positively correlated with her own mpi and previous cesd scores . for the spouses , we observed similar significant `` partner '' effects : the spouse s depression increases with the patient s mpi and previous cesd scores . the `` actor '' effects for the spouses are different from those for the patients . the spouse s depression correlates with his previous cesd scores but not the mpi level , whereas the patient s depression is related to both variables . based on these results , we can see that the patients and spouses are highly interdependent and influence each other s depression status . therefore , when designing a prevention program to reduce depression in patients , we may achieve better outcomes by targeting both patients and spouses simultaneously . as for the dropout process , the results in table [ tab4 ] suggest that the missing data for the patients are nonignorable because the probability of dropout is significantly associated with the patient s current ( missing ) cesd score . in contrast , the missing data for the spouse appears to be ignorable , as the probability of dropout does not depend on the spouse s current ( missing ) cesd score . for the variance components , the estimates of residuals variances for patients and spouses are @xmath141 [ 95% @xmath142 and @xmath143 [ 95% @xmath144 , respectively . the estimates of the variances for the random effects @xmath20 and @xmath59 are @xmath145 [ 95% @xmath146 and @xmath147 [ 95% @xmath148 , respectively , suggesting substantial variations across dyads . compared to the proposed approach , both the complete - case and available - case analyses fail to detect some `` partner '' effects . for example , for spouses , the complete - case and available - case analyses assert that the spouse s cesd is correlated with his own previous cesd scores only , whereas the proposed method suggested that the spouse s cesd is related not only to his own cesd but also to the patient s cesd and mpi level . in addition , for patients , the `` partner '' effect of the spouse s cesd is not significant under the complete - case and available - case analyses , but is significant under the proposed approach . these results suggest that ignoring the nonignorable dropouts could lead to a failure to detect important covariate effects . nonidentifiability is a common problem when modeling nonignorable missing data . in our approach , the observed data contain very limited information on the parameters that link the missing outcome with the dropout process , that is , @xmath149 and @xmath150 in the dropout model . the identification of these parameters is heavily driven by the untestable model assumptions [ @xcite ; @xcite ] . in this case , a sensible strategy is to perform a sensitivity analysis to examine how the inference changes with respect to the values of @xmath151 and @xmath152 [ daniels and hogan ( @xcite , @xcite ) ; @xcite ] . we conducted a bayesian sensitivity analysis by assuming informative normal prior distributions for @xmath149 and @xmath153 with a small variance of 0.01 and the mean fixed , successively , at various values . figures [ figsensitivitygibbs1 ] and [ figsensitivitygibbs2 ] show the parameter estimates of the measurement models when the prior means of @xmath149 and @xmath150 vary from @xmath154 to 3 . in general , the estimates were quite stable , except that the estimate of cancer stage in the measurement model of patient ( figure [ figsensitivitygibbs1 ] ) and the estimate of spouse s mpi in the measurement model of spouse ( figure [ figsensitivitygibbs2 ] ) demonstrated some variations . and @xmath150 with a mean varying from @xmath154 to 3 and a fixed variance of 0.01 . ] [ fig1 ] and @xmath150 with a mean varying from @xmath154 to 3 and a fixed variance of 0.01 . ] [ fig2 ] we conducted another sensitivity analysis on the prior distributions of @xmath155 , @xmath156 , @xmath90 and @xmath106 . we considered various inverse gamma priors , @xmath157 , by setting @xmath158 and 5 . as shown in table [ tablepriorvar ] , the estimates of the measurement model parameters were stable under different prior distributions , suggesting the proposed method is not sensitive to the priors of these parameters . [ seccon ] we have developed a selection - model - based approach to analyze longitudinal dyadic data with nonignorable dropouts . we model the longitudinal outcome process using a transition model and account for the correlation within dyads using random effects . in the model , we allow a subject s outcome to depend on not only his / her own characteristics but also the characteristics of the other member in the dyad . as a result , the parameters of the proposed model have appealing interpretations as `` actor '' and `` partner '' effects , which greatly facilitates the understanding of interdependence within a relationship and the design of more efficient prevention programs . to account for the nonignorable dropout , we adopt a discrete time survival model to link the dropout process with the longitudinal measurement process . we used the data augment method to address the complex missing data problem caused by dropout and interdependence within dyads . the simulation study shows that the proposed method yields consistent estimates with correct coverage rates . we apply our methodology to the longitudinal dyadic data collected from a breast cancer study . our method identifies more `` partner '' effects than the methods that ignore the missing data , thereby providing extra insights into the interdependence of the dyads . for example , the methods that ignore the missing data suggest that the spouse s cesd related only to his own previous cesd scores , whereas the proposed method suggested that the spouse s cesd related not only to his own cesd but also to the patient s cesd and mpi level . this extra information can be useful for the design of more efficient depression prevention programs for breast cancer patients . [ tab5 ] @lcd2.14d2.14d2.14@ & & & & + & intercept & 4.72 ( 3.32 , 6.11 ) & 5.00 ( 3.48 , 6.47)&5.02 ( 3.57 , 6.48 ) + & patient cesd & 0.87 ( 0.81 , 0.93 ) & 0.86 ( 0.80 , 0.92 ) & 0.88 ( 0.83 , 0.94 ) + & spouse cesd & 0.14 ( 0.09 , 0.19 ) & 0.14 ( 0.08 , 0.19 ) & 0.13 ( 0.08 , 0.18 ) + & patient mpi & 1.27 ( 0.84 , 1.71 ) & 1.12 ( 0.67 , 1.60 ) & 1.20 ( 0.85 , 1.57 ) + & spouse mpi & 0.71 ( 0.49 , 0.91 ) & 0.68 ( 0.46 , 0.87 ) & 0.61 ( 0.39 , 0.82 ) + & cancer stage & -0.03 ( -0.50 , 0.50)&0.18 ( -0.31 , 0.65)&-0.08 ( -0.57 , 0.40 ) + spouses & intercept & 6.40 ( 4.39 , 8.41 ) & 7.56 ( 5.35 , 9.93 ) & 7.52 ( 5.43 , 9.55 ) + & patient cesd & 0.67 ( 0.62 , 0.73 ) & 0.67 ( 0.62 , 0.72 ) & 0.69 ( 0.64 , 0.73 ) + & spouse cesd & 0.76 ( 0.71 , 0.80 ) & 0.75 ( 0.71 , 0.81 ) & 0.75 ( 0.71 , 0.80 ) + & patient mpi & 0.51 ( 0.32 , 0.71 ) & 0.54 ( 0.35 , 0.73 ) & 0.53 ( 0.34 , 0.72 ) + & spouse mpi & 0.79 ( -0.05 , 1.46 ) & 0.54 ( -0.03 , 1.06 ) & 0.45 ( -0.23 , 1.09 ) + & cancer stage & -0.41 ( -0.86 , 0.02 ) & -0.38 ( -0.81 , 0.03 ) & -0.48 ( -0.87 , 0.08 ) + in the proposed dropout model ( [ dropmodel ] ) , we assume that time - dependent covariates @xmath5 and @xmath159 , @xmath46 , have captured all important time - dependent factors that influence dropout . however , this assumption may not be always true . a more flexible approach is to include in the model a time - dependent random effect @xmath160 to represent all unmeasured time - variant factors that influence dropout . we can further put a hierarchical structure on @xmath160 to shrink it toward a dyad - level time - invariant random effect @xmath59 to account for the effects of unmeasured time - invariance factors on dropout . in addition , in ( [ dropmodel ] ) , in order to allow members in a dyad to drop out at different times , we specify separate dropout models for each dyadic member , linked by a common random effect . although the common random effect makes the members in a dyad more likely to drop out at the same time , it may not be the most effective modeling approach when dropout mostly occurs at the dyad level . in this case , a more effective approach is that , in addition to the dyad - level random effect , we further put hierarchical structure on the coefficients of common covariates ( in the two dropout models ) to shrink toward a common value to reflect that dropout is almost always at the dyad level . we would like to thank the referees , associate editor and editor ( professor susan paddock ) for very helpful comments that substantially improved this paper .

dyadic data are common in the social and behavioral sciences , in which members of dyads are correlated due to the interdependence structure within dyads . the analysis of longitudinal dyadic data becomes complex when nonignorable dropouts occur . we propose a fully bayesian selection - model - based approach to analyze longitudinal dyadic data with nonignorable dropouts . we model repeated measures on subjects by a transition model and account for within - dyad correlations by random effects . in the model , we allow subject s outcome to depend on his / her own characteristics and measure history , as well as those of the other member in the dyad . we further account for the nonignorable missing data mechanism using a selection model in which the probability of dropout depends on the missing outcome . we propose a gibbs sampler algorithm to fit the model . simulation studies show that the proposed method effectively addresses the problem of nonignorable dropouts . we illustrate our methodology using a longitudinal breast cancer study . .

arxiv

according to quantum electrodynamics , quantum fluctuations of electric and magnetic fields give rise to a zero - point energy that never vanishes , even in the absence of electromagnetic sources@xcite . in 1948 , h. b. g. casimir predicted that , as a consequence , two electrically neutral metallic parallel plates in vacuum , assumed to be perfect conductors , should attract each other with a force inversely proportional to the fourth power of separation@xcite . the plates act as a cavity where only the electromagnetic modes that have nodes on both the walls can exist . the zero - point energy ( per unit area ) when the plates are kept at close distance is smaller than when the plates are at infinite separation . the plates thus attract each other to reduce the energy associated with the fluctuations of the electromagnetic field . e. m. lifshitz , i. e. dzyaloshinskii , and l. p. pitaevskii generalized casimir s theory to isotropic dielectrics @xcite . in their theory the force between two uncharged parallel plates with arbitrary dielectric functions can be derived according to an analytical formula that relates the helmholtz free energy associated with the fluctuations of the electromagnetic field to the dielectric functions of the interacting materials and of the medium in which they are immersed@xcite . at very short distances ( typically smaller than a few nanometers ) , lifshitz s theory provides a complete description of the non - retarded van der waals force . at larger separations , retardation effects give rise to a long - range interaction that in the case of two ideal metals in vacuum reduces to casimir s result . lifshitz s equation also shows that two plates made out of the same material always attract , regardless of the choice of the intervening medium . for slabs of different materials , on the contrary , the sign of the force depends on the dielectric properties of the medium in which they are immersed@xcite . while the force is always attractive in vacuum , there are situations for which a properly chosen liquid will cause the two plates to repel each other@xcite . as mentioned above , one of the limitations of lifshitz s theory@xcite is the assumption that the dielectric properties of the interacting materials are isotropic . in 1972 v. a. parsegian and g. h. weiss derived an equation for the non - retarded van der waals interaction energy between two dielectrically anisotropic plates immersed in a third anisotropic material@xcite . one of the authors of the present paper ( y. b. ) analyzed a similar problem and found an equation for the helmholtz free energy ( per unit area ) of the electromagnetic field in which retardation effects are included@xcite . in the non - retarded limit , the two results are in agreement . both articles also show that a torque develops between two parallel birefringent slabs ( with in plane optical anisotropy , as shown in figure [ barash_config ] ) placed in a isotropic medium , causing them to spontaneously rotate towards the configuration in which their principal axes are aligned . this effect can be qualitatively understood by noting that the relative rotation of the two plates will result in a modification of the zero - point energy , because the reflection , transmission , and absorption coefficients of these materials depend on the angle between the wave vector of the virtual photons , responsible for the zero - point energy , and the optical axis . the anisotropy of the zero - point energy between the plates then generates the torque that makes them rotate toward configurations of smaller energy . the casimir - lifshitz force between isotropic dielectrics is receiving considerable attention in the modern literature . the theory has been verified in several high precision experiments , and although the investigation has been mainly focused on the interaction between metallic surfaces in vacuum , there are no doubts about its general validity@xcite ( for a review of previous measurements , see @xcite ; for a critical discussion on the precision of the most recent experiments , see @xcite ) . less precise measurements in liquids have been reported@xcite , and experimental evidence for repulsive van der waals forces between dielectric surfaces in different fluids has also been reported@xcite . finally , it has been pointed out that the casimir - lifshitz force might be a potentially relevant issue for the development of micro- and nanoelectromechanical systems@xcite . on the other hand , essentially no attention has been devoted to the torque between anisotropic materials predicted by parsegian , weiss and barash , with the exception of a theoretical derivation of a more simplified equation of the torque between two plates in a one dimension calculation@xcite and between engineered anisotropic surfaces ( two ellipsoids with anisotropic dielectric function@xcite and two dielectric slabs with different directions of conductivity@xcite ) . no experimental attempts to demonstrate the effect have ever been reported , and so far no numerical calculations to estimate its magnitude have been presented@xcite . in this paper we calculate the magnitude of the torque induced by quantum fluctuations for specific materials and discuss possible experimental validations of the effect . we consider a small birefringent disk ( diameter 40 @xmath3 m , thickness 20 @xmath3 m ) made out of either quartz or calcite placed parallel to a birefringent barium titanate ( batio@xmath8 ) plate in vacuum . using the dielectric properties for the materials reported in the literature , we show that the magnitude of the torque is within the sensitivity of available instrumentation , provided that the plate and the disk are kept at sub - micron distances . unfortunately , at such short separations the tendency of the two surfaces to stick together represents a major technical difficulty . therefore , the measurement of the rotation of the disk may seem to be an extremely challenging problem , which can be only addressed with the design of sophisticated mechanical systems . in this paper we propose a much simpler experimental approach , where the batio@xmath8 plate is immersed in liquid ethanol and the quartz or calcite disk is placed on top of it . in this case the retarded van der waals force between the two birefringent slabs is repulsive . the disk is thus expected to float on top of the plate at a distance of approximately 100 nm , where its weight is counterbalanced by the van der waals repulsion . because there is no contact between the two birefringent surfaces , the disk would be free to rotate in a sort of _ frictionless bearing _ , sensitive to even very small driving torques . let s consider two plates made of uniaxial birefringent materials kept parallel at a distance @xmath2 and immersed inside a medium with dielectric function @xmath9 . for sake of simplicity , let s assume that the two plates are oriented as in figure [ barash_config ] . the @xmath10-axis of our reference system is chosen to be orthogonal to the plates . the optical axis of one of the two crystals ( i.e. the threefold , fourfold , or sixfold axis of symmetry for rhombohedral , tetragonal , or hexagonal crystals , respectively@xcite ) is aligned with the @xmath11-axis . the optical axis of the second crystal is also in the @xmath12 plane but rotated by an angle @xmath1 with respect to the other . the dielectric tensors of the two plates are then described by the following matrices@xcite : @xmath13 where the subscripts @xmath14 and @xmath15 indicate the value of the dielectric tensor along the optical axis and along the plane orthogonal to the optical axis , respectively . it is important to stress that in equation [ ematr ] @xmath16 , @xmath17 , @xmath18 , and @xmath19 are functions of the angular frequency of the electromagnetic wave @xmath20 . the helmholtz free energy ( per unit area ) of the system is given by@xcite : @xmath21 where @xmath22 is the boltzmann constant , @xmath23 is the temperature of the system , and @xmath24+c \bigr\ } \biggr\ } \\ \end{split } \label{d}\ ] ] @xmath25 \\ & \cdot \biggl [ ( \epsilon_3^{(n)}\rho_2+\epsilon_{2\bot}^{(n)}\rho_3)- { \epsilon_{2\bot}^{(n)}(\widetilde{\rho}_2-\rho_2)[(r\cos\varphi\sin\theta+r\sin\varphi\cos\theta)^2 -\rho_2\rho_3 ] \over \rho_2 ^ 2 ( r\cos\varphi\sin\theta+r\sin\varphi\cos\theta)^2 } \biggr ] \\ \end{split } \label{eqgamma}\ ] ] @xmath26 \\ & \cdot [ ( \epsilon_3^{(n)}\rho_1+\epsilon_{1\bot}^{(n)}\rho_3)(\epsilon_3^{(n)}\rho_2 + \epsilon_{2\bot}^{(n)}\rho_3)-(\epsilon_3^{(n)}\rho_1-\epsilon_{1\bot}^{(n)}\rho_3)(\epsilon_3^{(n)}\rho_2- \epsilon_{2\bot}^{(n)}\rho_3)\exp(-2\rho_3d ) ] \\ & -{(\widetilde{\rho_1}-\rho_1)\epsilon_{1\bot}^{(n)}\over \rho_1 ^ 2-r^2\sin^2\varphi}\{(r^2\sin^2\varphi-\rho_1\rho_3 ) ( \epsilon_3^{(n)}\rho_2+\epsilon_{2\bot}^{(n)}\rho_3)(\rho_2+\rho_3)(\rho_1+\rho_3 ) \\ & + 2(\epsilon_{2\bot}^{(n)}-\epsilon_3^{(n)})[r^2\sin^2\varphi ( r^2\rho_1-\rho_2\rho_3 ^ 2)+\rho_1\rho_3 ^ 2(r^2 - 2r^2\sin^2\varphi+\rho_1\rho_2 ) ] \\ & \cdot \exp(-2\rho_3d)+(r^2\sin^2\varphi+\rho_1\rho_3 ) ( \epsilon_3^{(n)}\rho_2-\epsilon_{2\bot}^{(n)}\rho_3)(\rho_1-\rho_3)(\rho_2-\rho_3)\exp(-4\rho_3d)\ } \\ \end{split } \label{eqa}\ ] ] @xmath27 \\ & + { ( \widetilde{\rho}_1-\rho_1)\epsilon_{1\bot}^{(n ) } \over \rho_1 ^ 2-r^2\sin^2\varphi } \{-(r^2\sin^2\varphi-\rho_1\rho_3)(\rho_1+\rho_3)(\rho_2+\rho_3 ) \\ & + 2[r^2\sin^2\varphi(\rho_1\rho_2+\rho_3 ^ 2)-\rho_1 ^ 2\rho_3 ^ 2+\rho_1\rho_2\rho_3 ^ 2 ] \exp(-2\rho_3d)\\ & -(r^2\sin^2\varphi+\rho_1\rho_3)(\rho_1-\rho_3)(\rho_2-\rho_3)\exp(-4\rho_3d)\ } \\ \end{split } \label{eqb}\ ] ] @xmath28 \\ & + { ( \widetilde{\rho_1}-\rho_1)\epsilon_{1\bot}^{(n)}\over \rho_1 ^ 2-r^2\sin^2\varphi}\rho_2\rho_3\{(r^2\sin^2\varphi-\rho_1\rho_3 ) ( \rho_1+\rho_3)(\rho_2+\rho_3 ) \\ & + 2\rho_3[\rho_1 ^ 2\rho_2+\rho_1\rho_3 ^ 2+r^2\sin^2\varphi(\rho_1-\rho_2 ) ] \exp(-2\rho_3d ) \\ & -(r^2\sin^2\varphi+\rho_1\rho_3)(\rho_1-\rho_3)(\rho_2-\rho_3)\exp(-4\rho_3d)\ } \label{eqc } \end{split}\ ] ] @xmath29 @xmath30 @xmath31 @xmath32 where @xmath33 is the speed of light in vacuum . the prime in the summation in equation [ energy ] indicates that the first term ( @xmath34 ) must be multiplied by a factor @xmath35 . furthermore @xmath36 , @xmath37 , @xmath38 , @xmath38 , and @xmath39 represent the values of the dielectric functions of the interacting materials calculated at imaginary angular frequencies@xcite @xmath40 , where @xmath41 is the planck constant divided by @xmath42 . the torque @xmath43 induced on the two parallel birefringent plates is given by@xcite : @xmath44 where @xmath45 is the area of the interacting surfaces . the retarded van der waals ( or casimir - lifshitz ) force is given by : @xmath46 following reference @xcite , it is also possible to show that in the non - retarded limit the torque between two plates made out of slightly birefringent materials ( i.e. with @xmath47 ) reduces to : @xmath48 where @xmath49 is given by : @xmath50 ^ 2 } \label{omegabar}\ ] ] we recall that in equation [ omegabar ] @xmath16 , @xmath17 , @xmath18 , and @xmath19 must be evaluated at imaginary frequency @xmath51 . the integral over @xmath11 in equation [ omegabar ] can be solved analytically ; the equation for @xmath49 reads : @xmath52 in the above limits the torque is thus inversely proportional to the second power of @xmath2 and is proportional to @xmath0 . in the retarded theory , it is generally not possible to reduce the expression for the torque to a simplified analytical equation . in order to determine the dependence of @xmath43 on @xmath2 and @xmath1 , it is thus necessary to solve equations [ energy ] through [ eqtorque ] numerically . in the next section we will analyze a particular configuration and calculate the torque as a function of @xmath1 at a distance where retardation effects can not be neglected . we now focus on a particular experimental configuration . we consider a 20 @xmath3 m thick , 40 @xmath3 m diameter disk made out of either quartz or calcite , kept in vacuum parallel to a large batio@xmath8 plate at a distance @xmath2 . it is easy to recognize that to perform the numerical computation of the torque experienced by the disk as a function of @xmath1 and @xmath2 we only need to know the dielectric functions of the two plates at imaginary angular frequencies @xmath53 . the dielectric properties of many materials are well described by a multiple oscillator model ( the so - called ninham - parsegian representation)@xcite , which can be written as follows : @xmath54 in equation [ np ] , @xmath55 is given by @xmath56 , where @xmath57 is the oscillator strength and @xmath58 is the relaxation frequency multiplied by @xmath42 , while @xmath59 is the damping coefficient of the oscillator . for most inorganic materials , only two undamped oscillators are commonly used to describe the whole dielectric function@xcite : @xmath60 where @xmath61 and @xmath62 are the characteristic absorption angular frequencies in the infrared and ultraviolet range , respectively , and @xmath63 and @xmath64 are the corresponding absorption strengths . the two oscillator model does not always provide a complete description of the dielectric properties of materials ; however , in spite of its simplicity , when applied to dispersion effects it usually leads to rather precise results@xcite . we have thus used this model for our calculations . the limits of this choice will be discussed in section vii . the parameters that determine the dielectric properties of quartz , calcite , and batio@xmath8 in the limit of the two oscillator model ( equation [ twoosc ] ) are listed in table [ tabdf]@xcite . figure [ figepsi ] shows the calculated @xmath65 . using these functions , we have calculated the torque expected for different angles at @xmath66 nm , both in the quartz - batio@xmath8 and in the calcite - batio@xmath8 configurations . the results obtained for @xmath67 k are reported in figure [ qorcvb ] . calculations for smaller temperatures give rise to nearly identical values : at this distance the torque is solely generated by the fluctuations of the electromagnetic field associated with the zero - point energy , because contributions arising from thermal radiation can be neglected@xcite . the computational data were interpolated using a sinusoidal function with periodicity equal to @xmath68 ( @xmath69 ) ; its amplitude ( @xmath70 ) was adjusted by means of an unweighted fit . this curve , reported in figure [ qorcvb ] , well interpolates the numerical results . the maximum magnitude of the torque occurs at @xmath5 and @xmath71 . comparison of the results obtained with quartz and calcite shows that , as expected , materials with less pronounced birefringent properties ( such as quartz with respect to calcite ) give rise to a smaller torque . interestingly , the sign of the torque obtained for the quartz - batio@xmath8 configuration is opposite to the one obtained for calcite - batio@xmath8 . the reason for this behavior can be understood from the dependence of the dielectric functions on imaginary frequency ( figure [ figepsi ] ) . for quartz , @xmath72 at all frequencies , while for batio@xmath8 , @xmath73 is always larger than @xmath74 . for calcite , the two curves cross at @xmath75 rad / s ; however , the contribution to the torque arising from frequencies below this value are relatively small . therefore , the largest contribution to the torque comes from angular frequencies in the region where @xmath76 . the minimum zero - point energy corresponds to the situation in which the axes of the dielectric tensors with larger values of @xmath77 are aligned . for the quartz - batio@xmath8 combination , this situation is reached for @xmath78 : the torque is positive from @xmath79 to @xmath78 and negative from @xmath78 to @xmath80 . for calcite - batio@xmath8 , on the contrary , the minimum energy corresponds to @xmath79 : the sign of the torque is thus reversed with respect to the previous case . we have also calculated the magnitude of the torque at @xmath5 as a function of the distance between the disk and the plate . from the results reported in figure [ mdv ] one can clearly verify that it is not possible to infer a single power law dependence that describes the torque at all distances regardless of the choice of the interacting materials . for @xmath66 nm , the maximum magnitude of the torque is approximately equal to @xmath81 n@xmath7 m for the quartz disk and @xmath82 n@xmath7 m for the calcite disk . in 1936 r. a. beth performed an experiment where a torsional balance was used to measure the rotation of a macroscopic quartz disk induced by the transfer of angular momentum of light@xcite . he achieved a sensitivity of @xmath83 n@xmath7m@xcite . it is thus reasonable to ask if similar set - ups with today s improved technology could be used to observe the rotation between the @xmath84 @xmath3 m diameter disk and the batio@xmath8 plate induced by virtual photons associated with the zero - point energy . the main difficulty of this experiment , and the main difference with the measurement cited above , is that it is necessary to keep the disk freely suspended just above the other plate at separations where the two surfaces would tend to come into contact . from figure [ mdv ] , we can estimate that in order to observe the effect the two surfaces should be kept at least at sub - micron distances . one could argue that the plate used in beth s experiment was much larger than the disk that we have considered so far . because the torque is proportional to the area of the interacting surfaces , one could use a plate with a much larger diameter . for example , for a 1 cm diameter quartz disk kept parallel to a batio@xmath8 plate , the torque is larger than @xmath85 n@xmath86 m for @xmath87 @xmath3 m . however , other problems would arise in this case . the surface roughness and the curvature of the two birefringent plates should be much smaller than @xmath88 m over an area of several @xmath89 . this area should also be completely free from dust particles with diameter larger than a few hundreds of nm . furthermore , one should still design a mechanical set - up to keep the two slabs parallel without compromising the sensitivity of the instrument . we conclude that the use of a torsional balance for the measurement of the torque induced by quantum fluctuations would present several major technical problems . instead of discussing this possibility , we propose a simpler solution . let s consider again the microscopic disk described at the beginning of this section . we will show below that the retarded van der waals force between quartz or calcite and batio@xmath8 in liquid ethanol is repulsive ; thus , if the disk is placed on top of the plate , this repulsion can be used to counterbalance the weight of the quartz disk . in liquid ethanol , therefore , the disk would float parallel on top of the batio@xmath8 plate at a small distance . the static friction between the two birefringent plates would be virtually zero , and the disk would be free to rotate suspended in bulk liquid . if the torque induced by quantum fluctuations does not sensibly decrease after the introduction of liquid ethanol in the gap , and if the equilibrium distance is smaller than a few hundreds nanometer , the configuration proposed should allow the demonstration of the rotation of the disk in a reasonably straightforward experiment . the force between the disk and the plate was calculated according to equation [ eqforce ] . the parameters used to determine the dielectric properties of ethanol are reported in table [ tabdf]@xcite . the results of the calculations for @xmath90 are reported in figure [ force ] . similar calculations were carried out for @xmath79 and @xmath91 : the results differ from the curves represented in figure [ force ] by less than 10@xmath92 . for distances shorter than a few nanometers the force is attractive . however , at larger distances , where retardation effects start to play an important role in the interaction , the force switches to repulsive . the reason for this behavior can be understood by comparing the dielectric functions of quartz , calcite , batio@xmath8 , and ethanol , which are reported in the inset of figure [ force ] . for sake of simplicity , for each birefringent material , we show the average of @xmath73 and @xmath74 . the arguments discussed below refer to the average of @xmath73 and @xmath74 , but they can be similarly applied to the two principal components of the dielectric tensor separately . for @xmath93 , we have : @xmath94 while at higher frequencies @xmath95 is smaller than @xmath96 , @xmath97 , and @xmath98 . from lifshitz s theory for isotropic materials , it is possible to show that the force between two plates with dielectric functions @xmath99 and @xmath100 immersed in a medium with dielectric function @xmath9 is repulsive if , for imaginary frequencies , @xmath101 or @xmath102 , and it is attractive in all other cases@xcite . in our system there is a crossover from @xmath101 to @xmath103 at @xmath104 rad / s . the force is thus repulsive at large distances , where low imaginary frequencies give rise to the most important contribution to the force , and attractive for smaller separations , where higher frequencies are more relevant@xcite . the zero - point energy due to electromagnetic quantum fluctuations depends on the distance between the two interacting plates . if the condition @xmath101 ( or @xmath102 ) is satisfied , the zero - point energy per unit area is smaller at larger separation , which means that it is energetically more favorable for the liquid to stay inside the gap rather than outside . as a consequence , the net force between the plates is repulsive . for detailed energy balance considerations and a more rigorous proof of this statement , we refer the reader to the so - called _ hamaker theorem _ , discussed in @xcite . the net weight of the disk immersed in ethanol is given by : @xmath105 where @xmath45 is the surface of the disk , @xmath106 is its thickness , @xmath107 is the mass density of either quartz ( 2643 kg / m@xmath108 ) or calcite ( 2760 kg / m@xmath108 ) , @xmath109 is the mass density of ethanol ( 789 kg / m@xmath108 ) , and @xmath110 m / s@xmath111 . the arrow on the curves reported in figure [ force ] indicates the distance at which this force is counterbalanced by the retarded van der waals repulsion . in both the cases under investigation , the equilibrium separation is about 100 nm . this distance can be tailored to the experimental needs by changing the thickness of the disk . we have calculated the expected torque between the disk and the plate in liquid ethanol as a function of angle and distance . we have verified that also in this case the torque varies as @xmath0 and has maximum magnitude at @xmath90 and @xmath112 . the sign of the rotation that one should observe using a quartz disk is opposite with respect to what is expected for a calcite disk , and temperature corrections are negligible . figure [ mde ] shows the calculated magnitude of the torque at @xmath5 for different values of @xmath2 . at @xmath66 nm the torque is smaller by a factor of 2 with respect to the case of vacuum . note that at short distances the torque in ethanol is actually larger than in vacuum ( figure [ mde ] ) . at present , we do not have an intuitive explanation of this phenomenon . a schematic view of the proposed experimental set - up is shown in figure [ setup ] . a 40 @xmath3 m diamater , 20 @xmath3 m thick disk made out of quartz or calcite is placed on top of a batio@xmath8 plate immersed in ethanol . the optical axes of the birefringent crystals are oriented as in figure [ barash_config ] . according to the arguments in the previous section , the disk should levitate approximately 100 nm above the plate and should be free to rotate in a sort of _ frictionless bearing_. a 100 mw laser beam can be collimated onto the disk to rotate it by the transfer of angular momentum of light . a shutter can then block the beam to stop the light - induced rotation . the position of the disk can be monitored by means of a microscope objective coupled to a ccd - camera for imaging . using the laser , one can rotate the disk until @xmath90 . once the laser beam is shuttered , the disk is free to rotate back towards the configuration of minimum energy according to the following equation ( see appendix ) : @xmath113 where @xmath114 is the radius of the disk , @xmath115 is its momentum of inertia , @xmath116 is the viscosity of ethanol ( @xmath117 ns / m@xmath111 ) , @xmath2 is the distance between the disk and the plate , and @xmath118 is the torque due to quantum fluctuations . for an estimate of the time evolution , we have determined @xmath70 from figure [ mde ] and solved equation [ thetat ] for @xmath119 and @xmath120 . the results are reported in the inset of figure [ setup ] . note that the rotation is overdamped in both cases : the disk moves asymptotically and monotonically towards the equilibrium position . for the calcite disk , easily measurable rotations should be observed within a few minutes after the laser beam shutter is closed . the quartz disk would rotate much slower , and it is questionable whether its rotation could be detected or not . however , it is worth to stress that the set - up presented above is not yet optimized . disks with different dimensions and geometries might rotate faster . suitably engineered samples could also result in more favorable experimental configurations . for example , a thick layer of lead could be deposited on a portion of the disk to make it heavier : the disk would then float at a smaller distance , where the magnitude of the driving torque would be larger . furthermore , the use of a different liquid with optical properties similar to ethanol but with a smaller viscosity would significantly increase the angular velocity of the disk . finally , a more sophisticated optical set - up could be implemented for the measurement of small rotations . one could argue that it might be difficult to distinguish the cause of the rotation of the disk from other effects that could mimic the phenomenon under investigation . surface roughness , charge accumulation , and liquid motion could be typical reasons . however , there are two defining properties that should help experimenters rule out spurious effects : ( i ) the torque induced by quantum fluctuations has periodicity @xmath68 , and ( ii ) the sign of the torque depends on whether the experiment is performed with a quartz or a calcite disk . for example , a calcite disk would rotate clockwise if initially positioned at @xmath121 or @xmath122 , and anti - clockwise if initially positioned at @xmath123 or @xmath124 . for a quartz disk , one would obtain the opposite behavior . an experimental observation of the dependence of the rotation direction on the initial position and on the choice of the interacting materials would thus indicate that the disk is solely driven by the quantum fluctuations of the electromagnetic field . as an additional proof , the experiment could be repeated by placing the disk over a non - birefringent plate with @xmath125 . the retarded van der waals force would still be repulsive , but there would be no torque induced by quantum fluctuations . calculations were performed using mathematica ( version 5 , wolfram research ) . although integrals and summations are estimated to be exact with a level of accuracy @xmath126 , the overall results can not be considered equally precise . the model used for the dielectric function of the materials ( equation [ twoosc ] ) is in fact relatively unprecise . to give a sense of how much the model might affect the results , we will reconsider below the configuration of quartz - ethanol - batio@xmath8 . in our previous calculations we assumed @xmath127 . this value is the average of the two values available in the literature@xcite , @xmath128 and @xmath129 . at @xmath66 nm and @xmath5 , both the torque and the casimir - lifshitz force that one obtains using either @xmath130 or @xmath131 differs from the results obtained with the average by less than 14@xmath92 . this discrepancy is due to the fact that the dielectric function of batio@xmath8 is very large in the infrared . as a consequence , a large contribution to the summation in equation [ energy ] comes from the first few terms ( i.e. for small @xmath132 ) , which correspond to relatively large wavelengths . this means that if the model is not accurate enough in that region , larger errors can be introduced in the calculation . although it is obvious that in order to compare experiment to theory a deeper knowledge of the dielectric properties of the materials is needed , the use of slightly different values of the parameters in @xmath65 does not significantly influence the order of magnitude of our results . we have performed detailed numerical calculations of the mechanical torque between a 40 @xmath3 m diameter birefringent disk , made of quartz or calcite , and a batio@xmath8 birefringent plate . at separations of the order of a few hundreds of nanometers , the magnitude of the torque is of the order of @xmath133 n@xmath7 m . we have shown that a demonstration of the effect could be readily obtained if the birefringent slabs were immersed in liquid ethanol . in this case the disk would float on top of the plate at a distance where the repulsive retarded van der waals force balances gravity , giving rise to a mechanical bearing with ultra - low static friction . the disk , initially set in motion via transfer of angular momentum of light from a laser beam , would return to its equilibrium position solely driven by the torque arising from quantum fluctuations . enlightening discussions with l. levitov and h. stone are gratefully acknowledged . this work was partially supported by nsec ( nanoscale science and engineering center ) , under nsf contract number phy-0117795 . consider a disk of radius @xmath114 rotating at angular velocity @xmath134 parallel to a surface at distance @xmath2 and immersed in a liquid with viscosity @xmath116 . inside the gap , the velocity @xmath135 of the liquid is described by the following equation@xcite : where we have introduced cylindrical coordinates @xmath137 in the reference system reported in figure [ barash_config ] . the stress induced by the viscosity of the liquid on an infinitesimal area of the disk at coordinates @xmath138 is given by : it is interesting to note that the drag is dominated by the torque due to the liquid inside the gap . to estimate the contribution to the drag of the surrounding liquid , one can calculate the drag expected if the disk were rotating in bulk , far away from any other surfaces . in this case , the torque is given by @xmath141 ( see article cited in note @xcite ) , which , for the configuration chosen in our experiment , is a factor @xmath142 smaller than the one expected from equation [ app3 ] . note also that thermal fluctuations do not contribute to the rotation as long as the disk is far away from the equilibrium position . the potential energy due to quantum fluctuations is in fact much larger than @xmath143 for angles larger than a few degrees . in this context , it is worth mentioning a recent theoretical work on the van der waals force in 2-dimensionally anisotropic materials : r. r. degastine , d. c. prieve , and l. r. white , _ j. colloid interface sci . _ * 249 * , 78 ( 2002 ) . the paper , however , does not contain considerations on the quantum torque . in the calculation of the temperature dependence of the torque we have assumed that , regardless of the temperature , the dielectric functions of the interacting materials are equal to the ones described by equation [ twoosc ] and by the parameters of table [ tabdf ] , which refer to room temperature . this approximation does not affect the conclusion that the torque is solely generated by virtual photons . from an experimental point of view , the assumption is indeed correct if one performs experiments at @xmath144 k. however , it is worth to stress that , at @xmath145 k , batio@xmath8 crystals undergo a transition of their crystalline structure from tetragonal to orthorhombic ( m. e. lines and a. m. glass , _ principles and applications of ferroelectrics and related materials _ ( clarendon press , oxford , 1977 ) ) . therefore , results obtained in this paper are valid only for @xmath146 k. several groups recently demonstrated that small optically trapped particles can be rotated by means of angular momentum transfer of light ( see for example m. e. j. friese , t. a. niemenen , n. r. heckenberg , and h. rubinsztein - dunlop , _ nature _ * 394 * , 348 ( 1998 ) ) . the sensitivity achieved in this kind of experiments is generally better than @xmath6 n@xmath7 m .

we present detailed numerical calculations of the mechanical torque induced by quantum fluctuations on two parallel birefringent plates with in plane optical anisotropy , separated by either vacuum or a liquid ( ethanol ) . the torque is found to vary as @xmath0 , where @xmath1 represents the angle between the two optical axes , and its magnitude rapidly increases with decreasing plate separation @xmath2 . for a 40 @xmath3 m diameter disk , made out of either quartz or calcite , kept parallel to a barium titanate plate at @xmath4 nm , the maximum torque ( at @xmath5 ) is of the order of @xmath6 n@xmath7 m . we propose an experiment to observe this torque when the barium titanate plate is immersed in ethanol and the other birefringent disk is placed on top of it . in this case the retarded van der waals ( or casimir - lifshitz ) force between the two birefringent slabs is repulsive . the disk would float parallel to the plate at a distance where its net weight is counterbalanced by the retarded van der waals repulsion , free to rotate in response to very small driving torques . pacs numbers : 12.20.-m,07.10.pz,46.55.+d

arxiv

the study of complex networks has notably increased in the last years with applications to a variety of fields ranging from computer science@xcite and biology to social science@xcite and finance@xcite . a central problem in network science @xcite is the study of the random walks ( rw ) on a graph , and in particular of the relation between the topological properties of the network and the properties of diffusion on it . this subject is not only interesting from a purely theoretical perspective , but it has important implications to various scientific issues ranging from epidemics @xcite to the classification of web pages through pagerank algorithm @xcite . finally , rw theory is also used in algorithms for community detection @xcite . in this paper we set up a new framework for the study of topologically biased random walks on graphs . this allows to address problems of community detection and synchronization @xcite in the field of complex networks @xcite . in particular by using topological properties of the network to bias the rws we explore the network structure more efficiently . a similar approach but with different focus can be found in @xcite . in this research we are motivated by the idea that biased random walks can be efficiently used for community finding . to this aim we introduce a set of mathematical tools which allow us an efficient investigation of the `` bias parameters '' space . we apply this tools to uncover some details in the spectra of graph transition matrix , and use the relation between spectra and communities in order to introduce a novel methodology for an efficient community finding . the paper is organized as follows : in the first section we define the topologically biased random walks ( tbrw ) . we then develop the mathematical formalism used in this paper , specifically the perturbation methods and the parametric equations of motion , to track the behaviour of different biases . in the second section we focus on the behavior of spectral gap in biased random walks . we define the conditions for which such a spectral gap is maximal and we present numerical evidence that this maximum is global . in the third section we present an invariant quantity for the biased random walk ; such constant quantity depends only upon topology for a broad class of biased random walks . finally , in the fourth section we present a general methodology for the application of different tbrw in the community finding problems . we then conclude by providing a short discussion of the material presented and by providing an outlook on different possible applications of tbrw . rws on graphs are a sub - class of markov chains @xcite . the traditional approach deals with the connection of the _ unbiased _ rw properties to the spectral features of _ transition operators _ associated to the network @xcite . a generic graph can be represented by means of the adjacency matrix @xmath0 whose entries @xmath1 are @xmath2 if an edge connects vertices @xmath3 and @xmath4 and @xmath5 otherwise . here we consider undirected graphs so that @xmath0 is symmetric . the _ normal matrix _ @xmath6 is related to @xmath0 through @xmath7 , where @xmath8 is a diagonal matrix with @xmath9 , i.e. the degree , or number of edges , of vertex @xmath3 . in the following we use uppercase letters for non - diagonal matrices and lowercase letters for the diagonal ones . note that by definition @xmath10 . consequently @xmath11 with @xmath12 _ iif _ @xmath13 , i.e. if @xmath3 and @xmath4 are nearest neighbors vertices . the matrix @xmath14 defines the transition probabilities for an _ unbiased _ random walker to pass from @xmath4 to @xmath3 . in such a case @xmath15 has the same positive value for any of the neighbors @xmath3 of @xmath4 and vanishes for all the other vertices@xcite . in analogy to the operator defining the single step transition probabilities in general markov chains , @xmath6 is also called the transition _ matrix _ of the unbiased rw . a _ biased _ rw on a graph can be defined by a more general transition matrix @xmath16 where the element @xmath15 gives again the probability that a walker on the vertex @xmath4 of the graph will move to the vertex @xmath3 in a single step , but depending on appropriate weights for each pair of vertex @xmath17 . a genuine way to write these probabilities is to assign weights @xmath18 which represent the rates of jumps from vertex @xmath4 to vertex @xmath3 and normalize them : [ probabpasage ] t_ij=. in this paper we consider biases which are self - consistently related to graph topological properties . for instance @xmath18 can be a function of the vertex properties ( the network degree , clustering , etc . ) or some functions of the edge ones ( multiplicity or shortest path betweenness ) or any combination of the two . there are other choices of biases found in the literature such as for instance maximal entropy related biases @xcite . some of the results mentioned in this paper hold also for biases which are not connected to graph properties as will be mentioned in any such case . our focus on graph properties for biases is directly connected with application of biased random walks in examination of community structure in complex networks . let us start by considering a vertex property @xmath19 of the vertex @xmath3 ( it can be either local as for example the degree , or related to the first neighbors of @xmath3 as the clustering coefficient , or global as the vertex betweenness ) . we choose the following form for the weights : [ probbias ] w_ij = a_ije^x_i , where the parameter @xmath20 tunes the strength of the bias . for @xmath21 the unbiased case is recovered . by varying @xmath22 the probability of a walker to move from vertex @xmath4 to vertex @xmath3 will be enhanced or reduced with respect to the unbiased case according to the property @xmath19 of the vertex @xmath3 . for instance when @xmath23 , i.e. the degree of the vertex @xmath3 , for positive values of the parameter @xmath22 the walker will spend more time on vertices with high degree , i.e. it will be attracted by hubs . for @xmath24 it will instead try to `` avoid '' traffic congestion by spending its time on the vertices with small degree . the entries of the transition matrix can now be written as : [ transitionentries ] t_ij(,)=. for this choice of bias we find the following results : ( i ) we have a unique representation of any given network via operator @xmath25 , i.e. knowing the operator , we can reconstruct the graph ; ( ii ) for small @xmath26 we can use perturbation methods around the unbasied case ; ( iii ) this choice of bias permits in general also to visit vertices with vanishing feature @xmath27 , which instead is forbidden for instance for a power law @xmath28 ; ( iv ) this choice of biases is very common in the studies of energy landscapes , when biases represent energies @xmath29 ( see for example @xcite and references therein ) . in a similar way one can consider a _ edge property @xmath30 ( for instance edge multiplicity or shortest path betweenness ) as bias . in this case we can write the transition probability as : [ transitionentriesedge ] t_ij(,)=. the general case of some complicated multiparameter bias strategy can be finally written as [ transitionentriesmulti ] t_ij(,,)= . while we mostly consider biased rw based on vertex properties , as shown below , most of the results can be extended to the other cases . the transition matrix in the former case can also be written as : @xmath31 where the diagonal matrices @xmath32 and @xmath33 are such that @xmath34 and @xmath35 . the frobenius - perron theorem implies that the largest eigenvalue of @xmath36 is always @xmath37 @xcite . furthermore , the eigenvector @xmath38 associated to @xmath39 is strictly positive in a connected aperiodic graph . its normalized version , denoted as @xmath40 , gives the asymptotic stationary distribution of the biased rw on the graph . assuming for it the form @xmath41 , where @xmath42 and @xmath43 is a normalization constant , and plugging this in the equation @xmath44 we get : @xmath45 hence the equation holds _ iif _ @xmath46 . therefore the stable asymptotic distribution of vertex centerd biased rws is [ explidistro ] p_i()=()^-1 e^x_iz_i ( ) . for @xmath21 we have the usual form of the stationary distribution in an unbiased rw where @xmath47 and @xmath48 . for general @xmath22 it can be easily demonstrated that the asymptotic solution of edge biased rw is @xmath49 , while for multiparametric rw the solution is @xmath50 . using eqs . ( [ explidistro ] ) and ( [ transitionentries ] ) we can prove that the detailed balance condition @xmath51 holds . at this point it is convenient to introduce a different approach to the problem @xcite . we start by symmetrizing the matrix @xmath25 in the following way : @xmath52^{-1/2}\bsy{\hat{t}}(\bsy{x } , \beta ) [ \bsy{\hat{p}}(\beta)]^{1/2}\,,\ ] ] where @xmath53 is the diagonal matrix with the stationary distribution @xmath54 on the diagonal . the entries of the symmetric matrix for vertex centered case are given by @xmath55 the symmetric matrix @xmath56 shares the same eigenvalues with the matrix @xmath25 ; anyhow the set of eigenvectors is different and forms a complete orthogonal basis , allowing to define a meaningful distance between vertices . such distance can provide important additional information in the problem of community partition of complex networks . if @xmath57 is the @xmath58 eigenvector of the asymmetric matrix @xmath25 associated to the eigenvalue @xmath59 ( therefore @xmath60 ) , the corresponding eigenvector @xmath61 of the symmetric matrix @xmath56 , can always be written as @xmath62 . in particular for @xmath63 we have @xmath64 . the same transformation ( [ symmat ] ) can be applied to the most general multiparametric rw . in that case the symmetric operator is @xmath65 this form also enables usage of perturbation theory for hermitian linear operators . for instance , knowing the eigenvalue @xmath66 associated to eigenvector @xmath67 , we can write the following expansions at sufficiently small @xmath68 : @xmath69 and @xmath70 . it follows that for a vertex centered bias [ firstorderlambda ] _ ^(1)()=^s(1 ) ( , ) , where , @xmath71\ ] ] with @xmath72 being the anticommutator operator . operator @xmath73 and @xmath74 are diagonal matrices with @xmath75 and @xmath76 which is the expected value of @xmath27 that an random walker , will find moving from vertex @xmath3 to its neighbors . in the case of edge bias the change of symmetric matrix with parameter @xmath22 can be written as @xmath77 , and @xmath78 represents the schur - hadamard product i.e. element wise multiplication of matrix elements . the eigenvector components in @xmath79 at the first order of expansion in the basis of the eigenvectors at @xmath22 are given by ( for @xmath80 ) : [ firstordvecs ] = . for @xmath81 the product @xmath82 vanishes and eqs . ( [ derivation ] ) and ( [ firstordvecs ] ) hold only for non - degenerate cases . in general , usual quantum mechanical perturbation theory can be used to go to higher order perturbations or to take into account degeneracy of eigenvalues . we can also exploit further the formal analogy with quantum mechanics using parametric equations of motion ( pem ) @xcite to study the @xmath22 dependence of the spectrum of @xmath83 . if we know such spectrum for one value of @xmath22 , we can calculate it for any other value of @xmath22 by solving a set of differential equations corresponding to pem in quantum mechanics . they are nothing else the expressions of eqs . ( [ firstorderlambda ] ) and ( [ firstordvecs ] ) in an arbitrary complete orthonormal base @xmath84 . first the eigenvector is expanded in such a base : @xmath85 . we can then write @xmath86 where @xmath87 ( @xmath88 ) is a column ( row ) vector with entries @xmath89 and @xmath90 is the matrix with entries @xmath91 . let us now define the matrix @xmath92 whose rows are the copies of vector @xmath93 . the differential equation for the eigenvectors in the basis @xmath94 is then @xcite @xmath95 a practical way to integrate eqs . ( [ pemlambda ] ) and ( [ pemvector ] ) can be found in @xcite . in order to calculate parameter dependence of eigenvectors and eigenvalues , the best way to proceed is to perform an lu decomposition of the matrix @xmath96 as the product of a lower triangular matrix @xmath97 and an upper triangular matrix @xmath98 , and integrate differential equations of higher order which can be constructed in the same way as equations ( [ pemlambda ] ) and ( [ pemvector ] ) @xcite . a suitable choice for the basis is just the ordinary unit vectors spanned by vertices , i.e. @xmath99 . we found that for practical purposes , depending on the studied network , it is appropriate to use pem until the error increases to much and then diagonalize matrix again to get better precision . pem efficiently enables study of the large set of parameters for large networks due to its compatitive advantage over ordinary diagonalization . vs @xmath22 for networks of 10 communities with 10 vertices each ( the probability for an edge to be in a community is @xmath100 while outside of the community it is @xmath101 ) . solid points represent the solutions computed via diagonalization , while lines report the value obtained through integration of pem . different bias choice have been tested . circles ( blue ) are related to degree - based strategy , square ( red ) are related to clustering - based strategies , diamonds ( green ) multiplicity - based strategies . the physical quantities to get the variable @xmath27 in eq . ( [ probbias ] ) in these strategies have been normalized with respect to their maximum value.,width=302,height=245 ] a key variable in the spectral theory of graphs is the _ spectral gap _ @xmath102 , i.e. the difference between first unitary and the second eigenvalues . the spectral gap measures how fast the information on the rw initial distribution is destroyed and the stationary distribution is approached . the characteristic time for that is @xmath103\simeq 1/\mu$ ] @xcite . we show in fig . 1 the dependence of spectral gap of simulated graphs with communities for different strategies ( degree , clustering and multiplicity based ) at a given value of parameter @xmath22 . in all investigated cases the spectral gap has its well defined maximum , i.e. the value of parameter @xmath22 for which the random walker converges to stationary distribution with the largest rate . the condition of maximal spectral gap implies that it is a stationary point for the function @xmath104 , i.e. that its first order perturbation coefficient vanishes at this point : @xmath105\ket{v_2(\beta_m)},\end{aligned}\ ] ] where @xmath73 and @xmath106 are defined above . the squares of entries @xmath107 of the vector @xmath108 in the chosen basis @xmath109 , define a particular measure on the graph . equation ( [ maximalspectralgap ] ) can be written as @xmath110 . thus we conclude that the local spectral maximum is achieved if the average difference between property @xmath19 and its expectation @xmath111 , with respect to this measure , in the neighborhood of vertex @xmath3 vanishes . we have studied behavior of spectral gap for different sets of real and simulated networks ( barabsi - albert model with different range of parameters , erds - rnyi model and random netwroks with given community structure ) and three different strategies ( degree - based , clustering - based and multiplicity - based ) . although in general it is not clear that the local maximum of spectral gap is unique , we have found only one maximum in all the studied networks . this observation is interesting because for all cases the shapes of spectral gap _ vs. _ @xmath22 looks typically gaussian - like . in both limits @xmath112 the spectral gap of heterogeneous network is indeed typically zero , as the rw stays in the vicinity of the vertices with maximal or minimal value of studied property @xmath19 . a fundamental question in the theory of complex networks is how topology affects dynamics on networks . our choice of @xmath22-parametrized biases provides a useful tool to investigate this relationship . a central issue is , for instance , given by the search of properties of the transition matrix @xmath113 which are independent of @xmath22 and the chosen bias , but depend only on the topology of the network . an important example comes from the analysis the determinant of @xmath113 as a function of the bias parameters : @xmath114 for vertex centered bias using eq . ( [ derivation ] ) we have @xmath115 and using the diagonality of the @xmath73 and @xmath116 @xmath117 in other words the quantity @xmath118 is a topological constant which does not depend on the choice of parameters . for @xmath21 we get @xmath119 and it follows that this quantity does not depend on the choice of vertex biases @xmath19 either . it can be shown that such quantity coincides with the determinant of adjacency matrix which must be conserved for all processes . .,width=302,height=245 ] there are many competing algorithms and methods for community detection @xcite . despite a significant scientific effort to find such reliable algorithms , there is not yet agreement on a single general solving algorithm for the various cases . in this section instead of adding another precise recipe , we want to suggest a general methodology based on tbrw which could be used for community detection algorithms . to add trouble , the very definition of communities is not a solid one . in most of the cases we define communities as connected subgraphs whose density of edges is larger within the proposed community than outside it ( a concept quantified by modularity @xcite ) . scientific community is therefore thriving to find a benchmark in order to assess the success of various methods . one approach is to create synthetic graphs with assigned community structure ( benchmark algorithms ) and test through them the community detection recipes @xcite . the girvan - newman ( gn ) @xcite and lancichinetti - fortunato - radicchi ( lfr ) @xcite are the most common benchmark algorithms . in both these models several topological properties ( not only edge density ) are unevenly distributed within the same community and between different ones . we use this property to propose a novel methodology creating suitable tbrw for community detection . the difference between internal and external part of a community is related to the `` physical '' meaning of the graph . in many real processes the establishment of a community is facilitated by the subgraph structure . for instance in social networks agents have a higher probability of communication when they share a lot of friends . we test our approach on gn benchmark since in this case we can easily compute the expected differences between the frequency of biased variables within and outside the community . in this section we will describe how to use tbrw for community detection . for @xmath21 our method is rather similar to the one introduced by donetti and muoz @xcite . the most notable difference is that we consider the spectral properties of transition matrix instead of the laplacian one . we decide if a vertex belongs to a community according to the following ideas : _ ( i ) _ we expect that the vertices belonging to the same community to have similar values of eigenvectors components ; _ ( ii ) _ we expect relevant eigenvectors to have the largest eigenvalues . indeed , spectral gap is associated with temporal convergence of random walker fluctuations to the ergodic stationary state . if the network has well defined communities , we expect the random walker to spend some time in the community rather than escaping immediately out of it . therefore the speed of convergence to the ergodic state should be related to the community structure . therefore eigenvectors associated with largest eigenvalues ( except for the maximal eigenvalue 1 ) should be correlated with community structure . coming back to the above mentioned donetti and muoz approach here we use the fact that some vertex properties will be more common inside a community and less frequent between different communities . we then vary the bias parameters trying both to shrink the spectral gap in transition matrix and to maximize the separation between relevant eigenvalues and the rest of the spectra . , @xmath120 , @xmath121 , @xmath122 . there is a clear gap between `` community '' band and the rest of the eigenvalues . , width=302,height=245 ] as a function of parameter @xmath123 which biases rw according to degrees of the vertices and parameter @xmath124 which bias rw according to multiplicities of the edges . both degrees and multiplicity values are normalized with respect to the maximal degree and multiplicity ( therefore the largest value is one).,width=302,height=245 ] for example in the case of gn benchmark the network consist of @xmath125 communities each with @xmath120 vertices i.e. @xmath126 vertices all together . the probability that the two vertices which belong to the same community are connected is @xmath127 . the probability that the two vertices which belong to different communities are connected is @xmath128 . the fundamental parameter @xcite which characterizes the difficulty of detecting the structure is [ mu ] = , where @xmath129 is the mean degree related to inter - community connections and @xmath130 is the mean degree related to edges inside - community . as a rule of thumb we can expect to find well defined communities when @xmath131 , and observe some signature of communities even when @xmath132 @xcite . the probabilities @xmath127 and @xmath128 are related via the control parameter @xmath133 as @xmath134 . we now examine the edge multiplicity . the latter is defined as the number of common neighbors shared by neighbouring vertices . the expected multiplicity of an edge connecting vertices inter - community and inside - communities are respectively @xmath135 on fig . [ params ] we plot the ratio of the quantitites above defined , @xmath136 , _ vs. _ the parameter @xmath133 . we see that even for @xmath137 the ratio remains smaller than @xmath2 implying that the multiplicity is more common in the edges in the same community . based on this analysis for this particular example we expect that if we want to find well - defined communities via tbrw we have to increase bias with respect to the multiplicity . through numerical simulations we find that the number of communities is related to number of eigenvalues in the `` community band '' . namely one in general observes a gap between eigenvalues @xmath138 and the next eigenvalue evident in a network with a strong community structure ( @xmath139 ) . the explanation that we give for that phenomenon can be expressed by considering a network of @xmath140 separated graphs . for such a network there are @xmath140 degenerate eignevalues @xmath141 . if we now start to connect these graphs with very few edges such a degeneracy is broken with the largest eigenvalue remaining @xmath2 while the next @xmath142 eigenvalues staying close to it . the distance between any two of this set of @xmath142 eigenvalues will be smaller than the gap between this community band and the rest of the eigenvalues in the spectrum . therefore , the number of eigenvalues different of @xmath2 which are forming this `` community band '' is always equal to the number of communities minus one , at least for different gn - type networks with different number of communities and different sizes , as long as @xmath139 . for example in the case of 1000 gn networks described with parameters @xmath126 , @xmath120 , @xmath121 , @xmath122 , i.e. @xmath143 , the histograms of eigenvalues are depicted on figure [ band ] . for our purposes we used two parameters biased rw , in which topological properties are @xmath144 i.e. the normalized degree ( with respect to maximal degree in the network ) and @xmath145 i.e. the normalized multiplicity ( with respect to maximal multiplicity in the network ) . we choosed gn network whose parameters are @xmath126 , @xmath120 , @xmath146 and @xmath147 , for which @xmath148 . being @xmath149 the number of communities , as a criterion for good choice of parameters we decided to use the difference between @xmath150 and @xmath151 , i.e. , we decided to maximize the gap between `` community band '' and the rest of eigenvalues ; checking at the same time that the spectral gap shrinks . in fig . [ lambda3lambda4 ] , we plot such a quantity with respect to different biases . it is important to mention that for every single network instance there are different optimal parameters . this can be seen on figure [ lambdahistograms ] , where we show the difference between unbiased and biased eigenvalues for 1000 gn nets created with same parameters . as shown in the figure the difference between fourth and fifth eigenvalue is now not necessarily the optimal for this choice of parameters . every realization of the network should be independently analyzed and its own parameters should be carefully chosen . , @xmath152 , @xmath150 and @xmath151 for 1000 gn networks described with parameters @xmath126 , @xmath120 , @xmath146 and @xmath147 . with black colour we indicate the eigenvalues of nonbiased rw , while with red we indicate the eigenvalues of rw biased with parameters @xmath153 and @xmath154 . note how this choice of parameters does not maximize `` community gap '' for all the different realizations of monitored gn network.,width=302,height=245 ] in the figs . [ unbiased ] and [ biased ] we present instead the difference between unbiased and biased projection on three eigenvectors with largest nontrivial eigenvalues . using 3d view it is easy to check that communities are better separated in the biased case then in the non - biased case . and @xmath147 . for this choice of parameters @xmath148 . there is a strong dispersion between different vertices which belong to the same community.,width=302,height=245 ] and @xmath154 . different markers represent four different predefined communities . this is an example of the same gn graph realization with @xmath146 and @xmath147 as the one on the previous figure . for this choice of parameters @xmath148 . one can notice tetrahedral distribution of vertices in which vertices from the same community belong to the same branch of tetrahedron.,width=302,height=245 ] in this paper we presented a detailed theoretical framework to analyze the evolution of tbrw on a graph . by using as bias some topological property of the graph itself allows to use the rw as a tool to explore the environment . this method maps vertices of the graph to different points in the @xmath155-dimensional euclidean space naturally associated with the given graph . in this way we can measure distances between vertices depending on the chosen bias strategy and bias parameters . in particular we developed a perturbative approach to the spectrum of eigenvalues and eigenvectors associated with the transition matrix of the system . more generally we generalized the quantum pem approach to the present case . this led naturally to study the behavior of the gap between the largest and the second eigenvalue of the spectrum characterizing the relaxation to the stationary markov state . in numerical applications of such a theoretical framework we have observed a unimodal shape of the spectral gap _ vs. _ the bias parameter which is not an obvious feature of the studied processes . we have finally outlined a very promising application of topologically biased random walks to the fundamental problem of community finding . we described the basic ideas and proposed some criteria for the choice of parameters , by considering the particular case of gn graphs . we are working further in direction of this application , but the number of possible strategies ( different topological properties we can use for biasing ) and types of networks is just too large to be presented in one paper . furthermore , since in many dynamical systems as the www or biological networks , a feedback between function and form ( topology ) is evident , our framework may be a useful way to describe mathematically such an observed mechanism . in the case of biology , for instance , the shape of the metabolic networks can be triggered not only by the chemical properties of the compounds , but also by the possibility of the metabolites to interact . biased rw can be therefore the mechanism through which a network attains a particular form for a given function . by introducing such approach we can now address the problem of community detection in the graph . this the reason why here we have not introduced another precise method for community detection , but rather a possible framework to create different community finding methods with different _ ad hoc _ strategies . indeed in real situations we expect different types of network to be efficiently explored by use of different topological properties . this explains why we believe that tbrw could play a role in community detection problems , and we hope to stimulate further developments , in the network scientific community , of this promising methodology . * acknowledgments * vinko zlati wants to thanks mses of the republic of croatia through project no . 098 - 0352828 - 2836 for partial support . authors acknowledge support from ec fet open project `` foc '' nr . 255987 . l. lovasz , _ combinatorics _ * 2 * , 1 - 48 , ( 1993 ) d. aldous , j. fill , _ reversible markov chains and random walks on graphs _ , in press , r. pastor - satorras , a. vespignani _ phys . _ * 86 * , 3200 , ( 2001 ) .

we present a new approach of topology biased random walks for undirected networks . we focus on a one parameter family of biases and by using a formal analogy with perturbation theory in quantum mechanics we investigate the features of biased random walks . this analogy is extended through the use of parametric equations of motion ( pem ) to study the features of random walks _ vs. _ parameter values . furthermore , we show an analysis of the spectral gap maximum associated to the value of the second eigenvalue of the transition matrix related to the relaxation rate to the stationary state . applications of these studies allow _ ad hoc _ algorithms for the exploration of complex networks and their communities .

arxiv

the ability to gain control of a huge amount of internet hosts could be easily achieved by the exploitation of worms which self - propagate through popular internet applications and services . internet worms have already proven their capability of inflicting massive - scale disruption and damage to the internet infrastructure . these worms employ normal _ scanning _ as a strategy to find potential vulnerable targets , i.e. , they randomly select victims from the ip address space . so far , there have been many existing schemes that are effective in detecting such scanning worms @xcite , e.g. , by capturing the scanning events @xcite or by passively detecting abnormal network traffic activities @xcite . in recent years , peer - to - peer ( p2p ) overlay applications have experienced an explosive growth , and now dominate large fractions of both the internet users and traffic volume @xcite ; thus , a new type of worms that leverage the popular p2p overlay applications , called _ p2p worms _ , pose a very serious threat to the internet @xcite . generally , the p2p worms can be grouped into two categories : _ passive _ p2p worms and _ active _ p2p worms . the passive p2p worm attack is generally launched either by copying such worms into a few p2p hosts shared folders with attractive names , or by participating into the overlay and responding to queries with the index information of worms . unable to identify the worm content , normal p2p hosts download these worms unsuspectedly into their own shared folders , from which others may download later without being aware of the threat , thus passively contributing to the worm propagation . the passive p2p worm attack could be mitigated by current patching systems @xcite and reputation models @xcite . in this paper , we focus on another serious p2p worm : active p2p worm . the active p2p worms could utilize the p2p overlay applications to retrieve the information of a few vulnerable p2p hosts and then infect these hosts , or as an alternative , these worms are directly released in a hit list of p2p hosts to bootstrap the worm infection . since the active p2p worms have the capacity of gaining control of the infected p2p hosts , they could perform rapid _ topological self - propagation _ by spreading themselves to neighboring hosts , and in turn , spreading throughout the whole network to affect the quality of overlay service and finally cause the overlay service to be unusable . the p2p overlay provides an accurate way for worms to find more vulnerable hosts easily without probing randomly selected ip addresses ( i.e. , low connection failure rate ) . moreover , the worm attack traffic could easily blend into normal p2p traffic , so that the active p2p worms will be more deadly than scanning worms . that is , they do not exhibit easily detectable anomalies in the network traffic as scanning worms do , so many existing defenses against scanning worms are no longer effective @xcite . besides the above internal infection in the p2p overlay , the infected p2p hosts could again mount attacks to external hosts . in similar sense , since the p2p overlay applications are pervasive on today s internet , it is also attractive for malicious external hosts to mount attacks against the p2p overlay applications and then employ them as an ideal platform to perform future massive - scale attacks , e.g. , botnet attacks . in this paper , we aim to develop a _ holistic _ immunity system to provide the mechanisms of both _ internal defense _ and _ external protection _ against active p2p worms . in our system , we elect a small subset of p2p overlay nodes , _ phagocytes _ , which are immune with high probability and specialized in finding and `` eating '' active p2p worms . each phagocyte in the p2p overlay is assigned to manage a group of p2p hosts . these phagocytes monitor their managed p2p hosts connection patterns and traffic volume in an attempt to detect active p2p worm attacks . once detected , the local isolation procedure will cut off the links of all the infected p2p hosts . afterwards , the responsible phagocyte performs the contagion - based alert propagation to spread worm alerts to the neighboring phagocytes , and in turn , to other phagocytes . here , we adopt a threshold strategy to limit the impact area and enhance the robustness against the malicious alert propagations generated by infected phagocytes . finally , the phagocytes help acquire the software patches and distribute them to the managed p2p hosts . with the above four modules , i.e. , detection , local isolation , alert propagation and software patching , our system is capable of preventing internal active p2p worm attacks from being effectively mounted within the p2p overlay network . the phagocytes also provide the access control and filtering mechanisms for the connection establishment between the internal p2p overlay and the external hosts . firstly , the p2p traffic should be contained within the p2p overlay , and we forbid any p2p traffic to leak from the p2p overlay to external hosts . this is because such p2p traffic is generally considered to be malicious and it is possible that the p2p worms ride on such p2p traffic to spread to the external hosts . secondly , in order to prevent external worms from attacking the p2p overlay , we hide the p2p hosts ip addresses with the help of scalable distributed dns service , e.g. , codons @xcite . an external host who wants to gain access to the p2p overlay has no alternative but to perform an interaction towards the associated phagocyte to solve an adaptive computational puzzle ; then , according to the authenticity of the puzzle solution , the phagocyte can determine whether to process the request . we implement a prototype system , and evaluate its performance on a massive - scale testbed with realistic p2p network traces . the evaluation results validate the effectiveness and efficiency of our proposed holistic immunity system against active p2p worms . * outline*. we specify the system architecture in section [ sec : systemarchitecture ] . sections [ sec : internaldefenses ] and [ sec : externaldefenses ] elaborate the internal defense and external protection mechanisms , respectively . we then present the experimental design in section [ sec : exdesign ] , and discuss the evaluation results in section [ sec : exresults ] . finally , we give an overview of related work in section [ sec : relatedwork ] , and conclude this paper in section [ conclusions ] . current p2p overlay networks can generally be grouped into two categories @xcite : _ structured _ overlay networks , e.g. , chord @xcite , whose network topology is tightly controlled based on distributed hash table , and _ unstructured _ overlay networks , e.g. , gnutella @xcite , which merely impose loose structure on the topology . in particular , most modern unstructured p2p overlay networks utilize a two - tier structure to improve their scalability : a subset of peers , called _ ultra - peers _ , construct an unstructured mesh while the other peers , called _ leaf - peers _ , connect to the ultra - peer tier for participating into the overlay network . as shown in figure [ fig : lovers ] , the network architecture of our system is similar to that of the two - tier unstructured p2p overlay networks . in our system , a set of p2p hosts act as the phagocytes to perform the functions of defense and protection against active p2p worms . these phagocytes are elected among the participating p2p hosts in terms of the following metrics : high bandwidth , powerful processing resource , sufficient uptime , and applying the latest patches ( interestingly , the experimental result shown in section [ sec : exresults ] indicates that we actually do not need to have a large percentage of phagocytes applying the latest patches ) . as existing two - tier unstructured overlay networks do , the phagocyte election is performed periodically ; moreover , even if an elected phagocyte has been infected , our internal defense mechanism ( described in section [ sec : internaldefenses ] ) can still isolate and patch the infected phagocyte immediately . in particular , the population of phagocytes should be small as compared to the total overlay population , otherwise the scalability and applicability are questionable . as a result , each elected phagocyte covers a number of managed p2p hosts , and each managed p2p host will belong to one closest phagocyte . that is , the phagocyte acts as the proxy for its managed hosts to participate into the p2p overlay network , and has the control over the managed p2p hosts . moreover , a phagocyte further connects to several nearby phagocytes based on close proximity . our main interest is the unstructured p2p overlay networks , since most of the existing p2p worms target the unstructured overlay applications @xcite . naturally , due to the similar network architecture , our system can easily be deployed into the unstructured p2p overlay networks . moreover , for structured p2p overlay networks , a subset of p2p hosts could be elected to perform the functions of phagocytes . we aim not to change the network architecture of the structured p2p overlay networks ; however , we elect phagocytes to form an overlay to perform the defense and protection functions this overlay acts as a security wall in a separate layer from the existing p2p overlay , thus not affecting the original p2p operations . in the next two sections , we will elaborate in detail our mechanisms of internal defense and external protection against active p2p worms . in this section , we first describe the active p2p worm attacks , and then , we design our internal defense mechanism . generally , active p2p worms utilize the p2p overlay to accurately retrieve the information of a few vulnerable p2p hosts , and then infect these hosts to bootstrap the worm infection . on one hand , a managed p2p host clearly knows its associated phagocyte and its neighboring p2p hosts that are managed by the same phagocyte ; so now , an infected managed p2p host could perform the worm infection in several ways simultaneously . firstly , the infected p2p host infects its neighboring managed p2p hosts very quickly . secondly , the infected p2p host attempts to infect its associated phagocyte . lastly , the infected managed p2p host could issue p2p key queries with worms to infect many vulnerable p2p hosts managed by other phagocytes . on the other hand , a phagocyte could be infected as well ; if so , the infected phagocyte infects its managed p2p hosts and then its neighboring phagocytes . as a result , in such a topological self - propagation way , the active p2p worms spread through the whole system at extraordinary speed . since the active p2p worms propagate based on the topological information , and do not need to probe any random ip addresses , thus their connection failure rate should be low ; moreover , the p2p worm attack traffic could easily blend into normal p2p traffic . therefore , the active p2p worms do not exhibit easily detectable anomalies in the network traffic as normal scanning worms do . in our system , the phagocytes are those elected p2p hosts with the latest patches , and they can help their managed p2p hosts detect the existence of active p2p worms by monitoring these managed hosts connection transactions and traffic volume . if a managed p2p host always sends similar queries or sets up a large number of connections , the responsible phagocyte deduces that this managed p2p host is infected . another pattern the phagocytes will monitor is to determine if a portion of the managed p2p hosts have some similar behaviors such as issuing the similar queries , repeating to connect with their neighboring hosts , uploading / downloading the same files , etc . , then they are considered to be infected . concretely , a managed p2p host s _ latest _ behaviors are processed into a _ behavior sequence _ consisting of continuous @xmath0 hereafter . ] then , we can compute the behavior similarity between any two p2p hosts by using the _ levenshtein edit distance _ @xcite . without loss of generality , we suppose that there are two behavior sequences @xmath1 and @xmath2 , in which @xmath3 , where @xmath4 , and @xmath5 is the length of the behavior sequence . further , we can treat each behavior sequence @xmath6 as the combination of the _ operation sequence _ @xmath7 and the _ payload sequence _ @xmath8 . now , we simultaneously _ sort _ the operation sequence @xmath9 and the payload sequence @xmath10 of the behavior sequence @xmath2 to make the following similarity score @xmath11 be maximum . to obtain the optimal solution , we could adopt the _ maximum weighted bipartite matching _ algorithm @xcite ; however , for efficiency , we use the _ greedy _ algorithm to obtain the approximate solution as an alternative . @xmath12 here , @xmath13 denotes the sorted @xmath2 ; @xmath14 and @xmath15 denote the @xmath16 item of the sorted @xmath9 and @xmath17 , respectively ; @xmath18 is the levenshtein edit distance function . finally , we treat the maximum @xmath11 as the similarity score of the two behavior sequences . if the score exceeds a threshold @xmath19 , we consider the two p2p hosts perform similarly . these detection operations are also performed between phagocytes at the phagocyte - tier because they could be infected as well though with latest patches . the infected phagocytes could perform the worm propagation rapidly ; however , we have the local isolation , alert propagation and software patching procedures in place to handle these infected phagocytes after detected by their neighboring phagocytes with the detection module as described above . note that , our detection mechanism is _ not _ a substitution for the existing worm detection mechanisms , e.g. , the worm signature matching @xcite , but rather an effective p2p - tailored complement to them . specifically , some _ tricky _ p2p worms may present the features of mild propagation rate , polymorphism , etc . , so they may maliciously propagate in lower speed than the aggressive p2p worms ; here , our software patching module ( in section [ subsec : patching ] ) and several existing schemes @xcite can help mitigate such tricky worm attacks . moreover , a few elaborate p2p worms , e.g. , p2p-worm.win32.hofox , have recently been reported to be able to kill the anti - virus / anti - worm programs on p2p hosts @xcite ; at the system level , some local countermeasures have been devised to protect defense tools from being eliminated , and the arms race will continue . in this paper , we assume that p2p worms can not disable our detection module , and therefore , each phagocyte can perform the normal detection operations as expected ; so can the following modules . if a phagocyte discovers that some of its managed p2p hosts are infected , the phagocyte will cut off its connections with the infected p2p hosts , and ask these infected hosts to further cut off the links towards any other p2p hosts . also , if a phagocyte is detected ( by its neighboring phagocyte ) as infected , the detecting phagocyte immediately issues a message to ask the infected phagocyte to cut off the connections towards the neighboring phagocytes , and then to trigger the software patching module ( in section [ subsec : patching ] ) at the infected phagocyte ; after the software patching , these cut connections should be reestablished . with the local isolation module , our system has the capacity of self - organizing and self - healing . we utilize the local isolation to limit the impact of active p2p worms as quickly as possible . if a worm event has been detected , i.e. , any of the managed p2p hosts or neighboring phagocytes are detected as infected , the phagocyte propagates a worm alert to all its neighboring phagocytes . further , once a phagocyte has received the worm alerts from more than a threshold @xmath20 of its neighboring phagocytes , it also propagates the alert to all its neighboring phagocytes that did not send out the alert . in general , we should appropriately tune @xmath20 to limit the impact area and improve the robustness against the malicious alert propagation generated by infected phagocytes . the analytical study in @xcite implied that the effective software patching is feasible for an overlay network if combined with schemes to bound the worm infection rate . in our system , the security patches are published to the participating p2p hosts using the following two procedures : * periodical patching : * a patch distribution service provided by system maintainers periodically pushes the latest security patches to all phagocytes through the underlying p2p overlay , and then these phagocytes install and distribute them to all their managed p2p hosts . note that , we can utilize the periodical patching to help mitigate the tricky p2p worms ( in section [ subsec : detection ] ) which are harder to be detected . * urgent patching : * when a phagocyte is alerted of a p2p worm attack , it will immediately pull the latest patches from a system maintainer via the direct http connection ( for efficiency , not via the p2p overlay ) , and then install and disseminate them to all its managed p2p hosts . specifically , each patch must be signed by the system maintainer @xcite , so that each p2p host can verify the patch according to the signature . note that , the zero - day vulnerabilities are not fictional , thus installing the latest patches can not always guarantee the worm immunity . the attackers may utilize these vulnerabilities to perform deadly worm attacks . we can integrate our system with some other systems , e.g. , shield @xcite and vigilante @xcite , to defend against such attacks , which can be found in @xcite . as much as possible , the phagocytes provide the containment of p2p worms in the p2p overlay networks . further , we utilize the phagocytes to implement the p2p traffic filtering mechanism which forbids any p2p connections from the p2p overlay to external hosts because such p2p connections are generally considered to be malicious it is possible that the p2p worms ride on the p2p traffic to spread to the external hosts . we can safely make the assumption that p2p overlay traffic should be contained inside the p2p overlay boundary , and any leaked p2p traffic is abnormal . therefore , once this leakiness is detected , the phagocytes will perform the former procedures for local isolation , alert propagation and software patching . our external protection mechanism aims to protect the p2p overlay network against the external worm attacks . we hide the p2p hosts ip addresses to prevent external hosts from directly accessing the internal p2p resources . this service can be provided by a scalable distributed dns system , e.g. , codons @xcite . such dns system returns the associated phagocyte which manages the requested p2p host . then , the phagocyte is able to adopt our following proposed computational puzzle scheme to perform the function of access control over the requests issued by the requesting external host . we propose a _ novel _ adaptive and interaction - based computational puzzle scheme at the phagocytes to provide the access control over the possible external worms attacking the internal p2p overlay . since we are interested in how messages are processed as well as what messages are sent , for clarity and simplicity , we utilize the annotated alice - and - bob specification to describe our puzzle scheme . as shown in figure [ fig : puzzle ] , to gain access to the p2p overlay , an external host has to perform a lightweight interaction towards the associated phagocyte to solve an adaptive computational puzzle ; then , according to the authenticity of the puzzle solution , the phagocyte can determine whether to process the request . * step 1 . * the external host @xmath21 first generates a @xmath22-bit nonce @xmath23 as its session identifier @xmath24 . then , the external host stores @xmath24 and sends it to the phagocyte . * step 2 . * on receiving the message consisting of @xmath24 sent by the host @xmath21 , the phagocyte @xmath25 adaptively adjusts the puzzle difficulty @xmath26 based on the following two real - time statuses of the network environment . @xmath27 _ status of phagocyte _ : this status indicates the usage of the phagocyte s resources , i.e. , the ratio of consumed resources to total resources possessed by the phagocyte . the more resources a phagocyte has consumed , the harder puzzles the phagocyte issues in the future . @xmath27 _ status of external host _ : in order to mount attacks against p2p hosts effectively , malicious external hosts have no alternative but to perform the interactions and solve many computational puzzles . that is , the more connections an external host tries to establish , the higher the probability that this activity is malicious and worm - like . hence , the more puzzles an external host has solved in the recent period of time , the harder puzzles the phagocyte issues to the very external host . note that , since a malicious external host could simply spoof its ip address , in order to effectively utilize the status of external host , our computational puzzle scheme should have the capability of defending against ip spoofing attacks , which we will describe later . subsequently , the phagocyte @xmath25 simply generates a _ unique _ @xmath22-bit session identifier @xmath28 for the external host according to the host s ip address @xmath29 ( extracted from the ip header of the received message ) , the host s session identifier @xmath24 and the puzzle difficulty @xmath26 , as follows : @xmath30 here , the @xmath31 is a keyed hash function for message authentication , and the @xmath32 is a @xmath33-bit key which is _ periodically _ changed and only known to the phagocyte itself . such @xmath32 limits the time external hosts have for computing puzzle solutions , and it also guarantees that an external attacker usually does not have enough resources to pre - compute all possible solutions in step 3 . after the above generation process , the phagocyte replies to the external host at @xmath29 with the host s session identifier @xmath24 , the phagocyte s session identifier @xmath28 and the puzzle difficulty @xmath26 . once the external host has received this reply message , it first checks whether the received @xmath24 is really generated by itself . if the received @xmath24 is bogus , the external host simply drops the message ; otherwise , the host stores the phagocyte s session identifier @xmath28 immediately . such reply and checking operations can effectively defend against ip spoofing attacks . * step 3 . * the external host @xmath21 retrieves the @xmath34 pair as the global session identifier , and then tries to solve the puzzle according to the equation below : @xmath35 here , the @xmath36 is a cryptographic hash function , the @xmath37 is a hash value with the first @xmath26 bits being equal to @xmath38 , and the @xmath39 is the puzzle _ solution_. due to the features of hash function , the external host has no way to figure out the solution other than brute - force searching the solution space until a solution is found , even with the help of many other solved puzzles . the cost of solving the puzzle depends exponentially on the difficulty @xmath26 , which can be effortlessly adjusted by the phagocyte . after the brute - force computation , the external host sends the phagocyte a message including the global session identifier ( i.e. , the @xmath40 pair ) , the puzzle difficulty , the puzzle solution and the actual _ request_. once the phagocyte has received this message , it performs the following operations in turn : * _ a _ ) * check whether the session identifier @xmath41 is really fresh based on the database of the past global session identifiers . this operation can effectively defend against replay attacks . * _ b _ ) * check whether the phagocyte s session identifier @xmath28 can be correctly generated according to equation [ eqn : sip ] . specifically , this operation can additionally check whether the difficulty level @xmath26 reported by the external host is the original @xmath26 determined by the phagocyte . * _ c _ ) * check whether the puzzle solution is correct according to equation [ eqn : compute ] , which will also not incur significant overhead on the phagocyte . * _ d _ ) * store the global session identifier @xmath41 , and act as the overlay proxy to transmit the request submitted by the external host . note that , in our scheme , the phagocyte stores the session - specific data and processes the actual request only after it has verified the external host s puzzle solution . that is , the phagocyte does not commit its resources until the external host has demonstrated the sincerity . specifically in the above sequence of operations , if one operation succeeds , the phagocyte continues to perform the next ; otherwise , the phagocyte cancels all the following operations , and the entire interaction ends . more details about the puzzle design rationale can be found in @xcite . so far , several computational puzzle schemes @xcite have been proposed . however , most of them consider only the status of resource providers , so they can not reflect the network environment completely . recently , an ingenious puzzle scheme , portcullis @xcite , was proposed . in portcullis , since a resource provider gives priority to requests containing puzzles with higher difficulty levels , to gain access to the requested resources , each resource requester , no matter legitimate or malicious , has to compete with each other and solve hard puzzles under attacks . this may influence legitimate requesters experiences significantly . compared with existing puzzle schemes , our adaptive and interaction - based computational puzzle scheme satisfies the fundamental properties of a good puzzle scheme @xcite . it treats each external host _ distinctively _ by performing a lightweight interaction to flexibly adjust the puzzle difficulty according to the real - time statuses of the network environment . this guarantees that our computational puzzle scheme does not influence legitimate external hosts experiences significantly , and it also prevents a malicious external host from attacking p2p overlay without investing unbearable resources . in real - world networks , hosts computation capabilities vary a lot , e.g. , the time to solve a puzzle would be much different between a host with multiple fast cpus and a host with just one slow cpu . to decrease the computational disparity , some other kinds of puzzles , e.g. , memory - bound puzzle @xcite , could be complementary to our scheme . note that , with low probability , a phagocyte may also be compromised by external worm attackers , then they could perform the topological worm propagation ; here , our proposed internal defense mechanism could be employed to defend against such attacks . in our experiments , we first implement a prototype system , and then construct a massive - scale testbed to verify the properties of our prototype system . * internal defense . * we implement an internal defense prototype system including all basic modules described in section [ sec : internaldefenses ] . here , a phagocyte monitors each of its connected p2p hosts latest @xmath42 requests . firstly , if more than half of the managed p2p hosts perform similar behaviors , the responsible phagocyte considers that the managed zone is being exploited by worm attackers . secondly , if more than half of a phagocyte s neighboring phagocytes perform the similar operations , the phagocyte considers its neighboring phagocytes are being exploited by worm attackers . in particular , the similarity is measured based on the equation [ eqn : sim ] with a threshold @xmath19 of @xmath43 . then , in the local isolation module , if a phagocyte has detected worm attacks , the phagocyte will cut off the associated links between the infection zone and the connected p2p hosts . afterwards , in the alert propagation module , if a phagocyte has detected any worm attacks , it will broadcast a worm alert to all its neighboring phagocytes ; further , if a phagocyte receives more than half of its neighboring phagocytes worm alerts , i.e. , @xmath44 , the phagocyte will also broadcast the alert to all its neighboring phagocytes that did not send out the alert . finally , in the software patching module , the phagocytes acquire the patches from the closest one of the system maintainers ( i.e. , @xmath42 online trusted phagocytes in our testbed ) , and then distribute them to all their managed p2p hosts . we have not yet integrated the signature scheme into the software patching module of our prototype system . note that , in the above , we simply set the parameters used in our prototype system , and in real - world systems , the system designers should appropriately tune these parameters according to their specific requirements . * external protection . * we utilize our adaptive and interaction - based computational puzzle module to develop the external protection prototype system . in this prototype system , we use sha1 as the cryptographic hash function . generally , solving a puzzle with difficulty level @xmath26 will force an external host to perform @xmath45 sha1 computations on average . in particular , the difficulty level @xmath26 varies between @xmath38 and @xmath46 in our system this will cost an external host @xmath47 second ( @xmath48 ) to @xmath49 seconds ( @xmath50 ) on our power5 cpus . in addition , the change cycle of the puzzle - related parameters is set to @xmath51 minutes . yet , we have not integrated our prototype system with the scalable distributed dns system , and this work will be part of our future work . we use the realistic network traces crawled from a million - node gnutella network by the cruiser @xcite crawler . the dedicated massive scale gnutella network is composed of two tiers including the ultra - peer tier and leaf - peer tier . for historical reasons , the ultra - peer tier consists of not only modern ultra - peers but also some _ legacy - peers _ that reside in the ultra - peer tier but can not accept any leaf - peers . specifically , in our experiments , the ultra - peers excluding legacy - peers perform the functions of phagocytes , and the leaf - peers act as the managed p2p hosts . then , we adopt the widely accepted gt - itm @xcite to generate the transit - stub model consisting of @xmath52 routers for the underlying hierarchical internet topology . there are @xmath53 transit domains at the top level with an average of @xmath53 routers in each , and a link between each pair of these transit routers has a probability of @xmath43 . each transit router has an average of @xmath53 stub domains attached , and each stub has an average of @xmath53 routers , with the link between each pair of stub routers having a probability of @xmath54 . there are two million end - hosts uniformly assigned to routers in the core by local area network ( lan ) links . the delay of each lan link is set to @xmath51ms and the average delay of core links is @xmath55ms . now , the crawled gnutella networks can model the realistic p2p overlay , and the generated gt - itm network can model the underlying internet topology ; thus , we deploy the crawled gnutella networks upon the underlying internet topology to simulate the realistic p2p network environment . we do not model queuing delay , packet losses and any cross network traffic because modeling such parameters would prevent the massive - scale network simulation . as shown in table [ tab : trace ] , we list various gnutella traces that we use in our experiments with different node populations and/or different percentages of phagocytes . @xmath27 _ trace _ 1 : crawled by cruiser on sep . 27th , 2004 . @xmath27 _ trace _ 2 : crawled by cruiser on feb . 2nd , 2005 . @xmath27 _ trace _ 3 : based on trace 1 , we remove a part of phagocytes randomly ; then , we remove the _ isolated _ phagocytes , i.e. , these phagocytes do not connect to any other phagocytes ; finally , we further remove the isolated managed p2p hosts , i.e. , these managed p2p hosts do not connect to any phagocytes . @xmath27 _ trace _ 4 : based on trace 3 , we remove a part of managed p2p hosts randomly . @xmath27 _ trace _ 5 : based on trace 4 , we further remove a part of managed p2p hosts randomly . @xmath27 _ trace _ 6 : based on trace 1 , we use the same method as described in the generation process of trace 3 . in addition , we remove an extra part of managed p2p hosts . in our experiments , we characterize the performance under various different circumstances by using three metrics : @xmath27 _ peak infection percentage of all p2p hosts _ : the ratio of the maximum number of infected p2p hosts to the total number of p2p hosts . this metric indicates whether phagocytes can effectively defend against internal attacks . @xmath27 _ blowup factor of latency _ : this factor is the latency penalty between the external hosts and the p2p overlay via the phagocytes and direct routing . this indicates the efficiency of our phagocytes to filter the requests from external hosts to the p2p overlay . @xmath27 _ percentage of successful external attacks _ : the ratio of the number of successful external attacks to the total number of external attacks . this metric indicates the effectiveness of our phagocytes to prevent external hosts from attacking the p2p overlay . in our prototype system , we model a percentage of phagocytes and managed p2p hosts being initially _ immune _ , respectively ; except these immune p2p hosts , the other hosts are _ vulnerable_. moreover , there are a percentage of p2p hosts being initially _ infected _ , which are distributed among these vulnerable phagocytes and vulnerable managed p2p hosts uniformly at random . all the infected p2p hosts perform the active p2p worm attacks ( described in section [ subsec : threat_model ] ) , and meanwhile , our internal defense modules deployed at each participant try to defeat such attacks . with different experimental parameters described in table [ tab : ex_in ] , we conduct four different experiments to evaluate the internal defense mechanism . * experiment 1 impact of immune phagocytes : * with seven different initial percentages of immune phagocytes , we fix the initial percentage of immune managed p2p hosts to @xmath56 , and vary the number of initial infected p2p hosts so that these infected hosts make up between @xmath57 and @xmath58 of all the vulnerable p2p hosts . now , we can investigate the impact of immune phagocytes by calculating the peak infection percentage of all p2p hosts . the experimental result shown in figure [ fig : ex1 ] demonstrates that when the initial infection percentage of all vulnerable p2p hosts is low ( e.g. , @xmath59 ) , the phagocytes can provide a good containment of active p2p worms ; otherwise , the worm propagation is very fast , but the phagocytes could still provide the sufficient containment this property is also held in the following experiments . interestingly , the initial percentage of immune phagocytes does not influence the performance of our system significantly , i.e. , the percentage of phagocytes being initially immune has no obvious effect . this is a good property because we do not actually need to have high initial percentage of immune phagocytes . also , this phenomenon implies that increasing the number of immune phagocytes does not further provide much significant defense . thus , we can clearly conclude that the phagocytes are effective and scalable in performing detection , local isolation , alert propagation and software patching . * experiment 2 impact of immune managed p2p hosts : * in this experiment , for @xmath60 of phagocytes being initially immune , we investigate the performance of our system with various initial percentages of immune managed p2p hosts in steps of @xmath56 . the result shown in figure [ fig : ex2 ] is within our expectation . the peak infection percentage of all p2p hosts decreases with the growth of the initial percentage of immune managed p2p hosts . actually , in real - world overlay networks , even a powerful attacker could initially control tens of thousands of overlay hosts ( @xmath61 @xmath62 in the x - axis ) ; hence , we conclude that our phagocytes have the capacity of defending against active p2p worms effectively even in a highly malicious environment . * experiment 3 impact of network scale : * figure [ fig : ex3 ] plots the performance of our system in terms of different network scales . in traces 1 , 2 , 5 and 6 , there are different node populations , but the ratios of the number of phagocytes to the number of all p2p hosts are all around @xmath63 . the experimental result indicates that our system can indeed help defend against active p2p worms in various overlay networks with different network scales . furthermore , although the phagocytes perform more effectively in smaller overlay networks ( e.g. , traces 5 and 6 ) , they can still work quite well in massive - scale overlay networks with million - node participants ( e.g. , traces 1 and 2 ) . * experiment 4 impact of the percentage of phagocytes : * in our system , the phagocytes perform the functions of defending against p2p worms . in this experiment , we evaluate the system performance with different percentages of phagocytes but the same number of phagocytes . the result in figure [ fig : ex4 ] indicates that the higher percentage of phagocytes , the better security defense against active p2p worms . that is , as the percentage of phagocytes increases , we can persistently improve the security capability of defending against active p2p worms in the overlay network . further , the experimental result also implies that we do not need to have a large number of phagocytes to perform the defense functions around @xmath56 of the node population functioning as phagocytes is sufficient for our system to provide the effective worm containment . in this section , we conduct two more experiments in our prototype system to evaluate the performance of the external protection mechanism . * experiment 5 efficiency : * in this experiment , we show the efficiency in terms of the latency penalty between the external hosts and the p2p overlay via the phagocytes and direct routing . based on trace 1 , we have @xmath42 external hosts connect to every p2p host via the phagocytes and direct routing in turn . then , we measure the latencies for both cases . figure [ fig : ex5 ] plots the measurement result of latency penalty . we can see that , if routing via the phagocytes , about @xmath64 and @xmath65 of the connections between the external hosts and p2p hosts have the blowup factor of latency be less than @xmath66 and @xmath67 , respectively . figure [ fig : ex5a ] shows the corresponding absolute latency difference , from which we can further deduce that the average latency growth of more than half of these connections ( via the phagocytes ) is less than @xmath68ms . actually , due to the interaction required by our proposed computational puzzle scheme , we would expect some latency penalty incurred by routing via the phagocytes . with the puzzle scheme , our system can protect against external attacks effectively which we will illustrate in the next experiment . hence , there would be a tradeoff between the efficiency and effectiveness . * experiment 6 effectiveness : * in this experiment , based on trace 1 , we have @xmath42 external worm attackers flood all phagocytes in the p2p overlay . then , we evaluate the percentage of successful external attacks to show the effectiveness of our protection mechanism against external hosts attacking the p2p overlay . for other numbers of external worm attackers , we obtain the similar experimental results . in figure [ fig : ex6 ] , the x - axis is the attack frequency in terms of the speed of external hosts mounting worm attacks to the p2p overlay , and the y - axis is the percentage of successful external attacks . the result clearly illustrates the effectiveness of phagocytes in protecting the p2p overlay from external worm attacks . our adaptive and interaction - based computational puzzle module at the phagocytes plays an important role in contributing to this observation . even in an extremely malicious environment , our system is still effective . that is , to launch worm attacks , the external attackers have no alternative but to solve hard computational puzzles which will incur heavy burden on these attackers . from the figure [ fig : ex6 ] , we can also find that when the attack frequency decreases , the percentage of successful external attacks increases gradually . however , with a low attack frequency , the attackers can not perform practical attacks . even if a part of external attacks are mounted successfully , our internal defense mechanism can mitigate them effectively . p2p worms could exploit the perversive p2p overlays to achieve fast worm propagation , and recently , many p2p worms have already been reported to employ real - world p2p systems as their spreading platforms @xcite . the very first work in @xcite highlighted the dangers posed by p2p worms and studied the feasibility of self - defense and containment inside the p2p overlay . afterwards , several studies @xcite developed mathematical models to understand the spreading behaviors of p2p worms , and showed that p2p worms , especially the active p2p worms , indeed pose more deadly threats than normal scanning worms . recognizing such threats , many researchers started to study the corresponding defense mechanisms . specifically , yu _ et al . _ in @xcite presented a region - based active immunization defense strategy to defend against active p2p worm attacks ; freitas _ et al . _ in @xcite utilized the diversity of participating hosts to design a worm - resistant p2p overlay , verme , for containing possible p2p worms ; moreover , in @xcite , xie and zhu proposed a partition - based scheme to proactively block the possible worm spreading as well as a connected dominating set based scheme to achieve fast patch distribution in a race with the worm , and in @xcite , xie _ et al . _ further designed a p2p patching system through file - sharing mechanisms to internally disseminate security patches . however , existing defense mechanisms generally focused on the internal p2p worm defense without the consideration of external worm attacks , so that they can not provide a total worm protection for the p2p overlay systems . in this paper , we have addressed the deadly threats posed by active p2p worms which exploit the pervasive and popular p2p applications for rapid topological worm infection . we build an immunity system that responds to the active p2p worm infection by using _ phagocytes_. the phagocytes are a small subset of specially elected p2p hosts that have high immunity and can `` eat '' active p2p worms in the p2p overlay networks . each phagocyte manages a group of p2p hosts by monitoring their connection patterns and traffic volume . if any worm events are detected , the phagocyte will invoke the internal defense strategies for local isolation , alert propagation and software patching . besides , the phagocytes provide the access control and filtering mechanisms for the communication establishment between the p2p overlay and external hosts . the phagocytes forbid the p2p traffic to leak from the p2p overlay to external hosts , and further adopt a novel adaptive and interaction - based computational puzzle scheme to prevent external hosts from attacking the p2p overlay . to sum up , our holistic immunity system utilizes the phagocytes to achieve both internal defense and external protection against active p2p worms . we implement a prototype system and validate its effectiveness and efficiency in massive - scale p2p overlay networks with realistic p2p network traces .

active peer - to - peer ( p2p ) worms present serious threats to the global internet by exploiting popular p2p applications to perform rapid topological self - propagation . active p2p worms pose more deadly threats than normal scanning worms because they do not exhibit easily detectable anomalies , thus many existing defenses are no longer effective . we propose an immunity system with _ phagocytes _ a small subset of elected p2p hosts that are immune with high probability and specialized in finding and `` eating '' worms in the p2p overlay . the phagocytes will monitor their managed p2p hosts connection patterns and traffic volume in an attempt to detect active p2p worm attacks . once detected , local isolation , alert propagation and software patching will take place for containment . the phagocytes further provide the access control and filtering mechanisms for communication establishment between the internal p2p overlay and the external hosts . we design a novel adaptive and interaction - based computational puzzle scheme at the phagocytes to restrain external worms attacking the p2p overlay , without influencing legitimate hosts experiences significantly . we implement a prototype system , and evaluate its performance based on realistic massive - scale p2p network traces . the evaluation results illustrate that our phagocytes are capable of achieving a total defense against active p2p worms .

arxiv

debris disk systems provide a look at an intermediate stage of stellar system evolution . they represent the transition between the early formation of stars and planets in a primordial protoplanetary disk as seen toward pre - main sequence stars , and the mature stage of an evolved system , like our solar system , which is clear of all primordial material and retains only a hint of secondary products ( e.g. , zodiacal dust ) , the final remnants of the stellar and planetary formation process . although a debris disk has lost most of its primordial material , the observed infrared luminosity of circumstellar dust , caused by collisions of planetismals and other small bodies , is typically several orders of magnitude larger than estimated for the kuiper and asteroid belts in our solar system @xcite . ever since the detection of dusty circumstellar material around main sequence stars via infrared excesses @xcite , researchers have been looking for circumstellar gas phase absorption @xcite . of the initial major infrared excess main sequence stars , only @xmath0 pic showed gas phase absorption in optical absorption lines ( e.g. , and ) , due to its disk morphology and edge - on orientation @xcite . such on orientation provides a unique opportunity to simultaneously measure both the dust and gas components of a debris disk , at an interesting transition near the end of stellar and planetary formation . only a few other edge - on debris disks have been found since , including @xmath0 car @xcite , hd85905 @xcite , hr10 @xcite , and au mic ( @xcite @xcite ; @xcite @xcite ) . @xcite observed @xmath0 car , hd85905 , hr10 with the _ spitzer space telescope _ and did not find strong infrared excesses toward any of them , although an optical monitoring campaign showed clear signs of gas variability , as noted by researchers earlier . however , the magnitude of circumstellar absorption in these systems is lower than observed toward @xmath0 pic . long monitoring campaigns of @xmath0 pic ( e.g. , * ? ? ? * ) , find significant short - term absorption variability . this variability can be explained by gas clouds very close to the star , which are caused by evaporating , star - grazing , km - sized objects , simply referred to as , falling evaporating bodies ( feb s ; * ? ? ? . a strong `` stable '' component , at rest in the stellar reference frame , is also detected toward @xmath0 pic ( e.g. , * ? ? ? the distribution of gas in this component , contrary to the variable component located very close to the star , is dispersed throughout the extended dust disk @xcite . a `` stable '' absorption component in a gas phase resonance line can be caused by either intervening circumstellar or interstellar gas . measuring the interstellar medium ( ism ) along the line of sight and in the locality surrounding a circumstellar disk candidate , is critical to characterizing any `` contaminating '' ism absorption @xcite . in particular , the sun resides in a large scale ism structure known as the local bubble , whose boundary at @xmath4100pc is defined by a significant quantity of interstellar material @xcite . if a `` stable '' absorption component is observed at the stellar radial velocity , and similar absorption is not detected toward any proximate stars , it is likely that the absorption component is caused by circumstellar material . using near - infrared scattered light observations taken with the _ hubble space telescope _ , @xcite discovered that the debris disk surrounding hd32297 has an edge - on orientation . disk emission extends out to @xmath4400au in their observations , while radii @xmath633.6au are occulted by the coronagraphic obstacle . optical scattered light observations by @xcite confirmed this orientation and extended the range of disk emission to @xmath41680au . the edge - on orientation of hd32297 makes it an ideal target for gas phase absorption measurements . observations of the d doublet ( 5895.9242 and 5889.9510 ) toward hd32297 were made over several epochs . the doublet is among the strongest transitions in the optical wavelength band , appropriate for observing interstellar @xcite and circumstellar @xcite absorption toward nearby stars . in addition , several stars in close angular proximity to hd32297 were observed , in order to reconstruct the ism absorption profile along the line of sight . stellar parameters of the observed targets are given in table [ tab : basics ] , and the observational parameters are listed in table [ tab : fits ] . high resolution optical spectra were obtained using the coud spectrometer on the 2.7 m harlan j. smith telescope at mcdonald observatory . the spectra were obtained at a resolution of @xmath7240,000 , using the 2dcoud spectrograph @xcite in the cs21 configuration . the data were reduced using image reduction and analysis facility ( iraf ; * ? ? ? * ) and interactive data language ( idl ) routines to subtract the bias , flat field the images , remove scattered light and cosmic ray contamination , extract the echelle orders , calibrate the wavelength solution , and convert to heliocentric velocities . wavelength calibration images were taken using a th - ar hollow cathode before and after each target . numerous weak water vapor lines are commonly present in spectra around the doublet , and must be modeled and removed , in order to measure an accurate interstellar ( or circumstellar ) absorption profile . i use a forward modeling technique demonstrated by @xcite to remove telluric line contamination in the vicinity of the d lines , with a terrestrial atmosphere model ( at - atmospheric transmission program , from airhead software , boulder , co ) developed by erich grossman . with two absorption lines , it is straightforward to identify contaminating telluric absorption . all absorption lines were fit using standard methods ( e.g. , 2.2 in * ? ? ? * ) . gaussian absorption components are fit to both d lines simultaneously using atomic data from @xcite , and then convolved with the instrumental line spread function . fitting the lines simultaneously reduces the influence of systematic errors , such as continuum placement and contamination by weak telluric features . the free parameters are the central velocity ( @xmath8 ) , the line width or doppler parameter ( @xmath9 ) , and the column density ( @xmath10 ) of ions along the line of sight . the fits are shown in figure [ fig : hd32297_figna1 ] and fit parameters with 1@xmath11 statistical errors are listed in table [ tab : fits ] . in addition , the spectra were used to estimate the stellar radial velocity ( @xmath12 ) and projected stellar rotation ( @xmath13 ) for hd32297 , bd+07 777s , and bd+07 778 ( see table [ tab : basics ] ) , quantities not listed in simbad for these targets . the radial velocities of all 3 objects differ significantly , and therefore it is unlikely that they are physically associated . note that the radial velocity of hd32297 ( @xmath14 km s@xmath15 ) is measured from broad and h@xmath16 stellar absorption lines , and therefore is not tightly constrained . the left column of figure [ fig : hd32297_figna1 ] shows that absorption is clearly detected toward hd32297 in 5 observations over 5 months . two components are easily distinguished , a strong component at @xmath424.5 km s@xmath15 and a weaker component at @xmath420.5 km s@xmath15 . the spectral region for 5 stars in close angular proximity to hd32297 is also shown in figure [ fig : hd32297_figna1 ] . only a single ism component , at @xmath424.2 km s@xmath15 , is detected in the 3 distant neighbors , indicating that large scale interstellar material is located at a distance between 59.4112pc . all targets located beyond this material , including hd32297 , should have a similar ism absorption feature . this strong ism absorption is probably associated with the boundary material of the local bubble , which is estimated to be @xmath490 pc in this direction @xcite . if located at this distance , the physical separation of the interstellar material observed toward hd32297 and the material toward bd+07 777s ( @xmath17 0@xmath19 ) is 0.025pc , bd+07 778 ( 2@xmath14 ) is 0.064pc , and 18 ori ( 5@xmath181 ) is 8.1pc . toward hd32297 s two closest neighbors , the ism absorption is almost identical in projected velocity and column density to the strong absorption seen toward hd32297 , while toward 18 ori , the absorption differs slightly in both @xmath8 and @xmath10 , indicating that any small scale morphological variations in the local bubble shell are on scales @xmath20.1pc but @xmath68pc . small scale variations in the local bubble shell have been detected by @xcite on scales @xmath40.5pc . it is unlikely that the unique 20.5 km s@xmath15 feature observed toward hd32297 is caused by a small scale interstellar structure . although small ism structures ( 0.012.0pc ) have been observed ( e.g. , * ? ? ? * ; * ? ? ? * ) , it is more likely that the unique feature is due to absorption in the circumstellar environment surrounding hd32297 because ( 1 ) hd32297 is known to be an edge - on debris disk , ( 2 ) no similar absorption is detected in the very close neighboring sightlines ( 0.030.08pc ) , and ( 3 ) the absorption matches the stellar radial velocity . temporal variability is also a hallmark of circumstellar material ( e.g. , @xcite , @xcite , @xcite ) . to search for variability , figure [ fig : hd32297_diff ] shows difference spectra of all observations . some indication of temporal variability on time scales of months is detected . for example , between the 2005 sep and 2006 feb observations , the @xmath420.5 km s@xmath15 feature became stronger and the separation between the circumstellar and interstellar features became less distinct , despite the fact that the 2006 feb observations were made at a slightly higher resolving power . the redshifted variability seen between 2005 sep and 2006 feb is @xmath45@xmath11 above the standard deviation . the same pattern is seen in both lines , indicating that the telluric contamination is not causing the variation . slight changes in the resolving power of our instrument could mimic this variable behavior , differentially moving light from the cores of the line to the wings ( or vice versa ) . however , resolution variability should cause ( 1 ) symmetric features in the wings of the line whereas we see a feature only to the blue of the ism feature and not to the red , and ( 2 ) should have a stronger effect on stronger absorption features , whereas the feature is roughly identical in both lines , which could be caused if the absorbing material covers only a fraction of the stellar disk , as has been seen toward @xmath0 pic @xcite . this data alone provides only a subtle indication of temporal variation in , partially because any significant absorption toward the red , is masked by the strong ism feature . redshifted circumstellar absorption dominates the gas absorption variability toward @xmath0 pic ( e.g. , * ? ? ? * ) , while no temporal variability has ever been detected in toward @xmath0 pic , only the `` stable '' absorption component is seen in this ion @xcite . circumstellar variability in has been detected in other edge - on debris disks , e.g. , @xmath0 car , hd85905 , and hr10 @xcite . any redshifted absorption occuring in this object could cause fluctuations in the measured column density of the `` constant '' ism feature . little evidence for variability is found toward the blue . these observations indicate that hd32297 has the strongest circumstellar disk signature detected around a nearby main sequence debris disk star . even compared to @xmath0 pic , the prototypical edge - on debris disk with absorption column densities of @xmath19@xmath20 @xcite , the gas disk around hd32297 , with @xmath3 , has 5@xmath21 the column density . a crude estimate of the gas mass surrounding hd32297 can be made if it is assumed to have the same morphology and abundances as the stable gas around @xmath0 pic . although the observations of hd32297 indicate some red - shifted temporal variability , much of the gas is stable over all observations . using @xmath0 pic as a proxy , the variable gas is likely located very close to the star @xcite , while the stable gas at rest in the stellar frame , likely traces the bulk dust disk @xcite . for this calculation , i assume all the gas is in the stable component , and therefore this gas mass estimate should be considered an upper limit . the morphology of the disk is assumed to follow a broken power law density profile , as fit to the emission profile of the @xmath0 pic disk ( see equation 1 of @xcite ) , and assumed to extend out to the edge of the debris disk at @xmath41680au @xcite . the abundances in the hd32297 disk are assumed to be similar to @xmath0 pic @xcite , where the ratio @xmath22@xmath23@xmath24 , is based on @xmath0 pic measurements by @xcite and limits by @xcite . given these assumption , i calculate a gas mass , distributed through the bulk debris disk surrounding hd32297 at @xmath25 . future observations are planned to continue monitoring the temporal variability of the circumstellar gas toward hd32297 to determine the ratio of stable to variable gas , and measure the gas disk absorption , in order to independently measure the to ratio . a more definitive detection of temporal variability may require monitoring excited lines which will show circumstellar absorption , but not the strong interstellar feature . i present the first high resolution optical spectra of the doublet toward the debris disk hd32297 and stars in close angular proximity . a summary of results include : \(1 ) two absorption components are detected toward hd32297 , while only one is detected toward its proximate neighbors located at a comparable distance . the extra absorption component in the spectrum of hd32297 , which is also at rest in the stellar reference frame , is therefore likely caused by circumstellar material . + ( 2 ) the ism absorption is similar among hd32297 and its two closest neighbors , and is likely due to absorption from the shell that defines the boundary of the local bubble . some variation in local bubble absorption is detected toward 18 ori . + ( 3 ) radial velocities of hd32297 , bd+07 777s , and bd+07 778 are measured and differ significantly , indicating that they are likely not physically associated . + ( 4 ) some indication of temporal variability is detected over several epochs of observations . instrumental resolution variations and masking by the strong ism absorption , make a definitive detection of circumstellar variability difficult . + ( 5 ) the measured circumstellar feature toward hd32297 ( @xmath26 ) is the strongest such absorption measured toward any nearby main sequence debris disk , @xmath45 times greater than the column density of the prototypical edge - on debris disk , @xmath0 pic . + ( 6 ) if the morphology and abundances of the stable gas component around hd32297 are assumed to be similar to @xmath0 pic , i estimate an upper limit to the gas mass in the circumstellar disk surrounding hd32297 of @xmath40.3 @xmath5 . support for this work was provided by nasa through hubble fellowship grant hst - hf-01190.01 awarded by the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , for nasa , under contract nas 5 - 26555 . i would like to thank d. doss , g. harper , and a. brown , for their assistance with these observations . the insightful comments by the anonymous referee were very helpful . lllcllcclcc 32297 & bd+07 777 & a0 & 8.13 & @xmath4 + 20 & @xmath480 & 192.83 & 20.17 & 112@xmath27 & 0.0000 & 0.0 + & bd+07 777s & g0 & 10.2 & 55 & @xmath42 & 192.85 & 20.17 & & 0.0156 & 0.030 + 32304 & bd+07 778 & g5 & 6.87 & 1.4 & @xmath43.5 & 192.88 & 20.17 & 134@xmath28 & 0.0406 & 0.079 + 30739 & @xmath29 ori & a1vn & 4.35 & + 24 & 212 & 189.82 & 21.83 & 59.4@xmath30 & 3.2653 & 3.4 + 31295 & @xmath31 ori & a0v & 4.66 & + 13 & 120 & 189.35 & 20.25 & 37.0@xmath32 & 3.2748 & 2.1 + 34203 & 18 ori & a0v & 5.52 & 8.2 & 70 & 191.29 & 15.25 & 112.9@xmath33 & 5.1306 & 10.1 + lllcccccc 32297 & bd+07 777 & 2005 sep 15 & + 28.2 & 1.565 & 31 & 20.459 @xmath34 0.030 & 0.38 @xmath34 0.28 & 10.97@xmath35 + & & & & & & 24.546 @xmath34 0.012 & 0.742 @xmath34 0.063 & 12.388@xmath36 + 32297 & bd+07 777 & 2006 jan 26 & 22.8 & 1.253 & 17 & 20.50 @xmath34 0.27 & 1.23 @xmath34 0.55 & 11.384 @xmath34 0.077 + & & & & & & 24.30 @xmath34 0.12 & 0.84 @xmath34 0.21 & 12.16@xmath37 + 32297 & bd+07 777 & 2006 jan 28 & 23.4 & 1.266 & 22 & 20.48 @xmath34 0.23 & 1.24 @xmath34 0.42 & 11.449@xmath38 + & & & & & & 24.34 @xmath34 0.17 & 0.82 @xmath34 0.31 & 11.98@xmath39 + 32297 & bd+07 777 & 2006 jan 29 & 23.7 & 1.288 & 20 & 20.42 @xmath34 0.20 & 1.71 @xmath34 0.28 & 11.454@xmath40 + & & & & & & 24.424 @xmath34 0.086 & 0.71 @xmath34 0.23 & 12.02@xmath41 + 32297 & bd+07 777 & 2006 feb 15 & 27.5 & 1.199 & 39 & 20.51 @xmath34 0.26 & 1.14 @xmath34 0.41 & 11.297@xmath42 + & & & & & & 24.41 @xmath34 0.13 & 0.94 @xmath34 0.30 & 11.97@xmath43 + & bd+07 777s & 2006 feb 16 & 27.7 & 1.238 & 6 & 25.55 @xmath34 0.28 & 1.581 @xmath34 0.078 & 12.03@xmath44 + 32304 & bd+07 778 & 2006 feb 16 & 27.7 & 1.244 & 52 & 24.583 @xmath34 0.083 & 1.65 @xmath34 0.15 & 12.18@xmath37 + 30739 & @xmath29 ori & 2006 feb 16 & 28.3 & 1.213 & 133 & & & @xmath45 + 31295 & @xmath31 ori & 2004 oct 18 & + 21.1 & 1.761 & 100 & & & @xmath46 + 34203 & 18 ori & 2006 feb 15 & 27.3 & 1.189 & 117 & 22.464 @xmath34 0.022 & 1.150 @xmath34 0.036 & 11.403@xmath47 +

near - infrared and optical imaging of hd32297 indicate that it has an edge - on debris disk , similar to @xmath0 pic . i present high resolution optical spectra of the doublet toward hd32297 and stars in close angular proximity . a circumstellar absorption component is clearly observed toward hd32297 at the stellar radial velocity , which is not observed toward any of its neighbors , including the nearest only 0@xmath19 away . an interstellar component is detected in all stars @xmath290pc , including hd32297 , likely due to the interstellar material at the boundary of the local bubble . radial velocity measurements of the nearest neighbors , bd+07777s and bd+07778 , indicate that they are unlikely to be physically associated with hd32297 . the measured circumstellar column density around hd32997 , @xmath3 , is the strongest absorption measured toward any nearby main sequence debris disk , even the prototypical edge - on debris disk , @xmath0 pic . assuming that the morphology and abundances of the gas component around hd32297 are similar to @xmath0 pic , i estimate an upper limit to the gas mass in the circumstellar disk surrounding hd32297 of @xmath40.3 @xmath5 .

arxiv

gravity is a universal and fundamental force . anything which has energy creates gravity and is affected by it , although the smallness of newton s constant @xmath1 often means that the associated classical effects are too weak to be measurable . an important prediction of various theories of quantum gravity ( such as string theory ) and black hole physics is the existence of a minimum measurable length @xcite . the prediction is largely model - independent , and can be understood as follows : the heisenberg uncertainty principle ( hup ) , @xmath2 , breaks down for energies close to the planck scale , when the corresponding schwarzschild radius is comparable to the compton wavelength ( both being approximately equal to the planck length ) . higher energies result in a further increase of the schwarzschild radius , resulting in @xmath3 . at this point , it should be stressed that limits on the measurement of spacetime distances as well as on the synchronization of clocks were put in much earlier studies @xcite . these limitations showed up when quantum mechanics ( qm ) and general relativity ( gr ) were put together under simple arguments . it is more than obvious that in this context where one attempts to reconcile the principles of qm with those of gr there are several and even diverging paths to follow @xcite . in this framework , two of the authors ( sd and ecv ) tracked a new path and showed that certain effects of quantum gravity are universal , and can influence almost any system with a well - defined hamiltonian @xcite . although the resultant quantum effects are generically quite small , with current and future experiments , bounds may be set on certain parameters relevant to quantum gravity , and improved accuracies could even make them measurable @xcite . one of the formulations , among those existing in the literature , of the _ generalized uncertainty principle _ ( gup ) and which holds at all scales , is represented by @xcite x_i p_i [ 1 + ( ( p)^2 + < p>^2 ) + 2 ( p_i^2 + < p_i>^2 ) ] , i=1,2,3 [ uncert1 ] where @xmath4 , @xmath5 , @xmath6 planck mass , and @xmath7 planck energy @xmath8 . it is normally assumed that the dimensionless parameter @xmath9 is of the order of unity . however , this choice renders quantum gravity effects too small to be measurable . on the other hand , if one does not impose the above condition _ a priori _ , current experiments predict large upper bounds on it , which are compatible with current observations , and may signal the existence of a new length scale . note that such an intermediate length scale , @xmath10 can not exceed the electroweak length scale @xmath11 ( as otherwise it would have been observed ) . this implies @xmath12 . therefore , as stated above , quantum gravity effects influence all quantum hamiltonians @xcite . moreover , some phenomenological implications of this interesting result were presented in @xcite . the recently proposed _ doubly special relativity _ ( or dsr ) theories on the other hand ( which predict maximum observable momenta ) , also suggest a similar modification of commutators @xcite . the commutators which are consistent with string theory , black holes physics , dsr , _ and _ which ensure @xmath13=0=[p_i , p_j]$ ] ( via the jacobi identity ) under specific assumptions lead to the following form @xcite = i [ comm01 ] where @xmath14 . equation ( [ comm01 ] ) yields , in @xmath15-dimension , to @xmath16 x p [ uncert2 ] where the dimensional constant @xmath17 is related to @xmath18 that appears in equation ( [ uncert1 ] ) through dimensional analysis with the expression @xmath19 = [ \alpha^2]$ ] . however , it should be pointed out that it does not suffice to connect the two constants @xmath17 and @xmath18 through a relation of the form @xmath20 in order to reproduce equation ( [ uncert1 ] ) from ( [ uncert2 ] ) , or vice versa . equations ( [ uncert1 ] ) and ( [ uncert2 ] ) are quite different and , in particular , the most significant difference is that in equation ( [ uncert1 ] ) all terms appear to be quadratic in momentum while in equation ( [ uncert2 ] ) there is a linear term in momentum . commutators and inequalities similar to ( [ comm01 ] ) and ( [ uncert2 ] ) were proposed and derived respectively in @xcite . these in turn imply a minimum measurable length _ and _ a maximum measurable momentum ( to the best of our knowledge , ( [ comm01 ] ) and ( [ uncert2 ] ) are the only forms which imply both ) x & & ( x)_min _ 0_pl + p & & ( p)_max . it is normally assumed as in the case of @xmath9 that the dimensionless parameter @xmath21 is of the order of unity , in which case the @xmath22 dependent terms are important only when energies ( momenta ) are comparable to the planck energy ( momentum ) , and lengths are comparable to the planck length . however , if one does not impose this condition _ a priori _ , then using the fact that all quantum hamiltonians are affected by the quantum gravity corrections as was shown in @xcite and applying this formalism to measure a single particle in a box , one deduces that all measurable lengths have to be quantized in units of @xmath23 @xcite . in order to derive the energy - time uncertainty principle , we employ the equations x & ~ & c + p & ~ & where @xmath24 is a characteristic time of the system under study , and it is straightforward to get x p e . [ energytime1 ] substituting equation ( [ energytime1 ] ) in the standard hup , one gets the energy - time uncertainty principle e . [ energytime2 ] it should be stressed that the characteristic time @xmath24 is usually selected to be equal to the planck time @xmath25 in the context of cosmology . the scope of the present work is to investigate in a cosmological setup what corrections , if any , are assigned to physical quantities such as the mass and energy of the universe at the planck time . in particular , our present approach , regarding the quantum gravity corrections at the planck time , has been based on a methodology that presented in the book of coles and lucchin @xcite . simply , in our phenomenological formulation instead of using the standard hup for deriving the planck time ( as done in [ 11 ] ) , we let ourselves to utilize various versions of the gup and basically apply the methodology presented in the previously mentioned book . + the significance of planck time _ per se _ is due to the fact that it is really a `` turning point '' because from the birth of the universe till the planck time quantum gravity corrections are significant ( classical general relativity does not work at all ) while after that general relativity seems to work properly . in this section we investigate the effects of quantum gravity on the planck era of the universe . by employing the different versions of gup presented before , we evaluate the modifications to several quantities that characterize the planck era , i.e. planck time , length , mass , energy , density , effective number density and entropy . this will enhance our understanding of the consequences of the quantum gravity in the universe during the planck epoch and afterwards . without wanting to appear too pedagogical , we briefly present how one can derive some physical quantities at the planck epoch starting from the hup . following the methodology of @xcite ( see page 110 ) , we first write the hup in the form e [ hup1 ] and we adopt as characteristic time @xmath24 of the system under study the planck time , i.e. @xmath25 , for which the quantum fluctuations exist on the scale of planck length , i.e. @xmath26 . in addition , the uncertainty in energy can be identified with the planck energy and thus @xmath27 . thus , the hup as given in equation ( [ hup1 ] ) is now written as m_pl c^2 t_pl . [ hup2 ] since the universe at planck time can be seen as a system of radius @xmath28 , the planck mass can be written as m_pl _ pl _ pl^3 [ mass2 ] where by employing the first friedmann equation the planck density , on dimensional grounds , reads [ density1 ] substituting equations ( [ mass2 ] ) and ( [ density1 ] ) in equation ( [ hup2 ] ) , one gets _ pl^3 c^2 t_pl and therefore one can easily prove that the planck time is t_pl10 ^ -43sec . [ plancktime ] all the other parameters are defined in terms of the planck time modulus some constants . indeed , the planck length , density , mass , energy , temperature and effective number density are given by the following expressions _ pl1.710 ^ -33 cm , _ pl 410 ^ 93gcm^-3 , [ plancklen ] m_pl_pl l^3_pl2.510 ^ -5 g , e_plm_pl c^2 1.210 ^ 19gev , [ planckener ] t_pl k^-1_b 1.410 ^ 32k , n_pll^-3_pl ( ) ^3/2 10 ^ 98 cm^-3 . [ plancktep ] the corresponding energy - time gup of equation ( [ uncert1 ] ) e _ pl [ gup2 ] where we have kept only the first gup - induced term of order @xmath30 . note that the tilde denotes quantities with respect to the gup . as expected , for @xmath31 the gup boils down to the standard form dictated by the heisenberg result ( @xmath32 ) . from the previously presented formalism ( see section i ) , the uncertainty in energy @xmath29 at the planck time is of order of the modified planck energy , i.e. @xmath33 , where the modified planck mass lies inside the universe s horizon of scale of the modified planck length , i.e. @xmath34 , and expands as _ pl=_pl^ _ [ mass1 ] the modified planck density can be easily derived from the first friedmann equation ( or , from the definition of the dynamical time scale ) and be given by _ pl [ rho1 ] modulus some constants . substituting equations ( [ mass1 ] ) and ( [ rho1 ] ) in equation ( [ gup2 ] ) , one gets an equation for the planck time which now has been affected by the quantum gravity corrections , namely _ pl_pl^3c^2 _ pl & & + c^2 _ pl & & + ^2_pl & & . therefore , after some simple algebra , one gets the following equation ( ) ^2 ^ 4_pl - ( ) ^2_pl + 0 . [ equation1 ] it is easily seen that if we choose @xmath35 to be strictly equal to zero then the current solution of the above equation reduces practically to that of the standard planck time ( cf . ( [ plancktime ] ) ) . on the other hand , @xmath36 $ ] equation ( [ equation1 ] ) has two real solutions of the form _ pl & = & f_(_0 ) + & = & t_pl f_(_0 ) [ solution ] where f_(_0)=^1/2 . it is worth noting that for @xmath37 [ or , equivalently , @xmath38 we find @xmath39 , despite the fact that we have started from a completely different uncertainty principle . this implies that in this specific case , i.e. @xmath37 , the gup - induced effects in equation ( [ solution ] ) can not be observed . now we must first decide which is the important term when @xmath9 takes values in the set @xmath40 $ ] ; is it @xmath41 or @xmath42 ? using some basic elements from calculus , one can prove that the function @xmath41 is continuous and increases strictly in the range of @xmath43 which implies that @xmath44 $ ] , where @xmath45 . therefore , the modified planck time lies in the range t_pl _ pl t_pl . in the left panel of figure 1 , we present in a logarithmic scale the @xmath41 ( dashed line ) as a function of @xmath35 . practically speaking , the @xmath41 term has no effect on the planck time . on the other hand , following the latter analysis , we find that the @xmath42 function ( solid line ) decreases strictly in the range @xmath43 which means that @xmath46 , where @xmath47 . it becomes evident that as long as @xmath48 [ or , equivalently , @xmath49 the current gup can affect the planck quantities via the function @xmath42 . for example , in the case where @xmath50 we find that @xmath51 . from a cosmological point of view , the ratio of the modified planck density , i.e. @xmath52 , versus the measured dark energy , i.e. @xmath53 with @xmath54 ( for details see @xcite ) , is given as ( 10 ^ 119 - 10 ^ 121 ) . one now is interested to investigate if and how the main planck quantities related to the planck time are affected by the above quantum gravity corrections . the corresponding relations are [ quantities ] = = = ( ) ^1/2= ( ) ^1/3= f_+ . finally , it should be stressed that the dimensionless entropy enclosed in the cosmological horizon of size @xmath55 now reads _ pl _ pl . [ entropy1 ] it is evident that the entropic content of the universe behind the cosmological horizon at the planck time is unaltered when quantum gravity corrections are taken into account . therefore , the information remains unchanged : one particle " of planck mass is stored " in the planck volume of the universe at the planck time behind the cosmological horizon of size @xmath55 . the corresponding energy - time gup of equation ( [ uncert2 ] ) becomes e _ . [ gup22 ] as it is anticipated , for @xmath56 ( or , equivalently , @xmath57 since @xmath58 ) the gup boils down to the standard form dictated by hup . evidently , performing the same methodology as before ( see subsection b ) , we obtain the following equation [ new22 ] 2 ^ 2_0 ( ) ^2 ^ 4_pl- ( 1+_0)^2_pl+ 0 . in deriving equation ( [ new22 ] ) we have substituted the various terms as @xmath59 , @xmath60 , @xmath61 , and @xmath58 . in this framework , equation ( [ new22 ] ) has two real solutions @xmath62 $ ] . these are _ pl = t_pl f_(_0 ) [ solution22 ] where f_(_0)= ^1/2 . [ solution22 ] again it is routine to estimate the limiting values of @xmath63 @xmath64 therefore , the function @xmath65 does not play a significant role ( see the dashed line in the right panel of figure 1 ) , since the modified planck time tends to the usual value ( @xmath66 ) . on the contrary , if we consider the case of @xmath67 ( see the solid line in the right panel of figure 1 ) , then it becomes evident that for small values of @xmath68 the function @xmath69 goes rapidly to infinity . as an example for @xmath70 we find that @xmath71 . thus the ratio of the modified planck density versus the measured dark energy now reads ~o ( 10 ^ 115 - 10 ^ 119 ) . the main planck quantities related to the planck time are affected by the above quantum gravity corrections exactly in the same way as shown in equation ( [ quantities ] ) employing the current form of @xmath72 defined in equation ( [ solution22 ] ) . furthermore , it is interesting to point out that the dimensionless entropy enclosed in the cosmological horizon of size @xmath55 remains unaltered @xmath73 . this result is in accordance with the result derived in previous subsection ( see equation ( [ entropy1 ] ) ) . therefore , the information in the planck volume remains unchanged even if one takes into account the quantum gravity effects , i.e. one particle " of planck mass is stored " in the planck volume of the universe at the planck time behind the cosmological horizon of size @xmath55 . in this work we have investigated analytically the quantum gravity corrections at the planck time by employing a methodology that was introduced in the book of coles and lucchin @xcite . specifically , in this work instead of using the standard hup for deriving the planck time ( as done in [ 11 ] ) , we let ourselves to utilize various versions of the gup . from our analysis , it becomes evident that the planck quantities , predicted by the generalized uncertainty principle gup , extends nicely to those of the usual heisenberg uncertainty principle ( hup ) and connects smoothly to them . we also find that under of specific circumstances the modified planck quantities defined in the framework of the gup are larger by a factor of @xmath74 with respect to those found using the standard hup . these results indicate that we anticipate modifications in the framework of cosmology since changes in the planck epoch will be inherited to late universe through quantum gravity ( or quantum field theory ) . as an example , the calculation of the density fluctuations at the epoch of inflation sets an important limit on the potential of inflation . indeed , in the context of slow - roll approximation , one can prove that the density fluctuations are of the form @xmath75 where @xmath76 is the scalar field called inflaton , @xmath77 stands for the hubble parameter and @xmath78 is the potential energy of the scalar field . assuming that @xmath79 where @xmath80 is the inflaton mass , and @xmath81 , we obtain @xmath82 . in order to achieve inflation the scalar field has to satisfy the inequality @xmath83 @xcite . combining the above equations and utilizing the observational value @xmath84 , one gets @xmath85 . employing the quantum gravity corrections , the latter condition becomes @xmath86 . from the latter calculations it becomes evident that the quantum gravity corrections affect directly the main cosmological quantities ( such as the inflaton mass ) in the early universe , via the @xmath87 factor . due to the fact that the @xmath88 factor is the consequence of the quantum gravity _ per se _ ( based on the gup ) implies that it can not be re - absorbed by a redefinition of units . from the observational point of view one can study the latter corrections in the context of the primordial gravitational waves which can be detected with very sensitive measurements of the polarization of the cmb ( see page 5 in @xcite ) . it is interesting to mention here that the polarization of the cmb will be one of the main scientific targets of the next generation of the cmb data based on the planck satellite . therefore , if in the near future the observers measure such an effect in the cmb data then we may open an avenue in order to understand the transition from the mainly - quantum gravitational regime to the mainly - classical regime . we have already started to investigate theoretically the above possibility and we are going to present our results in a forthcoming paper . furthermore , it was also shown that the dimensionless entropy enclosed in the cosmological horizon does not feel " the quantum gravity corrections and thus the information remains unaltered . therefore , the entropic content of the universe at the planck time remains the same . this is quite important since entropy is the cornerstone for one of the basic principles of quantum gravity named holographic principle and for its incarnation known as ads / cft . + * acknowledgments * the authors thank the anonymous referee for useful comments and suggestions . sd was supported in part by the natural sciences and engineering research council of canada and by the perimeter institute for theoretical physics . d. amati , m. ciafaloni , g. veneziano , phys . b * 216 * ( 1989 ) 41 ; m. maggiore , phys . b * 304 * ( 1993 ) 65 [ arxiv : hep - th/9301067 ] ; m. maggiore , phys . d * 49 * ( 1994 ) 5182 [ arxiv : hep - th/9305163 ] ; m. maggiore , phys . b * 319 * ( 1993 ) 83 [ arxiv : hep - th/9309034 ] ; l. j. garay , int . j. mod . phys . a * 10 * ( 1995 ) 145 [ arxiv : gr - qc/9403008 ] ; f. scardigli , phys . lett . b * 452 * ( 1999 ) 39 [ arxiv : hep - th/9904025 ] ; s. hossenfelder , m. bleicher , s. hofmann , j. ruppert , s. scherer and h. stoecker , phys . b * 575 * ( 2003 ) 85 [ arxiv : hep - th/0305262 ] ; c. bambi and f. r. urban , class . quant . * 25 * ( 2008 ) 095006 [ arxiv:0709.1965 [ gr - qc ] ] ; r. banerjee and s. ghosh , phys . b * 688 * ( 2010 ) 224 [ arxiv:1002.2302 [ gr - qc ] ] . c. a. mead , phys . * 135 * ( 1964 ) b849 ; t. padmanabhan , class . grav . * 4 * ( 1987 ) l107 . g. amelino - camelia , mod . lett . a * 9 * ( 1994 ) 3415 [ arxiv : gr - qc/9603014 ] ; y. j. ng and h. van dam , mod . lett . a * 9 * ( 1994 ) 335 . s. das , e. c. vagenas , phys . * 101 * ( 2008 ) 221301 [ arxiv:0810.5333 [ hep - th ] ] . g. amelino - camelia , int . j. mod . d * 11 * ( 2002 ) 35 [ arxiv : gr - qc/0012051 ] . j. magueijo and l. smolin , phys . * 88 * ( 2002 ) 190403 [ arxiv : hep - th/0112090 ] ; j. magueijo and l. smolin , phys . d * 67 * ( 2003 ) 044017 [ arxiv : gr - qc/0207085 ] .

we investigate the effects of quantum gravity on the planck era of the universe . in particular , using different versions of the generalized uncertainty principle and under specific conditions we find that the main planck quantities such as the planck time , length , mass and energy become larger by a factor of order @xmath0 compared to those quantities which result from the heisenberg uncertainty principle . however , we prove that the dimensionless entropy enclosed in the cosmological horizon at the planck time remains unchanged . these results , though preliminary , indicate that we should anticipate modifications in the set - up of cosmology since changes in the planck era will be inherited even to the late universe through the framework of quantum gravity ( or quantum field theory ) which utilizes the planck scale as a fundamental one . more importantly , these corrections will not affect the entropic content of the universe at the planck time which is a crucial element for one of the basic principles of quantum gravity named holographic principle .

arxiv

to solve a conformal field theory ( cft ) amounts to finding its spectrum and 3-point correlation functions , since higher point functions may be obtained using the operator product expansion ( ope ) . in the former case this means finding the anomalous dimensions of the operators of the theory , while in the latter case it means finding the couplings in 3-point correlation functions , whose space - time dependence is otherwise fixed by conformal invariance . in the simplest case of scalar primary operators the 3-point function has the simple form _ a(0)o_b(x)o_c(y)= . where @xmath11 is the dimension of the operator @xmath2 , and so on . the definition of the couplings @xmath12 requires that the operators diagonalise the anomalous dimension matrix and depends on the choice of normalisation in the 2-point function of each operator . our main interest is to explore new methods to compute the couplings @xmath12 for certain single trace operators in @xmath4 sym . in recent years there have been great progresses in finding the spectrum of this theory , in the planar limit and for any value of the coupling constant , using integrability @xcite-@xcite . on the other hand , much remains to be done in the computation of the couplings @xmath12 . at weak coupling these may be evaluated , order by order in perturbation theory , by computing feynman diagrams @xcite-@xcite . although this approach is essential to uncover new structures and to verify new exact results , it is unpractical to obtain exact results for general operators . a more promising approach is to explore integrability of planar @xmath4 sym . however , how integrability will enter computations of the couplings @xmath12 remains unclear . one strategy to compute the couplings in a cft is to deform the theory from its fixed point with a marginal or irrelevant operator @xmath0 . we will show in section 2 that this deformation introduces new divergences in the renormalised operators of the critical theory , which are determined by the couplings @xmath1 . more precisely , to leading order in the deformation parameter , the entry of the deformed anomalous dimension matrix between operators @xmath2 and @xmath3 is determined by the coupling @xmath1 . thus , in planar @xmath4 sym , finding the action of such matrix on operators diagonalized by means of the bethe ansatz is a new method to compute the couplings @xmath1 . in practice , we will show in section 3 how to implement these ideas in the case of the coupling deformation , which is considerably easier since it is an exact deformation . another example , that is expected to work in a similar fashion is the @xmath13 deformation of @xmath4 @xcite . more general deformations may also be considered . whether this technique will be useful in unveiling new integrability structures in the perturbative computation of the couplings @xmath12 remains an open problem . at strong t hooft coupling we may use the ads / cft duality @xcite to compute the couplings @xmath12 . the duality relates the @xmath14 string partition function , computed with suitable boundary condition , to the generating functional for correlation functions of the gauge theory @xcite . however , in practice this relation has been useful only to compute , at strong coupling , correlation functions of chiral operators that are dual to the supergravity fields @xcite . recently , a path integral approach to compute the string theory partition function for a heavy string state propagating between two boundary points has been developed @xcite ( see also @xcite ) . in this case the string path integral is dominated by a classical saddle point , giving a new method to compute at strong coupling the 2-point function of single trace operators with a large number of basic fields @xcite . in section 4 we shall extend this computation to the case of a 3-point function with an additional chiral operator . the basic idea is that , taking into account the coupling between the heavy string worldsheet and the supergravity fields , the path integral giving the aforementioned 2-point function can be extended to include the interaction with light fields . in practice all one needs to do is to compute a witten diagram with a supergravity field propagating from the @xmath14 boundary to the heavy string worldsheet , which acts as a tadpole for this field . we will show how this computation works for the dilaton field and several heavy string configurations , obtaining couplings of the form @xmath15 , in complete agreement with the value predicted by renormalisation group arguments . we conclude in section 5 with comments and open problems . the goal of this section is to show how to relate the 3-point correlation function in a cft to the anomalous dimension matrix obtained from deforming the cft with a marginal or irrelevant operator @xmath0 of dimension @xmath16 at the cft fixed point . we emphasise that the results presented in this section are valid at a cft fixed point with coupling @xmath17 . we shall explore basic ideas given in @xcite ( see also appendix in @xcite ) . the example that we have in mind , and that we will work in detail in the following sections , is @xmath4 sym , so we shall stick to four dimensions . in this case , the dimension @xmath16 of the operator @xmath0 satisfies @xmath18 . in the case of @xmath4 sym we have a line of cft s parameterised by the coupling constant @xmath17 , so we may wish to take the coupling to be finite and large , or to expand to arbitrary order in the coupling constant . we may also wish to consider an operator @xmath0 of protected dimension , but that is not necessary . our starting point is a cft with action @xmath19 . we consider the deformed theory with action @xmath20 where @xmath21 is the dimensionless deformation parameter at the cut - off scale @xmath22 of dimension inverse length , and the operators that appear in this action are the renormalized operators of the undeformed theory . the beta function for the coupling @xmath21 has the form , @xmath23 where @xmath24 represents terms quadratic , or of higher powers , in the couplings to all operators around the fixed point . for what we are doing it will be sufficient to work to linear order in @xmath21 , so we keep only the first term in the beta function @xmath25 with @xmath16 computed at the fixed point . sending the cut - off to infinity , the coupling @xmath26 at a fixed scale @xmath27 is constant for @xmath28 ( marginal deformation ) and vanishes for @xmath29 ( irrelevant deformation ) . for simplicity we shall consider the operator @xmath0 to be a scalar primary . but this can be generalised to more operators , for instance , @xmath0 could be the energy - momentum tensor , in which case @xmath21 would be a tensor valued deformation parameter . for the sake of clarity , we shall consider in what follows the case of an operator @xmath0 with dimension @xmath28 at the fixed point . since we are interested in the case of @xmath4 sym at any value of the coupling , this means the operator has protected dimension . in the appendix we extend our results to the case of irrelevant deformations . we decided to separate the discussion because in the following sections we shall be working with the marginal case , therefore avoiding the duplication of formulae in the main text . a final introductory word about notation , we shall use the label @xmath21 to denote quantities computed in the deformed cft with action given by ( [ deformedaction ] ) . quantities without the label @xmath21 are computed at the undeformed theory for which @xmath30 . we now analyse the divergences that appear in the deformed theory , in terms of renormalized quantities of the undeformed theory . let @xmath2 be any renormalized operator of the undeformed theory . we shall denote its full dimension ( classical + quantum ) , at the fixed point , by @xmath11 . when computing the correlation function of this operator with any other operators , we obtain in the deformed theory to linear order in @xmath21 , @xmath31 where the right hand side of this equation is computed in the undeformed theory . in general new divergences can appear in equation ( [ expansionn - pf ] ) , that can be cancelled by renormalizing the operators @xmath32 , and that come from the behaviour of the correlation function involving @xmath33 , when @xmath34 approaches any of the positions of the other operators . the form of this divergences is entirely determined by the ope in the undeformed theory of the operator @xmath0 with the operators appearing in the correlation function . for the operator @xmath35 we have @xmath36 where the constants @xmath1 are precisely the couplings appearing in the 3-point function @xmath37 . we remark that for now we assume that the complete basis of operators @xmath38 is diagonal with unit norm , i.e. @xmath39 the physically meaningful couplings @xmath1 are defined with respect to operators satisfying this normalisation . using the ope expansion ( [ ope ] ) , we conclude that the divergence in the @xmath34 integral of ( [ expansionn - pf ] ) , arising from the region of integration @xmath40 , is given by @xmath41 hence , powerlike divergences arise from operators that enter the ope of @xmath2 and @xmath0 , and whose dimensions satisfy @xmath42 . by the unitarity bounds this is a finite number of operators , for instance , for scalar operators in four dimensions we must have @xmath43 . logarithmic divergences appear from operators in the ope with @xmath44 . we are now in position to define renormalized operators @xmath45 of the deformed theory , expressed in terms of renormalized operators of the undeformed theory , such that the general correlation function ( [ expansionn - pf ] ) is finite . this is quite simple , because there is a finite number of operators @xmath3 entering the ope ( [ ope ] ) and contributing to the divergences in ( [ divergences ] ) . we define the renormalized operators @xmath46 as usual , we see that operator mixing occurs for @xmath47 . in terms of bare operators of the undeformed theory , then , in a theory without dimensional couplings , mixing will only occur between operators of the same classical dimension , i.e. for @xmath48 . then , the last term in ( [ renormalized ] ) only concerns operators with different anomalous dimensions , since the power like divergence becomes logarithmic when expanding in the coupling @xmath17 . ] with this renormalization scheme , correlation functions @xmath49 computed at the fixed value of the coupling @xmath17 , and to linear order in @xmath21 for the theory with action ( [ deformedaction ] ) , are finite . of particular importance to us will be the case of 2-point functions . for operators @xmath2 and @xmath3 with the same dimension in the undeformed theory , it is simple to see that @xmath50 for @xmath51 this gives @xmath52 if there are different operators @xmath2 and @xmath3 of equal dimension , we see that the effect of turning on the deformation is to induce operator mixing , since the above 2-point function is no longer diagonal . it is also simple to see that the 2-point function for operators @xmath2 and @xmath3 of different dimension still vanishes . we now wish to better understand the basis of renormalized operators introduced in the previous section , by defining a deformed anomalous dimension matrix . we will then verify the callan - symanzik equation for correlation functions in the deformed theory . let us start by defining renormalized operators of the deformed theory using the usual renormalization matrix @xmath53 where we omitted the summation in @xmath54 . from ( [ renormalized ] ) we can read the entries of this matrix , @xmath55 it is now simple to compute the anomalous dimension matrix associated to the deformation , defined by @xmath56 its non - vanishing entries are _ ab = u 2 ^ 2 a_dab ^_a-_b , [ deformedgamma ] for @xmath57 . we remark that the anomalous dimension matrix @xmath58 is defined with respect to renormalized operators of the undeformed theory with total dimension given by @xmath11 at the fixed point . if we order the operators in blocks with descending value of dimension , the non - diagonal top - right blocks of the anomalous dimension matrix @xmath59 have zero entries . it is then clear that its eigenvalues @xmath60 are independent of the cut - off @xmath22 , although the eigenvectors do depend in general on @xmath22 ( when there is mixing between operators of different dimension ) . thus , in the diagonal basis we have , as usual , @xmath61 . an alternative way of deriving the relation between the anomalous dimension @xmath58 and the couplings @xmath62 is to verify the callan - symanzik equation . this is simpler for a marginal deformation , and to linear order in @xmath21 , because the beta function @xmath25 vanishes ( in the appendix we consider the case of irrelevant deformations ) . for the non - renormalized two - point function of the deformed theory , computed using renormalized operators of the cft at the fixed point , the callan - symanzik equation has the form @xmath63 using ( [ expansionn - pf ] ) and the form of the divergences given in ( [ divergences ] ) this equation is satisfied provided ( [ deformedgamma ] ) holds . for practical perturbative computations it is useful to relate the couplings @xmath62 to the anomalous dimension matrix computed with respect to bare operators of the cft ( not renormalized ) . let us denote a basis of such operators by @xmath64 . now assume that we manage to diagonalize the anomalous dimension matrix of the critical theory , so that in the basis @xmath64 we have @xmath65 , where @xmath66 are the eigenvalues ( for instance , in @xmath4 sym we can use integrability techniques to do that quite effectively ) . in this basis , and denoting by @xmath67 the classical dimension of operators , it is simple to see that the renormalization matrix @xmath68 relating bare operators to the renormalized operators of the deformed theory in the usual way , @xmath69 , has entries @xmath70 the corresponding deformed anomalous dimension matrix has entries ^u_ab = _ ab _ a + u 2 ^ 2 a_dab ^^0_a-^0_b . [ deformedh ] note that these are the entries of the matrix @xmath71 in the basis @xmath64 that diagonalizes the anomalous dimension matrix of the critical theory . again , it is important to realize that the structure of the matrix @xmath71 , given by ( [ deformedh ] ) , implies that its eigenvalues are independent of the cut - off @xmath22 , although the eigenvectors may depend on @xmath22 . let us show explicitly how the knowledge of the deformed anomalous dimension @xmath71 allows to relate the couplings @xmath1 to the deformed anomalous dimensions and renormalized operators @xmath45 expressed in terms of the bare quantities . first we write the anomalous dimension matrix as @xmath72 where @xmath73 is the anomalous dimension matrix of the critical theory , and @xmath74 is the term arising from the deformation which we treat as a perturbation . for simplicity we assume that the operators @xmath65 and @xmath75 do not have the same anomalous dimension at the critical point ( they may or may not have the same classical dimension ) . then , writing the eigenvalues and eigenvectors of @xmath71 respectively as @xmath76 basic quantum mechanics formulae gives @xmath77 where the matrix elements are computed in the basis @xmath38 with unit normalised operators . from the explicit form of the deformed anomalous dimension matrix in the basis @xmath64 given in ( [ deformedh ] ) , we conclude that @xmath78 note that ( [ final2 ] ) has the correct dependence in the cut - off @xmath22 to relate operators @xmath2 and @xmath79 of different dimension , as required by ( [ quantummechanics ] ) . it is now clear that if we have a way of determining the action of the perturbation matrix @xmath80 on the bare operators , we may then compute the corresponding couplings using ( [ final ] ) and ( [ final2 ] ) . this will be the case in the next section , where we consider coupling deformations of @xmath4 sym and the known form of the integrable anomalous dimension matrix at a given order in the coupling constant . we finish this section with a word on normalization of operators . in the next section it will actually be convenient to perform computations with operators that are not normalized to unit , i.e. after diagonalizing the eigenvectors of the undeformed theory we will have @xmath81 with this normalization , to obtain the physically meaningful couplings for the unit normalized operators , we need to divide the operators in equations ( [ final ] ) and ( [ final2 ] ) by their norm . in this section we consider the simplest case of @xmath4 sym deformed by the lagrangian operator , since this theory is actually a line of fixed points parametrised by the coupling constant . we shall use integrability to show how to compute the couplings in the 3-point function of the lagrangian with any two operators of the theory . for simplicity , we restrict our analysis to the @xmath82 scalar subsector , and consider in detail operators corresponding to two - magnon excitations in the spin chain language . we shall use the following convention for the @xmath4 sym action @xmath83 ^ 2 + { \rm fermions}\big)\ , , \label{action}\ ] ] where @xmath84 and the covariant derivative is defined by @xmath85 $ ] . all fields are in the adjoint representation and the @xmath86 generators are normalized with @xmath87 . we will be considering the @xmath82 sector with complex scalars @xmath88 now consider the theory at some fixed value of the t hooft coupling , defined by @xmath89 we will consider ( planar ) perturbation theory to some order in the coupling @xmath90 . we are therefore considering the cft at the fixed point with coupling @xmath90 . then , to deform the theory with @xmath91 , it is clear from ( [ action ] ) that we should write @xmath92 hence , making this replacement in the anomalous dimension matrix of @xmath4 , to a given order in @xmath90 , and then keeping only the linear terms in @xmath21 , we obtain the form of the deformed anomalous dimension matrix @xmath71 . we may then use the results ( [ final ] ) and ( [ final2 ] ) to compute the couplings . alternatively , we can also compute the derivative of @xmath93 or @xmath45 with respect to @xmath21 or , instead , the derivative @xmath94 of @xmath95 or @xmath2 to a given order in @xmath90 . finally , note that the two - point function of the lagrangian is ( dropping the @xmath96 correction ) @xmath97 we shall compute the couplings with respect to this normalization , but it is simple to re - scale with respect to the unit normalised lagrangean @xmath98 . in that case we would obtain that all the couplings @xmath99 computed in this paper are of order @xmath96 , for fixed t hooft coupling , as expected . as an example consider single trace operators made by @xmath100 fields of the @xmath82 sector and regard the fields @xmath101 as impurities in the vacuum state @xmath102 . for operators with @xmath103 impurities , we use the integers @xmath104 to indicate the position of the impurities in the corresponding spin chain , @xmath105 the anomalous dimension matrix is that of an integrable spin chain and may be diagonalized by solving the bethe equations @xcite . then the operator @xmath2 , with anomalous dimension @xmath95 , is given by @xmath106 where the wave function @xmath107 is parameterized by the momenta @xmath108 of the magnons , which in general can be complex , sum to zero mod @xmath109 , and depend on the t hooft coupling . then , the contribution of the @xmath110-th magnon to the anomalous dimension of the operator @xmath2 is given by @xcite @xmath111 this formula is believed to be correct to all orders in the t hooft coupling , provided wrapping effects , that become important at order @xmath112 , are neglected . the interactions between magnons are responsible for the dependence of their momenta on the t hooft coupling . this effect appears in the computation of the anomalous dimension at two - loop order , while is appears at one - loop in the computation of the coupling @xmath113 . thus , neglecting wrapping effects , ( [ final ] ) gives the all - loop result @xmath114 where prime denotes derivative with respect to @xmath90 . we remark that the normalised coupling @xmath115 scales with @xmath96 as expected . we shall compute in the next section this coupling up to order @xmath116 , in the simple case of operators with two - magnons . next we consider the dilute limit of @xmath117 . in this limit the magnons propagate freely on the spin chain and their momentum is trivially quantised as @xmath118 in this case the second term in the numerator of ( [ alllooplaa ] ) can be dropped . one may now study both weak and strong coupling limits . the leading order term in @xmath90 , which comes from the 1-loop correction to the anomalous dimension of @xmath2 , is given by @xmath119 and can be derived simply by doing wick contractions between @xmath120 , @xmath2 and @xmath121 . on the other hand , at strong coupling and neglecting wrapping effects , we have @xmath122 in section [ sec : strongcoup ] we shall confirm this computation of the coupling @xmath113 , by directly computing this 3-point function using the ads / cft duality in the gravity limit . we shall now illustrate how one can use integrability techniques and the general results given in ( [ final ] ) and ( [ final2 ] ) to compute the couplings of two operators , each with two magnons , and the lagrangian . we will compute 1-loop corrections to these couplings , which correspond to diagonalizing the anomalous dimension matrix at two - loop order . the corresponding spin chain hamiltonian includes next to neighbour interactions @xcite , @xmath123 where @xmath124 is the permutation operator . at this order , the bethe wave function includes a contact term and can be written in the following form @xcite @xmath125 with @xmath126 since for two magnons the total momentum vanishes , we have @xmath127 . we shall now write the formulae for the contact function and for the s - matrix in this simpler case @xcite . for the contact function @xmath128 , we have @xmath129 which satisfies @xmath130 . the s - matrix @xmath131 , can be written as @xmath132 with @xmath133 it is clear that @xmath134 . the momenta that solve the bethe equation @xmath135 are given by @xmath136 where @xmath137 is an integer . it is now a mechanical calculation to replace this expression in ( [ alllooplaa ] ) , to obtain @xmath138 + o(g^6)\ , . \label{twolooplaa}\ ] ] next we consider the coupling @xmath139 , where @xmath2 is an operator with two magnons of momenta @xmath140 and @xmath141 , and @xmath3 is an operator with two magnons of momenta @xmath142 and @xmath143 . this amounts to computing the matrix element @xmath144 . using the two - loop anomalous dimension matrix given in ( [ anomalousdimmatrix ] ) we have @xmath145 now we argue that the first two terms in this expression do not contribute to @xmath144 . first recall that @xmath146 and @xmath147 are eigenstates of @xmath73 , with terms of order @xmath148 and @xmath90 . since @xmath73 is diagonalized by these eigenstates , the contribution from the first term in @xmath80 vanishes . moreover , since the second term in @xmath80 starts at order @xmath116 , for this term we may consider the eigenstates @xmath146 and @xmath147 only at order @xmath148 . thus , this term is proportional to the hamiltonian @xmath73 at one - loop , and it will also give a vanishing contribution . we are therefore left with the contribution from the last term in ( [ derivativeh ] ) , which we can compute with the eigenstates @xmath146 and @xmath147 of order @xmath148 . a computation shows that @xmath149 since we are working with states with norm @xmath150 , after normalising to unit we obtain @xmath151 the example given in this section shows that one can use integrability of @xmath4 sym to compute quite effectively the couplings of operators to the lagrangian . of course one can try to compute these couplings to higher orders in the t hooft coupling @xmath90 , to consider operators with more magnons and also operators outside the @xmath82 sector . another generalization would be to consider the beta deformation of @xmath4 sym , computing the couplings involving the operator that generates such deformation . it would be very interesting to study the deformed anomalous dimension matrix associated to operators that are not exact , and whose deformation does not lead to an integrable theory . in particular , having a representation of @xmath80 acting on the spin chain associated to operators of the cft at the fixed point would allow for quite effective computations of the corresponding couplings . in this section we compute 3-point correlation functions of @xmath4 sym at strong coupling using the ads / cft duality . so far , computations of correlation functions in the gauge / gravity duality use the field theory limit of strings propagating in ads . in this case , the computation of witten diagrams involves only supergravity fields , giving correlation functions of chiral operators @xcite . on the other hand , here we shall compute 3-point correlation functions involving two insertions of an operator @xmath2 dual to a very massive string @xcite , with a chiral operator @xmath0 dual to a supergravity field . the corresponding witten diagram is given in figure [ wittendiagram ] , where a heavy string state propagates between boundary points at @xmath152 and @xmath153 , and interacts with a light field sourced at the boundary point @xmath34 . this computation can be done for the supergravity fields that couple to a heavy string worldsheet . clearly one can also generalise this computation to higher point functions with more supergravity fields . to compute the string partition function we shall use different approaches to treat the heavy and light string fields . for the heavy string state we shall consider the action for a string ( or particle ) in the first quantised theory and compute its contribution to the partition function by summing over classical trajectories , while for the light fields we shall use the supergravity approximation . it is therefore convenient to represent the source for the operator @xmath2 dual to the heavy string field by @xmath154 , and the source for the chiral operators @xmath0 dual to the supergravity fields by @xmath155 . the gauge theory generating functional for diagrams with insertions of @xmath2 at @xmath152 and at @xmath153 can then be written as @xmath156 by varying with respect to the sources @xmath155 we may compute correlation functions with many chiral operators . in this section we are interested in the simplest case of taking one such derivative to compute the 3-point function @xmath157 for some chiral operator @xmath0 . the ads / cft duality states that the gauge theory generating functional for correlation functions of local operators equals the string partition function with suitable boundary conditions @xcite . in particular , at strong coupling , the generating functional ( [ generatingfunctional ] ) can be approximated by @xmath158 + s_{sugra}[\phi ] \right)}\ , , \label{stringgeneratingfunctional}\ ] ] where we use the string polyakov action @xmath159 to describe the propagation of the heavy string state . the corresponding worldsheet starts and ends very close to the boundary , i.e. in poincar coordinates @xmath160 it obeys the boundary conditions @xmath161 where @xmath162 is a regulator . the effect of these boundary conditions is to generate two functional derivatives with respect to the source @xmath154 of the heavy field , justifying the identification between ( [ generatingfunctional ] ) and ( [ stringgeneratingfunctional ] ) . the supergravity fields in ( [ stringgeneratingfunctional ] ) are represented by @xmath163 and approximate the gauge theory sources @xmath164 near the boundary , in the sense that @xmath165 as @xmath166 . the propagation of the light fields is determined by the supergravity action around the @xmath167 vacuum , which we denote below by @xmath168 . the vacuum value for the ten - dimensional einstein metric @xmath169 is given by @xmath170 where the ads radius satisfies @xmath171 . then , it is simple to show that the five - dimensional supergravity action in the einstein s frame has the form @xmath172 where @xmath24 represents terms in the action other than the metric and dilaton fields . the gravitational coupling is given by @xmath173 . the propagation of the heavy string state , and its coupling to the supergravity fields , is determined from the polyakov action s_p[x , , ] = - g d^2 ^ _ x^a _ x^b g_ab e^/2 + , [ polyakov ] where @xmath169 is the ten - dimensional metric in the einstein frame , @xmath174 represents the fluctuations of the dilaton field and @xmath24 includes other terms like worldsheet fermions and other supergravity fields . the heavy string will have the worldsheet topology of a cylinder . working in the conformal gauge , the integration over worldsheet metrics becomes simply an integration over the modular parameter @xmath175 of the cylinder , i.e. [ gammatos ] d^2 ^ _ -s/2^s/2d_0 ^ 2 d^. to compute the generating functional ( [ stringgeneratingfunctional ] ) it is convenient to perform first the path integral over the supergravity fields . we write ( [ stringgeneratingfunctional ] ) as @xmath176 } \int d\phi \,e^{i \left(s_{sugra}[\phi ] + \int d^2\sigma\left . \frac{\delta s_{p}[x , s,\phi]}{\delta \phi}\right|_{\phi=0}\,\phi + \cdots\right ) } \ , . \label{stringgeneratingfunctional2}\ ] ] for a fixed off - shell string worldsheet , the supergravity functional can be computed with witten diagrams , after boundary sources for the supergravity fields are specified . the new ingredient are the extra terms localized along the string worldsheet that add to the supergravity action . these terms determine the coupling between the light fields and the heavy string state . in ( [ stringgeneratingfunctional2 ] ) we wrote just the leading term , which acts as a simple tadpole for the supergravity fields ( it comes from a cubic interaction in string field theory ) . these terms can be treated perturbatively and do not affect the free propagators of light fields . before computing diagrams with a supergravity field , let us recall the computation of the 2-point function for the operator @xmath2 dual to a heavy field , as done in @xcite . to obtain the correct scaling of the 2-point function it is necessary to convolute the generating functional ( [ stringgeneratingfunctional2 ] ) with the wave function of the classical field we are considering . in the wkb approximation this amounts to changing the measure in the string path integral such that the action determining the propagator of the heavy field is actually @xmath177 \ , , \label{newaction}\ ] ] where we use letters @xmath178 and @xmath179 respectively for the @xmath180 and @xmath181 indices . the worldsheet canonical momentum is @xmath182 , and @xmath183 and @xmath184 are the @xmath180 zero modes @xmath185 the arbitrariness in the definition of these zero modes requires a precise prescription . in @xcite it was proposed to use the embedding coordinates of @xmath180 , therefore preserving the @xmath186 symmetry of the conformal group . for a number of particular examples , it was shown in @xcite the following result , which is expected to be general , @xmath187 } \approx \frac{p}{|x_i - x_f|^{2\delta_a}}\,,\ ] ] where we absorbed the cut - off dependence in the measure . the path integral is dominated by the classical saddle point , which yields the correct conformal dependence for the 2-point function of the operator @xmath2 . the pre - factor @xmath124 , which is associated to the integration of fluctuations of the classical solution , will define the normalization of @xmath2 . since the 3-point function is defined with respect to unit normalized operators , we shall see below that to leading order we actually do not need to evaluate this pre - factor . next let us consider the 3-pt function @xmath188 , where @xmath189 is a chiral operator of dimension @xmath16 dual to some particular supergravity field @xmath190 . this field may have some tensor structure in @xmath180 and also some kk structure from the @xmath181 compactification . the functional integral for the supergravity fields in ( [ stringgeneratingfunctional2 ] ) can be computed using witten diagrams . if the field @xmath190 has a source at the boundary , ( [ stringgeneratingfunctional2 ] ) leads to [ 3pt - function ] . k_(x ( , ) ; y ) , [ i_integral ] and @xmath191 is the bulk - to - boundary propagator of the field @xmath190 . equation ( [ 3pt - function ] ) states that the 3-point function is simply the expectation value over the heavy string trajectories of the interaction term @xmath192 $ ] , weighted by the action @xmath193 . note that the measure used for the propagation of the heavy string is that defined by the computation of the 2-point function as described above , i.e. after the convolution with the heavy state wave function . on the other hand , the coupling @xmath192 $ ] is determined by the polyakov action @xmath159 . as usual , to compute this path integral one expands around the classical saddle point x ( , ) = |x ( , ) + , s=|s + s , where we rescaled the quantum fluctuations for @xmath194 so that the t hooft coupling @xmath90 does not enter in the quadratic terms arising from the expansion of the action @xmath193 around the saddle point solution @xmath195 . it is then clear that , after expanding the interaction term @xmath192 $ ] around this saddle point , the dominant contribution in ( [ 3pt - function ] ) for large @xmath196 is given by . i _ , where the pre - factor @xmath124 coincides precisely with that in the computation of the 2-point function of @xmath2 . thus , defining @xmath2 to have a unit normalised 2-point function , we conclude that at strong coupling [ result ] _ a(x_i)o_a(x_f)d_(y ) . equation ( [ result ] ) is one of the main results of this paper . the approximations that led to ( [ result ] ) assume that the initial and final heavy string states are the same . this means that interactions with supergravity fields that change conserved charges of the heavy string , such as @xmath197-charge or @xmath14 spin , are not taken into account . it would be interesting to consider a heavy string with different initial and final boundary conditions and include the effect of the light supergravity field on the string saddle point . to fix our conventions let us remark that in the simple case of a scalar field @xmath190 , normalised such that @xmath198 its bulk - to - boundary propagator is given by @xmath199 in this simple case the normalisation of the 2-point function of the operator @xmath189 appearing in ( [ result ] ) is given by _ ( x)d_(y ) = . [ 2pt - functiond ] in the remainder of this section we shall compute 3-point functions of the type @xmath200 . we will consider the simplest case where the operator @xmath0 is dual to the dilaton field , i.e. we will consider the operator @xmath201 . this will allow us to check our results since , as shown in sections 2 and 3 , this correlation function can be obtained from the derivative of @xmath202 with respect to the coupling constant . we need to be careful with normalisations , since the dilaton field in the sugra action has a factor of @xmath203 multiplying the canonical kinetic term . instead , we should compute the witten diagram with the canonically normalised field @xmath204 , whose propagator is given by ( [ kprop ] ) with @xmath28 . the final result should then be multiplied by @xmath205 , since @xmath206 . in practice , when computing @xmath207 in ( [ i_integral ] ) , this amounts to taking the derivative of the action @xmath159 with respect to @xmath174 , while using the normalised propagator @xmath208 as given in ( [ kprop ] ) . in what follows we shall refer to @xmath207 in ( [ i_integral ] ) with that abuse of notation . finally , let us remark that in our conventions the 2-point function of @xmath120 is given at large @xmath209 by ( [ normalisationl ] ) , which can also be verified at strong coupling using the duality . let us consider first the limit where the heavy string field dual to the operator @xmath2 can be approximated by a point - particle of mass @xmath210 . in the einstein frame , the nambu - goto action for a particle coupled to the dilaton takes the form @xmath211= - m \int_0 ^ 1 d\tau \ , e^{\phi/4 } \sqrt { - \dot{x}^a \dot{x}^b g_{ab } } \,,\ ] ] where dot denotes derivative with respect to the worldline parameter @xmath212 . on dimensional grounds one concludes that massive string states will have @xmath213 . we shall be working with the usual poincar coordinates @xmath214 and for simplicity assume only motion in the @xmath180 part of the space . the corresponding polyakov action , depending on both the particle trajectory and the einbein @xmath215 , is @xmath216= \frac{1}{2 } \int_0 ^ 1 d\tau \ , e^{\phi/4 } \left ( \frac{1}{{\rm e } } \,\dot{x}^a \dot{x}^b g_{ab } - { \rm e}\ , m^2 \right ) \,.\ ] ] the functional integration over the einbein @xmath215 can be substituted by a simple integration over the modular parameter @xmath175 , @xmath217= \frac{1}{2 } \int_{-s/2}^{s/2 } d\tau \ , e^{\phi/4 } \left(\dot{x}^a \dot{x}^b g_{ab } - m^2 \right ) \,,\ ] ] analogously to ( [ gammatos ] ) . we may now apply the procedure to obtain the 3-point function starting from ( [ stringgeneratingfunctional2 ] ) . for spacelike separation on the boundary along a direction @xmath218 , the particle action on the @xmath14 vacuum simplifies to [ spnodilaton ] s_p[x , s,=0 ] = _ -s/2^s/2 d ( - m^2 ) . the computation of the 2-point function for the point particle , using this action , was performed in @xcite . the procedure is as follows : ( i ) determine a solution @xmath219 to the particle equations of motion , @xmath220 ( ii ) impose that the endpoints of the motion approach the boundary , @xmath221 , which implies [ kappabc ] , where we have set @xmath222 and @xmath223 ; ( iii ) compute the action s_p[|x , s,=0 ] = ( ^2 - m^2 ) s ; ( iv ) perform the integration over the modular parameter @xmath175 by taking the saddle point , [ saddleparticle ] |s = - i , which corresponds to the `` virasoro constraint '' for the einbein . this computation leads to the correct dependence of the 2-point function , because at the saddle point e^i s_p[|x,|s,=0 ] = ( ) ^2_a , where we considered the large @xmath11 limit , for which @xmath224 . to compute the 3-point function @xmath225 , we need to evaluate @xmath226 $ ] , as given by ( [ i_integral ] ) . taking care of the correct normalization , we have i_= i _ -s/2^s/2 d ( - m^2 ) ( ) ^4 . for small @xmath162 , at the saddle point trajectory ( [ xzparticle ] ) we obtain i_= . at the modular parameter saddle point ( [ saddleparticle ] ) , this expression becomes simply i_=- . we conclude from ( [ result ] ) that _ a(0)o_a(x_f)l(y ) - . [ resultparticle ] this expression has the spacetime dependence required by conformal invariance . the coupling @xmath15 is determined for large @xmath11 , and it agrees with the expectation from the renormalization group result ( [ final ] ) . to see this , notice that since @xmath227 , we have 2 ^ 2 a_laa = - g^2 - . in agreement with ( [ resultparticle ] ) . the simplest example after the point particle is the circular rotating string with two equal spins @xcite , whose 2-point function was also computed in @xcite . we start with the polyakov action coupled to the metric and the dilaton field ( [ polyakov ] ) . the solution @xmath219 for the circular rotating string is given by ( [ xzparticle ] ) in the @xmath180 part of the geometry . in the @xmath181 part , with line element ds^2_s^5 = d^2 + ^2d_3 ^ 2 + ^2(d^2 + ^2d_1 ^ 2 + ^2 d_2 ^ 2 ) , it is given by = , _ 3=0 , = , _ 1=_2=. the conserved angular momenta of the solution are @xmath228 . this configuration is dual to an operator of the type @xmath229 . let us apply now the procedure in @xcite . we have @xmath230 & = g \int_{-s/2}^{s/2 } d\tau \int_{0}^{2\pi } d\sigma \left ( \frac{\dot{x}^2 + \dot{z}^2}{z^2 } - \psi'^2 + \cos^2 \psi \,\dot{\phi_1}^2 + \sin^2 \psi \,\dot{\phi_2}^2 \right ) \nonumber \\ & = 2 \pi g \left ( \frac{4}{s^2 } \log^2\frac{x_f}{\epsilon } + ( \omega^2 - 1 ) \right)s \,,\end{aligned}\ ] ] where prime denotes derivative with respect to @xmath231 , and we have imposed the relation ( [ kappabc ] ) . as detailed in @xcite , there is a subtlety in obtaining the string propagator , so that the classical solution for the cylinder coincides with the classical state . this amounts to considering ( [ newaction ] ) , _ p[|x , s,=0 ] = 2 g ( ^2 -(1+^2 ) ) s . the saddle point in the modular parameter @xmath175 is given by [ saddlecircular ] |s = - i . looking at ( [ kappabc ] ) this implies the virasoro constraint @xmath232 . we conclude that at the saddle point , we have e^i _ p[|x,|s,=0 ] = ( ) ^8 g . this gives the correct dimension @xmath233 . now we will obtain the 3-point function . first we evaluate @xmath234 & = i\ , \frac{3\ , g}{\pi^2 } \int_{-s/2}^{s/2 } d\tau \int_{0}^{2\pi } d\sigma \left ( \frac{\dot{x}^2 + \dot{z}^2}{z^2 } - \psi'^2 + \cos^2 \psi \,\dot{\phi_1}^2 + \sin^2 \psi \,\dot{\phi_2}^2 \right ) \times \nonumber \\ & \hspace{5 cm } \times \left ( \frac{z}{z^2 + ( x - y)^2}\right)^4 \nonumber \\ & = i\,\frac{g}{4\pi}\ , \displaystyle { \frac{\left ( \frac{4}{s^2}\log^2\frac{x_f}{\epsilon}+(\omega^2 - 1)\right)s}{\log\frac{x_f}{\epsilon}}\ ; \frac{x_f^4}{y^4\,(x_f - y)^4 } } \,,\end{aligned}\ ] ] at the saddle point ( [ saddlecircular ] ) , we have i_=- . therefore , we conclude that _ a(0 ) o_a(x_f)l(y ) - . as happened in the point particle case , the spacetime dependence is the one required by conformal invariance . the coupling @xmath15 determined in this way agrees with the expectation , since 2 ^2 a_laa = - g^2 - g^2 2 = - , where we kept the angular momentum @xmath235 fixed when taking the derivative . following the same steps , we move to the more complicated case of the giant magnon where the string rotates on a @xmath236 subspace of @xmath237 @xcite . we remark that although an operator with a single magnon is not gauge invariant , we are implicitly computing the contribution of a single magnon to the 3-point coupling @xmath113 involving an operator @xmath2 in the dilute limit . since the contribution of a magnon to the 2-point function of some operator was not computed in @xcite , we will present first this calculation and then concentrate on the 3-point function . let us start by writing the solution in poincar coordinates . the @xmath14 part is the same as in the two previous cases given in . parametrizing the @xmath181 as ds^2_s^5 = d^2 + ^2d^2 + ^2 d_3 ^ 2 , the giant magnon has non - trivial worldsheet fields in the @xmath238 part , given by @xmath239 where @xmath240 and @xmath241 is the momentum of the magnon . the virasoro constraints , which we will not impose at this stage , require @xmath242 , where @xmath243 given in characterises the @xmath14 motion . then we have @xmath244 & = g \int_{-s/2}^{s/2 } d\tau \int_{-l}^{l } d\sigma \left ( \frac{\dot{x}^2 + \dot{z}^2}{z^2 } + ( \dot{\theta}^2 - \theta'^2 ) + \sin^2 \psi \,(\dot{\varphi}^2 - \varphi'^2 ) \right ) \nonumber \\ & = g \int_{-\frac{s}{2}}^{\frac{s}{2}}d\tau \int_{-l}^{l } d\sigma \left[\kappa^2 + \omega^2 - 2 \omega^2 \cosh^{-2 } ( \omega u)\right]\ , . \end{aligned}\ ] ] using we convolute with respect to the wave function of the rotating string state , which will change the @xmath181 action into its energy . we obtain @xmath245= g \int_{-\frac{s}{2}}^{\frac{s}{2}}d\tau \int_{-l}^{l } d\sigma \left(\kappa^2 - \omega^2 \right ) = 2\ , g \ , s \left(\kappa^2 - \omega^2 \right ) l \ , .\ ] ] taking into account the condition for @xmath243 , it is possible to perform the remaining integration over the modular parameter @xmath175 by saddle point , with the result [ saddle - gm ] |s = -i . again , this corresponds to the virasoro constraint , which in this case reads @xmath242 , and leads to [ conv - polyakov - gm ] _ p[|x,|s,=0]= i 8 g l . it is convenient now to introduce the angular momentum , @xmath246 where we took the large @xmath100 approximation ( notice that the above saddle point defines a @xmath212 integration with @xmath247 real ) . substituting in and exponentiating , we obtain the expected behaviour for the 2-point function , [ 2p - gm ] e^i _ p[|x,|s,=0 ] = ( ) ^2 ( j + 4 g ) , in particular , @xmath248 , which agrees with ( [ gmall - loop ] ) . now it is straightforward to compute the 3-point function . we evaluate @xmath249 & = i\ , \frac{3\ , g}{\pi^2 } \int_{-\bar{s}/2}^{\bar{s}/2 } d\tau \int_{-l}^{l } d\sigma \big ( \kappa^2 + \omega^2 - 2 \omega^2 \cosh^{-2 } ( \omega u ) \big ) \left ( \frac{z}{z^2 + ( x - y)^2}\right)^4 \nonumber \\ & = \frac{12\,g}{\pi^2 } \,\sin \frac{p}{2}\ , \int_{-\bar{s}/2}^{\bar{s}/2 } d\tau \ ; \frac { \sinh{\left(2 \omega l \csc{\frac{p}{2 } } \right ) } } { [ \cos{\left(2 \omega\ , i\tau \cot{\frac{p}{2}}\right ) } + \cosh{\left(2 \omega l \csc{\frac{p}{2}}\right ) } ] } \times\\ & \phantom{aaaaaaaaaaaaa } \times \frac{x_f^{4 } } { \left[(2y^2 - 2y\ , x_f + x_f^2 ) \cosh(\omega\ , i\tau)+(x_f^2 - 2y x_f)\sinh(\omega\,i\tau)\right]^4}\ , . \nonumber \end{aligned}\ ] ] taking the large @xmath100 approximation , as in ( [ jmagnon ] ) , we obtain i_= - . therefore , we conclude that the one - magnon contribution to the 3-point function is @xmath250 which agrees with the expected result 2 ^ 2 a_laa = - g^2 - 2 g . the examples considered in the previous three sections only dealt with relatively simple string configurations with particle - like motion in the @xmath180 part . we are interested in testing our approach in a more general setup , where the bulk - to - boundary propagator for the supergravity field varies along the string worldsheet for fixed worlsheet time @xmath212 . we shall study the spinning string solution with angular momenta both in the @xmath181 and the @xmath180 factors @xcite , whose 2-point function was also computed in @xcite . the study of this solution is necessarily more intricate , because the way in which the string approaches the boundary depends non - trivially on the @xmath180 rotation . in this case it is convenient to use embedding coordinates . again , the starting point of this calculation is the polyakov action in the conformal gauge , coupled to the metric and dilaton , @xmath252 = -g \int_{-s/2}^{s/2 } d\tau \int_{0}^{2\pi } d\sigma\,e^{\phi/2}\,\big[\eta^{\alpha\beta } \partial_{\alpha } y^{a}\partial_{\beta } y^{b}g_{ab}+\eta^{\alpha\beta}\partial_{\alpha } x^{i } \partial_{\beta } x^{j}\ , g_{ij}+ \\ \tilde{\lambda } \,(y^2 + 1)+\lambda\ , ( x^2 - 1)\big]\,,\ \ \ \ \label{eq : embedding}\end{gathered}\ ] ] where , as before , we have set the @xmath180 length and the radius of the @xmath181 to unity and @xmath253 is the embedding metric . the classical solution representing a spinning string is given by @xmath254 , \nonumber \\ & y^2 + y^0= \cosh \rho _ 0 \left(\frac{r}{2}+\frac{e^{\kappa \tau } } { r}\right)\,,\qquad y^3-y^4 = e^{\kappa \tau } \cosh \rho _ 0\ , , \nonumber \\ & y^4 = \sinh \rho _ 0 \cos ( \tilde{\omega}\ , \tau+\sigma)\ , , \qquad y^1 = y^5 = 0\,,\qquad \tilde{\lambda } = -\kappa^2\ , , \nonumber\\ & x^1+i\ , x^2 = e^{i\ , ( \omega\,\tau-\sigma)}\ , , \qquad x^i = 0,\text { for } i > 2\ , , \qquad \lambda = \omega^2 - 1\,,\label{eq : tute}\end{aligned}\ ] ] where @xmath255 . the conserved charges of this solution can be readily obtained as functions of @xmath256 , @xmath257 and @xmath243 , and are given by @xmath258 , @xmath259 and @xmath260 , where @xmath235 , @xmath19 and @xmath261 are the angular momentum on the @xmath181 , angular momentum on the @xmath180 and energy , respectively . this solution has the required boundary conditions if we further identify @xmath262 and @xmath263 where , as in @xcite , we have conveniently absorbed a factor of @xmath264 in @xmath162 . in order to calculate the 2-point function , we have to apply the procedure described in @xcite , which amounts to considering ( [ newaction ] ) instead of @xmath265 $ ] defined in ( [ eq : embedding ] ) . note , however , that the @xmath180 part in ( [ eq : tute ] ) also depends on @xmath231 , which will lead to non - zero values of the zero - modes defined in ( [ zeromodes ] ) . a computation shows that ( [ newaction ] ) becomes @xmath266= 4 \pi g \left[\kappa-\frac{s \sqrt{1-\kappa ^2}}{2 \pi g\kappa } -\frac{1}{\kappa}\left(1+\frac{j^2}{16 \pi^2 g^2}\right)\right]\log\frac{x_f}{\epsilon}\,.\ ] ] it will be more convenient to evaluate the saddle point with respect to @xmath243 instead of the modular parameter @xmath175 . the saddle point in @xmath243 yields the following condition @xmath267 this condition is exactly the virasoro constraint for the spinning string ( [ eq : tute ] ) . at the saddle point we recover the usual 2-point function behaviour @xmath268 } = \left(\frac{\epsilon}{x_f}\right)^{8 \pi g \left(\kappa _ s+\frac{s \kappa _ s}{4 \pi g \sqrt{\kappa _ s^2 + 1}}\right)}\,,\ ] ] which gives the correct scaling dimension @xmath269 , where we have defined @xmath270 . note that in the @xmath271 limit we recover the circular spinning string of section 4.2 , as we should . now we will obtain the 3-point function . first we need to evaluate ( [ i_integral ] ) . in contrast to the other cases , the integrand in this case strongly depends on @xmath231 , and the integrals might appear to be very complicated . however , because we are integrating @xmath231 from @xmath272 to @xmath109 we can rewrite our integral as an integral in the complex plane by considering the complex variable @xmath273 . the integral can then be computed using the residues theorem . for fixed wordsheet time @xmath212 , the integrand has two poles , one of which has zero residue and the other gives the relevant contribution . the intermediate steps are too cumbersome to be presented here , so we just state the final result @xmath274 = -\frac{\delta_a}{4\pi^2 \kappa_s^2 } \left(1+\frac{s}{2\pi g \tilde{\omega}}\right)\frac{x_f^4}{y^4(x_f - y)^4}\,.\ ] ] therefore , we conclude that , to leading order in @xmath196 , _ a(0 ) o_a(x_f)l(y ) - . as happened in the previous cases , the spacetime dependence is the one required by conformal invariance . moreover , to leading order in @xmath196 , the coupling @xmath15 determined in this way agrees with the rg expectation , since 2 ^2 a_laa = - g^2 - , where we kept the angular momenta @xmath19 and @xmath235 fixed when taking the derivative . we used ( [ eq : saddlerot ] ) to determine @xmath275 , which leads to @xmath276 . one of the present challenges in the gauge / gravity duality is to search for new techniques to compute correlation functions of single trace operators in @xmath4 sym , therefore leading to the exact solution of this theory possibly using integrability , at least in the planar limit . this paper is focused on such new search , both at weak and strong coupling . we started by presenting generic arguments based on the renormalization of operators when a cft is deformed by a marginal or irrelevant operator @xmath0 . these arguments were independent of the value of the coupling at the cft fixed point ( a fixed line in the case of @xmath4 sym ) . to linear order in the deformation parameter @xmath21 , we can write the anomalous dimension matrix as ^u = h + u h. we showed that the matrix element @xmath277 for two operators @xmath2 and @xmath3 is determined by the coupling @xmath1 . for @xmath4 sym and at weak t hooft coupling , we can start by diagonalizing @xmath73 using integrability to some order in the coupling . then the problem of computing @xmath1 amounts to determining the matrix elements @xmath277 . we saw how to implement these ideas for the simplest case of the exact lagrangian deformation , where the action of the matrix @xmath278 on the basis of operators represented as a spin chain is known ( it is just the derivative with respect to the coupling of the anomalous dimension matrix at the critical point ) . in this case one needs to compute the matrix elements of @xmath278 between bethe roots . a very interesting open problem is to extend this procedure to other deformations , therefore allowing for a systematic computation of the couplings @xmath1 in perturbation theory using integrability . when @xmath0 is a chiral operator in the same multiplet of the lagrangian , for example the energy - momentum tensor , we expect that the action of @xmath279 on the basis represented as a spin chain can be obtain acting with the supersymmetry algebra on @xmath278 . it would be very interesting to see how far one can go with this type of approach . the gauge / gravity duality can be used to compute @xmath4 sym correlation functions at strong coupling , but in practice one is limited to the supergravity approximation which only includes chiral operators . we have improved on this limitation , by including two insertions of an operator dual to a heavy string state , and then studied the case of 3-point correlation functions with one extra chiral operator . we considered specific examples with the lagrangian operator , checking agreement with the expected result from renormalization group arguments , since in this case the coupling @xmath15 is simply related to the derivative with respect to the t hooft coupling of @xmath11 . this is an important check , since it gives confidence that the method can be applied to other chiral operators . our computation of the string theory partition function is based on a saddle point approximation to the string path integral that describes the propagation of the heavy state , generalising the analysis of the 2-point function introduced in @xcite . other chiral operators can be included because the heavy string acts as a tadpole for the supergravity fields , which may then propagate to the boundary of @xmath14 if sources are present therein . an alternative way to think of this computation is to realise that the supergravity fields act as sources in the equations of motion for the worldsheet fields of the heavy string worldsheet , therefore deforming it . in fact , the same occurs in the case of three large operators , whose 3-point function at strong coupling ought to be determined by a string worldsheet with fixed boundary condition on three points in the boundary of @xmath14 . in @xcite this was shown to yield the correct conformal dependence of the 3-point function on the boundary points , but the evaluation of the coupling @xmath12 , related to the area of the string worlsheet minimal surface , is still an open problem . it is expected that such computation will show a direct relation with integrability , but that remains to be seen . * acknowledgements * the authors are grateful to joo penedones and pedro vieira for helpful discussions . we also thank konstantin zarembo for sharing with us a draft of his work @xcite , which has some overlap with section 4 of this article . m.s.c . wishes to thank nordita for hospitality during the programme _ integrability in string and gauge theories ; ads / cft duality and its applications_. d.z . is funded by the fct fellowship sfrh / bpd/62888/2009 . this work was partially funded by the research grants ptdc / fis/099293/2008 and cern / fp/109306/2009 . centro de fsica do porto _ is partially funded by fct . in this appendix we re - write the formulae of section 2 for the case of irrelevant deformations . the main results for the renormalization of operators are essentially the same , so we shall be brief . the first modification is that we must be careful with the running of the coupling @xmath21 , so that correlation functions involving a renormalized operator of the undeformed theory @xmath2 become @xmath280 where we work to linear order in the deformation parameter @xmath21 , as defined at the cut - off scale @xmath22 . using the ope expansion between @xmath0 and @xmath2 given in ( [ ope ] ) , we conclude that the region of integration @xmath40 contributes with @xmath281 for @xmath57 . we renormalize the operators of the deformed theory according to @xmath282 so that the correlation functions @xmath283 are finite . with this prescription the two - point function between operators @xmath284 remains the same . this is expected because the deformation is irrelevant and therefore , to leading order in @xmath21 , it is not expected to change the anomalous dimension of operators . we remark that the constant renormalization in ( [ irr_renormalized ] ) for @xmath44 guarantees that this two - 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we study the effect of marginal and irrelevant deformations on the renormalization of operators near a cft fixed point . new divergences in a given operator are determined by its ope with the operator @xmath0 that generates the deformation . this provides a scheme to compute the couplings @xmath1 between the operator @xmath0 and two arbitrary operators @xmath2 and @xmath3 . we exemplify for the case of @xmath4 sym , considering the simplest case of the exact lagrangian deformation . in this case the deformed anomalous dimension matrix is determined by the derivative of the anomalous dimension matrix with respect to the coupling . we use integrability techniques to compute the one - loop couplings @xmath5 between the lagrangian and two distinct large operators built with magnons , in the su(2 ) sector of the theory . then we consider @xmath6 at strong coupling , and show how to compute it using the gauge / gravity duality , when @xmath0 is a chiral operator dual to any supergravity field and @xmath2 is dual to a heavy string state . we exemplify for the lagrangian and operators @xmath2 dual to heavy string states , showing agreement with the prediction derived from the renormalization group arguments . miguel s. costa@xmath7 , ricardo monteiro@xmath8 , jorge e. santos@xmath8 , dimitrios zoakos@xmath7 + _ @xmath9 centro de fsica do porto + departamento de fsica e astronomia + faculdade de cincias da universidade do porto + rua do campo alegre 687 , 4169007 porto , portugal _ + _ @xmath10damtp , centre for mathematical sciences + university of cambridge + wilberforce road , cambridge cb3 0wa , uk _

arxiv

there are many problems concerning electronic structure where attention is focussed on a small region of a larger system , at surfaces or defects in crystals being perhaps the most common . let us call this region i , figure [ fig:1 ] , and the rest of the system region ii . although not of primary interest region ii can not be ignored , since in general the electron wave functions in i will be sensitive to the contents of region ii . some time ago inglesfield @xcite derived an embedding scheme which enables the single - particle schrdinger equation to be solved explicitly only in region i. the influence of region ii is taken into account exactly by adding an energy - dependent non - local potential to the hamiltonian for region i , which constrains the solutions in i to match onto solutions in ii . this embedding method has been developed into a powerful tool most notably for surface electronic structure problems @xcite where it has found widespread application especially to situations where an accurate description of the spectrum of electron states is necessary . examples include studies of image states @xcite , surface states at metals surfaces @xcite , static and dynamic screening @xcite , atomic adsorption and scattering at surfaces @xcite , studies of surface optical response @xcite and field emission @xcite . recent applications to transport problems have also been described @xcite . for a review of the embedding method see inglesfield @xcite . = 80 mm in the case of materials containing heavier elements , relativistic effects can be significant @xcite and lead to important deviations from the electronic structure as predicted by the schrdinger equation shifts in inner core levels of 5d elements are typically several 100 or 1000 ev , valence bands shifts are on the ev scale and spin orbit splitting is often measured in tenths of ev . even ignoring the concomitant changes in electron wave functions these shifts can reorder levels and so affect calculated densities , fundamental to the determination of ground state properties within the density functional framework @xcite . for this reason most of the conventional electronic structure techniques developed for accurately solving the single - particle schrdinger equation in solids have subsequently been modified to deal with the dirac equation , including the relativistic augmented plane wave method @xcite , relativistic linear muffin - tin orbital method @xcite , relativistic augmented spherical wave method @xcite and the relativistic multiple - scattering method @xcite , and each has subsequently been used in studying a diverse range of problems . the last method alone has formed the basis of calculations of photoemission @xcite , magnetocrystalline anisotropy @xcite , hyperfine interactions @xcite and magnetotransport @xcite amongst other topics . inglesfield s embedding method has particular advantages that encourage its extension to the relativistic case . it permits the inclusion of extended substrates for surface and interface calculations , enables the study of isolated point defects in solids and being a basis set technique is highly flexible and permits full - potential studies with relative ease . at surfaces extended substrates ( as against the use of the supercell or thin - film approximation in which the crystal is approximated by a small number of layers , typically 5 - 7 ) enable the proper distinction between surface states , resonances and the continuum of bulk states @xcite . the behaviour of the w(110 ) surface @xcite where the addition of half a monolayer of li is observed to _ increase _ the spin - orbit splitting of a surface state by @xmath2 ev ( resulting in fermi surface crossings separated by @xmath3% of the brillouin zone dimension ) typifies a type of problem a relativistic embedding scheme could address . indeed each of the topics mentioned at the end of the previous paragraph are relevant at surfaces and/or interfaces , and could be usefully investigated within a relativistic embedding framework . in this paper we develop an embedding scheme for the dirac equation that parallels inglesfield s scheme for the schrdinger equation . inglesfield s starting point is the expectation value of the hamiltonian using a trial wave function which is continuous in amplitude but discontinuous in derivative across the surface @xmath1 separating i and ii . the first order nature of the dirac equation precludes the use of a similar trial function . instead , in the following section we use a trial function in which the large component is continuous and the small component discontinuous across @xmath1 . continuity in the small component is restored when the resulting equations are solved exactly . using the green function for region ii we are able to derive an expression for the expectation value purely in terms of the trial function in i. in section [ section : app ] the application of the method is illustrated by calculating the eigenstates of a hydrogen atom within a cavity and in section [ section : green ] we determine the green function for the embedded region . section [ section : monolayer ] briefly illustrates the method applied to a sandwich structure where relativistic effects are marked . we conclude with a brief summary and discussion . in this section we consider region i joined onto region ii ( figure [ fig:1 ] ) , and derive a variational principle for a trial wave function @xmath4 defined explicitly only within region i. we are primarily interested in the positive energy solutions of the dirac equation @xcite , and so we refer to the upper and lower spinors of the dirac bi - spinor solutions as the large and small components of the wave function respectively . we notionally extend @xmath4 into ii as @xmath5 , an exact solution of the dirac equation at some energy @xmath6 , with the large components of @xmath4 and @xmath5 ( @xmath7 and @xmath8 ) matching on the surface @xmath1 separating i and ii , but with no constraint upon the small components ( @xmath9 and @xmath10 ) , figure [ fig:2 ] . the expectation value for the energy @xmath11 is then @xmath12 } { \langle\varphi|\varphi\rangle_{\mathrm{i}}+ \langle\chi|\chi\rangle_{\mathrm{ii } } } \label{eqn : exp}\ ] ] where @xmath13 . ( for clarity we omit the interaction @xmath14 which appears in the relativistic density functional theory @xcite neglecting orbital and displacement currents , where @xmath15 is a `` spin - only '' effective magnetic field containing an external and exchange - correlation contribution . its inclusion has no consequences for the derivation . ) the first two terms in the numerator are the expectation value of the hamiltonian through regions i and ii , and the third the contribution due to the discontinuity in the small component of the wave function on @xmath1 ( in this and the following , surface normals are directed from i to ii ) . = 60 mm we eliminate reference to @xmath5 by introducing two relations . firstly , for @xmath16 , @xmath5 satisfies the dirac equation at energy @xmath6 @xmath17\chi=0 \label{eqn : dirii}\ ] ] and differentiating with respect to @xmath6 the energy derivative of @xmath5 , @xmath18 , satisfies @xmath19\dot{\chi}=\chi\qquad \bi{r}\in\mathrm{ii}. \label{eqn : eder}\ ] ] multiplying the hermitian conjugate of the first equation by @xmath20 from the right , multiplying the second from the left by @xmath21 , subtracting and integrating over region ii gives a relation between the normalisation of @xmath5 in ii and the amplitude on @xmath1 : @xmath22 we have assumed that @xmath5 vanishes sufficiently strongly at infinity . for the second relation we introduce the green function ( resolvant ) @xmath23 corresponding to equation ( [ eqn : dirii ] ) : @xmath24g=-\delta(\bi{r}-\bi{r } ' ) \qquad \bi{r},\bi{r}'\in \mathrm{ii}. \label{eqn : gf}\ ] ] multiplying the hermitian conjugate of this equation by @xmath5 from the right , and subtracting @xmath25 times equation ( [ eqn : dirii ] ) , integrating over region ii and then using the reciprocity of the green function gives @xmath26 we see that the green function relates the amplitude of the wave function on @xmath1 to the amplitude at any point within ii . in particular , we can obtain a relation between the large and small components of @xmath5 on s. writing the @xmath27 green function as @xmath28 where each entry is a @xmath29 matrix , substituting into equation ( [ eqn : chis ] ) , and rearranging the two equations coupling the small and large components of @xmath5 gives @xmath30 where @xmath31 it follows from ( [ eqn : chis ] ) that the green functions in ( [ eqn : gam ] ) are the limiting forms of @xmath32 as @xmath33 from within ii . equations ( [ eqn : norm ] ) and ( [ eqn : cgc ] ) are the desired results that enable us to express the expectation value @xmath11 in ( [ eqn : exp ] ) in terms of @xmath4 alone . after substitution and use of the continuity of the large components @xmath34 on @xmath1 we obtain @xmath35 } { \langle\varphi|\varphi\rangle_{\mathrm{i}}- c^2\hbar^2 \int_s \rmd\bi{r}_s\cdot \varphi_{\rm l}^\dag \bsigma \int_s \rmd\bi{r}'_s \cdot \dot{\gamma } \bsigma\ , \varphi_{\rm l } } . \label{eqn : exp2}\ ] ] this is an expression for the expectation value of the energy @xmath11 , given purely in terms of the trial function @xmath4 in region i and on the surface @xmath1 , with all details of region ii entering via @xmath36 and its energy derivative . following the convention in the non - relativistic embedding scheme we shall refer to @xmath36 as the embedding potential . to see what this variational principle means in practice , we consider variations in @xmath37 , whereby @xmath38 } { \langle\varphi|\varphi\rangle_{\mathrm{i}}- c^2\hbar^2\int_s \rmd\bi{r}_s\cdot \varphi_{\rm l}^\dag \bsigma \int_s \rmd\bi{r}'_s \cdot \dot{\gamma } \bsigma\ , \varphi_{\rm l } } , \nonumber\\ \label{eqn : de1}\end{aligned}\ ] ] so that solutions @xmath4 stationary with respect to arbitrary variations @xmath39 satisfy @xmath40 the first expression indicates @xmath4 is a solution of the dirac equation at energy @xmath11 in region i. comparing the second with ( [ eqn : cgc ] ) shows that @xmath4 also possesses the correct relationship between large and small components on @xmath1 , the surface separating i and ii , to match onto solutions in ii . the term @xmath41 provides a first order correction to @xmath42 so that the boundary condition is appropriate for energy @xmath11 . in practice expression ( [ eqn : exp2 ] ) may be used to obtain solutions of the dirac hamiltonian by inserting a suitably parameterised trial function and varying the parameters to obtain a stationary solution . this is conveniently achieved by expanding the trial solution in a finite basis of separate large and small component spinors @xmath43 + \sum_{n=1}^{n_{\rm s } } a_{{\rm s},n } \left[\begin{array}{c}0\\ \psi_{{\rm s},n}(\bi{r})\end{array}\right ] = \left[\begin{array}{cc } \bpsi_{\rm l}(\bi{r } ) & 0 \\ 0 & \bpsi_{\rm s}(\bi{r } ) \end{array}\right ] \left [ \begin{array}{c } \bi{a}_{\rm l } \\ \bi{a}_{\rm s } \end{array } \right].\ ] ] the matrix in the final expression is @xmath44 by @xmath45 , and the column vector contains the @xmath45 coefficients . substituting into ( [ eqn : exp2 ] ) we find states @xmath46 that are stationary with respect to variations in the expansion coefficients @xmath47 are then given by the eigenstates of a generalised eigenvalue problem of the form @xmath48 \left [ \begin{array}{c } \bi{a}_{\rm l } \\ \bi{a}_{\rm s } \end{array } \right ] = w \left [ \begin{array}{cc } { o}_{\rm ll } & 0 \\ 0 & { o}_{\rm ss } \end{array } \right ] \left [ \begin{array}{c } \bi{a}_{\rm l } \\ \bi{a}_{\rm s } \end{array } \right ] , \label{eqn : a1}\ ] ] where @xmath49_{nn'}= \int_i \psi_{{\rm l},n}^\dag(\bi{r } ) \left(v(\bi{r})+mc^2 \right ) \psi_{{\rm l},n'}(\bi{r } ) \rmd\bi{r } \nonumber \\ \fl \phantom{\left[{h}_{\rm ll}\right]_{nn'}=}{}+c^2\hbar^2\!\int_s \rmd\bi{r}_s\cdot \psi_{{\rm l},n}^\dag(\bi{r}_s)\bsigma\ ! \int_s \rmd\bi{r}'_s \!\cdot\ ! \left [ \gamma(\bi{r}_s,\bi{r}'_s;w)\ ! - \ ! w \dot{\gamma}(\bi{r}_s,\bi{r}'_s;w)\right ] \bsigma\psi_{{\rm l},n'}(\bi{r}'_s ) \label{eqn : h2}\\ \fl \left[{h}_{\rm ls}\right]_{nn'}= \int_i \psi_{{\rm l},n}^\dag(\bi{r } ) c\bsigma\cdot \widehat{\bi{p } } \psi_{{\rm s},n'}(\bi{r } ) \rmd\bi{r}+\rmi c\hbar\int_s \rmd\bi{r}_s\cdot \psi_{{\rm l},n}^\dag(\bi{r}_s)\bsigma\psi_{{\rm s},n'}(\bi{r}_s)\label{eqn : h3}\\ \fl \left[{h}_{\rm sl}\right]_{nn'}= \int_i \psi_{{\rm s},n}^\dag(\bi{r } ) c\bsigma\cdot\widehat{\bi{p } } \psi_{{\rm l},n'}(\bi{r } ) \rmd\bi{r } \label{eqn : h4}\\ \fl \left[{h}_{\rm ss}\right]_{nn'}= \int_i \psi_{{\rm s},n}^\dag(\bi{r } ) \left(v(\bi{r})-mc^2 \right ) \psi_{{\rm s},n'}(\bi{r } ) \rmd\bi{r } \label{eqn : h5}\\ \fl \left[{o}_{\rm ll}\right]_{nn'}= \int_i \psi_{{\rm l},n}^\dag(\bi{r } ) \psi_{{\rm s},n'}(\bi{r } ) \rmd\bi{r}\nonumber\\ \fl \phantom{\left[{o}_{\rm ll}\right]_{nn'}= } { } -c^2\hbar^2\int_s \rmd\bi{r}_s\cdot \psi_{{\rm l},n}^\dag(\bi{r}_s)\bsigma \int_s \rmd\bi{r}'_s \cdot \dot{\gamma}(\bi{r}_s,\bi{r}'_s;w ) \bsigma\psi_{{\rm l},n'}(\bi{r}'_s ) \label{eqn : h6}\\ \fl \left[{o}_{\rm ss}\right]_{nn'}= \int_i \psi_{{\rm s},n}^\dag(\bi{r } ) \psi_{{\rm s},n'}(\bi{r } ) \rmd\bi{r}.\label{eqn : h7}\end{aligned}\ ] ] of course the spectrum of the dirac hamiltonian is unbounded below , and care must be taken to prevent solutions collapsing to negative energies . this can be avoided through the use of a kinetically balanced basis @xcite in which there is a one - to - one relationship between large and small component spinors , @xmath50 , and where the small component spinors are given by @xmath51 the upper half of the spectrum of the @xmath52 eigenstates of ( [ eqn : a1 ] ) then provide approximations to the spectrum of electronic states . = 80 mm to illustrate the application of the relativistic embedding scheme we consider a model problem of a hydrogen atom within a spherical cavity , finding bound states of the dirac equation corresponding to the potential illustrated in figure [ fig:3 ] : @xmath53 where @xmath54 and @xmath55 . we choose this model as the bound states may also be found straightforwardly by alternative methods . region i , the region to be treated explicitly , is the sphere of radius @xmath56 centered on @xmath57 . the external region ii where @xmath58 is replaced by an embedding potential acting on the surface of the sphere . the value of the embedding potential is most readily evaluated from equation ( [ eqn : cgc ] ) . a general solution to the dirac equation at some energy @xmath6 in region ii and satisfying the appropriate boundary conditions is @xcite @xmath59 where @xmath60 , @xmath61 , @xmath62 a spin - angular function , @xmath63 a modified spherical bessel function of the third kind @xcite , @xmath64 , @xmath65 and @xmath66 the spherical symmetry of region ii means the the embedding potential @xmath36 may be expanded on @xmath1 as @xmath67 and substituting ( [ eqn : a2b ] ) and ( [ eqn : a2 ] ) into ( [ eqn : cgc ] ) leads to @xmath68 using ( [ eqn : gam ] ) with the green function for constant potential ( @xmath69 ) @xmath70 gives the same result but after rather more involved manipulations . @xmath71 is a modified spherical bessel function of the first kind . because of the spherical symmetry we can determine separately states with a given angular character @xmath72 . using as a basis set for the large component spinors @xmath73 so that the small component spinors ensuring kinetic balance are @xmath74 e^{-r } , \label{eqn : bas2}\ ] ] the matrix elements become @xmath75_{nn'}= \int_0^r g_{n}(r)\left[-\frac{\lambda}{r}+mc^2\right ] g_{n'}(r ) \rmd r \nonumber\\ + \hbar^2c^2r^2g_{n}(r)g_{n'}(r ) \left[\gamma_\kappa(w)-w\dot{\gamma}_\kappa(w)\right ] \\ \fl \left[{h}^{(\lambda)}_{\rm ss}\right]_{nn'}= \int_0^rf_{n\kappa}(r)\left[-\frac{\lambda}{r}-mc^2\right]f_{n'\kappa}(r ) \rmd r\\ \fl \left[{h}^{(\lambda)}_{\rm ls}\right]_{nn'}=-\hbar c \int_0^r g_{n}(r)\left[\frac{\rmd f_{n'\kappa}(r)}{\rmd r}- \frac{\kappa}{r}f_{n'\kappa}(r)\right ] \rmd r + \hbar c g_{n}(r)f_{n'\kappa}(r)\\ \fl \left[{h}^{(\lambda)}_{\rm sl}\right]_{nn'}=\hbar c \int_0^r f_{n\kappa}(r)\left[\frac{\rmd g_{n'}(r)}{\rmd r}+ \frac{\kappa}{r}g_{n'}(r)\right ] \rmd r \\ \fl \left[{o}^{(\lambda)}_{\rm ll}\right]_{nn'}= \int_0^r g_n(r)g_{n'}(r ) \rmd r -\hbar^2c^2r^2g_n(r)g_{n'}(r)\dot{\gamma}_\kappa(w ) \\ \fl \left[{o}^{(\lambda)}_{\rm ss}\right]_{nn'}= \int_0^r f_{n\kappa}(r)f_{n'\kappa}(r ) \rmd r\end{aligned}\ ] ] the eigenvalues only depend upon the quantum number @xmath76 . in table [ table:1 ] the lowest two eigenvalues of @xmath77 symmetry ( corresponding to the @xmath78 and @xmath79 of free hydrogen ) are shown as a function of basis set size and for different values of the energy @xmath6 at which the embedding potential is evaluated , for the case @xmath80 , @xmath81 . for comparison also given are the values found by matching the external solution ( [ eqn : a2 ] ) to the regular internal solution , which can be expressed in terms of confluent hypergeometric functions@xcite . for a given fixed @xmath6 the eigenvalues converge from above to values that are equal or above the exact values . the further @xmath6 lies from the eigenvalue , the larger the difference between the limiting value for large basis sets and the correct value . however , the influence of the @xmath82 terms in ( [ eqn : exp2 ] ) means the error is relatively small . when @xmath83 , the lowest eigenvalue found with @xmath84 is @xmath85 ha and in error by only 0.0000044 ha , a factor @xmath86 smaller than the error in @xmath6 . @cccc @xmath87 & @xmath88 & @xmath83 & @xmath89 + 2 & -0.4111620 , 1.6980995 & -0.4111527 , 1.6949300 & -0.4111624 , 1.6896482 + 4 & -0.4451482 , 0.9129418 & -0.4451439 , 0.9126817 & -0.4396204 , 0.9714775 + 6 & -0.4455519 , 0.8914789 & -0.4455477 , 0.8912219 & -0.4455520 , 0.8910268 + 8 & -0.4455532 , 0.8912708 & -0.4455488 , 0.8910141 & -0.4455532 , 0.8908194 + `` exact '' & -0.4455532 , 0.8908194 & -0.4455532 , 0.8908194 & -0.4455532 , 0.8908194 + differentiating ( [ eqn : exp2 ] ) with respect to the trial energy @xmath6 shows the expectation value is stationary at @xmath90 . in this case @xmath11 is given by the solutions of @xmath91 } { \langle\varphi|\varphi\rangle_{\mathrm{i } } } . \label{eqn : exp3}\ ] ] eigenfunctions @xmath4 solving this equation satisfy the dirac equation within i and the relationship between small and large components on @xmath1 ( [ eqn : de2 ] ) is exact . the final column in table [ table:1 ] shows the lowest two positive energy eigenvalues of ( [ eqn : exp3 ] ) , again as a function of basis set size . the eigenvalues again converge from above , and by @xmath84 reproduce the exact values by at least 7 significant figures . it is worth noting that with this particular basis set increasing @xmath87 much further leads to some numerical difficulties due to overcompleteness . for more accurate work a more suitable basis set should be used . it should also be noted that conventional finite basis set calculations using a basis satisfying kinetic balance can given eigenvalues that lie below exact limiting values by an amount of order @xmath92 @xcite , and similar behaviour is expected in this embedding scheme . most practical applications of the schrdinger embedding scheme have actually used the green function of the embedded system . this is a more convenient quantity when dealing with systems where the spectrum is continuous , such as at surfaces or defects in solids . we therefore consider the green function for the embedded dirac system . differentiating ( [ eqn : exp2 ] ) with respect to @xmath6 shows @xmath11 is stationary when @xmath90 , as would be expected . in this case stationary solutions satisfy the embedded dirac equation @xmath93 where , introducing @xmath94 , the component of @xmath95 in the direction normal to the surface @xmath1 ( from i to ii ) at @xmath96 , the additional term @xmath97 enforcing the embedding is @xmath98 . \label{eqn : gf2}\end{aligned}\ ] ] the corresponding green function satisfies @xmath99 for @xmath100 . a similar line of argument to that given by inglesfield @xcite for the embedded schrdinger equation shows that this green function is identical for @xmath100 to the green functions @xmath101 for the entire system i@xmath102ii . for simplicity assuming i@xmath102ii constitute a finite system so that the spectrum is discrete , the green function @xmath101 is given by @xmath103 where @xmath104 is the eigenvalue corresponding to eigenstate @xmath105 of the entire system , normalised to unity over i@xmath102ii . for a given @xmath11 , the green function solving ( [ eqn : gf3 ] ) can be expanded in terms of the eigenstates @xmath106 of the corresponding homogeneous equation @xmath107 normalised to unity over i , as @xmath108 clearly @xmath109 has poles at @xmath110 . at these energies ( [ eqn : gf5 ] ) becomes the exact embedded dirac equation ( [ eqn : gf1 ] ) so as we have seen the poles will occur at eigenstates of the entire system and the spectrum of @xmath109 and @xmath101 coincide . it remains to show the poles of @xmath109 have the appropriate weight . the residue of @xmath109 at @xmath111 is @xmath112 the second term in the denominator is precisely the additional factor necessary to correctly normalise the states ( see ( [ eqn : norm ] ) , ( [ eqn : cgc ] ) ) so that @xmath113 the residues of the green function of the embedded system and those of the entire system are identical . hence the two green functions are identical for @xmath114 i. for practical calculations the green function can be expanded using a double - basis of separate large and small component spinors : @xmath115{g}(w ) \left[\begin{array}{cc } \bpsi_{\rm l}(\bi{r } ' ) & 0 \\ 0 & \bpsi_{\rm s}(\bi{r } ' ) \end{array}\right]^\dag.\ ] ] the matrix elements of the matrix of coefficients @xmath116 may be found by substituting into ( [ eqn : gf3 ] ) , multiplying from the right by the vector of basis functions , multiplying from the left by the hermitian transpose of the vector of basis functions , and integrating over region @xmath117 . this leads to @xmath118^{-1}\ ] ] where the overlap and hamiltonian matrices have their previous definitions ( [ eqn : h2]-[eqn : h7 ] ) with @xmath119 . as an illustration we calculate the local density of states for the confined hydrogen model at energies above @xmath120 where the spectrum is continuous . integrating over the embedded region this is given by @xmath121 figure [ fig:4 ] shows the @xmath122 wave local density of states for @xmath80 , @xmath123 , calculated with varying number of basis functions . the basis functions ( [ eqn : bas1 ] ) , ( [ eqn : bas2 ] ) are not particularly appropriate for representing the continuum wave solutions , and so convergence is only achieved using a relatively large set ; however the results serve to illustrate the systematic improvement that accompanies an increasing number of basis functions . the local density of states shows two resonances , the precursors of bound states that exist when any of @xmath56 , @xmath124 or @xmath120 are increased sufficiently . as a further example , one that provides a test of the relativistic embedding scheme when applied to a more challenging problem , we use it to calculate the local density of states on a silver monolayer in a au(001)/ag / au(001 ) sandwich structure . using the embedding scheme only the region occupied by the ag monolayer is explicitly treated . this is region i , with the two au halfspaces to either side entering the calculation via embedding potentials expanded on planar surfaces . then , using bloch s theorem the calculation is performed within a unit cell containing one atom . the full technical details will be described elsewhere , but briefly the green function at two - dimensional wave vector @xmath125 is expanded in a set of linearised augmented relativistic plane waves . we use large component basis functions @xmath126\omega_\lambda(\widehat{\bi{r } } ) & & \bi{r } \in \mbox{muffin - tin } \end{array } \right.\ ] ] where @xmath127 is a pauli spinor , @xmath128 , with @xmath129 a two - dimensional reciprocal lattice vector and @xmath130 , @xmath131 and where @xmath132 exceeds the width of the embedded region ensuring variational freedom in the basis . the function @xmath133 is the large component of the wavefunction that satisfies the radial dirac equation for the spherically symmetric component of the potential at some pivot energy ; @xmath134 is the energy derivative of @xmath135 . the matching coefficients @xmath136 , @xmath137 ensure continuity of the basis function in amplitude and derivative at the muffin - tin radius . the small component basis functions are chosen to satisfy kinetic balance . overlap and hamiltonian matrix elements follow directly from these basis functions . the embedding potential is obtained from ( [ eqn : cgc ] ) using the general expression for a wavefunction outside a surface at wave vector @xmath125 . this gives for the embedding potential describing the left au half space @xmath138_{\bi{g}\sigma\bi{g}'\sigma'}\nonumber\\ \times \exp(\rmi(\bi{k}+\bi{g})\cdot \bi{r}_s ) \exp(-\rmi(\bi{k}+\bi{g}')\cdot \bi{r}'_s ) \,\varphi_\sigma \otimes \varphi_{\sigma ' } \label{eqn : embpot}\end{aligned}\ ] ] with @xmath139 the reflection matrix @xmath140 is found using standard layer - scattering methods @xcite . a similar approach may be used to obtain an embedding potential for the right half space , which unlike the non - relativistic case differs from that for the left half space . figure [ fig:5 ] compares the local density of states calculated using the relativistic embedding technique for an embedded ag monolayer using embedding potentials corresponding to au(001 ) with that found for an au(001)/ag / au(001 ) sandwich geometry using relativistic scattering theory @xcite . the same au and ag potentials has been used in each case , and the local density of states found within the same muffin - tin volume . therefore the results obtained with the two methods should be comparable , and we find that they are indistinguishable . this confirms that the embedding potential ( [ eqn : embpot ] ) imposes the correct variational constraint upon wave functions for the embedded ag monolayer so that they replicate the behaviour of an extended au(001)/ag / au(001 ) sandwich structure . the inset in figure [ fig:5 ] shows the local density of states in the non - relativistic limit ( @xmath141 ) , indicating the significant relativistic effects on the electronic structure which are correctly reproduced with this dirac - embedding scheme . we have outline above an embedding scheme for the dirac equation . it enables the dirac equation to be solved within a limited region i when this region forms part of a larger system , i@xmath102ii . region ii is replaced by an additional term added to the hamiltonian for region i , and which acts on the surface @xmath1 separating i and ii . the embedding scheme is derived using a trial function in which continuity in the small component across @xmath1 is imposed variationally . expanding the wave function in a basis set of separate large and small component spinors , the problem of variational collapse is avoided by using a basis satisfying kinetic balance . calculating the spectrum of a confined hydrogen atom , the method is shown to be stable and converge to the exact eigenvalues . we have also derived the green function for the embedded hamiltonian and illustrated its use in the continuum regime of the same confined hydrogen system and an au / ag / au sandwich structure . these are demonstration calculations future applications are likely to be to defects and surfaces of materials containing heavier ( typically 5@xmath142 ) elements , within the framework of density functional theory . it is worthwhile to discuss further the use of a trial function that is discontinuous in the small component , since such a wave function gives rise to a discontinuous probability density and so would normally be dismissed in quantum theory . in non - relativistic quantum mechanics discontinuous trial functions are not permitted , since they possess infinite energy . however the dirac equation is first order in @xmath143 and as we have seen a perfectly regular expectation value of @xmath0 results . exploiting this freedom , the embedding scheme outlined above leads to solutions that are continuous in both large and small component _ only _ when the embedding potential @xmath144 is evaluated at the same energy @xmath6 as the energy @xmath11 that appears in the dirac equation itself , for then the relationship between small and large components on @xmath1 inside ( equation [ eqn : de2 ] ) and outside ( equation [ eqn : cgc ] ) coincide , the large components matching by construction . this may be achieved for example via the iterative scheme used in connection with equation ( [ eqn : exp3 ] ) and the final column of table [ table:1 ] , or explicitly when determining the green function as in section [ section : green ] . these are the methods in which the non - relativistic embedding scheme has been most widely used . when @xmath6 and @xmath11 do not coincide , the solutions obtained via this embedding scheme will retain small components that are discontinuous across @xmath1 . this may be unacceptable for certain applications , but the solutions continue to be valid approximations at least in as much as they provide estimates of the energies of the solutions of the dirac equation , and so could suffice e.g. for interpreting spectroscopic measurements . this embedding scheme places no greater emphasis on a discontinuity in the small amplitude at @xmath1 than on an incorrect ( but continuous ) amplitude elsewhere within the embedded region . it aims merely to optimise the energy of the state , and will retain a discontinuity in the small component if in doing so it can better ( in terms of energy ) approximate the solution inside the embedded region . in the non - relativistic embedding scheme the discontinuous derivative of the trial function implies a probability current ( and electric current ) that is discontinuous across the embedding surface . this is similarly unphysical , yet numerous applications such as those cited above have demonstrated the utility and accuracy of the method . indeed , there have been many applications in which this scheme has been used to determine currents and or transport properties , such as in relation to surface optical response @xcite and electron transport in electron waveguides or through domain walls @xcite . the reason for the success of these calculations is that they employed schemes in which the embedding potential was evaluated at the correct energy , ensuring that the derivative of the wavefunction was continuous across @xmath1 . in practise there have been few calculations using the non - relativistic embedding scheme in which the energies did not coincide . there are a number of aspects of the method which are worthy of further consideration . we started with a trial function in which by construction the large component was continuous and the small component discontinuous across the surface @xmath1 dividing i and ii . we could have reversed these conditions , leading to a similar embedded dirac equation but with a modified embedding term . the particular choice was motivated by the wish to have a theory which behaves reasonably in the limit @xmath141 when the small component becomes negligible a discontinuous amplitude is not permissible in trial solutions to the schrdinger equation . however , the behaviour of the alternative formulation should be investigated . perhaps in connection with this there is the question of the spectrum of negative energy solutions , to which we have paid scant attention . exploring the @xmath141 limit it might be possible to identify how to embed a relativistic region i within a region ii treated non - relativistically a 5@xmath142 overlayer on a simple metal substrate might be a physical system where such a treatment is appropriate . there could be benefits in terms of computational resources expended if the embedding potential could be determined within the framework of a non - relativistic calculation , and there might also be useful insights in terms of simple models . finally , in terms of implementation for realistic systems , some of the novel schemes for deriving embedding potentials @xcite could certainly be adapted to the relativistic case . it would also be worthwhile to consider whether it is possible to use a restricted electron - like basis , in which the large and small component spinors are combined . this is common practice in most relativistic electronic structure calculations for solids when using basis set techniques ( e.g. @xcite ) , and would result in significant computational efficiencies .

an embedding scheme is developed for the dirac hamiltonian @xmath0 . dividing space into regions i and ii separated by surface @xmath1 , an expression is derived for the expectation value of @xmath0 which makes explicit reference to a trial function defined in i alone , with all details of region ii replaced by an effective potential acting on @xmath1 and which is related to the green function of region ii . stationary solutions provide approximations to the eigenstates of @xmath0 within i. the green function for the embedded hamiltonian is equal to the green function for the entire system in region i. application of the method is illustrated for the problem of a hydrogen atom in a spherical cavity and an au(001)/ag / au(001 ) sandwich structure using basis sets that satisfy kinetic balance .

arxiv

the analysis of dynamic processes taking place in complex networks is a major research area with a wide range of applications in social , biological , and technological systems @xcite . the spread of information in online social networks , the evolution of an epidemic outbreak in human contact networks , and the dynamics of cascading failures in the electrical grid are relevant examples of these processes . while major advances have been made in this field , most modeling and analysis techniques are specifically tailored to study dynamic processes taking place in static networks . however , empirical observations in social @xcite , biological @xcite , and financial networks @xcite illustrate how real - world networks are constantly evolving over time @xcite . unfortunately , the effects of temporal structural variations in the dynamics of networked systems remain poorly understood . in the context of temporal networks , we are specially interested in the interplay between the dynamics on networks ( i.e. , the dynamics of processes taking place in the network ) and the dynamics of networks ( i.e. , the temporal evolution of the network structure ) . although the dynamics on and of networks are usually studied separately , there are many cases in which the evolution of the network structure is heavily influenced by the dynamics of processes taking place in the network . one of such cases is found in the context of epidemiology , since healthy individuals tend to avoid contact with infected individuals in order to protect themselves against the disease a phenomenon called _ social distancing _ @xcite . as a consequence of social distancing , the structure of the network adapts to the dynamics of the epidemics taking place in the network . similar adaptation mechanisms have been studied in the context of power networks @xcite , biological and neural networks @xcite and on - line social networks @xcite . despite the relevance of network adaptation mechanisms , their effects on the network dynamics are not well understood . in this research direction , we find the seminal work by gross et al . in @xcite , where a simple adaptive rewiring mechanism was proposed in the context of epidemic models . in this model , a susceptible node can cut edges connecting him to infected neighbors and form new links to _ any _ randomly selected susceptible nodes without structural constraint in the formation of new links . despite its simplicity , this adaptation mechanism induces a complex bifurcation diagram including healthy , oscillatory , bistable , and endemic epidemic states @xcite . several extensions of this work can be found in the literature @xcite , where the authors assume homogeneous infection and recovery rates in the network . another model that is specially relevant to our work is the adaptive susceptible - infected - susceptible ( asis ) model proposed in @xcite . in this model , edges in a given contact network can be temporarily removed in order to prevent the spread of the epidemic . an interesting feature of the asis model is that , in contrast with gross model , it is able to account for arbitrary contact patterns , since links are constrained to be part of a given contact graph . despite its modeling flexibility , analytical results for the asis model @xcite are based on the assumption of homogeneous contact patterns ( i.e. , the contact graph is complete ) , as well as homogeneous node and edge dynamics ( i.e. , nodes present the same infection and recovery rates , and edges share the same adaptation rates ) . as a consequence of the lack of tools to analyze network adaptation mechanisms , there is also an absence of effective methodologies for actively utilizing adaptation mechanisms for containing spreading processes . although we find in the literature a few attempts in this direction , most of them rely on extensive numerical simulations @xcite , on assuming a homogeneous contact patterns @xcite , or a homogeneous node and edge dynamics @xcite . in contrast , while controlling epidemic processes over static networks , we find a plethora of tools based on game theory @xcite or convex optimization @xcite . in this paper , we study adaptation mechanisms over arbitrary contact networks . in particular , we derive an explicit expression for a lower bound on the epidemic threshold of the asis model for arbitrary networks , as well as heterogeneous node and edge dynamics . in the case of homogeneous node and edge dynamics , we show that the lower bound is proportional to the epidemic threshold of the standard sis model over a static network @xcite . furthermore , based on our results , we propose an efficient algorithm for optimally tuning the adaptation rates of an arbitrary network in order to eradicate an epidemic outbreak in the asis model . we confirm the tightness of the proposed lower bonds with several numerical simulations and compare our optimal adaptation rates with popular centrality measures . in this section , we describe the adaptive susceptible - infected - susceptible ( asis ) model over _ arbitrary _ networks with _ heterogeneous _ node and edge dynamics ( heterogeneous asis model for short ) . we start our exposition by considering a spreading process over a time - varying contact graph @xmath0 , where @xmath1 is the set of nodes and @xmath2 is the time - varying set of edges . for any @xmath3 , @xmath4_{i , j}$ ] corresponds to the adjacency matrix of @xmath5 , and the neighborhood of node @xmath6 at time @xmath7 is defined as @xmath8 . in the standard sis epidemic model , the state of node @xmath6 at time @xmath7 is described by a bernoulli random variable @xmath9 , where node @xmath6 is said to be _ susceptible _ if @xmath10 , and _ infected _ if @xmath11 . when the contact graph evolves over time , the evolution of @xmath12 is described by a markov process with the following transition probabilities : @xmath13[.8\linewidth ] { p}(x_i(t+h ) = 1 \mid x_i(t ) = 0 ) = \beta_i\ \sum_{\mathclap{k \in { \pazocal{n}}_i(t)}}\ x_k(t)\,h + o(h ) , \end{multlined } \nonumber \\ & { p}(x_i(t+h ) = 0 \mid x_i(t ) = 1 ) = \delta_i h + o(h ) , \label{eq : recovery}\end{aligned}\ ] ] where the parameters @xmath14 and @xmath15 are called the _ infection _ and _ recovery _ rates of node @xmath6 . in the heterogeneous asis model , the epidemics takes place over a time - varying network that we model as a continuous - time stochastic graph process @xmath16 , described below . let @xmath17 be an initial connected contact graph with adjacency matrix @xmath18_{i , j}$ ] . we assume that @xmath19 is strongly connected . edges in the initial graph @xmath19 appear and disappear over time according to the following markov processes : @xmath13[.9\linewidth ] { p}(a_{ij}(t+h ) = 0 \mid a_{ij}(t ) = 1)=\vspace{.1 cm } \\ \phi_{ij}x_i(t ) h+ \phi_{ji}x_j(t ) h + o(h ) , \end{multlined}\label{eq : cut } \\ & { p}(a_{ij}(t+h ) = 1 \mid a_{ij}(t ) = 0 ) = a_{ij}(0)\psi_{ij } h + o(h ) , \label{eq : rewire}\end{aligned}\ ] ] where the parameters @xmath20 and @xmath21 are called the _ cutting _ and _ reconnecting _ rates . notice that the transition rate in depends on @xmath12 and @xmath22 , inducing an adaptation mechanism of the network structure to the state of the epidemics . the transition probability in can be interpreted as a protection mechanism in which edge @xmath23 is stochastically removed from the network if either node @xmath6 or @xmath24 is infected . more specifically , because of the first summand ( respectively , the second summand ) in , whenever node @xmath6 ( respectively , node @xmath24 ) is infected , edge @xmath25 is removed from the network according to a poisson process with rate @xmath26 ( respectively , rate @xmath27 ) . on the other hand , the transition probability in describes a mechanism for which a ` cut ' edge @xmath25 is ` reconnected ' into the network according to a poisson process with rate @xmath28 ( see figure [ fig : adaptive ] ) . notice that we include the term @xmath29 in to guarantee that only edges present in the initial contact graph @xmath19 can be added later on by the reconnecting process . in other words , we constrain the set of edges in the adaptive network to be a part of the arbitrary contact graph @xmath19 . in this section , we derive a lower bound on the epidemic threshold for the heterogeneous asis model . for @xmath30 , let @xmath31 denote a poisson counter with rate @xmath32 @xcite . in what follows , we assume all poisson counters to be stochastically independent . then , from the two equations in , the evolution of the nodal states can be described by the following set of stochastic differential equations @xmath33 for all @xmath34 . similarly , from and , the evolution of the edges can be described by the following set of stochastic differential equations : @xmath35 for all @xmath36 . by , the expectation @xmath37 $ ] obeys the differential equation @xmath38 = -\delta_i e[x_i ] + \beta_i \ \sum_{\mathclap{k\in{\pazocal{n}}_i(0)}}\ e[(1-x_i)a_{ik}x_k].\ ] ] let @xmath39 $ ] and @xmath40 $ ] . then , it follows that @xmath41 where @xmath42\ ] ] contains positive higher - order terms . in what follows , we derive a set of differential equations to describe the evolution of @xmath43 . from and , we obtain the following equation using it rule for jump processes ( see , e.g. , @xcite ) @xmath44 therefore , @xmath45 for all @xmath46 , where @xmath47\\ & + \beta_i\ \sum_{\mathclap{k\in{\pazocal{n}}_i(0)}}\ e\bigl [ x_i(t)x_k(t)a_{ik}(t ) \\ & + ( 1-a_{ij}(t))a_{ik}(t)x_k(t)\bigr],\end{aligned}\ ] ] which contains positive higher - order terms . the differential equations and describe the joint evolution of the spreading process and the network structure . for further analysis , it is convenient to express the differential equations and using vectors and matrices . for this purpose , let us introduce the following notation . let @xmath48 and @xmath49 be , respectively , the @xmath50 identity matrix and the @xmath51-dimensional column vector of all ones . given two matrices @xmath52 and @xmath53 , their kronecker product @xcite is denoted by @xmath54 . given a sequence of matrices @xmath55 , their direct sum , denoted by @xmath56 , is defined as the block diagonal matrix having @xmath55 as its block diagonals @xcite . if @xmath55 have the same number of columns , then the matrix obtained by stacking @xmath55 in vertical ( @xmath57 on top ) is denoted by @xmath58 . based on this notation , we define the vector - variable @xmath59 , which contains the infection probabilities of all the nodes in the graph . similarly , let @xmath60 and define the column vector @xmath61 . define @xmath62 as the unique row - vector satisfying @xmath63 note that the length of the row vector @xmath62 and the column vector @xmath64 equals @xmath65 , where @xmath66 is the number of the edges in the initial graph @xmath19 . using this notation , we define the following matrices : @xmath67 where @xmath68 denotes the degree of node @xmath6 in the initial graph @xmath19 . furthermore , we also define the following matrices @xmath69 stacking the set of @xmath70 differential equations in into a single vector equation , and ignoring the negative higher - order term @xmath71 , we obtain the following entry - wise vector inequality for the probabilities of infection : @xmath72 also , stacking the set of differential equations in with respect to @xmath73 , and ignoring the negative term @xmath74 , we obtain the following entry - wise vector inequality : @xmath75 where @xmath76 . we can further stack the above inequalities with respect to the index @xmath6 to obtain the following entry - wise vector inequality : @xmath77 combining this inequality and , we obtain @xmath78/dt } \frac{d}{dt}\begin{bmatrix } p q \end{bmatrix } \leq m \begin{bmatrix } p \\ q \end{bmatrix},\ ] ] where @xmath79 is an irreducible matrix ( see appendix [ appx : pf : irreducibility ] for the proof of irreducibility ) defined as @xmath80 therefore , the evolution of the joint vector variable @xmath81 is upper - bounded by the linear dynamics given by the matrix @xmath79 . moreover , the upper bound is tight around the origin , since both @xmath82 and @xmath83 consist of higher - order terms . from , we conclude that the epidemics dies out exponentially fast in the heterogeneous asis model if @xmath84 where @xmath85 is defined as the maximum among the real parts of the eigenvalues of @xmath79 . furthermore , since @xmath79 is a metzler matrix ( i.e. , has nonnegative off - diagonals ) and irreducible , there is a real eigenvalue of @xmath79 equal to @xmath85 @xcite . in the homogeneous case , where all the nodes share the same infection rate @xmath86 and recovery rate @xmath87 , and all the edges share the same cutting rate @xmath88 and reconnecting rate @xmath89 , the condition reduces to the following inequality : @xmath90 where @xmath91 is the spectral radius of the initial graph @xmath19 and @xmath92 which we call the _ effective cutting rate_. the proof of the extinction condition is given in appendix [ appx : pf : thm : stbl : homo ] . we remark that , in the special case when the network does not adapt to the prevalence of infection , i.e. , when @xmath93 , we have that @xmath94 and , therefore , the condition in is identical to the extinction condition @xmath95 corresponding to the homogeneous networked sis model over a static network @xcite . it is worth comparing the condition in with the epidemic threshold @xmath96 given in @xcite for the case in which @xmath19 is the complete graph : @xmath97 where @xmath98 is the link - creating rate , @xmath99 is the link - breaking rate , and @xmath100 is a positive and `` slowly varying '' function depending on the metastable long - time average of the number of infected nodes ( for details , see @xcite ) . we first notice that our lower bound in can be checked directly from the parameters of the model , namely , the adjacency matrix of the initial graph @xmath19 and the relevant rates of the model . this is in contrast with the threshold in , since it depends on the metastable average of the number of infected nodes and , thus , can only be computed via numerical simulations . we also remark that the lower bound on the epidemic threshold in and the epidemic threshold in both exhibit affine dependence on the effective link - breaking rates @xmath101 and @xmath102 , respectively . finally , we see that the recovery rate @xmath103 appears in different places in the two conditions , namely , inside the expression of @xmath101 in and inside the function @xmath100 in . however , the consequences of this difference are not obvious , since @xmath100 is defined via the metastable state and , therefore , does not allow an analytical investigation . [ cols="^,^,^,^ " , ] once the cost functions are selected , we must solve the problem of finding the optimal tuning investment to achieve a desired exponential decay rate in the probabilities of infection . from the inequality in , the infection probabilities @xmath104 , @xmath105 , @xmath106 decay exponentially at a rate ( at least ) @xmath107 if @xmath108 since @xmath79 is an irreducible and metzler matrix , we can use perron - frobenius theory @xcite to prove that is satisfied if there exists an entry - wise positive vector @xmath109 satisfying the following entry - wise vector inequality ( see @xcite for more details ) : @xmath110 therefore , problem [ prb : ] can be reduced to the following equivalent optimization problem : @xmath111 as we show in appendix [ appx : gp ] , we can equivalently transform this optimization problem into a geometric program @xcite , which can be efficiently solved using standard optimization software . the computational complexity of solving the resulting geometric program is @xmath112 , where @xmath70 is the number of nodes and @xmath66 is the number of edges in the initial network @xmath19 . in the rest of this section , we compute the optimal tuning profiles for three different graphs and compare our results with several network centralities . in our simulations , we consider the following three graphs with @xmath113 nodes : _ 1 _ ) an erds - rnyi graph with @xmath114 edges , _ 2 _ ) a barabsi - albert random graph with @xmath115 edges , and _ 3 _ ) a social subgraph ( obtained from facebook ) with @xmath116 edges . for simplicity in our simulations , we assume that all nodes share the same recovery rate @xmath117 and infection rate @xmath118 , where @xmath91 denotes the spectral radius of each initial graph . since @xmath119 , the extinction condition indicates that the infection process does not necessarily die out without adaptation , i.e. , when @xmath120 . the rest of parameters in our simulations are set as follows : we let @xmath121 , @xmath122 , and @xmath123 . also , the parameters in the cost functions are @xmath124 and @xmath125 . the desired exponential decay rate is chosen to be @xmath126 . using this set of parameters , we solve the optimization problem following the procedure described in appendix [ appx : gp ] . our numerical results are illustrated in various figures included in table [ table : ] . each figure is a scatter plot where each point corresponds to a particular edge @xmath46 ; the ordinate of each point corresponding to its optimal cutting rate @xmath26 , and the abscissa of each point corresponds to a particular edge - centrality measure . in these figures , we use three different edge - centrality measures : a ) the product of the degrees of nodes @xmath6 and @xmath24 ( left column ) , b ) the product of the eigenvector - centralities of nodes @xmath6 and @xmath24 ( center column ) , and c ) the betweenness centralities of edge @xmath25 ( right column ) . in our numerical results , we observe how both degree - based and eigenvector - based edge - centralities are good measures for determining the amount of investment in tuning cutting rates . in contrast , betweenness centrality does not show a significant dependency on the optimal cutting rates . in particular , for the synthetic networks in rows 1 ) and 2 ) in table [ table : ] , we observe an almost piecewise affine relationship with the centrality measures in columns a ) and b ) . in particular , in these subplots we observe how edges of low centrality require no investment , while for higher - centrality edges , the tuning investment tends to increase affinely as the centrality of the edge increases as expected . for the real social network in row c ) , the relationship between investment and centralities is still strong although not as clear as in synthetic networks . in the scatter plot corresponding to the relationship between the optimal investment and the eigenvector - based centrality in the real social network ( lower center figure in table [ table : ] ) , we observe a collection of several stratified parallel lines . we conjecture that each line corresponds to a different community inside the social network ; in other words , the relationship between centrality and optimal investment is almost affine inside each community . we have studied an _ adaptive _ susceptible - infected - susceptible ( asis ) model with heterogeneous node and edge dynamics and arbitrary network topologies . we have derived an explicit expression for a lower bound on the epidemic threshold of this model in terms of the maximum real eigenvalue of a matrix that depends explicitly on the network topology and the parameters of the model . for networks with homogeneous node and edge dynamics , the lower bound turns out to be a constant multiple of the epidemic threshold in the standard sis model over static networks ( in particular , the inverse of the spectral radius ) . furthermore , based on our results , we have proposed an optimization framework to find the cost - optimal adaptation rates in order to eradicate the epidemics . we have confirmed the accuracy of our theoretical results with several numerical simulations and compare cost - optimal adaptation rates with popular centrality measures in various networks . 43ifxundefined [ 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 10.1038/30918 [ * * , ( ) ] @noop _ _ ( , ) @noop _ _ ( , ) link:\doibase 10.1007/s00779 - 005 - 0046 - 3 [ * * , ( ) ] @noop * * ( ) in link:\doibase 1102.0629v1 [ _ _ ] ( ) pp . link:\doibase 10.1093/bib / bbp057 [ * * , ( ) ] link:\doibase 10.1038/nature02555 [ * * , ( ) ] link:\doibase 10.1038/nbt.1522 [ * * , ( ) ] link:\doibase 10.1140/epjb / e20020151 [ * * , ( ) ] link:\doibase 10.1016/j.physrep.2012.03.001 [ * * , ( ) ] link:\doibase 10.3201/eid1201.051371 [ * * , ( ) ] link:\doibase 10.1098/rsif.2010.0142 [ * * , ( ) ] link:\doibase 10.1209/epl / i2004 - 10533 - 6 [ * * , ( ) ] link:\doibase 10.1038/304158a0 [ * * , ( ) ] link:\doibase 10.1161/01.atv.0000069625.11230.96 [ * * , ( ) ] link:\doibase 10.1186/s40649 - 015 - 0023 - 6 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.96.208701 [ * * , ( ) ] link:\doibase 10.1098/rsif.2007.1229 [ * * , ( ) ] link:\doibase 10.1007/s10867 - 008 - 9060 - 9 [ * * , ( ) ] link:\doibase 10.1103/physreve.82.036116 [ * * , ( ) ] link:\doibase 10.1103/physreve.83.026102 [ * * , ( ) ] link:\doibase 10.1088/1742 - 5468/2012/08/p08018 [ * * , ( ) ] link:\doibase 10.1007/s00285 - 012 - 0555 - 4 [ * * , ( ) ] link:\doibase 10.1103/physreve.90.022801 [ * * , ( ) ] link:\doibase 10.1103/physreve.88.042802 [ * * , ( ) ] link:\doibase 10.1103/physreve.92.030801 [ * * , ( ) ] link:\doibase 10.1103/physreve.88.042801 [ * * , ( ) ] link:\doibase 10.1186/1471 - 2458 - 12 - 679 [ * * , ( ) ] link:\doibase 10.1103/physreve.85.036108 [ * * , ( ) ] link:\doibase 10.1109/tcns.2015.2426755 [ * * , ( ) ] link:\doibase 10.1016/j.jcss.2006.02.003 [ * * , ( ) ] link:\doibase 10.1109/mcs.2015.2495000 [ * * , ( ) ] link:\doibase 10.1109/tcns.2014.2310911 [ * * , ( ) ] link:\doibase 10.1109/tnet.2008.925623 [ * * , ( ) ] @noop _ _ ( , ) @noop _ _ ( , ) @noop _ _ ( , ) link:\doibase 10.1103/physreve.87.062816 [ * * , ( ) ] link:\doibase 10.1007/s11081 - 007 - 9001 - 7 [ * * , ( ) ] in link:\doibase 10.1109/cdc.2013.6761078 [ _ _ ] ( ) pp . @noop _ _ ( , ) ( ) we show that the matrix @xmath79 defined in is irreducible , that is , there is no similarity transformation that transforms @xmath79 into a block upper - triangular matrix . for this purpose , define @xmath127 where @xmath128 , @xmath129 , and @xmath130 . since the rates @xmath131 and @xmath28 are positive , if @xmath132 , then @xmath133 for all distinct @xmath6 and @xmath24 . therefore , to prove the irreducibility of @xmath79 , it is sufficient to show that @xmath134 is irreducible . in order to show that @xmath134 is irreducible , we shall show that the directed graph @xmath135 on the nodes @xmath136 , defined as the graph having adjacency matrix @xmath137 , is strongly connected . we identify the nodes @xmath138 , @xmath105 , @xmath139 and variables @xmath104 , @xmath105 , @xmath106 , @xmath140 ( @xmath141 ) , @xmath105 , @xmath142 ( @xmath143 ) . then , the upper - right block @xmath144 of the matrix @xmath134 indicates that the graph @xmath135 contains the directed edge @xmath145 for all @xmath146 and @xmath147 . similarly , from the matrices @xmath148 and @xmath149 in @xmath79 , we see that @xmath135 contains the directed edges @xmath150 and @xmath151 for all @xmath146 and @xmath152 . using these observations , let us first show that @xmath135 has a directed path from @xmath153 to @xmath154 for all @xmath155 . since @xmath19 is strongly connected , it has a path @xmath156 such that @xmath157 and @xmath158 . therefore , from the above observations , we see that @xmath135 contains the directed path @xmath159 . in the same way , we can show that @xmath135 also contains the directed path @xmath160 for every @xmath161 . these two types of directed paths in @xmath135 guarantees that @xmath135 is strongly connected . hence the matrix @xmath134 is irreducible , as desired . in the homogeneous case , the matrix @xmath79 takes the form @xmath162 in what follows , we show that @xmath163 if and only if holds true . since @xmath19 is strongly connected by assumption , its adjacency matrix @xmath164 is irreducible and therefore has an entry - wise positive eigenvector @xmath109 corresponding to the eigenvalue @xmath91 ( see @xcite ) . define the positive vector @xmath165 . then , the definition of @xmath62 in shows @xmath166 and therefore @xmath167 . in the same manner , we can show that @xmath168 . moreover , it is straightforward to check that @xmath169 . therefore , for a real number @xmath170 , it follows that @xmath171 hence , if a real number @xmath172 satisfies the following equations : @xmath173 then @xmath174 is an eigenvector of @xmath79 . since @xmath79 is irreducible ( shown in appendix [ appx : pf : irreducibility ] ) , by perron - frobenius theory @xcite , if @xmath175 then @xmath176 ( see ( * ? ? ? * theorem 17 ) ) . this , in particular , shows that @xmath163 if and only if there exist @xmath175 and @xmath177 such that holds . the two equations in have two pairs of solutions @xmath178 and @xmath179 such that @xmath180 , and @xmath181 . therefore , we need to show @xmath182 if and only if holds true . to see this , we notice that @xmath183 and @xmath184 are the solutions of the quadratic equation @xmath185 following from . since @xmath186 , we have @xmath187 if and only if the constant term @xmath188 of the quadratic equation is positive , which is indeed equivalent to . this completes the proof of the extinction condition stated in . we first give a brief review of geometric programs @xcite . let @xmath189 , @xmath105 , @xmath190 denote positive variables and define @xmath191 . we say that a real function @xmath192 is a _ monomial _ if there exist @xmath193 and @xmath194 such that @xmath195 . also , we say that a function @xmath196 is a _ posynomial _ if it is a sum of monomials of @xmath197 ( we point the readers to @xcite for more details ) . given a collection of posynomials @xmath198 , @xmath105 , @xmath199 and monomials @xmath200 , @xmath105 , @xmath201 , the optimization problem @xmath202 is called a _ geometric program_. a constraint of the form @xmath203 with @xmath196 being a posynomial is called a posynomial constraint . although geometric programs are not convex , they can be efficiently converted into equivalent convex optimization problems @xcite . in the following , we rewrite the optimization problem into a geometric program using the new variables @xmath204 and @xmath205 . by , these variables should satisfy the constraints @xmath206 also , using these variables , the cost function @xmath207 can be written as @xmath208 , where @xmath209 are posynomials . in order to rewrite the constraint , we first define the matrices @xmath210 we also introduce the positive constants @xmath211 , @xmath212 , @xmath213 . now , adding @xmath214 to both sides of , we equivalently obtain @xmath215 where @xmath216 is given by summarizing , we have shown that the optimization problem is equivalent to the following optimization problem with ( entry - wise ) positive variables : @xmath218 in this optimization problem , the objective function is a posynomial in the variables @xmath219 , @xmath220 , and @xmath221 . also , the box constraints , , and can be written as posynomial constraints @xcite . finally , since each entry of the matrix @xmath222 is a posynomial in the variables @xmath219 , @xmath220 , and @xmath221 , the vector - constraint yields @xmath139 posynomial constraints . therefore , the optimization problem is a geometric program , as desired . furthremore , a standard estimate on the computational complexity of solving geometric program ( see , e.g. , ( * ? ? ? * proposition 3 ) ) shows that the computational complexity of solving the optimization problem in is given by @xmath112 . finally we remark that @xmath222 contains both the terms @xmath28 and @xmath223 so that we can not use @xmath28 as the decision variable in the geometric program due to the positivity constraint on decision variables . by this reason , we can not design the reconnecting rates @xmath28 under the framework presented in this paper . this issue is left as an open problem .

in this paper , we study the dynamics of epidemic processes taking place in adaptive networks of arbitrary topology . we focus our study on the adaptive susceptible - infected - susceptible ( asis ) model , where healthy individuals are allowed to temporarily cut edges connecting them to infected nodes in order to prevent the spread of the infection . in this paper , we derive a closed - form expression for a lower bound on the epidemic threshold of the asis model in arbitrary networks with heterogeneous node and edge dynamics . for networks with homogeneous node and edge dynamics , we show that the resulting lower bound is proportional to the epidemic threshold of the standard sis model over static networks , with a proportionality constant that depends on the adaptation rates . furthermore , based on our results , we propose an efficient algorithm to optimally tune the adaptation rates in order to eradicate epidemic outbreaks in arbitrary networks . we confirm the tightness of the proposed lower bounds with several numerical simulations and compare our optimal adaptation rates with popular centrality measures .

arxiv

open dielectric resonators have received great attention due to numerous applications @xcite , e.g. , as microlasers @xcite or as sensors @xcite , and as paradigms of open wave - chaotic systems @xcite . the size of dielectric microcavities typically ranges from a few to several hundreds of wavelengths . wave - dynamical systems that are large compared to the typical wavelength have been treated successfully with semiclassical methods . these provide approximate solutions in terms of properties of the corresponding classical system . in the case of dielectric cavities , the corresponding classical system is an open dielectric billiard . inside the billiard rays travel freely while , when impinging the boundary , they are partially reflected and refracted to the outside according to snell s law and the fresnel formulas . the field distributions of resonance states of dielectric cavities can be localized on the periodic orbits ( pos ) of the corresponding billiard @xcite and the far - field characteristics of microlasers can be predicted from its ray dynamics @xcite . semiclassical corrections to the ray picture due to the goos - hnchen shift @xcite , fresnel filtering @xcite , and curved boundaries @xcite are under investigation for a more precise understanding of the connections between ray and wave dynamics @xcite . one of the most important tools in semiclassical physics are trace formulas , which relate the density of states of a quantum or wave - dynamical system to the pos of the corresponding classical system @xcite . recently , a trace formula for two - dimensional ( 2d ) dielectric resonators was developed @xcite . the trace formula was successfully tested for resonator shapes with regular classical dynamics in experiments with 2d dielectric microwave resonators @xcite and with polymer microlasers of various shapes @xcite . however , typical microlasers like those used in refs . @xcite are three - dimensional ( 3d ) systems . while trace formulas for closed 3d electromagnetic resonators have been derived @xcite and tested @xcite , hitherto there is practically no investigation of the trace formula for 3d dielectric resonators . the main reason is the difficulty of the numerical solution of the full 3d maxwell equations for real dielectric cavities . the case of flat microlasers is special since their in - plane extensions are large compared to the typical wavelength , whereas their height is smaller than or of the order of the wavelength . even in this case complete numerical solutions are rarely performed . in practice , flat dielectric cavities are treated as 2d systems by introducing a so - called effective index of refraction @xcite ( see below ) . this approximation has been used in refs . @xcite and a good overall agreement between the experiments and the theory was found . however , it is known @xcite that this 2d approximation ( called the @xmath0 model in the following ) introduces certain uncontrolled errors . even the separation between transverse electric and transverse magnetic polarizations intrinsic in this approach is not , strictly speaking , valid for 3d cavities @xcite . to the best of the authors knowledge , no _ a priori _ estimates of such errors are known even when the cavity height is much smaller than the wavelength . the purpose of the present work is the careful comparison of the experimental length spectra and the trace formula computed within the 2d @xmath0 approximation . furthermore , the effect of the dispersion of the effective index of refraction on the trace formula is investigated as well as the need for higher - order corrections of the trace formula due to , e.g. , curvature effects . the experiments were performed with two dielectric microwave resonators of circular shape and different thickness like in ref . these are known to be ideal testbeds for the investigation of wave - dynamical chaos @xcite and have been used , e.g. , in refs . the results of these microwave experiments can be directly applied to dielectric microcavities in the optical frequency regime if the ratio of the typical wavelength and the resonator extensions are similar . the paper is organized as follows . the experimental setup and the measured frequency spectrum are discussed in . section [ sec : theo ] summarizes the @xmath0 model for flat 3d resonators , the semiclassical trace formula for 2d resonators and how these are combined here . the experimental length spectra are compared to this model in and concludes with a discussion of the results . two flat circular disks made of teflon were used as microwave resonators . the first one , disk a , has a radius of @xmath1 mm and a thickness of @xmath2 mm so @xmath3 . its index of refraction is @xmath4 . a typical frequency of @xmath5 ghz corresponds to @xmath6 , where @xmath7 is the wave number and @xmath8 the speed of light . the second one , disk b , has @xmath9 mm , @xmath10 mm ( @xmath11 ) , and @xmath12 , with @xmath13 ghz corresponding to @xmath14 . the values of the indices of refraction @xmath15 of both disks were measured independently ( see ref . @xcite ) and validated by numerical calculations @xcite . they showed negligible dispersion in the considered frequency range .. ] figure [ sfig : setupphoto ] shows a photograph of the experimental setup . the circular teflon disk is supported by three pillars arranged in a triangle . the prevalent modes observed experimentally are whispering gallery modes ( wgms ) that are localized close to the boundary of the disk @xcite . therefore , the pillars perturb the resonator only negligibly because their position is far away from the boundary . additionally , @xmath16 cm of a special foam ( rohacell 31ig by evonik industries @xcite ) with an index of refraction of @xmath17 is placed between the pillars and the disk as isolation [ see ] . the total height of the pillars is @xmath18 mm so the resonator is not influenced by the optical table . two vertical wire antennas are placed diametrically at the cylindrical sidewalls of the disk ( cf . they are connected to a vectorial network analyzer ( pna n5230a by agilent technologies ) with coaxial rf cables . the network analyzer measures the complex scattering matrix element @xmath19 , where @xmath20 is the ratio between the powers @xmath21 coupled out via antenna @xmath22 and @xmath23 coupled in via antenna @xmath24 for a given frequency @xmath25 . plotting @xmath26 versus @xmath25 yields the frequency spectrum . the measured frequency spectrum of disk b is shown in . it consists of several series of roughly equidistant resonances . the associated modes can be labeled with their polarization and the quantum numbers of the circle resonator , which are indicated in . here , @xmath27 are the azimuthal and radial quantum number , respectively ; tm denotes transverse magnetic polarization with the @xmath28 component of the magnetic field , @xmath29 , equal to zero ; and te denotes transverse electric polarization with the @xmath28 component of the electric field , @xmath30 , equal to zero . each series of resonances consists of modes with the same polarization and radial quantum number . the free spectral range ( @xmath31 ) for each series is in the range of @xmath32@xmath33 mhz . only modes with @xmath34 , that is , wgms , are observed in the experiment . the quantum numbers were determined from the intensity distributions , which were measured with the perturbation body method ( see ref . @xcite and references therein ) . to determine the polarization of the modes a metal plate was placed parallel to the resonator at a variable distance @xmath35 [ see ] . figure [ fig : polmeas ] shows a part of the frequency spectrum with two resonances for different distances of the metal plate to the disk . the metal plate induces a shift of the resonance frequencies , where the magnitude of the frequency shift increases with decreasing distance @xmath35 . notably , the direction of the shift depends on the polarization of the corresponding mode : te modes are shifted to higher frequencies and tm modes to lower ones , so the polarization of each mode can be determined uniquely . this behavior is attributed to the different boundary conditions for the @xmath30 field ( tm modes ) and the @xmath29 field ( te modes ) at the metal plates . the former obeys dirichlet , the latter neumann boundary conditions . a detailed explanation is given in appendix [ sec : polmeascalc ] . the open dielectric resonators are described by the vectorial helmholtz equation @xmath36 \left\ { \begin{array}{c } \vec{e } \\ \vec{b } \end{array } \right\ } = \vec{0}\ ] ] with outgoing - wave boundary conditions , where @xmath37 and @xmath38 are the electric and the magnetic field , respectively , and @xmath39 is the index of refraction at the position @xmath40 . though all components of the electric and magnetic fields obey the same helmholtz equation , they are not independent but rather coupled in the bulk and at the boundaries as required by the maxwell equations . the eigenvalues @xmath41 of are complex and the real part of @xmath41 corresponds to the resonance frequency @xmath42 of the resonance @xmath43 , while the imaginary part corresponds to the resonance width @xmath44 ( full width at half maximum ) . for the infinite slab geometry ( see ) , the vectorial helmholtz equation can be simplified by separating the wave vector @xmath45 into a vertical @xmath28 component , @xmath46 , and a component parallel to the @xmath47-@xmath48 plane , @xmath49 . thus , @xmath50 , and the angle of incidence on the top and bottom surface of the resonator is @xmath51 . for a resonant wave inside the slab the wave vector component @xmath52 must obey the resonance condition @xmath53 where @xmath54 is the fresnel reflection coefficient . the solutions of yield the quantized values of @xmath52 . the effective index of refraction @xmath0 is defined as @xmath55 and corresponds to the phase velocity with respect to the @xmath47-@xmath48 plane . in the experiments only modes trapped due to total internal reflection ( tir ) are observed . in this case , the reflection coefficient @xmath54 can be written as @xmath56 with @xmath57 where @xmath58 for tm modes and @xmath59 for te modes . with these definitions , the quantization condition for @xmath52 can be reformulated as an implicit equation for the determination of @xmath0 , @xmath60\ ] ] with @xmath61 being the order of excitation in the @xmath28 direction @xcite . the @xmath62 term in corresponds to the fresnel phase due to the reflections and the @xmath63 term to the geometrical phase . in the framework of the @xmath0 model the flat resonator is treated as a dielectric slab waveguide and the vectorial helmholtz equation [ ] is accordingly reduced to the 2d scalar helmholtz equation @xcite by replacing @xmath15 by the effective index of refraction @xmath0 , @xmath64 where the wave function @xmath65 inside , respectively , outside of the resonator corresponds to @xmath30 in the case of tm modes and to @xmath29 in the case of te modes . the boundary conditions at the boundary of the resonator in the @xmath47-@xmath48 plane ( i.e. , the cylindrical sidewalls ) , @xmath66 , are @xmath67 where @xmath68 is the unit normal vector for @xmath66 , @xmath69 for tm modes , and @xmath70 for te modes . equation ( [ eq : helmholtzscal ] ) can be solved analytically for a circular dielectric resonator @xcite . however , it should be stressed that is not exact for flat 3d cavities . it defines the 2d @xmath0 approximation whose accuracy is unknown analytically but which has been determined experimentally in ref . our purpose is to investigate the precision of this approximation for the length spectra of simple 3d dielectric cavities . the effective index of refraction for the tm modes with the lowest @xmath28 excitation @xmath71 of disk a and b is shown in . obviously , @xmath0 depends strongly on the frequency , and this dispersion plays a crucial role in the present work . it should be noted that also te modes and modes with higher @xmath28 excitation exist in the considered frequency range , however , in the following we focus on tm modes . the density of states ( dos ) in a dielectric resonator is given by @xcite @xmath72 ^ 2 + [ { \mathrm{im}\left(k_j\right)}]^2 } \ , , \ ] ] where the summation runs over all resonances @xmath43 . the dos can be separated into a smooth , average part @xmath73 and a fluctuating part , @xmath74 . the smooth part is well described by the weyl formula given in ref . @xcite and depends only on the area , the circumference and the index of refraction of the resonator . the fluctuating part , on the other hand , is related to the pos of the corresponding classical dielectric billiard . for a 2d dielectric resonator with regular classical dynamics , the semiclassical approximation for @xmath74 is @xcite @xmath75 where @xmath76 is the length of the po , @xmath77 is the product of the fresnel reflection coefficients for the reflections of the rays at the dielectric interfaces , @xmath78 denotes the phase changes accumulated at the reflections [ i.e. , @xmath79 and at the conjugate points of the corresponding po , and the amplitude @xmath80 is proportional to @xmath81 , where @xmath82 is the area of the billiard covered by the family of the po . it should be noted that this semiclassical formula fails to accurately describe contributions of pos with angle of incidence close to the critical angle for tir , @xmath83 , as concerns the amplitude . consequently , higher - order corrections to the trace formula need to be developed for these cases @xcite . we restrict the discussion to the experimentally observed tm modes and compare the results with the trace formula obtained in ref . @xcite . to select the tm modes with @xmath71 the polarization and the quantum numbers of all resonances had to be determined experimentally as described in . the trace formula for 2d resonators is applied to the flat 3d resonators considered here by inserting the frequency - dependent effective index of refraction @xmath84 instead of @xmath15 into . to test the accuracy of the resulting trace formula we computed the fourier transform ( ft ) of @xmath74 . in ref . @xcite it was shown that it is essential to fully take into account the dispersion of @xmath0 in the ft for a meaningful comparison of the resulting length spectrum with the geometric lengths of the pos . therefore , we define @xmath85 where the quantity @xmath86 is a geometrical length and @xmath87 is , thus , called the length spectrum . we will compare it to the ft as defined in using @xmath84 instead of @xmath15 of the trace formula , @xmath88 . the resonance parameters @xmath41 are obtained by fitting lorentzians to the measured frequency spectra , and @xmath89 correspond to the frequency range considered . since in a circle resonator the resonance modes with @xmath90 are doubly degenerate , the measured resonances are counted twice each . note that even though only the most long - lived resonances ( i.e. , the wgms ) are observed experimentally , and these comprise only a fraction of all resonances , a comparison of the experimental length spectrum with the trace formula is meaningful @xcite . figure [ fig : lspektsclhkreis ] shows the experimental length spectrum evaluated using and the ft of the semiclassical trace formula , , for disk a. a total of @xmath91 measured tm modes with radial quantum numbers @xmath92@xmath93 from @xmath94 ghz to @xmath95 ghz was used . the pos in the circle billiard are denoted by their period @xmath96 and their rotation number @xmath97 and have the shape of polygons and stars ( see insets in ) . their lengths are indicated by the solid arrows . the pos with @xmath98 and @xmath99 were used to compute the trace formula . only pos with @xmath100 are indicated in for the sake of clarity . the pos with @xmath101 only add small contributions to the right shoulder of the peak corresponding to the @xmath102 and @xmath103 orbits . the amplitudes are @xmath104 and the phases are @xmath105}\ ] ] with the angles of incidence of the pos ( with respect to the surface normal ) being @xmath106 the overall agreement between the experimental length spectrum and the semiclassical trace formula is good and the major peaks in the length spectrum are close to the lengths of the @xmath107@xmath103 orbits . however , no clear peaks are observed at the lengths of the @xmath108 and the @xmath109 orbit in the experimental length spectrum . note that in the experimental length spectrum only orbits that are confined by tir are observed ( cf . this is not the case for the @xmath108 and the @xmath109 orbits in the frequency range of interest , where their angle of incidence @xmath110 is smaller than the critical angle @xmath111 as depicted in . the length spectrum of disk b is shown in . altogether @xmath112 resonances with @xmath92@xmath93 from @xmath113 ghz to @xmath114 ghz were used . the semiclassical trace formula was again computed for @xmath98 and @xmath99 . the agreement of the experimental length spectrum and the ft of the trace formula is good and comparable to that obtained for disk a. as in the case of disk a , the experimental length spectrum exhibits no peaks for the @xmath108 orbit , whose length is not within the range depicted in , and for the @xmath109 orbit since they are not confined by tir in the considered frequency range . a closer inspection of figs . [ fig : lspektsclhkreis ] and [ fig : lspektsclbkreis ] shows two unexpected effects . first , the peak positions of the ft of the semiclassical trace formula deviate slightly from the lengths of the pos . this can be seen best in the bottom parts of figs . [ fig : lspektsclhkreis ] and [ fig : lspektsclbkreis ] , where the contributions of the individual pos to the trace formula are depicted . second , there is a small but systematic difference between the peak positions of the experimental length spectrum and those of the ft of the trace formula . we will demonstrate that the first effect is related to the dispersion of @xmath0 and the second effect to the systematic error of the @xmath0 model . cccc @xmath115 & @xmath76 ( m ) & @xmath116 ( m ) & @xmath117 ( m ) + disk a + @xmath107 & @xmath118 & @xmath119 & @xmath120 + @xmath121 & @xmath122 & @xmath123 & @xmath124 + @xmath102 & @xmath125 & @xmath126 & @xmath127 + @xmath103 & @xmath128 & @xmath129 & @xmath130 + @xmath131 & @xmath132 & @xmath133 & @xmath134 + @xmath135 & @xmath136 & @xmath137 & @xmath138 + + disk b + @xmath107 & @xmath139 & @xmath140 & @xmath141 + @xmath121 & @xmath142 & @xmath143 & @xmath144 + @xmath102 & @xmath145 & @xmath146 & @xmath147 + @xmath103 & @xmath148 & @xmath149 & @xmath150 + @xmath131 & @xmath151 & @xmath152 & @xmath153 + @xmath135 & @xmath154 & @xmath155 & @xmath151 + the difference between the peak positions of the trace formula and the lengths of the pos can be understood by considering the exponential term in the ft of the semiclassical trace formula , which for a single po is @xmath156 \}}\ ] ] with @xmath78 given by . the crucial point is that the phase @xmath78 is frequency dependent because it contains the phase of the fresnel coefficients , which , in turn , depends on @xmath84 . the modulus of the ft will be largest for that length @xmath86 for which the exponent in is stationary , i.e. , its derivative with respect to @xmath157 vanishes . this leads to the following estimate @xmath116 for the peak position , @xmath158 where @xmath159 is related to the fresnel coefficients via . the wave number @xmath160 at which is evaluated is the center of the relevant wave number / frequency interval , which is @xmath161 with @xmath162 . the frequency @xmath163 is defined by @xmath164 , i.e. , it corresponds to the minimum frequency at which the considered po is confined by tir ( cf . ) . below @xmath165 the fresnel phase vanishes . the estimated peak positions @xmath116 are indicated by the dashed arrows in figs . [ fig : lspektsclhkreis ] and [ fig : lspektsclbkreis ] and agree well with the peak positions @xmath117 of the individual po contributions in the bottom parts of the figures . a list of the lengths of the pos , the peak positions of the single po contributions , and the estimates @xmath116 according to for disks a and b is provided in . in general , the estimate @xmath116 deviates only by @xmath24@xmath22 mm from the actual peak position @xmath117 ( about @xmath166 of @xmath117 ) . the @xmath108 and the @xmath109 orbits are not confined by tir . therefore , for these orbits the fresnel phase vanishes and accordingly @xmath167 . furthermore , their contributions to the length spectrum are symmetric with respect to @xmath168 , while those of the other pos are asymmetric with an oscillating tail to the left ( see bottom parts of figs . [ fig : lspektsclhkreis ] and [ fig : lspektsclbkreis ] ) . these tails are attributed to the frequency dependence of the fresnel phase . they can lead to interference effects , as can be seen for example in . there , e.g. , the peak positions of the semiclassical trace formula ( dashed line in the top part ) for the @xmath107 and the @xmath121 orbit deviate from the peak positions @xmath117 of the corresponding single orbit contributions ( dashed and dotted lines in the bottom part ) due to interferences between the contributions of a po and the side lobes of those of the other pos . in order to identify such interferences , it is generally instructive to compare the ft of the semiclassical trace formula with those of its single orbit contributions . it should be noted that the effect discussed in this paragraph also occurs for any 2d resonator made of a dispersive material . it was shown in the previous paragraph that the dispersion of @xmath0 plays an important role . furthermore , the semiclassical trace formula is known to be imprecise for pos with angles of incidence close to the critical angle . this is especially crucial here since several pos are close to the critical angle in at least a part of the considered frequency regime ( see ) . these deficiencies of the semiclassical trace formula indicate the necessity to implement modifications of it . to pursue this presumption we will compare the experimental length spectrum with the ft of the exact trace formula for the 2d dielectric circle resonator using a frequency - dependent index of refraction @xmath169 in order to investigate the deviations between it and the ft of the semiclassical trace formula . the trace formula is called exact since it is derived directly from the quantization condition for the dielectric circle resonator and without semiclassical approximations . it is given by @xmath170 with the definitions @xmath171 @xmath172 @xmath173 @xmath174 and @xmath175 here , @xmath176 , @xmath177 are the hankel functions of the first and second kinds , respectively , and the prime denotes the derivative with respect to the argument . equation ( [ eq : trformexact ] ) is essentially eq . ( 67 ) of ref . @xcite with an additional factor @xmath178 in the term @xmath179 . a detailed derivation is given in appendix [ sec : exacttraceform ] . in the semiclassical limit , the term @xmath180 in turns into the product of the fresnel reflection coefficients , the term @xmath181 into the oscillating term @xmath182 , and @xmath183 contributes to the amplitude @xmath80 . for pos close to the critical angle , i.e. , when the stationary point of the integrand in is @xmath184 , the term @xmath180 varies rapidly with @xmath185 , whereas it is assumed to change slowly in the stationary phase approximation used to derive the semiclassical trace formula @xcite . therefore , including curvature corrections in the fresnel coefficients does not suffice for an accurate calculation of the contributions of these pos to @xmath74 . consequently , we compute the integral entering in numerically . figure [ fig : lspektexacthkreis ] shows the comparison of the experimental length spectrum ( solid line ) , the ft of the semiclassical trace formula ( dashed line ) , , and that of the exact trace formula ( dotted line ) , , for disk a. the latter was computed for @xmath186 and @xmath99 . the other two curves are the same as in the top part of . the main difference between the semiclassical and the exact trace formula are the larger peak amplitudes of the exact trace formula , where this is mainly attributed to the additional @xmath187 factor . the differences between the peak amplitudes of the experimental length spectrum and those of the exact trace formula are actually expected since the measured resonances comprise only a part of the whole spectrum ( cf . @xcite ) , though they are not very large . on the other hand , the peak positions @xmath188 of the exact trace formula differ only slightly from those of the semiclassical one , and still deviate from those of the experimental length spectrum , @xmath189 ( see inset of ) . the difference @xmath190 is in the range of @xmath191@xmath16 mm , i.e. , about @xmath192 of the periodic orbit length @xmath76 . similar effects are observed in , which shows the experimental length spectrum and the ft of the semiclassical trace formula and that of the exact trace formula ( computed for @xmath186 and @xmath99 ) for disk b. the peak amplitudes of the exact trace formula are , again , somewhat larger than those of the experimental length spectrum , and the peak positions @xmath188 of the exact trace formula differ from those of the experimental length spectrum by @xmath193 mm for the @xmath107 and the @xmath121 orbits . the relative error @xmath194 is , thus , slightly smaller than in the case of disk a. since the trace formula itself is exact , the only explanation for these deviations is the systematic error of the @xmath0 model . therefore , we compare the measured ( @xmath195 ) and the calculated ( @xmath196 ) resonance frequencies for disk a in . the resonance frequencies were calculated by solving the 2d helmholtz equation [ ] for the circle as in ref . the difference between the measured and the calculated frequencies in ( a ) is as large as half a fsr ( i.e. , the distance between two resonances with the same radial quantum number ) and slowly decreases with increasing frequency . this is in accordance with the result that the fsr of the calculated resonances in ( b ) is slightly larger than that of the measured ones . since the frequency spectrum consists of series of almost equidistant resonances , the effect of this systematic error on the peak positions in the length spectrum can be estimated by considering a simple 1d system with equidistant resonances whose distance equals @xmath31 . the peak position in the corresponding length spectrum is @xmath197 , and a deviation of @xmath198 leads to an error , @xmath199 of the peak position . with @xmath200 mhz compared to @xmath201 mhz we expect a deviation of @xmath202 or @xmath203 mm in the peak positions , which agrees quite well with the magnitude of the deviations found in . for disk b , the comparison between the measured and the calculated fsr ( not shown here ) yields @xmath204 , which also agrees well with the deviations of @xmath194 found in . thus , we may conclude that the deviations between the peak positions of the experimental length spectrum and that of the trace formula indeed arise from the systematic error of the @xmath0 model . unfortunately , we know of no general method to estimate the magnitude of this systematic error beforehand . it should be noted that the exact magnitude of the systematic error contributing to the deviations found in figs . [ fig : lspektexacthkreis ] and [ fig : lspektexactbkreis ] depends on the index of refraction used in the calculations . still , it was shown in ref . @xcite and also checked here that deviations remain regardless of the value of @xmath15 used , which is known with per mill precision for the disks a and b @xcite . furthermore , ( b ) demonstrates that the index of refraction of a disk can not be determined without systematic error from the measured @xmath31 even if the dispersion of @xmath0 is fully taken into account . the resonance spectra of two circular dielectric microwave resonators were measured and the corresponding length spectra were investigated . in contrast to previous experiments with 2d resonators @xcite , flat 3d resonators were used . the length spectra were compared to a combination of the semiclassical trace formula for 2d dielectric resonators proposed in ref . @xcite and a 2d approximation of the helmholtz equation for flat 3d resonators using an effective index of refraction ( in accordance with ref . the experimental length spectra and the trace formula showed good agreement , and the different contributions of the pos to the length spectra could be successfully identified . the positions of the peaks in the experimental length spectrum are , however , slightly shifted with respect to the geometrical lengths of the pos . we found that this shift is related to two different effects , which are , first , the frequency dependence of the effective index of refraction and , second , a systematic inaccuracy of the @xmath0 approximation . in the examples considered here , the former effect is as large as @xmath205 of the po length while the latter effect is as large as @xmath192 of @xmath76 , and the two effects cancel each other in part . the results and methods presented here provide a refinement of the techniques used in refs . @xcite and allow for the detailed understanding of the spectra of realistic microcavities and -lasers in terms of the 2d trace formula . furthermore , many of the effects discussed here also apply to 2d systems made of a dispersive material . some open problems remain , though . the comparison of the semiclassical trace formula with the exact one for the circle showed that the former needs to be improved for pos close to the critical angle . furthermore , there are some deviations between the experimental length spectra and the trace formula predictions due to the systematic error of the effective index of refraction approximation . its effect on the length spectra proved to be rather small and , thus , allowed for the identification of the different po contributions . however , the computation of the resonance frequencies of flat 3d resonators based on the combination of the 2d trace formula and the @xmath0 approximation would lead to the same systematic deviations from the measured ones as in ref . another problem with this systematic error is that its magnitude can not be estimated _ a priori_. the comparison of the results for disk a and disk b seems to indicate that it gets smaller for @xmath206 , but there are not enough data to draw final conclusions yet , especially since the value of @xmath207 is of similar magnitude for both disks . in fact , ref . @xcite rather indicates that the systematic error of the @xmath0 model increases with decreasing @xmath207 . this could be attributed to diffraction effects at the boundary of the disks that become more important when @xmath208 gets smaller compared to the wavelength . on the other hand , the exact 2d case is recovered for @xmath209 , i.e. , an infinitely long cylinder . in conclusion the accuracy of the @xmath0 model in the limit @xmath206 remains an open problem . an analytical approach to the problem of flat dielectric cavities that is more accurate than the @xmath0 model would be of great interest . another perspective direction is to consider 3d dielectric cavities with the size of all sides having the same order of magnitude and to develop a trace formula for them similar to those for metallic 3d cavities @xcite . the authors wish to thank c. classen from the department of electrical engineering and computer science of the tu berlin for providing numerical calculations to validate the measured resonance data . this work was supported by the dfg within the sonderforschungsbereich 634 . the placement of a metal plate parallel to the dielectric disk influences the effective index of refraction , thus , leading to a shift of the resonance frequencies . in order to determine the change of @xmath0 , we , first , calculate the reflection coefficient @xmath210 for a wave traveling inside the dielectric and being reflected at the dielectric - air interface with the metal plate at a distance @xmath35 . the geometry used here is depicted in . the ansatz for the @xmath30 field ( tm polarization ) and , respectively , the @xmath29 field ( te polarization ) is @xmath211 where @xmath212 fulfills @xmath213 , @xmath214 is the angular frequency , @xmath215 , @xmath216 are constants , and @xmath54 is the reflection coefficient . the different wave numbers are connected by @xmath217 for the case of tir considered here , @xmath96 is real , and the penetration depths @xmath218 of the field intensity into region ii is @xmath219 . for the case of tm polarization the electric field is @xmath220 } e^{-i \omega t}\ ] ] since it obeys neumann boundary conditions at the metal plate , where @xmath221 is a constant . the boundary conditions at the interface between region i and ii are that @xmath222 and @xmath223 are continuous , which yields @xmath224 for the case of te polarization the magnetic field in region ii is @xmath225 } e^{-i \omega t}\ ] ] since it obeys dirichlet boundary conditions at the metal plate . with the condition that @xmath29 and @xmath226 are continuous at the dielectric interface , @xmath227 is obtained . analogous to we define @xmath228}$ ] and obtain @xmath229 } = \nu \sqrt{\frac{{n_{\mathrm{eff}}}^2 - 1}{n^2 - { n_{\mathrm{eff}}}^2 } } h[{d}/ ( 2 { l})]\ ] ] with @xmath230 for @xmath231 , is recovered . this explains why the optical table does not disturb the resonator . by inserting @xmath232 into we obtain @xmath233 as condition for @xmath234 , where @xmath235 refers to . in order to determine the effect of a change of @xmath35 on the effective index of refraction , we compute @xmath236 where @xmath237 } h'[{d}/(2 { l } ) ] \ , .\ ] ] the derivative of @xmath238 is @xmath239 which approaches @xmath240 for @xmath241 with @xmath242 for tm and @xmath243 for te polarization . then , for @xmath231 , @xmath244 with @xmath245 given as @xmath246 accordingly , for large distances of the metal plate , @xmath247 i.e. , for tm modes @xmath0 increases for decreasing distance @xmath35 of the metal plate from the resonator and for te modes it gets smaller . this qualitative behavior is found for the whole range of @xmath35 , even when is no longer valid . since the resonance frequencies of the disk are to first - order approximation @xmath248 , the tm modes are shifted to lower and the te modes to higher frequencies as observed in . the dos for the tm modes of the 2d circular dielectric resonator is @xmath249 where the eigenvalues @xmath250 are the roots of @xcite @xmath251\ ] ] with @xmath252 . the factor @xmath47 ensures that @xmath253 has no poles . informally one can write @xmath254 , where @xmath255 is a certain smooth function without zeros and poles . this means that the dos can be written as @xmath256 the smooth term contributes to the weyl expansion and requires a separate treatment @xcite . in the following we ignore all such terms . the derivative @xmath257 in contains two terms , @xmath258 the first term is @xmath259 where the second derivatives of the bessel and hankel functions were resolved via the bessel differential equation . the second term is @xmath260 \end{array}\ ] ] where @xmath261 was replaced by @xmath262 since @xmath263 . we approximate @xmath264 and obtain @xmath265 it was checked numerically that this approximation is very precise , which is why we still call the result an exact trace formula . combining both terms yields @xmath266 with @xmath267 the dos is , thus , @xmath268 we replace the bessel functions @xmath269 in @xmath270 by @xmath271 and extract a term @xmath272 $ ] with @xmath273 defined in to obtain @xmath274 \ , , \ ] ] where @xmath275 , @xmath276 , and @xmath277 are defined in eqs . ( [ eq : trerm ] ) , ( [ eq : tream ] ) , and ( [ eq : trebm ] ) , respectively . with the help of the geometric series @xmath278 is rewritten as @xmath279 with @xmath280 using the wronskian @xcite @xmath281 = \frac{4 i}{\pi z}\ ] ] this simplifies to @xmath282 and we obtain @xmath283 + \mathrm{c.c . } \ , . \end{array}\ ] ] the first term , @xmath284 , corresponds to the smooth part of the dos . since we are only interested in the fluctuating part , we drop the term and apply the poisson resummation formula to the rest to obtain @xmath285 with @xmath183 defined in . replacing @xmath286 with @xmath287 and ignoring those @xmath115 combinations that are not related to any pos and , thus , do not give significant contributions finally yields . taking the semiclassical limit as described in ref . @xcite results in with an additional factor of @xmath187 . this means that the dispersion of @xmath15 leads to slightly higher amplitudes in the semiclassical limit . 43ifxundefined [ 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 @noop ( ) \doibase doi:10.1063/1.106688 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1364/ol.20.002093 [ * * , ( ) ] link:\doibase 10.1103/physreva.76.023830 [ * * , ( ) ] link:\doibase 10.1063/1.1807967 [ * * , ( ) ] link:\doibase 10.1364/ol.31.001896 [ * * , ( ) ] link:\doibase 10.1038/nnano.2011.99 [ * * , ( ) ] link:\doibase 10.1007/978 - 3 - 540 - 85859 - 1_23 [ * * , ( ) ] link:\doibase 10.1364/ol.27.000824 [ * * , ( ) ] link:\doibase 10.1103/physreve.67.015207 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physreve.78.016201 [ * * , ( ) ] link:\doibase 10.1364/ol.19.001693 [ * * , ( ) ] link:\doibase 10.1038/385045a0 [ * * , ( ) ] link:\doibase 10.1103/physreva.79.013830 [ * * , ( ) ] link:\doibase 10.1002/andp.19474360704 [ * * , ( ) ] link:\doibase 10.1364/ol.27.000007 [ * * , ( ) ] link:\doibase 10.1103/physreve.65.045603 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.88.094102 [ * * , ( ) ] link:\doibase 10.1209/0295 - 5075/84/10008 [ * * , ( ) ] link:\doibase 10.1103/physreve.82.026202 [ * * , ( ) ] \doibase doi:10.1063/1.1665328 [ * * , ( ) ] link:\doibase 10.1063/1.1665596 [ * * , ( ) ] @noop _ _ ( , , ) link:\doibase 10.1103/physreve.78.056202 [ * * , ( ) ] link:\doibase 10.1088/1751 - 8113/44/15/155305 [ * * , ( ) ] link:\doibase 10.1103/physreve.81.066215 [ * * , ( ) ] link:\doibase 10.1103/physreve.83.036208 [ * * , ( ) ] link:\doibase 10.1016/0003 - 4916(77)90334 - 7 [ * * , ( ) ] link:\doibase 10.1103/physreve.53.4166 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.89.064101 [ * * , ( ) ] link:\doibase 10.1109/jstqe.2005.853848 [ * * , ( ) ] link:\doibase 10.1103/physreva.80.023825 [ * * , ( ) ] link:\doibase 10.1364/josab.22.002295 [ * * , ( ) ] in @noop _ _ , , vol . , ( , , ) pp . @noop _ _ ( , , ) \doibase doi:10.1088/1367 - 2630/8/3/046 [ * * , ( ) ] link:\doibase 10.1088/1367 - 2630/13/2/023013 [ * * , ( ) ] in link:\doibase 10.1109/iceaa.2010.5653856 [ _ _ ] ( ) p. link : www.rohacell.com [ ] link:\doibase 10.1103/physreve.66.056207 [ * * , ( ) ] @noop _ _ , edited by , vol . ( , , )

the length spectra of flat three - dimensional dielectric resonators of circular shape were determined from a microwave experiment . they were compared to a semiclassical trace formula obtained within a two - dimensional model based on the effective index of refraction approximation and a good agreement was found . it was necessary to take into account the dispersion of the effective index of refraction for the two - dimensional approximation . furthermore , small deviations between the experimental length spectrum and the trace formula prediction were attributed to the systematic error of the effective index of refraction approximation . in summary , the methods developed in this article enable the application of the trace formula for two - dimensional dielectric resonators also to realistic , flat three - dimensional dielectric microcavities and -lasers , allowing for the interpretation of their spectra in terms of classical periodic orbits .

arxiv

the merger of two neutron stars , two black holes or a black hole and a neutron star are among the most important sources of gravitational waves ( gw ) , due to the huge energy released in the process . in particular , the coalescence of double neutron stars ( dns ) may radiate about 10@xmath6 erg in the last seconds of their inspiral trajectory , at frequencies up to 1.4 - 1.6 khz , range covered by most of the ground - based laser interferometers like virgo @xcite , ligo @xcite , geo @xcite or tama @xcite . besides the amount of energy involved in these events , the rate at which they occur in the local universe is another parameter characterizing if these mergings are or not potential interesting sources of gw . in spite of the large amount of work performed in the past years , uncertainties persist in estimates of the dns coalescence rate . in a previous investigation , we have revisited this question @xcite , taking into account the galactic star formation history derived directly from observations and including the contribution of elliptical galaxies when estimating the mean merging rate in the local universe . based on these results , we have predicted a detection rate of one event every 125 and 148 years by initial ligo and virgo respectively and up to 6 detections per year in their advanced configurations . besides the emission produced by the coalescence of the nearest dns , the superposition of a large number of unresolved sources at high redshifts will produce a stochastic background of gw . in the past years , different astrophysical processes susceptible to generate a stochastic background have been investigated . on the one hand , distorted black holes @xcite , bar mode emission from young neutron stars @xcite are examples of sources able to generate a shot noise ( time interval between events large in comparison with duration of a single event ) , while supernovas or hypernovas @xcite are expected to produce an intermediate popcorn " noise . on the other hand , the contribution of tri - axial rotating neutron stars @xcite , including magnetars @xcite , constitutes a truly continuous background . populations of compact binaries as , for instance , the cataclismic variables are responsible for the existence of a galactic background of gw in the mhz domain , which could represent an important source of confusion noise for space detectors as lisa @xcite . these investigations have been extended recently to the extra - galactic contribution . @xcite , @xcite and @xcite considered cosmological populations of double and mixed systems involving black holes , neutron stars and white dwarfs , while close binaries originated from low and intermediate mass stars were discussed by @xcite . in this work , using the dns merging rate estimated in our precedent study , we have estimated the gravitational wave background spectrum produced by these coalescences . numerical simulations based on monte carlo methods were performed in order to determine the critical redshift @xmath7 beyond which the duty cycle condition required to have a continuous background ( @xmath8 ) is satisfied . unlike previous studies which focus their attention on the early low frequency inspiral phase covered by lisa @xcite , here we are mainly interested in the few thousand seconds before the last stable orbit is reached , when more than 96% of the gravitational wave energy is released . the signal frequency is in the range 10 - 1500 hz , covered by ground based interferometers . the paper is organized as follows . in 2 , the simulations are described ; in 3 the contribution of dns coalescences to the stochastic background is calculated ; in 4 the detection possibility with laser beam interferometers is discussed and , finally , in 5 the main conclusions are summarized . in order to simulate by monte carlo methods the occurrence of merging events , we have adopted the following procedure . the first step was to estimate the probability for a given pair of massive stars , supposed to be the progenitors of dns , be formed at a given redshift . this probability distribution is essentially given by the cosmic star formation rate @xcite , normalized in the redshift interval @xmath9 , e.g. , @xmath10 the normalization factor in the denominator is essentially the rate at which massive binaries are formed in the considered redshift interval , e.g. , @xmath11 which depends on the adopted cosmic star formation rate , as we shall see later . the formation rate of massive binaries per redshift interval is @xmath12 in the equation above , @xmath13 is the cosmic star formation rate ( sfr ) expressed in m@xmath14 mpc@xmath15yr@xmath16 and @xmath17 is the mass fraction converted into dns progenitors . hereafter , rates per comoving volume will always be indicated by the superscript * , while rates with indexes z@xmath18 " or z@xmath19 " refer to differential rates per redshift interval , including all cosmological factors . the ( 1+z ) term in the denominator of eq . [ eq - frate ] corrects the star formation rate by time dilatation due to the cosmic expansion . in the present work we assume that the parameter @xmath20 does not change significantly with the redshift and thus it will be considered as a constant . in fact , this term is the product of three other parameters , namely , @xmath21 where @xmath22 is the fraction of binaries which remains bounded after the second supernova event , @xmath23 is the fraction of massive binaries formed among all stars and @xmath24 is the mass fraction of neutron star progenitors . according to the results by @xcite,@xcite , @xmath22 = 0.024 and @xmath23 = 0.136 , values which will be adopted in our calculations . assuming that progenitors with initial masses above 40 m@xmath14 will produce black holes and considering an initial mass function ( imf ) of the form @xmath25 , with @xmath26 = 2.35 ( salpeter s law ) , normalized within the mass interval 0.1 - 80 m@xmath14 such as @xmath27 = 1 , it results finally @xmath28 and @xmath29 m@xmath30 . the evaluation of the parameters @xmath22 and @xmath23 depends on different assumptions , which explain why estimates of the coalescence rate of dns found in the literature may vary by one or even two orders of magnitude . the evolutionary scenario of massive binaries considered in our calculations ( see @xcite for details ) is similar to that developed by @xcite , in which none of the stars ever had the chance of being recycled by accretion . besides the evolutionary path , the resulting fraction @xmath22 of bound ns - ns binaries depends on the adopted velocity distribution of the natal kick . the imparted kick may unbind binaries which otherwise might have remained bound or , less probably , conserve bound systems which without the kick would have been disrupted . the adopted value for @xmath22 corresponds to a 1-d velocity dispersion of about 80 km / s . this value is smaller than those usually assumed for single pulsars , but consistent with recent analyses of the spin period - eccentricity relation for ns - ns binaries @xcite . had we adopted a higher velocity dispersion in our simulations ( 230 km / s instead of 80 km / s ) , the resulting fraction of bound systems is reduced by one order of magnitude , e.g. , @xmath22 = 0.0029 . if , on the one hand the fraction of bound ns - ns systems after the second supernova event depends on the previous evolutionary history of the progenitors and on the kick velocity distribution , on the other hand estimates of the fraction @xmath23 of massive binaries formed among all stars depends on the ratio between single and double ns systems in the galaxy and on the value of @xmath22 itself @xcite . we have estimated relative uncertainties of about @xmath31 and @xmath32 , leading to a relative uncertainty in the parameter @xmath20 of about @xmath33 . however , we emphasize that these are only formal uncertainties resulting from our simulations , which depend on the adopted evolutionary scenario for the progenitors . a comparison with other estimates can be found in @xcite . the element of comoving volume is given by @xmath34 with @xmath35^{1/2}\ ] ] where @xmath36 and @xmath37 are respectively the present values of the density parameters due to matter ( baryonic and non - baryonic ) and vacuum , corresponding to a non - zero cosmological constant . a flat " cosmological model ( @xmath38 ) was assumed . in our calculations , we have taken @xmath36 = 0.30 and @xmath37 = 0.70 , corresponding to the so - called concordance " model derived from observations of distant type ia supernovae @xcite and the power spectra of the cosmic microwave background fluctuations @xcite . the hubble parameter h@xmath39 was taken to be @xmath40 . @xcite provide three models for the cosmic sfr history up to redshifts @xmath41 . differences among these models are mainly due to various corrections applied , in particular those due to extinction by the cosmic dust . in our computations , we have considered the second model , labelled sfr2 @xcite but numerical results using sfr1 @xcite will be also given for comparison . both rates increase rapidly between @xmath42 and peak at @xmath43 , but sfr1 decreases gently after @xmath44 while sfr2 remains more or less constant ( figure [ fig - sfr ] ) . the next step consists to estimate the redshift @xmath45 at which the progenitors have already evolved and the system is now constituted by two neutron stars . this moment fixes also the beginning of the inspiral phase . if @xmath46 ( @xmath47 yr ) is the mean lifetime of the progenitors ( average weighted by the imf in the interval 9 - 40 m@xmath14 ) then @xmath48 once the beginning of the inspiral phase is established , the redshift at which the coalescence occurs is estimated by the following procedure . the duration of the inspiral phase depends on the orbital parameters just after the second supernova and on the neutron star masses . the probability for a given dns system to coalesce in a timescale @xmath49 was initially derived by @xcite , confirmed by subsequent simulations @xcite and is given by @xmath50 simulations indicate a minimum coalescence timescale @xmath51 yr but a considerable number of systems have a coalescence timescale higher than the hubble time . the normalized probability in the range @xmath52 yr up to 20 gyr implies @xmath53 . therefore , the redshift @xmath54 at which the coalescence occurs after a timescale @xmath49 is derived from the equation @xmath55 which was solved in our code by an iterative method . the resulting distribution of the number of coalescences as a function of @xmath54 is shown in figure [ fig - dist_coalescence ] for both star formation rates sfr1 and sfr2 , while the corresponding coalescence rate per redshift interval , @xmath56 , is shown in figure [ fig - sfr ] . in the same figure , for comparison , we have plotted the formation rate @xmath57 ( eq . [ eq - frate ] ) . notice that the maximum of @xmath56 is shifted towards lower redshifts with respect to the maximum of @xmath57 , reflecting the time delay between the formation of the progenitors and the coalescence event . the coalescence rate @xmath56 , does not fall to zero at @xmath58 because a non negligible fraction of coalescences ( @xmath59 for sfr2 and @xmath60 for sfr1 ) occurs later than @xmath58 . the nature of the background is determined by the duty cycle defined as the ratio of the typical duration of a single burst @xmath61 to the average time interval between successive events , e.g. , @xmath62 the critical redshift @xmath7 at which the background becomes continuous is fixed by the condition @xmath63 . since we are interested in the last instants of the inspiral , when the signal is within the frequency band of ground based interferometers , we took @xmath64 s , duration which includes about @xmath65 of the total energy released ( see table [ tbl - duration ] ) . from our numerical experiments and imposing d = 1 , one obtains @xmath66 when the sfr2 is used and @xmath67 for the sfr1 . about 96% ( 94% in the case of sfr1 ) of coalescences occur above such a redshift , contributing to the production of a continuous background . sources in the redshift interval @xmath68 ( sfr2 ) or @xmath69 ( sfr1 ) correspond to a duty cycle d = 0.1 and they are responsible for a cosmic popcorn " noise . the gravitational fluence ( given here in erg @xmath70hz@xmath16 ) in the observer frame produced by a given dns coalescence is : @xmath71 where @xmath72 is the distance luminosity , @xmath73 is the proper distance , which depends on the adopted cosmology , @xmath74 is the gravitational spectral energy and @xmath75 the frequency in the source frame . in the quadrupolar approximation and for a binary system with masses @xmath76 and @xmath77 in a circular orbit : @xmath78 where the fact that the gravitational wave frequency is twice the orbital frequency was taken into account . then @xmath79 assuming @xmath80 one obtains @xmath81 erg hz@xmath82 . the spectral properties of the stochastic background are characterized by the dimensionless parameter @xcite : @xmath83 where @xmath84 is the wave frequency in the observer frame , @xmath85 is the critical mass density needed to close the universe , related to the hubble parameter @xmath86 by , @xmath87 @xmath88 is the gravitational wave flux ( given here in erg @xmath70hz@xmath16s@xmath16 ) at the observer frequency @xmath89 , integrated over all sources at redshifts @xmath90 , namely @xmath91 instead of solving analytically the equation above by introducing , for instance , an adequate fit of the cosmic coalescence rate , we have calculated the integrated gravitational flux by summing individual fluences ( eq . [ eq - fluence ] ) , scaled by the ratio between the expected number of events per unit of time and the number of simulated coalescences or , in other words , the ratio between the total formation rate of progenitors ( eq . [ eq - nevents ] ) and the number of simulated massive binaries , e.g. , @xmath92 the number of runs ( or @xmath93 ) in our simulations was equal to @xmath94 , representing an uncertainty of @xmath95 0.1% in the density parameter @xmath4 . using the sfr2 , the derived formation rate of progenitors is @xmath96 , whereas for the sfr1 one obtains @xmath97 . for each run , the probability distribution @xmath98 defines , via monte carlo , the redshift at which the massive binary is formed . the beginning of the inspiral phase at @xmath45 is fixed by eq . [ eq - zb ] . then , in next step , the probability distribution of the coalescence timescale and eq . [ eq - h0_tau ] define the redshift @xmath54 at which the merging occurs . the fluence produced by this event is calculated by eq . [ eq - fluence ] , stored in different frequency bins in the observer frame and added according to the equation above . figure [ fig - omega ] shows the density parameter @xmath4 as a function of the observed frequency derived from our simulations . the density parameter @xmath4 increases as @xmath99 at low frequencies , reaches a maximum amplitude of about @xmath100 around 670 hz in the case of sfr2 and a maximum of @xmath101 ) around 630 hz , in the case of sfr1 . a high frequency cut - off at @xmath102 hz ( @xmath103 hz for sfr1 ) is observed , corresponding approximately to the frequency of the last stable orbit at the critical redshift @xmath104 ( @xmath105 for sfr1 ) . calculations performed by @xcite , in spite of the similar local merging rates , indicate that the maximum occurs at lower frequencies ( @xmath106 100 hz ) with an amplitude ( scaled to the hubble parameter adopted in this work ) lower by a factor of seven . however , as those authors have stressed , their calculations are expected to be accurate in the frequency range 10 @xmath107hz up to 1 hz , since they have set the value of the maximum frequency @xmath108 to about that expected at a separation of three times the last stable orbit " ( lso ) , e.g. , @xmath109 . thus , a direct comparison with our results is probably meaningless . a more conservative estimate can be obtained if one adopts a higher duty cycle value , namely , @xmath110 , corresponding to sources located beyond @xmath111 ( @xmath112 for sfr1 ) . our results are plotted in fig . [ fig - omega ] , which includes , for comparison , the popcorn " noise contribution arising from sources between @xmath113 ( @xmath69 for sfr1 ) , corresponding to the intermediate zone between a shot noise ( @xmath114 ) and a continuous background ( @xmath115 ) . when increasing the critical redshift ( or removing the nearest sources ) , the amplitude of @xmath4 decreases and the spectrum is shifted toward lower frequencies . the amplitude of the popcorn " background is about one order of magnitude higher than the continuous background , with a maximum of about @xmath116 ( @xmath117 for sfr1 ) around a frequency of @xmath118 khz . since some authors use , instead of @xmath4 , the gravitational strain @xmath119 defined by @xcite as @xmath120 we show this quantity in figure [ fig - sh ] . because the background obeys a gaussian statistic and can be confounded with the instrumental noise background of a single detector , the optimal detection strategy is to cross - correlate the output of two ( or more ) detectors , assumed to have independent spectral noises . the cross correlation product is given by @xcite : @xmath121 where @xmath122 is a filter that maximizes the signal to noise ratio ( @xmath123 ) . in the above equations , @xmath124 and @xmath125 are the power spectral noise densities of the two detectors and @xmath126 is the non - normalized overlap reduction function , characterizing the loss of sensitivity due to the separation and the relative orientation of the detectors . the optimized @xmath127 ratio for an integration time @xmath128 is given by @xcite : @xmath129 in the literature , the sensitivity of detector pairs is usually given in terms of the minimum detectable amplitude for a flat spectrum ( @xmath4 equal to constant ) @xcite , e.g. , @xmath130 the expected minimum detectable amplitude for the main pair of detectors in the world , after one year integration , are given in table [ tbl - sensitivity ] , for a detection rate @xmath131 and a false alarm rate @xmath132 . the power spectral densities expressions used for the present calculation can be found in @xcite . @xmath133 is of the order of @xmath134 for the first generation of interferometers combined as ligo / ligo and ligo / virgo . their advanced counterparts will permit an increase of two or even three orders of magnitude in sensitivity ( @xmath135 ) . the pair formed by the co - located and co - aligned ligo hanford detectors , for which the overlap reduction function is equal to one , is potentially one order of magnitude more sensitive than the hanford / livingston pair , provided that instrumental and environmental noises could be removed . however , because the spectrum of dns coalescences _ is not flat _ and the maximum occurs out of the optimal frequency band of ground based interferometers , which is typically around @xmath136 hz , as shown in figure [ fig - snr ] , the s / n ratio is slightly reduced . considering the co - located and co - aligned ligo interferometer pair , we find a signal - to - noise ratio of @xmath137 ( @xmath138 ) for the initial ( advanced ) configuration . unless the coalescence rate be substantially higher than the present expectations , our results indicate that their contribution to the gravitational background is out of reach of the first and the second generation of interferometers . on the other hand , the sensitivity of the future third generation of detectors , presently in discussion , could be high enough to gain one order of magnitude in the expected s / n ratio . examples are the large scale cryogenic gravitational wave telescope ( lcgt ) , sponsored by the university of tokyo and the european antenna ego ( sathyaprakash , private communication ) . ego will incorporate signal recycling , diffractive optics on silicon mirrors , cryo - techniques and kw - class lasers , among other technological improvements . a possible sensitivity for this detector is shown in figure [ fig - sh ] , compared to the expected sensitivity of advanced ligo . around 650 hz , the planned strain noise @xmath139 is about @xmath140 hz@xmath141 for the advanced ligo configuration while at this frequency , the planned strain noise for ego is @xmath142 hz@xmath141 , which represents a gain by a factor of @xmath143 . considering two interferometers located at the same place , we find a signal - to - noise ratio @xmath144 . on the other hand , the popcorn noise contribution could be detected by new data analysis techniques currently under investigation , such as the search for anisotropies @xcite that can be used to create a map of the gw background @xcite , the maximum likelihood statistic @xcite , or methods based on the probability event horizon " concept @xcite , which describes the evolution , as a function of the observation time , of the cumulated signal throughout the universe . the peh of the gw signal evolves fastly from contributions of high redshift populations , forming a real continuous stochastic background , to low redshift and less probable sources that can be resolved individually , while the peh of the instrumental noise is expected to evolve much slower . consequently , the gw signature could be distinguished from the instrumental noise background . in this work , we have performed numerical simulations using monte carlo techniques to estimate the occurrence of double neutron star coalescences and the gravitational stochastic background produced these events . since the coalescence timescale obeys a well defined probability distribution ( @xmath145 ) , derived from simulations of the evolution of massive binaries @xcite , the cosmic coalescence rate does not follow the cosmic star formation rate and presents necessarily a time - lag . in the case where the sources are supernovae or black holes , the gravitational burst is produced in a quite short timescale after the the formation of the progenitors . therefore , the time - lag is negligible and the comoving volume where the progenitors are formed is practically the same as that where the gravitational wave emission occurs , introducing a considerable simplification in the calculations . this is not the case when ns - ns coalescences are considered , since timescales comparable or even higher than the hubble timescale have non negligible probabilities . the maximum probability to form a massive binary occurs at @xmath146 , depending slightly on the adopted cosmic star formation rate , whereas the maximum probability to occur a coalescence is around @xmath147 . we have found that a truly continuous background is formed only when sources located beyond @xmath148 ( @xmath149 for the sfr1 case ) , including 96% ( 94% for sfr1 ) of all events and the critical redshift corresponds to the condition @xmath8 . sources in the redshift interval @xmath113 ( @xmath69 for sfr1 ) produce a popcorn " noise . our computations indicate that the density parameter @xmath4 has a maximum around 670 hz ( 630 hz for sfr1 ) , attaining an amplitude of about of @xmath100 ( @xmath150 for sfr1 ) . the low frequency cutoff around 1.2 khz corresponds essentially to the gravitational redshifted wave frequency associated to last stable orbit of sources located near the maximum of the coalescence rate . the computed signal is below the sensitivity of the first and the second generation of detectors . however , using the planned sensitivity of third generation interferometers , we found that after one year of integration , the cross - correlation of two ego like coincident antennas , gives an the optimized signal - to - noise of @xmath151 . the popcorn " contribution is one order of magnitude higher with a maximum of @xmath152 ( @xmath117 for sfr1 ) at @xmath153 khz . this signal , which is characterized by the spatial and temporal evolution of the events as well as by its signature , can be distinguished from the instrumental noise background and adequate data analysis strategies for its detection are currently under investigation @xcite .

in this work , numerical simulations were used to investigate the gravitational stochastic background produced by coalescences occurring up to @xmath0 of double neutron star systems . the cosmic coalescence rate was derived from monte carlo methods using the probability distributions for forming a massive binary and to occur a coalescence in a given redshift . a truly continuous background is produced by events located only beyond the critical redshift @xmath1 . events occurring in the redshift interval @xmath2 give origin to a popcorn " noise , while those arising closer than @xmath3 produce a shot noise . the gravitational density parameter @xmath4 for the continuous background reaches a maximum around 670 hz with an amplitude of @xmath5 , while the popcorn " noise has an amplitude about one order of magnitude higher and the maximum occurs around a frequency of 1.2 khz . the signal is below the sensitivity of the first generation of detectors but could be detectable by the future generation of ground based interferometers . correlating two coincident advanced - ligo detectors or two ego interferometers , the expected s / n ratio are respectively 0.5 and 10 .

arxiv

diffusive acceleration at newtonian shock fronts is an extensively studied phenomenon . detailed discussions of the current status of the investigations can be found in some recent excellent reviews @xcite . while much is by now well understood , some issues are still subjects of much debate , for the theoretical and phenomenological implications that they may have . the most important of these is the backreaction of the accelerated particles on the shock : the violation of the _ test particle approximation _ occurs when the acceleration process becomes sufficiently efficient to generate pressures of the accelerated particles which are comparable with the incoming gas kinetic pressure . both the spectrum of the particles and the structure of the shock are changed by this phenomenon , which is therefore intrinsically nonlinear . at present there are three viable approaches to account for the backreaction of the particles upon the shock : one is based on the ever - improving numerical simulations @xcite that allow a self - consistent treatment of several effects . the second approach is a _ fluid _ approach , and treats cosmic rays as a relativistic second fluid . this class of models was proposed and discussed in @xcite . these models allow one to obtain the thermodynamics of the modified shocks , but do not provide information about the spectrum of accelerated particles . the third approach is analytical and may be very helpful to understand the physics of the nonlinear effects in a way that sometimes is difficult to achieve through simulations , due to their intrinsic complexity . in ref . @xcite a perturbative approach was adopted , in which the pressure of accelerated particles was treated as a small perturbation . by construction this method provides an answer only for weakly modified shocks . an alternative approach was proposed in @xcite , based on the assumption that the diffusion of the particles is sufficiently energy dependent that different parts of the fluid are affected by particles with different average energies . the way the calculations are carried out implies a sort of separate solution of the transport equation for subrelativistic and relativistic particles , so that the two spectra must be somehow connected at @xmath0 _ a posteriori_. recently , in @xcite , the effects of the non - linear backreaction of accelerated particles on the maximum achievable energy were investigated , together with the effects of geometry . the maximum energy of the particles accelerated in supernova remnants in the presence of large acceleration efficiencies was also studied in @xcite . the need for a _ practical _ solution of the acceleration problem in the non - linear regime was recognized in @xcite , where a simple analytical broken - power - law approximation of the non - linear spectra was presented . recently , some promising analytical solutions of the problem of non - linear shock acceleration have appeared in the literature @xcite . these solutions seem to avoid many of the limitations of previous approaches . numerical simulations have been instrumental to identify the dramatic effects of the particles backreaction : they showed that even when the fraction of particles injected from the thermal gas is relatively small , the energy channelled into these few particles can be an appreciable part of the kinetic energy of the unshocked fluid , making the test particle approach unsuitable . the most visible effects of the backreaction of the accelerated particles on the shock appear in the spectrum of the accelerated particles , which shows a peculiar flattening at the highest energies . the analytical approaches reproduce well the basic features arising from nonlinear effects in shock acceleration . while several calculations exist of the nonlinear effects in the shock acceleration of quasi - monochromatic particles injected at a shock surface , there is no description at present of how these effects appear , if they do , when the shock propagates in a medium where ( pre)accelerated particles already exist . the linear theory of this phenomenon was developed by bell @xcite , but has never been generalized to its nonlinear extension . in fact , the backreaction can severely affect the process of re - energization of preaccelerated particles : bell already showed that for strong shocks the energy content of a region where cosmic rays were present could be easily enhanced by a factor @xmath1 at each shock passage , so that equipartition could be readily reached . in these conditions the backreaction of the accelerated particles should be expected . we report here on the first analytical treatment of the shock acceleration in the presence of seed nonthermal particles , with the inclusion of their nonlinear backreaction on the shock . our approach is a generalization of the analytical method introduced in @xcite to describe the nonlinear shock acceleration with monochromatic injection of quasi - thermal particles . in fact , we present here also a general calculation that accounts for both thermal particles and seed nonthermal particles . this case may be of interest for the study of supernova shocks propagating through the interstellar medium ( ism ) where pressure balance exists between gas and cosmic rays . nonlinear effects in shock acceleration of thermal particles result in the appearance of multiple solutions in certain regions of the parameter space . this behaviour resembles that of critical systems , with a bifurcation occurring when some threshold is reached . in the case of shock acceleration , it is not easy to find a way of discriminating among the multiple solutions when they appear . neverthless , in @xcite , a two fluid approach has been used to demonstrate that when three solutions appear , the one with intermediate efficiency for particle acceleration is unstable to corrugations in the shock structure and emission of acustic waves . plausibility arguments may be put forward to justify that the system made of the shock plus the accelerated particles may sit at the critical point , but the author is not aware of any real proof that this is what happens . the physical parameters that play a role in this approach to criticality are the maximum momentum achievable by the particles in the acceleration process , the mach number of the shock , and the injection efficiency , namely the fraction of thermal particles crossing the shock that are accelerated to nonthermal energies . the last of them , the injection efficiency , hides a crucial physics problem by itself , and may play an important role in establishing the level of shock modification . this efficiency parameter in reality is defined by the microphysics of the shock and should not be a free parameter of the problem . unfortunately , our poor knowledge of such microphysics , in particular for collisionless shocks , does not allow us to establish a clear and universal connection between the injection efficiency and the macroscopic shock properties . the paper is structured as follows : in [ sec : nonlin ] we describe the effect of non linearity on shock acceleration and our mathematical approach to describe it . in particular we generalize previous calculations to the case in which seed particles exist in the region where the shock is propagating ; in [ sec : gasdyn ] we describe the gas dynamics in the presence of a non - negligible pressure of accelerated particles . in [ sec : results ] we describe our results , with particular attention for the onset of the particle backreaction and for the appearance of multiple solutions . we conclude in [ sec : conclusions ] . in this section we solve the diffusion - convection equation for the cosmic rays in the most general case in which particles are injected according to some function @xmath2 and the shock propagates within a region where some cosmic ray distribution exists , that we denote as @xmath3 , spatially homogeneous before the shock crossing ( upstream ) . for simplicity we limit ourselves to the case of one - dimensional shocks , but the introduction of different geometrical effects is relatively simple , and in fact many of our conclusions should not be affected by geometry . the equation that describes the diffusive transport of particles in one dimension is @xmath4 - u \frac{\partial f ( x , p)}{\partial x } + \frac{1}{3 } \frac{d u}{d x}~p~\frac{\partial f(x , p)}{\partial p } + q(x , p ) = 0 , \label{eq : trans}\ ] ] where we assumed stationarity ( @xmath5 ) . the @xmath6 axis is oriented from upstream to downstream , as in fig . the presence of pre - existing cosmic rays is introduced as a boundary condition at upstream infinity , by requiring that @xmath7 . one should keep in mind that the common picture of a fluid whose speed is constant ( @xmath8 ) until it hits the shock surface is not appropriate for modified shocks . in fact , in this case , the pressure of the accelerated particles may become large enough to slow down the fluid before it crosses the shock surface . therefore in general at upstream infinity the gas flows at speed @xmath9 , different from @xmath8 ( fluid speed immediately upstream of the shock ) . the two quantities are approximately equal only when the accelerated particles do not have dynamical relevance . the injection term is taken in the form @xmath10 . as a first step , we integrate eq . [ eq : trans ] around @xmath11 , from @xmath12 to @xmath13 , denoted in fig . 1 as points `` 1 '' and `` 2 '' respectively , so that the following equation can be written : @xmath14_2 - \left [ d \frac{\partial f}{\partial x}\right]_1 + \frac{1}{3 } p \frac{d f_0}{d p } ( u_2 - u_1 ) + q_0(p)= 0,\ ] ] where @xmath8 ( @xmath15 ) is the fluid speed immediately upstream ( downstream ) of the shock and @xmath16 is the particle distribution function at the shock location . by requiring that the distribution function downstream is independent of the spatial coordinate ( homogeneity ) , we obtain @xmath17_2=0 $ ] , so that the boundary condition at the shock can be rewritten as @xmath14_1 = \frac{1}{3 } p \frac{d f_0}{d p } ( u_2 - u_1 ) + q_0(p ) . \label{eq : boundaryshock}\ ] ] we can now perform the integration of eq . ( [ eq : trans ] ) from @xmath18 to @xmath12 ( point `` 1 '' ) , in order to take into account the boundary condition at upstream infinity . ( [ eq : boundaryshock ] ) we obtain @xmath19 we can now introduce a quantity @xmath20 defined as @xmath21 whose physical meaning is instrumental to understand the nonlinear backreaction of particles . the function @xmath20 is the average fluid velocity experienced by particles with momentum @xmath22 while diffusing upstream away from the shock surface . in other words , the effect of the average is that , instead of a constant speed @xmath8 upstream , a particle with momentum @xmath22 experiences a spatially variable speed , due to the pressure of the accelerated particles that is slowing down the fluid . since the diffusion coefficient is in general @xmath22-dependent , particles with different energies _ feel _ a different compression coefficient , higher at higher energies if , as expected , the diffusion coefficient is an increasing function of momentum . the role of @xmath20 can also be explained as follows : the distribution function @xmath23 at a distance @xmath6 from the shock surface can be written as @xcite @xmath24,\ ] ] where @xmath25 is the local slope of @xmath26 and the diffusion coefficient @xmath27 has been assumed independent of the location @xmath6 . in first approximation , we can assume that the exponential factor remains important when it is of order unity , namely when its argument is much less than unity . we can therefore introduce a distance @xmath28 , which is the distance at which the exponential equals one . this means that @xmath29\approx u(x_p),\ ] ] so that the speed @xmath20 can be interpreted as the fluid speed at the point @xmath28 where the particles with momentum @xmath22 reverse their motion in the upstream fluid and return to the shock . note that the diffusion coefficient enters the calculation of the distance @xmath28 but does not enter directly the calculation of @xmath20 . in other words , different diffusion coefficients may move the point where the fluid speed is @xmath20 closer to or farther from the shock surface , but do not affect the value of @xmath20 . this approach is similar to that introduced in @xcite . with the introduction of @xmath20 , eq . ( [ eq : step ] ) becomes @xmath30 + u_0 f_\infty + q_0(p ) = 0 , \label{eq : step1}\ ] ] where we used the fact that @xmath31.\ ] ] eq . ( [ eq : step1 ] ) can be written in a way that resembles more the equation for shocks with no particles backreaction but in the presence of seed particles with distribution @xmath3 and injection function @xmath32 : @xmath33 - u_0 f_\infty -q_0(p ) \right\}. \label{eq : transport}\ ] ] the solution of this equation can be written in the following implicit form : @xmath34 @xmath35}{u_{\bar p } - u_2 } \exp\left\{-\int_{\bar p}^p \frac{dp'}{p ' } \frac{3}{u_{p ' } - u_2}\left[u_{p'}+\frac{1}{3}p ' \frac{du_{p'}}{d p'}\right ] \right\}. \label{eq : solut}\ ] ] in the case of monochromatic injection with momentum @xmath36 at the shock surface , we can write @xmath37 where @xmath38 is the gas density immediately upstream ( @xmath12 ) and @xmath39 parametrizes the fraction of the particles crossing the shock which are going to take part in the acceleration process . the injection term in eq . ( [ eq : solut ] ) becomes @xmath40\right\}=\ ] ] @xmath41\right\}. \label{eq : inje}\ ] ] here we introduced the two quantities @xmath42 and @xmath43 , which are respectively the compression factor at the gas subshock and the total compression factor between upstream infinity and downstream . the two compression factors would be equal in the test particle approximation . for a modified shock , @xmath44 can attain values much larger than @xmath45 and more in general , much larger than @xmath46 , which is the maximum value achievable for an ordinary strong non - relativistic shock . the increase of the total compression factor compared with the prediction for an ordinary shock is responsible for the peculiar flattening of the spectra of accelerated particles that represents a feature of nonlinear effects in shock acceleration . in terms of @xmath45 and @xmath44 the density immediately upstream is @xmath47 . in eq . ( [ eq : inje ] ) we can introduce a dimensionless quantity @xmath48 so that @xmath49 introducing the same formalism also for the reacceleration term in eq . ( [ eq : solut ] ) , we obtain the general expression @xmath50 @xmath51 the solution of the problem is known if the velocity field @xmath48 is known . the nonlinearity of the problem reflects in the fact that @xmath52 is in turn a function of @xmath16 as it is clear from the definition of @xmath20 . in order to solve the problem we need to write the equations for the thermodynamics of the system including the gas , the reaccelerated cosmic rays , the cosmic rays accelerated from the thermal pool and the shock itself . we write and solve these equations in the next section . the velocity , density and thermodynamic properties of the fluid can be determined by the mass and momentum conservation equations , with the inclusion of the pressure of the accelerated particles and of the preexisting cosmic rays . we write these equations between a point far upstream ( @xmath18 ) , where the fluid velocity is @xmath9 and the density is @xmath53 , and the point where the fluid upstream velocity is @xmath20 ( density @xmath54 ) . the index @xmath22 denotes quantities measured at the point where the fluid velocity is @xmath20 , namely at the point @xmath28 that can be reached only by particles with momentum @xmath55 . the mass conservation implies : @xmath56 conservation of momentum reads : @xmath57 where @xmath58 and @xmath59 are the gas pressures at the points @xmath18 and @xmath60 respectively , and @xmath61 is the pressure in accelerated particles at the point @xmath28 ( we used the symbol @xmath62 to mean _ cosmic rays _ , in the sense of _ accelerated particles _ ) . the mass flow in accelerated particles has reasonably been neglected . our basic assumption , similar to that used in @xcite , is that the diffusion is @xmath22-dependent and more specifically that the diffusion coefficient @xmath27 is an increasing function of @xmath22 . therefore the typical distance that a particle with momentum @xmath22 moves away from the shock is approximately @xmath63 , larger for high energy particles than for lower energy particles increases with @xmath22 faster than @xmath20 does , therefore @xmath64 is a monotonically increasing function of @xmath22 . ] . as a consequence , at each given point @xmath28 only particles with momentum larger than @xmath22 are able to affect appreciably the fluid . strictly speaking the validity of the assumption depends on how strongly the diffusion coefficient depends on the momentum @xmath22 . the cosmic ray pressure at @xmath28 is the sum of two terms : one is the pressure contributed by the adiabatic compression of the cosmic rays at upstream infinity , and the second is the pressure of the particles accelerated or reaccelerated at the shock ( @xmath65 ) and able to reach the position @xmath28 . since only particles with momentum @xmath66 can reach the point @xmath60 , we can write @xmath67 @xmath68 where @xmath69 is the velocity of particles with momentum @xmath22 , @xmath70 is the maximum momentum achievable in the specific situation under investigation , and @xmath71 is the adiabatic index for the accelerated particles . let us consider separately the case of a strongly modified and weakly modified shock , in order to determine the best choice for @xmath71 . in the case of strongly modified shocks , we will show that most energy is piled up in the region @xmath72 , therefore in this case we can safely adopt @xmath73 , appropriate for a relativistic gas . for weakly modified shocks , the accelerated particles have an approximately power law spectrum with a slope @xmath74 . it can be shown that in this case @xmath75 , so that the relativistic result @xmath73 still applies for @xmath76 ( strong shocks ) . for steeper spectra ( @xmath77 ) a larger adiabatic index should be adopted , but in those cases the solution is basically independent of the choice of @xmath71 because of the weak backreaction of the particles . for the purpose of carrying out our numerical calculation we will therefore always take @xmath73 . the pressure of cosmic rays at upstream infinity is simply @xmath78 where @xmath79 is some minimum momentum in the spectrum of seed particles . for simplicity , we assume that @xmath80 , namely the minimum momentum of the seed particles coincides with the momentum at which particles are injected in the shock and are accelerated . from eq . ( [ eq : pressure ] ) we can see that there is a maximum distance , corresponding to the propagation of particles with momentum @xmath70 such that at larger distances the fluid is unaffected by the accelerated particles and @xmath81 . we will show later that for strongly modified shocks the integral in eq . ( [ eq : cr ] ) is dominated by the region @xmath82 . this improves even more the validity of our approximation @xmath83 . this also suggests that different choices for the diffusion coefficient @xmath27 may affect the value of @xmath70 , but at fixed @xmath70 the spectra of the accelerated particles should not change in a significant way . assuming an adiabatic compression of the gas in the upstream region , we can write @xmath84 where we used mass conservation , eq . ( [ eq : mass ] ) . the gas pressure far upstream is @xmath85 , where @xmath86 is the ratio of specific heats for the gas ( @xmath87 for an ideal gas ) and @xmath88 is the mach number of the fluid far upstream . we introduce now a parameter @xmath89 that quantifies the relative weight of the cosmic ray pressure at upstream infinity compared with the pressure of the gas at the same location , @xmath90 . using this parameter and the definition of the function @xmath52 , the equation for momentum conservation becomes @xmath91 + \frac{1}{\rho_0 u_0 ^ 2 } \frac{d{\tilde p}_{cr}}{dp } = 0.\ ] ] using the definition of @xmath92 and multiplying by @xmath22 , this equation becomes @xmath93 = \frac{4\pi}{3 \rho_0 u_0 ^ 2 } p^4 v(p ) f_0(p ) , \label{eq : eqtosolve}\ ] ] where @xmath16 depends on @xmath52 as written in eq . ( [ eq : laeffe ] ) . ( [ eq : eqtosolve ] ) is therefore an integral - differential nonlinear equation for @xmath52 . the solution of this equation also provides the spectrum of the accelerated particles . the last missing piece is the connection between @xmath45 and @xmath44 , the two compression factors appearing in eq . ( [ eq : solut ] ) . the compression factor at the gas shock around @xmath11 can be written in terms of the mach number @xmath94 of the gas immediately upstream through the well known expression @xmath95 on the other hand , if the upstream gas evolution is adiabatic , then the mach number at @xmath12 can be written in terms of the mach number of the fluid at upstream infinity @xmath88 as @xmath96 so that from the expression for @xmath45 we obtain @xmath97^{\frac{1}{\gamma_g+1}}. \label{eq : rsub_rtot}\ ] ] now that an expression between @xmath45 and @xmath44 has been found , eq . ( [ eq : eqtosolve ] ) basically is an equation for @xmath45 , with the boundary condition that @xmath98 . finding the value of @xmath45 ( and the corresponding value for @xmath44 ) such that @xmath98 also provides the whole function @xmath52 and , through eq . ( [ eq : solut ] ) , the distribution function @xmath26 for the particles resulting from acceleration and reacceleration in the nonlinear regime . if the backreaction of the accelerated particles is small , the _ test particle _ solution must be recovered . in this section we investigate the shock modification due to the backreaction of the accelerated particles . we split the discussion in two parts : in [ subsec : seed ] we consider the case of pre - existing seed particles populating the region where the shock is propagating , and re - energized by the shock . we find the conditions for the onset of the nonlinear regime in which the shock gets modified by the re - accelerated particles . in particular we show that the critical behaviour found for shock acceleration and manifesting itself through the appearance of three solutions , takes place also in this case . in [ subsec : thermal ] we explore the more complicated situation in which a shock accelerates a new population of particles while possibly reaccelerating a pre - existing population of seed particles . here we assume that a fluid is moving with speed @xmath9 in a region where the temperature is @xmath99 . the mach number is some pre - defined value @xmath88 , which can be easily related to @xmath9 and @xmath99 . in the region of interest we assume that a population of seed particles is present with energy per particle already higher than some injection energy necessary for the particles to _ feel _ the shock as a discontinuity . for simplicity we assume that these seed particles have a spectrum which is a power law in momentum in the form : @xmath100 in this section we explore the situation in which the shock reacceleration of a pre - existing population of cosmic rays can modify the shock structure , but there is no acceleration of particles different from those that are already present as seed particles . this is equivalent to turn off the injection term ( @xmath101 in eq . ( [ eq : laeffe ] ) ) . in the next section we discuss what happens when both components are present . the crucial difference between the two components is that the seed particles are by definition already above the threshold for _ injection _ , so that there is no injection efficiency that instead represents such a crucial parameter for the case of acceleration of particles extracted from the thermal pool . having in mind the case of shocks propagating in the interstellar medium of our galaxy , we consider here the case in which the inflowing gas and the seed particles are in pressure equilibrium , namely @xmath102 . in terms of the parameter @xmath103 this implies @xmath104 . the results can then be easily repeated for a generic value of @xmath89 . ( [ eq : eqtosolve ] ) gives the quantity @xmath52 as a function of the momentum @xmath22 for any choice of @xmath45 and @xmath44 , while the acceptable solutions are those with the right matching conditions at @xmath70 . for simplicity , let us assume that the @xmath70 in eq . ( [ eq : f_infty ] ) is the same maximum momentum that particles injected at the shock would achieve : this value only depends on the environmental conditions ( energy losses of the particles ) and/or on the geometry of the shock , which may allow the escape of the particles . the momentum @xmath70 is the same as in the distribution of the pre - exisiting seed particles if , for instance , the seed particles have been accelerated by a shock identical to the one we are considering . in any case , this assumption is not needed for the validity of our conclusions and may be easily relaxed , it simply serves to avoid parameter proliferation . within this assumption , particles re - energized at the shock are simply redistributed in the momentum range between @xmath36 and @xmath70 . the solution , namely the right pair of values for the compression parameters @xmath45 and @xmath44 [ related through eq . ( [ eq : rsub_rtot ] ) ] , is obtained when the solution corresponding to @xmath98 is selected . in fig . [ fig : m150 ] , we plot @xmath105 as a function of @xmath44 for a shock having mach number at infinity @xmath106 . the different lines are labelled by a number representing the @xmath107 of @xmath70 in units of @xmath108 . in other words the parameter @xmath70 changes between @xmath109 and @xmath110 . the physical solutions are found by determining the intersections of each curve with the horizontal line corresponding to @xmath98 . the minimum momentum of the seed particles is taken as @xmath111 . the intersection at @xmath112 is close to the well known linear solution , namely the solution that one would obtain in the test particle approximation . the energy channelled into the nonthermal particles for this solution is very small , and the shock remains approximately unmodified . increasing the value of @xmath70 , the solution moves toward slightly larger values of @xmath44 ( namely the shock becomes more modified ) . for some values of @xmath70 , multiple solutions appear . in particular three regions of @xmath70 can be identified : \1 ) @xmath113 : in this region ( low values of @xmath70 ) the only solution is very close to the one obtained in the test particle approximation . for the values of the parameters in fig . [ fig : m150 ] , @xmath114 . \2 ) @xmath115 : in this region ( intermediate values of @xmath70 three solutions appear , two of which imply a strong modification of the shock , namely an appreciable part of the energy flowing through the shock is converted into energy of the accelerated particles . for the values of the parameters in fig . [ fig : m150 ] , @xmath116 . \3 ) @xmath117 : in this region ( high values of @xmath70 ) the solution becomes one again , and the shock is always strongly modified ( @xmath118 ) . this critical behaviour appears also when one fixes the maximum momentum and uses the mach number of the shock as the order parameter . the results are shown in fig . [ fig : pmax10_5 ] , where mach numbers between 10 and 500 have been considered , at fixed @xmath119 . one can see that for mach numbers below @xmath120 there is only one solution . for mach numbers between @xmath120 and @xmath121 three solutions appear , one of which roughly corresponds to an unmodified shock . for larger mach numbers , only this linear solution remains , and the shock is always only weakly modified . the same situation is plotted in fig . [ fig : pmax10_5rsub ] , where instead of @xmath44 on the x - axis there is @xmath122 , and @xmath45 is the compression coefficient at the gas subshock . for unmodified shocks one expects @xmath123 . again three regions can be identified , in the parameter @xmath88 . \1 ) for @xmath124 only one solution exists and does not necessarily correspond to unmodified shocks . in fact one can see that for @xmath125 , the compression coefficient at the subshock is @xmath126 , and @xmath127 , for the situation plotted in figs . [ fig : pmax10_5 ] and [ fig : pmax10_5rsub ] . even smaller values of @xmath88 do not give a perfectly unmodified shock . \2 ) for @xmath128 three solutions appear , two of which imply a strong modification of the shock . for the parameters used in figs . [ fig : pmax10_5 ] and [ fig : pmax10_5rsub ] , @xmath129 . \3 ) for @xmath130 only the nearly unmodified solution exists . in order to emphasize the fact that the multiple solutions actually correspond to physical solutions with very different spectral characteristics for the accelerated particles we plot in fig . [ fig : spectra ] the spectra for @xmath131 ( for which there is only one solution ) and @xmath106 ( for which there are three solutions ) . in both cases we used @xmath132 . the solution corresponding to mach number @xmath131 has @xmath133 and @xmath134 , resulting in a total pressure of the accelerated particles @xmath135 ( solid line in fig . [ fig : spectra ] ) . the three dashed lines in fig . [ fig : spectra ] are for @xmath106 and represent the spectra for the three solutions . in numbers , these solutions ( from top to bottom in fig . [ fig : spectra ] ) are summarized as follows : _ first solution : _ @xmath136 , @xmath137 , @xmath138 _ second solution : _ @xmath139 , @xmath140 , @xmath141 _ third solution : _ @xmath142 , @xmath143 , @xmath144 in the cases of strong shock modification , the asymptotic spectra for @xmath145 are @xmath146 , while the linear theory , in case of strong shocks ( @xmath147 ) , would predict @xmath148 . in the region of very low energies , the spectra of the reaccelerated particles tend to zero , as known from the linear theory as well . nonlinear effects in shock acceleration were first investigated in the case when a fraction of the thermal gas crossing the shock surface is energized to nonthermal energies . in this section we wish to apply the analytical method discussed in [ sec : nonlin ] and [ sec : gasdyn ] to the most general case in which thermal particles are accelerated but seed particles may already be present in the environment . we start our discussion with the case of acceleration of particles from the thermal distribution , in order to show that the multiple solutions already found in other analytical approaches @xcite are also obtained by adopting the approach illustrated here . in fig . [ fig : noseed ] , we plot the @xmath105 as a function of @xmath44 for the case in which a fraction @xmath39 of the particles ( as indicated in the plots ) crossing the shock is actually accelerated to suprathermal energies . the mach number is chosen to be @xmath106 and the maximum momentum is taken as @xmath149 . one can easily see that @xmath150 , the value for unmodified shocks , for small values of @xmath39 ( low efficiencies ) , while @xmath44 increases above 4 for @xmath151 . three solutions appear for intermediate efficiencies , while the solution for the accelerated particles always predicts a strongly modified shock for high efficiencies , @xmath152 . an asymptotic value of @xmath153 is achieved for the parameters used here . this example demonstrates that the critical behaviour shown to appear for high values of @xmath39 in several analytical or semi - analytical calculations is in fact also predicted by the approach proposed here . next question to ask is however whether the presence of seed particles can change the critical behavior of shocks . in order to answer this question we consider a situation in which particles are accelerated from the thermal pool with efficiency @xmath39 , and at the same time the shock propagates in a medium where the preshock pressure in seed particles equals the gas pressure . this situation is considered to resemble that of a supernova explosion in our galaxy , where cosmic rays fill the volume remaining in quasi - equipartition with the gas . in order to test the critical structure of the shock we calculate @xmath105 as a function of the compression factor @xmath44 between upstream infinity and downstream . our results are plotted in fig . [ fig : crseed ] . the lines refer to the cases @xmath154 from bottom to top , as in fig . [ fig : noseed ] . it is clear from this plot that the shock may be modified by the backreaction of the accelerated particles even for very low values of @xmath39 , because of the presence of seed particles . in other words , the nonlinearity of the shock can well be dominated by the presence of preaccelerated particles rather than by the acceleration of particles from the thermal pool . clearly this depends however on the values of the parameters ( injection momentum , @xmath39 , maximum momentum achievable , mach number ) . the injection momentum , in principle , could be related to @xmath39 in order to reduce the number of free parameters of the problem . this is possible in the assumption that the particles downstream keep a thermal distribution . simulations suggest that the particles injected in the accelerator are the ones with momentum a few times the thermal momentum of the particles downstream , so that @xmath39 can be simply calculated by integration of the maxwell - boltzmann ( mb ) distribution above this minimum momentum . unfortunately , we do not know whether the particle distribution downstream is in fact maxwell - boltzmann - like . moreover , we do not know exactly what is the threshold to impose on the injection momentum , which is a delicate issue because the number of particles taking part to the acceleration process would result from the integral of the mb distribution over its exponentially decreasing part . therefore we preferred here to keep @xmath39 and the injection momentum as separate free parameters . we proposed a semi - analytical approach to show that the backreaction of particles accelerated at a shock is able to affect the shock itself , in such a way that the shock and the accelerated particles become parts of a nonlinear system . in particular , for the first time we included in this kind of calculations the seed particles that may be present in the region where the shock is propagating and that can be re - energized by the shock . while a test - particle approach to this problem was first presented in @xcite , a nonlinear treatment was never investigated . in the pioneering work of ref . @xcite , it was recognized that the energy of the seed particles could be enhanced by about one order of magnitude at each shock passage , and that after an infinite number of strong shocks passing through the region , the spectrum of the particles would tend to the asymptotic spectrum @xmath155 . two comments are in order . first , the continuous increase of the cosmic ray energy due to the re - energization of seed particles leads unavoidably the shock to be modified by the nonthermal pressure , unless one starts from an uninterestingly small pressure of seed particles at the beginning . second , the fact that the spectrum becomes flatter than @xmath156 , which is the result for a strong non - relativistic shock , implies that most of the nonthermal energy is pushed to the highest energies , therefore the shock again can be more easily modified . both these points suggest that a nonlinear treatment of shock re - acceleration is required . in our galaxy , cosmic rays are observed to be in rough equipartition with the gas pressure and with magnetic fields , therefore supernova shocks or shocks generated in other environments propagate in a medium in which the seed particles ( cosmic rays ) are non - negligible . in these circumstances the non - linear effects may be very important . we showed here that in fact for some regions of the parameter space , the shock is modified mainly by the backreaction of the seed particles rather than by the cosmic rays accelerated at the shock from the thermal pool . the spectra of the re - accelerated particles have also been calculated . the interesting phenomenon of the appearance of multiple shock solutions , already known for the case of shock acceleration , appears also for the case of reacceleration . this puzzling phenomenon may suggest that the shock behaves as a self - regulating system settling on the critical point , as proposed in @xcite . on the other hand , it is possible that the multiple solutions may be the artifact of some of the assumptions used in the analytical approach , in particular the request for time independent ( stationary ) solutions and the fact that the role of the self - generated waves on the diffusion coefficient is not taken into account . further investigation , in particular in the direction of a detailed comparison of our results with numerical simulations of shock acceleration is required in order to unveil the physical meaning of the multiple solutions for modified shocks . from the phenomenological point of view , it would be certainly worth to study the implications of nonlinear shock reacceleration on the nonthermal activity in astrophysical environments where the effect is expected to play an important role , in particular in the case of supernova remnants . in particular , as pointed out by the referee , the suggested dominance of reaccelerated ambient seed particles over freshly injected particles may have serious implications for the spectra of secondary nuclei resulting from spallation processes . the author gratefully acknowledges useful discussions with m. baring , d. ellison and m. vietri . the author is also grateful to the referee , l.oc . drury for his very useful remarks . this work was partially supported through grant cofin 2002 at the arcetri astrophysical observatory .

particles crossing repeatedly the surface of a shock wave can be energized by first order fermi acceleration . the linear theory is successful in describing the acceleration process as long as the pressure of the accelerated particles remains negligible compared to the kinetic pressure of the incoming gas ( the so - called test particle approximation ) . when this condition is no longer fulfilled , the shock is modified by the pressure of the accelerated particles in a nonlinear way , namely the spectrum of accelerated particles and the shock structure determine each other . in this paper we present the first description of the nonlinear regime of shock acceleration when the shock propagates in a medium where seed particles are already present . this case may apply for instance to supernova shocks propagating into the interstellar medium , where cosmic rays are in equipartition with the gas pressure . we find that the appearance of multiple solutions , previously found in alternative descriptions of the nonlinear regime , occurs also for the case of reacceleration of seed particles . moreover , for parameters of concern for supernova shocks , the shock is likely to turn nonlinear mainly due to the presence of the pre - existing cosmic rays , rather than due to the acceleration of new particles from the thermal pool . we investigate here the onset of the nonlinear regime for the three following cases : 1 ) seed particles in equipartition with the gas pressure ; 2 ) particles accelerated from the thermal pool ; 3 ) combination of 1 ) and 2 ) . cosmic rays , high energy , acceleration

arxiv

cosmological observations coming from type ia supernovae @xcite , cosmic microwave background radiation @xcite and the large scale structure @xcite , provide evidences that the universe is currently in an accelerating phase . this result is , in general , ascribed to the existence of a sort of dark energy ( de ) sector in the universe , an exotic energy source characterized by a negative pressure . at late times , the dark - energy sector eventually dominates over the cold dark matter ( cdm ) , and drives the universe to the observed accelerating expansion . the simplest candidate for de is the cosmological constant @xmath2 , which has an equation - of - state parameter @xmath3 . although this model is in agreement with current observations , it is plagued by some difficulties related to the small observational value of de density with respect to the expected one arising from quantum field theories ( the well known cosmological constant problem @xcite ) . moreover , the @xmath2cdm paradigm , where cold dark matter ( cdm ) is considered into the game , may also suffer from the age problem , as it was shown in @xcite , while the present data seem to slightly favor an evolving de with the equation - of - state parameter crossing @xmath3 from above to below in the near cosmological past @xcite . over the past decade several de models have been proposed , such as quintessence @xcite , phantom @xcite , k - essence @xcite , tachyon @xcite , quintom @xcite , chaplygin gas @xcite , generalized chaplygin gas ( gcg ) @xcite , holographic de @xcite , new agegraphic de @xcite , ricci de @xcite etc . on the other hand , there are also numerous models that induce an effective dark energy which arises from modifications of the gravitational sector itself , such as @xmath4 gravity @xcite ( this class is very efficient in verifying observational and theoretical constraints and explain the universe acceleration and phantom crossing @xcite ) , or gravity with higher curvature invariants @xcite , by coupling the ricci scalar to a scalar field @xcite , by introducing a vector field contribution @xcite , or by using properties of gravity in higher dimensional spacetimes @xcite ( for a review see @xcite ) . a possibility that can be explored to explain the accelerated phase of the universe is to consider a theory of gravity based on the weitzenbck connection , instead of the levi - civita one , which deduces that the gravitational field is described by the torsion instead of the curvature tensor . in such theories , the torsion tensor is achieved from products of first derivatives of tetrad fields , and hence no second derivatives appear . this _ teleparallel _ approach @xcite , is closely related to general relativity , except for `` boundary terms '' @xcite that involve total derivatives in the action , and thus one can construct the teleparallel equivalent of general relativity ( tegr ) , which is completely equivalent with general relativity at the level of equations but is based on torsion instead of curvature . teleparallel gravity possesses a number of attractive features related to geometrical and physical aspects @xcite . hence , one can start from tegr and construct various gravitational modifications based on torsion , with @xmath1 gravity being the most studied one @xcite . in particular , it may represent an alternative to inflationary models without the use of the inflaton , as well as to effective de models , in which the universe acceleration is driven by the extra torsion terms @xcite ( for a detailed review , see @xcite ) . the main advantage of @xmath1 gravity is that the field equations are 2nd - order ones , a property that makes these theories simpler if compared to the dynamical equations of other extended theories of gravity , such as @xmath4 gravity . the aim of this paper is to explore the implications of @xmath1 gravity to the formation of light elements in the early universe , i.e. to the big bang nucleosynthesis ( bbn ) . on the other hand , we want to explore the possibility to constrain @xmath1 gravity by bbn observatio nal data . bbn has occurred between the first fractions of second after the big bang , around @xmath5 sec , and a few hundreds of seconds after it , when the universe was hot and dense ( indeed bbn , together with cosmic microwave background radiation , provides the strong evidence about the high temperatures characterizing the primordial universe ) . it describes the sequence of nuclear reactions that yielded the synthesis of light elements @xcite , and therefore drove the observed universe . in general , from bbn physics , one may infer stringent constraints on a given cosmological model . hence , in this work , we shall confront various @xmath1 gravity models with bbn calculations based on current observational data on primordial abundance of @xmath0 , and we shall extract constraints on their free parameters . the layout of the paper is as follows . in section [ revmodel ] we review @xmath1 gravity and the related cosmological models . in section [ bbnanal ] we use bbn calculations in order to impose constraints on the free parameters of specific @xmath1 gravity models . conclusions are reported in section [ conclusions ] . finally , in the appendix we summarize the main notions of bbn physics . let us briefly review @xmath1 gravity , and apply it in a cosmological framework . in this formulation , the dynamical variable is the vierbein field @xmath6 , @xmath7 , which forms an orthonormal basis in the tangent space at each point @xmath8 of the manifold , i.e. @xmath9 , with @xmath10 the minkowsky metric with signature @xmath11 : @xmath12 . denoting with @xmath13 , @xmath14 the components of the vectors @xmath15 in a coordinate basis @xmath16 , one can write @xmath17 . as a convection , here we use the latin indices for the tangent space , and the greek indices for the coordinates on the manifold . the dual vierbein allows to obtain the metric tensor of the manifold , namely @xmath18 . in teleparallel gravity , one adopts the curvatureless weitzenbck connection ( contrarily to general relativity which is based on the torsion - less levi - civita connection ) , which gives rise to the non - null torsion tensor : @xmath19 remarkably , the torsion tensor ( [ torsion ] ) encompasses all the information about the gravitational field . the lagrangian density is built using its contractions , and hence the teleparallel action is given by @xmath20 with @xmath21 , and where the torsion scalar @xmath22 reads as @xmath23 here , it is @xmath24 with @xmath25 the contorsion tensor which gives the difference between weitzenbck and levi - civita connections . finally , the variation of action ( [ action0 ] ) in terms of the vierbiens gives rise to the field equations , which coincide with those of general relativity . that is why the above theory is called the teleparallel equivalent of general relativity ( tegr ) . one can now start from tegr , and generalize action ( [ action0 ] ) in order to construct gravitational modifications based on torsion . the simplest scenario is to consider a lagrangian density that is a function of @xmath22 , namely @xmath26},\ ] ] that reduces to tegr as soon as @xmath27 . considering additionally a matter lagrangian @xmath28 , variation with respect to the vierbein gives the field equations @xcite @xmath29- e_i^\lambda { t^\rho}_{\mu\lambda}{s_\rho}^{\nu\mu}[1+f']\nonumber\\ & & + e^\rho_i { s_\rho}^{\,\,\mu\nu}(\partial_\mu t)f '' + \frac{1}{4}e^\nu_i [ t+f]=4\pi g\,{e_i}^\rho\ , { \theta_\rho}^\nu\,,\end{aligned}\ ] ] where @xmath30 , @xmath31 and @xmath32 is the energy - momentum tensor for the matter sector . in order to explore the cosmological implications of @xmath1 gravity , we focus on homogeneous and isotropic geometry , considering the usual choice for the vierbiens , namely @xmath33 which corresponds to a flat friedmann - robertson - walker ( frw ) background metric of the form @xmath34 where @xmath35 is the scale factor . equations ( [ torsion ] ) , ( [ lagrangian ] ) , ( [ s ] ) and ( [ contorsion ] ) allow to derive a relation between the torsion @xmath22 and the hubble parameter @xmath36 , namely @xmath37 hence , in the case of frw geometry , and assuming that the matter sector corresponds to a perfect fluid with energy density @xmath38 and pressure @xmath39 , the @xmath40 component of ( [ equations ] ) yields @xmath41+[t+f]=16\pi g\rho,\ ] ] while the @xmath42 component gives @xmath43-(t - f)=16\pi gp.\ ] ] the equations close by considering the equation of continuity for the matter sector , namely @xmath44 . one can rewrite ( [ friedmann ] ) and ( [ acceleration ] ) in the usual form @xmath45 @xmath46 where @xmath47,\\ p_t&=&\frac{1}{16\pi g}\ , \frac{f - t f'+ 2t^2f''}{1+f'+ 2{t}f '' } \ , , \label{pt}\end{aligned}\ ] ] are the effective energy density and pressure arising from torsional contributions . one can therefore define the effective torsional equation - of - state parameter as @xmath48 in these classes of theories , the above effective torsional terms are responsible for the accelerated phases of the early or / and late universe @xcite . let us present now three specific @xmath1 forms , which are the viable ones amongst the variety of @xmath1 models with two parameters out of which one is independent , i.e which pass the basic observational tests @xcite . 1 . the power - law model by bengochea and ferraro ( hereafter @xmath49cdm ) @xcite is characterized by the form @xmath50 where @xmath51 and @xmath52 are the two model parameters . inserting this @xmath1 form into friedmann equation ( [ friedmann ] ) at present , we acquire @xmath53 where @xmath54 is the matter density parameter at present , and @xmath55 is the current hubble parameter value . the best fit on the parameter @xmath52 is obtained taking the @xmath56 observational data , and it reads @xcite @xmath57 clearly , for @xmath58 the present scenario reduces to @xmath2cdm cosmology , namely @xmath59 , with @xmath60 . 2 . the linder model ( hereafter @xmath61cdm ) @xcite arises from @xmath62 with @xmath63 and @xmath39 ( @xmath64 ) the two model parameters . in this case ( [ friedmann ] ) gives that @xmath65 the @xmath56 observational data imply that the best fit of @xmath64 is @xcite @xmath66 as we can see , for @xmath67 the present scenario reduces to @xmath2cdm cosmology . 3 . motivated by exponential @xmath4 gravity @xcite , bamba et . al . introduced the following @xmath1 model ( hereafter @xmath68cdm ) @xcite : @xmath69 with @xmath63 and @xmath39 ( @xmath64 ) the two model parameters . in this case we acquire @xmath70 for this model , and using @xmath56 observational data , the best fit is found to be @xcite @xmath71 similarly to the previous case we can immediately see that @xmath68cdm model tends to @xmath2cdm cosmology for @xmath67 . the above @xmath1 models are considered viable in literature because pass the basic observational tests @xcite . they are characterized by two free parameters . actually there are two more models with two free parameters , namely the logarithmic model @xcite , @xmath72 and the hyperbolic - tangent model @xcite , @xmath73 nevertheless since these two models do not possess @xmath2cdm cosmology as a limiting case and since they are in tension with observational data @xcite , in this work we do not consider them . finally , let us note that one could also construct @xmath1 models with more than two parameters , for example , combining the above scenarios . however , considering many free parameters would be a significant disadvantage concerning the corresponding values of the information criteria . in the section , we examine the bbn in the framework of @xmath1 cosmology . as it is well known , bbn occurs during the radiation dominated era . the energy density of relativistic particles filling up the universe is given by @xmath74 , where @xmath75 is the effective number of degrees of freedom and @xmath76 the temperature ( in the appendix we review the main features related to the bbn physics ) . the neutron abundance is computed via the conversion rate of protons into neutrons , namely @xmath77 and its inverse @xmath78 . the relevant quantity is the total rate given by @xmath79 explicit calculations of eq . ( [ lambda ] ) lead to ( see ( [ lambdafinapp ] ) in the appendix ) @xmath80 where @xmath81 is the mass difference of neutron and proton , and @xmath82gev@xmath83 . the primordial mass fraction of @xmath84 can be estimated by making use of the relation @xcite @xmath85 here @xmath86 , with @xmath87 the time of the freeze - out of the weak interactions , @xmath88 the time of the freeze - out of the nucleosynthesis , @xmath89 the neutron mean lifetime given in ( [ rateproc3 ] ) , and @xmath90 is the neutron - to - proton equilibrium ratio . the function @xmath91 is interpreted as the fraction of neutrons that decay into protons during the interval @xmath92 $ ] . deviations from the fractional mass @xmath93 due to the variation of the freezing temperature @xmath94 are given by @xmath95 \frac{\delta { \cal t}_f}{{\cal t}_f}\,,\ ] ] where we have set @xmath96 since @xmath97 is fixed by the deuterium binding energy @xcite . the experimental estimations of the mass fraction @xmath93 of baryon converted to @xmath98 during the big bang nucleosynthesis are @xcite @xmath99 inserting these into ( [ deltayp ] ) one infers the upper bound on @xmath100 , namely @xmath101 during the bbn , at the radiation dominated era , the scale factor evolves as @xmath102 , where @xmath103 is cosmic time . the torsional energy density @xmath104 is treated as a perturbation to the radiation energy density @xmath38 . the relation between the cosmic time and the temperature is given by @xmath105 ( or @xmath106mev ) . furthermore , we use the entropy conservation @xmath107 . the expansion rate of the universe is derived from ( [ modfri ] ) , and can be rewritten in the form @xmath108 where @xmath109 ( @xmath110 is the expansion rate of the universe in general relativity ) . thus , from the relation @xmath111 , one derives the freeze - out temperature @xmath112 , with @xmath113 mev ( which follows from @xmath114 ) and @xmath115 from which , in the regime @xmath116 , one obtains : @xmath117 with @xmath118gev@xmath83 . in what follows we shall investigate the bounds that arise from the bbn constraints , on the free parameters of the three @xmath1 models presented in the previous section . these constraint will be determined using eqs . ( [ deltat / tboundg ] ) and ( [ rhot ] ) . moreover , we shall use the numerical values @xmath119 where @xmath120 is the present value of cmb temperature . 1 . @xmath121cdm model . + for the @xmath121cdm model of ( [ eq : ftmyrzabis ] ) relation ( [ rhot ] ) gives @xmath122 \nonumber \\ & = & \frac{3h_0 ^ 2}{8\pi g}\ , \omega_{m0}\left(\frac{{\cal t}}{{\cal t}_0}\right)^{4n } \label{rhoth } \ , , \end{aligned}\ ] ] and then ( [ deltat / tboundg ] ) yields @xmath123 in fig . [ deltatf1 ] we depict @xmath124 from ( [ deltat / tboundgfin ] ) vs @xmath52 , as well as the upper bound from ( [ deltat / tbound ] ) . as we can see , constraints from bbn require @xmath125 . remarkably , this bound is in agreement with the best fit for @xmath52 of ( [ bestfit1 ] ) , namely @xmath126 , that was obtained using @xmath56 observational data in @xcite . + _ @xmath124 from ( [ deltat / tboundgfin ] ) vs @xmath52 ( thick line ) for the @xmath121cdm model of ( [ eq : ftmyrzabis ] ) , and the upper bound for @xmath124 from ( [ deltat / tbound ] ) ( dashed line ) . as we can see , constraints from bbn require @xmath125._,width=278,height=216 ] 2 . @xmath127cdm model . + in the case of @xmath128cdm model of ( [ modf2 ] ) and @xmath129cdm model of ( [ modf3 ] ) , and for the purpose of this analysis , we can unified their investigation parameterizing them as @xmath130\,,\ ] ] with @xmath131 where @xmath132 for model @xmath128cdm and @xmath133 for model @xmath129cdm . inserting ( [ f12insieme ] ) into ( [ deltat / tboundg ] ) we acquire @xmath134e^{-p({\cal t}_f/{\cal t}_0)^{4 m } } \ ! -\!\frac{1}{2}\right\}.\end{aligned}\ ] ] hence , using this relation we can calculate the value of @xmath135 for various values of @xmath136 that span the order of magnitude of the best fit values ( [ bestfit1 ] ) and ( [ bestfit2 ] ) that were obtained using @xmath56 observational data in @xcite , and we present our results in table [ tab1 ] . as we can see , in all cases the value of @xmath135 is well below the bbn bound ( [ deltat / tbound ] ) . hence , bbn can not impose constraints on the parameter values of @xmath128cdm and @xmath129cdm models . .@xmath135 from ( [ delta12insieme ] ) for different values of @xmath136 , for @xmath137 ( @xmath128cdm model ) and @xmath138 ( @xmath129cdm model ) . [ cols="^,^,^,^,^",options="header " , ] in this work we have investigated the implications of @xmath1 gravity to the formation of light elements in the early universe , i.e. to the bbn . in particular , we have examined the three most used and well studied viable @xmath1 models , namely the power law , the exponential and the square - root exponential , and we have confronted them with bbn calculations based on current observational data on primordial abundance of @xmath0 . hence , we were able to extract constraints on their free parameters . concerning the power - law @xmath1 model , the obtained constraint on the exponent @xmath52 , is @xmath125 . remarkably , this bound is in agreement with the constraints obtained using @xmath56 observational data @xcite . concerning the exponential and the square - root exponential , we showed that , for realistic regions of free parameters , they always satisfy the bbn bounds . this means that , in these cases , bbn can not impose strict constraints on the values of free parameters . in summary , we showed that viable @xmath1 models , namely those that pass the basic observational tests , can also satisfy the bbn constraints . this feature acts as an additional advantage of @xmath1 gravity , which might be a successful candidate for describing the gravitational interaction . as discussed in @xcite , this kind of constraints could contribute in the debate of fixing the most realistic picture that can be based on curvature or torsion . this article is based upon work from cost action ca15117 `` cosmology and astrophysics network for theoretical advances and training actions '' ( cantata ) , supported by cost ( european cooperation in science and technology ) . in this appendix we briefly review the main features of big bang nucleosynthesis following @xcite . in the early universe , the primordial @xmath0 was formed at temperature @xmath139 mev . the energy and number density were formed by relativistic leptons ( electron , positron and neutrinos ) and photons . the rapid collisions maintain all these particles in thermal equilibrium . interactions of protons and neutrons were kept in thermal equilibrium by means of their interactions with leptons @xmath140 the neutron abundance is estimated by computing the conversion rate of protons into neutrons , i.e. @xmath141 , and its inverse @xmath78 . thus , the weak interaction rates ( at suitably high temperature ) are given by @xmath142 the rate @xmath143 is the sum of the rates associated to the processes ( [ proc1])-([proc3 ] ) , namely @xmath144 finally , the rate @xmath143 is related to the rate @xmath145 as @xmath146 , with @xmath147 the mass difference of neutron and proton . during the freeze - out stage , one can use the following approximations @xcite : ( i ) the temperatures of particles are the same , i.e. @xmath148 . ( ii ) the temperature @xmath76 is lower than the typical energies @xmath149 that contribute to the integrals entering the definition of the rates ( one can therefore replace the fermi - dirac distribution with the boltzmann one , namely @xmath150 ) . ( iii ) the electron mass @xmath151 can be neglected with respect to the electron and neutrino energies ( @xmath152 ) . having these in mind , the interaction rate corresponding to the process ( [ proc1 ] ) is given by @xmath153 where @xmath154\ , , \label{wa}\\ { \cal p } & \equiv & p_n+p_{\nu_e}-p_p - p_e\ , , \\ { \cal m } & = & \left(\frac{g_w}{8m_w}\right)^2 [ { \bar u}_p\omega^\mu u_n][{\bar u}_e\sigma_\mu v_{\nu_e}]\ , , \label{m } \\ \omega^\mu & \equiv & \gamma^\mu(c_v - c_a \gamma^5)\ , , \\ \sigma^\mu & \equiv & \gamma^\mu(1-\gamma^5 ) \ , . \end{aligned}\ ] ] in ( [ m ] ) we have used the condition @xmath155 , where @xmath156 is the mass of the vector gauge boson @xmath157 , with @xmath158 the transferred momentum . from eq . ( [ rateproc1 ] ) it follows that @xmath159 where @xmath160 and where @xmath161d\epsilon,\ ] ] with @xmath162 lastly , for the neutron decay ( [ proc3 ] ) one obtains @xmath166 hence , in the calculation of ( [ sumprocess ] ) we can safely neglect the above interaction rate of the neutron decay , i.e. during the bbn the neutron can be considered as a stable particle . the above approximations ( i)-(iii ) lead to @xcite @xmath167 thus , inserting ( [ auxilirel ] ) into ( [ sumprocess ] ) , and then into ( [ lambdaa ] ) , allows to derive the expression for @xmath168 , namely @xmath169 which using ( [ ne - pnu - fin ] ) leads to @xmath170 99 a. g. riess , et al . , astron . j. * 116 * , 1009 ( 1998 ) . s. perlmutter , et al . , astrophys . j. * 517 * , 565 ( 1999 ) . d. n. spergel , et al . , apjs . * 148 * , 175 ( 2003 ) . d. n. spergel , et al . , apjs . * 170 * , 377 ( 2007 ) . m. tegmark , et al . d * 69 * , 103501 ( 2004 ) . d. j. eisenstein , et al . , astrophys . j. * 633 * , 560 ( 2005 ) . s. m. carroll , living rev . rel . * 4 * , 1 ( 2001 ) . e. j. copeland , m. sami and s. tsujikawa , int . d. * 15 * , 1753 ( 2006 ) . r. j. yang and s. n. zhang , mon . not . . soc . * 407 * , 1835 ( 2010 ) . b. feng , x. l. wang and x. m. zhang , phys . b. * 607 * , 35 ( 2005 ) . r. r. caldwell , r. dave and r. j. steinhardt , phys . 80 * , 1582 ( 1998 ) . r. r. caldwell , phys . b. * 545 * , 23 ( 2002 ) . c. armendariz - picon , v. mukhanov and p. j. steinhardt , phys . d * 63 * , 103510 ( 2001 ) . t. padmanabhan , phys . rev . d. * 66 * , 021301 ( 2002 ) . a. sen , phys . t. * 117 * , 70 ( 2005 ) . y. f. cai , e. n. saridakis , m. r. setare and j. q. xia , phys . rept . * 493 * , 1 ( 2010 ) . e. elizalde , s. nojiri and s. d. odintsov , phys . d. * 70 * , 043539 ( 2004 ) . a. kamenshchik , u. moschella and v. pasquier , phys . b. * 511 * , 265 ( 2001 ) . m. c. bento , o. bertolami and a. a. sen , phys . d. * 66 * , 043507 ( 2002 ) . s. capozziello and m. de laurentis , phys . * 509 * , 167 ( 2011 ) . a. einstein , sitz . p. 217 ; ibid a. einstein ( 2005 ) , translations of einstein papers by a. unzicker and t. case , ( arxiv : physics/0503046 ) . s. bahamonde and c. g. bhmer , eur.phys.j . c * 76 * , 578 ( 2016 ) .

we use bbn observational data on primordial abundance of @xmath0 to constrain @xmath1 gravity . the three most studied viable @xmath1 models , namely the power law , the exponential and the square - root exponential are considered , and the bbn bounds are adopted in order to extract constraints on their free parameters . for the power - law model , we find that the constraints are in agreement with those acquired using late - time cosmological data . for the exponential and the square - root exponential models , we show that for realiable regions of parameters space they always satisfy the bbn bounds . we conclude that viable @xmath1 models can successfully satisfy the bbn constraints .

arxiv

in a bidirectional relay network , two users exchange information via a relay node @xcite . several protocols have been proposed for such a network under the practical half - duplex constraint , i.e. , a node can not transmit and receive at the same time and in the same frequency band . the simplest protocol is the traditional two - way relaying protocol in which the transmission is accomplished in four successive point - to - point phases : user 1-to - relay , relay - to - user 2 , user 2-to - relay , and relay - to - user 1 . in contrast , the time division broadcast ( tdbc ) protocol exploits the broadcast capability of the wireless medium and combines the relay - to - user 1 and relay - to - user 2 phases into one phase , the broadcast phase @xcite . thereby , the relay broadcasts a superimposed codeword , carrying information for both user 1 and user 2 , such that each user is able to recover its intended information by self - interference cancellation . another existing protocol is the multiple access broadcast ( mabc ) protocol in which the user 1-to - relay and user 2-to - relay phases are also combined into one phase , the multiple - access phase @xcite . in the multiple - access phase , both user 1 and user 2 simultaneously transmit to the relay which is able to decode both messages . generally , for the bidirectional relay network without a direct link between user 1 and user 2 , six transmission modes are possible : four point - to - point modes ( user 1-to - relay , user 2-to - relay , relay - to - user 1 , relay - to - user 2 ) , a multiple access mode ( both users to the relay ) , and a broadcast mode ( the relay to both users ) , where the capacity region of each transmission mode is known @xcite , @xcite . using this knowledge , a significant research effort has been dedicated to obtaining the achievable rate region of the bidirectional relay network @xcite-@xcite . specifically , the achievable rates of most existing protocols for two - hop relay transmission are limited by the instantaneous capacity of the weakest link associated with the relay . the reason for this is the fixed schedule of using the transmission modes which is adopted in all existing protocols , and does not exploit the instantaneous channel state information ( csi ) of the involved links . for one - way relaying , an adaptive link selection protocol was proposed in @xcite where based on the instantaneous csi , in each time slot , either the source - relay or relay - destination links are selected for transmission . to this end , the relay has to have a buffer for data storage . this strategy was shown to achieve the capacity of the one - way relay channel with fading @xcite . moreover , in fading awgn channels , power control is necessary for rate maximization . the highest degree of freedom that is offered by power control is obtained for a joint average power constraint for all nodes . any other power constraint with the same total power budget is more restrictive than the joint power constraint and results in a lower sum rate . therefore , motivated by the protocols in @xcite and @xcite , our goal is to utilize all available degrees of freedom of the three - node half - duplex bidirectional relay network with fading , via an adaptive mode selection and power allocation policy . in particular , given a joint power budget for all nodes , we find a policy which in each time slot selects the optimal transmission mode from the six possible modes and allocates the optimal powers to the nodes transmitting in the selected mode , such that the sum rate is maximized . adaptive mode selection for bidirectional relaying was also considered in @xcite and @xcite . however , the selection policy in @xcite does not use all possible modes , i.e. , it only selects from two point - to - point modes and the broadcast mode , and assumes that the transmit powers of all three nodes are fixed and identical . although the selection policy in @xcite considers all possible transmission modes for adaptive mode selection , the transmit powers of the nodes are assumed to be fixed , i.e. , power allocation is not possible . interestingly , mode selection and power allocation are mutually coupled and the modes selected with the protocol in @xcite for a given channel are different from the modes selected with the proposed protocol . power allocation can considerably improve the sum rate by optimally allocating the powers to the nodes based on the instantaneous csi especially when the total power budget in the network is low . moreover , the proposed protocol achieves the maximum sum rate in the considered bidirectional network . hence , the sum rate achieved with the proposed protocol can be used as a reference for other low complexity suboptimal protocols . simulation results confirm that the proposed protocol outperforms existing protocols . finally , we note that the advantages of buffering come at the expense of an increased end - to - end delay . however , with some modifications to the optimal protocol , the average delay can be bounded , as shown in @xcite , which causes only a small loss in the achieved rate . the delay analysis of the proposed protocol is beyond the scope of the current work and is left for future research . [ c][c][0.75]@xmath0 [ c][c][0.75]@xmath1 [ c][c][0.75]@xmath2 [ c][c][0.75]@xmath3 [ c][c][0.75]@xmath4 [ c][c][0.75]@xmath5 [ c][c][0.75]@xmath6 [ c][c][0.75]@xmath7 [ c][c][0.75]@xmath8 [ c][c][0.75]@xmath9 [ c][c][0.5]@xmath0 [ c][c][0.5]@xmath1 [ c][c][0.5]@xmath2 [ c][c][0.75]@xmath10 [ c][c][0.75]@xmath11 [ c][c][0.75]@xmath12 [ c][c][0.75]@xmath13 [ c][c][0.75]@xmath14 [ c][c][0.75]@xmath15 in this section , we first describe the channel model . then , we provide the achievable rates for the six possible transmission modes . we consider a simple network in which user 1 and user 2 exchange information with the help of a relay node as shown in fig . we assume that there is no direct link between user 1 and user 2 , and thus , user 1 and user 2 communicate with each other only through the relay node . we assume that all three nodes in the network are half - duplex . furthermore , we assume that time is divided into slots of equal length and that each node transmits codewords which span one time slot or a fraction of a time slot as will be explained later . we assume that the user - to - relay and relay - to - user channels are impaired by awgn with unit variance and block fading , i.e. , the channel coefficients are constant during one time slot and change from one time slot to the next . moreover , in each time slot , the channel coefficients are assumed to be reciprocal such that the user 1-to - relay and the user 2-to - relay channels are identical to the relay - to - user 1 and relay - to - user 2 channels , respectively . let @xmath3 and @xmath4 denote the channel coefficients between user 1 and the relay and between user 2 and the relay in the @xmath16-th time slot , respectively . furthermore , let @xmath17 and @xmath18 denote the squares of the channel coefficient amplitudes in the @xmath16-th time slot . @xmath19 and @xmath20 are assumed to be ergodic and stationary random processes with means @xmath21 and @xmath22 in expectations for notational simplicity . ] , respectively , where @xmath23 denotes expectation . since the noise is awgn , in order to achieve the capacity of each mode , nodes have to transmit gaussian distributed codewords . therefore , the transmitted codewords of user 1 , user 2 , and the relay are comprised of symbols which are gaussian distributed random variables with variances @xmath24 , and @xmath7 , respectively , where @xmath25 is the transmit power of node @xmath26 in the @xmath16-th time slot . for ease of notation , we define @xmath27 . in the following , we describe the transmission modes and their achievable rates . in the considered bidirectional relay network only six transmission modes are possible , cf . fig . [ figmodes ] . the six possible transmission modes are denoted by @xmath28 , and @xmath29 , denotes the transmission rate from node @xmath30 to node @xmath31 in the @xmath16-th time slot . let @xmath8 and @xmath9 denote two infinite - size buffers at the relay in which the received information from user 1 and user 2 is stored , respectively . moreover , @xmath32 , denotes the amount of normalized information in bits / symbol available in buffer @xmath33 in the @xmath16-th time slot . using this notation , the transmission modes and their respective rates are presented in the following : @xmath34 : user 1 transmits to the relay and user 2 is silent . in this mode , the maximum rate from user 1 to the relay in the @xmath16-th time slot is given by @xmath35 , where @xmath36 . the relay decodes this information and stores it in buffer @xmath8 . therefore , the amount of information in buffer @xmath8 increases to @xmath37 . @xmath38 : user 2 transmits to the relay and user 1 is silent . in this mode , the maximum rate from user 2 to the relay in the @xmath16-th time slot is given by @xmath39 , where @xmath40 . the relay decodes this information and stores it in buffer @xmath9 . therefore , the amount of information in buffer @xmath9 increases to @xmath41 . @xmath42 : both users 1 and 2 transmit to the relay simultaneously . for this mode , we assume that multiple access transmission is used , see @xcite . thereby , the maximum achievable sum rate in the @xmath16-th time slot is given by @xmath43 , where @xmath44 . since user 1 and user 2 transmit independent messages , the sum rate , @xmath45 , can be decomposed into two rates , one from user 1 to the relay and the other one from user 2 to the relay . moreover , these two capacity rates can be achieved via time sharing and successive interference cancelation . thereby , in the first @xmath46 fraction of the @xmath16-th time slot , the relay first decodes the codeword received from user 2 and considers the signal from user 1 as noise . then , the relay subtracts the signal received from user 2 from the received signal and decodes the codeword received from user 1 . a similar procedure is performed in the remaining @xmath47 fraction of the @xmath16-th time slot but now the relay first decodes the codeword received from user 1 and treats the signal of user 2 as noise , and then decodes the codeword received from user 2 . therefore , for a given @xmath48 , we decompose @xmath45 as @xmath49 and the maximum rates from users 1 and 2 to the relay in the @xmath16-th time slot are @xmath50 and @xmath51 , respectively . @xmath52 and @xmath53 are given by @xmath54 the relay decodes the information received from user 1 and user 2 and stores it in its buffers @xmath8 and @xmath9 , respectively . therefore , the amounts of information in buffers @xmath8 and @xmath9 increase to @xmath37 and @xmath41 , respectively . @xmath55 : the relay transmits the information received from user 2 to user 1 . specifically , the relay extracts the information from buffer @xmath9 , encodes it into a codeword , and transmits it to user 1 . therefore , the transmission rate from the relay to user 1 in the @xmath16-th time slot is limited by both the capacity of the relay - to - user 1 channel and the amount of information stored in buffer @xmath9 . thus , the maximum transmission rate from the relay to user 1 is given by @xmath56 , where @xmath57 . therefore , the amount of information in buffer @xmath9 decreases to @xmath58 . @xmath59 : this mode is identical to @xmath55 with user 1 and 2 switching places . the maximum transmission rate from the relay to user 2 is given by @xmath60 , where @xmath61 and the amount of information in buffer @xmath8 decreases to @xmath62 . @xmath63 : the relay broadcasts to both user 1 and user 2 the information received from user 2 and user 1 , respectively . specifically , the relay extracts the information intended for user 2 from buffer @xmath8 and the information intended for user 1 from buffer @xmath9 . then , based on the scheme in @xcite , it constructs a superimposed codeword which contains the information from both users and broadcasts it to both users . thus , in the @xmath16-th time slot , the maximum rates from the relay to users 1 and 2 are given by @xmath64 and @xmath65 , respectively . therefore , the amounts of information in buffers @xmath8 and @xmath9 decrease to @xmath62 and @xmath58 , respectively . our aim is to develop an optimal mode selection and power allocation policy which in each time slot selects one of the six transmission modes , @xmath28 , and allocates the optimal powers to the transmitting nodes of the selected mode such that the average sum rate of both users is maximized . to this end , we introduce six binary variables , @xmath66 , where @xmath67 indicates whether or not transmission mode @xmath68 is selected in the @xmath16-th time slot . in particular , @xmath69 if mode @xmath68 is selected and @xmath70 if it is not selected in the @xmath16-th time slot . furthermore , since in each time slot only one of the six transmission modes can be selected , only one of the mode selection variables is equal to one and the others are zero , i.e. , @xmath71 holds . in the proposed framework , we assume that all nodes have full knowledge of the csi of both links . thus , based on the csi and the proposed protocol , cf . theorem [ adaptprot ] , each node is able to individually decide which transmission mode is selected and adapt its transmission strategy accordingly . in this section , we first investigate the achievable average sum rate of the network . then , we formulate a maximization problem whose solution is the sum rate maximizing protocol . we assume that user 1 and user 2 always have enough information to send in all time slots and that the number of time slots , @xmath72 , satisfies @xmath73 . therefore , using @xmath67 , the user 1-to - relay , user 2-to - relay , relay - to - user 1 , and relay - to - user 2 average transmission rates , denoted by @xmath74 , @xmath75 , @xmath76 , and @xmath77 , respectively , are obtained as lll[ratreg123 ] & & |r_1r = _ i = 1^n + & & |r_2r = _ i = 1^n + & & |r_r1 = _ i = 1^n \{c_r1(i),q_2(i-1 ) } + & & |r_r2 = _ i = 1^n \{c_r2(i),q_1(i-1)}. the average rate from user 1 to user 2 is the average rate that user 2 receives from the relay , i.e. , @xmath77 . similarly , the average rate from user 2 to user 1 is the average rate that user 1 receives from the relay , i.e. , @xmath76 . in the following theorem , we introduce a useful condition for the queues in the buffers of the relay leading to the optimal mode selection and power allocation policy . [ queue ] the maximum average sum rate , @xmath78 , for the considered bidirectional relay network is obtained when the queues in the buffers @xmath8 and @xmath9 at the relay are at the edge of non - absorbtion . more precisely , the following conditions must hold for the maximum sum rate lll[ratregapp456-buffer ] |r_1r=|r_r2 = _ i = 1^n c_r2(i ) + where @xmath74 and @xmath75 are given by ( [ ratreg123]a ) and ( [ ratreg123]b ) , respectively . please refer to ( * ? ? ? * appendix a ) . using this theorem , in the following , we derive the optimal transmission mode selection and power allocation policy . the available degrees of freedom in the considered network in each time slot are the mode selection variables , the transmit powers of the nodes , and the time sharing variable for multiple access . herein , we formulate an optimization problem which gives the optimal values of @xmath67 , @xmath25 , and @xmath48 , for @xmath79 , @xmath80 , and @xmath81 , such that the average sum rate of the users is maximized . the optimization problem is as follows cll[adaptprob ] & |r_1r+|r_2r + & : |r_1r=|r_r2 + & : |r_2r=|r_r1 + & : |p_1+|p_2+|p_r p_t + & : _ k = 1 ^ 6 q_k ( i ) = 1 , i + & : q_k(i ) [ 1-q_k(i ) ] = 0 , i , k + & : p_j(i)0 , i , j + & : 0t(i)1 , i where @xmath82 is the total average power constraint of the nodes and @xmath83 , and @xmath84 denote the average powers consumed by user 1 , user 2 , and the relay , respectively , and are given by lll |p_1= _ i = 1^n ( q_1(i)+q_3(i))p_1 ( i ) + in the optimization problem given in ( [ adaptprob ] ) , constraints @xmath85 and @xmath86 are the conditions for sum rate maximization introduced in theorem [ queue ] . constraints @xmath87 and @xmath88 are the average total transmit power constraint and the power non - negativity constraint , respectively . moreover , constraints @xmath89 and @xmath90 guarantee that only one of the transmission modes is selected in each time slot , and constraint @xmath91 specifies the acceptable interval for the time sharing variable @xmath48 . furthermore , we maximize @xmath92 since , according to theorem 1 ( and constraints @xmath85 and @xmath86 ) , @xmath93 and @xmath94 hold . in the following theorem , we introduce a protocol which achieves the maximum sum rate . [ adaptprot ] assuming @xmath73 , the optimal mode selection and power allocation policy which maximizes the sum rate of the considered three - node half - duplex bidirectional relay network with awgn and block fading is given by lll[seleccrit ] q_k^*(i)= 1 , & k^*= \{_k(i ) } + 0 , & where @xmath95 is referred to as _ selection metric _ and is given by lll[selecmet ] _ 1(i ) = ( 1-_1)c_1r(i ) - p_1(i ) |_p_1(i)=p_1^_1(i ) + _ 2(i ) = ( 1-_2)c_2r(i ) - p_2(i)|_p_2(i)=p_2^_2(i ) + _ 3(i ) = ( 1-_1)c_12r(i)+(1-_2)c_21r(i ) + - ( p_1(i)+p_2(i))|_p_2(i)=p_2^_3(i)^p_1(i)=p_1^_3(i ) + _ 6(i ) = _ 1 c_r2(i)+_2 c_r1(i ) - p_r(i)|_p_r(i)=p_r^_6(i ) where @xmath96 denotes the optimal transmit power of node @xmath30 for transmission mode @xmath68 in the @xmath16-th time slot and is given by lll[optpower ] p_1^_1 ( i ) = ^+ + p_2^_2 ( i ) = ^+ + p_1^_3 ( i ) = ^+ , _ 1_2 + ^+ , + p_2^_3 ( i ) = ^+ , _ 1 _ 2 + ^+ , + p_r^_6 ( i ) = ^+ where @xmath97^+=\max\{x,0\}$ ] , @xmath98 , and @xmath99 . the thresholds @xmath100 and @xmath101 are chosen such that constraints @xmath85 and @xmath86 in ( [ adaptprob ] ) hold and threshold @xmath102 is chosen such that the total average transmit power satisfies @xmath87 in ( [ adaptprob ] ) . the optimal value of @xmath48 in @xmath52 and @xmath53 is given by lll t^*(i ) = 0 , & _ 1 _ 2 + 1 , & _ 1 < _ 2 please refer to appendix [ appkkt ] . we note that the optimal solution utilizes neither modes @xmath13 and @xmath14 nor time sharing for any channel statistics and channel realizations . the mode selection metric @xmath95 introduced in ( [ selecmet ] ) has two parts . the first part is the instantaneous capacity of mode @xmath68 , and the second part is the allocated power with negative sign . the capacity and the power terms are linked via thresholds @xmath100 and/or @xmath101 and @xmath102 . we note that thresholds @xmath100 , @xmath101 , and @xmath102 depend only on the long term statistics of the channels . hence , these thresholds can be obtained offline and used as long as the channel statistics remain unchanged . to find the optimal values for the thresholds @xmath100 , @xmath101 , and @xmath102 , we need a three - dimensional search , where @xmath103 and @xmath104 . adaptive mode selection for bidirectional relay networks under the assumption that the powers of the nodes are fixed is considered in @xcite . based on the average and instantaneous qualities of the links , all of the six possible transmission modes are selected in the protocol in @xcite . however , in the proposed protocol , modes @xmath13 and @xmath14 are not selected at all . moreover , the protocol in @xcite utilizes a coin flip for implementation . therefore , a central node must decide which transmission mode is selected in the next time slot . however , in the proposed protocol , all nodes can find the optimal mode and powers based on the full csi . in this section , we evaluate the average sum rate achievable with the proposed protocol in the considered bidirectional relay network in rayleigh fading . thus , channel gains @xmath19 and @xmath20 follow exponential distributions with means @xmath105 and @xmath106 , respectively . all of the presented results were obtained for @xmath107 and @xmath108 time slots . in fig . [ mixed ] , we illustrate the maximum achievable sum rate obtained with the proposed protocol as a function of the total average transmit power @xmath82 . in this figure , to have a better resolution for the sum rate at low and high @xmath82 , we show the sum rate for both log scale and linear scale @xmath109-axes , respectively . the lines without markers in fig . [ mixed ] represent the achieved sum rates with the proposed protocol for @xmath110 . we observe that as the quality of the user 1-to - relay link increases ( i.e. , @xmath105 increases ) , the sum rate increases too . however , for large @xmath105 , the bottleneck link is the relay - to - user 2 link , and since it is fixed , the sum rate saturates . [ c][t][0.75]@xmath111 [ c][b][0.75]@xmath112 [ l][c][0.45]@xmath113 [ l][c][0.45]@xmath114 [ l][c][0.45]@xmath115 [ l][c][0.45]@xmath116}\,(\omega_1=1)$ ] [ l][c][0.45]@xmath117}\,(\omega_1=1)$ ] [ l][c][0.45]@xmath118 [ l][c][0.45]@xmath119 [ c][c][0.6]@xmath120 [ c][c][0.6]@xmath121 [ c][c][0.6]@xmath122 [ c][c][0.6]@xmath123 [ c][c][0.6]@xmath124 [ c][c][0.6]@xmath125 [ c][c][0.6]@xmath126 [ c][c][0.6]@xmath127 [ c][c][0.6]@xmath128 [ c][c][0.6]@xmath129 [ c][c][0.6]@xmath130 [ c][c][0.6]@xmath131 [ c][c][0.6]@xmath132 [ c][c][0.6]@xmath133 [ c][c][0.30]@xmath124 [ c][c][0.30]@xmath134 [ c][c][0.30]@xmath135 [ c][c][0.30]@xmath136 [ c][c][0.30]@xmath137 [ c][c][0.30]@xmath138 [ c][c][0.30]@xmath139 [ c][c][0.30]@xmath140 [ c][c][0.30]@xmath141 [ c][c][0.30]@xmath142 [ c][c][0.30]@xmath125 [ c][c][0.30]@xmath120 [ c][c][0.30]@xmath121 [ c][c][0.30]@xmath122 [ c][c][0.30]@xmath123 [ c][c][0.30]@xmath124 [ c][c][0.30]@xmath125 [ c][c][0.30]@xmath126 [ c][c][0.30]@xmath127 [ c][c][0.30]@xmath128 for different protocols.,title="fig:",width=3 ] as performance benchmarks , we consider in fig . [ mixed ] the sum rates of the tdbc protocol with and without power allocation @xcite and the buffer - aided protocols presented in @xcite and @xcite , respectively . for clarity , for the benchmark schemes , we only show the sum rates for @xmath143 . for the tdbc protocol without power allocation and the protocol in @xcite , all nodes transmit with equal powers , i.e. , @xmath144 . for the buffer - aided protocol in @xcite , we adopt @xmath145 and @xmath146 is chosen such that the average total power consumed by all nodes is @xmath82 . we note that since @xmath143 and @xmath147 , the protocol in @xcite only selects modes @xmath12 and @xmath15 . moreover , since @xmath143 , we obtain @xmath148 in the proposed protocol . thus , considering the optimal power allocation in ( [ optpower]c ) and ( [ optpower]d ) , we obtain that either @xmath149 or @xmath150 is zero . therefore , for the chosen parameters , only modes @xmath151 , and @xmath15 are selected , i.e. , the same modes as used in @xcite , the protocol in @xcite is optimal for given fixed node transmit powers . ] . hence , we can see how much gain we obtain due to the adaptive power allocation by comparing our result with the results for the protocol in @xcite . on the other hand , the gain due to the adaptive mode selection can be evaluated by comparing the sum rate of the proposed protocol with the result for the tdbc protocol with power allocation . from the comparison in fig . [ mixed ] , we observe that for high @xmath82 , a considerable gain is obtained by the protocols with adaptive mode selection ( ours and that in @xcite ) compared to the tdbc protocol which does not apply adaptive mode selection ( around @xmath152 db gain ) . however , for high @xmath82 , power allocation is less beneficial , and therefore , the sum rates obtained with the proposed protocol and that in @xcite converge . on the other hand , for low @xmath82 , optimal power allocation is crucial and , therefore , a considerable gain is achieved by the protocols with adaptive power allocation ( ours and tdbc with power allocation ) . we have derived the maximum sum rate of the three - node half - duplex bidirectional buffer - aided relay network with fading links . the protocol which achieves the maximum sum rate jointly optimizes the selection of the transmission mode and the transmit powers of the nodes . the proposed optimal mode selection and power allocation protocol requires the instantaneous csi of the involved links in each time slot and their long - term statistics . simulation results confirmed that the proposed selection policy outperforms existing protocols in terms of average sum rate . in this appendix , we solve the optimization problem given in ( [ adaptprob ] ) . we first relax the binary condition for @xmath67 , i.e. , @xmath153=0 $ ] , to @xmath154 , and later in appendix [ appbinrelax ] , we prove that the binary relaxation does not affect the maximum average sum rate . in the following , we investigate the karush - kuhn - tucker ( kkt ) necessary conditions @xcite for the relaxed optimization problem and show that the necessary conditions result in a unique sum rate and thus the solution is optimal . to simplify the usage of the kkt conditions , we formulate a minimization problem equivalent to the relaxed maximization problem in ( [ adaptprob ] ) as follows cll[adaptprobmin ] & -(|r_1r+|r_2r ) + & : |r_1r-|r_r2=0 + & : |r_2r-|r_r1=0 + & : |p_1+|p_2+|p_r - p_t0 + & : _ k = 1 ^ 6 q_k ( i ) - 1 = 0 , i + & : q_k(i)-1 0 , i , k + & : -q_k(i ) 0 , i , k + & : -p_j(i ) 0 , i , k + & : t(i)-1 0 , i + & : -t(i ) 0 , i. the lagrangian function for the above optimization problem is provided in ( [ kkt function ] ) at the top of the next page where @xmath155 , and @xmath156 are the lagrange multipliers corresponding to constraints @xmath157 , and @xmath158 , respectively . the kkt conditions include the following : l[kkt function ] = + - ( |r_1r+|r_2r ) + _ 1(|r_1r-|r_r2 ) + _ 2(|r_2r-|r_r1 ) + ( |p_1+|p_2+|p_r - p_t ) + + _ i = 1^n ( i ) ( _ k = 1 ^ 6 q_k ( i ) - 1 ) + _ i = 1^n _ k = 1 ^ 6 _ k ( i ) ( q_k ( i ) - 1 ) - _ i = 1^n _ k = 1 ^ 6 _ k ( i ) q_k ( i ) + -_i = 1^n + _ i = 1^n _ 1(i ) ( t(i)-1 ) - _ i = 1^n _ 0(i ) t(i ) * 1 ) * stationary condition : the differentiation of the lagrangian function with respect to the primal variables , @xmath159 , and @xmath160 , is zero for the optimal solution , i.e. , cccl[stationary condition ] & = & 0 , & i , k + & = & 0 , & i , j + & = & 0 , & i. * 2 ) * primal feasibility condition : the optimal solution has to satisfy the constraints of the primal problem in ( [ adaptprobmin ] ) . * 3 ) * dual feasibility condition : the lagrange multipliers for the inequality constraints have to be non - negative , i.e. , lll[dual feasibility condition ] _ k(i)0 , & i , k + _ k(i)0,&i , k + 0 , & + _ j(i ) 0 , & i , j + _ l(i ) 0 , & i , l . * 4 ) * complementary slackness : if an inequality is inactive , i.e. , the optimal solution is in the interior of the corresponding set , the corresponding lagrange multiplier is zero . thus , we obtain lll[complementary slackness ] _ k ( i ) ( q_k ( i ) - 1 ) = 0,&i , k + _ k ( i ) q_k ( i ) = 0 , & i , k + ( |p_1+|p_2+|p_r - p_t ) = 0 & + _ j(i ) p_j(i ) = 0 , & i , j + _ 1(i ) ( t(i)-1 ) = 0 , & i + _ 0(i ) t(i ) = 0 , & i. a common approach to find a set of primal variables , i.e. , @xmath161 and lagrange multipliers , i.e. , @xmath162 , which satisfy the kkt conditions is to start with the complementary slackness conditions and see if the inequalities are active or not . combining these results with the primal feasibility and dual feasibility conditions , we obtain various possibilities . then , from these possibilities , we obtain one or more candidate solutions from the stationary conditions and the optimal solution is surely one of these candidates . in the following subsections , with this approach , we find the optimal values of @xmath163 and @xmath164 . in order to determine the optimal selection policy , @xmath165 , we must calculate the derivatives in ( [ stationary condition]a ) . this leads to lll[stationary mode ] = - ( 1-_1)c_1r(i)+(i)+_1(i)-_1(i ) + p_1(i)=0 + = -(1-_2)c_2r(i)+(i)+_2(i)-_2(i ) + p_2(i)=0 + = -[(1-_1)c_12r(i)+(1-_2)c_21r(i)]+(i ) + _ 3(i)-_3(i)+(p_1(i)+p_2(i))=0 + = - _ 2 c_r1(i)+(i)+_4(i)-_4(i)+p_r(i)=0 + = -_1 c_r2(i)+(i)+_5(i)-_5(i)+p_r(i)=0 + = -[_1 c_r2(i)+_2 c_r1(i)]+(i)+_6(i)-_6(i)+p_r(i)=0 . without loss of generality , we first obtain the necessary condition for @xmath166 and then generalize the result to @xmath167 . if @xmath168 , from constraint @xmath89 in ( [ adaptprobmin ] ) , the other selection variables are zero , i.e. , @xmath169 . furthermore , from ( [ complementary slackness ] ) , we obtain @xmath170 and @xmath171 . by substituting these values into ( [ stationary mode ] ) , we obtain lll[met ] ( i)+_1(i ) = ( 1-_1)c_1r(i ) -p_1(i ) _ 1(i ) + ( i)-_2(i ) = ( 1-_2)c_2r(i ) -p_2(i ) _ 2(i ) + ( i)-_3(i ) = ( 1-_1)c_12r(i)+(1-_2)c_21r(i ) -(p_1(i)+p_2(i ) ) _ 3(i ) + ( i)-_4(i ) = _ 2 c_r1(i ) -p_r(i ) _ 4(i ) + ( i)-_5(i ) = _ 1 c_r2(i ) -p_r(i ) _ 5(i ) + ( i)-_6(i ) = _ 1 c_r2(i)+_2 c_r1(i ) -p_r(i ) _ 6(i ) , where @xmath95 is referred to as selection metric . by subtracting ( [ met]a ) from the rest of the equations in ( [ met ] ) , we obtain rcl[eq_2_1 ] _ 1(i ) - _ k(i ) = _ 1(i)+_k(i ) , k=2,3,4,5,6 . from the dual feasibility conditions given in ( [ dual feasibility condition]a ) and ( [ dual feasibility condition]b ) , we have @xmath172 . by inserting @xmath172 in ( [ eq_2_1 ] ) , we obtain the necessary condition for @xmath166 as lll _ 1(i ) \ { _ 2(i ) , _ 3(i ) , _ 4(i ) , _ 5(i ) , _ 6(i ) } . repeating the same procedure for @xmath167 , we obtain a necessary condition for selecting transmission mode @xmath173 in the @xmath16-th time slot as follows lll[optmet ] _ k^*(i ) \{_k(i ) } , where the lagrange multipliers @xmath174 , and @xmath102 are chosen such that @xmath175 , and @xmath87 in ( [ adaptprobmin ] ) hold and the optimal value of @xmath48 in @xmath52 and @xmath53 is obtained in the next subsection . we note that if the selection metrics are not equal in the @xmath16-th time slot , only one of the modes satisfies ( [ optmet ] ) . therefore , the necessary conditions for the mode selection in ( [ optmet ] ) is sufficient . moreover , in appendix [ appbinrelax ] , we prove that the probability that two selection metrics are equal is zero due to the randomness of the time- continuous channel gains . therefore , the necessary condition for selecting transmission mode @xmath68 in ( [ optmet ] ) is in fact sufficient and is the optimal selection policy . in order to determine the optimal @xmath25 , we have to calculate the derivatives in ( [ stationary condition]b ) . this leads to [ stationary power ] @xmath177 \nonumber \\ & + \gamma \frac{1}{n } ( q_1(i)+q_3(i ) ) - \nu_1(i ) = 0 \quad \tag{\stepcounter{equation}\theequation}\\ \frac{\partial\mathcal{l}}{\partial p_2(i ) } { \hspace{-0.7mm}=\hspace{-0.7mm}}&-\frac{1}{n\mathrm{ln}2 } \big [ \big \ { ( 1{\hspace{-0.7mm}-\hspace{-0.7mm}}\mu_2)q_2(i ) { \hspace{-0.7mm}+\hspace{-0.7mm}}(1{\hspace{-0.7mm}-\hspace{-0.7mm}}t(i))(\mu_1{\hspace{-0.7mm}-\hspace{-0.7mm}}\mu_2)q_3(i ) \big\ } \nonumber \\ & \times\frac{s_2(i)}{1{\hspace{-0.7mm}+\hspace{-0.7mm}}p_2(i)s_2(i ) } { \hspace{-0.7mm}+\hspace{-0.7mm}}\big\{t(i)(\mu_1{\hspace{-0.7mm}-\hspace{-0.7mm}}\mu_2){\hspace{-0.7mm}+\hspace{-0.7mm}}1{\hspace{-0.7mm}-\hspace{-0.7mm}}\mu_1 \big\}q_3(i ) \nonumber \\ & \times\frac{s_2(i)}{1+p_1(i)s_1(i)+p_2(i)s_2(i ) } \big]\nonumber \\ & + \gamma \frac{1}{n } ( q_2(i)+q_3(i ) ) - \nu_2(i)=0 \quad\tag{\stepcounter{equation}\theequation}\\ \frac{\partial\mathcal{l}}{\partial p_r(i ) } = & -\frac{1}{n\mathrm{ln}2 } \big [ \mu_2\left(q_4(i)+q_6(i)\right)\frac{s_1(i)}{1+p_r(i)s_1(i)}\nonumber \\ & + \mu_1\left(q_5(i)+q_6(i)\right)\frac{s_2(i)}{1+p_r(i)s_2(i ) } \big]\nonumber \\ & + \gamma \frac{1}{n } ( q_4(i)+q_5(i)+q_6(i ) ) - \nu_r(i)=0 \tag{\stepcounter{equation}\theequation}\end{aligned}\ ] ] the above conditions allow the derivation of the optimal powers for each transmission mode in each time slot . for instance , in order to determine the transmit power of user 1 in transmission mode @xmath10 , we assume @xmath166 . from constraint @xmath89 in ( [ adaptprobmin ] ) , we obtained that the other selection variables are zero and therefore @xmath178 . moreover , if @xmath10 is selected then @xmath179 and thus from ( [ complementary slackness]d ) , we obtain @xmath180 . substituting these results in ( [ stationary power]a ) , we obtain lll [ eq_11 ] p_1^_1 ( i ) = ^+ , where @xmath97^+=\max\{0,x\}$ ] . in a similar manner , we obtain the optimal powers for user 2 in mode @xmath11 , and the optimal powers of the relay in modes @xmath13 and @xmath14 as follows : lll[p245 ] p_2^_2 ( i ) = ^+ + p_r^_4 ( i ) = ^+ + p_r^_5 ( i ) = ^+ in order to obtain the optimal powers of user 1 and user 2 in mode @xmath12 , we assume @xmath181 . from @xmath89 in ( [ adaptprobmin ] ) , we obtain that the other selection variables are zero , and therefore @xmath182 and @xmath183 . we note that if one of the powers of user 2 and user 1 is zero mode @xmath12 is identical to modes @xmath10 and @xmath11 , respectively , and for that case the optimal powers are already given by ( [ eq_11 ] ) and ( [ p245]a ) , respectively . for the case when @xmath179 and @xmath184 , we obtain @xmath180 and @xmath185 from ( [ complementary slackness]d ) . furthermore , for @xmath181 , we will show in appendix [ appkkt].c that @xmath48 can only take the boundary values , i.e. , zero or one , and can not be in between . hence , if we assume @xmath186 , from ( [ stationary power]a ) and ( [ stationary power]b ) , we obtain lll[powerm3 ] & - + = 0 + & - + = 0 by substituting ( [ powerm3]a ) in ( [ powerm3]b ) , we obtain @xmath187 and then we can derive @xmath188 from ( [ powerm3]a ) . this leads to lll [ pm3-t0 ] p_1^_3 ( i ) = p_1^_1 ( i ) , s_2 + ^+ , + p_2^_3 ( i ) = p_2^_2 ( i ) , s_2 s_1 + ^+ , similarly , if we assume @xmath189 , we obtain lll [ pm3-t1 ] p_1^_3 ( i ) = p_1^_1 ( i ) , s_2 s_1 + ^+ , + p_2^_3 ( i ) = p_2^_2 ( i ) , s_2 + ^+ , we note that when @xmath190 , we obtain @xmath191 which means that mode @xmath12 is identical to mode @xmath10 . thus , there is no difference between both modes so we select @xmath10 . in figs [ figsregion ] a ) and [ figsregion ] b ) , the comparison of @xmath192 , and @xmath193 is illustrated in the space of @xmath194 . moreover , the shaded area represents the region in which the powers of users 1 and 2 are zero for @xmath151 , and @xmath12 . [ c][c][0.6]@xmath19 [ c][c][0.6]@xmath20 [ c][c][0.6]@xmath195 [ c][c][0.6]@xmath196 [ c][c][0.6]@xmath195 [ c][c][0.6]@xmath196 [ c][c][0.6]@xmath197 [ c][c][0.6]@xmath198 [ c][c][0.75]@xmath199 [ c][b][0.7]@xmath200 [ c][c][0.7]@xmath201 [ c][c][0.75]@xmath202 [ c][c][0.75]@xmath203 [ c][c][0.75]@xmath204 [ c][c][0.65]@xmath200 [ c][t][0.65]@xmath201 [ c][c][0.75]@xmath202 [ c][c][0.75]@xmath205 [ c][c][0.75]@xmath206 [ c][c][0.75]a ) @xmath207 [ c][c][0.75]b ) @xmath208 [ c][c][0.75]c ) for mode @xmath15 , we assume @xmath209 . from constraint @xmath89 in ( [ adaptprobmin ] ) , we obtain that the other selection variables are zero and therefore @xmath210 and @xmath211 . moreover , if @xmath209 then @xmath212 and thus from ( [ complementary slackness]d ) , we obtain @xmath213 . using these results in ( [ stationary power]c ) , we obtain lll[powerm6 ] _ 2 + _ 1 = 2 the above equation is a quadratic equation and has two solutions for @xmath7 . however , since we have @xmath214 , we can conclude that the left hand side of ( [ powerm6 ] ) is monotonically decreasing in @xmath7 . thus , if @xmath215 , we have a unique positive solution for @xmath7 which is the maximum of the two roots of ( [ powerm6 ] ) . thus , we obtain lll[pm6 ] p_r^_6 ( i ) = ^+ , where @xmath216 , and @xmath99 . in fig . [ figsregion ] c ) , the comparison between selection metrics @xmath217 and @xmath218 is illustrated in the space of @xmath194 . we note that @xmath219 and @xmath220 hold and the inequalities hold with equality if @xmath221 and @xmath222 , respectively , which happen with zero probability for time - continuous fading . to prove @xmath219 , from ( [ met ] ) , we obtain rcl _ 6(i ) & = & _ 1c_r2(i ) + _ 2c_r1(i ) -p_r(i ) |_p_r(i)=p_r^_6(i ) + & & _ 1c_r2(i ) + _ 2c_r1(i ) -p_r(i)|_p_r(i)=p_r^_4(i ) + & & _ 2c_r1(i ) -p_r(i ) |_p_r(i)=p_r^_4(i ) = _ 4(i ) , where @xmath223 follows from the fact that @xmath224 maximizes @xmath218 and @xmath225 follows from @xmath226 . the two inequalities @xmath223 and @xmath225 hold with equality only if @xmath221 which happens with zero probability in time - continuous fading or if @xmath227 . however , in appendix [ appmuregion ] , @xmath227 is shown to lead to a contradiction . therefore , the optimal policy does not select @xmath13 and @xmath14 and selects only modes @xmath228 , and @xmath15 . to find the optimal @xmath48 , we assume @xmath230 and calculate the stationary condition in ( [ stationary condition]c ) . this leads to lll [ stationary t ] = & -(_1-_2 ) + & + _ 1(i)-_0(i)=0 now , we investigate the following possible cases for @xmath229 : * case 1 : * if @xmath231 then from ( [ complementary slackness]e ) and ( [ complementary slackness]f ) , we have @xmath232 . therefore , from ( [ stationary t ] ) and @xmath233 , we obtain @xmath148 . then , from ( [ stationary power]a ) and ( [ stationary power]b ) , we obtain rcl[contradict ] - ( 1-_1 ) + = 0 + - ( 1-_1 ) + = 0 in appendix [ appmuregion ] , we show that @xmath234 , therefore , the above conditions can be satisfied simultaneously only if @xmath235 , which , considering the randomness of the time - continuous channel gains , occurs with zero probability . hence , the optimal @xmath48 takes the boundary values , i.e. , zero or one , and not values in between . * case 2 : * if @xmath236 , then from ( [ complementary slackness]e ) , we obtain @xmath237 and from ( [ dual feasibility condition]e ) , we obtain @xmath238 . combining these results into ( [ stationary t ] ) , the necessary condition for @xmath186 is obtained as @xmath239 . * case 3 : * if @xmath240 , then from ( [ complementary slackness]f ) , we obtain @xmath241 and from ( [ dual feasibility condition]e ) , we obtain @xmath242 . combining these results into ( [ stationary t ] ) , the necessary condition for @xmath189 is obtained as @xmath243 . we note that if @xmath148 , we obtain either @xmath244 or @xmath245 . therefore , mode @xmath12 is not selected and the value of @xmath48 does not affect the sum rate . moreover , from the selection metrics in ( [ met ] ) , we can conclude that @xmath207 and @xmath208 correspond to @xmath246 and @xmath247 , respectively . therefore , the optimal value of @xmath48 is given by lll t^*(i ) = 0 , & _ 1 _ 2 + 1 , & _ 1 < _ 2 now , the optimal values of @xmath159 , and @xmath160 are derived based on which theorem [ adaptprot ] can be constructed . this completes the proof . in this appendix , we prove that the optimal solution of the problem with the relaxed constraint , @xmath248 , selects the boundary values of @xmath67 , i.e. , zero or one . therefore , the binary relaxation does not change the solution of the problem . if one of the @xmath249 , adopts a non - binary value in the optimal solution , then in order to satisfy constraint @xmath89 in ( [ adaptprob ] ) , there has to be at least one other non - binary selection variable in that time slot . assuming that the mode indices of the non - binary selection variables are @xmath250 and @xmath251 in the @xmath16-th time slot , we obtain @xmath252 from ( [ complementary slackness]a ) , and @xmath253 and @xmath254 from ( [ complementary slackness]b ) . then , by substituting these values into ( [ stationary mode ] ) , we obtain lll[binrelax ] ( i ) = _ k(i ) + ( i)= _ k(i ) + ( i)-_k(i ) = _ k(i ) , kk , k . from ( [ binrelax]a ) and ( [ binrelax]b ) , we obtain @xmath255 and by subtracting ( [ binrelax]a ) and ( [ binrelax]b ) from ( [ binrelax]c ) , we obtain rcl _ k(i ) - _ k(i ) & = & _ k(i ) , kk , k + _ k(i ) - _ k(i ) & = & _ k(i ) , kk , k . from the dual feasibility condition given in ( [ dual feasibility condition]b ) , we have @xmath256 which leads to @xmath257 . however , as a result of the randomness of the time - continuous channel gains , @xmath258 holds for some transmission modes @xmath259 and @xmath260 , if and only if we obtain @xmath261 or @xmath262 which leads to a contradiction as shown in appendix [ appmuregion ] . this completes the proof . in this appendix , we find the intervals which contain the optimal value of @xmath100 and @xmath101 . we note that for different values of @xmath100 and @xmath101 , some of the optimal powers derived in ( [ eq_11 ] ) , ( [ p245 ] ) , ( [ pm3-t0 ] ) , ( [ pm3-t1 ] ) , and ( [ pm6 ] ) are zero for all channel realizations . for example , if @xmath263 , we obtain @xmath264 from ( [ eq_11 ] ) . [ figmuregion ] illustrates the set of modes that can take positive powers with non - zero probability in the space of ( @xmath174 ) . in the following , we show that any values of @xmath100 and @xmath101 except @xmath265 and @xmath266 can not lead to the optimal sum rate or violate constraints @xmath85 or @xmath86 in ( [ adaptprobmin ] ) . * case 1 : * sets @xmath267 and @xmath268 lead to selection of either the transmission from the users to the relay or the transmission from the relay to the users , respectively , for all time slots . this leads to violation of constraints @xmath85 and @xmath86 in ( [ adaptprobmin ] ) and thus the optimal values of @xmath100 and @xmath101 are not in this region . * case 2 : * in set @xmath269 , both modes @xmath13 and @xmath15 need the transmission from user 2 to the relay which can not be realized in this set . thus , this set leads to violation of constraint @xmath86 in ( [ adaptprobmin ] ) . similarly , in set @xmath270 , both modes @xmath14 and @xmath15 require the transmission from user 1 to the relay which can not be selected in this set . thus , this region of @xmath100 and @xmath101 leads to violation of constraint @xmath85 in ( [ adaptprobmin ] ) . * case 3 : * in set @xmath271 , there is no transmission from user 2 to the relay . therefore , the optimal values of @xmath100 and @xmath101 have to guarantee that modes @xmath13 and @xmath15 are not selected for any channel realization . however , from ( [ met ] ) , we obtain where @xmath223 follows from the fact that @xmath224 maximizes @xmath218 and @xmath225 follows from @xmath272 . the two inequalities @xmath223 and @xmath225 hold with equality only if @xmath222 which happens with zero probability for time - continuous fading , or @xmath273 which is not included in this region . therefore , mode @xmath15 is selected in this region which leads to violation of constraint @xmath86 in ( [ adaptprobmin ] ) . a similar statement is true for set @xmath274 . thus , the optimal values of @xmath100 and @xmath101 can not be in these two regions . where inequality @xmath223 comes from the fact that @xmath276 and the equality holds when @xmath221 which happens with zero probability , or @xmath227 . inequality @xmath225 holds since @xmath277 maximizes @xmath278 and holds with equality only if @xmath279 and consequently @xmath227 . if @xmath280 , mode @xmath15 is not selected and there is no transmission from the relay to user 2 . therefore , the optimal values of @xmath100 and @xmath101 have to guarantee that modes @xmath10 and @xmath12 are not selected for any channel realization . thus , we obtain @xmath263 which is not contained in this region . if @xmath227 , from ( [ met ] ) , we obtain where both inequalities @xmath223 and @xmath225 hold with equality only if @xmath281 . if @xmath282 , modes @xmath13 and @xmath15 are not selected thus , there is no transmission from the relay to the users which leads to violation of @xmath85 and @xmath86 in ( [ adaptprobmin ] ) . if @xmath281 , we obtain @xmath283 , thus mode @xmath11 can not be selected and either @xmath244 or @xmath284 , thus mode @xmath12 can not be selected either . since both modes @xmath13 and @xmath15 require the transmission from user 2 to the relay , and both modes @xmath11 and @xmath12 are not selected , constraint @xmath86 in ( [ adaptprobmin ] ) is violated and @xmath227 and @xmath281 can not be optimal . a similar statement is true for set @xmath285 . therefore , the optimal values of @xmath100 and @xmath101 are not in this region . s. j. kim , n. devroye , p. mitran , and v. tarokh , `` achievable rate regions and performance comparison of half duplex bi - directional relaying protocols , '' _ ieee trans . inf . theory _ , 57 , no . 6405 6418 , oct . 2011 . n. zlatanov and r. schober , `` capacity of the state - dependent half - duplex relay channel without source - destination link , '' _ submitted ieee transactions on information theory _ , 2013 . [ online ] . available : http://arxiv.org/abs/1302.3777 v. jamali , n. zlatanov , a. ikhlef , and r. schober , `` adaptive mode selection in bidirectional buffer - aided relay networks with fixed transmit powers , '' _ submitted in part to eusipco13 _ , 2013 . [ online ] . available : http://arxiv.org/abs/1303.3732

in this paper , we consider the problem of sum rate maximization in a bidirectional relay network with fading . hereby , user 1 and user 2 communicate with each other only through a relay , i.e. , a direct link between user 1 and user 2 is not present . in this network , there exist six possible transmission modes : four point - to - point modes ( user 1-to - relay , user 2-to - relay , relay - to - user 1 , relay - to - user 2 ) , a multiple access mode ( both users to the relay ) , and a broadcast mode ( the relay to both users ) . most existing protocols assume a fixed schedule of using a subset of the aforementioned transmission modes , as a result , the sum rate is limited by the capacity of the weakest link associated with the relay in each time slot . motivated by this limitation , we develop a protocol which is not restricted to adhere to a predefined schedule for using the transmission modes . therefore , all transmission modes of the bidirectional relay network can be used adaptively based on the instantaneous channel state information ( csi ) of the involved links . to this end , the relay has to be equipped with two buffers for the storage of the information received from users 1 and 2 , respectively . for the considered network , given a total average power budget for all nodes , we jointly optimize the transmission mode selection and power allocation based on the instantaneous csi in each time slot for sum rate maximization . simulation results show that the proposed protocol outperforms existing protocols for all signal - to - noise ratios ( snrs ) . specifically , we obtain a considerable gain at low snrs due to the adaptive power allocation and at high snrs due to the adaptive mode selection .

arxiv

in standard cosmology , the acoustic oscillations imprinted in the matter power spectrum at recombination have a length scale that can be accurately calculated based on measurements of the cmb anisotropy power spectrum @xcite . it should then be possible to measure this `` standard ruler '' scale at low redshifts , for example in large galaxy redshift surveys , and thereby constrain the matter and energy content of the universe @xcite . however , if the cmb measurements were misled by some new physics , e.g. a new undetected relativistic particle , then the misinterpretation could potentially spread to the low - redshift application and bias the inferences . here , we show that the interpretation of the low - redshift acoustic oscillations are robust if the cmb correctly tells us the baryon - to - photon ratio and the epoch of matter - radiation equality . these quantities are robustly measured in the cmb . the actual densities of matter and radiation drop out of the calculation ; only their ratio matters . the result is that even if the physical matter density @xmath0 is misinterpreted from the cmb due to undetected relativistic components , the inferences for dark energy from the combined cmb and low - redshift survey data sets are unchanged . knowledge of actual densities , e.g. @xmath1 , translates into improved constraints on the hubble constant , @xmath2 . the acoustic peak method depends upon measuring the sound horizon , which is the comoving distance that a sound wave can travel between the end of inflation and the epoch of recombination @xcite . nearly all of this distance is accumulated just prior to the epoch of recombination at @xmath3 . the sound horizon integral depends only on the hubble parameter @xmath4 and the sound speed @xmath5 in the baryon - photon plasma . if we assume dark energy is sub - dominant at @xmath6 , then @xmath7 where @xmath8 is epoch of matter - radiation equality . the sound speed depends only on the baryon - to - photon ratio and is @xmath9 with @xmath10 . these two produce the sound horizon @xmath11 where ` rec ' and ` eq ' refer to recombination and equality respectively . one sees that the aside from a prefactor of @xmath12 , the sound horizon depends only on the baryon - to - photon ratio and the redshift of equality . the epoch of recombination , being controlled by atomic physics , is very insensitive to the cosmology . for reasonable changes in the early universe and our current uncertainties of the theory of recombination @xcite , any shift in @xmath13 is negligible . the baryon - to - photon ratio is also exquisitely well measured in the cmb power spectrum by both the ratios of the odd and even acoustic peaks and by the silk damping tail @xcite . the former effect depends only on the gravitational inertia of the baryons driven by the potentials near the epoch of recombination . thus the modulation gives us a precise measurement of the baryon - to - photon ratio @xmath14 , which with our precise knowledge of @xmath15 fixes @xmath16 . moreover , for the established value of @xmath17 near @xmath18 , the effect on the sound horizon is already small . it seems very likely that the cmb will determine the baryon - to - photon ratio to sufficient accuracy for this portion of the sound horizon computation @xcite . information about matter - radiation equality is encoded in the amplitudes of the peaks through the manner in which the potentials evolve as they cross the horizon : the potential envelope @xcite . measurements of the potential envelope thus robustly constrain equality . normally , one interprets this constraint as the matter density @xmath1 , on the assumption that the photons and standard neutrino background are the full radiation density . however , one could imagine other relativistic components , and in this case , measuring the redshift of equality does not imply the matter density @xmath1 ( we continue to assume that the extra components are `` undetected '' in the cmb and return to this point in the next section ) . as we can see from eq . ( [ eq : rsound ] ) , the dependence of @xmath19 on @xmath20 is relatively small since @xmath21 , thus even a crude determination suffices to fix @xmath19 up to an overall factor of @xmath12 , i.e. , @xmath22 is very well measured . the sound horizon decreases by only 5% if @xmath20 is lowered by @xmath23 ! understanding the acoustic oscillations at @xmath3 allows us to translate knowledge of the sound horizon into knowledge of wavelength of the baryonic features in the mass power spectrum up to the same normalization uncertainty . we then wish to consider the measurement of this scale at lower redshift , such as could be accomplished in large galaxy surveys . measuring the scale along and across the line of sight , as a redshift or angular scale , constrains @xmath24 and @xmath25 , respectively . for dark energy with ( constant ) equation of state @xmath26 , the low - redshift quantities can be written as @xmath27^{-1/2},\ ] ] and ( for zero curvature ) @xmath28^{-1/2}.\ ] ] because @xmath29 is well constrained , we find that the observations actually constrain @xmath30 and @xmath31 @xcite , which contain only the terms that depend on the bare @xmath32 values , , where @xmath33 is equation [ eq : da ] . but we can write @xmath34 , where @xmath35 is the usual curvature term . this substitution shows that @xmath36 depends only on the bare @xmath32 s . ] i.e. , @xmath37 , @xmath38 , etc . in other words , the prefactors of @xmath12 have canceled out between @xmath19 and the low - redshift distances . we can thus reliably predict the distance _ ratios _ between low @xmath39 and @xmath3 as a function of redshift and hence constrain cosmology . what has happened is simply that the overall scale of the universe does nt affect any of the distance ratios . usually this scale is labeled as the hubble constant , such that @xmath40 drops out . we have rephrased the scale as @xmath1 , despite the fact that this would appear to include @xmath37 in addition to @xmath40 . another way of saying this is that the standard ruler defined by the cmb at constant redshift of equality actually scales as @xmath41 . hence , even @xmath42 redshift - space measurements of this ruler do not measure @xmath40 , but instead measure @xmath43 . parameter estimation for acoustic oscillations based on standard models @xcite will be unchanged by the presence of undetectable new radiation provided that the redshift of equality is measured correctly . only the hubble constant would be incorrect . in the context of cmb parameter estimation , it would be more useful to report @xmath44 rather than @xmath19 itself , as this is the quantity that isolates the cosmological densities at low redshift . massive neutrinos should be counted as radiation at @xmath45 but would be counted as matter at low redshift . that does create a small error in the dark energy inferences . one would be computing @xmath46 from the cmb , but the low - redshift distances in equations ( [ eq : h ] ) and ( [ eq : da ] ) would still have prefactors of @xmath47 . as the observations constrain @xmath25 and @xmath24 , the evaluation of these quantities in terms of bare @xmath32 s will be altered by @xmath48 . for the massive neutrino fraction inferred for non - degenerate neutrino species and the atmospheric neutrino mass splitting @xcite this is a 0.2% correction ; however , it could be a few percent correction at the upper limit of the allowed masses for degenerate neutrino species @xcite . fortunately , such neutrino masses should be detectable in upcoming data due to their suppression of the late - time matter power spectrum @xcite . because the integral for the sound horizon extends to early times ( essentially to the time when the cosmic perturbations were established ) , alterations to the hubble parameter ( eq . [ [ eq : h ] ] ) could alter the sound horizon even at fixed redshift of equality . for example , there might exist a non - relativistic particle that decays into relativistic unseen particles sometime prior to recombination @xcite . however , such decays create alterations to the gravitational potentials that would be detectable in the cmb if the horizon scale at that epoch is visible in the primary anisotropies ( i.e. , wavenumbers @xmath49 ) . this makes decays at @xmath50 difficult to hide . furthermore , such decays alter the transfer function and would affect the amplitude of the late - time matter power spectrum . precision measurement of the matter power spectrum out to @xmath51 from the lyman - alpha forest could push the limits on particle decays to earlier times . in the standard theory , the first 1% of the sound horizon integral is contributed by @xmath52 . it therefore seems very unlikely that a decaying particle could significantly affect the sound horizon ( relative to the standard theory at fixed redshift of equality ) and escape detection in the cmb or large - scale structure . as a corollary , it is interesting to note that when one measures the acoustic oscillation scale at @xmath13 with the cmb and at some low redshift @xmath53 in a galaxy survey , one can construct an observable ( the difference of the suitably scaled angular wavenumbers of the acoustic peaks ) that isolates the cosmology between @xmath53 and @xmath13 , i.e. , @xmath54 this integral is trivial when the universe is completely matter - dominated . hence , if we can perform a galaxy survey at some suitably high redshift , e.g. , @xmath55 , where the dark energy is supposed to be negligible , we could search for dynamical shenanigans at @xmath56 . a photometric - redshift survey over large amount of sky would be an economical route to this goal , as one can acquire very large survey volumes @xcite . we have argued that only the ratios of the radiation and matter densities enter the acoustic oscillation standard ruler method , but of course it is interesting to measure the actual values of the densities as they might reveal new relativistic components or unexpected evolution . the cmb is sensitive to the non - photon radiation density primarily through its effect on equality and the evolution of the potentials . this encodes a sensitivity to both the amount of radiation and its quadrupole moment . in general any particle that free streams rather than behaves as a fluid will have a local quadrupole that will affect the evolution of the potentials and thus the anisotropy in the cmb this breaks the degeneracy in the cmb between changes in @xmath1 and @xmath57 @xcite and allows us to constrain a `` conventional '' neutrino at the @xmath58 level @xcite . at the other extreme , extra radiation which is of the perfect fluid form leads to an increase in the small - scale power which is easily discerned from a change in @xmath1 ( see fig . [ fig : cl ] ) . only a radiation component whose higher moments track closely that of the traditional mix of photons and neutrinos could be confused with a change in @xmath1 . altering the radiation and matter density while holding the baryon ( and photon ) densities fixed would alter the baryon fraction @xmath59 and would produce significant offsets in the late - time matter power spectrum ( e.g. @xcite ) . higher baryon fractions suppress power on small scales compared to large , with the transition occurring near the sound horizon at @xmath60 . the acoustic oscillations shortward of this scale would also be increased . the small - scale amplitude ( e.g. , @xmath61 ) would be reduced . these changes are observable in galaxy surveys , weak lensing surveys , and cluster abundance measurements . of course one can also directly measure the hubble constant @xcite . in short , although the acoustic peak method depends only on the redshift of equality , other cosmological measurements both at @xmath62 and at @xmath63 are well poised to measure the actual densities and thereby constrain the presence of unknown relativistic species . we have shown that the standard ruler defined by the acoustic oscillations prior to recombination can deal gracefully with uncertainties in the matter density @xmath64 provided that the redshift of matter - radiation equality is well measured . the anisotropies of the cmb have very good leverage on this quantity , and so the acoustic peak method of probing dark energy is robust . in addition , it is likely that the cmb , perhaps in combination with other probes , will be able to constrain the actual matter and radiation densities . acknowledgements : we thank eric linder and nikhil padmanabhan for useful discussions and the lawrence berkeley laboratory for the hospitality of its summer workshop program where this work was begun . dje is supported by national science foundation grant ast-0098577 and by an alfred p. sloan research fellowship . mw is supported by the nsf and nasa . peebles , j.t . yu , astrophys . j. , 162 , 815 ( 1970 ) j.r . bond , g. efstathiou , astrophys . j. , 285 , l45 ( 1984 ) j.a . holtzman , astrophys . , 71 , 1 ( 1989 ) d.j . eisenstein , w. hu , m. tegmark , astrophys . j. , 504 , l57 ( 1998 ) a. meiksin , m. white , j.a . peacock , mnras , 304 , 851 ( 1999 ) d.j . eisenstein , in `` wide field multi - object spectroscopy '' , asp conference series , 2003 , ed . a. dey , p. 35 c. blake , k. glazebrook , astrophys . j. , 594 , 665 ( 2003 ) e.v . linder , phys . d , 68 , 083504 ( 2003 ) w. hu , z. haiman , phys . d. , 68 , 063004 ( 2003 ) h .- j . seo , d.j . eisenstein , astrophys . j. , 598 , 720 ( 2003 ) w. hu , s. dodelson , ann . astrophys . , 40 , 171 ( 2002 ) w. hu , d.j . eisenstein , m. tegmark , m. white , phys . d59 , 023512 ( 1999 ) s. seager , d.d . sasselov , d. scott , astrophys . j. lett . , 523 , 1 ( 1999 ) ; s. seager , d.d . sasselov , d. scott , astrophys . j. supp . , 128 , 407 ( 2000 ) w. hu , m. white , astrophys . j. , 471 , 30 ( 1996 ) w. hu , m. white , astrophys . j. , 479 , 568 ( 1997 ) w. hu , d. scott , n. sugiyama , m. white , phys . rev . , d52 , 5498 ( 1995 ) y. fukuda et al . , 81 , 1562 ( 1998 ) ; m. sanchez et al . d68 , 113004 ( 2003 ) ; y. ashie et al . , lett . , submitted ( 2004 ) , hep - ex/0404034 d.n . spergel et al . , astrophys . , 148 , 175 ( 2003 ) ; o. elgaroy & o. lahav , j. cosm . phys . , 04 , 004 ( 2003 ) ; s. hannestad , j. cosm . , 05 , 004 ( 2003 ) ; m. tegmark et al . , d69 , 103501 ( 2004 ) w. hu , d.j . eisenstein , & m. tegmark , phys . lett . , 80 , 5255 ( 1998 ) ; w. hu & m. tegmark , astrophys . j. , 514 , l65 ( 1999 ) ; r.a.c . croft , w. hu , & r. dav , phys . lett . , 83 , 1092 ( 1999 ) j. bardeen , j.r . bond , g. efstathiou , astrophys . j. , 321 , 28 ( 1987 ) ; j.r . bond , g. efstathiou , phys . , b265 , 245 ( 1991 ) e.j . chun , h.b . kim , j.e . kim , phys . , 72 , 1956 ( 1994 ) m. white , g. gelmini , j. silk , phys . d51 , 2669 ( 1995 ) r. bowen , s.h . hansen , a. melchiorri , j. silk , mon . not . soc . , 334 , 760 ( 2002 ) ; e. pierpaoli , mon . not . , 342 , l63 ( 2003 ) ; p. crotty , j. lesgourgues , pastro , s. , phys . , d69 , 123007 ( 2004 ) k. nishikawa , presented at the xxi int . on lepton and photon interactions at high energies ( lepton photon 2003 ) , fermilab , august , 2003 . eisenstein & w. hu , astrophys . j. 496 , 605 ( 1998 ) w.l . freedman , et al . , astrophys . j. , 553 , 47 ( 2001 )

we discuss the systematic uncertainties in the recovery of dark energy properties from the use of baryon acoustic oscillations as a standard ruler . we demonstrate that while unknown relativistic components in the universe prior to recombination would alter the sound speed , the inferences for dark energy from low - redshift surveys are unchanged so long as the microwave background anisotropies can measure the redshift of matter - radiation equality , which they can do to sufficient accuracy . the mismeasurement of the radiation and matter densities themselves ( as opposed to their ratio ) would manifest as an incorrect prediction for the hubble constant at low redshift . in addition , these anomalies do produce subtle but detectable features in the microwave anisotropies .

arxiv

in the last few years there has been another wave of excitement regarding the question of neutrino masses . this is largely due to the many new experiments testing neutrino oscillations , most notably the positive indications obtained by super kamiokande on atmospheric neutrino oscillations @xcite . similar indications come from other experiments @xcite . the solar neutrino experiments have for many years provided independent evidence for neutrino oscillations @xcite . accelerator and reactor experiments have also played an important role . they have furnished strict bounds on neutrino oscillation parameters @xcite . in the case of the lsnd experiment @xcite at los alamos evidence for @xmath2 oscillation has been reported . see refs @xcite for recent reviews . it is hoped that new experimental results can be used to determine the neutrino squared mass differences and mixing angles . in turn , these may help to infer the neutrino mass matrix . this is presumably a possible gateway to a more fundamental theory beyond the standard model . of course this is a highly speculative area , and even though there are many imaginative proposals @xcite , it seems fair to say that the the true answer is essentially unknown . in order to make progress in this direction , it seems useful to investigate various plausible ansatze for the neutrino mass matrix . from this point of view we propose the ansatz for the 3 generation neutrino mass matrix , @xmath3 : @xmath4 and investigate its consequences . we are considering the neutrinos to be represented by 2-component spinors so that , in the most general situation , @xmath3 is an arbitrary symmetric complex matrix . as we will see in section ii , eq . ( 1.1 ) can be motivated from an so(10 ) grand unified model @xcite , in which it may be derived with some assumptions . physically , eq . ( 1.1 ) corresponds to the well known approximate signature of grand unification that @xmath5 . furthermore we will see in sections iv and v that eq . ( 1.1 ) can be straightforwardly combined with experimental information to get an idea of the neutrino masses themselves as well as the `` texture '' of @xmath6 . relevant matters of notation are discussed in section iii while a summary is presented in section vi . in the so(10 ) grand unification model each generation contains one light massive two component neutrino and also a very heavy one which is `` integrated out '' according to the `` seesaw mechanism '' @xcite . the effective @xmath7 neutrino mass matrix takes the form : @xmath8 where @xmath9 , @xmath10 and @xmath11 are respectively the mass matrices of the light neutrinos , heavy neutrinos and heavy - light mixing ( or `` dirac matrix '' ) . generally the second , seesaw , term is considered to dominate . here however we shall assume the first term to be the dominant one . this is necessary for the present derivation of eq . ( 1.1 ) to hold . also , a rough order of magnitude estimate for the second term would be @xmath12 or about @xmath13 ev . thus , the seesaw term could be negligible if neutrino masses turn out to be appreciably larger than this value . now in so(10 ) , higgs mesons belonging to the 10 , 120 and 126 representations can contribute to the fermion masses at tree level . one has @xcite for the down quark , charged lepton and light neutrino mass matrices , @xmath14 where @xmath15 , @xmath16 , @xmath17 , @xmath18 , @xmath19 are numbers representing higgs meson vacuum values . s(10 ) , a(120 ) and s(126 ) are the matrices of the yukawa type constants which couple the fermions to the 10 , 120 and 126 higgs mesons respectively ; the matrices s(10 ) and s(126 ) must be symmetric while a(120 ) is antisymmetric . finally , @xmath20 is a renormalization factor for comparing the quark masses with the charged lepton masses at a low energy scale rather than at the grand unified scale ; @xmath21 is a similar factor for the neutrino masses . with the stated assumption that the @xmath22 term dominates in eq . ( 2.1 ) we get @xmath23 which clearly also holds when any number of 10 s or 120 s are present but only a single 126 . the matrices appearing in eq . ( 2.3 ) are so far essentially unrestricted complex ones . to proceed , we make the further assumption that the matrices are hermitian . then @xmath24 and @xmath25 may each be brought to diagonal form by unitary transformations . thus the right hand side of eq . ( 2.3 ) may be evaluated to yield approximately , @xmath26 according to a well known numerical success , based on the observation that @xmath27 , of grand unification @xcite . note that we have not needed to assume that the mass matrix has any zero elements . where , in addition , a special combined fritzsch - stech ansatz was used . here we are not making any special ansatz of this type for the mass matrices . ] even if the cancellation on the right hand side of eq . ( 2.4 ) is not perfect , it should still be a good approximation . in an so(10 ) model where the mass matrices are hermitian , @xmath28 will be real symmetric . we will investigate this case and also the possibility that the more general case holds . our plan is to combine the ansatz eq . ( 1.1 ) with experimentally obtained results on neutrino oscillations in order to learn more about @xmath29 itself . for this purpose it may be helpful to set down our notation @xcite for the pieces of the effective @xmath30 theory involving neutrinos and to make some related remarks . the free lagrangian containing three two component massive fields is : @xmath31 where @xmath32 is the ( not yet diagonalized ) neutrino mass matrix of the underlying theory to be identified with the matrix in eq . note that we are free to multiply the first mass term in eq . ( [ lfree ] ) by an overall arbitrary phase which is a matter of convention . it is possible@xcite to find a unitary matrix @xmath33 which brings @xmath0 to real , positive , diagonal form in the following way : @xmath34 the mass diagonal fields @xmath35 are then @xmath36 similarly , the column vector of left handed negatively charged leptons in the underlying theory , @xmath37 is related to the mass diagonal fields @xmath38 by @xmath39 where @xmath40 . combining factors from eq . ( 3.3 ) and eq . ( 3.4 ) we obtain the unitary mixing matrix , @xmath41 for the charged current weak interaction , @xmath42 this appears in the lagrangian term , @xmath43 where a conventional four component dirac notation with @xmath44 diagonal is being employed and @xmath35 has only the first two components non zero . next we parameterize @xmath41@xcite . it is possible to restrict @xmath45 by adjusting an overall phase which can be absorbed in @xmath46 . then we write @xmath47 where @xmath48 with @xmath49 and , for example @xmath50 . \label{3.9}\ ] ] eq . ( 3.7 ) contains the eight parameters needed to characterize an arbitrary unitary unimodular matrix . from the standpoint of eq . ( 3.6 ) it can be further simplified by using the freedom to rephase @xmath51 without changing the free part of the charged lepton lagrangian . on the other hand , the form of the mass terms in eq . ( [ lfree ] ) shows that the neutrino fields can not be rephased . thus a suitable minimal parameterization for @xmath41 in ( 3.6 ) is @xmath52 involving three angles " , @xmath53 and three phases " , @xmath54 . note the identity @xmath55 this identity may be used to transfer two of the phases @xmath54 in eq . ( 3.7 ) to a diagonal matrix on the right of @xmath41 as , for example , @xmath56 where @xmath57 , which may be used instead of eq . ( 3.10 ) . we also need the formula for the amplitude of neutrino oscillation . for the case when a neutrino , produced by a charged lepton of type @xmath15 , oscillates " to make at time @xmath58 , a charged lepton of type @xmath16 , we have @xmath59 where the sum goes over the neutrinos of definite mass , @xmath60 . inserting the parameterization eq . ( 3.12 ) into eq . ( 3.13 ) shows that the effect of the factor @xmath61 cancels out . thus for ordinary oscillations , @xmath41 is parameterized by three angles and one cp violating phase as for the ckm quark mixing matrix . on the other hand , the two additional cp violating phases @xmath62 and @xmath63 show up if one considers neutrino - antineutrino oscillations @xcite or neutrinoless double beta decay @xcite . the formula for the probability , @xmath64 is gotten by taking the squared magnitude of eq . ( 3.13 ) and replacing the exponential factor @xmath65 by @xmath66 , where @xmath67 is the neutrino energy and @xmath68 is the oscillation distance . for practical reasons it is very important to take account of the experimental uncertainties in @xmath67 and @xmath68 . the simplest approximation@xcite is to define @xmath69 and assume that one can smear @xmath64 with a gaussian distribution in @xmath16 . @xmath70 is defined as the mean value and @xmath71 as the standard deviation appropriate to the particular physical setup . then we find for the smeared probability @xmath72 , \label{3.14}\end{aligned}\ ] ] where @xmath73 and @xmath74 . notice that when @xmath75 , @xmath76 is independent of the sign of @xmath77 . since @xmath78 provides only two real equations for 12 real parameters , it is clear that it has a relatively small amount of predictivity . in particular it can not say much about the texture ( e.g. possible zeroes ) of @xmath0 which is suppposed to derive from a deeper theory than the standard model . on the other hand , we shall see that our ansatz is complementary to the results which should emerge from analysis of neutrino oscillation experiments . together , they should enable us to actually ( with some conditions ) reconstruct @xmath0 . first , in this section , we shall consider @xmath0 to be hermitian so that the argument in favor of @xmath78 presented in section ii holds without any further assumptions . since @xmath0 is symmetric it must be real . it can be brought to real diagonal form via a real rotation @xmath79 as @xmath80 . however there is no guarantee that all eigenvalues of @xmath0 will be positive . we can make them all positive by rephasing the diagonal fields with negative eigenvalues by a factor @xmath81 ( see eqs . ( 3.1 ) and ( 3.2 ) ) . this means that the general diagonalizing matrix @xmath33 in eq . ( 3.2 ) now takes the form @xmath82 where @xmath83 with @xmath84 for a positive eigenvalue and @xmath85 for a negative eigenvalue . we notice that eq . ( 4.1 ) is of the form eq . ( 3.12 ) for which we already noticed that the factor @xmath86 cancels out in the neutrino oscillation formula eq . ( 3.14 ) . furthermore only the square of the mass is relevant in eq . ( 3.14 ) . thus we choose to work in this section with some negative masses and no factor @xmath86 in eq . ( 4.1 ) . to avoid confusion , we remark that the factor @xmath86 in eq . ( 4.1 ) does _ not _ introduce any cp violation in the theory @xcite since @xmath0 is real in any event . now let us suppose that an experimental analysis of all neutrino oscillation experiments is made based on a formula like eq . ( 3.14 ) ( or one which treats the experimental uncertainties in a more sophisticated way ) . furthermore assume that the cp violating phase @xmath87 in eq . ( 3.12 ) is negligible . then we should know the _ magnitudes _ of the squared neutrino mass differences @xmath88 where @xmath89 and @xmath90 can be either positive or negative . then , assuming the leptonic theory to be cp conserving , our ansatz would imply @xmath91 where eq . ( 3.2 ) was used . ( 4.2 ) and ( 4.3 ) comprise three equations for the three neutrino masses @xmath92 , @xmath93 and @xmath94 . we can solve to get : @xmath95 , \\ \nonumber m_2 & = & \frac { { b } - { \left ( m_1 \right)}^2}{2m_1 } , \\ \nonumber m_3 & = & - m_1 - m_2 . \label{4.4}\end{aligned}\ ] ] this leads to a limited number of solutions , depending on sign choices . if we make the further assumption that the charged lepton mixing matrix @xmath96 in eq . ( 3.4 ) is approximately the unit matrix ( this is expected to be a reasonable but not perfect approximation ) we can identify @xmath97 which would be obtained from experiment . then , using the masses found in eq . ( 4.4 ) , we could reconstruct @xmath0 as @xmath98 to proceed , we need only insert the experimental results for @xmath89 , @xmath90 and @xmath41 in ( 4.4 ) and ( 4.5 ) . of course , it is presumably the task of the next decade to solidify the experimental determination of these quantities . we can , at the moment , only give a preliminary discussion . for this purpose we will use the results of a recent preliminary analysis of all neutrino experiments by ohlsson and snellman @xcite . these authors found , by a least square analysis , a best fit for ( our notation ) @xmath99 , @xmath100 and the leptonic mixing matrix @xmath41 . they used the formula eq . ( 3.14 ) with a suitable choice of @xmath70 and @xmath101 for each experiment . furthermore , they made the simplifying assumption that @xmath41 is real . finally they only searched for a fit in the range @xmath102 , @xmath103 . this range corresponds to mass difference choices for which the msw effect @xcite for solar and atmospheric neutrinos is not expected to be important and so greatly simplifies the analysis . thus there is no guarantee that the solution of @xcite is unique . altogether they fit sixteen different solar neutrino , atmospheric neutrino , accelerator and reactor experiments , including lsnd . the best fit is : @xmath104 for the squared mass differences and @xmath105 \label{4.7}\ ] ] for the lepton mixing matrix @xmath41 . as discussed above we will identify @xmath106 here , keeping @xmath41 real but allowing for negative masses . the best fit matrix k was obtained to be similar but not identical to the `` bimaximal mixing '' matrix @xcite . with the best fit squared mass differences in eq . ( 4.6 ) , our model predicts , from the first of eq . ( 4.4 ) , eight different possibilities . these correspond to four different sign configurations for @xmath89 and @xmath90 times the two possible signs for @xmath92 . however , only two of these eight are essentially different ; these are @xmath107 the other solutions correspond to interchanging whichever of @xmath108 and @xmath109 is greater ( which has only a negligible effect since they are almost degenerate ) or reversing the signs of all masses . physically it is clear what is happening : the smallness of @xmath110 compared to @xmath111 in eq . ( 4.2 ) forces @xmath112 . then we have either @xmath113 with , using the constraint eq . ( 4.3 ) , @xmath114 or @xmath115 with @xmath94 very small . since we have assumed the neutrinos to be of majorana type for our plausibility argument in section ii , their interactions will violate lepton number . then they should mediate neutrinoless double beta decay @xmath116 @xcite . such a process has not yet been observed and an upper bound has been set for the relevant quantity @xmath117 the best upper bound at present is @xcite @xmath118 ev , reflecting some uncertainty in the estimation of the needed nuclear matrix elements . substituting the best fit for the matrix k from eq . ( 4.7 ) together with our results in eqs . ( 4.8 ) and ( 4.9 ) into eq . ( 4.10 ) yields predictions for the two cases : @xmath119 both solutions seem to be acceptable , the type i case marginally but the type ii case definitely . note that the small value for @xmath120 in the type ii case is due to the best fit prediction @xcite @xmath121 and also to the fact that @xmath93 is negative . the same value would clearly result if we made @xmath93 positive and set @xmath122 as discussed around eq . ( 4.1 ) above . finally , let us reconstruct the underlying neutrino mass matrices for each of the two cases . we use eq . ( 4.5 ) based on the assumption that @xmath0 is real and also our ansatz to find ( in units of ev ) : @xmath123 , \\ { \rm type } \hskip 0.2 cm { \rm ii } : \quad m_\nu & = & m_\nu^t \approx \left [ \begin{array } { c c c } 6.7 \times 10^{-5 } & -0.9199&0.5078\\ -0.9199&0.0654&0.0410\\ 0.5078&0.0410&-0.0654\\ \end{array } \right ] . \label{4.13}\end{aligned}\ ] ] the type i matrix does not have an excellent candidate for a `` texture '' zero . however the small value of @xmath124 in the type ii case is certainly suggestive . these matrices lead to neutrino masses and a mixing matrix which give a best fit to all present data . it will be interesting to see if either of them hold up in the future . incidentally , on comparing eqs . ( 4.13 ) and ( 4.14 ) it is amusing to observe the large difference in two mass matrices `` generated '' in the same way except with respect to how @xmath125 is satisfied . it seems interesting to also investigate the ansatz @xmath126 when @xmath0 is no longer restricted to be real . this also raises the problem of constructing the unitary diagonalizing matrix @xmath33 in eq.(3.2 ) , in terms of the experimentally measured lepton mixing matrix @xmath127 . for simplicity , as before , we will make the approximation that the charged lepton diagonalizing matrix @xmath96 is the unit matrix . apart from an overall ( conventional ) phase we may write @xmath128 where the 2-parameter quantity @xmath129 was defined in eq . ( 3.8 ) . since @xmath127 has four parameters @xmath33 in eq . ( 5.1 ) is described by eight parameters . as mentioned before , the two parameters in @xmath61 are not measurable in neutrino oscillation experiments but show up when one considers @xmath130 . the two parameters in @xmath131 may be eliminated , for experimental purposes , by rephasing the charged leptons . however , for the theoretical purpose of reconstructing the underlying neutrino mass matrix @xmath0 , their existence can not be ruled out . ( they also do not contribute to @xmath132 . ) for the purpose of relating the ansatz on @xmath0 to the physical neutrino masses in @xmath133 , we note @xmath134 for further simplicity of the analysis we adopt the special case @xmath135 and also identify @xmath127 with the real best fit in eq . ( 4.7 ) ; our ansatz now reads @xmath136 with the redefinitions @xmath137 and @xmath138 , eq . ( 5.3 ) becomes @xmath139 this may be conveniently visualized as the vector triangle shown in fig . [ trianglefig ] . combining eqs . ( 5.4 ) and ( 4.2 ) gives four real equations for the five unknown quantities ( @xmath140 ) . thus we have ( for each set of ( @xmath141 ) sign choices ) a one parameter family of solutions . it is convenient to choose this parameter to be @xmath94 . then @xmath92 and @xmath93 may be found from the equations ( 4.2 ) , provided that solutions exist . in this way all three sides of the triangle in fig.[trianglefig ] are determined . the angles may finally be found as @xmath142 we also need to investigate the constraint arising from the non - observation of @xmath132 . ( 4.10 ) now becomes , with eq . ( 5.1 ) as the mixing matrix @xmath143 using the ansatz constraint eq . ( 5.4 ) , eq . ( 5.6 ) may be rewritten as @xmath144 m_2 e^{-i\beta_2 } + \left [ ( k_{exp\hskip .2 cm 13})^2 - ( k_{exp\hskip .2 cm 11})^2 \right ] m_3 \right|.\ ] ] this form is very convenient when identifying @xmath127 with the best fit solution in eq . ( 4.7 ) . in the present context such an identification corresponds to cp violation for the @xmath132 process but not for usual neutrino oscillations . since the ( 11 ) and ( 12 ) matrix elements are equal in eq . ( 4.7 ) we find the simple result @xmath145 thus if the upper bound on @xmath146 is conservatively identified as in the @xmath147 ev range , we should have in this case @xmath148 in the present complex case there is a continuum of possible solutions labelled by those values of @xmath94 satisfying eq . ( 5.9 ) , rather than just the two possibilities found in eq . ( 4.8 ) and eq . ( 4.9 ) . actually , the continuum separates roughly into two classes similar to either eq . ( 4.8 ) or eq . ( 4.9 ) . in the generalized type i class , @xmath94 is of the order @xmath149 ev while @xmath92 and @xmath93 are related to nearly oppositely directed vectors in fig . [ trianglefig ] and are also of the order @xmath90 . in the generalized type ii class , @xmath90 is negative ; @xmath92 and @xmath93 correspond to vectors of order @xmath150 which are oppositely directed to each other , while @xmath94 ranges from very small to order @xmath100 . given the bound eq . ( 5.9 ) from the non - observation of @xmath130 , there are important limitations on the allowed @xmath94 values for type i solutions . in this case @xmath90 is positive so the equation @xmath151 will only allow solutions for @xmath152 ev . this range is barely compatible with eq . thus the type ii case where @xmath153 and @xmath154 seems most probable . as an example of a solution for complex @xmath0 , consider choosing @xmath155 ev . then a solution is obtained with ( compare with fig . [ trianglefig ] ) @xmath156 the matrix @xmath33 , which diagonalizes @xmath0 is obtained from eq . ( 5.1 ) , with the approximation @xmath157 , and with now : @xmath158 this factor introduces cp violation in the @xmath132 process but not in ordinary neutrino oscillations . finally the underlying neutrino mass matrix , @xmath159 is `` reconstructed '' as ( in units of ev ) : @xmath160.\ ] ] this is structurally similar to the real type ii solution displayed in eq . ( 4.14 ) , although the suppression of the ( 11 ) element is not so pronounced . notice that @xmath94 is considerably smaller than the almost degenerate pair @xmath92 and @xmath93 . furthermore @xmath92 and @xmath93 are large enough to possibly play some role in astrophysics . we investigated the ansatz @xmath78 for the underlying ( pre - diagonal ) three generation neutrino mass matrix . it was motivated by noting that in an so(10 ) grand unified model where @xmath0 was taken to be real ( cp conserving ) , it corresponds to the well known unification of b quark and @xmath161 lepton masses . while not very predictive by itself it yields information complementary to what would be gotten from a complete three flavor analysis of all lepton number conserving neutrino oscillation experiments . specifically from the specification of the magnitudes of two neutrino squared mass differences and also of the leptonic mixing matrix we can , with some assumptions , find the neutrino masses themselves and `` reconstruct '' @xmath0 . this determination can be sharpened by consideration of the constraints imposed by non - observation of neutrinoless double beta decay . for the purpose of testing our ansatz we employed the results of a reasonable best fit to all present neutrino experiments ( including lsnd ) by ohlsson and snellmann @xcite . this fit will inevitably be improved in the next few years as new experiments are completed . they were able to fit the data without assuming any cp violation . this agrees with assuming @xmath0 to be real . we found two essentially different solutions in that case . the first features two neutrinos having approximately equal mass 0.608 ev and a third neutrino of mass 1.217 ev . this solution is on the borderline of being ruled out by non - observation of @xmath162 . the second solution has two neutrinos with approximately degenerate mass 1.054 ev and a third neutrino with a mass @xmath163 ev . this solution is very safe from being ruled out by @xmath164 experiments . it also features a reconstructed @xmath0 which has an extremely small ( 11 ) element . note that , for both solutions , even though @xmath0 is real there are some ( cp conserving ) factors of @xmath81 in the mixing matrix when all masses are taken to be positive . alternatively one may have no @xmath81 s in the mixing matrix while allowing some masses to be negative . the latter form is useful for seeing intuitively how @xmath165=0 is possible . the case of matching the above best fit data to a complex @xmath0 was also considered . in this situation there are cp violating phases in the lepton mixing matrix which affect the @xmath132 process but do not affect ordinary total lepton number conserving neutrino oscillations . such phases could also be measurable in principle with the observation of a decay like @xmath166 . the case of complex @xmath0 allows a larger number of solutions . with a simplifying assumption there is a one parameter family of allowed neutrino mass sets . roughly , these fall into one of the two types already encountered for real @xmath0 . a question of some interest is whether the neutrinos are massive enough to play a role in cosmology . the relevant criterion @xcite for this to occur is usually stated as @xmath167 ev . for the type ii solutions with complex @xmath0 we have found the largest mass sum to be about 4.5 ev corresponding to @xmath168 ev and @xmath169 ev . however this is on the very border of acceptability for non observation of @xmath132 . super - kamiokande collaboration , y. fukuda _ et al . * 81 * , 1562 ( 1998 ) . super - kamiokande and kamiokande collaborations , t. kajita , in _ proceedings of the xviiith international conference on neutrino physics and astrophysics ( neutrino 98 ) _ , takayama , japan , 1998 [ nucl . phys . b ( proc . suppl . ) * 77 * , 123 ( 1999 ) ] . soudan collaboration , w.w.m . et al . _ , b * 391 * , 491 ( 1997 ) ; kamiokande collaboration , y. fukuda _ et al . lett . * 77 * , 1683 ( 1996 ) ; chooz collaboration , m. apollonio _ et al . _ , b * 420 * , 397 ( 1998 ) ; chooz collaboration , c. bemporad , in proceedings of the xviiith international conference on neutrino physics and astrophysics ( neutrino 98 ) , nucl . b ( proc . suppl . ) * 77 * 159 ( 1999 ) ; e. k. akhmedov , hep - ph/0001264 ; j. n. bahcall , nucl . . suppl . * 48 * , 309(1996 ) ; s. bilenky , hep - ph/990246 ; p. ficher , b. kayser and k. s. mcfarland , anual review of nuclear and particle physics , * 49 * ( 1999 ) ; d. r. o. morrison , part . world * 3 * , 30(1992 ) ; a. rubbia , acta . b * 30 * , 2351(1999 ) ; a. y. smirnov , hep - ph/9901208 ; j. w. f. valle , hep - ph/9911224 , hep - ph/9906539 ; j. d. vergados , phys . * 133 * ( 1986 ) . c. h. albright and s. m. barr , phys . lett . b**461 * * , 218(1999 ) ; v. barger , s. pakvasa , t. j. weiler and k.whisnant , phys . lett . b**437 * * , 107(1998 ) ; h. b. benaoum and s. nasri ; phys . rev . d**60 * * , 113003(1999 ) ; m. jezabek and y. sumino , phys . lett . b**457 * * , 139(1999 ) ; s. k. kang and c. s. kim , phys . rev . d**59 * * , 091302(1999 ) ; r. n. mohapatra and s. nussinov , phys . d**60 * * , 013002(1999 ) , r. n. mohapatra , a. perez - lorenzano and c .a . de sousa pires , phys . b**474 * * , 355(2000 ) ; h. georgi and s. l. glashow , phys . rev . d*61 * , 097301 ( 2000 ) . t. yanagida , proc . of the workshop on unified theory and baryon number in the universe , ed . by o. sawada and a. sugamato ( kek report 79 - 18,1979 ) , p 95 ; m. gell - mann , p. ramond and r. slansky in supergravity , eds p. van niewenhuizen and d. z. freedman ( north holland , 1979 ) ; r. n. mohapatra and g. senjanovic , phys . * 44 * , 912 ( 1980 ) . see for example k. matsuda , t. fukuyama and h. nishiura , phys . d * 61 * , 053001(2000 ) ; a. bottino , c. w. kim , h. nishiura and w. k. tse , phys . d * 34 * , 862 ( 1986 ) ; k. s. babu and r. n. mohapatra , phys . rev * 70 * , 2845 ( 1993 ) .

we propose a simple ansatz for the three generation neutrino mass matrix @xmath0 which is motivated from an so(10 ) grand unified theory . the ansatz can be combined with information from neutrino oscillation experiments and bounds on neutrinoless double beta decay to determine the neutrino masses themselves and to reconstruct , with some assumptions , the matrix @xmath1 . = 17pt # 1=#1pt = by -by -makecaption#1#2 tempboxa tempboxa > = = # 1 # 2 to = 13pt su-4240 - 719 + hep - ph/0004105 + * complementary ansatz for the neutrino mass matrix * = 12pt deirdre black amir h. fariborz + 0.5 cm salah nasri joseph schechter + _ department of physics , syracuse university , syracuse , ny 13244 - 1130 , usa . _ + 0.5 cm * abstract *

arxiv

the contact pattern among individuals in a population is an essential factor for the spread of infectious diseases . in deterministic models , the transmission is usually modelled using a contact rate function , which depends on the contact pattern among individuals and also on the probability of disease transmission . the contact function among individuals with different ages , for instance , may be modelled using a contact matrix @xcite or a continuous function @xcite . however , using network analysis methods , we can investigate more precisely the contact structure among individuals and analyze the effects of this structure on the spread of a disease . the degree distribution @xmath1 is the fraction of vertices in the network with degree @xmath2 . scale - free networks show a power - law degree distribution @xmath3 where @xmath4 is a scaling parameter . many real world networks @xcite are scale - free . in particular , a power - law distribution of the number of sexual partners for females and males was observed in a network of human sexual contacts @xcite . this finding is consistent with the preferential - attachment mechanism ( ` the rich get richer ' ) in sexual - contact networks and , as mentioned by liljeros et al . @xcite , may have epidemiological implications , because epidemics propagate faster in scale - free networks than in single - scale networks . epidemic models such as the susceptible infected ( si ) and susceptible infected susceptible ( sis ) models have been used , for instance , to model the transmission dynamics of sexually transmitted diseases @xcite and vector - borne diseases @xcite , respectively . many studies have been developed about the dissemination of diseases in scale - free networks @xcite and in small - world and randomly mixing networks @xcite . scale - free networks present a high degree of heterogeneity , with many vertices with a low number of contacts and a few vertices with a high number of contacts . in networks of human contacts or animal movements , for example , this heterogeneity may influence the potential risk of spread of acute ( e.g. influenza infections in human and animal networks , or foot - and - mouth disease in animal populations ) and chronic ( e.g. tuberculosis ) diseases . thus , simulating the spread of diseases on these networks may provide insights on how to prevent and control them . in a previous publication @xcite , we found that networks with the same degree distribution may show very different structural properties . for example , networks generated by the barabsi - albert ( ba ) method @xcite are more centralized and efficient than the networks generated by other methods @xcite . in this work , we studied the impact of different structural properties on the dynamics of epidemics in scale - free networks , where each vertex of the network represents an individual or even a set of individuals ( for instance , human communities or animal herds ) . we developed routines to simulate the spread of acute ( short infectious period ) and chronic ( long infectious period ) infectious diseases to investigate the disease prevalence ( proportion of infected vertices ) levels and how fast these levels would be reached in networks with the same degree distribution but different topological structure , using si and sis epidemic models . this paper is organized as follows . in section [ sec : hypothetical ] , we describe the scale - free networks generated . in section [ sec : model ] , we show how the simulations were carried out . the results of the simulations are analyzed in section [ sec : results ] . finally , in section [ sec : conclusions ] , we discuss our findings . we generated scale - free networks following the barabsi - albert ( ba ) algorithm @xcite , using the function barabasi.game(@xmath5 , @xmath6 , directed ) from the r package igraph @xcite , varying the number of vertices ( @xmath5 = @xmath7 , @xmath8 and @xmath9 ) , the number of edges of each vertex ( @xmath6 = 1 , 2 and 3 ) and the parameter that defines if the network is directed or not ( directed = true or false ) . for each combination of @xmath5 and @xmath6 , 10 networks were generated . then , in order to guarantee that all the generated networks would follow the same degree distribution and that the differences on the topological structure would derive from the way the vertices on the networks were assembled , we used the degree distribution from ba networks as input , to build the other networks following the method a ( ma ) @xcite , method b ( mb ) @xcite , molloy - reed ( mr ) @xcite and kalisky @xcite algorithms , all of which were implemented and described in detail in ref . as mentioned above , these different networks have distinct structural properties . in particular , the networks generated by mb are decentralized and with a larger number of components , a smaller giant component size , and a low efficiency when compared to the centralized and efficient ba networks that have all vertices in a single component . the other three models ( ma , mb and kalisky ) generate networks with intermediate characteristics between mb and ba models . the element @xmath10 of the adjacency matrix of the network , @xmath11 , is defined as @xmath12 if there is an edge between vertices @xmath13 and @xmath14 and as @xmath15 , otherwise . we also define the elements of the vector of infected vertices , @xmath16 . if vertex @xmath13 is infected , then @xmath17 , and , if it is not infected , @xmath18 . the result of the multiplication of the vector of infected vertices , @xmath16 , by the adjacency matrix , @xmath11 , is a vector , @xmath19 , whose element @xmath13 corresponds to the number of infected vertices that are connected to the vertex @xmath13 and may transmit the infection @xmath20 using matlab , the spread of the diseases with hypothetical parameters along the vertices of the network was simulated using the following algorithm : 1 . a proportion ( @xmath21 ) of the vertices is randomly chosen to begin the simulation infected . for our simulations , @xmath22 , since we are interested in the equilibrium state and this proportion guarantees that the disease would not disappear due to the lack of infected vertices at the beginning of the simulations . 2 . in the sis ( susceptible infected susceptible ) epidemic model , a susceptible vertex can get infected , returning , after the infectious period , to the susceptible state . for each time step : 1 . we calculate the probability ( @xmath23 ) of a susceptible vertex @xmath13 , that is connected to @xmath24 infected vertices , to get infected , using the following equation : @xmath25 where @xmath26 is the probability of disease spread . [ item2.b ] we determine which susceptible vertices were infected in this time step : if @xmath27uniform(0,1 ) @xmath28 , the susceptible vertex becomes infected . for each vertex infected , we generate the time ( @xmath29 ) that the vertex will be infected following a uniform distribution : @xmath30 uniform ( @xmath31 , @xmath32 ) , where @xmath31 and @xmath32 are , respectively , the minimum and the maximum time of the duration of the disease . 3 . decrease in 1 time step the duration of the disease on the vertices that were already infected , verifying if any of them returned to the susceptible state ; 4 . update the status of all vertices . for the si ( susceptible infected ) epidemic model , we chose @xmath29 in order to guarantee that an infected vertex remains infected until the end of the simulation . varying the values of the parameter @xmath29 , we simulated the behaviour of hypothetical acute and chronic diseases , using different values of @xmath26 , considering that once a vertex gets infected it would remain in this state during an average fixed time ( an approach that can be used when we lack more accurate information about the duration of the disease in a population ) or that there would be a variation in this period , representing more realistically the process of detection and treatment of individuals ( table [ tab : table1 ] shows the diseases simulated and the values of @xmath29 assumed ) . [ cols="^,^,^,^,^ " , ] we adopted a total time of simulation ( @xmath33 ) of 1000 arbitrary time steps . for each spreading model , we carried out 100 simulations for each network , calculating the prevalence of the simulated disease for each time step . then we calculated the average of the prevalence of these simulations on each network . after that we grouped the simulations by network algorithm . finally , we calculated the average prevalence of these network models . on the undirected networks , we observed that the disease spreads independently of the value of @xmath26 used , and that an increase in @xmath26 leads to an increase in the prevalence of the infection ( figure [ fig : figure1 ] ) . also , we observed that the prevalence tends to stabilize approximately in the same level despite the addition of vertices ( figure not shown ) . when we increase the number of edges of each vertex , there is an increase in the prevalence of the infection . a result that stands out is that , when @xmath34 , there is a great difference in the equilibrium level of the prevalence in each network . however , as we increase the value of @xmath6 , the networks tend to show closer values of equilibrium ( figure [ fig : figure3 ] ) . among the undirected networks , the networks generated using the mb algorithm presented the lowest values of prevalence in the spreading simulations ( figure [ fig : figure4 ] ) . on the directed networks , we observed that , despite the simulations of acute diseases , the disease spreads independently of the value of @xmath26 used and , as in the undirected networks , an increase in @xmath26 leads to an increase in the prevalence of the infection ( figure [ fig : figure5 ] ) . also , similarly to what was observed for undirected networks , the prevalence tends to stabilize approximately in the same level despite the addition of vertices ( figure not shown ) . when we increase the number of edges of each vertex , there is an increase in the prevalence of the infection ( figure [ fig : figure7 ] ) . a result that stands out is that , when @xmath35 , the prevalence in the kalisky networks tend to stabilize in a level a little bit higher than the other ones . among the networks , those generated using the ba algorithm presented the lowest values of prevalence in the spreading simulations ( figure [ fig : figure8 ] ) . to compare the numerical results of the simulation with a theoretical approach , it is possible to deduce , for an undirected scale - free network assembled following the ba algorithm , the equilibrium prevalence , given by @xcite @xmath36 . \label{eq:3}\ ] ] this expression applies to the ba undirected network with @xmath37 and a fixed infectious period of one time unit . for instance , for @xmath38 and @xmath39 , we obtain @xmath40 . in figure [ fig : figure9 ] , we observe that the equilibrium prevalence in the simulation reaches the value predicted by equation ( [ eq:3 ] ) . our approach , focusing on different networks with the same degree distribution , allows us to show how the topological features of a network may influence the dynamics on the network . analyzing the results of the spreading simulations , we have , as expected , that the variation in the number of vertices of the hypothetical networks had little influence in the prevalence of the diseases simulated , a result that is consistent with the characteristics of the scale - free complex networks as observed by pastor - satorras and vespignani @xcite . with respect to the effect of the increase of the probability of spreading on the prevalence in undirected networks ( figure [ fig : figure1 ] ) , we observed that the prevalence reaches a satured level . for undirected networks , if the probability of infection is high , there is a saturation of infection on the population for chronic diseases and therefore no new infections can occur . regarding the variation in the number of edges , the increase in the prevalence was also expected since it is known that the addition of edges increases the connectivity on the networks studied , allowing a disease to spread more easily . about the effect of considering the networks directed or undirected , we have that the diseases tend to stay in the undirected networks independently of the spreading model and the value of @xmath26 considered @xcite , while in the directed cases due to the limitations imposed by the direction of the movements , when @xmath34 , the acute diseases tend to disappear on some of the networks . also due to the direction of the links , in directed networks , the disease may not reach or may even disappear from parts of the network , explaining to some degree why the prevalence in directed networks ( figure [ fig : figure3 ] ) is smaller than in undirected networks ( figure [ fig : figure7 ] ) . in the si simulations , we could observe what would be the average maximum level of prevalence of a disease on a network and how fast this level would be reached . in the sis simulations , the oscillations on the stability of the prevalence observed result from the simultaneous recovery of a set of vertices . with a fixed time of infection , the set of vertices that simultaneously recover is greater than in the case of a variable time of infection , since in the latter , due to the variability of the disease duration , the vertices form smaller subsets that will recover in different moments of the simulation . a result that called attention is that , in the directed networks with @xmath34 , when we simulated the chronic diseases using a fixed time , the equilibrium levels achieved were lower than the ones achieved when we used a variable time . examining the results of the simulations on each network model , we have that among the undirected ones , the mb network has the lowest prevalence , with a plausible cause for this being how this network is composed , since there is a large number of vertices that are not connected to the most connected component of the network @xcite . among the directed networks , the ba network has the lowest prevalence observed , what is also probably due to the topology of this network , since it is composed of many vertices with outgoing links only and a few vertices with many incoming and few outgoing links , thus preventing the spread of a disease . using the methodology of networks , it is possible to analyze more clearly the effects that the heterogeneity in the connections between vertices have on the spread of infectious diseases , since we observed different prevalence levels in the networks generated with the same degree distribution but with different topological structures . moreover , considering that the increase in the number of edges led to an increase in the prevalence of the diseases on the networks , we have indications that the intensification of the interaction between vertices may promote the spread of diseases . so , as expected , in cases of sanitary emergency , the prevention of potentially infectious contacts may contribute to control a disease . this work was partially supported by fapesp and cnpq .

the transmission dynamics of some infectious diseases is related to the contact structure between individuals in a network . we used five algorithms to generate contact networks with different topological structure but with the same scale - free degree distribution . we simulated the spread of acute and chronic infectious diseases on these networks , using si ( susceptible infected ) and sis ( susceptible infected susceptible ) epidemic models . in the simulations , our objective was to observe the effects of the topological structure of the networks on the dynamics and prevalence of the simulated diseases . we found that the dynamics of spread of an infectious disease on different networks with the same degree distribution may be considerably different . * adv . studies theor . phys . , vol . 7 , 2013 , no . 16 , 759 - 771 * * hikari ltd , www.m-hikari.com * * http://dx.doi.org/10.12988/astp.2013.3674 * * raul ossada , jos h. h. grisi - filho , * * fernando ferreira and marcos amaku * faculdade de medicina veterinria e zootecnia universidade de so paulo so paulo , sp , 05508 - 270 , brazil copyright @xmath0 2013 raul ossada et al . this is an open access article distributed under the creative commons attribution license , which permits unrestricted use , distribution , and reproduction in any medium , provided the original work is properly cited . * keywords : * scale - free network , power - law degree distribution , dynamics of infectious diseases

arxiv

to provide physical meaning to solutions of einstein equations , is an endeavour whose relevance deserves to be emphasized @xcite . this is particularly true in the case of the levi - civita ( lc ) spacetime @xcite which after many years and a long list of works dedicated to its discussion still presents serious challenges to its interpretation ( @xcite , @xcite-@xcite , and references therein ) . this metric has two essential constants , usually denoted by @xmath3 and @xmath0 . one of them , @xmath4 has to do with the topology of spacetime and , more specifically , refers to the deficit angle . it may accordingly be related to the gravitational analog of aharonov bohmm effect @xcite , @xcite . it is however @xmath0 , the parameter which presents the most serious obstacles to its interpretation . indeed , for small @xmath0 @xmath5 lc describes the spacetime generated by an infinite line mass , with mass @xmath0 per unit coordinate length . when @xmath6 the spacetime is flat @xcite . however , circular timelike geodesics exits only for @xmath7 becoming null when @xmath8 and being spacelike for @xmath9 furthermore , as the value of @xmath0 increases from @xmath10 to @xmath11 the corresponding kretschmann scalar diminishes monotonically , vanishing at @xmath12 , and implying thereby that the space is flat also when @xmath13 still worse , if @xmath14 the spacetime admits an extra killing vector which corresponds to plane symmetry @xcite ( also present of course in the @xmath12 case ) . thus , the obvious question is : what does lc represents for values of @xmath15 outside the range ( 0,@xmath10 ) ? the absence of circular test particle orbits for @xmath16 , and the fact that most of the known material sources for lc , @xcite , @xcite , @xcite , @xcite require @xmath17 , led to think that lc describes the field of a cylinder only if @xmath0 ranges within the ( 0,@xmath10 ) interval . however , interior solutions matching to lc exist,@xcite , @xcite,@xcite , @xcite with @xmath16 . furthermore , the absence of circular test particle orbits for @xmath16 may simply be interpreted , as due to the fact that the centrifugal force required to balance the gravitational attraction implies velocities of the test particle larger than 1 ( speed of light ) @xcite . this last argument in turn , was objected in the past on the basis that kretschmann scalar decreases as @xmath0 increases from @xmath10 to @xmath11 , suggesting thereby that the gravitational field becomes weaker @xcite , @xcite . however , as it has been recently emphasized @xcite , @xcite , kretschmann scalar may not be a good measure of the strength of the gravitational field . instead , those authors suggest that the acceleration of the test particle represents more suitably the intensity of the field . parenthetically , this acceleration increases with @xmath0 in the interval ( @xmath10,@xmath11 ) . on the basis of the arguments above and from the study of a specific interior solution matched to lc @xcite , bonnor @xcite proposes to interpret lc as the spacetime generated by a cylinder whose radius increases with @xmath0 , and tends to infinity as @xmath0 approaches @xmath11 . this last fact suggests that when @xmath1 , the cylinder becomes a plane . this interpretation of the @xmath1 case was already put forward by gautreau and hoffman in @xcite ( observe that theirs @xmath0 is twice ours ) , though based on different considerations . however , in our opinion , the question is not yet solved . indeed , the interior solution analyzed in @xcite is not valid when @xmath12 . therefore the vanishing of the normal curvatures of the coordinate lines on the bounding surface when @xmath18 , suggests but does not prove that the exterior solution with @xmath1 has a plane source . the lc spacetime has no horizons . according to our present knowledge of the formation of black holes , this seems to indicate that there is an upper limit to the mass per unit length of the line sources , and this limit has to be below the critical linear mass , above which horizons are expected to be formed @xcite . the anisotropic fluid @xcite with @xmath19 matched to lc , produces an effective mass per unit length that has maximum at @xmath20which might explain the inexistence of horizons . furthermore , this fact might support too the previous acceleration representation of the field intensity . it agrees with the result that the tangential speed @xmath21 of a test particle @xcite in a circular geodesics increases with @xmath0 , attaining @xmath22 for @xmath23 the source studied in @xcite remains cylindrical for @xmath20 producing a cosmic string with finite radius . however , the effective mass density by increasing up to @xmath24 and then decreasing for bigger values of @xmath0 , raises a disturbing situation of a cylindrical distribution mass not curving spacetime exactly at its maximum value . on the other hand , there exists a puzzling asymmetry between the negative and the positive mass case , for the plane source . the point is that , as mentioned before , the @xmath14 case posseses plane symmetry and furthermore test particles are repelled by the singularity . therefore lc with @xmath14 , has been interpreted as the gravitational field produced by an infinite sheet of negative mass density @xcite ( though there are discrepancies on this point @xcite ) . however in this case ( @xmath14 ) the space is not flat , unlike the @xmath1 case . in other words , if we accept both interpretations , i.e. @xmath1 @xmath25 represents the field produced by an infinite plane with positive ( negative ) mass density , then we have to cope with the strange fact that the negative mass plane curves the spacetime , whereas the positive mass plane does not . this asymmetry is , intuitively , difficult to understand . in favor of the plane interpretation for the @xmath1 case , point the arguments presented in @xcite , although as already mentioned , they are not conclusive . furthermore , even if we admit the arguments based on the principle of equivalence , leading to the plane interpretation of the @xmath1 case , there is a problem with the localization of the source itself ( the plane ) . indeed , it seems reasonable to assume , according to the equivalence principle , that the physical components of curvature tensor of an homogeneous static field , vanish everywhere , except on the source ( the plane ) , where they should be singular . however , when @xmath1 the space is flat everywhere ( everywhere meaning the region covered by the patch of coordinates under consideration ) , and therefore a pertinent question is : where is the source ? in the @xmath14 case , the plane interpretation is supported by the plane symmetry of the spacetime , although objections to this interpretation have been raised , on the basis that the proper distance between neighbouring paths of test particles changes with time @xcite . however , see a comment on this point , below eq.(10 ) . also , in this case , the physical components of the curvature tensor , and the cartan scalars , are singular at @xmath26 , revealing the existence of a source , however they do not vanish ( except in the limit @xmath27 ) and therefore the pertinent question here is : why does the arguments based on the equivalence principle , mentioned above , do not apply , if @xmath28 corresponds to a plane ? so , unless additional arguments are presented , we are inclined to think that either of the interpretations ( or both ) are wrong . in order to delve deeper into these questions , and with the purpose of bringing forward new arguments , we propose here to analyze some gedanken experiments with a gyroscope circumventing the axis of symmetry . the obtained expression for the total precession per revolution , @xmath29 , will depend on @xmath0 . then , analyzying the behaviour of @xmath29 as function of different physical variables we shall be able to provide additional elements for the interpretation of lc . in relation to this , we shall consider also the c - metric ( see@xcite-@xcite and references therein ) , which , as it is well known , describes , in the limit of vanishing mass parameter , the flat space as seen by an accelerated observer ( as the @xmath1 case ) . as it will be seen below , the discussion presented here does not lead to conclusive answers to the raised issues , but provides hints reinforcing some already given interpretations and , in some cases , creating doubts abouts formerly accepted points of view . in particular it appears that the interpretation of the coordinate @xmath2 as an angle coordinate seems to be untenable in some cases . a fact already brought out in @xcite . at any rate , it is our hope that the results and arguments here presented , will stimulate further discussions on this interesting problem . the paper is organized as follows . in the next section we describe the lc spacetime and the c - metric . in section 3 we give the expression for the total precession per revolution of a gyroscope circumventing the axis of symmetry and display figures indicating its dependence upon different variables . finally , results are discussed in the last section . we shall first describe the lc line element , together with the notation and conventions used here . next we shall briefly describe the c - metric . the lc metric can be written as @xcite , @xcite @xmath30 where @xmath3 and @xmath0 are constants . the coordinates are numbered @xmath31 and their range are @xmath32 with the hypersurface @xmath33 and @xmath34 being identified . as stressed in @xcite neither @xmath3 nor @xmath0 can be removed by coordinate transformations , and therefore they have to be considered as essential parameters of the lc metric . as mentioned before , @xmath3 has to do with the topology of spacetime , giving rise to an angular deficit @xmath35 equal to @xcite @xmath36 also , as commented in the introduction , the spacetime becomes flat if @xmath37 is @xmath38 or @xmath11 . in the first case , @xmath39 the line element ( 2 ) , adopts the usual form of the minkoswski interval in cylindrical coordinates ( except for the presence of @xmath3 ) in the second case , @xmath1 , the line element becomes @xmath40 this last expression corresponding to the flat spacetime described by an uniformly accelerated observer with a topological defect associated to @xmath3 . indeed , putting @xmath41 for simplificity , the transformation @xmath42 casts ( 6 ) into @xmath43 then , the components of the four - acceleration of a particle at rest in the frame of ( 6 ) ( @xmath44 , @xmath45 @xmath46 ) as measured by an observer at rest in the minkowski frame of ( 8) are @xmath47 and therefore @xmath48 thereby indicating that such a particle is accelerated , with proper acceleration @xmath49 . it is perhaps worth noticing that due to ( 4 ) and ( 7 ) , the range of the minkowski coordinate @xmath50 is rather unusual . also observe that bodies located at different points , undergo different accelerations . this implies in turn that two bodies undergoing the same proper acceleration do not maintain the same proper distance ( see p.176 in @xcite for details ) . this metric was discovered by levi - civita @xcite , and rediscovered since then by many authors ( see a detailed account in @xcite ) . it may be written in the form @xmath51 with @xmath52 where @xmath53 and @xmath54 are the two constant parameters of the solution . introducing retarded coordinates @xmath55 and @xmath56 , defined by @xmath57 @xmath58 the metric takes the form @xmath59 with @xmath60 if @xmath61 , and @xmath62 , the c - metric becomes schwarzschild . but , if @xmath63 and @xmath64 then ( 15 ) may be written , with @xmath65 @xmath66 as @xmath67 @xmath68 which can be casted into the minkowski line element by @xmath69 @xmath70 @xmath71 @xmath72 now , for a particle at rest in the ( @xmath73 ) frame ( @xmath74 , @xmath75 @xmath46 ) the components of the four - acceleration as measured by an observer at rest in the ( @xmath76 , @xmath77 , @xmath50 , @xmath78 ) frame , are @xmath79 @xmath80 then , the absolute value of the four acceleration vector for such particle is @xmath81 indicating that the locus @xmath82 is accelerated with constant proper acceleration @xmath54 . observe that in this case the ( @xmath73 ) coordinates are only restricted by @xmath83 let us consider a gyroscope circumventing the symmetry axis along a circular path ( not a geodesic ) , with angular velocity @xmath84 . then it can be shown that the total precession per revolution is given by ( see @xcite for details ) @xmath85 with @xmath86 . the tangential velocity of particles along circular trayectories ( not necessarily geodesics ) on the plane ortogonal to the symmetry axis , is given by the modulos of the four - vector ( see @xcite , @xcite,@xcite ) @xmath87 ^{-1}v^\mu , \label{25}\ ] ] with @xmath88 then , for a particle in lc spacetime @xmath89 in terms of @xmath21 , the expression for @xmath90 becomes @xmath91 . \label{28}\ ] ] next , due to the similarity of interpretation , mentioned before , between the @xmath1 case and the c - metric with @xmath63 , we shall also calculate the total precession per revolution of a gyroscope circumventing the axis of symmetry , in the space - time of the c - metric . using the rindler - perlick method @xcite , and writing the c - metric in the form @xcite @xmath92 @xmath93 with @xmath94 @xmath95 and @xmath96 one obtains , @xmath97 @xmath98 with @xmath99 if @xmath100 we recover the usual thomas precession in a minkowski spacetime . if @xmath63 and @xmath64 on the @xmath101 plane , @xmath102 ^{-1/2}\right\ } \label{33}\ ] ] which is the thomas precession modified by the acceleration factor @xmath54 ; while if @xmath62 , @xmath61 , we recover the usual fokker - de sitter expression for precession of a gyroscope in the schwarzschild metric @xcite , @xmath103 in the general case @xmath62 , @xmath104 ( on the @xmath101 plane ) , we have from ( 31 ) that , either @xmath105 or @xmath106 . in the first case ( @xmath105 ) we obtain @xmath107 ^{-1/2}\right\ } \label{35}\ ] ] whereas in the case @xmath106 , the result is @xmath108 @xmath109 ^{-1/2}\right\ } \label{36}\ ] ] however , this last case implies @xmath110 , for otherwise @xmath111 , what would change the signature of the metric . finally , the tangential velocity of the gyroscope on the circular orbit calculated from ( 23 ) for the c - metric yields @xmath112 then replacing @xmath84 by @xmath21 with ( 37 ) , into ( 35 ) , we obtain ( @xmath113 ) @xmath114 where @xmath115 and @xmath116 where @xmath117 in the last case however , remember that @xmath53 must be negative . if @xmath63 , ( 32 ) may be written ( with ( 37 ) ) as @xmath118 with @xmath119 indicating that the precession is retrograde for any @xmath120 and @xmath121 in the next section we shall discuss about the meaning of lc in the light of the information provided by ( 28 ) and ( 38 ) . let us now analyze some figures obtained from ( 28 ) and ( 38 ) . for @xmath124 ( @xmath16 ) the precession is always forward ( @xmath125 ) as it obvious from ( 28 ) . however for @xmath126 ( @xmath127 ) it may be retrograde ( @xmath128 ) depending on @xmath129 and @xmath21 , as indicated in figure(2 ) , figure(3 ) . thus the cases @xmath130 ( @xmath1 ) and @xmath131 ( @xmath14 ) induce very different behaviours on gyroscopes . this fact , together with the assymmetry mentioned in the introduction , reinforces our doubts about the simultaneous interpretation of both cases ( @xmath132 ) as due to infinite sheet of either positive or negative mass density . next , let us consider the c - metric in the @xmath63 case . figure(4 ) shows the behaviour of the gyroscope as function of the acceleration . observe that the precession is retrograde , in contrast with the @xmath130 case , for which @xmath133 is always positive . this behaviour is the opposite for lc and @xmath134 equal to @xmath135 ( see fig.(5 ) ) , and reinforces further the difficulty of interpreting @xmath2 ( in lc with @xmath130 ) as the usual azhimutal angle . still worse , in this later case , @xmath90 always exceed @xmath136 indicating that the precession is forward even in the rotating frame . now , since both cases ( c - metric with @xmath63 and lc with @xmath130 ) represent the same physical situation ( i.e. flat space described by an uniformly accelerated observer ) then we have to conclude that the meaning of @xmath2 in lc with @xmath130 , is different from its usual interpretation ( as an angle ) . this also becomes apparent from the definition of @xmath21 given by ( 27 ) ( the tangential velocity decreases as @xmath137 ) . also observe that in the case of the c - metric with @xmath63 , we recover the thomas precession in the limit @xmath138 . this however is impossible in the lc case with @xmath130 . in the same order of ideas it is worth noticing that in the case @xmath131 , the meaning of @xmath2 seems to correspond ( qualitatively ) to that of an azimuthal angle . on the other hand , it is clear that in the case of a plane source we should not expect @xmath2 to behave like an angle coordinate ( see also @xcite on this point ) . therefore , on the basis of all comments above , we are inclined to think ( as in @xcite ) that the @xmath139 @xmath11 case corresponds to an infinite plane . the absence of singularities in the physical components of the curvature tensor , remaining unexplained , although ( probably ) related to the restrictions on the covering , of the coordinate system . by the same arguments it should be clear that the interpretation of the @xmath131 case as due to a plane , seems to be questionable .

the physical meaning of the levi - civita spacetime , for some critical values of the parameter @xmath0 , is discussed in the light of gedanken experiments performed with gyroscopes circumventing the axis of symmetry . the fact that @xmath1 corresponds to flat space described from the point of view of an accelerated frame of reference , led us to incorporate the c - metric into discussion . the interpretation of @xmath2 as an angle coordinate for any value of @xmath0 , appears to be at the origin of difficulties .

arxiv

quantum information processing has attracted a lot of interest in recent years , following deutsch s investigations @xcite concerning the potentiality of a quantum computer , i.e. , a computer where information is stored and processed in quantum systems . their application as quantum information carriers gives rise to outstanding possibilities , like secret communication ( quantum cryptography ) and the implementation of quantum networks and quantum algorithms that are more efficient than classical ones @xcite . many investigations concern the transmission of quantum information from one party ( usually called alice ) to another ( bob ) through a communication channel . in the most basic configuration the information is encoded in qubits . if the qubits are perfectly protected from environmental influence , bob receives them in the same state prepared by alice . in the more realistic case , however , the qubits have a nontrivial dynamics during the transmission because of their interaction with the environment @xcite . therefore , bob receives a set of distorted qubits because of the disturbing action of the channel . up to now investigations have focused mainly on two subjects : determination of the channel capacity @xcite and reconstruction schemes for the original quantum state under the assumption that the action of the quantum channel is known @xcite . here we focus our attention on the problem that precedes , both from a logical and a practical point of view , all those schemes : the problem of determining the properties of the quantum channel . this problem has not been investigated so far , with the exception of very recent articles @xcite . the reliable transfer of quantum information requires a well known intermediate device . the knowledge of the behaviour of a channel is also essential to construct quantum codes @xcite . in particular , we consider the case when alice and bob use a finite amount @xmath0 of qubits , as this is the realistic case . we assume that alice and bob have , if ever , only a partial knowledge of the properties of the quantum channel and they want to estimate the parameters that characterize it . the article is organized as follows . in section [ generaldescript ] we shall give the basic idea of quantum channel estimation and introduce the notation as well as the tools to quantify the quality of channel estimation protocols . we shall then continue with the problem of parametrizing quantum channels appropriately in section [ parametrization ] . then we are in a position to envisage the estimation protocol for the case of one parameter channels in section [ oneparameter ] . in particular , we shall investigate the optimal estimation protocols for the depolarizing channel , the phase damping channel and the amplitude damping channel . we shall also give the estimation scheme for an arbitrary qubit channel . in section [ qubitpauli ] we explore the use of entanglement as a powerful nonclassical resource in the context of quantum channel estimation . section [ quditpauli ] deals with higher dimensional quantum channels before we conclude in section [ conclude ] . the determination of all properties of a quantum channel is of considerable importance for any quantum communication protocol . in practice such a quantum channel can be a transmission line , the storage for a quantum system , or an uncontrolled time evolution of the underlying quantum system . the behaviour of such channels is generally not known from the beginning , so we have to find methods to gain this knowledge . this is in an exact way only possible if one has infinite resources , which means an infinite amount of well prepared quantum systems . the influence of the channel on each member of such an ensemble can then be studied , i.e. , the corresponding statistics allows us to characterize the channel . in a pratical application , however , such a condition will never be fulfilled . instead we have to come along with low numbers of available quantum systems . we therefore can not determine the action of a quantum channel perfectly , but only up to some accuracy . we therefore speak of channel estimation rather than channel determination , which would be the case for infinite resources . a quantum channel describes the evolution affecting the state of a quantum system . it can describe effects like decoherence or interaction with the environment as well as controlled or uncontrolled time evolution occuring during storage or transmission . in mathematical terms a quantum channel is a completely positive linear map @xmath1 ( cp - map ) @xcite , which transforms a density operator @xmath2 to another density operator @xmath3 each quantum channel @xmath1 can be parametrized by a vector @xmath4 with @xmath5 components . for a specific channel we shall therefore write @xmath6 throughout the paper . depending on the initial knowledge about the channel , the number of parameters differs . the goal of channel estimation is to specify the parameter vector @xmath4 . the protocol alice and bob have to follow in order to estimate the properties of a quantum channel is depicted in figure [ figurescheme ] . alice and bob agree on a set of @xmath0 quantum states @xmath7 , which are prepared by alice and then sent through the quantum channel @xmath8 . therefore , bob receives the @xmath0 states @xmath9 . he can now perform measurements on them . from the results he has to deduce an estimated vector @xmath10 which should be as close as possible to the underlying parameter vector @xmath4 of the quantum channel . quantum state @xmath11 to bob . the channel maps these states onto the states @xmath12 , on which bob can perform arbitrary measurements . note that bob s measurements are designed with the knowledge of the original quantum states @xmath11 . his final aim will be to present an estimated vector @xmath10 being as close as possible to the underlying parameter vector @xmath4.,width=302 ] how can we quantify bob s estimation ? to answer this we introduce two errors or cost functions which describe how good the channel is estimated . the first obvious cost function is the _ statistical error _ c_s(n,)_=1^l ( _ -_^est(n))^2 [ ciesse ] in the estimation of the parameter vector @xmath4 . note that the elements of the estimated parameter vector @xmath13 strongly depend on the available resources , i.e. the number @xmath0 of systems prepared by alice . we also emphasize that @xmath14 describes the error for one single run of an estimation protocol . however , we are not interested in the single run error ( [ ciesse ] ) but in the average error of a given protocol . therefore , we sum over all possible measurement outcomes @xmath15 to get the _ mean statistical error _ @xmath16 while keeping the number @xmath0 of resources fixed . though this looks as a good benchmark to quantify the quality of an estimation protocol it has a major drawback . the cost function @xmath14 strongly depends on the parametrization of the quantum channel . while this is not so important if one compares different protocols using the same parametrization it anyhow would be much better if we could give a cost function which is independent of any specifications . we define such a cost function with the help of the average overlap ( c_1,c_2 ) _ [ channelfidelity ] between two quantum channels @xmath17 and @xmath18 , where we average the fidelity @xcite f(_1,_2)^2 [ mixedstatefidelity ] between two mixed states @xmath19 and @xmath20 over all possible pure quantum states @xmath21 . since the fidelity ranges from zero to one , the _ fidelity error _ is given by c_f(n,)1-f(c_,c_^est(n ) ) [ costfct_f ] which now is zero for identical quantum channels . again we average over all possible measurement outcomes to get the _ mean fidelity error _ _ f(n,)c_f(n,)_j [ costfct_fav ] which quantifies the whole protocol and not a specific single run . in the first part of this paper we are only dealing with qubits described by the density operator 12(1+s ) [ blochvector ] with bloch vector @xmath22 and the pauli matrices @xmath23 . the action of a channel is then completely described by the function s=c(s ) , [ mapbloch ] which maps the bloch vector @xmath24 to a new bloch vector @xmath25 . in particular , the mixed state fidelity equation ( [ mixedstatefidelity ] ) , for two qubits with bloch vectors @xmath26 and @xmath27 , see equation ( [ blochvector ] ) , then simplifies to @xcite f(s_1,s_2)=12[fidss1 ] which leads to an average channel overlap , equation ( [ channelfidelity ] ) , ( c_1,c_2 ) f(c_1(n),c_2(n ) ) [ fidss2 ] for the two qubit channels @xmath17 and @xmath18 . as emphasized above we only average over pure input states with unit bloch vector @xmath28 and bloch sphere element @xmath29 . by inserting equations ( [ fidss1 ] ) and ( [ fidss2 ] ) into equations ( [ costfct_f ] ) and ( [ costfct_fav ] ) we get a cost function for the comparison of two qubit channels which is independent of the chosen parametrization . as we have already mentioned in section [ generaldescript ] , the qubits that alice sends to bob are fully characterized by their bloch vector . therefore , the disturbing action of the channel modifies the initial bloch vector @xmath30 according to equation ( [ mapbloch ] ) . it has been shown that for completely positive operators the action of the quantum channel @xmath31 is given by an affine transformation @xcite @xmath32 where @xmath33 denotes a @xmath34 invertible matrix and @xmath35 is a vector . the transformation is thus described by 12 parameters , 9 for the matrix @xmath33 and 3 for the vector @xmath35 . these 12 parameters have to fullfill some constraints to guarantee the complete positivity of @xmath36 @xcite . the definition of the parameters is somewhat arbitrary . we will not use the parameters as defined in @xcite , but adopt a different parametrization @xmath37 in terms of the parameter vector @xmath38 . in general , the choice of the best parametrization of quantum channels depends on the relevant features of the channel and also on which observables can be measured . the reason for the choice of the parametrization , equation ( [ ssprime ] ) , will become clear at the end of the section [ oneparameter ] , in which a protocol for the general channel is described . although the characterization of the general quantum channel requires the determination of 12 parameters , only a smaller number of parameters must be determined in practice for a given class of quantum channels . indeed , the knowledge of the properties of the physical devices used for quantum communication gives information on some parameters and allows to reduce the number of parameters to be estimated . we shall now examine in detail some known channels described by only one parameter . the first channel we consider is the depolarizing channel . the relation ( [ eq12 ] ) between the bloch vectors @xmath30 and @xmath39 of alice s and bob s qubit , respectively , reduces to the simple form @xmath40 where @xmath41 is the only parameter that describes this quantum channel . this channel is a good model when quantum information is encoded in the photon polarization that can change along the transmission fiber via random rotations of the polarization direction . if alice prepares the qubit in the pure state @xmath42 , the qubit bob receives is described by the state @xmath43 where @xmath44 denotes the state orthogonal to @xmath45 . we note that the depolarizing channel has no preferred basis . therefore , its action is isotropic in the direction of the input state . this means that the action of the channel is described by equation ( [ dep ] ) even after changing the basis of states . bob must estimate @xmath46 , which ranges between 0 ( noiseless channel ) and 1/2 ( total depolarization ) . the estimation protocol is the following : alice prepares @xmath0 qubits in the pure state @xmath47 which denotes spin up in the direction @xmath48 , equation ( [ enne ] ) , in terms of eigenvectors @xmath49 and @xmath50 of @xmath51 . the state @xmath52 is sent to bob through the channel . bob knows the direction @xmath48 and measures the spin of the qubit he receives along @xmath48 . the outcome probabilities are @xmath53 since alice sends a finite number of qubits , bob can only determine frequencies of measurements instead of probabilities . after bob has measured @xmath54 qubits with spin down and the remaining @xmath55 with spin up , his estimate of @xmath46 is @xmath56 . note that in a single run of the probabilistic estimation method we can get results @xmath57 , which is outside the range of the depolarizing channel . however , for the calculation of the average errors we do not have to take that into account . the cost function @xmath58 , equation ( [ ciesse ] ) , is thus @xmath59 the average error is easily obtained when one considers that bob s frequencies of measurements occur according to a binomial probability distribution , since each of the @xmath60 measurements of spin down occurs with probability @xmath46 and each of the spin up measurements occurs with probability @xmath61 . therefore , the mean statistical error reads @xmath62 this function , shown in figure [ depolarizingfigure ] a ) , scales with the available finite resources @xmath0 . it vanishes when the channel faithfully preserves the polarization , @xmath63 . the largest average error occurs for @xmath64 , when the two actions of preserving and changing the polarization have the same probability to occur . equation ( [ eq16 ] ) , and b ) the fidelity mean error @xmath65 , equation ( [ eqdepolarizingcf ] ) , for a depolarizing channel with parameter @xmath46 , are shown for different values of the number of qubits n.,width=453 ] the cost function @xmath66 is obtained from equations ( [ costfct_f ] ) and ( [ fidss2 ] ) @xmath67 ^ 2,\ ] ] which leads to the fidelity mean error @xmath68 which is shown in figure [ depolarizingfigure ] b ) . although @xmath69 does not show a simple @xmath70dependency , it clearly decreases for increasing values of the number of qubits @xmath0 . for a given quality @xmath71 or @xmath72 one can use figure [ depolarizingfigure ] and read off the number @xmath0 of needed resources . the phase - damping channel acts only on two components of the bloch vector , leaving the third one unchanged : @xmath73 here @xmath74 is the damping parameter . this channel , contrarily to the depolarizing channel , has a preferred basis . in terms of density matrices , it transforms the initial state @xmath75 where @xmath76 , etc . , into @xmath77 we note that here the parameter @xmath46 only appears in the off - diagonal terms . for this reason , the phase damping channel is a good model to describe decoherence @xcite . indeed , a repeated application of this channel leads to a vanishing of the off diagonal terms in @xmath78 , whereas its diagonal terms are preserved . since the parameter @xmath46 of the phase damping channel appears in the off - diagonal elements of @xmath78 , the protocol is the following : alice sends @xmath0 qubits with the bloch vector in the @xmath79 plane ; for instance , qubits in the state @xmath80 can be used . this would correspond to a density operator whose matrix elements are all equal to 1/2 , can be used . the density matrix of bob s qubit is then given by @xmath81 now he measures the spin in the @xmath82 direction . the theoretical probabilities are @xmath83 and we denote the frequency of a spin down measurement as @xmath84 , leading to @xmath56 . the mean statistical error has again the form @xmath85 as for the depolarizing channel . for the fidelity , equation ( [ fidss1 ] ) , we obtain @xmath86 \left(s_{1}^{2}+s_{2}^{2}\right)\end{aligned}\ ] ] and after averaging over the bloch surface , according to equation ( [ fidss2 ] ) , @xmath87 therefore the fidelity mean error @xmath72 reads @xmath88 \nonumber \\ = \frac{4}{3}\left [ \lambda \left ( 1-\lambda\right ) -\frac{1}{n}\sum_{i=0}^{n } { n \choose i } \lambda^{i}\left ( 1-\lambda \right)^{n - i } \sqrt{\lambda(1-\lambda)i(n - i ) } \right ] \label{eqphasecf}\end{aligned}\ ] ] which is shown in figure [ phasedampingfigure ] b ) . this mean error is very similar to that obtained for the depolarizing channel . , equation ( [ eq7 ] ) , and b ) the fidelity mean error @xmath65 , equation ( [ eqphasecf ] ) , for a phase damping channel with parameter @xmath46 , are shown for different values of the number of qubits n.,width=453 ] the amplitude damping channel affects all components of the bloch vector according to @xmath89 where @xmath90 is the damping parameter . the density matrix , equation ( [ denmat ] ) , is transformed into @xmath91 this channel is a good model for spontaneous decay @xcite from an atomic excited state @xmath92 to the ground state @xmath50 . repeated applications of this channel cause all elements but one of the density matrix to vanish . now the parameter @xmath46 appears in all the elements of @xmath78 and the channel clearly possesses a preferred basis . if alice and bob know that they are using an amplitude damping channel , alice sends all @xmath0 qubits in the state @xmath93 . the density operator of the qubit received by bob is @xmath94 he measures the spin along the @xmath95 direction ( the diagonal elements of @xmath78 are the probabilities to find spin down and spin up , respectively ) . we denote the frequency of spin down measurements with @xmath84 . the statistical cost function is again @xmath96 using equation ( [ eq28 ] ) we obtain @xmath97s_{3 } \nonumber \\ \left . + \lambda \lambda^{\rm est}+ ( 1-s_{3})^2\sqrt{\lambda(1-\lambda ) } \sqrt{\lambda^{\rm est}(1-\lambda^{\rm est } ) } \right]\end{aligned}\ ] ] for the fidelity cost function , equation ( [ fidss1 ] ) , and @xmath98\end{aligned}\ ] ] for the averaged fidelity ( [ fidss2 ] ) . thus @xmath99 \nonumber \\ = \frac{1}{3 } \left [ 1+\lambda(1 - 2\lambda)-\sqrt{\frac{1-\lambda}{n } } \sum_{i=0}^{n } { n \choose i } \lambda^{n - i}(1-\lambda)^{i}\sqrt{i } \right . \nonumber \\ \left . -\frac{2}{n}\sqrt{\lambda(1-\lambda ) } \sum_{i=0}^{n } { n \choose i } \lambda^{i}(1-\lambda)^{n - i}\sqrt{i(n - i ) } \right ] \label{eqamplitudecf}\end{aligned}\ ] ] which is illustrated by figure [ amplitudedampingfigure ] . , equation ( [ eq9 ] ) , and b ) the fidelity mean error @xmath65 , equation ( [ eqamplitudecf ] ) , for the amplitude damping channel with parameter @xmath46 , are shown for different values of the number of qubits n.,width=453 ] we see from the figures ( [ depolarizingfigure])([amplitudedampingfigure ] ) that for a fixed number of resources the mean error @xmath72 has a local maximum for small values of the parameter @xmath46 . for the amplitude damping channel there is also a second local maximum for large values of @xmath46 . we now come to the problem of estimating the 12 parameters @xmath100 of the general quantum channel . the protocol is summarized in table 1 : alice prepares 12 sets of qubits , divided into 4 groups . bob measures the spin of the three sets in each group along @xmath82 , @xmath101 , and @xmath95 , respectively . from the measurements on each set he gets an estimate of one of the parameters @xmath102 . the parametrization ( [ ssprime ] ) has been chosen for this purpose . the statistical cost function for the general channel is thus a generalization of the cost functions for one parameter channels , @xmath103 \sum_{j=1}^{12}\left ( \lambda_{j}-\lambda_{j}^{\rm{est } } \right)^2 \nonumber \\ & = & \frac{1}{n/12}\sum_{j=1}^{12}\lambda_{j}(1-\lambda_{j}).\end{aligned}\ ] ] the mean fidelity error can be calculated numerically . however , we do not give the expression here as it gives no particular further insight . up to now we have only considered estimation methods based on measuring single qubits sent through the quantum channel . however , we are not restricted to these estimation schemes . instead of sending single qubits we could use entangled qubit pairs @xcite . in this section we will demonstrate the superiority of such entanglement - based estimation methods for the estimation of the so called pauli channel by comparing both schemes ( for a different use of entanglement as a powerful resource when using pauli channels , see @xcite ) . the pauli channel is widely discussed in the literature , especially in the context of quantum error correction @xcite . its name originates from the error operators of the channel . these error operators are the pauli spin matrices @xmath104 , @xmath105 and @xmath51 . the operators define the quantum mechanical analogue to bit errors in a classical communication channel since @xmath104 causes a bit flip ( @xmath106 transforms into @xmath107 and vice versa ) , @xmath51 causes a phase flip ( @xmath108 transforms into @xmath109 ) and @xmath105 results in a combined bit and phase flip . the pauli channel is described by three probabilities @xmath110 for the occurrence of the three errors . if an initial quantum state @xmath2 is sent through the channel , the pauli channel transforms @xmath2 into @xmath111 thus we see that the density operator @xmath2 remains unchanged with probability @xmath112 , whereas with probability @xmath113 the qubit undergoes a bit flip , with probability @xmath114 there occurs a phase flip , and with probability @xmath115 both a bit flip and a phase flip take place . we will first describe the single qubit estimation scheme for the pauli channel , before we switch to the entanglement - based one in the next subsection . following the general estimation scheme presented in section [ generaldescript ] the protocol to estimate the parameters of the pauli channel , equation ( [ pau ] ) , requires the preparation of three different quantum states with spin along three orthogonal directions . alice sends ( i ) @xmath116 qubits in the state @xmath117 , ( ii ) @xmath118 qubits in the state @xmath119 , and ( iii ) @xmath118 qubits in the state @xmath120 . bob measures their spins along the direction of spin - preparation , namely the @xmath95 , @xmath82 , and @xmath101 axes , respectively . the measurement probabilities of spin down along those three directions are then given by @xmath121 the estimated parameter values can be calculated via @xmath122 \\ \lambda_{2}^{\rm{est } } & = & \frac{1}{2}\left [ \frac{i_{1}}{m}-\frac{i_{2}}{m } + \frac{i_{3}}{m}\right ] \\ \lambda_{3}^{\rm{est } } & = & \frac{1}{2}\left [ \frac{i_{2}}{m}-\frac{i_{3}}{m } + \frac{i_{1}}{m}\right]\end{aligned}\ ] ] where @xmath123 , @xmath124 , denote the frequencies of spin down results along the directions @xmath82 , @xmath101 and @xmath95 . we note that , although the probabilities @xmath102 are positive or vanish , their estimated values @xmath125 may be negative . this occurs because in the present case the measured frequencies are not the estimates of the parameters . nonetheless , the average cost functions can always be evaluated . for the statistical cost function we find @xmath126 \nonumber \\ & = & \frac{9}{2n}\left [ \lambda_{1}(1-\lambda_{1}-\lambda_{2})+\lambda_{2}(1-\lambda_2-\lambda_3 ) \right . \nonumber \\ & & \left . + \lambda_{3}(1-\lambda_3-\lambda_1)\right ] . \label{ave}\end{aligned}\ ] ] for the average error @xmath127 of the estimation with separable qubits @xcite . for fixed @xmath0 the average error has a maximum at @xmath128 , in which case all acting operators occur with the same probability . on the other hand , the average error vanishes when faithful transmission or one of the errors occurs with certainty . instead of using the statistical cost function @xmath14 we can also use the fidelity based cost function @xmath129 ( [ costfct_f ] ) . the average error @xmath130 of the estimation via separable qubits is then given by @xmath131 the cost function @xmath129 can not be calculated analytically . we shall compare the results ( [ ave ] ) and ( [ eqpaulicfsep ] ) with the same mean errors for a different estimation scheme , where entangled pairs are used , that we are now going to illustrate . all the protocols that we have considered so far are based on single qubits prepared in pure states . these qubits are sent through the channel one after another . however , one can envisage estimation schemes with different features . an interesting and powerful alternative scheme is based on the use of entangled states @xcite . in this case the estimation scheme requires alice and bob to share entangled qubits in the @xmath132 bell state . thus from @xmath0 initial qubits alice and bob can prepare @xmath133 bell states . alice sends her @xmath133 qubits of the entangled pairs through the pauli channel , which transforms the entangled state into the mixed state latexmath:[\ ] ] if we use the fidelity based cost function @xmath129 we can also write down the average error @xmath140 for the entangled estimation scheme . it reads @xmath141 and can be evaluated numerically . in figure [ deltafigure ] we show the difference @xmath142 between the average error obtained with single qubits and entangled qubits . we compare the two average errors when the same number of qubits are used . the figure shows that the use of entangled pairs always leads to an enhanced estimation and therefore we can consider entanglement as a nonclassical resource for this application . equation ( [ eqdelta ] ) between the fidelity mean errors @xmath143 , equation ( [ eqpaulicfsep ] ) , and @xmath144 , equation ( [ eqpaulicfent ] ) for the qubit pauli channel . @xmath145 is plotted as a function of @xmath146 and @xmath147 , while keeping @xmath148 and @xmath149 fixed . , width=302 ] before ending this section we want to add some comments about our findings . we have found that the mean statistical error has the form @xmath150 whenever the estimates of the parameters @xmath102 are directly given by the frequencies of measurements . this occurs also for the relevant case of the entanglement based protocol for the pauli channel , but not for the qubit based protocol . although the parametrization we have used is better indicated since the @xmath102 represent the probabilities of occurence of the logic errors , one might be tempted to use a different parametrization @xmath151 that gives again the expression ( [ univ ] ) . indeed , this can be done and actually the errors for the @xmath151 are smaller . nonetheless , one can show that in spite of this improvement , the scheme based on entangled pairs still gives an enhanced estimation even with the new parameters @xmath151 . this is not the case of the one parameter channels , where the use of entangled pairs does not lead to enhanced estimation . in the previous section we have proposed an entanglement based method for estimating the parameters which define the pauli channel . as we shall show in this section , this method can be easily extended to the case of quantum channels defined on higher dimensional hilbert spaces . let us start by considering the most general possible trace preserving transformation of a quantum system described initially by a density operator @xmath2 . this transformation can be written in terms of quantum operations @xmath152 as @xcite @xmath153 with @xmath154 . according to the stinespring theorem @xcite this is the most general form of a completely positive linear map . the set of operators @xmath155 can be interpreted as error operators which characterize the action of a given quantum channel onto a quantum system and the set of parameters @xmath156 as probabilities for the action of error operators @xmath157 . in particular , we can consider the action of the quantum channel , equation ( [ general quantum channel ] ) , onto only one , say the second , of the subsystems of a bipartite quantum system . for sake of simplicity we assume that the two subsystems are @xmath158level systems . if only the second particle is affected by the quantum channel , equation ( [ general quantum channel ] ) , then the final state is given by @xmath159 where @xmath160 is the identity operator . in the special case of an initially pure state , i. e. @xmath161 , the final state @xmath162 becomes @xmath163 where we have defined @xmath164 . our aim is the estimation of the parameter values @xmath165 which define the quantum channel . this can be done by projecting the state of the composite quantum system onto the set of states @xmath166 . as in the previous protocols , the probabilities @xmath165 can be inferred from the relative frequencies of each state @xmath167 and the quality of the estimation can be quantified with the help of a cost function . however , these states can be perfectly distinguished if and only if they are mutually orthogonal . hence , we impose the condition @xmath168 on the initial state @xmath169 . it means that the initial state @xmath169 of the bipartite quantum system should be chosen in such a way that it is mapped onto a set of mutually orthogonal states @xmath170 by the error operators @xmath157 . it is worth to remark the close analogy between this estimation strategy and quantum error correction . a non - degenerate quantum error correcting code @xcite corresponds to a hilbert subspace which is mapped onto mutually orthogonal subspaces under the action of the error operators . in this sense , a state @xmath169 satisfying the condition ( [ condition ] ) is a one - dimensional non - degenerated error correcting code . a necessary but not sufficient condition for the existence of a state @xmath171 satisfying equation([condition ] ) is given by the inequality @xmath172 where @xmath173 is the total number of error operators which describe the quantum channel and @xmath174 is the dimension of the hilbert space . this inequality shows why the use of an entangled pair enhances the estimation scheme . although the first particle is not affected by the action of the noisy quantum channel , it enlarges the dimension of the total hilbert space in such a way that the condition ( [ condition ] ) can be fulfilled . it is also clear from this inequality that if the state @xmath175 is a separable one , a set of orthogonal states can only be designed in principle if the number of error operators @xmath173 is smaller than @xmath174 . for instance , in the case of the pauli channel for qubits considered in the previous section , condition ( [ bound ] ) does not hold if we use single qubits for the estimation . the action of the pauli channel is defined by a set of @xmath176 error operators ( including the identity ) acting onto qubits ( @xmath177 ) . let us now apply our consideration to the case of a generalized pauli channel . the action of this channel is given by @xmath178 where the @xmath179 kraus operators @xmath180 are defined by @xmath181 the operation @xmath182 is the generalization of a phase flip in @xmath158 dimensions . the @xmath158dimensional bit flip is given by @xmath183 these errors occur with probabilities @xmath184 , which are normalized , @xmath185 . the states @xmath186 form a @xmath158-dimensional basis of the hilbert space of the quantum system . this extension of the pauli channel to higher dimensional hilbert spaces has been studied previously in the context of quantum error correction @xcite , quantum cloning machines @xcite and entanglement @xcite . now we will apply the estimation strategy based on condition ( [ condition ] ) to this particular channel . thus , we must look for a state of the bipartite quantum system which satisfies the set of conditions ( [ condition ] ) , for the error operators @xmath180 , i.e. @xmath187 it can be easily shown that such a state is @xmath188 where @xmath189 and @xmath190 , @xmath191 , are basis states for the two particles . in fact , the action of the operators @xmath192 onto state @xmath193 generates the basis latexmath:[\[|\psi_{\alpha,\beta}\rangle\equiv { \bf 1}\otimes u_{\alpha,\beta}|\psi_{0,0}\rangle= \frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}e^{i\frac{2\pi}{d}\alpha k } of @xmath179 maximally entangled states for the total hilbert space . thereby , the action of the generalized pauli channel yields @xmath195 . the particular values of the coefficients @xmath184 can now be obtained by projecting onto the states @xmath196 . the quality of this estimation strategy according to the statistical error is given by @xmath197 with @xmath133 the number of pairs of quantum systems used in the estimation . we have examined the problem of determining the parameters that describe a noisy quantum channel with finite resources . we have given simple protocols for the determination of the parameters of several classes of quantum channels . these protocols are based on measurements made on qubits that are sent through the channel . we have also introduced two cost functions that estimate the quality of the protocols . in the most simple protocols measurements are performed on each qubit . we have also shown that more complex schemes based on entangled pairs can give a better estimate of the parameters of the pauli channel . our investigations stress once more the usefulness of entanglement in quantum information . we acknowledge support by the dfg programme `` quanten - informationsverarbeitung '' , by the european science foundation qit programme and by the programmes `` qubits '' and `` quest '' of the european commission . m a c wishes to thank v buek for interesting discussions and remarks . 99 for recent books on the topic , see deutsch d 1985 _ proc . a _ * 400 * 97 gruska j 1999 _ quantum computing _ ( london : mcgraw hill ) ; nielsen m a and chuang i l 2000 _ quantum computation and quantum information _ ( cambridge : cambridge university press ) ; bouwmeester d , ekert a and zeilinger a 2000 _ the physics of quantum information _ ( berlin : springer ) ; g. alber _ et al_. 2001 _ quantum information _ ( berlin : springer ) schumacher b 1996 _ phys . rev . a _ * 54 2614 ; schumacher b and nielsen m a 1996 _ phys . rev . a _ * 54 2629 ; bennett c h , divincenzo d p , smolin j a and wootters w k 1996 _ phys . a _ * 54 3824 ; lloyd s 1997 _ phys . rev . a _ * 55 1613 ; bennett c h , divincenzo d p , and smolin j a 1997 _ phys . lett . _ * 78 3217 ; schumacher b and westmoreland m d 1997 _ phys . a _ * 56 131 ; adami c and cerf n j 1997 _ phys . rev . a _ * 56 3470 ; barnum h , nielsen m a and schumacher b 1998 _ phys . a _ * 57 4153 ; divincenzo d p , shor p w and smolin j a 1998 _ phys . rev . a _ * 57 830 mack h , fischer d g and freyberger m 2000 _ phys . rev . a _ * 62 * 042301 fujiwara a 2001 _ phys . a _ * 63 * 042304 fischer d g , mack h , cirone m a and freyberger m 2001 _ phys . a _ * 64 * 022309 shor p w 1995 _ phys . a _ * 52 r2493 ; calderbank a r and shor p w 1996 _ phys . rev . a _ * 54 1098 ; laflamme r , miquel c , paz j p and zurek w h 1996 _ phys . lett . _ * 77 198 ; steane a m 1996 _ phys . rev . lett . _ * 77 793 knill e and laflamme r 1997 _ phys . rev . a _ * 55 900 kraus k 1983 _ states , effects , and operations _ , lecture notes in physics vol . 190 ( berlin : springer ) stinespring w f 1955 _ proc . soc . _ * 6 * 211 jozsa r 1994 _ j. mod * 41 * 2315 fujiwara a and algoet p 1999 _ phys . a _ * 59 * 3290 for recent experiments on decoherence , see brune m , hagley e , dreyer j , maitre x , maali a , wunderlich c , raimond j m and haroche s 1996 _ phys . lett . _ * 77 * 4887 ; myatt c j , king b e , turchette q a , sackett c a , kielpinski d , itano w m , monroe c and wineland d j 2000 _ nature _ * 403 * 269 ; kokorowski d a cronin a d roberts t d and pritchard d e 2001 _ phys . rev . lett . _ * 86 * 2191 ; see also giulini d _ et al_. 1996 decoherence and the appearance of a classical world in quantum theory ( berlin : springer ) carmichael h 1993 _ an open system approach to quantum optics _ ( berlin : springer ) ; carmichael h j 1999 _ statistical methods in quantum optics i _ ( berlin : springer ) ; macchiavello c and palma g m 2001 _ preprint _ quant - ph/0107052 bennett c h and wiesner s j 1992 _ phys . lett . _ * 69 * 2881 knill e 1998 _ preprint _ quant ph/9808049 gottesmann d 1998 _ preprint _ quant ph/9802007 cerf n j 2000 _ j. mod _ * 47 * 187 fivel d i 1995 _ phys . * 74 * 835 * * * * * * * * * * * * * *

we investigate the problem of determining the parameters that describe a quantum channel . it is assumed that the users of the channel have at best only partial knowledge of it and make use of a finite amount of resources to estimate it . we discuss simple protocols for the estimation of the parameters of several classes of channels that are studied in the current literature . we define two different quantitative measures of the quality of the estimation schemes , one based on the standard deviation , the other one on the fidelity . the possibility of protocols that employ entangled particles is also considered . it turns out that the use of entangled particles as a new kind of nonclassical resource enhances the estimation quality of some classes of quantum channel . further , the investigated methods allow us to extend them to higher dimensional quantum systems . # 1|#1 # 1#1| # 1#2#1|#2 # 1#2#3#1|#2|#3

arxiv

the hadronic final state of collisions at 200 gev may provide a reference for other high - energy nuclear collisions at the relativistic heavy ion collider ( rhic ) and large hadron collider ( lhc ) . claims for novel physics at higher energies or in , or collision systems should be based on an accurate and self - consistent phenomenology for conventional processes at 200 gev . however , current theoretical and experimental descriptions of high - energy collisions appear to be incomplete . several unresolved aspects of collisions are notable : ( a ) the role of collision centrality in relation to the low-@xmath3 gluon transverse structure of the proton @xcite , ( b ) the nature and systematics of the _ underlying event _ ( ue ) defined as complementary to contributions from an event - wise - triggered high - energy dijet @xcite , ( c ) the systematics of _ minimum - bias _ ( mb ) dijet ( minijet ) production manifested in spectra and angular correlations @xcite and ( d ) possible existence and phenomenology of a _ nonjet azimuth quadrupole _ component in 2d angular correlations previously studied in collisions ( as quantity @xmath4 ) @xcite , especially in connection with a claimed same - side ridge observed in lhc angular correlations @xcite . a more detailed discussion of those issues is presented in sec . [ issues ] . in the present study we establish a more complete mathematical model for phenomenology based on the @xmath5 dependence of single - particle ( sp ) @xmath6 spectra and -integral @xmath2 densities and -integral 2d angular correlations . we confront several issues : is there any connection between @xmath5 and centrality ? is centrality a relevant concept ? a nonjet ( nj ) quadrupole component in collisions is the complement to jet - related and projectile - fragment correlations . is there an equivalent phenomenon in collisions , and what might a nj quadrupole component reveal about centrality or ue structure ? the phenomenological model should offer a conceptual context with two manifestations : ( a ) as a mathematical framework to represent data systematics efficiently , and ( b ) as a theoretical framework to provide physical interpretation of model elements via comparisons between data structures and qcd theory . preliminary responses to such questions were presented in ref . they are supplemented here by new sp density and 2d angular - correlation measurements . we emphasize the @xmath5 dependence of angular correlations from collisions , extending the two - component model ( tcm ) to include a nj quadrupole component previously extrapolated from measurements in collisions @xcite and now obtained directly from 2d angular correlations . we establish @xmath5-dependent phenomenology for soft ( projectile proton dissociation ) , hard ( parton fragmentation to mb dijets ) and nj quadrupole components in a _ three_-component model and explore possible correspondence with centrality , ue structure , dijet production and the partonic structure of projectile protons . we also present a tcm for the systematics of hadron densities on pseudorapidity @xmath2 . this article is arranged as follows : section [ issues ] summarizes open issues for collisions . section [ methods ] reviews analysis methods for two - particle correlations . section [ pptcm ] describes a two - component model for hadron production in collisions . section [ ppangcorr ] presents measured 2d angular correlations for 200 gev collisions . section [ modelfits ] summarizes the parametric results of 2d model fits to those correlation data . section [ jetcorr1 ] describes jet - related data systematics . section [ njcorr ] describes nonjet data systematics . section [ etadensity ] presents a two - component model for @xmath2 densities and the @xmath2-acceptance dependence of transverse - rapidity spectra . section [ syserr ] discusses systematic uncertainties . section [ ridgecms ] reviews same - side `` ridge '' properties and a proposed mechanism . sections [ disc ] and [ summ ] present discussion and summary . we present a summary of issues introduced in sec . i including centrality in relation to a conjectured underlying event , manifestations of mb dijets in spectra and correlations and existence and interpretation of a nj quadrupole component of 2d angular correlations . item ( a ) of sec . [ intro ] relates to interpretations of deep - inelastic scattering ( dis ) data to indicate that low-@xmath3 gluons are concentrated within a transverse region of the proton substantially smaller than its overall size . it is argued that a high-@xmath6 dijet trigger may select more - central collisions with greater soft - hadron production @xcite . the soft ( nonjet ) multiplicity increase should be observed most clearly within a narrow azimuth _ transverse region _ ( tr ) centered at @xmath8 and thought to _ exclude contributions from the triggered jets _ centered at 0 and @xmath9 . item ( b ) relates to measurements of charge multiplicity @xmath10 within the tr vs trigger condition @xmath11 and @xmath12 spectra employed to characterize the ue @xcite . substantial increase of @xmath10 with higher @xmath11 relative to a minimum - bias or non - single - diffractive ( nsd ) value is interpreted to reveal novel contributions to the ue , including _ multiple parton interactions _ ( mpi ) corresponding to a high rate of dijet production @xcite . monte carlo collision models such as pythia @xcite are tuned to accommodate such results @xcite . in ref . @xcite items ( a ) and ( b ) were considered in the context of a two - component ( soft+hard ) model ( tcm ) of hadron production as manifested in yields and spectra . it was observed that imposing a trigger condition on events does lead to selection for _ hard events _ ( containing at least one dijet ) but that the soft component of the selected events is not significantly different from a mb event sample , in contrast to expectations from ref . @xcite that increased dijet frequency should correspond to more - central collisions and therefore to a larger soft component from low-@xmath3 gluons . since apparently determines dijet rates directly @xcite it might also control centrality , but ref . @xcite concluded that further correlation measurements are required to explore that possibility . the present study responds with the dependence of mb dijet correlations and nj quadrupole systematics that speak to the issue of centrality and ue systematics . item ( c ) relates to the role of mb dijets in yields , spectra and various types of two - particle correlations . the contribution of mb dijets ( minijets @xcite ) to sp spectra was established in refs . @xcite , and the contribution of minijets to 2d angular correlations was identified in refs . however , further effort is required to establish a complete and self - consistent description of mb dijets in and yields , spectra and correlations . in ref . @xcite item ( c ) was addressed with 2d model fits applied to angular correlations from collisions at 62 and 200 gev to isolate several correlation components , including structures attributed to mb dijets and a nj quadrupole , with emphasis on the former in that study . the systematics of two components ( soft + hard ) are consistent with the tcm . the dijet ( hard - component ) trend on centrality exhibits a _ sharp transition _ near 50% fractional cross section below which collisions appear to be simple linear superpositions of binary collisions ( transparency ) and above which _ quantitative _ changes in the dijet component appear , but not in the nj quadrupole component @xcite . the mb dijet interpretation has been questioned variously , for more - central collisions @xcite or for all nuclear collisions @xcite . we wish to confirm the role of mb dijets as such via a self - consistent description of _ and _ collisions based on qcd theory . although a tcm for yields and spectra vs has been established @xcite the systematics of mb dijet production in collisions is incomplete . mb 2d angular correlations for collisions have been decomposed into soft and hard components via a single cut ( at 0.5 gev / c ) @xcite , but a tcm for angular correlations vs has not been available . in ref . @xcite mb jet - related correlation structure vs centrality was related quantitatively to spectrum hard components ( dijets ) to establish a direct link of both data formats with pqcd predictions . in the present study we carry out a similar analysis of vs trends . we also extend the tcm established on marginal to the 2d @xmath13 system to determine the distribution of minijets on the full sp momentum space . the extension to @xmath2 may provide further evidence that a mb dijet interpretation of the inferred tcm hard component is _ necessary _ as a distinct element of hadron production . item ( d ) relates to the possibility of a significant amplitude for a unique azimuth quadrupole in collisions . ( the nj quadrupole should be distinguished from the quadrupole component of a _ jet - related _ 2d peak projected onto 1d azimuth . ) measurements of a nj quadrupole component of angular correlations in collisions ( conventionally represented by parameter @xmath4 ) are found to be consistent with a simple universal trend on centrality and collision energy extrapolating to a nonzero value for collisions @xcite . the extrapolation is consistent with a qcd - theory prediction for @xmath4 in collisions @xcite . the nj quadrupole may be related to a same - side `` ridge '' reported in collisions at 7 tev ( with special cuts on and imposed ) @xcite . it has been suggested that the same - side ridge arises from the same mechanism proposed for collisions based on collective motion ( flows ) coupled to initial - state collision geometry . systematics of a possible nj quadrupole in collisions have thus emerged as an important new topic . reference @xcite considered extrapolation of nj quadrupole centrality systematics in 200 gev collisions to collisions , and further extrapolation to lhc energies based on measured rhic energy dependence . in that scenario the same - side ridge observed in lhc collisions corresponds to one lobe of the nj quadrupole . the other lobe is obscured by the presence of a dominant away - side ( as ) 1d jet peak . quantitative correspondence was observed in ref . @xcite suggesting that the nj quadrupole may play a significant role in collisions , but no direct quadrupole measurements existed . this study offers a response to that issue . measurement of nj quadrupole trends may shed light on the question of centrality [ item ( a ) ] by analogy with quadrupole systematics wherein the nj quadrupole measured by a _ per - particle _ variable first increases rapidly with centrality and then falls sharply toward zero with decreasing eccentricity , as described by a glauber model based on the eikonal approximation . since dijet production vs in collisions suggests that the eikonal approximation is not valid for that system @xcite the nj quadrupole trend could provide a critical test of the eikonal assumption for collisions . we review technical aspects of two - particle angular - correlation analysis methods applied to collisions at the rhic . further method details appear in refs . @xcite . high - energy nuclear collisions produce final - state hadrons as a distribution within cylindrical 3d momentum space @xmath14 , where @xmath6 is transverse momentum , @xmath2 is pseudorapidity and @xmath15 is azimuth angle . transverse mass is @xmath16 with hadron mass @xmath17 . pseudorapidity is @xmath18 $ ] ( @xmath19 is polar angle relative to collision axis @xmath20 ) , and @xmath21 near @xmath22 . to improve visual access to low-@xmath6 structure and simplify description of the -spectrum hard component ( defined below ) we present spectra on transverse rapidity @xmath23 $ ] . for unidentified hadrons @xmath1 , with pion mass assumed ( about 80% of hadrons ) , serves as a regularized logarithmic @xmath6 measure . a typical detector acceptance @xmath24 gev / c corresponds to @xmath25 . correlations are observed in two - particle momentum space @xmath26 . autocorrelation _ on angular subspace @xmath27 ( where @xmath28 or @xmath15 ) is derived by averaging pair density @xmath29 along diagonals on @xmath27 parallel to the sum axis @xmath30 @xcite . the averaged pair density @xmath31 on defined _ difference variable _ @xmath32 is then an autocorrelation . the notation @xmath33 rather than @xmath34 for difference variables is adopted to conform with mathematical notation conventions and to retain @xmath34 as a measure of a detector acceptance on parameter @xmath3 . for correlation structure approximately independent of @xmath35 over some limited acceptance @xmath34 ( stationarity , typical over @xmath36 azimuth and within some limited pseudorapidity acceptance @xmath37 ) angular correlations remain undistorted ( no information is lost in the projection by averaging ) . -integral 2d angular autocorrelations are thus lossless projections of 6d two - particle momentum space onto angle difference axes @xmath38 . the @xmath39 axis is divided into _ same - side _ ( ss , @xmath40 ) and _ away - side _ ( as , @xmath41 ) intervals . for collisions between two composite projectiles the collision final state ( fs ) may depend on the transverse separation of the collision partners ( impact parameter @xmath42 ) and the phase - space distribution of constituents within each projectile , collectively the initial - state ( is ) geometry . we wish to determine how the is geometry relates to an observable derived from fs hadrons and how the is influences fs hadron yields , spectra and correlations . for a - a collisions the projectile constituents are nucleons @xmath43 all sharing a common lab velocity ( modulo fermi motion ) and distributed over a nuclear volume . based on a glauber model of collisions ( assuming the eikonal approximation ) nucleons are classified as participants ( total number @xmath44 ) or spectators , and the mean number of binary encounters @xmath45 is estimated . the relation @xmath46 is a consequence of the eikonal approximation . parameters @xmath44 and @xmath45 , depending on impact parameter @xmath42 , are in turn related to macroscopic fs observable within some angular acceptance via the mb cross - section distribution @xmath47 . for collisions the constituents are partons distributed on the transverse configuration space of projectile protons _ and _ on longitudinal - momentum fraction @xmath3 ( fraction of proton momentum carried by a parton ) . one could apply a similar glauber approach to is geometry , including assumed eikonal approximation as in the description ( e.g. default pythia @xcite ) . as noted in the introduction , it is conjectured that imposing a dijet trigger should favor more - central collisions and therefore a substantial increase in soft - hadron production from low-@xmath3 gluons @xcite . however , some aspects of collision data appear to be inconsistent with such a description , specifically parton transverse position and a impact parameter . nevertheless , fs measures for number of participant low-@xmath3 partons and their binary encounters may be relevant and experimentally accessible @xcite . @xmath49 represents a basic pair density on 6d pair momentum space . the event - ensemble - averaged pair density @xmath50 derived from sibling pairs ( pairs drawn from single events ) includes the correlation structures to be measured . @xmath51 is a density of mixed pairs drawn from different but similar events . @xmath52 denotes a minimally - correlated reference - pair density derived from ( a ) a mixed - pair density or ( b ) a cartesian product of sp angular densities @xmath53 via a factorization assumption . differential correlation structure is determined by comparing a sibling - pair density to a reference - pair density in the form of difference @xmath54 representing a correlated - pair density or _ covariance _ density . _ per - particle _ measure @xmath55 has the form of pearson s normalized covariance @xcite wherein the numerator is a covariance and the denominator is approximately the geometric mean of marginal variances . in the poisson limit a marginal variance may correspond to @xmath56 . since @xmath57 it follows that the geometric mean of variances is given by @xmath58 and the normalized covariance density is a per - particle measure @xcite . the number of final - state charged hadrons @xmath59 in the denominator can be seen as a place holder . other particle degrees of freedom may be more appropriate for various physical mechanisms ( e.g. number of participant nucleons in collisions , number of participant low-@xmath3 partons in collisions ) as described below . we define @xmath60 where pair ratio @xmath61 cancels instrumental effects . that per - particle measure is not based on a physical model . in some analyses a correlation amplitude is defined as @xmath62 with @xmath63 @xcite , but such an amplitude then relies on a specific detector acceptance , is not `` portable . '' to assess the relation of data to is geometry we convert per - charged - hadron model - fit results to @xmath64 $ ] , the quantity in square brackets representing the _ number of correlated pairs _ within the detector acceptance . for this analysis we assume the soft - component density @xmath65 is an estimator for is @xmath66 ( low-@xmath3 parton participants ) . given the simplified notation @xmath67 we plot @xmath68 to convert `` per - particle '' from fs hadrons to is low-@xmath3 partons and obtain a more interpretable per - particle measure . correlations on two - particle momentum space @xmath69 can be factorized into distributions on 2d transverse - momentum space @xmath70 or transverse - rapidity space @xmath71 @xcite and on 4d angle space @xmath72 reducible with negligible information loss to autocorrelations on difference variables @xmath38 @xcite . in this study we focus on minimum - bias ( @xmath1-integral ) 2d angular correlations . each of the several features appearing in 2d angular correlations ( a correlation _ component _ ) can be modeled within acceptance @xmath37 by a simple functional form ( a model _ element _ ) , including 1d and 2d gaussians and azimuth sinusoids uniform on @xmath73 . the cosine elements @xmath74 represent _ cylindrical multipoles _ with pole number @xmath75 , e.g. , dipole , quadrupole and sextupole for @xmath76 . angular correlations can be formed separately for like - sign ( ls ) and unlike - sign ( us ) charge combinations , as well as for charge - independent ( ci = ls + us ) and charge - dependent ( cd = ls @xmath77 us ) combinations @xcite . the azimuth quadrupole ( @xmath78 fourier ) component is a prominent feature of angular correlations , represented there by symbol @xmath79 . the mean value is nominally relative to an estimated reaction plane @xcite . @xmath4 data are conventionally interpreted to represent elliptic flow , a hydrodynamic ( hydro ) response to is asymmetry in non - central collisions @xcite . if 2d angular correlations are projected onto 1d azimuth _ any _ resulting distribution can be expressed exactly in terms of a fourier series . the density of correlated pairs is then [ nf ] ( _ ) & = & |_sib - _ ref = v_0 ^ 2 + 2_m=1^ v_m^2 ( m _ ) , defining the _ power - spectrum _ elements @xmath80 of autocorrelation density @xmath81 . the corresponding _ per - pair _ correlation measure is the ratio & = & v_0 ^ 2 + 2_m=1^v_m^2 ( m _ ) . some fourier amplitudes from analysis of 1d azimuth projections may include contributions from more than one mechanism . for example , @xmath4 data from conventional 1d analysis may include contributions from jet - related ( `` nonflow '' ) as well as nj ( `` flow '' ) mechanisms @xcite . in contrast , a complete model of 2d angular correlations _ with @xmath73-dependent elements _ permits isolation of several production mechanisms including a nj quadrupole component @xcite . fourier components from 2d correlation analysis are denoted by @xmath82 or @xmath83 , and only the @xmath84 and 2 ( dipole and quadrupole ) fourier terms are _ required _ by 2d data histograms ( see sec . [ ppangcorr ] ) @xcite . for measure @xmath55 data derived from model fits to 2d angular correlations the quadrupole component is denoted by @xmath85 since the factorized reference density is @xmath86 . in the present study of 2d angular correlations we admit the possibility that a significant nj quadrupole component may persist in high - energy collisions ( not necessarily of hydro origin ) and retain the corresponding model element in the 2d data model function eq . ( [ modelfunc ] ) . the measured -integral nj quadrupole data for collisions are represented above 13 gev by @xcite [ loglog ] a_q\{}(b , ) & & |_0(b ) v_2 ^ 2\{}(b , ) + & = & c r ( ) n_bin(b ) _ 2,opt^2(b ) , where @xmath87 , the energy - dependence factor is @xmath88 , @xmath45 is the estimated number of binary encounters in the glauber model , and @xmath89 is the @xmath78 overlap eccentricity assuming a continuous ( optical - model ) nuclear - matter distribution . equation ( [ loglog ] ) describes measured @xmath1-integral azimuth quadrupole data in heavy ion collisions for all centralities down to collisions and energies above @xmath90 gev and represents factorization of energy and centrality dependence for the nj quadrupole . the 2d quadrupole data are also consistent with @xmath91 @xcite , a centrality trend that , modulo the is eccentricity , _ increases much faster than the dijet production rate_. a non - zero value @xmath92 from eq . ( [ loglog ] ) extrapolated to collisions agrees with a qcd color - dipole prediction @xcite . as one aspect of the present correlation study we confirm extrapolation of the nj quadrupole centrality trend to collisions and determine the dependence of @xmath93 . nj quadrupole systematics may help clarify the concept of centrality : is an is eccentricity relevant for collisions ; if so how does it vary with ? the two - component ( soft+hard ) model ( tcm ) of hadron production in high energy nuclear collisions has been reviewed in refs . @xcite for collisions and refs . @xcite for collisions . the tcm serves first as a mathematical framework for data description and then , after comparisons with theory , as a basis for physical interpretation of data systematics . the tcm has been interpreted to represent two main sources of final - state hadrons : longitudinal projectile - nucleon dissociation ( soft ) and large - angle - scattered ( transverse ) parton fragmentation ( hard ) . in collisions the two processes scale respectively proportional to @xmath44 ( participant nucleons @xmath43 ) and @xmath45 ( binary encounters ) . analogous scalings for collisions were considered in ref . @xcite . the ( soft + hard ) tcm accurately describes most fs hadron yield and spectrum systematics @xcite , whereas there is no significant manifestation of the nj quadrupole in yields and spectra @xcite . in contrast , the nj quadrupole plays a major role in 2d angular correlations and is measurable as such even for collisions ( per this study ) . the tcm previously applied to yields and spectra must therefore be extended to include the nj quadrupole as a third component of all high - energy nuclear collisions . an effective tcm should be complete and self - consistent , capable of describing all aspects of data from any collision system . the joint single - charged - particle 2d ( azimuth integral ) density on @xmath1 and @xmath2 is denoted by @xmath95 . the @xmath2-averaged ( over @xmath37 ) spectrum is @xmath96 . the @xmath1-integral mean angular density is @xmath97 averaged over acceptance @xmath37 ( @xmath2 averages are considered in more detail in sec . [ etadensity ] ) . the @xmath5 dependence of @xmath1 spectra was determined in ref . @xcite , and mb spectra were decomposed into soft and hard components according to the tcm . in collisions soft and hard spectrum components have fixed forms but their relative admixture varies with @xmath5 @xcite . the relation of the hard component to pqcd theory was established in ref . @xcite . in collisions the soft component retains its fixed form but the hard - component form changes substantially with centrality , reflecting quantitative modification of jet formation @xcite . identification of the hard component with jets in and more - peripheral collisions is supported by data systematics and comparisons with pqcd theory @xcite . in more - central collisions a jet interpretation for the tcm hard component has been questioned @xcite . the two - component decomposition of @xmath1 spectra conditional on uncorrected @xmath98 integrated over angular acceptance @xmath37 within @xmath36 azimuth is denoted by @xcite [ ppspec ] |_0(y_t;n_ch ) & = & s(y_t;n_ch ) + h(y_t;n_ch ) + & = & |_s(n_ch ) s_0(y_t ) + |_h(n_ch ) h_0(y_t ) , where @xmath99 and @xmath100 are corresponding @xmath2-averaged soft and hard components with corrected @xmath101 ( see sec . [ etadensity ] ) . the inferred soft and hard @xmath1 spectrum shapes [ unit normal @xmath102 and @xmath103 are independent of @xmath98 and are just as defined in refs . @xcite . the fixed hard - component spectrum shape @xmath104 ( gaussian plus power - law tail ) is predicted quantitatively by measured fragmentation functions convoluted with a measured 200 gev minimum - bias jet spectrum @xcite . figure [ fig1a ] ( left ) shows @xmath1 spectra for several @xmath98 classes . the spectra ( uncorrected for tracking inefficiencies ) are normalized by corrected - yield soft component @xmath105 . a common @xmath1-dependent inefficiency function is introduced for comparison of this analysis with corrected spectra in ref . @xcite , indicated below @xmath106 by the ratio of the two dash - dotted curves representing uncorrected @xmath107 and corrected @xmath102 soft - component models . the data spectra are represented by spline curves rather than individual points to emphasize systematic variation with @xmath98 . left : normalized spectra for six multiplicity classes of 200 gev collisions ( @xmath108 see table [ multclass ] ) . @xmath102 is the soft - component model function for corrected ( upper dash - dotted ) and uncorrected ( lower dash - dotted ) data . @xmath105 is the soft - component multiplicity assuming @xmath109 ( see text ) . the spectra are averaged over acceptance @xmath110 . right : spectrum hard components in the form @xmath111 from eq . ( [ ppspec ] ) compared to hard - component model function @xmath112 ( dashed ) . bars and carets are omitted from the figure labels . , title="fig:",width=158,height=153 ] left : normalized spectra for six multiplicity classes of 200 gev collisions ( @xmath108 see table [ multclass ] ) . @xmath102 is the soft - component model function for corrected ( upper dash - dotted ) and uncorrected ( lower dash - dotted ) data . @xmath105 is the soft - component multiplicity assuming @xmath109 ( see text ) . the spectra are averaged over acceptance @xmath110 . right : spectrum hard components in the form @xmath111 from eq . ( [ ppspec ] ) compared to hard - component model function @xmath112 ( dashed ) . bars and carets are omitted from the figure labels . , title="fig:",width=158,height=153 ] figure [ fig1a ] ( right ) shows normalized spectra from the left panel in the form @xmath113 / \bar \rho_s \approx \alpha \hat h_0(y_t)$ ] with @xmath114 . from ref . @xcite we infer @xmath115 . given that empirical relation and @xmath116 as simultaneous equations we can obtain @xmath117 , @xmath105 and @xmath118 for any @xmath98 and @xmath37 ( see details in sec . [ etayt ] ) . note that although the data hard - component shapes for the lowest two @xmath98 values deviate significantly from the @xmath112 model the integrals on remain close to the value @xmath119 . the amplitude 0.33 of unit - normal @xmath120 corresponds to the maximum of the dashed curve @xmath121 . these spectrum results are consistent with those from ref . @xcite with @xmath122 ( see sec . [ etadep ] ) . in ref . @xcite the tcm spectrum hard - component yield @xmath123 within @xmath122 was observed to vary as @xmath124 , with uncorrected @xmath125 . a refined analysis provided the more precise density relation @xmath126 . as noted , the tcm soft - component density @xmath105 is then defined by the simultaneous equations @xmath127 , and @xmath128 for some @xmath129 , consistent with pqcd plus ref . @xcite . in ref . @xcite @xmath105 is associated with the number of _ participant low-@xmath3 partons _ ( gluons ) and dijet production , proportional to the number of participant - parton binary encounters , is then observed to scale @xmath130 . based on a dijet interpretation for the hard component we define @xmath131 , where @xmath132 is the dijet frequency per collision and per unit @xmath2 , @xmath133 $ ] is the average fraction of a dijet appearing in acceptance @xmath37 , and @xmath134 is the mb mean dijet fragment multiplicity in @xmath135 . dijet fraction @xmath136 should be distinguished from initial - state eccentricity @xmath137 associated with the nj quadrupole . we also distinguish among number of dijets , number of jets , dijet mean fragment multiplicity and jet mean fragment multiplicity ( their values integrated over @xmath135 vs within some limited detector acceptance @xmath37 ) . the definition of @xmath118 above separates factors @xmath138 $ ] and @xmath139 that were combined in previous studies . the @xmath139 values estimated here are thus approximately a factor 2 ( i.e. @xmath140 ) larger than previous estimates @xcite . for 200 gev nsd collisions with @xmath141 and dijet mean fragment multiplicity @xmath142 derived from measured jet systematics the inferred jet frequency @xmath143 is inferred from spectrum data within @xmath122 . that value can be compared with the pqcd prediction @xmath144 for 200 gev collisions @xcite based on a measured jet spectrum and nsd cross section corresponding to mean - value parton distribution functions . thus , the observed nsd spectrum hard - component yield @xmath123 @xcite is quantitatively consistent with measured dijet systematics derived from event - wise jet reconstruction @xcite . if a non - nsd event sample with arbitrary mean @xmath98 is selected we employ empirical trends consistent with ref . @xcite and having their own pqcd implications , as discussed in ref . we assume for the present study that the dijet frequency varies with soft multiplicity as [ nj1 ] f(n_ch ) & & 0.027 ^2 with @xmath145 for 200 gev collisions . we define @xmath146 as the _ dijet number _ within some angular acceptance @xmath37 . for the @xmath98 range considered in this study the fraction of hard hadrons @xmath147 is not more than about 15% . the final state is never dominated by hard processes but mb dijet production does play a major role , especially for @xmath148 ( @xmath149 gev / c ) where _ most jet fragments appear_. mb two - particle correlations have been studied extensively for nsd collisions @xcite and collisions @xcite . a correspondence between jet - related correlations and @xmath1-spectrum hard components has been established quantitatively in refs . angular - correlation structure is consistent with extrapolation of the centrality systematics of angular correlations from collisions @xcite . both correlations on transverse rapidity @xmath150 and 2d angular correlations on @xmath38 from collisions are described by the tcm . @xmath150 correlations for 200 gev collisions are fully consistent with the sp spectrum results described above and in ref . the hard component corresponds ( when projected onto 1d @xmath1 ) to the hard component of eq . ( [ ppspec ] ) . figure [ ppcorr ] ( left panel ) shows @xmath150 correlations for 200 gev approximately nsd collisions . the logarithmic interval @xmath151 $ ] corresponds to @xmath152 $ ] gev / c . the two peak features correspond to tcm soft and hard components . the 2d hard component with mode near = 2.7 ( @xmath153 gev / c ) corresponds quantitatively to the 1d sp spectrum hard component modeled by @xmath104 in ref . the soft component ( us pairs only ) is consistent with longitudinal fragmentation ( dissociation ) of projectile nucleons manifesting local charge conservation . ( color online ) ( a ) minimum - bias correlated - pair density on 2d transverse - rapidity space @xmath150 from 200 gev collisions showing soft ( smaller ) and hard ( larger ) components as peak structures . ( b ) correlated - pair density on 2d angular difference space @xmath38 . hadrons are selected with @xmath154 gev / c ( @xmath155 ) . nevertheless , features expected for dijets are observed : ( i ) same - side 2d peak representing intrajet correlations and ( ii ) away - side 1d peak on azimuth representing interjet ( back - to - back jet ) correlations @xcite . , title="fig:",width=158,height=134 ] ( -85,92 ) * ( a ) * ( color online ) ( a ) minimum - bias correlated - pair density on 2d transverse - rapidity space @xmath150 from 200 gev collisions showing soft ( smaller ) and hard ( larger ) components as peak structures . ( b ) correlated - pair density on 2d angular difference space @xmath38 . hadrons are selected with @xmath154 gev / c ( @xmath155 ) . nevertheless , features expected for dijets are observed : ( i ) same - side 2d peak representing intrajet correlations and ( ii ) away - side 1d peak on azimuth representing interjet ( back - to - back jet ) correlations @xcite . , title="fig:",width=158,height=134 ] ( -85,92 ) * ( b ) * figure [ ppcorr ] ( right panel ) shows 2d angular correlations on difference variables @xmath38 . the hadron values for that plot are constrained to lie near 0.6 gev / c ( just above = 2 ) , corresponding to the saddle between soft and hard peaks in the left panel . although the hadron is very low the structures expected for jet angular correlations are still clearly evident : a ss 2d peak at the origin representing intrajet correlations and a 1d peak on azimuth at @xmath156 corresponding to interjet correlations between back - to - back jet pairs . the volume of the ss 2d peak corresponds quantitatively to the hard component of the total hadron yield inferred from @xmath1 spectrum data and to pqcd calculations @xcite . the soft component , a narrow 1d gaussian on @xmath73 including only us charge pairs , is excluded by the @xmath157 gev / c cut @xcite . angular correlation systematics have been compared to the qcd monte carlo pythia @xcite , and general qualitative agreement is observed @xcite . the correlation measure @xmath55 proportional to the number of correlated pairs per final - state hadron @xcite is analogous to ratio @xmath158 given @xmath159 correlated - pair number . in the present study we extend the results by measuring systematic variations of 2d angular correlations with parameter @xmath98 . ( -120,115 ) * ( a ) * ( -120,115 ) * ( b ) * ( -120,115 ) * ( c ) * + ( -120,115 ) * ( d ) * ( -120,115 ) * ( e ) * ( -120,115 ) * ( f ) * two - particle angular correlations are obtained with the same basic methods as employed in refs . @xcite . data for this analysis were obtained from a mb sample of collisions at @xmath160 gev . charged particles were detected with the star time projection chamber ( tpc ) . the acceptance was @xmath36 azimuth , pseudorapidity @xmath161 , and @xmath24 gev / c . the _ observed _ ( uncorrected ) charge multiplicity within the @xmath2 acceptance is denoted by @xmath98 , whereas the efficiency - corrected and @xmath6-extrapolated _ true _ event multiplicity in the acceptance is denoted by @xmath5 with corrected mean angular density @xmath162 within acceptance @xmath37 . seven event classes indexed by the observed charged - particle multiplicity are defined in table [ multclass ] . the range of corrected particle density @xmath163 is approximately 2 - 20 particles per unit @xmath2 . this analysis is based on 6 million ( m ) events , compared to 3 m events for the @xmath6-spectrum study in ref . @xcite . .multiplicity classes based on the observed ( uncorrected ) multiplicity @xmath98 falling within acceptance @xmath161 or @xmath110 . the efficiency - corrected density is @xmath162 . event numbers are given in millions ( m = @xmath164 ) . tcm parameters include @xmath109 and @xmath165 . [ cols="^,^,^,^,^,^,^,^",options="header " , ] alteration of the ss 2d peak is also notable . visually the peak appears to narrow dramatically on @xmath39 with increasing . however , 2d model fits reveal that the fitted peak width decreases only slightly . the _ apparent _ narrowing is due to superposition of the nj quadrupole component onto the 2d peak structure . the change from left to right panel is dominated by the ten - fold increase of curvature ratio @xmath166 , as indicated in fig . [ curvature ] ( right ) . in ref . @xcite the systematics of 2d angular correlations derived from 62 and 200 gev collisions were extrapolated first to peripheral collisions ( as a proxy for nsd collisions ) and then to 7 tev for comparisons with cms data . the question posed : are ss 2d peak , as dipole and nj quadrupole systematics at and below 200 gev consistent with those at 7 tev and especially with the emergence of a ss `` ridge '' structure for certain conditions imposed at that energy ? as to estimates , at 200 gev the nsd values of @xmath167 and @xmath168 from the present study are numerically consistent with the extrapolation to collisions . the value of @xmath169 for 200 gev collisions was overestimated by a factor 2 by addition of a conjectured quadrupole contribution from the as 1d peak modeled as a 1d gaussian . according to the present study the as peak for collisions is actually well described by a single dipole element . it was demonstrated that dijet production at 7 tev is consistent with extrapolation from 200 gev using factor @xmath170 ( derived from comparison of 62 and 200 gev data ) interpreted to describe the increase of participant low-@xmath3 gluons with increasing collision energy . that factor applies to the _ per - particle _ ss 2d peak amplitude @xmath167 ( intrajet correlations ) , whereas increase of as dipole amplitude @xmath168 ( jet - jet correlations ) is considerably less ( consistent with no amplitude increase from 62 to 200 gev @xcite . nj quadrupole measurements at rhic suggest that @xmath169 also increases by factor 2.3 . as to multiplicity trends for collisions , in the present study the corrected charge density for multiplicity class @xmath171 is @xmath172 , 5.5 times the mb value 2.5 , whereas at 7 tev the cms @xmath173 multiplicity class corresponds to corrected @xmath174 , 4.8 times the mb value 5.8 . in this study the @xmath171 class corresponds to measured four - fold increase of @xmath168 and @xmath167 and fifteen - fold increase of @xmath169 over their mb values . as to responses to cuts we note that about half of all mb jet fragments appear below the mode of the 200 gev spectrum hard component at 1 gev / c @xcite . in contrast , the mode of @xmath175 on is close to 3 gev / c @xcite . the cms cut @xmath176 $ ] gev / c is effectively a lower limit at 1 gev / c , since the hadron spectrum falls rapidly with . we thus expect that a lower limit imposed at 1 gev / c should reduce the as dipole substantially more than the nj quadrupole . we have reduced @xmath168 by factor 1/2 and @xmath169 by factor 2/3 to estimate the effect of cuts . table [ table ] summarizes the various estimates . the first two rows report results from the present study and the curvature ratios shown in fig . [ curvature ] . as noted in the text the 7 tev mb values are obtained by multiplying @xmath167 and @xmath169 by 2.3 and @xmath168 by 1.4 . the values for @xmath177 are obtained with factors 4 and 15 applied as for the 200 gev values for @xmath171 ( ignoring the small difference in ratios to mb multiplicities between rhic and lhc energies ) . the effect of the cut is estimated by factors 1/2 and 2/3 as noted above . results from the present study describe the reported cms 7 tev 2d angular correlations quantitatively and are generally consistent with ref . @xcite but also provide insight into the physical origins of the reported ss `` ridge . '' the large collision - energy increase combined with imposed @xmath6 and multiplicity cuts increases the nj quadrupole amplitude eight - fold relative to the as 1d jet peak , changing the ss curvature sign and producing an apparent ss ridge . in effect , the ss azimuth curvature functions as a comparator , switching from valley to ridge as one amplitude increases relative to another . a quantitative curvature change is transformed to a qualitative shape change ( mis)interpreted as emergence of a novel phenomenon at a higher energy . several open issues for high - energy collisions were summarized in sec . [ issues ] : ( a ) the role of collision centrality , ( b ) the definition and nature of the underlying event or ue , ( c ) the systematics of mb dijet production and ( d ) confirmed existence and characteristics of a nj quadrupole component in angular correlations . we return to those points in light of results from this study . the measured hard components of spectra , @xmath2 densities and 2d angular correlations presented in this study complete a unified experimental and theoretical picture of mb dijet production ( no cuts ) established previously for collisions @xcite . there were no previous measurement of a nj quadrupole component for collisions . the combined dijet and quadrupole results from the present study convey important implications for claims of collectivity ( flows ) , centrality , ue studies and the mechanism of the cms ridge . the term `` collective '' or `` collectivity '' ( e.g. as recently applied to small collision systems at the lhc ) is ambiguous , since jet formation is a form of `` collectivity '' as is the nj quadrupole whatever its production mechanism . introducing the term `` collectivity '' as synonymous with `` flow '' may produce confusion . there are certainly collective aspects of collisions although it is unlikely that hydrodynamic flow ( in the sense of fluid motion resulting from particle rescattering ) plays a role . dijet production and the nj quadrupole amplitude follow characteristic trends on @xmath105 precisely over a large range of amplitudes ( 100-fold for dijets , 1000-fold for quadrupole as correlated - pair numbers ) while the underlying particle ( participant - gluon ) density varies 10-fold . how can a small collision system with extremely low particle density support a hydrodynamic phenomenon that conspires to follow the same trends over such a large density interval ? the notion of centrality ( impact parameter ) for collisions is ambiguous in principle . concerning centrality several questions arise : what does `` is geometry '' mean for collisions ? is an impact parameter relevant ? how are total , triggered dijets , transverse low-@xmath3 parton ( gluon ) density and centrality correlated ? how do those factors relate to measured ensemble - mean pdfs , _ event - wise _ participant - parton distributions on @xmath3 and initial momentum transfer ? there are certainly large event - wise fluctuations in soft - hadron and dijet production , but whether those correspond to fluctuations of is transverse geometry or some other collision aspect is in question . the need for comprehensive study of the dependence of angular correlations in relation to centrality was one motivation for the present study . results from this study support two arguments against a major role for a centrality concept : ( a ) dijet production scales as @xmath178 but the eikonal approximation implies binary - collision scaling as @xmath179 ( as for collisions ) . the observed dijet trend is consistent with encounters between all possible participant - gluon pairs in each collision , not a smaller subset determined by an impact parameter . ( b ) the nj quadrupole amplitude scales as @xmath180 over a large range consistent with _ part _ of the @xmath181 trend observed for collisions , but there is no significant reduction with a decreasing eccentricity . the combined trends suggest that is geometry is not a determining factor for either phenomenon . instead , the event - wise depth of penetration on momentum fraction @xmath3 of the projectile wave function may be the main source of variation for soft , hard and quadrupole components . as noted in sec . [ issues ] ue studies rely on several assumptions : ( a ) concentration of low-@xmath3 gluons at small radius in the proton ( inferred from dis data ) provides a correlation among centrality , soft hadron production and dijet production , ( b ) the conventionally - defined tr on azimuth includes no contribution from a triggered dijet and ( c ) multiple - parton interactions ( mpi ) may occur in jet - triggered collisions . the integrated tr yield denoted by @xmath10 is observed to increase with increasing jet - trigger condition and is interpreted to represent a soft background increasing with centrality . reference @xcite addressed part of that narrative with simulations based on the tcm for hadron production from collisions . it concluded that most dijets ( what would result from an applied trigger at lower ) emerge from _ low_-multiplicity collisions . spectrum studies already indicated that higher - multiplicity collisions do produce dijets at higher rates but are few in number and so contribute only a small fraction of the -triggered event population . a @xmath182 condition can not significantly alter the soft component or collision centrality ( if relevant ) . the present study adds the following new information : ( a ) 2d angular correlations confirm a strong contribution from mb dijets _ within the tr_. ( b ) the dijet production trend @xmath130 suggests that the eikonal approximation is invalid and that centrality is not a useful concept for collisions . ( c ) monotonic increase of the nj quadrupole @xmath183 over a large range also suggests that centrality , as manifested in a varying is eccentricity , is not a useful concept . those factors confirm the conclusions of ref . @xcite and lead to the following scenario for @xmath10 variation with a trigger : as the trigger condition is increased from zero the integrated spectrum soft component increases @xmath10 from zero to a plateau on @xmath11 . the hard - component ( jet ) contribution to @xmath10 is similarly integrated up to a plateau . @xmath10 thus has both soft and hard components exhibiting plateau structures slightly displaced from one another on @xmath11 . almost all events satisfying an increased trigger condition contain a dijet ( are hard events ) but remain characteristic of a mb population with smaller soft multiplicity , not the expected more - central population with larger soft multiplicity . given the observed -dependent structure of 2d angular correlations we arrive at three conclusions : ( a ) all dijets include a large - angle base that strongly overlaps the tr . that base dominates minimum - bias jets but may persist within all higher - energy ( e.g. -triggered ) dijets . ( b ) the region with a minimal dijet contribution that might suffice for ue studies is defined by @xmath184 and @xmath185 ( see fig . [ quadcomp ] ) . an immediate example of novel ue structure that might be discovered there is provided by the cms `` ridge , '' a manifestation of the nj quadrupole that was not expected in collisions . ( c ) the likelihood of multiple dijet production in -triggered events ( which retain a low soft multiplicity as noted ) is small whereas the likelihood of multiple dijets in high - multiplicity events approaches unity . the usual interpretation of @xmath10 trends in terms of mpi may be misleading . arguments against the tcm have been presented since commencement of rhic operation . it has been noted that the hijing monte carlo @xcite ( based on pythia @xcite ) fails to describe rhic and lhc data . that failure as been expressed as `` too slow increase '' of hadron production with centrality and energy @xcite . hijing is assumed to represent the tcm and its failure is then confused with failure of the tcm itself , of which hijing is only a specific theory implementation . the problems with hijing are traceable to the eikonal - model assumption included in default pythia @xcite . such arguments typically rely on data from a centrality range covering only the more - central 40 - 50% of the cross section @xcite . the critical centrality region extending from or collisions to the _ sharp transition _ in jet formation @xcite is not considered . alternative models that seem to describe the more - central data are actually falsified by more - peripheral data @xcite . an alternative argument is based on assuming that tcm agreement with data is accidental , that a _ constituent - quark _ model ( soft only , excluding jets ) is more fundamental and describes data as well @xcite , but that argument is questionable @xcite . the tcm invoked in this study is based on previous analysis of spectrum @xmath5 and centrality dependence @xcite , fluctuations and correlations @xcite , transverse - rapidity @xmath150 correlations @xcite and minimum - bias 2d number angular correlations @xcite . in each case model elements were determined quantitatively by systematic analysis without regard to physical mechanisms . only after the tcm was so established were connections with theory and physical interpretations introduced . in the present study we extend the tcm to describe the dependence of 200 gev @xmath2 densities and 2d angular correlations . in the latter we observe for the first time a significant nj quadrupole component and its dependence as a novel nonjet phenomenon within the ue . we find that the extended tcm remains fully self - consistent and provides accurate and efficient representation of a large body of data . we report measurements of the charge - multiplicity dependence of single - particle ( sp ) densities on transverse rapidity ( as spectra ) and pseudorapidity @xmath2 and 2d angular correlations on @xmath186 from 200 gev collisions . the sp densities are described accurately by a two - component ( soft + hard ) model ( tcm ) of hadron production . the inferred -spectrum tcm is consistent with a previous study . the result for @xmath2 densities newly reveals the distribution on @xmath2 of minimum - bias ( mb ) jet fragments . 2d angular correlations are fitted with a multi - element fit model previously applied to data from 62 and 200 gev collisions . fit residuals are consistent with statistical uncertainties in all cases . trends for several 2d correlation model parameters are simply expressed in terms of tcm soft - component multiplicity @xmath187 or mean density @xmath188 ( @xmath37 is a detector acceptance ) . correlated - pair numbers for soft component ( projectile dissociation ) scale @xmath189 , for hard component ( dijet production ) scale @xmath130 and for nonjet ( nj ) quadrupole scale @xmath183 . the nj quadrupole amplitude is quite significant for higher - multiplicity collisions . the dijet production trend is inconsistent with an eikonal approximation for collisions ( which would require dijets @xmath190 ) , and the monotonically - increasing nj quadrupole trend is inconsistent with an initial - state eccentricity determined by impact parameter . the two trends combined suggest that centrality is not a useful concept for collisions . fluctuations may instead depend on the event - wise depth of penetration on momentum fraction @xmath3 of the projectile wave functions and in turn on the number of participant low-@xmath3 partons . 2d angular - correlation data are in conflict with assumptions relating to the underlying event ( ue , the complement to a triggered dijet ) . the azimuth transverse region ( tr ) bracketing @xmath191 is assumed to contain no contribution from a triggered dijet , but minimum - bias dijets are observed to make a strong contribution there . the region with minimal jet contribution is defined by @xmath184 near the azimuth origin that excludes the same - side 2d jet peak and most of the away - side 1d jet peak . the presence of a significant nj quadrupole component and its multiplicity trend have several implications : ( a ) initial - state transverse geometry does not appear to be a useful concept for collisions as noted above , ( b ) the appearance of a nj quadrupole component in a small system with negligible particle density contradicts the concept of a hydro phenomenon based on particle rescattering and large energy / particle density gradients and ( c ) the same - side `` ridge '' observed in collisions at the large hadron collider ( lhc ) , interpreted by some to suggest `` collectivity '' ( flows ) in small systems , results from an interplay of the jet - related away - side 1d peak and the nj quadrupole that together determine the curvature on azimuth near the origin . when that curvature transitions from positive to negative ( depending on collision energy and other applied cuts ) a same - side `` ridge '' appears . in a hydro narrative the nj quadrupole component interpreted as elliptic flow should represent azimuth modulation of radial flow detected as a modification of sp spectra . but no corresponding modification is observed in spectra despite precise differential analysis . t. a. trainor , d. t. kettler , d. j. prindle and r. l. ray , j. phys . g * 42 * , no . 2 , 025102 ( 2015 ) . d. t. kettler , d. j. prindle and t. a. trainor , phys . c * 91 * , 064910 ( 2015 ) . d. t. kettler ( star collaboration ) , eur . j. c * 62 * , 175 ( 2009 ) . d. kettler ( star collaboration ) , j. phys . . ser . * 270 * , 012058 ( 2011 ) . t. a. trainor and d. t. kettler , phys . c * 84 * , 024910 ( 2011 ) . t. a. trainor , phys . d * 87 * , no . 5 , 054005 ( 2013 ) . t. a. trainor , mod . a * 23 * , 569 ( 2008 ) . t. a. trainor , int . j. mod . e * 17 * , 1499 ( 2008 ) . t. a. trainor , phys . c * 80 * , 044901 ( 2009 ) . g. agakishiev , _ et al . _ ( star collaboration ) , phys . c * 86 * , 064902 ( 2012 ) . t. a. trainor and d. t. kettler , phys . c * 83 * , 034903 ( 2011 ) . s. gavin , l. mclerran and g. moschelli , phys . c * 79 * , 051902 ( 2009 ) . t. a. trainor , phys . c * 91 * , no . 4 , 044905 ( 2015 ) . b. z. kopeliovich , a. h. rezaeian and i. schmidt , phys . d * 78 * , 114009 ( 2008 ) . cms collaboration , jhep * 1009 * , 091 ( 2010 ) . t. a. trainor , phys . c * 90 * , no . 2 , 024909 ( 2014 ) t. a. trainor , phys . c * 92 * , no . 2 , 024915 ( 2015 ) . j. adams _ et al . _ ( star collaboration ) , phys . c * 73 * , 064907 ( 2006 ) . j. adams _ et al . _ ( star collaboration ) , phys . b * 634 * , 347 ( 2006 ) . t. a. trainor , r. j. porter and d. j. prindle , j. phys . g * 31 * , 809 ( 2005 ) . r. j. porter and t. a. trainor ( star collaboration ) , acta phys . b * 36 * , 353 ( 2005 ) . t. a. trainor , phys . d * 89 * , no . 9 , 094011 ( 2014 ) . t. a. trainor , j. phys . g * 37 * , 085004 ( 2010 ) . t. a. trainor , phys . c * 81 * , 014905 ( 2010 ) . ya . i. azimov , yu . l. dokshitzer , v. a. khoze , s. i. troyan , z. phys . c * 27 * , 65 ( 1985 ) , z. phys . c * 31 * , 213 ( 1986 ) . g. wolschin , phys . c * 91 * , 014905 ( 2015 ) . t. a. trainor , j. phys . g * 40 * , 055104 ( 2013 ) . t. a. trainor , d. j. prindle and r. l. ray , phys . c * 86 * , 064905 ( 2012 ) . m. b. de kock , h. c. eggers and t. a. trainor , phys . c * 92 * , no . 3 , 034908 ( 2015 ) . b. i. abelev _ et al . _ ( star collaboration ) , phys . c * 80 * , 064912 ( 2009 ) . t. a. trainor , j. phys . g * 39 * , 095102 ( 2012 ) m. gyulassy and l. mclerran , nucl . a * 750 * , 30 ( 2005 ) . b. b. back _ et al . _ ( phobos collaboration ) , phys . b * 578 * , 297 ( 2004 ) . d. kharzeev and e. levin , phys . b * 523 * , 79 ( 2001 ) . s. s. adler _ et al . _ ( phenix collaboration ) , phys . c * 89 * , no . 4 , 044905 ( 2014 ) .

an established phenomenology and theoretical interpretation of @xmath0-@xmath0 collision data at lower collision energies should provide a reference for @xmath0-@xmath0 and other collision systems at higher energies , against which claims of novel physics may be tested . the description of @xmath0-@xmath0 collisions at the relativistic heavy ion collider ( rhic ) has remained incomplete even as claims for collectivity and other novelties in data from smaller systems at the large hadron collider ( lhc ) have emerged recently . in this study we report the charge - multiplicity dependence of two - dimensional ( 2d ) angular correlations and of single - particle ( sp ) densities on transverse rapidity @xmath1 and pseudorapidity @xmath2 from 200 gev @xmath0-@xmath0 collisions . we define a comprehensive and self - consistent two - component ( soft + hard ) model ( tcm ) for hadron production and report a significant @xmath0-@xmath0 nonjet ( nj ) quadrupole component as a third ( angular - correlation ) component . our results have implications for @xmath0-@xmath0 centrality , the underlying event ( ue ) , collectivity in small systems and the existence of flows in high - energy nuclear collisions .

arxiv

since the early years of quantum mechanics @xcite the principle of quantum superposition has been recognized to play a prominent role in the theory and its applications . the destruction and preservation of these superpositions of quantum states occupy a central place in issues such as the quantum - to - classical transition @xcite and potential technological applications in quantum information , computation and cryptography @xcite . from a physical standpoint the loss of coherence in quantum systems is rooted on the pervasive action of the environment upon the system . this environmental action has received a careful mathematical treatment ( cf . @xcite and multiple references therein ) going from a constructive approach based on disregarding the degrees of freedom of the environment due to their lack of control by the experimenter ( `` tracing - out '' methods ) to an axiomatic approach based on the initial setting of physically motivated axioms to derive an appropiate evolution ( master ) equation for the system @xcite . + most of these master equations ( me s hereafter ) satisfy the markov approximation ( semigroup condition ) and can be put into the lindblad form : @xmath0+\frac{1}{2}\sum_{j}\left\{[v_{j}\rho(t),v_{j}^{\dagger}]+[v_{j},\rho(t)v_{j}^{\dagger}\right\}\ ] ] where @xmath1 is the hamiltonian of the system and @xmath2 are operators ( so - called lindblad operators ) containing the effect of the environment upon the system . indeed in the axiomatic approach the markov approximation is posed as an initial hypothesis @xcite , thus rendering highly difficult a generalization to nonmarkovian situations . + in this work we develop a novel attempt to derive me s both in the markovian and the nonmarkovian regimes using stochastic methods @xcite jointly with well - known operator techniques commonly used in quantum mechanics @xcite . the main idea consists of building _ random _ evolution operators ( evolution operators with one or several stochastic parameters in it ) which contains the decohering effect of the environment and then taking the stochastic expectaction value with respect to this ( uncontrollable ) randomness . the paper is organized as follows . in section [ lindasrand ] we state and prove our main ( though still somewhat partial ) result , namely that any lindblad - type me , either markovian or nonmarkovian , with selfadjoint lindblad operators can be understood as an averaged random unitary evolution . in section [ analcons ] we discuss some first mathematical consequences of this result such as a very fast method to solve me s provided the unitary solutions are known ; we illustrate this by solving the phase - damping me for the multiphoton resonant jaynes - cummings model in the rotating - wave approximation @xcite ( section [ soldampjcm ] ) . we then comment in section [ nonmarkevol ] two immediate consequences , namely both markovian and nonmarkovian regimes are attainable under the same mathematical formalism and the lindbladian structure with selfadjoint lindblad operators is shown to have an origin independent of the markov approximation . in section [ intrdecoh ] we show how the flexibility of the mathematical language employed can easily generalize some intrinsic decoherence models present in the literature @xcite . in section [ rabiqed ] we discuss the previous main result under a more physical spirit by proposing a slight generalization of the jaynes - cummings model ( section [ stojcm ] ) , comparing this proposal with experimental results in optical cavities experiments ( section [ qed ] ) and finally ( section [ iondecay ] ) showing how the proposed formalim can account for reported exponential decays of rabi oscillations in ion traps . we include in section [ discuss ] some important comments regarding a brief comparison with existing models of stochastic evolution in hilbert space , the possibility of intrinsic decoherence phenomena and future prospects . conclusions and a short appendix close the paper . the main result whose consequences are to be discussed below is the following : _ every lindblad evolution with selfadjoint lindblad operators can be understood as an averaged random unitary evolution_. we will analyse this proposition in detail . the objective is to reproduce the lindblad equation . ] @xmath3+\frac{1}{2}\sum_{i=1}^{n}\{[v_{i}\rho(t),v_{i}]+[v_{i},\rho(t)v_{i}]\}\nonumber\\ \label{lindeq}&=&-i[h,\rho(t)]-\frac{1}{2}\sum_{i=1}^{n}[v_{i},[v_{i},\rho(t)]]\end{aligned}\ ] ] by adequately modifying chosen parameters in the original evolution operator . for simplicity let us start by considering the case @xmath4 . we will first study the case where the hamiltonian @xmath1 and the ( selfadjoint ) lindblad operator @xmath5 commute . it is very convenient to introduce the following notation . the commutator between an operator @xmath6 and @xmath7 will be denoted by @xmath8\equiv[g , x]$ ] . thus the von neumann - liouville operator will be @xmath9 , where @xmath1 denotes the hamiltonian ( @xmath10 ) . then the lindblad equation with @xmath4 can be arrived at by 1 . adding a stochastic term @xmath11 to the argument of the evolution operator : + @xmath12 + where @xmath13 denotes standard real brownian motion @xcite . 2 . taking the stochastic average with respect to @xmath13 in the density operator deduced from @xmath14 : + @xmath15\ ] ] + where @xmath16 denotes the expectation value with respect to the probability measure of @xmath13 . the proof of this result is nearly immediate . taking advantage of the commutativity of @xmath1 and @xmath5 and making use of theorem 3 in @xcite ( cf . appendix ; relation ) we may write for the density operator : @xmath17[\rho(0)]\ ] ] thus all we have to do is to calculate the expectation value of the random superoperator @xmath18 . developing the exponential into a power series and recalling @xcite @xmath19=\frac{(2n)!}{2^{n}n!}t^{n}$ ] if @xmath20 is even and @xmath19=0 $ ] otherwise , we arrive at @xmath21\nonumber\\ & = & \exp(-t\mathcal{l}-\frac{t}{2}\mathcal{c}_{v}^{2})[\rho(0)]\end{aligned}\ ] ] which produces the desired master equation : @xmath22-\frac{1}{2}[v,[v,\rho(t)]]\ ] ] when the hamiltonian @xmath1 and the lindblad operator @xmath5 do not commute , the previous method is not suitable , since the calculation of the expectation value can not be performed in the same way . a way to circumvent this problem is to proceed in the same way as before but in the heisenberg picture . thus let @xmath23 be the original evolution operator in the schrdinger picture . the corresponding evolution operator in the heisenberg picture will trivially be @xmath24 . as before we proceed in steps : 1 . [ nonc1 ] add a stochastic term to the argument of the evolution operator : @xmath25 + where @xmath26 denotes time - ordering and @xmath27 is the operator @xmath5 in the heisenberg representation . [ nonc2 ] take the stochastic average with respect to @xmath13 in the corresponding density operator : + @xmath28\ ] ] 3 . [ nonc3 ] finally to arrive at change to the schrdinger representation . a comment should be made . the stochastic term added to the original evolution operator in is a natural generalization of the one added in . the ito integration now appears as a consequence of the time dependence of the operator to be added : @xmath27 . note that when @xmath29=0 $ ] , @xmath30 and the stochastic term reduces to @xmath31 as before + now to perform the previous tasks is a bit more involved . firstly combining relation and the @xmath32operation , the step [ nonc1 ] can be carried over : @xmath33=\nonumber\\ \label{expvalnonc}&=&\mathcal{t}\mathbb{e}[e^{-i\int_{0}^{t}\mathcal{c}_{v_{h}(s)}d\mathcal{b}_{s}}][\rho(0)]\end{aligned}\ ] ] the expectation value can be evaluated by resorting to functional techniques @xcite . recall that the characteristic functional of a stochastic process @xmath34 is defined as @xmath35=\mathbb{e}\left[\exp\left(i\int k(t)\chi_{t}dt\right)\right]\ ] ] where @xmath36 is an arbitrary real - valued function . in particular , for white noise @xmath37 ( cf . @xcite ) @xmath38&=&\mathbb{e}[\exp\left(i\int k(s)d\mathcal{b}_{s}\right)]=\nonumber\\ & = & e^{-\frac{1}{2}\int_{0}^{t}k^{2}(s)ds}\end{aligned}\ ] ] where @xmath39 , @xmath40 and @xmath41=\delta(t - s)$ ] have been used . from this it is then clear that can be written as @xmath42\ ] ] back to the schrdinger picture , the master equation derived from is @xmath22-\frac{1}{2}[v,[v,\rho(t)]]\ ] ] the generalization to many lindblad operators is elementary : all we have to do is to use the n - dimensional standard real brownian motion @xcite @xmath43 . the strategy is the same . the first consequences one can derive from the previous result are of analytical fashion . as an immediate aplication we will show how the jaynes - cummings model with phase damping in the rotating - wave approximation can be solved provided we know the solution to the original jaynes - cummings model . as a second consequence we will discuss how the previous result can be generalized to nonmarkovian situations , thus providing a common language for both markovian and nonmarkovian evolutions . finally it is shown how existing intrinsic decoherence models are naturally generalized using this formalism . the jaynes - cummings model ( jcm hereafter ) @xcite shows an undoubtable relevance in the study of quantum systems in different fields such as quantum optics , nuclear magnetic resonance or particle physics . it is an exactly solvable model which allows us to study specifically quantum properties of nature such as electromagnetic field quantization or periodic collapses and revivals in atomic population . the jcm describes the evolution of a two - level quantum system ( the atom ) interacting with a mode of the electromagnetic field under certain approximations ( rotating wave approximation , dipole approximation , etc . @xcite for details ) . usually in normal experimental conditions this will be an idealization and the environment should be taken into account , the effect of which can be very appropiately treated introducing a phase - damping term @xcite . thus the master equation for this system will read @xmath44-\frac{\gamma}{2}[h,[h,\rho(t)]]\ ] ] where @xmath1 is the hamiltonian of the system and @xmath45 is a damping constant . here we will show how can be very easily solved when @xmath1 is the resonant multiphoton jc hamiltonian ( cf . @xcite for an alternative approach ) , i.e. when @xmath46 where @xmath47 denotes the frequency of the field mode , @xmath48 is the atomic transition frequency , @xmath49 is the atom - field coupling constant , @xmath50 and @xmath51 are the mode creation and annihilation operators respectively , @xmath52 is the atomic - inversion operator and @xmath53 are the atomic `` raising '' and `` lowering '' operators . an exact resonance is assumed , thus @xmath54 . + we will focus in two quantities of relevant physical meaning , namely the atomic inversion @xmath55 $ ] and the photon number distribution at time @xmath56 : @xmath57 . to compare with methods found in the literature @xcite we will restrict to the case in which initially the atom is in its excited state @xmath58 and the electromagnetic field is in a coherent state @xmath59 , with @xmath60 . the unitary evolution ( @xmath61 ) provides the following expressions for these quantities : @xmath62\\ p_{n}(t)&=&|q_{n}|^2\cos^{2}\left[\lambda t\sqrt{\frac{(n+m)!}{n!}}\right]+\nonumber\\ & + & |q_{n - m}|^{2}\sin^{2}\left[\lambda t\sqrt{\frac{(n+m)!}{n!}}\right]\end{aligned}\ ] ] the objective is to calculate these same quantities when the phase damping term is present in , i.e. when @xmath63 . we will make use of the result proved in the previous section and note that the equation can be obtained by adding a stochastic term to the original evolution operator and then performing the stochastic average . in our case , @xmath64 , which obviously commutes with the hamiltonian , thus we are in the first case . the original evolution operator is promoted to @xmath65 equivalently we may think that it is @xmath56 which is promoted @xmath66 . thus to arrive at the desired `` phase - damped '' expressions @xmath67 and @xmath68 all we have to do is to add a stochastic term to the time variable @xmath56 and then perform the average : @xmath69\right]\nonumber\\ & & \\ p_{n}^{pd}(t)&=&\mathbb{e}\left[|q_{n}|^2\cos^{2}\left[\lambda(t+\gamma^{1/2}\mathcal{b}_{t})\sqrt{\frac{(n+m)!}{n!}}\right]\right.+\nonumber\\ & + & \left.|q_{n - m}|^{2}\sin^{2}\left[\lambda(t+\gamma^{1/2}\mathcal{b}_{t})\sqrt{\frac{(n+m)!}{n!}}\right]\right]\nonumber\\\end{aligned}\ ] ] using the linearity property of the expectation value and recalling the moments of the standard real brownian motion ( cf . above and @xcite ) , the previous calculations can be carried over elementarily using ( see appendix [ appendix ] ) : @xmath70&=&e^{-2\gamma\lambda^{2}t\frac{(n+m)!}{n!}}\times\nonumber\\ & \times&\cos\left[2\lambda t\sqrt{\frac{(n+m)!}{n!}}\right]\nonumber\\ & & \end{aligned}\ ] ] hence @xmath71\nonumber\\ & & \\ p_{n}^{pd}(t)&=&\frac{1}{2}|q_{n}|^{2}\left\{1+e^{-2\gamma\lambda^{2}t\frac{(n+m)!}{n!}}\right\}\cos\left[2\lambda t\sqrt{\frac{(n+m)!}{n!}}\right]+\nonumber\\ & + & \frac{1}{2}|q_{n - m}|^{2}\left\{1-e^{-2\gamma\lambda^{2}t\frac{(n+m)!}{n!}}\right\}\cos\left[2\lambda t\sqrt{\frac{(n+m)!}{n!}}\right]\nonumber\\\end{aligned}\ ] ] which exactly coincides with equations ( 41 ) and ( 43 ) in @xcite and eqs . ( 3.26 ) and ( 3.27 ) in @xcite for @xmath72 . we encourage the reader to compare this method with those used in @xcite . + obviously this formalism can also be used to solve the equation with any arbitrary hamiltonian provided we already know the solution when @xmath61 . a second consequence of the formalism depicted above is its immediate generalization to nonmarkovian situations . the result in section [ lindasrand ] can be readily generalized to the following : _ any lindbladian master equation , whether markovian or nonmarkovian , but with selfadjoint lindblad operators can be obtained as the stochastic average of a random unitary evolution_. the generalization stems out from the single fact that whereas in the markovian regime we necessarily have to add a stochastic term of the form @xmath73 , in the nonmarkovian case this restriction drops out and then we may add a term like @xmath74 , where @xmath75 is an arbitrary real - valued function which encodes e.g. the time response of the environment to the system evolution . is a stochastic process . in this case , since we are interested in physical properties which are obtained after averaging , this does not suppose any actual gain . ] under these circumstances , the previous procedure ( for simplicity s sake we will only care about the commuting case ; the noncommuting case is similar ) drives us to @xmath76[\rho(0)]\ ] ] now developing the exponential again into a power series , calculating the expectation value of each term with some elementary ito calculus and resumming the series , one arrives at @xmath22-\frac{\gamma(t)}{2}[v,[v,\rho(t)]]\ ] ] where @xmath77 . this is clearly a lindbladian nonmarkovian master equation . the extension to more than one lindblad operator is again trivial . this result casts some light into the origin of the lindbladian structure of master equations with selfadjoint lindblad operators , independently of their markovian or nonmarkovian character , something beyond reach of the original axiomatic approach of @xcite . + in this sense the result proven here generalizes previous derivation of lindblad evolution using stochastic calculus @xcite by dropping out the semigroup condition . note that this generalization allows us to conclude that since @xmath77 the decoherence process is irreversible , i.e. no coherence can be recovered within the domain of validity of the phase - damping me as an evolution equation for the quantum system . + the time dependence of the decoherence factor also suggests a classification of different kinds of environments depending on the rate at which the environment decoheres the system ( cf . it remains open what the physical conditions should be to have the different decoherence factors . a third advantage appears as a natural generalization of intrinsic decoherence models already present in the literature @xcite . these two models propose an intrinsic mechanism of decoherence based on the random nature of time evolution ( we will not enter into the discussion of the physical justification of this hypothesis see original references for discussion , we will only show how they can be generalized ) , which basically drives us to the evolution equation . the starting hypotheses ( apart from the random nature of time evolution and the usual representation of a quantum system by a density operator ) are a specific probability distribution @xcite or a semigroup condition ( markovianity ) @xcite for the time evolution . as a result we obtain a _ nondissipative markovian _ master equation in both cases . + the formalism presented here dispenses with any of these specific conditions , something which allows us to obtain more general master equations , i.e. both markovian or nonmarkovian and dissipative or nondissipative . + the result comes from the combination of ito calculus and the spectral representation theorem for unitary operators @xcite . let @xmath78 be an unitary evolution operator . by means of the spectral decomposition theorem @xcite it can be written as @xmath79 where @xmath80 denotes the spectral measure of the evolution operator . now we perform the stochastic promotion as before by substituting @xmath81 where @xmath82 is real stochastic process . then ( see @xcite for details ) 1 . the markovian nondissipative master equation appearing in @xcite and @xcite is obtained if @xmath83 . but note that now the lindblad operators are not fixed by any initial assumption . if e.g. @xmath84 with correlation function @xmath85 we arrive at a lindblad equation + @xmath3-\nonumber\\ & -&\frac{\gamma}{2}\left(h^{2}\rho(t)+\rho(t)h^{2}-2he^{-\tau^2\mathcal{c}_{h}^{2}}[\rho(t)]h\right)\nonumber\\\end{aligned}\ ] ] + which is clearly different from the phase - damping master equation . thus even restricting ourselves to the same range of assumptions ( markovianity and nondissipation ) we can obtain more master equations . 2 . a nonmarkovian nondissipative master equation like e.g. + @xmath22-\frac{\lambda(t)}{2}[h,[h,\rho(t)]]\ ] ] + is obtained if @xmath86 and @xmath87 . more general equations can be readily obtained by appropiately combining the correlation properties of @xmath88 and the time dependency of @xmath89 . this allows these models to be used to explain a wider range of phenomena than that originally considered . in previous sections we have developed a method to adequately modify the original evolution operator of a quantum system to finally arrive at a lindbladian master equation . now we find legitimate to proceed the other way around , i.e. what physical predictions are derived from the assumption that a parameter in the evolution operator of a quantum system is random ? to be concrete we will focus upon two different physical systems , namely a rydberg atom in an optical cavity and a linear rf ( paul ) ion trap . we will confront the previous hypothesis with experimental results . the jcm model describes the interaction between an atom and the electromagnetic field under very special conditions @xcite . different generalizations have been proposed to take the model closer to experimental reality keeping its solvability . among these one can find the inclusion of dissipation ( often modelled by coupling the field oscillator to a reservoir of external modes ) and/or damping ( as a consequence of spontaneous emission ) , multi - atom , multi - level atom , generalized - interaction and multiple - mode generalizations ( see @xcite for references ) . + here we want to introduce a novel proposal , which states that jcm predictions can be rendered more realistic by noticing that the coupling constant @xmath49 between the atom and the field mode should have a stochastic part which contains part of the effects of the approximations assumed in constructing the original model . since these effects are not under control , to make physical predictions we must average on the introduced random parameters . to illustrate the idea let us consider the original jcm with hamiltonian @xmath90 , where @xmath47 denotes the frequency of the field mode , @xmath51 and @xmath50 their corresponding creation and destruction operators , @xmath48 the frequency difference between the two energy atomic levels , @xmath52 the atomic population operator , @xmath49 the atom - field coupling constant and @xmath53 the energy raising / lowering atomic operators . we claim that the evolution stemming out from @xmath1 should be modified by inserting a random part @xmath34 , where @xmath91 is a real stochastic process which contains the departure from the original ideal situation . the connection with the previous formalism is established by noticing that the evolution operator ( in interaction picture ) must then be : @xmath92=\nonumber\\ & = & \exp\left[-i(\lambda t+\int_{0}^{t}\chi_{s}ds)h^{int}\right]\end{aligned}\ ] ] where @xmath93 is the interaction hamiltonian and @xmath94 is the interaction hamiltonian in interaction picture ( for simplicity s shake exact resonance has been assumed @xmath95 ) . now the expression @xmath96 is a real stochastic process itself which can always be expressed in the form @xcite @xmath97+\int_{0}^{t}v_{s}d\mathcal{b}_{s}\ ] ] where @xmath98 is a real stochastic process uniquely determined by @xmath99 . then the density operator in interaction picture will then be given by @xmath100+v_{s}d\mathcal{b}_{s}\right)\mathcal{c}_{h^{int}_{i}(s)}\right)ds\right][\rho(0)]\ ] ] instead of giving the general form of the expectation value in ( which will be difficult to obtain in full generality ) , we will propose some physical choices based on @xmath34 . if @xmath101 , i.e. the original deterministic evolution is randomly perturbed by a white noise coming from a stochastic perturbation of the coupling constant , then @xmath102 and reduces to @xmath103\nonumber\\ & = & \mathbb{e}[\exp\left(-i\left(\lambda t+\gamma^{1/2}\mathcal{b}_{t}\right)\mathcal{c}_{h_{i}}\right)][\rho(0)]\end{aligned}\ ] ] which yields a density operator in schrdinger picture given by @xmath104[\rho(0)]\ ] ] this relationship means that to obtain the physical predictions of this proposal , all one has to do is to make the sustitution @xmath105 in the original expressions and calculate the expectation value . + note that this proposal allows us to embrace nonmarkovian ( though lindbladian ) situations with little extra effort , e.g. by claiming that the random perturbation is time - dependent @xmath106 . more general options are also possible . the particular choice for @xmath34 relies upon the specific system under study . notice also that the generalization proposed here is compatible with the ones quoted above , i.e. one may combine both type of generalizations . let us consider the situation depicted in @xcite , which appears as the first direct ( in time domain ) experimental evidence of field quantization . the system consists of a rydberg atom in a high - q optical cavity with the atom initially excited and the electromagnetic field in a coherent state @xmath107 . this system is accurately described using the jcm , thus rabi oscillations are expected and concordantly experimentally measured . the theoretical prediction for the probability @xmath108 to find the atom at a time @xmath56 in its ground state is @xmath109 . however an exponential damping in these oscillations are detected ( see @xcite for details ) . the stochastic jcm accounts for this damping ( see fig . [ qedrabi ] ) assuming @xmath101 which produces @xmath110 the physical interpretation under this assumption is rather clear : the ideal coupling assumed in the original jcm does not hold any longer and departures from this ideality should be considered . darks counts and decoherence caused by collisions with background gas have been considered as candidates to explain the damping behaviour and even more radical proposals appear in the literature @xcite . except for the latter which reveals an original intrinsic process , all of them resort to external agents . here note that we do not need to do so , since the departure from ideal atom - field mode coupling can be justified within the domain of the jcm assumptions themselves , i.e. the assumption of coupling between a unique field mode and two levels of the atom can be relaxed by adding in a natural way a random background in this coupling . this decay is also present in the case of arbitrary initial conditions . if @xmath111 is the orthodox prediction , then the previous recipe drives us to @xmath112 where @xmath113 depends on the actual initial conditions of both the atom and field mode . note that not only can any actual exponential decay ( with arbitrary time dependency ) be obtained with the substitution @xmath114 , but also possible changes in the argument of the cosine function could be accounted for by making @xmath115 . see appendix [ appendix ] for details . + in this way we have obtained a decohering system ( oscillations coming from quantum superpositions are progressively supressed ) without necessarily resorting to the action of the environment and keeping quantum principles untouched ( see discussion in section [ discuss ] later on ) . this opens new possibilities to discuss possible sources of decoherence . the previous experimental supression of quantum coherence has also been detected in a linear paul ion trap @xcite . the physical situation is formally similar to that of the rydberg atom coupled to a field mode : the laser field is operated upon the trap in such a way that it can be considered that only two internal energy levels of the ions are coupled to the center - of - mass ( com ) mode of the set of ions ( see @xcite ) . in the dipole and rotating - wave approximations , the interaction hamiltonian ( in interaction picture ) is @xmath116\right)+\textrm{h.c.}\ ] ] where @xmath117 is the lamb - dicke parameter ( @xmath118 and @xmath119 ) , @xmath53 denote the raising / lowering operators for the internal levels , @xmath51 and @xmath50 denote the destruction and creation operators for the com mode , @xmath120 denotes the frequency of the harmonic trap for the com , @xmath121 with @xmath47 the frequency of the laser mode and @xmath48 denotes the difference between the two internal energy levels of the ions . + the statevector can then be written as @xmath122 where @xmath123 and @xmath124 denote the ( time - independent ) internal and motional eigenstates . in the conditions of interest , i.e. in resonant transitions ( @xmath125 with @xmath126 integers ) , the coefficients @xmath127 satisfy the equations @xcite @xmath128 where @xmath129 is given by @xmath130 ( @xmath131 and @xmath132 is the operator for the com motion ) . from these one can predict the well - known rabi oscillations of the system . for concreteness shake let us focus upon the first blue sideband case , i.e. when @xmath133 . if the trap is prepared in the initial state @xmath134 , then the probability of finding a single ion in the @xmath135 state at time @xmath56 is @xmath136 . however experimentally an exponential decay is obtained . as before , one can argue that ideality should be restricted and both a substitution @xmath137 and the corresponding averaging should be performed on @xmath138 . this would drives us to the relation @xmath139 where @xmath140 . this way the exponential decay would have been obtained . physically this recipe can be justified by taking into account intensity fluctuations in the laser field ( see @xcite notice that some generality is gained with respect to this work ) . + however experimental data for the com initially in an arbitrary state and the ion in the ground state @xmath135 are better fit by @xmath141\ ] ] where @xmath113 is an @xmath20-dependent quantity which relies upon the initial conditions of the ion s motion and @xmath142 is a phenomenologically decoherence rate @xcite . the peculiar exponent @xmath143 in @xmath144 renders the previous physical explanation insufficient . more involved schemes to account for this exponent can already be found in the literature @xcite . here we propose a new one based on the previously introduced random evolution schemes . + the main problem attains the peculiar @xmath20-dependency of the argument of the exponential decaying function . in the mathematical realm the necessary flexibility comes from a combination of stochastic calculus and the spectral theorem for the evolution operator @xcite and in the physical one from realizing that not all energy levels of the com mode can be equally affected by a stochastic perturbation . this idea , in a different context in which the trap is coupled to a boson reservoir to account for the detected decoherence , has already been paid attention @xcite . + let s start by considering the spectral decomposition @xcite of the evolution operator generated by the hamiltonian when the laser is tuned to the first blue sideband , i.e. when the hamiltonian is given by @xmath145\ ] ] then the evolution operator will be decomposed as follows : @xmath146 where @xmath147 , @xmath148 ( @xmath149 ) and @xmath150 denote the eigenvalues and eigenstates of respectively and @xmath151 is the projector - valued measure associated to . the stochastic promotion is performed by substituting @xmath152 in and calculating @xmath153 $ ] . here note that the different energy levels are perturbed in a distinct fashion determined both by the deterministic functions @xmath154 and the standard real brownian motions @xmath155 . the latter show correlation properties expressed by the functions @xmath156 : @xmath157 the density operator in interaction picture will then be given by @xmath158p_{n}^{m_{z}}\rho(0)p_{m}^{m_{z}^{'}}\ ] ] the expectation value in can be calculated with the same techniques as before ( cf . also appendix [ appendix ] ) and drives us to : @xmath159 where @xmath160 with @xmath161 . the expression already contains the necessary ingredients to arrive at the detected behaviour , since if the ion trap is initially set in a fock state for the com mode and the ground state for the internal levels , i.e. @xmath162 , then the probability @xmath163 in this scheme is @xmath164\nonumber\\\end{aligned}\ ] ] before making physical assumptions let us notice that since the brownian motions are standard , @xmath165 for all @xmath20 ( the brackets mean that both superscripts must be equal ) and thus @xmath166 for all @xmath20 . now we pose the most important physical hypothesis , namely the stochastic perturbation depends exclusively upon the energy level of the com mode ( at least up to the order of detection we are nowadays capable ) . as a first consequence we then can claim that @xmath167 for all @xmath20 , and then @xmath168 and @xmath166 and also @xmath169 , hence reduces to @xmath170=\nonumber\\ & = & \frac{1}{2}\left[1+e^{-\frac{\lambda_{n+1,n+1}^{+,-}(t)}{2}}\cos(2\eta\omega t\sqrt{n+1})\right]\end{aligned}\ ] ] second since for fixed @xmath20 the internal levels are equally affected , we can also write @xmath171 for each @xmath20 . finally instead of discussing upon absolute energy values , it is physically more reasonable to talk about energy differences and we propose that the stochastic perturbations be introduced in such a way as to have @xmath172 where @xmath173 is an arbitrary exponent , @xmath174 a constant and where we have assumed for simplicity that the random perturbation is a white noise . note however that it is possible to use more general expressions . notice the different behaviour of the added term for each distinct subspace of constant com energy in agreement with the physical hypothesis assumed above . under these hypotheses @xmath175 and after elementary calculations finally reduces to @xmath176 this expression shows a clear resemblance to @xmath177 written above . we believe that both @xmath174 and @xmath173 depends sensitively upon the particular physical system under study . + for completeness we also include the expression for @xmath163 when the com mode has an initial state with diagonal density - matrix elements @xmath113 : @xmath178 this expression has the same structure as the experimental ones shown in @xcite . the whole scheme can obviously be applied to the carrier , first red sideband and successive excitation too . we include in figs . [ fockgraph ] , [ thermalgraph ] and [ cohergraph ] the predictions in the orthodox and the above formalism in the cases when the com mode is in a fock state , thermal state and coherent state and the internal state is the ground state . the use of stochastic methods in hilbert space is of course not new ( cf . e.g. @xcite ) . the idea of representing open quantum systems by means of stochastic processes already appeared in the literature some years ago @xcite and it has been widely used in quantum optics @xcite and in the foundations of quantum mechanics @xcite . here we pursue the line initiated in @xcite stepping forward by randomizing not just the ( thus stochastic ) state vector of the open quantum system , but its evolution operator . we find at least three reasons to do that . firstly when one write a random evolution equation for the state vector ( thus an ito stochastic differential equation ) an extra term must be included , namely the ito correction . consider for example the following evolution @xmath179 where the operator @xmath180 commutes with the hamiltonian @xmath1 ( just for simplicity ) . from a physical point of view we find little intuitive the origin of the ito correction term , which however appears in a natural way by applying ito s formula to the evolution operator with the stochastic modification @xmath181 . secondly the use of random evolution operators emphasizes the idea that it is the evolution which is random and there is nothing random about the hamiltonian ( and thus the energy levels of the system ) , something which may misleadingly be understood from equation . finally the use of operators rather than just state vectors opens the possibility of trying to employ group - representation techniques @xcite and thus of rooting the random nature of the evolution upon possible stochastic symmetries . + this proposal is not intended to solve the so - called _ macroobjectification _ problem ( better known as the measurement problem ) by means of a random dynamical reduction process . indeed it can be readily shown that the state driven by a random evolution operator @xmath182 is never reduced in clear contrast to these models ( see @xcite ) . for the case of the previous random evolution operator this can be readily proven . let @xmath181 be the random evolution operator of a quantum system ( @xmath183=0 $ ] for simplicity ) . then the ito differential equation for the state vector is . now to check whether this evolution produces dynamical state - reduction or not it is sufficient to study the stochastic process ( cf . @xcite ) @xmath184 where @xmath185 . since @xmath186 and @xmath187 , the random evolution is unitary almost surely and then @xmath188 thus there is no actual state - vector reduction process around the eigenvectors of @xmath180 . this of course differs radically for the behaviour of @xmath189 in stochastic state vector reduction models , where the evolution equation is typically written as @xcite @xmath190 the nonlinear terms appear as a consequence of normalization conditions @xcite and play no significant role in the reduction process . note the singular difference between eqs . and : despite the fact that both of them produce the same master equation ( already noted in @xcite ) , only the second one ensures a reduction process taking place and this is because of the @xmath191 factor appearing with the wiener differential @xmath192 . it is an open question whether there exists a physical process or not introducing this phase factor in the evolution equation for @xmath189 . + the great utility of stochastic processes to account for the decoherence suffered by a quantum system is its versatility to also account for possible intrinsic decohering effects . by this we mean not a fundamental modification of quantum principles , as e.g. in @xcite but just the idea that when a quantum system is described ( e.g. an atom interacting with the electromagnetic field ) some approximations must necessarily be made to be able to analytically handle with it and we claim that some of these approximations may hidingly induce decohering effects upon the approximated model . in this sense this decoherence can be called _ intrinsic _ since no environmental effect is taken into account . undoubtedly this does not deny in any way the possibility of having a system decohered by its environment . notice the special relevance of such a hypothetical effect in quantum systems designed to implement quantum - computational and quantum - information - processing tasks , since they usually possess certain degree of complexity which forces us to seek for adequate approximations to describe them . a possible relationship of this intrinsic decoherence with scalability of quantum - informational and quantum computational systems would also have important consequences to find robust mechanisms to process information in a quantum way . + besides possible new physical interpretations supporting the use of stochastic processes in quantum mechanics , its utility to solve certain me as shown above justifies the search to extend the main result reported here . currently the way to drop the condition of selfadjointness of lindblad operators is under study . + finally a mathematical remark should be made regarding the proof of the previous result . one may wonder why the same procedure as in the commuting case , i.e. adding a stochastic term @xmath193 , is not used in the noncommuting case . the reason jointly rests upon the ito s formula and the lack of some derivatives in the case of noncommuting operators . formula @xcite basically states that if the real stochastic process @xmath194 satisfies the ito sde @xmath195 , then the real stochastic process defined as @xmath196 satisfies the ito sde given by @xmath197 this means that both the first and second partial derivatives of @xmath198 must exist to be able to apply this formula . if the stochastic process is operator - valued as e.g. @xmath199 with @xmath29\neq 0 $ ] , then ito s formula can not be directly applied since the partial derivates of @xmath200 can not be found . this difficulty has been circumvented by changing to the heisenberg picture before introducing the stochastic modifications . however it would be desirable to have an ito s formula for operator - valued stochastic processes valid both for commuting and noncommuting cases . we have proven how any lindblad evolution with selfadjoint lindblad operators can be understood as an averaged random evolution operator . the proof included here allows us to extend the previous result to nonmarkovian situations , though keeping the lindbladian structure of the master equations , and as a result we have provided a straightforward method to solve this kind of master equations . the conjunction of stochastic methods and the spectral representation theorem has also allowed us to generalize intrinsic decoherence models already present in the literature . the mathematical versatility of stochastic methods has also permitted us to propose a generalization to the jaynes - cummings model , and compare its predictions with experimental results in cavity qed as well as in ion traps . we have argued that the main physical advantage stems from the possibility of studying decohering effects not necessarily rooted on the environmental action upon the system . a comparison with dynamical collapse models reveals that the random unitary operators used here do not produce any kind of state vector reduction . one of us ( d.s . ) acknowledges the support of madrid education council under grant bocam 20 - 08 - 1999 . though they are elementary we include some useful relations in order to render the text self - contained . first we will show how the moments of arbitrary order @xmath20 of the stochastic process @xmath201 @xmath202 a real function are calculated . let us denote the stochastic process and their @xmath20th moments respectively as @xmath203 and @xmath204^{n}$ ] . it is evident that @xmath205 . @xmath194 satisfies the stochastic differential equation @xmath206 . applying ito s formula @xcite to @xmath207 with @xmath208 and taking expectation values one readily arrives at

it is shown how any lindbladian evolution with selfadjoint lindblad operators , either markovian or nonmarkovian , can be understood as an averaged random unitary evolution . both mathematical and physical consequences are analyzed . first a simple and fast method to solve this kind of master equations is suggested and particularly illustrated with the phase - damped master equation for the multiphoton resonant jaynes - cummings model in the rotating - wave approximation . a generalization to some intrinsic decoherence models present in the literature is included . under the same philosophy a proposal to generalize the jaynes - cummings model is suggested whose predictions are in accordance with experimental results in cavity qed and in ion traps . a comparison with stochastic dynamical collapse models is also included .

arxiv

during the last decades , a new type of artificial materials , the so - called left - handed metamaterials ( lh ) , have attracted a great deal of attention . they present negative indices of refraction for some wavelengths @xcite , with considerable applications in modern optics and microelectronics @xcite . metamaterials can resolve images beyond the diffraction limit @xcite , act as an electromagnetic cloak @xcite , enhance the quantum interference @xcite or yield to slow light propagation @xcite . regarding the localization length in disordered systems , the presence of negative refraction in one - dimensional ( 1d ) disordered metamaterials strongly suppresses anderson localization @xcite . as a consequence , an unusual behavior of the localization length @xmath0 at long - wavelengths @xmath5 has been observed . et al . _ reported a sixth power dependence of @xmath0 with @xmath5 under refractive - index disorder @xcite instead of the well - known quadratic asymptotic behavior @xmath10 @xcite . recently , mogilevtsev _ et al . _ @xcite have also found a suppression of anderson localization of light in 1d disordered metamaterials combining oblique incidence and dispersion while torres - herrera _ et al . _ @xcite have developed a fourth order perturbation theory to resolve the problem of non - conventional anderson localization in bilayered periodic - on - average structures . the effects of polarization and oblique incidence on light propagation in disordered metamaterials were also studied in ref . @xcite . in this article , we calculate numerically the localization length of light @xmath0 for a one - dimensional arrangement of layers with index of refraction @xmath1 and thickness @xmath2 alternating with layers of index of refraction @xmath3 and thickness @xmath4 . in order to introduce disorder in our system , we change the position of the layer boundaries with respect to the periodic values maintaining the same values of the refraction indices @xmath1 and @xmath3 . this is the case of positional disorder , in contrast to the compositional disorder where there exist fluctuations of the index of refraction @xcite . two structures will be analyzed in detail : homogeneous stacks ( h ) , composed entirely by the traditional right - handed materials ( rh ) with positive indices of refraction , and mixed stacks ( m ) with alternating layers of left- and right- handed materials . for the sake of simplicity , the optical path in both layers will be the same , that is , the condition @xmath9 is satisfied in most of the work . these periodic - on - average bilayered photonic systems have already been studied analytically by izrailev _ these authors have developed a perturbative theory up to second order in the disorder to derive an analytical expression for the localization length for both h and m stacks . in our case , we have obtained two equations for the localization length @xmath0 as a function of the wavelength @xmath5 from our numerical results . for h stacks , a quadratic dependence of @xmath0 for long - wavelengths is found , as previously reported in the literature . on the other hand , the localization length saturates for lower values of @xmath5 . an exhaustive study of @xmath0 in the allowed and forbidden bands ( gaps ) of weakly disordered systems will be carried out . we will show that the localization length is modulated by the corresponding bands and this modulation decreases as the disorder increases . for low - disordered m stacks and wavelengths of several orders of magnitude greater than the grating period @xmath11 , the localization length @xmath0 depends linearly on @xmath5 with a slope inversely proportional to the modulus of the reflection amplitude between alternating layers . the plan of the work is as follows . in sec . ii we carry out an exhaustive description of our one - dimensional disordered system and the numerical method used in our localization length calculations . a detailed analysis of @xmath0 in the allowed bands and gaps of homogeneous stacks is performed in sec . iii where a practical expression for the localization length as a function of @xmath5 and the disorder is derived . in sec . iv we calculate @xmath0 for mixed stacks of alternating lh and rh layers . a linear dependence of the localization length at long - wavelengths is found for low - disordered m stacks . finally , we summarize our results in sec . v. let us consider a one - dimensional arrangement of layers with index of refraction @xmath1 alternating with layers of index of refraction @xmath3 . the width of each one is the sum of a fixed length @xmath12 for @xmath13 and a random contribution of zero mean and a given amplitude . the wave - numbers in layers of both types are @xmath14 , where @xmath15 is the frequency and @xmath16 the vacuum speed of light . as previously mentioned , the grating period of our system @xmath17 is defined as the sum of the average thicknesses @xmath2 and @xmath4 of the two types of layers , that is , @xmath18 . we have introduced the optical path condition @xmath19 for simplicity ( in the case of left - handed layers @xmath20 , so the absolute value has been written to consider these type of materials ) . without disorder , each layer would be limited by two boundaries @xmath21 and @xmath22 where @xmath23 is the total number of boundaries . the periodic part of the system considered is schematically represented in fig . [ fig1 ] . and thickness @xmath2 alternating with layers of index of refraction @xmath3 and thickness @xmath4 . the grating period is @xmath11 . ] in the presence of disorder , the position of the corresponding boundaries are @xmath24 except for the first and the last boundary , so as to maintain the same total length @xmath25 . the parameters @xmath26 are zero - mean independent random numbers within the interval @xmath27 $ ] . throughout all our calculations , we have chosen values of the disorder parameter @xmath28 less than @xmath2 and @xmath4 . for each @xmath25 , we calculate the transmission coefficient of our structure @xmath29 and average its logarithm , @xmath30 , over 800 disorder configurations . then , we obtain numerically the localization length @xmath0 via a linear regression of @xmath30 @xcite @xmath31 here , the angular brackets @xmath32 stand for averaging over the disorder . we choose 6 values of the total length @xmath25 to perform the linear regression of eq.([linearregr ] ) . the localization length @xmath0 is evaluated as a function of the disorder parameter @xmath28 and the frequency of the incident photon @xmath15 . we calculate the transmission coefficient of our system via the characteristic determinant method , firstly introduced by aronov _ this is an exact and non perturbative method that provides the information contained in the green function of the whole system . in our case , the characteristic determinant @xmath33 can be written as @xcite @xmath34 where the index @xmath35 runs from 1 to @xmath23 and the coefficients @xmath36 and @xmath37 can be written as @xmath38 and @xmath39 the parameters @xmath40 , which are the reflection amplitudes between media @xmath41 and @xmath35 , are given by @xmath42 where @xmath43 corresponds to the impedance of layer @xmath35 and can be be expressed for normal incidence in terms of its dielectric permittivity @xmath44 and magnetic permeability @xmath45 @xmath46 the quantity @xmath47 entering eqs . ( [ aj ] ) and ( [ bj ] ) is a phase term @xcite @xmath48.\ ] ] here @xmath49 is the wave - number in a layer with boundaries @xmath50 and @xmath51 . this recurrence relation facilitates the numerical computation of the determinant . the initial conditions are the following @xmath52 the transmission coefficient of our structure @xmath29 is given in terms of the determinant @xmath53 by @xmath54 before dealing with mixed stacks , we present results for low - disordered homogeneous systems with underlying periodicity , which has not been previously studied . in this section we perform a detailed analysis of the localization length @xmath0 in the allowed bands and in the forbidden gaps of disordered h stacks as a function of the disorder @xmath28 , the incident wavelength @xmath5 and the reflection coefficient between alternating layers @xmath55 . as it is well known , in the absence of disorder the transmission spectrum of right - handed systems presents allowed and forbidden bands whose position can be easily determined via the following dispersion relation obtained from the bloch - floquet theorem @xcite @xmath56 where @xmath57 is the block wave - vector . when the modulus of the right - hand side of eq . ( [ beta ] ) is greater than 1 , @xmath57 has to be taken as imaginary . this situation corresponds to a forbidden band . taking into account the condition @xmath7 , eq . ( [ beta ] ) reduces to @xmath58 on the other hand , when @xmath59 is equal to unity , the incident frequency @xmath15 is located at the center of the @xmath60-th allowed band , @xmath61 . after some algebra , we obtain from eq . ( [ betasimpl ] ) @xmath62 let us first consider a periodic h stack formed by 50 layers of length @xmath63 52.92 nm and index of refraction @xmath64 1.58 alternating with 49 layers of length @xmath65 39.38 nm and @xmath66 2.12 . the total size of our structure is 4.57 @xmath67 m and the reflection coefficient between alternating layers 0.05259 . fig.[fig2](a ) represents the transmission coefficient @xmath29 as a function of the frequency @xmath15 to illustrate its behavior . also shown are the center of each allowed band calculated via eq.([omegacenter ] ) . there are 99 peaks in each band so they can hardly been resolved on the scale used . moreover , in fig.[fig2](b ) the parameter @xmath59 is plotted versus the frequency @xmath15 for this periodic system . the first gap and the first allowed band have been shown for a better comprehension . and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the homogeneous periodic system described in the text ( 99 layers ) . ] a systematic numerical simulation of a realistic system with 50000 layers has been carried out . the parameters are the same as in the previous example . in fig . [ fig3 ] we represent the localization length @xmath0 versus the wavelength @xmath5 for different values of the disorder parameter @xmath28 ( shown in the legend of the figure ) . the dashed line corresponds to total disorder , that is , an arrangement of layers with random boundaries and alternating indices of refraction @xmath1 and @xmath3 . several features are evident in the figure . for long - wavelengths , one observes a quadratic asymptotic behavior , as can be compared with the dotted line @xcite . an in - deep numerical analysis of the coefficient characterizing this dependence has been performed . to this aim , 20 different h stacks were considered and the following expression for the localization length was found @xmath68 where @xmath69 is the optical path across one grating period @xmath17 . all the lengths in eq.([longlochinf ] ) are expressed in units of @xmath17 . in the opposite limit of short @xmath5 , the localization length @xmath0 saturates to a constant value @xcite . our numerical results have shown that this constant is proportional to the inverse of the reflection coefficient between alternating layers @xmath70 , that is , @xmath71 izrailev _ et al . _ @xcite have developed a perturbative theory up to second order in the disorder to calculate analytically the localization length in both homogeneous and mixed stacks . this model is quite general and is valid for both quarter stack medium ( mainly considered in our work ) and systems with different optical widths . assuming uncorrelated disorder and random perturbations with the same amplitude in both layers ( the main considerations in our numerical calculations ) one can easily derive the following analytical expression for @xmath0 at long - wavelengths from izrailev s formulation @xmath72 for similar values of the layer impedances @xmath73 , the first term in eq.([izragen ] ) can be approximated by @xmath74 and @xmath75 , so eq.([izragen ] ) reduces to @xmath76 which is similar to our numerical expression eq . ( [ longlochinf ] ) . the randomness only affects partially the periodicity of the system , which manifests in the existence of bands and gaps . the localization length depends on the position in the band and on the disorder . the modulation of @xmath0 by the bands can be clearly appreciated in fig . these results are consistent with other published works on this topic @xcite . recently , mogilevtsev _ et al . _ @xcite have reported that the photonic gaps of the corresponding periodic structure are not completely destroyed by the presence of disorder while luna - acosta _ et al . _ @xcite have shown that the resonance bands survive even for relatively strong disorder and large number of cells . versus the wavelength @xmath5 for different values of the disorder parameter @xmath28 . the h stack corresponds to the arrangement represented in fig . [ fig2 ] but now 50000 layers have been considered . the dashed line stands for the total disorder case . all lengths are expressed in units of the grating period @xmath17 . ] having a close look into the first gap in fig . [ fig3 ] , one observes that the localization length is practically independent of the disorder @xmath28 . in order to visualize this effect , fig.[fig4 ] represents ( a ) the first and ( b ) the second gaps depicted in fig . [ fig3 ] . as mentioned , the dependence of @xmath0 with the disorder is almost negligible in the first gap . when the wavelength is similar to the grating period @xmath17 , the influence of the disorder is greater , as can be easily deduced from simple inspection of fig . [ fig4](b ) . versus the wavelength @xmath5 for ( a ) the first and ( b ) the second gaps depicted in fig . [ fig3 ] . ] let us now focus on the allowed bands and study in detail the behavior of the localization length in these regions . to this aim , three - dimensional ( 3d ) graphs of @xmath0 versus the wavelength @xmath5 and the disorder @xmath28 have been plotted in fig.[fig5 ] for ( a ) the first and ( b ) the third allowed bands ( see again fig . [ fig2 ] ) . all this magnitudes have been normalized to the grating period @xmath17 . the localization length @xmath0 is enhanced in a small region around the center of each allowed band . a similar result was found by hernndez - herrejn _ et al . _ @xcite who obtained a resonant effect of @xmath0 close to the band center in the kronig - penney model with weak compositional and positional disorder . this increase in the localization length is due to emergence of the fabry perot resonances associated with multiple reflections inside the layers from the interfaces @xcite . in particular , for homogeneous quarter stack systems , the fabry perot resonances arise exactly in the middle of each allowed band where @xmath57 vanishes @xcite . the saturation of @xmath0 for short - wavelengths is also appreciated in these 3d images . versus the wavelength @xmath5 and the disorder @xmath28 for ( a ) the first and ( b ) the third allowed bands . the h stack is the same as in fig . all lengths are expressed in units of the grating period @xmath17.,scaledwidth=47.0% ] up to now , h stacks with the same optical path in layers of both types have been considered , that is , arrangements verifying the condition @xmath9 in the absence of disorder . as a consequence , the transmission spectrum @xmath29 of the corresponding periodic system presented a symmetric distribution of allowed bands and gaps ( as previously shown in fig . [ fig2 ] ) . what happens in the case of a non - symmetric band distribution , that is , when the condition @xmath9 is not satisfied ? to answer this question , we have plotted the transmission coefficient @xmath29 ( fig . [ fig6](a ) ) and the parameter @xmath77 ( fig . [ fig6](b ) ) versus the frequency @xmath15 for a periodic h stack formed by 50 layers of length @xmath63 52.92 nm and index of refraction @xmath64 1.58 alternating with 49 layers of length @xmath65 28.80 nm and @xmath66 2.12 . note that the condition @xmath7 is no longer held , so the band structure is asymmetric . accordingly , the localization length @xmath0 shown in fig.[fig6](c ) presents an irregular form in the allowed and forbidden bands . as in the symmetric case , no band modulation exists for high disorders and the quadratic asymptotic behavior for long - wavelengths is also verified . moreover , the peaks in the localization length due to fabry perot resonances still can be appreciated , although they are no longer in the center of the bands @xcite . a total number of 50000 layers was considered in our localization length calculations . and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the asymmetric periodic h stack described in the main text ( 99 layers ) and ( c ) the corresponding localization length @xmath0 versus the wavelength @xmath5 for different disorder parameters @xmath28 ( 50000 layers ) . once analyzed in detail the behavior of the localization length @xmath0 for homogeneous systems , let us now deal with m stacks composed of alternating lh and rh layers . in our numerical calculations we have considered a periodic m stack formed by 50 layers of length @xmath63 52.92 nm and index of refraction @xmath64 -1.58 alternating with 49 layers of length @xmath65 39.38 nm and @xmath66 2.12 . again , the condition @xmath19 has been imposed . note that this arrangement has similar parameters than the one depicted in sec . iii , but now @xmath1 is negative . this change of sign results in a severe modification of the transmission coefficient @xmath29 , as we will show immediately . for this the periodic system , fig . [ fig7 ] represents ( a ) the transmission coefficient @xmath29 and ( b ) the parameter @xmath59 versus the frequency @xmath15 of the incident light . unlike the h stack case , no allowed bands exist and practically the entire transmission spectrum is formed by gaps . a set of periodically distributed lorentzian resonances is found instead . the position of the center of each resonance is given by eq . ( [ omegacenter ] ) , that is , the center of the allowed bands in homogeneous systems . and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the mixed periodic system described in the text ( 99 layers ) . ] in respect to the localization length , positional disorder was introduced as explained in sec . as previously considered , the total number of layers in our numerical calculations was 50000 and the number of disordered configurations to average the logarithm of the transmission coefficient was 800 . the result is shown in fig . [ fig8 ] where the localization length @xmath0 is represented versus the wavelength @xmath5 for different values of the disorder parameter @xmath28 . the dashed line corresponds to the total disorder case . again , for long - wavelengths a quadratic asymptotic behavior of @xmath0 is found , but now a region where the localization length is proportional to @xmath5 exists . we will turn to this point in the next figure to quantify the slope of this linear dependence . as it is noticed , the lorentzian resonances associated with multiple reflections in the layers modulate the shape of @xmath0 and this modulation decreases as the disorder increases . moreover , the saturation of the localization length for low - wavelengths can also be appreciated . as in the h stack case , the constant where @xmath0 saturates is proportional to the inverse of the reflection coefficient between alternating layers @xmath70 . versus the wavelength @xmath5 for different disorder parameters @xmath28 . the m stack corresponds to the one represented in fig . [ fig7 ] but here 50000 layers have been considered . the dashed line stands for the total disorder case . ] the linear dependence of @xmath0 with the wavelength @xmath5 has been exhaustively studied by our group to find a simple analytical expression for the localization length in this region . more than 30 different m stacks have been simulated and we have arrived at the following empirical equation @xmath78 where @xmath0 , @xmath5 and @xmath79 are expressed in units of the grating period @xmath17 . in fig . [ fig9 ] , our numerical calculations of the slope @xmath80 versus @xmath81 have been plotted for several values of @xmath79 , triangles ( 1.25 ) , squares ( 3.25 ) and circles ( 7.55 ) . the solid lines correspond to the results obtained via eq . ( [ longlocm ] ) . one notices a good degree of validity for a wide range of @xmath81 values . versus @xmath81 for several values of @xmath79 ( expressed in units of the grating period @xmath17 ) . the solid lines correspond to the results obtained via eq.([longlocm ] ) . ] finally , let us now consider an asymmetrical m stack where the condition @xmath9 is no longer satisfied . in fig.[fig10 ] we have represented ( a ) the transmission coefficient @xmath29 and ( b ) the parameter @xmath59 versus the frequency @xmath15 for a periodic m stack formed by 50 layers of length of length @xmath63 52.92 nm and index of refraction @xmath64 -1.58 alternating with 49 layers of length @xmath65 28.80 nm and @xmath66 2.12 . note the strong difference between this transmission spectrum and the symmetrical one ( see fig . [ fig7](a ) ) where a set of periodically distributed lorenztian resonances exists . despite this fact , the localization length @xmath0 shown in fig.[fig10](c ) presents a region of linear dependence with the wavelength , as in the symmetric case . however , eq.([longlocm ] ) can not be used to evaluate the localization length in this region . and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the asymmetric periodic m stack described in the main text ( 99 layers ) and ( c ) the corresponding localization length @xmath0 versus the wavelength @xmath5 for different disorder parameters @xmath28 ( 50000 layers ) . we have analyzed numerically the localization length of light @xmath0 for homogeneous and mixed stacks of layers with index of refraction @xmath82 and thickness @xmath2 alternating with layers of index of refraction @xmath83 and thickness @xmath4 . the positions of the layer boundaries have been randomly shifted with respect to ordered periodic values . the refraction indices @xmath1 and @xmath3 present no disorder . for h stacks , the parabolic behavior of the localization length in the limit of long - wavelengths , previously found in purely disordered systems @xcite , has been recovered . on the other hand , the localization length @xmath0 saturates for very low values of @xmath5 . the transmission bands modulate the localization length @xmath0 and this modulation decreases with increasing disorder . moreover , the localization length is practically independent of the disorder @xmath28 at the first gap , that is , it has a very low tendency in this region . we have also characterized @xmath0 in terms of the reflection coefficient of alternating layers @xmath70 and the optical path across one grating period @xmath79 . eq.([longlochinf ] ) has been proved to be valid for a wide range of @xmath70 values , that is , from transparent to opaque h stacks . it has also been shown ( see fig . [ fig5 ] ) that the localization length @xmath0 is enhanced at the center of each allowed band . when left - handed metamaterials are introduced in our system , the localization length behavior presents some differences with respect to the traditional stacks , formed exclusively by right - handed materials . for low - disordered m stacks and wavelengths of several orders of magnitude greater than the grating period @xmath17 , the localization length @xmath0 depends linearly on @xmath5 with a slope inversely proportional to the modulus of the reflection amplitude between alternating layers @xmath81 ( see eq . ( [ longlocm ] ) ) . as in the h case , @xmath0 saturates for low - wavelengths , being this saturation constant proportional to the inverse of @xmath70 . if we take into account losses , there is an absorption term whose absorption length @xmath84 is @xcite @xmath85 where @xmath86 is an absorption coefficient . the inverse of the total decay length is the sum of the inverse of the localization length @xmath0 plus the inverse of the absorption length @xmath87 . note that @xmath84 is proportional to @xmath5 , so , for low - disordered m stacks and weak absorption metamaterials , the final expression for the localization length @xmath0 in the linear region can be written as @xmath88 in the case of both homogeneous and mixed stacks with non - symmetric band distribution , that is , when the condition @xmath89 is not satisfied , the localization length @xmath0 presents an irregular form in all the transmission spectrum . these changes in @xmath0 are more sensitives in mixed stacks than in homogeneous structures . a. yariv and p. yeh , optical waves in crystals , propagation and control of laser radiation ( wiley , new york , 1984 ) ; o. del barco , m. ortuo and v. gasparian , phys . a * 74 * , 032104 ( 2006 ) ; o. del barco and m. ortuo , phys . a * 81 * , 023833 ( 2010 ) .

we have analyzed numerically the localization length of light @xmath0 for nearly periodic arrangements of homogeneous stacks ( formed exclusively by right - handed materials ) and mixed stacks ( with alternating right and left - handed metamaterials ) . layers with index of refraction @xmath1 and thickness @xmath2 alternate with layers of index of refraction @xmath3 and thickness @xmath4 . positional disorder has been considered by shifting randomly the positions of the layer boundaries with respect to periodic values . for homogeneous stacks , we have shown that the localization length is modulated by the corresponding bands and that @xmath0 is enhanced at the center of each allowed band . in the limit of long - wavelengths @xmath5 , the parabolic behavior previously found in purely disordered systems is recovered , whereas for @xmath6 a saturation is reached . in the case of nearly periodic mixed stacks with the condition @xmath7 , instead of bands there is a periodic arrangement of lorenztian resonances , which again reflects itself in the behavior of the localization length . for wavelengths of several orders of magnitude greater than @xmath8 , the localization length @xmath0 depends linearly on @xmath5 with a slope inversely proportional to the modulus of the reflection amplitude between alternating layers . when the condition @xmath9 is no longer satisfied , the transmission spectrum is very irregular and this considerably affects the localization length .

arxiv

dans le modle cosmologique , dit de concordance car il est en conformit avec tout un ensemble de donnes observationnelles , la matire ordinaire do nt sont constitus les toiles , le gaz , les galaxies , etc . ( essentiellement sous forme baryonique ) ne forme que 4% de la masse - nergie totale , ce qui est dduit de la nuclosynthse primordiale des lments lgers , ainsi que des mesures de fluctuations du rayonnement du fond diffus cosmologique ( cmb ) le rayonnement fossile qui date de la formation des premiers atomes neutres dans lunivers . nous savons aussi quil y a 23% de matire noire sous forme _ non baryonique _ et do nt nous ne connaissons pas la nature . et les 73% qui restent ? et bien , ils sont sous la forme dune mystrieuse nergie noire , mise en vidence par le diagramme de hubble des supernovae de type ia , et do nt on ignore lorigine part quelle pourrait tre sous la forme dune constante cosmologique . le contenu de lunivers grandes chelles est donc donn par le `` camembert '' de la figure [ fig1 ] do nt 96% nous est inconnu ! la matire noire permet dexpliquer la diffrence entre la masse dynamique des amas de galaxies ( cest la masse dduite du mouvement des galaxies ) et la masse de la matire lumineuse qui comprend les galaxies et le gaz chaud intergalactique . mais cette matire noire ne fait pas que cela ! nous pensons quelle joue un rle crucial dans la formation des grandes structures , en entranant la matire ordinaire dans un effondrement gravitationnel , ce qui permet dexpliquer la distribution de matire visible depuis lchelle des amas de galaxies jusqu lchelle cosmologique . des simulations numriques trs prcises permettent de confirmer cette hypothse . pour que cela soit possible il faut que la matire noire soit non relativiste au moment de la formation des galaxies . on lappelera matire noire _ froide _ ou cdm selon lacronyme anglais , et il y a aussi un nom pour la particule associe : un wimp pour `` weakly interacting massive particle '' . il ny a pas dexplication pour la matire noire ( ni pour lnergie noire ) dans le cadre du modle standard de la physique des particules . mais des extensions au - del du modle standard permettent de trouver des bons candidats pour la particule ventuelle de matire noire . par exemple dans un modle de super - symtrie ( qui associe tout fermion un partenaire super - symtrique qui est un boson et rciproquement ) lun des meilleurs candidats est le _ neutralino _ , qui est un partenaire fermionique super - symtrique dune certaine combinaison de bosons du modle standard . , qui fut introduit dans une tentative pour rsoudre le problme de la violation cp en physique des particules , est une autre possibilit . il y a aussi les tats de kaluza - klein prdits dans certains modles avec dimensions suplmentaires . quant lnergie noire , elle apparat comme un milieu de densit dnergie _ constante _ au cours de lexpansion , ce qui implique une violation des `` conditions dnergie '' habituelles avec une pression ngative . lnergie noire pourrait tre la fameuse constante cosmologique @xmath0 queinstein avait introduite dans les quations de la relativit gnrale afin dobtenir un modle dunivers statique , puis quil avait abandonne lorsque lexpansion fut dcouverte . depuis zeldovich on interprte @xmath0 comme lnergie du vide associe lespace - temps lui - mme . le problme est que lestimation de cette nergie en thorie des champs donne une valeur @xmath1 fois plus grande que la valeur observe ! on ne comprend donc pas pourquoi la constante cosmologique est si petite . malgr lnigme de lorigine de ses constituents , le modle @xmath0-cdm est plein de succs , tant dans lajustement prcis des fluctuations du cmb que dans la reproduction fidle des grandes structures observes . une leon est que la matire noire apparat forme de particules ( les wimps ) grande chelle . la matire noire se manifeste de manire clatante dans les galaxies , par lexcs de vitesse de rotation des toiles autour de ces galaxies en fonction de la distance au centre cest la clbre courbe de rotation ( voir la figure [ fig2 ] ) . les mesures montrent qu partir dune certaine distance au centre la courbe de rotation devient pratiquement plate , cest - - dire que la vitesse devient constante . daprs la loi de newton la vitesse dune toile sur une orbite circulaire ( keplerienne ) de rayon @xmath2 est donne par @xmath3 o @xmath4 est la masse contenue dans la sphre de rayon @xmath2 . pour obtenir une courbe de rotation plate il faut donc supposer que la masse crot proportionnellement @xmath2 ( et donc que la densit dcrot comme @xmath5 ) , ce qui nest certainement pas le cas de la matire visible . on est oblig dinvoquer lexistence dun gigantesque halo de matire noire invisible ( qui ne rayonne pas ) autour de la galaxie et do nt la masse dominerait celle des toiles et du gaz . cette matire noire peut - elle tre faite de la mme particule que celle suggre par la cosmologie ( un wimp ) ? des lments de rponse sont fournis par les simulations numriques de cdm en cosmologie qui sont aussi valables lchelle des galaxies , et qui donnent un profil de densit universel pour le halo de matire noire . a grande distance ce profil dcroit en @xmath6 soit plus rapidement que ce quil faudrait pour avoir une courbe plate , mais ce nest pas trs grave car on peut supposer que la courbe de rotation est observe dans un rgime intermdiaire avant de dcrotre . plus grave est la prdiction dun pic central de densit au centre des galaxies , o les particules de matire noire tendent sagglomrer cause de la gravitation , avec une loi en @xmath7 pour @xmath2 petit . or les courbes de rotation favorisent plutt un profil de densit sans divergence , avec un coeur de densit constante . dautres problmes rencontrs par les halos simuls de cdm sont la formation dune multitude de satellites autour des grosses galaxies , et la loi empirique de tully et fisher qui nest pas explique de faon naturelle . cette loi montre dans la figure [ fig3 ] relie la luminosit des galaxies leur vitesse asymptotique de rotation ( qui est la valeur du plateau dans la figure [ fig2 ] ) par @xmath8 . noter que cette loi ne fait pas rfrence la matire noire ! la vitesse et la luminosit sont bien sr celles de la matire ordinaire , et la matire noire semble faire ce que lui dicte la matire visible . mais le dfi le plus important de cdm est de pouvoir rendre compte dune observation tonnante appele _ loi de milgrom _ @xcite , selon laquelle la matire noire intervient uniquement dans les rgions o le champ de gravitation ( ou , ce qui revient au mme , le champ dacclration ) est plus _ faible _ quune certaine acclration critique mesure la valeur `` universelle '' @xmath9 . tout se passe comme si dans le rgime des champs faibles @xmath10 , la matire ordinaire tait acclre non par le champ newtonien @xmath11 mais par un champ @xmath12 donn simplement par @xmath13 . la loi du mouvement sur une orbite circulaire donne alors une vitesse _ constante _ et gale @xmath14 . ce rsultat nous rserve un bonus important : puisque le rapport masse - sur - luminosit @xmath15 est approximativement le mme dune galaxie lautre , la vitesse de rotation doit varier comme la puissance @xmath16 de la luminosit @xmath17 , en accord avec la loi de tully - fisher ! pour avoir une rgle qui nous permette dajuster les courbes de rotation des galaxies il nous faut aussi prendre en compte le rgime de champ fort dans lequel on doit retrouver la loi newtonienne . on introduit une fonction dinterpolation @xmath18 dpendant du rapport @xmath19 et qui se ramne @xmath20 lorsque @xmath10 , et qui tend vers 1 quand @xmath21 . notre rgle sera donc @xmath22 ici @xmath12 dsigne la norme du champ de gravitation @xmath23 ressenti par les particules dpreuves . une formule encore plus oprationnelle est obtenue en prenant la divergence des deux membres de ce qui mne lquation de poisson modifie , o @xmath24 est le laplacien et @xmath25 le potentiel newtonien local . loprateur @xmath26 appliqu une fonction scalaire est le gradient , appliqu un vecteur cest la divergence : @xmath27 . par convention , on note les vecteurs en caractres gras . ] : @xmath28 = -4 \pi \ , g\,\rho_\text{b } \,,\ ] ] do nt la source est la densit de matire baryonique @xmath29 ( le champ gravitationnel est irrotationnel : @xmath30 ) . on appellera lquation la formule mond pour `` modified newtonian dynamics '' . le succs de cette formule ( on devrait plus exactement dire cette _ recette _ ) dans lobtention des courbes de rotation de nombreuses galaxies est impressionnant ; voir la courbe en trait plein dans la figure [ fig2 ] . cest en fait un ajustement un paramtre libre , le rapport @xmath15 de la galaxie qui est donc _ mesur _ par notre recette . on trouve que non seulement la valeur de @xmath15 est de lordre de 1 - 5 comme il se doit , mais quelle est remarquablement en accord avec la couleur observe de la galaxie . beaucoup considrent la formule mond comme `` exotique '' et reprsentant un aspect mineur du problme de la matire noire . on entend mme parfois dire que ce nest pas de la physique . bien sr ce nest pas de la physique _ fondamentale _ cette formule ne peut pas tre considre comme une thorie fondamentale , mais elle constitue de lexcellente physique ! elle capture de faon simple et puissante tout un ensemble de faits observationnels . au physicien thoricien dexpliquer pourquoi . la valeur numrique de @xmath31 se trouve tre trs proche de la constante cosmologique : @xmath32 . cette concidence cosmique pourrait nous fournir un indice ! elle a aliment de nombreuses spculations sur une possible influence de la cosmologie dans la dynamique locale des galaxies . face la `` draisonnable efficacit '' de la formule mond , trois solutions sont possibles . 1 . la formule pourrait sexpliquer dans le cadre cdm . mais pour rsoudre les problmes de cdm il faut invoquer des mcanismes astrophysiques compliqus et effectuer un ajustement fin des donnes galaxie par galaxie . 2 . on est en prsence dune modification de la loi de la gravitation dans un rgime de champ faible @xmath10 . cest lapproche traditionnelle de mond et de ses extensions relativistes . la gravitation nest pas modifie mais la matire noire possde des caractristiques particulires la rendant apte expliquer la phnomnologie de mond . cest une approche nouvelle qui se prte aussi trs bien la cosmologie . la plupart des astrophysiciens des particules et des cosmologues des grandes structures sont partisans de la premire solution . malheureusement aucun mcanisme convainquant na t trouv pour incorporer de faon naturelle la constante dacclration @xmath31 dans les halos de cdm . dans la suite nous considrerons que la solution 1 . est dores et dj exclue par les observations . les approches 2 . de gravitation modifie et 3 . que lon peut qualifier de _ matire noire modifie _ croient toutes deux dans la pertinence de mond , mais comme on va le voir sont en fait trs diffrentes . notez que dans ces deux approches il faudra expliquer pourquoi la matire noire semble tre constitue de wimps lchelle cosmologique . cette route , trs dveloppe dans la littrature , consiste supposer quil ny a pas de matire noire , et que reflte une violation fondamentale de la loi de la gravitation . cest la proposition initiale de milgrom @xcite un changement radical de paradigme par rapport lapproche cdm . pour esprer dfinir une thorie il nous faut partir dun lagrangien . or il est facile de voir que dcoule dun lagrangien , celui - ci ayant la particularit de comporter un terme cintique non standard pour le potentiel gravitationnel , du type @xmath33 $ ] au lieu du terme habituel @xmath34 , o @xmath35 est une certaine fonction que lon relie la fonction @xmath18 . ce lagrangien a servi de point de dpart pour la construction des thories de la gravitation modifie . on veut modifier la relativit gnrale de faon retrouver mond dans la limite non - relativiste , cest - - dire quand la vitesse des corps est trs faible par rapport la vitesse de la lumire @xmath36 . en relativit gnrale la gravitation est dcrite par un champ tensoriel deux indices appel la mtrique de lespace - temps @xmath37 . cette thorie est extrmement bien vrifie dans le systme solaire et dans les pulsars binaires , mais peu teste dans le rgime de champs faibles qui nous intresse ( en fait la relativit gnrale est le royaume des champs gravitationnels forts ) . la premire ide qui vient lesprit est de promouvoir le potentiel newtonien @xmath25 en un champ scalaire @xmath38 ( sans indices ) et donc de considrer une thorie _ tenseur - scalaire _ dans laquelle la gravitation est dcrite par le couple de champs @xmath39 . on postule , de manire _ ad - hoc _ , un terme cintique non standard pour le champ scalaire : @xmath40 o @xmath41 est reli @xmath18 , et on choisit le lagrangien deinstein - hilbert de la relativit gnrale pour la partie concernant la mtrique @xmath37 . tout va bien pour ce qui concerne le mouvement des toiles dans une galaxie , qui reproduit mond . mais notre thorie tenseur - scalaire est une catastrophe pour le mouvement des photons ! en effet ceux - ci ne ressentent pas la prsence du champ scalaire @xmath38 cens reprsenter la matire noire . dans une thorie tenseur - scalaire toutes les formes de matire se propagent dans un espace - temps de mtrique _ physique _ @xmath42 qui diffre de la mtrique deinstein @xmath37 par un facteur de proportionalit dpendant du champ scalaire , soit @xmath43 . une telle relation entre les mtriques est dite conforme et laisse invariants les cnes de lumire de lespace - temps . les trajectoires de photons seront donc les mmes dans lespace - temps physique que dans lespace - temps deinstein ( cela se dduit aussi de linvariance conforme des quations de maxwell ) . comme on observe dnormes quantits de matire noire grce au mouvement des photons , par effet de lentille gravitationnelle , la thorie tenseur - scalaire est limine . pour corriger cet effet dsastreux du mouvement de la lumire on rajoute un nouvel lment notre thorie . puisque cest cela qui cause problme on va transformer la relation entre les mtriques @xmath42 et @xmath37 . une faon de le faire est dy insrer ( encore de faon _ ad - hoc _ ) un nouveau champ qui sera cette fois un vecteur @xmath44 avec un indice . on aboutit donc une thorie dans laquelle la gravitation est dcrite par le triplet de champs @xmath45 . cest ce quon appelle une thorie _ tenseur - vecteur - scalaire _ ( teves ) . la thorie teves a t mise au point par bekenstein et sanders @xcite . comme dans la thorie tenseur - scalaire on aura la partie deinstein - hilbert pour la mtrique , plus un terme cintique non standard @xmath40 pour le champ scalaire . quant au champ vectoriel on le munit dun terme cintique analogue celui de llectromagntisme , mais dans lequel le rle du potentiel lectromagntique @xmath46 est tenu par notre champ @xmath44 . la thorie teves rsultante est trs complique et pour linstant non relie de la physique microscopique . il a t montr que cest un cas particulier dune classe de thories appeles thories einstein-_ther _ dans lesquelles le vecteur @xmath44 joue le rle principal , en dfinissant un rfrentiel priviligi un peu analogue lther postul au xix@xmath47 sicle pour interprter la non - invariance des quations de maxwell par transformation de galile . si elle est capable de retrouver mond dans les galaxies , la thorie teves a malheureusement un problme dans les amas de galaxies car elle ne rend pas compte de toute la matire noire observe . cest en fait un problme gnrique de toute extension relativiste de mond . cependant ce problme peut tre rsolu en supposant lexistence dune composante de matire noire _ chaude _ sous la forme de neutrinos massifs , ayant la masse maximale permise par les expriences actuelles soit environ @xmath48 . rappelons que toute la matire noire ne peut pas tre sous forme de neutrinos : dune part il ny aurait pas assez de masse , et dautre part les neutrinos tant relativistes auraient tendance lisser lapparence des grandes structures , ce qui nest pas observ . nanmoins une pince de neutrinos massifs pourrait permettre de rendre viables les thories de gravitation modifie . de ce point de vue les expriences prvues qui vont dterminer trs prcisment la masse du neutrino ( en vrifiant la conservation de lnergie au cours de la dsintgration dune particule produisant un neutrino dans ltat final ) vont jouer un rle important en cosmologie . teves a aussi des difficults lchelle cosmologique pour reproduire les fluctuations observes du cmb . l aussi une composante de neutrinos massifs peut aider , mais la hauteur du troisime pic de fluctuation , qui est caractristique de la prsence de matire noire sans pression , reste difficile ajuster . une alternative logique la gravit modifie est de supposer quon est en prsence dune forme particulire de matire noire ayant des caractristiques diffrentes de cdm . dans cette approche on a lambition dexpliquer la phnomnologie de mond , mais avec une philosophie nouvelle puisquon ne modifie pas la loi de la gravitation : on garde la relativit gnrale classique , avec sa limite newtonienne habituelle . cette possibilit merge grce lanalogue gravitationnel du mcanisme physique de polarisation par un champ extrieur et quon va appeler `` polarisation gravitationnelle '' @xcite . la motivation physique est une analogie frappante ( et peut - tre trs profonde ) entre mond , sous la forme de lquation de poisson modifie , et la physique des milieux dilectriques en lectrostatique . en effet nous apprenons dans nos cours de physique lmentaire que lquation de gauss pour le champ lectrique ( cest lune des quations fondamentales de maxwell ) , est modifie en prsence dun milieu dilectrique par la contribution de la polarisation lectrique ( voir lappendice ) . de mme , mond peut - tre vu comme la modification de lquation de poisson par un milieu `` digravitationnel '' . explicitons cette analogie . on introduit lanalogue gravitationnel de la susceptibilit , soit @xmath49 qui est reli la fonction mond par @xmath50 . la `` polarisation gravitationnelle '' est dfinie par @xmath51 la densit des `` masses de polarisation '' est donne par la divergence de la polarisation soit @xmath52 . avec ces notations lquation devient @xmath53 qui apparat maintenant comme une quation de poisson ordinaire , mais do nt la source est constitue non seulement par la densit de matire baryonique , mais aussi par la contribution des masses de polarisation @xmath54 . il est clair que cette criture de mond suggre que lon est en prsence non pas dune modification de la loi gravitationnelle , mais dune forme nouvelle de matire noire de densit @xmath54 , cest - - dire faite de moments dipolaires aligns dans le champ de gravitation . ltape suivante serait de construire un modle microscopique pour des diples gravitationnels @xmath55 ( tels que @xmath56 ) . lanalogue gravitationnel du diple lectrique serait un vecteur @xmath57 sparant deux masses @xmath58 . on se heurte donc un problme svre : le milieu dipolaire gravitationnel devrait contenir des masses ngatives ! ici on entend par masse lanalogue gravitationnel de la charge , qui est ce quon appelle parfois la masse grave . ce problme des masses ngatives rend _ a priori _ le modle hautement non viable . nanmoins , ce modle est intressant car il est facile de montrer que le coefficient de susceptibilit gravitationnelle doit tre ngatif , @xmath59 , soit loppos du cas lectrostatique . or cest prcisment ce que nous dit mond : comme la fonction @xmath18 interpole entre le rgime mond o @xmath60 et le rgime newtonien o @xmath61 , on a @xmath62 et donc bien @xmath59 . il est donc tentant dinterprter le champ gravitationnel plus intense dans mond que chez newton par la prsence de `` masses de polarisation '' qui _ anti - crantent _ le champ des masses gravitationnel ordinaires , et ainsi augmentent lintensit effective du champ gravitationnel ! dans le cadre de ce modle on peut aussi se convaincre quun milieu form de diples gravitationnels est intrinsquement instable , car les constituants microscopiques du diple devraient se repousser gravitationnellement . il faut donc introduire une force interne dorigine _ non - gravitationnelle _ , qui va supplanter la force gravitationnelle pour lier les constituants dipolaires entre eux . on pourrait qualifier cette nouvelle interaction de `` cinquime force '' . pour retrouver mond , on trouve de faon satisfaisante que ladite force doit dpendre du champ de polarisation , et avoir en premire approximation la forme dun oscillateur harmonique . par leffet de cette force , lquilibre , le milieu dipolaire ressemble une sorte d``ther statique '' , un peu limage du dilectrique do nt les sites atomiques sont fixes . les arguments prcdents nous laissent penser que mond a quelque chose voir avec un effet de polarisation gravitationnelle . mais il nous faut maintenant construire un modle cohrent , reproduisant lessentiel de cette physique , et _ sans _ masses graves ngatives , donc respectant le principe dquivalence . il faut aussi bien sr que le modle soit _ relativiste _ ( en relativit gnrale ) pour pouvoir rpondre des questions concernant la cosmologie ou le mouvement de photons . on va dcrire le milieu comme un fluide relativiste de quadri - courant @xmath63 ( o @xmath64 est la densit de masse ) , et muni dun quadri - vecteur @xmath65 jouant le rle du moment dipolaire . le vecteur de polarisation est alors @xmath66 . on dfinit un principe daction pour cette matire dipolaire , que lon rajoute laction deinstein - hilbert , et la somme des actions de tous les champs de matire habituels ( baryons , photons , etc ) . on inclue dans laction une fonction potentielle dpendant de la polarisation et cense dcrire une force interne au milieu dipolaire . par variation de laction on obtient lquation du mouvement du fluide dipolaire , ainsi que lquation dvolution de son moment dipolaire . on trouve que le mouvement du fluide est affect par la force interne , et diffre du mouvement godsique dun fluide ordinaire . ce modle ( propos dans @xcite ) reproduit bien la phnomnologie de mond au niveau des galaxies . il a t construit pour ! mais il a t aussi dmontr quil donne satisfaction en cosmologie o lon considre une perturbation dun univers homogne et isotrope . en effet cette matire noire dipolaire se conduit comme un fluide parfait sans pression au premier ordre de perturbation cosmologique et est donc indistinguable du modle cdm . en particulier le modle est en accord avec les fluctuations du fond diffus cosmologique ( cmb ) . en ce sens il permet de rconcilier laspect particulaire de la matire noire telle quelle est dtecte en cosmologie avec son aspect `` modification des lois '' lchelle des galaxies . de plus le modle contient lnergie noire sous forme dune constante cosmologique @xmath0 . il offre une sorte dunification entre lnergie noire et la matire noire _ la _ mond . en consquence de cette unification on trouve que lordre de grandeur naturel de @xmath0 doit tre compatible avec celui de lacclration @xmath31 , cest - - dire que @xmath67 , ce qui est en trs bon accord avec les observations . le modle de matire noire dipolaire contient donc la physique souhaite . son dfaut actuel est de ne pas tre connect de la physique microscopique fondamentale ( _ via _ une thorie quantique des champs ) . il est donc moins fondamental que cdm qui serait motiv par exemple par la super - symtrie . ce modle est une description effective , valable dans un rgime de champs gravitationnels faibles , comme la lisire dune galaxie ou dans un univers presque homogne et isotrope . lextrapolation du modle au champ gravitationnel rgnant dans le systme solaire nest pas entirement rsolue . dun autre ct le problme de comment tester ( et ventuellement falsifier ) ce modle en cosmologie reste ouvert . m. milgrom , astrophys . j. * 270 * , 365 ( 1983 ) . bekenstein , phys . rev . d * 70 * , 083509 ( 2004 ) . sanders , mon . not . 363 * , 459 ( 2005 ) . l. blanchet , class . * 24 * , 3529 ( 2007 ) . l. blanchet and a. le tiec , phys . d * 78 * , 024031 ( 2008 ) ; and submitted , arxiv:0901.3114 ( 2009 ) . un dilectrique est un matriau isolant , qui ne laisse pas passer les courants , car tous les lectrons sont rattachs des sites atomiques . nanmoins , les atomes du dilectrique ragissent la prsence dun champ lectrique extrieur : le noyau de latome charg positivement se dplace en direction du champ lectrique , tandis que le barycentre des charges ngatives cest - - dire le nuage lectronique se dplace dans la direction oppose . on peut modliser la rponse de latome au champ lectrique par un diple lectrique @xmath68 qui est une charge @xmath69 spare dune charge @xmath70 par le vecteur @xmath71 , et align avec le champ lectrique . la densit des diples nous donne la polarisation @xmath72 . le champ cre par les diples se rajoute au champ extrieur ( engendr par des charges extrieures @xmath73 ) et a pour source la densit de charge de polarisation qui est donne par la divergence de la polarisation : @xmath74 . ainsi lquation de gauss ( qui scrit normalement @xmath75 ) devient en prsence du dilectrique @xmath76 en utilisant les conventions habituelles , avec @xmath77 . on introduit un coefficient de susceptibilit lectrique @xmath78 qui intervient dans la relation de proportionalit entre la polarisation et le champ lectrique : @xmath79 , ainsi : @xmath80 . la susceptibilit est positive , @xmath81 , ce qui implique que le champ dans un dilectrique est plus faible que dans le vide . cest leffet d_crantage _ de la charge par les charges de polarisation . ainsi garnir lespace intrieur aux plaques dun condensateur avec un matriau dilectrique diminue lintensit du champ lectrique , et donc augmente la capacit du condensateur pour une tension donne .

pour lastrophysicien qui aborde le puzzle de la matire noire , celle - ci apparat sous deux aspects diffrents : dune part en cosmologie , cest - - dire trs grandes chelles , o elle semble tre forme dun bain de particules , et dautre part lchelle des galaxies , o elle est dcrite par un ensemble de phnomnes trs particuliers , qui paraissent incompatibles avec sa description en termes de particules , et qui font dire certains que lon est en prsence dune modification de la loi de la gravitation . rconcilier ces deux aspects distincts de la matire noire dans un mme formalisme thorique reprsente un dfi important qui pourrait peut - tre conduire une physique nouvelle en action aux chelles astronomiques .

arxiv

given their ubiquity in nature , long chain macromolecules have been the subject of considerable study . whereas there is now a reasonably firm basis for understanding the physical properties of homopolymers@xcite , considerably less is known about the heteropolymers of biological significance . from a biologist s perspective , it is the specific properties of a particular molecule that are of interest . after all the genetic information is coded by very specific sequences of nucleic acids , which are in turn translated to the chain of amino acids forming a protein@xcite . the energy of the polymer is determined by the van der waals , hydrogen bonding , hydrophobic / hydrophilic , and coulomb interactions between its constituent amino acids . in accord to these interactions , the protein folds into a specific shape that is responsible for its activity . given the large number of monomers making up such chains , and the complexity of their interactions , finding the configuration of a particular molecule is a formidable task . by contrast , a physicist s approach is to sacrifice the specificity , in the hope of gleaning some more general information from simplified models@xcite . there are in fact a number of statistical descriptions of _ ensembles _ of molecules composed of a random linear sequence of elements with a variety of interactions that determine their final shapes@xcite . these simple models of heteropolymers are of additional interest as examples of disordered systems with connections to spin glasses @xcite , with the advantage of faster relaxation @xcite . there are a number of recent experimental studies of solutions@xcite and gels@xcite of polymers that incorporate randomly charged groups . as statistical approaches only provide general descriptions of such heteropolymers , we focus on simple models which include the essential ingredients . the overall size and shape of a polymer with charged groups is most likely controlled by the coulomb interactions that are the strongest and with the longest range . we shall consider the typical properties of a model _ polyampholyte _ ( pa)@xcite : a flexible chain in which each of the @xmath5 monomers has a fixed charge @xmath0 selected from a well defined ensemble of quenches . the polymer has a characteristic microscopic length @xmath6 ( such as range of the excluded volume interaction , or nearest neighbor distance along the chain ) . in the numerical studies we further simplify the model by considering only self avoiding walk ( saw ) configurations on a cubic lattice with lattice constant @xmath6 . the long range nature of the coulomb interactions , combined with the randomness of the charge sequence , produces effects quite distinct from systems with short range interactions . in section [ secgend ] we use the knowledge accumulated in previous studies@xcite to explore the phase diagrams of quenched pas in @xmath7 dimensions . in particular , we show that for @xmath8 , the behavior of pas is similar to that of random chains with short range interactions , while for @xmath9 the spatial conformations of a pa strongly depend on its excess charge @xmath10 . in every space dimension @xmath9 , there is a critical charge @xmath11 such that pas with @xmath12 can not form a compact state . the probability of a randomly charged pa to have such an excess charge depends on both @xmath7 and its length . in the @xmath13 limit the excess charge will always ( i.e. with probability 1 ) be `` small '' for @xmath14 and `` big '' for @xmath15 . thus investigation of the `` borderline '' three dimensional case provides valuable insight into the behavior of the system in general space dimensions . in section [ secgen ] we summarize previous results for pas in @xmath16 : analytical arguments and monte carlo ( mc ) studies indicate that the pa undergoes a transition from a dense ( `` globular '' ) to a strongly stretched configuration as @xmath1 exceeds @xmath17 . the mc simulations@xcite were performed for polymer sizes up to @xmath18 and in a wide range of temperatures . they , however , could not provide information on the energy spectrum of pas , and on very low temperature properties . in this work we undertake a complete enumeration study of pas for all possible quenches up to @xmath19 , and are thus able to present very detailed results regarding energetics and spatial conformations of short pas . the details of the enumeration procedure are explained in section [ secenum ] , while the results are described in sections [ secenspec ] and [ secshape ] . the majority of these results add further support to the predictions of mc studies , and provide some details which could not be measured by mc ( e.g. , density of states , condensation energy , and surface tension in the globular phase ) . we also find some indication that pas with small @xmath1 may undergo a phase transition between two dense states . no signs of this transition could be detected in the mc studies , because it occurs at temperatures too low for that procedure to equilibrate . it is helpful to view the problem in the more general context of a variable space dimension @xmath7 . let us consider a continuum limit in which configurations of the pa are described by a function @xmath20 . the continuous index @xmath21 is used to label the monomers along the chain , while @xmath22 is the position of the monomer in @xmath7dimensional embedding space . the corresponding probabilities of these configurations are governed by the boltzmann weights of an effective hamiltonian , @xmath23\over t } & = & { k\over2}\int dx\left({d\vec{r}\over dx}\right)^2 + { v\over2}\int dxdx'\delta^d(\vec{r}(x)-\vec{r}(x ' ) ) \nonumber\\ & & + { 1\over 2t}\int dxdx'{q(x)q(x')\over & \equiv & h_0+h_v+h_q\ .\end{aligned}\ ] ] in this equation @xmath24 represents the entropic properties of the connected chain ( ideal polymer ) , @xmath25 is the continuum description of the excluded volume interactions , while @xmath26 represents the @xmath7dimensional electrostatic energy . for each pa , there is a specific ( quenched ) function @xmath27 representing the charges along the chain . ( in this work we set @xmath28 and measure @xmath2 in energy units . ) in the simplest ensemble of quenches , each monomer takes a charge @xmath0 independent of all the others ; i.e. @xmath29 , where the overline indicates averaging over quenches . while the average charge of such pas is zero , a `` typical '' sequence has an excess charge of about @xmath30 , with @xmath31 . this statement , as well as the definition of @xmath32 , are unrelated to the embedding dimension @xmath7 . however , the importance of charge fluctuations ( both for the overall polymer , or large segments of it ) does depend on the space dimension . the electrostatic energy of the excess charge , spread over the characteristic size of an ideal polymer ( @xmath33 ) , grows as @xmath34 . this simple dimensional argument shows that for @xmath8 weak electrostatic interactions are irrelevant . ( the excluded volume effects are also irrelevant in @xmath8 . ) thus , at high temperatures the pa behaves as an ideal polymer with an entropy dominated free energy of the order of @xmath35 . however , on lowering temperature it collapses into a dense state , taking advantage of a condensation energy of the order of @xmath36 . this collapse is similar to the well known @xmath37transition of polymers with short range interactions and will be discussed later in this section . for @xmath9 , electrostatic interactions are relevant and the high temperature phase is no longer a regular self avoiding walk . at high temperatures the behavior of the polymer can be studied perturbatively . for the above ensemble of uncorrelated charges , the lowest order ( @xmath38 ) correction to the quench averaged @xmath39 vanishes@xcite . however , if we restrict the ensemble of quenches to sequences with a fixed overall excess charge of @xmath1 , there is a lowest order correction term proportional to @xmath40 . thus pas with @xmath1 less than @xmath32 contract while those with larger charges expand . this trend appears in any space dimension @xmath7 , and is indicated by the vertical line at the top of fig . it should be noted that restricting the ensemble to yield fixed @xmath1 , slightly modifies the quench averaged charge charge correlations . in particular , the two point correlation function becomes @xmath41 for @xmath42 . this small ( order of @xmath43 ) correction to the correlation function may cause a significant change in @xmath39 due to the long range nature of the coulomb interaction . the above discussion can be extended to pas with short range correlations along the sequence : if neighboring charges satisfy @xmath44@xcite where @xmath45 , with no further restrictions , then @xmath46 . the resulting ensembles continuously interpolate between the deterministic extremes of an alternating sequence ( @xmath47 ) and a uniformly charged polyelectrolyte ( @xmath48 ) . as in the case of uncorrelated charges , we can impose an additional constraint on the overall charge , resulting in correlations @xmath49 where @xmath50 . we note that the variance of @xmath1 in such a correlated sequence also becomes @xmath51 . thus the proportion of quenches with @xmath1 above or below @xmath52 is independent of @xmath53 . all the results for uncorrelated sequences remain valid if we substitute @xmath52 for @xmath32 . as @xmath54 , the behavior of the pa crosses over from that of a random sequence to the deterministic ( alternating or homogeneous ) one . however , the crossover occurs only for @xmath55 . as typical of the qualitative behavior outside this narrow interval , we concentrate on the uncorrelated case of @xmath56 . a short distance cutoff @xmath6 , such as the range of the excluded volume interaction , introduces a temperature scale @xmath57 . for @xmath8 the electrostatic interactions in random pas are effectively short ranged . previous results on a random short range interaction model@xcite ( rsrim ) in @xmath16 indicate that , as long as the positive and negative charges are approximately balanced , the polymer assumes spatial conformations where the interactions are predominantly attractive . to maximize this attraction , the chain undergoes a transition from an expanded to a collapsed ( dense ) state at a @xmath37transition . for truly short range interactions , the @xmath37transition disappears only for a rather strong charge imbalance of @xmath58 . even if as a result of the relevant coulomb interactions in @xmath9 , the high temperature phase of uncorrelated pas turns out to be compact , we can not exclude the possibility of a transition into another dense ( possibly glassy ) state when @xmath2 decreases below a critical @xmath59 . such a potential `` @xmath59transition '' , indicated by the horizontal dashed line in fig . [ figa ] , must be different from a regular collapse since the lower density phase is not a self avoiding walk . the compact phase can also be destroyed by increasing the net charge as described in the following paragraph . a dense globular pa droplet of radius @xmath60 has a surface energy of @xmath61 , where @xmath62 is the surface tension . for small @xmath1 , the surface tension keeps the pa in an approximately spherical shape . however , as shown in appendix [ secrayleigh ] , at sufficiently large @xmath1 electrostatic forces destabilize the droplet . comparing the electrostatic ( @xmath63 ) and surface energies indicates that the droplet shape is controlled by the parameter @xmath64 , where @xmath65 is the _ rayleigh charge_. for a large enough @xmath66 a spherical shape is unstable ( a charged liquid droplet disintegrates ) . the rayleigh charge in @xmath16 is proportional to @xmath67 , while for @xmath14 ( @xmath15 ) it increases faster ( slower ) than @xmath32 . the solid vertical line at the bottom of fig . [ figa ] shows the position of this instability in @xmath16 . clearly , any @xmath59transition ( if at all present ) must also terminate at @xmath11 . only a negligible fraction of random quenches in @xmath14 have @xmath1 exceeding @xmath11 , and thus a typical pa is a spherical droplet at low @xmath2 . conversely , in @xmath15 almost all pas have charges larger than @xmath11 and the dense phase does not exist . the borderline case of @xmath16 , where a finite fraction of pas have @xmath1 exceeding @xmath11 , is the most controversial : an analogy with uniformly charged polyelectrolytes@xcite suggests@xcite that the pa is fully stretched ( @xmath68 ) in this case@xcite . by contrast , a debye hckel inspired theory@xcite predicts that low@xmath2 configurations are compact . partial resolution of this contradiction comes from the observation@xcite that pas in @xmath16 are extremely sensitive to the excess charge @xmath1 . in the following section we shall briefly review the main features of three dimensional pas obtained by mc simulations@xcite . numerical simulations are performed on a discretized version of eq . ( [ continuumh ] ) . configurations of a polymer are specified by listing the position vectors @xmath69 ( @xmath70 ) of its monomers . the shape and spatial extent of the polymer are then characterized by the tensor , @xmath71 with the greek indices labeling the various components . thermal averages of the eigenvalues @xmath72 of this tensor ( sometimes referred to as moments of inertia ) are used to describe the mean size and shape ; their sum is the squared radius of gyration , @xmath73 . since we are dealing with sequences of quenched disorder , these quantities must also be averaged over different realizations of @xmath74 . in three dimensions , uniform uncharged polymers in good solvents are swollen ; their @xmath75 scaling as @xmath76 with @xmath77 as in self avoiding walks . polymers in poor solvents are `` compact '' , i.e. described by @xmath78 . in previous work@xcite we used monte carlo ( mc ) simulations ( along with analytical arguments ) to establish the following properties for pas immersed in a good solvent : \(a ) the radius of gyration strongly depends on the total excess charge @xmath1 , and is weakly influenced by other details of the random sequence . \(b ) a @xmath38expansion indicates that the size of a pa tends to decrease upon lowering temperature if @xmath1 is less than a critical charge @xmath31 , and increases otherwise . this behavior is confirmed by mc simulations . \(c ) at low temperatures , neutral polymers ( @xmath79 ) are compact in the sense that their spatial extent in any direction grows as @xmath80 , where @xmath6 is a microscopic length scale . \(d ) the low@xmath2 size of the pa exhibits a sharp dependence on its charge : @xmath75 is almost independent of @xmath1 for @xmath81 , and grows rapidly beyond this point . this increase becomes sharper as the temperature is lowered , or as the length of the chain is made longer . we interpret the low temperature results by an analogy to the behavior of a charged liquid drop . the energy ( or rather the quench averaged free energy ) of the pa is phenomenologically related to its shape by @xmath82 the first term is a condensation energy proportional to the volume ( assumed compact ) , the second term is proportional to the surface area @xmath83 ( with a surface tension @xmath84 ) , while the third term represents the long range part of the electrostatic energy due to an excess charge @xmath1 ( @xmath85 is a dimensionless constant of order unity ) . the optimal shape is obtained by minimizing the overall energy . the first term is the same for all compact shapes , while the competition between the surface and electrostatic energies is controlled by the dimensionless parameter @xmath86 here @xmath87 and @xmath88 are the radius and volume of a spherical drop of @xmath5 particles , and we have defined the _ rayleigh charge _ @xmath11 . ( see eq . ( [ eqr ] ) of the appendix for the definition of @xmath11 in a general dimension @xmath7 . ) the dimensionless parameter @xmath66 controls the shape of a charged drop : a spherical drop becomes unstable and splits into two equal droplets for @xmath89 . we argued in ref.@xcite that the quenched pa has a similar instability at the vertical line in the bottom of fig . [ figa ] : charged beyond a critical @xmath66 , the pa splits to form a _ necklace _ of blobs connected by strands . from the definition of @xmath11 it is clear that it is proportional to @xmath32 ; the dimensionless prefactor relating the two depends on @xmath84 and is estimated in this work . \(e ) while confirming the general features of fig . [ figa ] , the mc simulations provide no indication of the suggested @xmath59 transition . however , these simulations are not reliable at very low temperatures@xcite due to the slowing down of the equilibration process . all the above results were obtained by mc simulations for pas of between 16 and 128 monomers . since the mc procedure does not provide good equilibration at low @xmath2 , we could not determine the properties of the ground states ( although some conclusions were drawn from the low@xmath2 data ) . systems with coulomb interactions are particularly poorly equilibrated , even at densities of only 10% the maximal value . to remedy these difficulties , in this work we resorted to complete enumeration of all possible spatial conformations , and all possible quenched charge sequences . while such an approach enables us to obtain exact results , and detailed information not available in mc studies , the immensity of configuration space restricts the calculation to chains of at most 12 steps ( 13 monomers ) . we considered self avoiding walks ( saws ) on a simple cubic lattice of spacing @xmath6 . an @xmath90step saw has @xmath91 sites ( atoms ) , and the randomly charged polymer is defined by assigning a fixed sequence of charges ( @xmath92 ) to its monomers . the charge sequence is considered to be quenched , i.e. it remains unchanged when the spatial conformation of the walk changes . the energy of any particular configuration is given by @xmath93 , where @xmath94 is the position of the @xmath95th atom on the three dimensional lattice . thermal averages of various quantities are calculated by summing over all conformations with the boltzmann weights of this energy . the resulting average is quench specific . we then obtain quenched averages by summing over all possible realizations of the sequence , possibly with certain restrictions such as on the total excess charge @xmath96 . our calculation consists of three steps : ( a ) generating lists of all spatial conformations and quench sequences ; ( b ) using the lists to calculate various thermodynamic quantities for each quenched configuration ; ( c ) averaging of results over restricted ensembles of quenches , and analyzing the data . since the calculational procedure is extremely time consuming , we used precalculated lists of all saws , and of all possible sequences of charges along the chains . other programs then used these two lists as input . since the first list of all @xmath90step saws with @xmath97 is extremely large , we tried to reduce it by including only `` truly different '' configurations and listing their degeneracies . as the actual position of a walk in space is not important , we disregard it and only give the _ directions _ of the @xmath90 steps . as the energy of a configuration is independent of its overall orientation , we assume that the first step is taken in the @xmath98 direction . the above trivial symmetries are not included in our counting ; e.g. we assign a completely straight line the degeneracy of @xmath99 . all saws , except for the straight line , have a four fold degeneracy related to rotation around the direction of the initial step . we shall therefore assume that the first step which is not in @xmath98 direction is taken along the @xmath100 axis , and attribute a degeneracy @xmath101 to all walks which are not straight lines . every non planar walk has an additional degeneracy due to reflection in the @xmath102 plane , leading to a total degeneracy of @xmath103 . thus the list of all 12step saws consists of the directions of 11 successive steps , along with a degeneracy factor . the total number of 4,162,866 chains consists of one straight line ( @xmath99 ) , 40,616 planar saws ( @xmath101 ) , and 4,122,249 non planar saws ( @xmath103 ) . accounting for degeneracies reduces the length of the list and the time needed to calculate various quantities by almost a factor of eight . some chains possess additional symmetries ; e.g. , by inverting the sequence of steps we may get some other saw in the list . we did not take advantage of this symmetry because the distribution of quenched charges on the chain is not necessarily symmetric under interchange of its two ends . in any case , we use the end to end exchange symmetry in the listing of all possible quenches . a second input list contains all possible charge sequences ; each quench " for an @xmath90step walk has @xmath91 charges . since the energy is unchanged by reversing the signs of all charges , we considered only configurations in which the total charge is @xmath104 . this shortens the list by almost a factor of 2 . the majority of quenches are not symmetric under order reversal , i.e. the sequence does not coincide with itself when listed backwards . since we are exploring all spatial configurations , without accounting for the end reversal symmetry mentioned in the previous paragraph , the list can be reduced by almost another factor of 2 by considering only one of each pair of such quenches ( keeping track of the degeneracy ) . after accounting for both charge and sequence reversal symmetries , the list for @xmath105 ( @xmath19 ) has only 2080 entries . for any sequence , the number of computer operations required to calculate the energy of a single configuration grows as @xmath106 . the total number of saws grows as @xmath107 , where @xmath108 is the effective coordination number for saws on a cubic lattice , and @xmath109 . the number of quenches " grows as @xmath110 . these factors limit the size of chains which can be investigated to @xmath105 steps . at this @xmath90 we needed 4 weeks of cpu time on a silicon graphics r4000 workstation . an increase of @xmath90 by a single unit multiplies the calculation time by an order of magnitude . thus it is impractical to employ our procedure for chains that are much longer than 12 steps . if , instead of completely enumerating all possible charge configurations we confine ourselves to sampling a few hundred quenches , the calculations can be extended to @xmath111 , but not much further . the order in which the calculations were performed is as follows : for each saw configuration we calculated the radius of gyration , squared end to end distance , _ and _ the energies of all possible quenched charge configurations along the backbone . these energies were then used to update histogram tables ( a separate histogram of possible energies for each quench ) . due to the long range nature of the coulomb interaction , the allowed energies form almost a continuous spectrum which we discretized in units of @xmath112 . this discretization was sufficient to accurately reproduce properties of the system on the temperature scales of interest . ( for @xmath113 , we used a finer division of the histograms to verify that the discretization process does not distort calculation of such properties as the specific heat , except at extremely low temperatures . ) in addition to histograms , we also collected data about the energy , the radius of gyration , and the end to end distance at the ground state of each quench . the issue of the multiplicity of the ground state is of much interest , and hotly debated in the context of models of proteins@xcite . in the presence of coulomb interactions , due to the quasi continuous nature of the energy spectrum , the ground state is almost never degenerate ( except for the trivial degeneracies mentioned above ) . it is quite likely that , for sufficiently long polymers and specific quenches , there may be exactly degenerate ground states which are not related by symmetry operations . however , for @xmath114 such cases are extremely rare . moreover , the distance between the ground state and the second lowest energy state remains of the order of @xmath112 for all @xmath90s in our calculation , even though at higher energies the densities of the states increase very rapidly with @xmath90 . we used this data to obtain averages over quenches ( with or without a constraint on the net charge ) . it should be mentioned that creation of histograms , as well as calculation of the thermal averages , required the correct accounting of degeneracies of spatial conformations , while averages over quenches needed proper care of the sequence degeneracies . for few selected quenches we also performed a calculation of the density of states as a function of two variables , the energy and the squared radius of gyration . due to large amount of data , we could not do such detailed studies for all possible quenches . fig . [ figb ] depicts ground states of four cases of quenched charges with different excess charges @xmath1 . it provides qualitative support for the conclusions previously obtained for mc simulations : fig . [ figb]a depicts the ground state configuration of an almost neutral pa which is quite compact . the pa in fig . [ figb]b has @xmath1 slightly smaller than @xmath32 ; while the configuration is still compact we see the beginning of a stretching . figs . [ figb]c and [ figb]d show strongly stretched configurations in cases where @xmath1 exceeds @xmath32 . in the following sections we will quantify this qualitative observations . as a by product of the above procedure we also obtained similar data for the model with short range interactions : in the random short range interaction model ( rsrim ) , a quenched sequence of dimensionless charges @xmath115 is defined along the chain . the interaction energy is @xmath116 , where @xmath117 if @xmath118 , and @xmath119 , otherwise . while we shall compare and contrast several properties of short and long range models in this paper , detailed results for rsrim can be found in ref.@xcite . the additional data were gathered without a substantial increase in the total execution time of the programs . there are a few minor differences in the data collection process in the two models : ( a ) as the energies of the rsrim are naturally discretized , the resulting histograms are exact . ( b ) the ground state of most quenches is highly degenerate . this required keeping track of the degeneracy , and obtaining the @xmath39 in the ground state as an average lowest energy configurations . we begin our analysis by testing the validity of eq . ( [ edefenerg ] ) for the ground states of the polymers . obviously , the exact value of each ground state energy depends on the details of the charge sequence . however , eq . ( [ edefenerg ] ) implies that the effect of the overall charge can be ( approximately ) separated ; the remaining parts of the energy depending only weakly on the details of the sequence . the basic energy unit of our model is @xmath120 , and a useful system for comparison is the regular crystal formed by alternating charges ( the `` sodium chloride '' structure ) . the condensation energy per atom of such a crystal ( @xmath121 ) is much smaller than the interaction energy per atom between the nearest neighbors ( @xmath122 ) . this demonstrates the importance of the long range coulomb interaction : although the system is locally neutral , the ground state energy depends on an extended neighborhood . similarly , the surface tension @xmath123 of the crystal is quite small . our first observation is that , for a fixed @xmath1 , the ground state energy is quite insensitive to the details of the sequence : fig . [ figc ] depicts the ground state energies of _ all _ 2080 possible quenches for 12step ( 13atom ) chains . ( the horizontal axis represents an arbitrary numbering of the quenches . ) the energies are clearly separated into 7 bands , corresponding to excess charges of @xmath124 , 3 , 5 , @xmath125 , 13 . ( there is only one quench with @xmath126 . ) while each band has a finite width , we see that the energy of a pa can be determined rather accurately by only specifying its net charge @xmath1 ! _ this is not the case for short range interactions : _ a comparison of histograms of ground state energies between ( a ) pas and ( b ) rsrim of length @xmath127 in fig . [ figd ] clearly shows the importance of long range interactions . there is a rather clear separation of energies into ` bands ' with fixed values of @xmath1 for the pas , which is almost absent in the rsrim . of course , the finite width of each ` band ' shows that the details of the sequence can not be completely neglected , although their influence on the ground state energy is rather small . using a debye hckel approximation@xcite , wittmer _ et al_@xcite have performed a systematic study of the dependence of the free energy of neutral pas ( @xmath79 ) at high @xmath2 on the correlations between neighboring charges along the chain ( see eq . ( [ qqcorr ] ) ) . they obtain an expression which smoothly interpolates between the free energy densities of a completely random sequence ( @xmath128 , where @xmath129 is the debye screening length ) , and non - random alternating sequence ( @xmath130 ) . ( the latter model was also studied in ref.@xcite . ) these results exclude the electrostatic self interaction energy , which is infinite in the continuum model used in ref.@xcite . thus the free energy of the alternating chain is roughly 20 times smaller than the random one . while these results can not be directly extended to the ground states , we may attempt to obtain crude estimates by setting @xmath131 and @xmath132 . however , our results indicate that the ground state energy of alternating polymers ( @xmath133 ) is only smaller by about 16% than the mean condensation energy of unrestricted sequences . such inconsistency is partially explained by the fact that the alternating pa has negative mean electrostatic energy ( approximately @xmath134 per atom ) even at @xmath135 , while such an energy for a completely random pa ( averaged over all quenches ) vanishes . thus , only about 1/4 of the ground state energy of an alternating pa is its condensation energy . ( this part of the energy explicitly depends on the discreteness of the chain and is not accounted for in ref.@xcite . ) this argument brings the approximate conclusions based on ref.@xcite in better qualitative agreement with our exact enumeration results . the dependence of the quench averaged ground state energies on the length of the chain is depicted in fig . a restricted average is performed at each value of @xmath1 ( indicated next to each line ) . the scaling of the axes is motivated by the re - casting of eq . ( [ edefenerg ] ) in the form @xmath136 where @xmath137 , with a prefactor @xmath138 depending on the average shape . the small number of data points makes an accurate determination of @xmath139 ( and hence the surface tension ) rather difficult . the value used in fig . [ fige ] is @xmath140 , for which the curves with different @xmath1 extrapolate to approximately the same value , giving a condensation energy of @xmath141 . for this choice of @xmath139 the slopes of the curves with @xmath142 approximately scale as @xmath143 . the condensation energy @xmath144 is surprisingly close to that of a regular crystal ( @xmath121 ) , despite the fact that in a random chain on average one neighbor ( along the chain ) has the `` wrong '' sign ( compared to the alternating arrangement ) , costing an energy of the order of @xmath145 . this again confirms our contention that the ground state energy is determined by very extended neighborhoods of each particle . if the ground state configuration has approximately cubic or spherical shape , then @xmath146 , while for the slightly elongated objects that we obtain , @xmath138 can be somewhat larger ( @xmath147 ) . therefore , we estimate @xmath148 . the error bars indicate our uncertainty in the values of @xmath138 and @xmath139 , and disregard possible systematic errors in attempts to evaluate surface tension from such small clusters . using these numbers we estimate that the rayleigh charge of the model pa is approximately the same as @xmath32 , since @xmath149 ( the relation @xmath150 assumes pas of maximum possible density . ) from fig . [ fige ] it is not clear that the ( charge unconstrained ) average energies ( indicated by the @xmath151 symbols ) of all quenches , also extrapolate to the same condensation energy of @xmath144 . this apparent inconsistency can be understood by noting that since the quench averaged @xmath143 is equal to @xmath152 , the last term in eq . ( [ eqnscale ] ) scales as @xmath153 . thus , the linear approach ( in the variables used in fig . [ fige ] ) to asymptotic value ( as @xmath154 ) is replaced by a very small power law . such a slow decay can not be detected for the small values of @xmath5 used in our enumeration study . since our model is defined on a discrete lattice , the allowed energies are discrete . however , as the length of the chain increases the separations between the states are reduced . the density of states becomes quasi continuous and can be described by a function @xmath155 . [ figf ] depicts @xmath156 , where the overline denotes averaging over all quenches with a fixed @xmath1 . ( note that this quantity is _ not _ the quench averaged free energy as the average is performed on @xmath157 rather than on @xmath158 . ) not surprisingly , the densities of states for different @xmath1s are shifted with respect to each other . for every quench the density of states is very high near the middle of the band and decreases towards the edges . we find that almost all pas have a unique ground state ( up to trivial symmetry transformations ) . this is not the case for short range interactions@xcite and may be an important clue to the problem of protein folding . ( for ease of calculation , most studies of similar random copolymers have focused on short - range interactions , and typically find highly degenerate ground states . ) furthermore , the gaps to the second lowest energy states typically remain of order of @xmath159 ( up to the studied size of @xmath105 ) , while most interstate separations decrease with @xmath90 . in the @xmath160 limit , the density of lowest energy excitations of our model pas appears to decay faster than a power law . ( of course our lattice model does not include any vibrational modes . ) this decay manifests itself in a vanishing heat capacity in the @xmath161 limit , as depicted in fig . the solid lines represent the quench averaged heat capacities per degree of freedom @xmath162 , of pas with @xmath79 at low temperatures . ( since the energy fluctuations of a polymer depend only on changes of its shape , and are independent of its overall position and orientation , we assumed that an @xmath5atom pa has @xmath163 degrees of freedom , where @xmath164 represents subtraction of translational and rotational degrees of freedom . such a choice decreases the bias in the @xmath5dependence of @xmath162 which would appear for very small values of @xmath5 . ) the vanishing heat capacity was _ not _ observed in mc studies@xcite , where poor equilibration at low @xmath2 hinders measurement of @xmath162 . it is instructive to compare and contrast the behavior of random pas with vanishing excess charge to that of an ordered alternating sequence ; the latter is a highly atypical member of the ensemble with @xmath79 . numerical investigations of alternating charge sequences by victor and imbert@xcite show that such polymers undergo a collapse transition , similar to saws with _ short range _ attractive interactions . this is because the exact compensation in the charges of any pair of neighboring monomers leads to large scale properties determined by dipole dipole ( and faster decaying ) interactions . thus coulomb interactions are irrelevant in the high temperature phase of the alternating chain that consequently behaves as a saw . by contrast , even though we consider a sub ensemble of quenches with @xmath79 , the charge fluctuations can not be neglected in random pas and control the long distance behavior of the chain . such pas are compact at _ any _ temperature . the attractive dipole dipole interactions eventually cause the collapse of the alternating charge sequence to a compact state at temperatures below a @xmath37point . of course , the ground state of such a chain is the ordered nacl crystal discussed earlier . however , it is not clear if the state of the chain immediately below the @xmath37 temperature is the ordered crystal . another possibility is that the initial collapse is into a molten globular " ( liquid like ) state @xcite , which then crystallizes at a lower temperature . we singled out the alternating pas in our complete ensemble of quenches ; the dashed lines in fig . [ figg ] depict the heat capacity of this sequence . the presence of a phase transition manifests itself in the peak in @xmath162 at @xmath165 ( for @xmath166 ) which grows ( and slightly shifts towards higher temperatures ) as @xmath90 increases . [ figg ] shows that the average heat capacity of random pas with @xmath79 also has a peak at @xmath167 ( for @xmath166 ) . as the high temperature phase is no longer swollen ( for @xmath79 ) , there are again two possible interpretations of this heat capacity peak . one is that it represents a crossover remnant of the @xmath37 transition , with an increase in the density of the compact polymer . indeed , the peak is lower and broader than that of alternating chains . another possibility is that there is a glass " transition in which the molten globule " freezes into its ` ground state ' . the proximity of the peak temperature to the energy gap for the first excited state supports the latter conclusion . no corresponding anomaly was observed in the mc simulations@xcite . since finite size effects are extremely important in such small systems , the heat capacity peak should be regarded only as a suggestion for the presence of a @xmath59transition " . as indicated by the dashed line in fig . [ figa ] , the location of such a transition may depend on @xmath1 , disappearing at @xmath168 , consistent with other features of the phase diagram . this behavior is analogous to that of the @xmath37point in the rsrim@xcite , although in that case the limiting charge scales linearly with @xmath5 . additional , studies are needed to establish the @xmath59transition . the contour plots in fig . [ figh ] depict the number of states as a function of both @xmath39 and @xmath169 , for three sequences of @xmath127 with charges @xmath124 ( a ) , 5 ( b ) , and 11 ( c ) . at high temperatures the typical configurations correspond to the highest densities . in all three cases these configurations are located in the middle of the diagram , and behave essentially as saws . on lowering temperature the polymer seeks out states of lowest energy which are very different in the three cases . the approximately neutral chain of fig . [ figh]a assumes a very compact shape represented by the lower left corner of the contours . the presence of a specific heat peak is consistent with the shape of this contour plot . while @xmath39 increases monotonically with @xmath2 , the chains are too short to permit a quantitative test for the presence of a @xmath59point from the scaling of @xmath39 . the lowest energy contour of the chain with @xmath170 ( fig . [ figh]b ) is almost horizontal . hence , upon lowering temperature the chain will not collapse , maintaining an extended shape . thus a putative transition must disappear for larger @xmath1 . finally , the fully charged polymer in fig . [ figh]c expands from a saw to the completely stretched configurations represented by the lower right corner of the contour plot . the low temperature results from mc simulations suggest that @xmath39 of a pa strongly depends on its charge , crossing over from compact configurations at small @xmath1 to extended states for larger @xmath1 . this is qualitatively supported by the ground state shapes in fig . [ figb ] , and will be more quantitatively examined here . [ figi]a depicts the @xmath90dependence of @xmath39 for several choices of @xmath1 . the vertical axis is scaled so that compact , i.e. fixed density , structures are represented by horizontal lines . since @xmath1 is fixed , the influence of the excess charge diminishes as the length of the polymer is increased , and thus all curves must asymptotically converge to the same horizontal line . there is some indication of this in fig . [ figi]a , although the crossover is rather delayed for larger values of @xmath1 . since the unrestricted ensemble ( solid circles ) includes a large range of @xmath1s , it is not surprising that the corresponding averages are not compact . the chains are too short to extract a meaningful value for the exponent @xmath4 . nevertheless , the effective slope of @xmath171 , strongly suggests that the average over an unrestricted ensemble is not compact . by comparison , the corresponding results for the rsrim in fig . [ figi]b clearly indicate that the averages both at fixed and varying @xmath1 have similar fixed density ground states . since the quench averaged @xmath39 of the unrestricted ensemble scales differently from the sub ensembles of fixed @xmath1 , the former set must contain a non negligible portion of non compact configurations for every @xmath90 . it is natural to assume that the borderline between compact and stretched states is controlled by @xmath67 . in previous work@xcite we argued that pas undergo a transition to an expanded state when @xmath1 exceeds @xmath11 ( @xmath172 ) : the transition is more pronounced for larger @xmath90 and lower @xmath2 . in the mc simulations@xcite we were able to use long pas , but were restricted to finite , albeit small , temperatures which slightly smeared the transition in @xmath75 with increasing @xmath1 . in this study we know the exact ground states but are limited to small @xmath90s where the difference between @xmath75 of compact and stretched states is less visible . the sum of all eigenvalues of the shape tensor , @xmath39 is somewhat insensitive to an expansion since the increase in the largest eigenvalue is partially compensated by the decrease of the other two eigenvalues . a clearer view is provided by the ratios of the mean eigenvalues of the shape tensor as depicted in fig . these ratios for different @xmath90s can be collapsed after scaling the charges by @xmath32 , consistent with the mc simulations . [ figk ] depicts ( on a logarithmic scale ) the distribution of values of @xmath39 in the ground states of all quenches for @xmath111 . the distribution is peaked near the smallest possible value of @xmath39 , but has a broad ( possibly power law ) tail . if the tail falls off sufficiently slowly , it will determine the asymptotic value of the exponent @xmath4 : as @xmath90 increases the very large values of @xmath39 of the ( minority ) stretched configurations will eventually dominate the total average . we thus expect @xmath173 to increase with @xmath90 , and the value of @xmath173 extracted from the slope of the solid line on fig . [ figi]a , probably underestimates the true asymptotic value . to get further insight into the behavior for larger @xmath90 , we performed separate averages for the 80% of configurations which have the smallest @xmath39 , and for the remaining top 20% . these averages are depicted in fig . the vertical axis is again scaled so that compact structures are represented by horizontal lines . the bottom 80% indeed scale as compact chains while the top 20% , which stand for the tail of the distribution , have radii that grow with @xmath90 with an effective exponent of @xmath174 . we thus conclude that the @xmath39 of the unrestricted ensemble increases with @xmath90 at least as fast as a saw . as a byproduct of our study , since we have access to the complete set of quenches , we can find which particular sequence , restricted only by its net charge , has the lowest energy . as this is the sequence that is selected in a model in which the charges are free to change positions along the chain , we shall refer to the results as describing the ground states of _ annealed pas_. for long chains , neither the sequence , nor its spatial conformation , need to be unique . however , for the sizes considered here , we always found a single ground state , several of which are shown in fig . [ figm ] for @xmath111 and different values of @xmath1 . it appears that the optimal configurations correspond to a uniform distribution of excess charge along the backbone . in particular , for small @xmath1 the preferred arrangement is the alternating sequence which then folds into a nacl structure . previously@xcite we suggested that annealed pas expel their excess charge ( provided @xmath175 ) into highly charged `` fingers '' . as a result of such `` charge expulsion '' the spanning length of annealed pas should increase dramatically ( @xmath176 ) . however , since most of the mass remains in a compact globule , @xmath39 is not substantially modified ( as long as @xmath177 ) . the chains used in our study are too short to exhibit an increased spanning length with no change in @xmath75 . moreover , the effects of lattice discreteness are much more pronounced for annealed pas where ground states correspond to a single sequence . in the quenched case , averaging over all sequences partially smoothens out lattice effects . as partial evidence we note that plots for the charge dependence of ratios of eigenvalues of the shape tensor ( analogous to fig . [ figj ] ) exhibit better collapse with the variable @xmath178 than with @xmath179 . however , given the scatter of the few data points , the evidence for the appearance of `` fingers '' is not really any more convincing than any conclusions drawn from inspection of the ground states in fig . [ figm ] . as noted earlier , we expect the ground state of a sufficiently long annealed pa with fixed @xmath1 to be the nacl structure . to test the approach to this limit , in fig . [ fign ] we plot the energies per atom of the ground states . as in the case of quenched pas ( fig . [ fige ] ) , we check for finite size corrections proportional to the surface area . ( unlike the case of quenched pas , each point in this figure represents a _ single _ configuration . ) here we used a value of @xmath180 although the results are rather insensitive to this choice , and we estimate the accuracy of this quantity as @xmath181 . the point of intersection with the @xmath182 axis is close to the known value of the @xmath145 . furthermore , @xmath183 corresponds to a surface tension of @xmath184 which is also consistent with the known value of @xmath185 . these consistency checks add further confidence to the values of @xmath144 and @xmath84 deduced for quenched pas . this work was supported by the us israel bsf grant no . 9200026 , by the nsf through grants no . dmr9400334 ( at mit s cmse ) , dmr 9115491 ( at harvard ) , and the pyi program ( mk ) . in this appendix we discuss instabilities of charged @xmath7dimensional drops . a detailed discussion of the three dimensional case can be found in appendices b and c of ref.@xcite , which also provides other references to the subject . the energy of a charged _ conducting _ ( hyper)sphere of radius @xmath87 with charge @xmath1 is given by @xmath186 where the first term is the surface energy ( @xmath84 is the surface tension , and @xmath187 denotes the @xmath7dimensional solid angle ) , while the second term is the electrostatic energy . ( we have used units such that , in @xmath7 dimensions , the electrostatic potential at a distance @xmath188 from a charge @xmath189 is @xmath190 ; and @xmath191 in @xmath192 . ) for small @xmath1 , the sphere is stable with respect to infinitesimal shape perturbations . however , when the electrostatic and surface energies are comparable , the drop becomes unstable . to explore this instability we differentiate eq . ( [ eeq ] ) with respect to @xmath87 to find the pressure difference between the interior and the exterior of the drop as @xmath193 the pressure difference vanishes when @xmath1 equals the _ rayleigh charge _ @xmath11 , where @xmath194 for @xmath195 a ( hyper)spherical shape is unstable to small perturbations ; initially the drop becomes distorted and subsequently it disintegrates . note that @xmath196 , where @xmath88 is the volume of the drop . when applied to pas , up to a dimensionless prefactor , @xmath197 we can regard the first term in eq . ( [ eeq ] ) as setting the overall energy scale , while the shape of the drop is determined by the dimensionless ratio @xmath198 while from the above argument we conclude that the spherical shape is ( locally ) unstable for @xmath199 , even for @xmath200 , the energy of the drop can be lowered by splitting into smaller droplets . in particular , we may split away from the original drop a large number @xmath157 , of small droplets of radius @xmath201 and charge @xmath202 , and remove them to infinity . it can be directly verified that for @xmath203 , the total electrostatic energy , total surface area of the small droplets , as well as their total volume , vanishes in the @xmath204 limit . thus the energy of any charged conducting drop can be lowered to that of an uncharged drop by expelling a large number of `` dust particles '' which carry away the entire charge . ( of course this argument neglects the finite size of any particles making up the drop ! ) the globular phase of a quenched random pa is better represented by a drop of immobile charges . therefore , we next consider a drop in which the charges are _ uniformly _ distributed over the volume . the sum of the surface and electrostatic energies is now given by @xmath205 for sufficiently large @xmath1 , the drop can lower its energy by splitting into two droplets of equal size . this will occur when @xmath66 exceeds a critical value of @xmath206 which is equal to 0 , 0.293 , 0.323 , 0.322 , and @xmath207 , for @xmath192 , 3 , 4 , 5 , and @xmath208 , respectively . as the value of @xmath66 increases further , the drop splits into a larger number of droplets . by examining the energy of a system of @xmath157 equal spherical droplets , we find that the optimal number is proportional to @xmath209 . if the typical @xmath143 is proportional to @xmath5 ( as happens in unrestricted pas ) , while @xmath210 is given by eq . ( [ edefqrind ] ) , the number of droplets scales as @xmath211 . thus @xmath16 is a special dimension , above which a typical pa prefers to stay in a single globule . m. mzard , g. parisi , and m. a. virasoro , _ spin glass theory and beyond _ , world scientific , singapore ( 1987 ) . yu , a. tanaka , k. tanaka , and t. tanaka , j. chem . phys . * 97 * , 7805 ( 1992 ) ; yu x .- h . , ph . d. thesis , mit ( 1993 ) . m. annaka and t. tanaka , nature * 355 * , 430 ( 1992 ) . m. scouri , j.p . munch , s.f . candau , s. neyret , and f. candau , macromol . * 27 * , 69 ( 1994 ) . y. kantor and m. kardar , europhys . * 27 * , 643 ( 1994 ) . y. kantor and m. kardar , phys . rev . * e51 * , in press ( 1995 ) . j. wittmer , a. johner and j.f . joanny , europhys . lett . * 24 * , 263 ( 1993 ) . y. kantor and m. kardar , europhys . * 28 * , 169 ( 1994 ) . p. pfeuty , r.m . velasco , and p.g . de gennes , j. phys . ( paris ) lett . * 38 * , l5 ( 1977 ) . the coulomb potential used in refs.@xcite was slightly distorted at short distances , with different short range cutoff and intermonomer distances . therefore , there is no exact equivalence between @xmath2 of this work and that in refs.@xcite . an approximate correspondence is obtained by multiplying the @xmath2s in refs.@xcite by a factor of 2 to 3 . see , e.g. , h. frauenfelder and p.g . wolynes , physics today * 47*(2 ) , 58 ( 1994 ) . landau and e.m . lifshitz , _ statistical physics _ , part 1 , pergamon , ny ( 1981 ) . j.m . victor and j.b.imbert , europhys . lett . * 24 * , 189 ( 1993 ) . in all figures in this paper lengths are measured in units of @xmath6 , charges in units of @xmath212 , temperatures and energies in units of @xmath120 . for the rsrim , charges are dimensionless , and energies are instead measured in units of @xmath213 .

we perform an exact enumeration study of polymers formed from a ( quenched ) random sequence of charged monomers @xmath0 . such polymers , known as polyampholytes , are compact when completely neutral and expanded when highly charged . our exhaustive search included all spatial conformations and quenched sequences for up to 12step ( 13site ) walks . we investigate the behavior of the polymer as a function of its overall excess charge @xmath1 , and temperature @xmath2 . at low temperatures there is a phase transition from compact to extended configurations when the charge exceeds @xmath3 . there are also indications of a transition for small @xmath1 between two compact states on varying temperature . numerical estimates are provided for the condensation energy , surface tension , and the critical exponent @xmath4 .

arxiv

half - metallic ferromagnets with high ( room temperature and above ) curie temperatures @xmath1 are ideal for spintronics applications , and as such , much experimental and theoretical@xcite effort has been devoted in recent years to the designing of and search for such materials . among these , cr - doped dilute magnetic semiconductors ( dms ) @xcite or cr - based alloys and in particular cras and crsb @xcite in zinc blende(zb ) structure have attracted particular attention , not only because of the possibility of complete spin polarization of the carriers at the fermi level , but also for their possible high @xmath1 . akinaga @xcite were able to grow zb thin films of cras on gaas ( 001 ) substrates by molecular beam epitaxy , which showed ferromagnetic behavior at temperatures in excess of 400 k and magnetic moments of 3@xmath2 per cras unit . theoretical calculations by akinaga @xcite and several other theoretical calculations since then @xcite have verified the half - metallic character of cras . the high value of @xmath1 has also been supported by some of these studies@xcite . thin films of crsb grown by solid - source molecular beam epitaxy on gaas , ( al , ga)sb , and gasb have been found to be of zb structure and ferromagnetic with @xmath1 higher than 400 k@xcite . galanakis and mavropoulos@xcite , motivated by the successful fabrication of zb cras , crsb and mnas@xcite , have examined the possibility of half - metallic behavior in ordered zb compounds of transition metals v , cr and mn with the @xmath3 elements n , p , as , sb , s , se and te . their theoretical study shows that the half - metallic ferromagnetic character of these compounds is preserved over a wide range of lattice parameters . they also found that the half - metallic character is maintained for the transition element terminated ( 001 ) surfaces of these systems . yamana @xcite have studied the effects of tetragonal distortion on zb cras and crsb and found the half - metallicity to survive large tetragonal distortions . of course , the ground states of many of these compounds in the bulk are known to be different from the zb structure , the most common structure being the hexagonal nias - type . zhao and zunger@xcite have argued that zb mnas , cras , crsb , and crte are epitaxially unstable against the nias structure , and zb crse is epitaxially stable only for lattice constants higher than 6.2 , remaining half - metallic at such volumes . they also find that even though the ground state of crs is zb , it is antiferromagnetic at equilibrium lattice parameter , and thus not half - metallic . these results reveal the challenge experimentalists face in synthesizing these compounds in zb structure . however , the possibility remains open that such difficulties will be overcome with progress in techniques of film - growth and materials preparation in general . recently , deng @xcite were successful in increasing the thickness of zb - crsb films to @xmath4 3 nm by molecular beam epitaxy using ( in , ga)as buffer layers , and li @xcite were able to grow @xmath4 4 nm thick zb - crsb films on nacl ( 100 ) substrates . in view of the above situation regarding the state of experimental fabrication of these compounds and available theoretical results , it would be appropriate to study the variation of magnetic properties , particularly exchange interactions and the curie temperature , of cr - based pnictides and chalcogenides as a function of the lattice parameter . towards this goal , we have carried out such calculations for the compounds crx ( x = as , sb , s , se and te ) and the mixed alloys cras@xmath0x@xmath0 with x = sb , s , se and te . essentially we study the effect of anion doping by choosing elements of similar atomic sizes ( neighboring elements in the periodic table ) , one of which , namely sb , is isoelectronic to as , while the others ( s , se , te ) bring one more valence electron to the system . the mixed pnictide - chalcogenide systems offer further opportunity to study the effects of anion doping . the alloying with other 3d transition metals ( both magnetic , e.g. fe or mn , or non - magnetic , e.g. v ) on cation sublattice would also change the carrier concentration and bring about strong d - disorder which can additionally modify the shape of the fermi surface . this , however , is not the subject of the present paper . almost all theoretical studies on these alloys so far address aspects of electronic structure and stability of these alloys only . although a few theoretical estimates of exchange interactions and the curie temperature for cras at equilibrium lattice parameter have appeared in the literature , a detailed study of the volume dependence of these quantities is missing . for the other alloys , crsb , crs , crse and crte , no theoretical results for the exchange interaction , curie temperature and their volume dependence exist at present . the mixed pnictide - chalcogenide systems offer the possibility of not only creating these alloys over a larger range of the lattice parameter , but also with a larger variation in the exchange interactions . this is because at low values of the lattice parameter the dominant cr - cr exchange interactions in the chalcogenides can be antiferromagnetic , while for the pnictides they are ferromagnetic . the pnictide - chalcogenide alloying is important from the experimental viewpoint of stabilizing the zb structure on a given substrate , via the matching of the lattice parameter of the film with that of the latter . although the present study is confined to the zb structure only , we hope that it will provide some guidance to the experimentalists in their search and growth of materials suitable for spintronic devices . electronic and magnetic properties of crx ( x = as , sb , s , se and te ) and cras@xmath0x@xmath0 ( x= sb , s , se and te ) were calculated for lattice parameters varying between 5.45 and 6.6 , appropriate for some typical ii - vi and iii - v semiconducting substrates . calculations were performed using the tb - lmto - cpa method@xcite and the exchange - correlation potential given by vosko , wilk and nusair@xcite . in our lmto calculation we optimize the asa ( atomic sphere approximation ) errors by including empty spheres in the unit cell . we use the fcc unit cell , with cr and x ( as , sb , s , se and te ) atoms located at ( 0,0,0 ) and ( 0.25,0.25,0.25 ) , respectively , and empty spheres at locations ( 0.5,0.5,0.5 ) and ( -0.25,-0.25,-0.25 ) . for several cases , we have checked the accuracy of the lmto - asa electronic structures against the full - potential lmto results@xcite and found them to be satisfactory . for the mixed alloys cras@xmath0x@xmath0 ( x= sb , s , se and te ) , the as - sublattice of the zb cras structure is assumed to be randomly occupied by equal concentration of as and x atoms . the disorder in this sublattice is treated under the coherent potential approximation ( cpa)@xcite . our spin - polarized calculations assume a collinear magnetic model . in the following we will present results referred to as fm and dlm . the fm results follow from the usual spin - polarized calculations , where self - consistency of charge- and spin - density yields a nonzero magnetization per unit cell . although we call this the fm result , our procedure does not guarantee that the true ground state of the system is ferromagnetic , with the magnetic moments of all the unit cells perfectly aligned . this is because we have not explored non - collinear magnetic states , nor all antiferromagnetic ( afm ) states attainable within the collinear model . indeed , our results for the exchange interactions in some cases do suggest the ground states being of afm or complex magnetic nature . for lack of a suitable label , we refer to all spin - polarized calculations giving a nonzero local moment as fm state calculations . within the stoner model , a nonmagnetic state above the curie temperature @xmath1 would be characterized by the vanishing of the local moments in magnitude . it is well - known and universally accepted that the neglect of the transversal spin fluctuations in the stoner model leads to an unphysical picture of the nonmagnetic state and a gross overestimate of @xmath1 . an alternate description of the nonmagnetic state is provided by the disordered local moment ( dlm ) model , where the local moments remain nonzero in magnitude above @xmath1 , but disorder in magnitude as well as their direction above @xmath1 causes the global magnetic moment to vanish . combining aspects of the stoner model and an itinerant heisenberg - like model , heine and co - workers@xcite have developed a suitable criterion for a dlm state to be a more appropriate description of the nonmagnetic state than what is given by the stoner model . within the collinear magnetic model , where all local axes of spin - quantization point in the same direction , dlm can be treated as a binary alloy problem and thus described using the coherent potential approximation ( cpa)@xcite . we have carried out such dlm calculations , assuming the cr - sublattice to be randomly occupied by an equal number of cr atoms with oppositely directed magnetic moments . the object for performing the dlm calculations is two - fold . if the total energy in a dlm calculation is lower than the corresponding fm calculation , we can safely assume that the ground state ( for the given lattice parameter and structure ) is not fm , albeit of unknown magnetic structure . the exchange interactions in the dlm state can also be used to compute estimates of @xmath1 , and such estimates of @xmath1 may be considered as estimates from above the magnetic - nonmagnetic transition . @xmath1 computed from exchange interactions in the fm state are estimates from below the transition . of course , if the ground state is known to be ferromagnetic , then estimates of @xmath1 based on exchange interactions in the fm reference state are the appropriate ones to consider . in some cases where the fm results point to the possibility of the ground state magnetic structure being afm or of complex nature , we have carried out a limited number of afm calculations to provide some insight into this problem ( see section [ subsec : stabilityjq ] ) . we have computed the spin - resolved densities of states ( dos ) for all the alloys for varying lattice parameters , and for both the fm and dlm configurations . the fm calculations show half - metallic character , due to the formation of bonding and antibonding states involving the @xmath5 orbitals of the cr - atoms and the @xmath3 orbitals of the neighboring pnictogen ( as , sb ) or chalcogen ( s , se , te ) . the hybridization gap is different and takes place in different energy regions in the two spin channels . the critical values of the lattice parameters above which the fm calculations show half - metallic character agree well with those reported by galanakis and mavropoulos@xcite . the dos for the alloys of the type crx ( x = sb , s , se , te ) have been presented by several other authors@xcite and thus will not be shown here . in figs.[fig1 ] and [ fig2 ] we show the dos for the mixed alloys cras@xmath0sb@xmath0 and cras@xmath0se@xmath0 , for lattice parameters above and below the critical values for the half - metallic character . according to galanakis and mavropoulos@xcite , half - metallicity in zb cras appears between the lattice parameters of 5.45 and 5.65 . the latter corresponds to the lattice parameter of the gaas substrate . for crsb half - metallicity appears at a lattice parameter between 5.65 and 5.87 . the mixed alloy cras@xmath0sb@xmath0 , as shown in fig.[fig1 ] , is not quite half - metallic at the lattice parameter of 5.65 , and fully half - metallic at the lattice parameter of 5.76 . replacing sb with se in the above alloy , i.e. for cras@xmath0se@xmath0 , brings the critical lattice parameter down slightly . as shown in fig.[fig2 ] , at a lattice parameter of 5.65 , cras@xmath0se@xmath0 is half - metallic , although barely so . in our calculation crs and crse are half - metallic at a lattice parameter of 5.65 , and not so at a lattice parameter of 5.55 . crte is not half - metallic at a lattice parameter of 5.76 , but at a lattice parameter of 5.87 . for both crs and crse the critical value should be close to 5.65 , and for crte it should be close to 5.87 . note that in general the half - metallic gap is larger in the chalcogenides than in the pnictides . this is due to larger cr - moment ( see section [ sec : mag.mom . ] ) for the chalcogenides , which results in larger exchange splitting . this explains the difference in the half - metallic gaps in figs . [ fig1 ] and [ fig2 ] for similar lattice parameters . fig . [ fig3 ] compares the total dos of cras for the fm and dlm calculations for the equilibrium lattice parameter 5.65 . higher dos at the fermi level for the dlm calculation , compared with the fm calculation , is an indication that the band energy is lower in the fm state . indeed , as indicated in table [ table1 ] , compared with the dlm state the total energy for zb cras is lower in the fm state for the lattice parameters from 5.44 to 5.98 . in fact , this holds for lattice parameters up to 6.62 , showing the robustness of ferromagnetism in cras over a wide range of the lattice parameter . this is also true for crsb . sb@xmath0 for lattice parameters ( a ) 5.55 , ( b ) 5.65 , ( c ) 5.76 @xmath6 and ( d ) 5.87 , respectively . , width=360 ] se@xmath0 for lattice parameters ( a ) 5.55 , ( b ) 5.65 , ( c ) 5.76 @xmath6 and ( d ) 5.87 , respectively . , width=360 ] for the dlm and fm states.,width=360 ] in table [ table1 ] we show the variation of total energies per atom in ry with the lattice parameter for crx ( x = as , sb , s , se and te ) in the dlm and fm states . the purpose of tabulating these energies is not to determine the bulk equilibrium lattice parameters in the zb structure , as this has already been done by several authors@xcite . our results for equilibrium zb phase lattice parameters agree with those found by galanakis and mavropoulos@xcite . the important point is that for crs and crse at low values of lattice parameters the dlm energies are lower than the fm energies , showing clearly that the fm configuration is unstable . the result for crs is in line with the observation by zhao and zunger@xcite , who find zb crs to be antiferromagnetic with an equilibrium lattice parameter of 5.37 . as shown later ( section [ sec : exchange ] ) , the exchange coupling constants for the cr atoms in the fm calculations are negative , indicating the instability of the ferromagnetic spin alignment . the tendency to antiferromagnetism in crse at compressed lattice parameters is also revealed in a study by sasaio@xmath7lu @xcite for crte at lower lattice parameters the fm energy is lower than the dlm energy , but the exchange constants for the cr - atoms in the fm calculations are still negative ( see discussion in section [ sec : exchange ] ) , signaling the possibility of the ground states in crte at low values of the lattice parameter being neither dlm nor fm . note that in our discussion ground state implies the lowest energy state in zb structure . for crs , crse and crte the ground states at low lattice parameters can be of an antiferromagnetic ( afm ) or complex magnetic structure . a lower total energy may also mean a lower band energy , and in some cases , the latter may be reflected in a lower density of states at the fermi level . this is shown in fig.[fig4 ] , where for crs at the lowest lattice parameter of 5.44 @xmath6 the dos at the fermi level is lower in the dlm state than in the fm state . the deviation from ferromagnetism at low values of the lattice parameter for crs , crse and crte is also revealed by our study of the lattice fourier transform of the exchange interaction between the cr atoms in the fm state ( section [ sec : exchange ] ) . the search for an antiferromagnetic state with lower energy is possible within our collinear magnetic model by enlarging the unit cell in various ways . we have pursued this issue to a limited extent , by considering 001 , 111 afm configurations for crs , crse and crte at low values of the lattice parameter ( see discussion in section[sec : exchange ] ) . a satisfactory resolution of such issues is possible only by going beyond the collinear model . lcccccc lattice parameter ( ) & 5.44 & 5.55 & 5.65 & 5.76 & 5.87 & 5.98 + * cras * + dlm energy & -1653.4039 & -1653.4021 & -1653.3995 & -1653.3960 & -1653.3920 & -1653.3876 + fm energy & -1653.4068 & -1653.4055 & -1653.4034 & -1653.40026 & -1653.3966 & -1653.3922 + * crsb * + dlm energy & -3762.4738 & -3762.4781 & -3762.4807 & -3762.4821 & -3762.4822 & -3762.4814 + fm energy & -3762.4770 & -3762.4813 & -3762.4841 & -3762.4857 & -3762.4862 & -3762.4856 + + + * crs * + dlm energy & -723.4504 & -723.4468 & -723.4426 & -723.4381 & -723.4332 & -723.4280 + fm energy & -723.4491 & -723.4464 & -723.4434 & -723.4395 & -723.4351 & -723.4302 + * crse * + dlm energy & -1737.9146 & -1737.9139 & -1737.9123 & -1737.9100 & -1737.9071 & -1737.9036 + fm energy & -1737.9139 & -1737.9132 & -1737.9124 & -1737.9110 & -1737.9087 & -1737.9057 + * crte * + dlm energy & -3918.9074 & -3918.9124 & -3918.9158 & -3918.9179 & -3918.9189 & -3918.9188 + fm energy & -3918.9078 & -3918.9128 & -3918.9161 & -3918.9181 & -3918.9194 & -3918.9201 + our spin - polarized calculations for the fm reference states lead to local moments not only on the cr atoms , but also on the other atoms ( as , sb , s , se , and te ) as well as the empty spheres . sandratskii @xcite have discussed the problem associated with such induced moments in case of the heusler alloy nimnsb and the hexagonal phase of mnas . usually such systems can be divided into sublattices with robust magnetic moments and sublattices where moment is induced under the influence of the former . these authors argue that the treatment of the induced moments as independent variables in a heisenberg hamiltonian may lead to artificial features in the spin - wave spectra , but these artificial features do not drastically affect the calculated curie temperatures of the two alloys , nimnsb and hexagonal mnas . clearly , in our case the sublattice with the robust magnetic moment is the cr - sublattice . among the three other sublattices , the magnitudes of the induced moments decrease in the following order for the two robust ferromagnets cras and crsb : x - sublattice ( x = as , sb ) , sublattice es-1 ( the sublattice of empty spheres that is at the same distance with respect to the cr - sublattice as the x - sublattice ) , sublattice es-2 ( sublattice of empty spheres further away from the cr - sublattice ) . this trend is particularly valid for low values of the lattice parameter . the induced moments originate from the tails of the orbitals ( primarily @xmath8 ) on the nearby cr - atoms . this is particularly true for the moments induced on the empty spheres . the magnitudes of the induced moments on the two empty sphere sublattices decrease as the lattice parameter increases , and so do the differences in their magnitudes . the signs of the moments on es-1 and es-2 may be the same for small lattice parameters , but are opposite for large lattice parameters . the sign of the moment on the x - sublattice is opposite to that on the cr - sublattice and the magnitudes of the moments on the two sublattices increase with increasing lattice parameters , due to decreased hybridization between cr-@xmath8 and x-@xmath3 orbitals . above a critical value of the lattice parameter , the moment per formula unit ( f.u . ) saturates at a value of 3.0 @xmath2 , as the half - metallic state is achieved , while the local moments on the cr- and x - sublattices increase in magnitude , remaining opposite in sign . the maximum ratio between the induced moment on x ( x = as , sb ) and the moment on cr is 0.18 for cras and 0.15 for crsb , occurring at the highest lattice parameter of 6.62 @xmath6 studied . the maximum ratio between the induced moment on es-1 and that on cr is 0.06 , occurring at the lowest lattice parameter of 5.44 @xmath6 studied . magnetic moments of cras and crsb per formula unit ( f.u . ) as well as the local moment at the cr site are shown in fig . [ fig5 ] , where we compare the two compounds with each other for their magnetic moments in the fm and dlm states . the same results are presented in fig . [ fig6 ] , comparing the moment per f.u.in the fm state with the cr local moment in the fm and dlm states separately for each compound . it is to be noted that there are no induced moments for the dlm reference states , i.e. the moments on the non - cr sublattices are several orders of magnitude smaller than the robust moment on the cr atoms . the total moment per formula unit in the dlm state is zero by construction . the local moment on the cr atom for the dlm reference state is usually less than the corresponding fm value for smaller lattice parameters , and larger for larger lattice parameters ( fig.[fig6 ] ) . similar trends in the variation of the local moment on cr and the induced moments on the other sublattices for the fm reference states as a function of lattice parameter are revealed for crx ( x = s , se , te ) , except that the moments on es-1 are always an order of magnitude larger than those on es-2 . i addition , the induced moments on es-2 are @xmath9 times larger than those on x sublattice for smaller values of the lattice parameter , with the two becoming comparable in magnitude for larger lattice parameters . the induced moments on es-1 and x - sublattices are never larger than @xmath4 5% of the moment on the cr atoms . the induced moments for the dlm reference states are several orders of magnitude smaller than the cr - moments , and can be safely assumed to be zero . results for zb crs , crse and crte are presented in figs.[fig7 ] and [ fig8 ] . the moment per f.u . reaches the saturation values of 4@xmath2 for crs , crse , and crte in the half - metallic state , as discussed in detail by galanakis and mavropoulos@xcite . the saturation values of the moments for all these alloys ( crx , x = as , sb , s , se , and te ) satisfy the so - called `` rule of 8 '' : @xmath10 , where @xmath11 is the total number of valence electrons in the unit cell . the number 8 accounts for the fact that in the half - metallic state the bonding @xmath12 bands are full , accommodating 6 electrons and so is the low - lying band formed of the @xmath13 electrons from the @xmath3 atom , accommodating 2 electrons . the magnetic moment then comes from the remaining electrons filling the @xmath8 states , first the @xmath14 states and then the @xmath5 . the saturation value of 3@xmath2/f.u . , or the half - metallic state , appears for a larger critical lattice constant in crsb than in cras . similarly , the critical lattice constants for the saturation magnetic moment of 4@xmath2/f.u . are in increasing order for crs , crse and crte . the local moment on the cr atom can be less / more than the saturation value , depending on the moment induced on the non - cr atoms and empty spheres . -[fig8 ] ) , fm calculations produce induced moments on non - cr spheres representing the x - atoms ( x = as , sb , s , se , te ) , and one set of empty spheres . the dlm calculations produce no such induced moments , i.e. , the moments reside on the cr - atoms only . see text for discussion.,width=360 ] fig.[fig9 ] shows the variation of the magnetic moment with the lattice parameter for the random alloys cras@xmath0x@xmath0 ( x = sb , s , se , te ) , where 50% of the as - sublattice is randomly occupied by x - atoms . the saturation moment per f.u . for cras@xmath0sb@xmath0 in the half - metallic state is 3@xmath2 , with the results falling between those for cras and crsb shown in fig.[fig6 ] . for cras@xmath0x@xmath0 ( x = s , se , te ) , the saturation moment per f.u . is 3.5@xmath2 . the local cr - moment deviates from the saturation value in the half - metallic state , being higher than the saturation value for all lattice parameters above 6.1 . from figs.[fig5]-[fig9 ] it is clear that the magnetic moment per formula unit is closer to the magnetic moment of the cr atoms in the fm calculations than in the dlm calculations . local cr - moments in the dlm calculations are suppressed w.r.t . the fm results for low lattice parameters and enhanced for larger lattice parameters . as shown in table [ table1 ] the total energy of the fm state is lower than that of the corresponding dlm state in almost all cases , except for some compressed lattice parameters for crs and crse . however , the consideration of the dlm state does provide an advantage in that there are no associated induced moments , i.e. , the dlm calculations produce moments that reside on the robust magnetic sublattice only . mapping of the total energy on to a heisenberg hamiltonian , therefore , does not result in exchange interactions involving atoms / spheres with induced moments and all associated artificial / non - physical features referred to by sandratskii @xcite currently , most _ first - principles _ studies of the thermodynamic properties of itinerant magnetic systems proceed via mapping @xcite the system energy onto a classical heisenberg model : @xmath15 where @xmath16 are site indices , @xmath17 is the unit vector pointing along the direction of the local magnetic moment at site @xmath18 , and @xmath19 is the exchange interaction between the moments at sites @xmath18 and @xmath20 . the validity of this procedure is justified on the basis of the adiabatic hypothesis- the assumption that the magnetic moment directions are slow variables on all the characteristic electronic time scales relevant to the problem , and thus can be treated as classical parameters . the energy of the system for a given set of magnetic moment directions is usually calculated via methods based on density functional theory ( dft ) . one of the most widely used mapping procedures is due to liechtenstein @xcite it involves writing the change in the energy due to the deviation of a single spin from a reference state in an analytic form using the multiple scattering formalism and by appealing to the magnetic variant of the andersen force theorem@xcite . the force theorem , derived originally for the change of total energy due to a deformation in a solid , dictates that the differences in the energies of various magnetic configurations can be approximated by the differences in the band energies alone@xcite . the energy of a magnetic excitation related to the rotation of a local spin - quantization direction can be calculated from the spinor rotation of the ground state potential . no self - consistent calculation for the excited state is necessary . a second approach is based on the total energy calculations for a set of collinear magnetic structures , and extracting the exchange parameters by mapping the total energies to those coming from the heisenberg model given by eq.([e1 ] ) . such calculations can be done using any of the standard dft methods . however , unlike the magnetic force theorem method , where the exchange interactions can be calculated directly for a given structure and between any two sites , several hypothetical magnetic configurations and sometimes large supercells need to be considered to obtain the values of a modest number of exchange interactions . in addition , some aspects of environment - dependence of exchange interactions are often simply ignored . the difference between these two approaches is , in essence , the same as that between the generalized perturbation method ( gpm)@xcite and the connolly - williams method@xcite in determining the effective pair interactions in ordered and disordered alloys . a third approach is a variant of the second approach , where the energies of the system in various magnetic configurations corresponding to spin - waves of different wave - vectors are calculated by employing the generalized bloch theorem for spin - spirals@xcite . the inter - atomic exchange interactions can be calculated by equating these energies to the fourier transforms of the classical heisenberg - model energies . this approach , known as the frozen magnon approach , is similar to the frozen phonon approach for the study of lattice vibrations in solids . in this work , we have used the method of liechtenstein , which was later implemented for random magnetic systems by turek , using cpa and the tb - lmto method@xcite . the exchange integral in eq.([e1 ] ) is given by @xmath21 dz \ ; , \ ] ] where @xmath22 represents the complex energy variable , @xmath23 , and @xmath24 , representing the difference in the potential functions for the up and down spin electrons at site @xmath18. in the present work @xmath25 represents the matrix elements of the green s function of the medium for the up and down spin electrons . for sublattices with disorder , this is a configurationally averaged green s function , obtained via using the prescription of cpa . the integral in this work is performed in the complex energy plane , where the contour includes the fermi energy @xmath26 . the quantity @xmath19 given by eq . ( [ eq - jij ] ) includes direct- , indirect- , double - exchange and superexchange interactions , which are often treated separately in model calculations . the negative sign in eq.([e1 ] ) implies that positive and negative values of @xmath19 are to be interpreted as representing ferromagnetic and antiferromagnetic interactions , respectively . a problem with the mapping of the total energy to a classical heisenberg hamiltonian following the approach of liechtenstein @xcite is that it generates exchange interactions between sites , where one or both may carry induced moment(s ) . of course this problematic scenario appears only for the fm reference states , as the dlm reference states do not generate induced moments . in the present work the liechtenstein mapping procedure , applied to fm reference sates , generates exchange interactions between the cr atoms , between cr and other atoms x ( x = as , sb , s , se , te ) , and also between cr atoms and the nearest empty spheres es-1 . depending on the lattice parameter , this latter interaction is either stronger than or at least comparable to that for the cr - x pairs . the exchange interactions between cr atoms and the furthest empty spheres es-2 are always about one or two orders of magnitude smaller than the cr - es1 interactions and can be neglected . in cras , the ratio of the nearest neighbor cr - es1 to cr - cr interaction varies from 0.2 - 0.25 at low lattice parameters to 0.06 - 0.07 at high values of the lattice parameter . in crsb , these ratios are smaller , varying between 0.14 and 0.05 . the cr - es1 exchange interactions are also relatively strong in magnitude in crs , crse and crte . one important point is that while these interactions are positive for nearest neighbors for all lattice parameters , cr - cr nearest neighbor interaction is negative for low values of lattice parameters in crs and crse . in crte , this interaction changes sign from positive to negative and then back , as the lattice parameter is varied in the range 5.44 - 6.62 . as mentioned earlier , the calculation for the dlm reference states do not produce induced moments , and thus no exchange interactions other than those between the cr atoms . sandratskii@xcite have discussed the case when , in addition to the interaction between the strong moments , there is one secondary , but much weaker , interaction between the strong and one induced moment . in this case , the curie temperature , calculated under the mean - field approximation ( mfa ) , seems to be enhanced due to this secondary interaction , irrespective of the sign of the secondary interaction . in other words , the curie temperature would be somewhat higher than that calculated by considering only the interaction between the strong moments . the corresponding results under the random phase approximation ( rpa ) have to be obtained by solving two equations simultaneously . one can assume that the rpa results for the curie temperature follow the trends represented by the mfa results , being only somewhat smaller , as observed in the absence of induced moments . in our case , since there are at least two secondary interactions ( cr - x and cr - es1 ) to consider in addition to the main cr - cr interaction , the influence of these secondary interactions is definitely more complex . in view of the above - described situation involving secondary interactions between cr- and the induced moments for the fm reference states , we have adopted the following strategy . since no induced moments appear in calculations for the dlm reference states , the curie temperature @xmath1 for these can be calculated as usual from the exchange interaction between the cr - atoms , i.e. the strong moments . for these cases the calculation of @xmath1 can proceed in a straightforward manner by making use of the mean - field approximation ( mfa ) or the more accurate random - phase approximation ( rpa)@xcite . one can obtain the mfa estimate of the curie temperature from @xmath27 where the sum extends over all the neighboring shells . an improved description of finite - temperature magnetism is provided by the rpa , with @xmath28 given by @xmath29^{-1 } \ , .\ ] ] here @xmath30 denotes the order of the translational group applied and @xmath31 is the lattice fourier transform of the real - space exchange integrals @xmath32 . it can be shown that @xmath33 is always smaller than @xmath34 @xcite . it has been shown that the rpa curie temperatures are usually close to those obtained from monte - carlo simulations @xcite . as shown by sandratskii @xcite the calculation of @xmath1 using rpa is considerably more involved even for the case where only one secondary interaction needs to be considered , in addition to the principal interaction between the strong moments . the complexity of the problem increases even for mfa , if more than one secondary interaction is to be considered . the same comment applies to stability analysis using the lattice fourier transform of the exchange interactions . the deviation of the nature of the ground state from a collinear and parallel alignment of the cr moments in the fm reference states could be studied by examining the lattice fourier transform of the exchange interaction between the cr atoms : @xmath35 , if all the secondary interactions could be ignored . this is definitely not possible for many of our fm results , where several pairs of interaction need to be considered , and @xmath36 is a matrix bearing a complicated relationship to the energy as a function of the wave - vector * q*. thus , in the following the results for @xmath1 will be presented mostly for the dlm reference states . for comparison , in a small number of cases we will present @xmath1 calculated for the fm reference states using only the cr - cr exchange interactions as the input . of course , this will be done with caution only for cases where we have reason to believe that the results are at least qualitatively correct . some fm results will also be included towards the stability analysis based on @xmath36 derived from cr - cr interactions only . again , this will be done with caution , only if the corresponding results can be shown to be meaningful via additional calculations . the cr - cr exchange interactions for all the alloys studied and for both fm and dlm reference states become negligible as the inter - atomic distance reaches about three lattice parameters or , equivalently , thirty neighbor shells . the same applies to the cr - x and cr - es interactions for the fm cases , these interactions in general being somewhat smaller . the cr - cr interactions for the dlm reference states are more damped compared with the corresponding fm results , showing less fluctuations in both sign and magnitude . the distance dependence of the exchange interactions between the cr atoms in cras is shown in fig.[fig10 ] for several lattice parameters . although the nearest neighbor interaction is always positive ( i.e , of ferromagnetic nature ) , the interactions with more distant neighbors are sometimes antiferromagnetic . such antiferromagnetic interactions are more common in cras for lower lattice parameters . with increasing lattice parameter , interactions become predominantly ferromagnetic , and by the time the equilibrium lattice parameter of 5.52 @xmath6 is reached , antiferromagnetic interactions mostly disappear . we have calculated such interactions up to the 405th neighbor shell , which amounts to a distance of roughly 8 lattice parameters . although the interactions themselves are negligible around and after the 30@xmath37 neighbor shell , their influence on the lattice sums continues up to about 100 neighbor shells . by about the neighbor 110@xmath37 shell ( a distance of @xmath4 5 lattice parameters ) the interactions fall to values small enough so as not to have any significant effect on the calculated lattice fourier transform of the exchange interaction and the curie temperature ( see below ) . it is clear from fig.[fig10 ] that ferromagnetism in cras is robust and exists over a wide range of lattice parameters . the distance dependence of the cr - cr exchange interactions in crsb is very similar to that in cras for both fm and dlm reference states . for crs , crse , and crte the situation is somewhat different . for crs and crse , the fm reference states for some low lattice parameters yield cr - cr interactions that are antiferromagnetic even at the nearest neighbor separation . for crte , at the lowest lattice parameter studied ( 5.44 ) the nearest neighbor interaction for the fm reference state is ferromagnetic , but becomes antiferromagnetic with increasing lattice parameter , changing back to ferromagnetic at higher lattice parameters . for all three compounds , the interactions are predominantly ferromagnetic at higher lattice parameters . figs.[fig11 ] and [ fig12 ] show the distance dependence of the exchange interactions calculated for the fm reference states in crse and crte , respectively , for several lattice parameters . predominant nearest neighbor antiferromagnetic interactions between the cr atoms result in negative values of the curie temperature , when calculated via eqs . ( [ e2 ] ) or ( [ e3 ] ) . these results for the curie temperature for the fm reference states can be discarded as being unphysical on two grounds : because of the neglect of the interactions involving the induced moments and also because they point to the possibility that the ground state is most probably antiferromagnetic or of complex magnetic structure . the antiferromagnetic cr - cr interactions mostly disappear , when calculated for the dlm reference states . this could be interpreted as being an indication that the actual magnetic structure of the ground states for these low lattice parameters in case of crs , crse and crte is closer to a dlm state than to an fm state . in fig.[fig13 ] we show the cr - cr exchange interactions for the dlm reference states in case of crse for the same lattice parameters as those considered for fig.[fig11 ] . a comparison of the two figures shows that all interactions have moved towards becoming more ferromagnetic for the dlm reference states , the nearest neighbor interaction for the lowest lattice parameter staying marginally antiferromagnetic . between the cr atoms in cras for various lattice parameters @xmath38 , calculated for the fm and dlm reference states . the distance between the cr atoms is given in units of the lattice parameter @xmath38 ( the same applies to figs.[fig11]-[fig13 ] ) . the main plot in fig.[fig10 ] shows the distance dependence up to 2.25@xmath38 , while the inset shows the values between 2.25@xmath38 and 5@xmath38 . although the individual values of @xmath19 are small beyond about 2.25@xmath38 , their cumulative effects on the total exchange constant and the curie temerature can not be neglected ( see text for details ) . comparison of the insets for the fm and dlm cases shows that the interactions are more damped for the dlm case , being at least an order of magnitude smaller for distances beyond @xmath42.25 - 2.5@xmath38 or 15 - 20 neighbor shells . similar comments apply to the interactions presented in figs.[fig11]-[fig13].,width=379 ] . , width=336 ] . , width=336 ] . , width=336 ] the deviation of the nature of the ground state from the reference state can be studied by examining the lattice fourier transform of the corresponding exchange interactions between the cr atoms : @xmath35 . as pointed out earlier , for the fm reference states this procedure suffers from the drawback of neglecting the effects of all other interactions involving the induced moments . for the dlm reference states there are no induced moments , so the relationship between the energy and @xmath36 is simpler , but a physical picture of the spin arrangement corresponding to a particular wave - vector @xmath39 is harder to visualize . for the fm reference states , if there were no moments other than those on the cr atoms , a maximum in @xmath36 at @xmath40 would imply that the ground state is ferromagnetic with collinear and parallel cr magnetic moments in all the unit cells . a maximum at symmetry points other than the @xmath41-point would imply the ground state being antiferromagnetic or a spin - spiral state . a maximum at a wave - vector @xmath39 that is not a symmetry point of the bz would imply the ground state being an incommensurate spin spiral . the presence of induced moments and the consequent interactions involving non - cr atoms and empty spheres spoil such interpretations based on @xmath36 derived from cr - cr interactions alone . however , the tendencies they reveal might still be useful . it is for this reason that we study the fourier transform @xmath36 , defined above , for both fm and dlm reference states . in fig.[fig14 ] we have plotted this quantity for cras . the results for crsb are quite similar . the maximum in @xmath36 at the @xmath41-point for all lattice parameters and for both fm and dlm reference states can be taken as an indication that the ground state magnetic structure is ferromagnetic for cras for all the lattice parameters studied . the same comment applies to crsb . the apparent lack of smoothness in @xmath36 shown for the fm reference states is a consequence of the fact that there are other additional bands ( involving induced moments ) , which are supposed to cross the band shown , but have not been computed . for crs , crse , and crte ( see figs.[fig15]-[fig16 ] ) , the deviation of the ground state for low lattice parameters from the parallel arrangement of cr moments is reflected in the result that the maximum moves away from the @xmath41-point for the fm reference states . at high values of the lattice parameter the maximum returns to the @xmath41-point . the curves for crse are similar to those for crs and have therefore not been shown . the fact that the maximum for the dlm reference sates lies at the @xmath41-point in most cases is again an indication that the ground state magnetic structure is closer to the dlm state than to the fm state . the conclusions based on the fm reference state results in figs.[fig15 ] and[fig16 ] may be suspect on ground of neglecting the interactions involving the induced moments . however , to explore whether they do carry any relevant information we have carried out additional calculations for the three compounds crs , crse and crte for two commonly occurring antiferromagnetic configurations : afm[001 ] , afm[111 ] . note that another commonly occurring afm configuration afm[110 ] is not unique , i.e. there are several configurations that could be seen as an afm[110 ] arrangement ( see fig3 . of ref . [ ] , table 2 of ref . the simplest among these is actually equivalent to afm[100 ] . the results for the total energy for the two afm calculations are shown in table [ table2 ] and compared with the corresponding fm and dlm total energies . for crs , the lowest energy state for lattice parameters 5.44 and 5.55 @xmath6 is afm[111 ] , exactly as suggested by the maximum in @xmath36 appearing at the l - point in fig.[fig15 ] for the fm reference state and for these two lattice parameters . as the lattice parameter increases beyond 5.55 , antiferromagnetic interactions diminish . for the next higher lattice parameter 5.66 @xmath6 in table [ table2 ] , the lowest energy state is dlm . this may suggest that the ground state has a complex magnetic structure , which remains to be explored . for higher lattice parameters the fm state has the lowest energy . for crse , afm[111 ] state has the lowest energy up to the lattice parameter 5.66 , as is also supported by the maximum of @xmath36 at l - point . the @xmath36 curves for crse are similar to those of crs , and have not been shown . for crte , at the lowest lattice parameter of 5.44 @xmath6 the lowest energy state is fm , as is also indicated by the maximum of @xmath36 at the @xmath41-point . for higher lattice parameters 5.65 and 5.76 , even though the @xmath36 curves point to the possibility of an afm[111 ] ground state , the fm state energy turns out to be the lowest among the configurations studied . it could be concluded that in this case a proper relationship between the energy and @xmath36 , obtained without the neglect of the induced moments , would point to the ground state being fm . for these three chalcogenides , for lattice parameters above 5.65 - 5.7 @xmath6the ground state should be fm . [ cols="<,^,^,^,^,^,^ " , ] we determine the curie temperature using eqs . ( [ e2 ] ) and ( [ e3 ] ) . for the dlm reference states , these produce estimates of @xmath1 from above the ferromagnetic@xmath42paramagnetic transition , and are free from errors due to induced moments . however , these estimates are high compared with properly derived values of @xmath1 from below the transition . the latter estimates would require the use of fm reference states ( where the ground states are known to be fm ) and thus a proper treatment of the induced moments . for cras and crsb , the magnetic state is ferromagnetic for all the lattice parameters considered . hence , for the sake of comparison we have calculated the @xmath1 for the fm reference states using eqs . ( [ e2 ] ) and ( [ e3 ] ) as well . according to the results of sandratskii @xcite the correctly calculated @xmath1 values , in the presence of interactions involving all the induced moments , would be higher . thus , the correct estimates of @xmath1 should lie somewhere between the dlm results and the fm results obtained with the neglect of the induced moments . in fig . [ fig17 ] we show these results for cras , crsb . we have used up to 111 shells in the evaluation of eq . ( [ e2 ] ) and for the lattice fourier transform of @xmath43 in eq . ( [ e3 ] ) , after having tested the convergence with respect to the number of shells included . the estimated computational error corresponding to the chosen number of shells used in these calculations is below @xmath44 . for comparison we also include the results for the mixed alloy cras@xmath0sb@xmath0 , for which the calculated @xmath1 values fall , as expected , in between those of cras and crsb . since rpa values are more accurate than mfa values , our best estimates of @xmath1 for cras range from somewhat higher than 500 k at low values of the lattice parameter , increasing to 1000 - 1100 k around the mid lattice parameter range ( 5.75 - 5.9 ) and then decreasing to around 600 k for higher lattice parameters ( 6.5 @xmath6 and above ) . for crsb these estimates are consistently higher than those for cras : 1100 k , 1500 k and 1200 k , respectively . the estimates for cras are similar to those provided by sasaiolglu @xcite for crs , crse , the results obtained with the fm reference states would clearly be wrong , in particular , for the low values of the lattice parameters , for which we have shown the ground state to be antiferromagnetic within our limited search . there is a possibility that the ground state for certain lattice parameters might have a complex magnetic structure . for crte , even though the ground state appears to be ferromagnetic , there are considerable antiferromagnetic spin fluctuations , making the fm estimates unreliable . in fig . [ fig18 ] we show the @xmath1 values for crs , crse and crte for the dlm reference states . the values for lattice parameters for which the ground state has been shown to be antiferromagnetic in the preceding section should be discarded as being inapplicable . similar results for the alloys cras@xmath0x@xmath0 ( x = s , se and te ) are shown in fig . [ fig19 ] for dlm reference states . for these , the ground state is ferromagnetic for all lattice parameters . however , because of the neglect of the induced moments related effects , our results for the curie temperatures for the fm reference states are lower than the properly calculated values . thus in fig.[fig19 ] we show the dlm results only , which are devoid of the induced moment effects and provide us with estimates of @xmath1 from above the transition . these are expected to be somewhat higher than the properly computed values for fm reference states . thus , for these alloys the trend revealed in fig . [ fig19 ] for the variation of @xmath1 with lattice parameter is correct . the estimates themselves are qualitatively correct , albeit somewhat higher than the correct values . only the rpa values are plotted in fig . [ fig19 ] , which are more reliable than the mfa values . for comparison , we also show the results for the pnictides cras , crsb , and cras@xmath0sb@xmath0 , which are isoelectronic among themselves , but have half an electron per unit cell less than the mixed alloys cras@xmath0x@xmath0 ( x = s , se and te ) sb@xmath0 are also shown . , width=360 ] x@xmath0 alloys with x = s , se and te . for comparison , the results for cras , crsb and cras@xmath0sb@xmath0 are also shown . all results shown are for dlm reference states , and as such , should be considered as upper limits for @xmath1 . , width=336 ] the differences between the results for the pnictides , chalcogenides and the mixed pnictide - chalcogenides can be summarized as follows . the pnictides , cras , crsb , and cras@xmath0sb@xmath0 , are strong ferromagnets at all the lattice parameters studied ( 5.44 - 6.62 ) . in the dlm description , their @xmath1 stays more or less constant ( apart from a minor increase ) as the lattice parameter increases from 5.4 /aa to 6.1 , and then decreases beyond ( figs . [ fig17 ] and [ fig19 ] . the chalcogenides are antiferromagnetic or have complex magnetic structure for low lattice parameters . in the dlm description , their @xmath1 in the ferromagnetic state increases and then becomes more or less constant as the lattice parameter increases ( fig . [ fig18 ] ) . the mixed alloys cras@xmath0x@xmath0 ( x = s , se , te ) are ferromagnetic at all the lattice parameters studied . in the dlm description , their @xmath1 rises and then falls as the lattice parameter is increased from 5.44 to 6.62 . a comparison of the results presented in figs . [ fig17]-[fig19 ] shows that large changes in @xmath1 take place by changing the number of carriers . changes due to isoelectronic doping are small compared with changes brought about by changing carrier concentration . our _ ab initio _ studies of the electronic structure , magnetic moments , exchange interactions and curie temperatures in zb crx ( x = as , sb , s , se and te ) and cras@xmath0x@xmath0 ( x = sb , s , se and te ) reveal that half - metallicity in these alloys is maintained over a wide range of lattice parameters . the results for the exchange interaction and the curie temperature show that these alloys have relatively high curie temperatures , i.e. room temperature and above . the exceptions occur for the alloys involving s , se and te at some low values of lattice parameters , where significant inter - atomic antiferromagnetic exchange interactions indicate ground states to be either antiferromagnetic or of complex magnetic nature . a comparison of total energies for the fm , dlm , and two zb antiferromagnetic configurations ( afm[001 ] and afm[111 ] ) show the lowest energy configuration to be afm[111 ] for crs and crse for compressed lattice parameters ( table[table2 ] ) . the possibility of afm ground states for compressed lattice parameters for crs was noted by zhao and zunger@xcite and for crse by sasioglu @xcite . our search for the antiferromagnetic ground states is more thorough than what was reported in these two studies . an extensive study of several antiferromagnetic configurations as well as ferrimagnetic and more complex magnetic structures for crs , crse and crte is currently underway . the mixed pnictide - chalcogenide alloys cras@xmath0x@xmath0 ( x= s , se , te ) do not show any tendency to antiferromagnetic spin fluctuations for the entire range of the lattice parameter studied . presumably the pnictogens suppress antiferromagnetic tendencies . such alloys may play an important role in fabricating stable zb half - metallic materials , as the concentration of the pnictogens and the chalcogens may be varied to achieve lattice - matching with a given substrate . as long as the concentration of as or sb is higher than the chalcogen concentration , half - metallic ferromagnetic state can be achieved . there is a large variation in the curie temperature of these alloys ( fig . [ fig19 ] ) as the lattice parameter varies from the low ( @xmath4 5.4 ) to the mid ( @xmath4 6.1 ) range of the lattice parameters studied . this variation is much smaller for the isoelectronic alloys cras , crsb and cras@xmath0sb@xmath0 ( fig . [ fig17 ] ) over this range of lattice parameters . note that most ii - vi and iii - v zb semiconductors have lattice parameters in this range . large changes in @xmath1 can be brought about by changing the carrier concentrations . the pnictides in general have a higher @xmath1 than the chalcogenides . our results for the curie temperature , the lattice fourier transform of the exchange interactions , and the resulting stability analysis are based on the exchange interactions between the cr atoms only . for the fm reference states this causes some errors due to the neglect of the effects of the induced moments . the dlm results are free from such errors . it is expected that the present study will provide both qualitative and quantitative guidance to experimentalists in the field . 99 k. sato and h. katayama - yoshida , semicond . * 17 * , 367 ( 2002 ) . see k. sato , t. fukushima and h. katayama - yoshida , j. phys . : matter * 19 * , 365212 ( 2007 ) , and references therein . see b. belhadji , l. bergqvist , r. zeller , p.h . dederichs , k. sato and h. katayama - yoshida , j. phys . : condens . matter * 19 * , 436227 ( 2007 ) , and references therein . h. saito , v. zayets , s. yamagata , and k. ando , phys . lett * 90 * , 207202 - 1 ( 2003 ) . k. sato and h. katayama - yoshida , jpn . . phys . * 40 * , l651 ( 2001 ) . h. akinaga , t. manago , and m. shirai , jpn . j. appl . phys.*39 * , l1118 ( 2000 ) . s. li , j - g duh , f. bao , k - x liu , c - l kuo , x. wu , liya l , z. huang , and y du , j. phys . phys . * 41 * 175004 ( 2008 ) . m. shirai , j. appl . phys . * 93 * , 6844 ( 2003 ) . h. akinaga , m. mizuguchi , k. nagao , y. miura , and m. shirai in _ springer lecture notes in physics _ * 676 * , 293 - 311 ( springer - verlag , berlin 2005 ) . k. yamana , m. geshi , h. tsukamoto , i. uchida , m. shirai , k. kusakabe , and n. suzuki , j. phys . : condens . matter * 16 * , s5815 ( 2004 ) . l. kahal , a. zaoul , m. ferhat , j. appl . phys . * 101 * , 093912 ( 2007 ) . i. galanakis and p. mavropoulos , , 104417 ( 2003 ) ; 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we present calculations of the exchange interactions and curie temperatures in cr - based pnictides and chalcogenides of the form crx with x = as , sb , s , se and te , and the mixed alloys cras@xmath0x@xmath0 with x = sb , s , se , and te . the calculations are performed for zinc blende ( zb ) structure for 12 values of the lattice parameter between 5.44 and 6.62 , appropriate for some typical ii - vi and iii - v semiconducting substrates . electronic structure is calculated via the linear muffin - tin - orbitals ( lmto ) method in the atomic sphere approximation ( asa ) , using empty spheres to optimize asa - related errors . whenever necessary , the results have been verified using the full - potential version of the method , fp - lmto . the disorder effect in the as - sublattice for cras@xmath0x@xmath0 ( x = sb , s , se , te ) alloys is taken into account via the coherent potential approximation ( cpa ) . exchange interactions are calculated using the linear response method for the ferromagnetic ( fm ) reference states of the alloys , as well as the disordered local moments ( dlm ) states . these results are then used to estimate the curie temperature from the low and high temperature side of the ferromagnetic / paramagnetic transition . estimates of the curie temperature are provided , based on the mean field and the more accurate random phase approximations . dominant antiferromagnetic exchange interactions for some low values of the lattice parameter for the fm reference states in crs , crse and crte prompted us to look for antiferromagnetic ( afm ) configurations for these systems with energies lower than the corresponding fm and dlm values . results for a limited number of such afm calculations are discussed , identifying the afm[111 ] state as a likely candidate for the ground state for these cases .

arxiv

x - ray reflection off the surface of cold disks in active galactic nuclei ( agn ) and galactic black holes ( gbhs ) has been an active field of research since the work of @xcite . in early studies , the illuminated material was assumed to be cold and non - ionized @xcite . it was soon realized , however , that photoionization of the disk can have a great impact on both the reflected continuum and the iron fluorescence lines . detailed calculations were then carried out by @xcite and @xcite . however , in all of these papers , the density of the illuminated material was assumed to be constant along the vertical direction . this assumption applies only to the simplest version of radiation - dominated shakura - sunyaev disks @xcite , and only for the portion where viscous dissipation is the dominating heating process . for the surface layers , however , photoionization and compton scattering are the major heating sources . therefore the approximation of constant density is not appropriate . moreover , thermal instability allows the coexistence of gas at different phases . these different phases have very different temperatures , and hence different densities to keep the gas in pressure balance . recently @xcite relaxed the simplifying assumption of constant gas density . they determined the gas density from hydrostatic balance solved simultaneously with ionization balance and radiative transfer . they made an important observation that the thomson depth of the hot coronal layer can have great influence on the x - ray reprocessing produced by the deeper , and much cooler disk . in order to simplify the calculation of the vertical structure , though , they ignored thermal conduction and the effects of transition layers between the different stable phases . a discontinuous change in temperature was allowed whenever an unstable phase was encountered . they argued that such transition layers are of little importance because their thomson depths are negligibly small . however , without taking into account the role of thermal conduction , their method of connecting two different stable layers is rather _ ad hoc_. moreover , even though the thomson depths of these transition layers are small , it does not guarantee that the x - ray emission and reflection from such layers are negligible . because the temperature regime where the transition layers exist is not encountered in the stable phases , some of the most important lines can have appreciable emissivity only in these layers . also , since resonance line scattering has much larger cross section than thomson scattering , the optical depths in resonance lines can be significant . including thermal conduction in the self - consistent solution of the vertical structure presents a serious numerical challenge . the difficulties are due to the coupling between hydrostatic balance , radiative transfer and heat conduction . @xcite first studied the phase equilibrium of a gas heated by cosmic rays and cooled by radiation . they found that taking into account heat conduction in the boundary layer allows one to obtain a unique solution of the stable equilibrium . @xcite calculated the full temperature profile for a compton - heated corona , and @xcite calculated the static conditions of the plasma for different forms of heating and cooling . but they did not include much discussion of the spectroscopic signatures resulting from the derived vertical structure . in this paper , we first calculate the temperature structure in the layers above the accretion disk , then calculate the emission lines via radiative recombination ( rr ) and reflection due to resonance line scattering from the derived layers . certain illuminating continua spectra allow more than two stable phases to coexist , with two transition layers connected by an intermediate stable layer . for the transition layer , since the thomson depth is small , the ionizing continuum can be treated as constant ; and since its geometric thickness is smaller than the pressure scale height , the pressure can be treated as constant as well . we can thus obtain semi - analytic solution of the temperature profile by taking into account thermal conduction . for the intermediate stable layer , its thickness is determined by the condition of hydrostatic equilibrium . in our model , the normally incident continuum has a power - law spectrum with an energy index of @xmath0 . we also assume a plane - parallel geometry and that the resonance line scattering is isotropic . the structure of this paper is as follows : in [ sec_structure ] we discuss the existence of the thermal instability and compute the thermal ionization structure of the transition layers ; in [ sec_spectrum ] we calculate the recombination emission lines and the reflection due to resonance line scattering ; in [ sec_summary ] we summarize the important points of the calculations , the validity of various approximations made in the calculations , and the detectability of the recombination emission and reflection lines . the vertical structure of an x - ray illuminated disk at rest is governed by the equations of hydrostatic equilibrium and of energy conservation @xmath1 in the first equation , @xmath2 is the force density due to gravity and radiation pressure . the dependence of the force on the plasma density is included explicitly through the hydrogen density @xmath3 . in the second equation , a time independent state is assumed , @xmath4 is the thermal conductivity , and @xmath5 is the net heating rate depending on the gas state and the incident flux @xmath6 ( differential in energy ) . we neglect the effects of magnetic field and adopt the spitzer conductivity appropriate for a fully ionized plasma , @xmath7 erg @xmath8 s@xmath9 k@xmath9 @xcite . we have used the classical heat flux , @xmath10 , in equation ( [ eq_transition ] ) because the electron mean free path is short compared to the temperature height scale . since the continuum flux may change along the vertical height , in principle , the above two equations must be supplemented by an equation for radiative transfer . a self - consistent solution of such equations is difficult to obtain . in the following , we invoke a few physically motivated approximations , which make the problem tractable . first , in thermally stable regions , the gas temperature is slowly varying , the heat conduction term in the energy balance equation can be neglected . therefore , the temperature can be determined locally with the condition @xmath11 . it is well known @xcite that the dependence of @xmath12 on the gas pressure @xmath13 and the illuminating continuum @xmath6 can be expressed in the form of @xmath14 , where @xmath15 is the electron density , @xmath16 is the net cooling rate per unit volume , and @xmath17 is an ionization parameter defined by @xmath18 where @xmath19 is the total flux of the continuum , and @xmath20 is the speed of light . in figure [ fig_scurve ] , we show the local energy equilibrium curve @xmath21 versus @xmath17 , at @xmath22 calculated with the photoionization code xstar @xcite . these curves are commonly referred to as `` s curves '' due to their appearance . the illuminating continuum is assumed to be a power law with energy index @xmath0 . the solid line labeled with `` s - curve 1 '' corresponds to a low energy cutoff at 1 ev and a high energy cutoff at 150 kev , while the dashed line `` s - curve 2 '' corresponds to a high energy cutoff at 200 kev . the choice of such incident spectra is based on their common appearance in many agns and gbhcs . the region is thermally `` unstable '' where the `` s - curve '' has a negative slope , and `` stable '' where the slope is positive , as indicated in figure [ fig_scurve ] . in the thermally unstable regions , we have @xmath23 where the derivative is taken while the energy balance is satisfied , i.e. , @xmath24 . this condition was shown @xcite to be equivalent to the instability condition discovered by @xcite @xmath25 the thermal instability allows the gas to coexist at different phases . the gas temperature may change by orders of magnitude over a geometric thin region whenever an unstable phase separates two stable ones . this results in enormous temperature gradients and heat conduction . therefore the heat conduction in the energy balance equation should be included in such transition layers between stable phases . on the other hand , the thicknesses of these transition layers are usually smaller than the pressure scale height , so one can safely treat the gas pressure as constant in these regions . moreover , the continuum radiative transfer can be neglected because the compton optical depth is found to be small . the vertical structure of the transition regions is then solely determined by the energy balance equation with heat conduction . the thomson optical depth @xmath26 of such regions is readily estimated by @xmath27 where @xmath28 is the thomson scattering cross section . the transition layer solution is not arbitrary under the steady - state conditions , i.e. , where there is no mass exchange between the two stable phases which the transition layer connects . a similar problem in the context of interstellar gas heated by cosmic rays was well studied by @xcite . we follow their procedure here and define @xmath29 , in order to rewrite equation ( [ eq_transition ] ) in the form : @xmath30 a steady - state requires vanishing heat flux at both boundaries of the transition layer , or @xmath31 where @xmath32 and @xmath33 are the temperatures of the two stable phases which are connected by a transition layer . this condition determines a unique ionization parameter @xmath17 for the transition layer in question , and the integration of equation ( [ eq_dy2 ] ) along the vertical height gives the detailed temperature profile as a function of optical depth . if the disk does not realize the steady - state solution , there are additional enthalpy terms in equation ( [ eq_transition ] ) which require that there be mass flow through the transition region i.e. the cool material in the disk evaporates , or the hot material in the corona condenses @xcite . physically , this corresponds to a movement of the transition layer up or down through the vertical disk structure . however , since the density increases monotonically toward the center of the disk , this `` motion '' stops where the transition layer reaches the steady state value of @xmath17 . thus , in the absence of disk winds or continuous condensation from a disk corona , the steady state solution should generally be obtained . there is a complication for the `` s - curves '' shown in figure [ fig_scurve ] , because in each curve there exist two unstable regions and therefore there should be two transition layers . for `` s curve 1 '' , condition [ eq_y2 ] can be met for both transition layers with resulting ionization parameters @xmath34 , where @xmath35 and @xmath36 are associated with the transition layer which connects to the lowest temperature phase and highest temperature phase , respectively . for `` s curve 2 '' , the resulting ionization parameters for two transition layers , however , satisfy @xmath37 . such a situation is unphysical , where the ionization parameter of the upper transition layer is smaller than that of the lower one , because in the context of accretion disks , the upper layer receives more ionizing flux and has lower pressure . so in practice , for `` s curve 2 '' the intermediate stable region is skipped and a transition layer connects the lowest temperature phase to the highest temperature phase directly . the ionization parameter of this transition layer is determined the same way by applying equation ( [ eq_y2 ] ) . there is then an intermediate stable layer , of nearly uniform temperature , in between these two transition layers for `` s curve 1 '' , as indicated with bc in figure [ fig_scurve ] . the thickness of the intermediate layer should in principle be obtained by solving the coupled equations of hydrostatic equilibrium , energy balance , and radiative transfer . however , unlike the stable phase at the disk base or that of the corona , which may be compton thick , this intermediate stable layer is generally optically thin , because its optical depth is restricted by the difference between @xmath35 and @xmath36 . furthermore , the temperature in this layer is slowly varying , and therefore heat conduction can be neglected . we shall make another approximation that the variation of the force density @xmath2 in the hydrostatic equation can also be neglected . this may not be a good approximation . however , since our main purpose is to investigate qualitatively the effects of an intermediate stable layer , such a simplifying procedure does capture the proper scaling relations of the problem , and has the advantage of less specific model dependence . writing equation ( [ eq_hydro ] ) in a dimensionless form and parameterizing the force factor by a dimensionless parameter @xmath38 , we obtain @xmath39 the parameter a defined here is identical to the gravity parameter in @xcite in the absence of radiation pressure . the integration of this equation from @xmath35 to @xmath36 gives the thomson depth of the intermediate stable layer . assuming radiation pressure can be neglected , the force factor at a given radius @xmath40 of the disk can be estimated as @xmath41 where @xmath42 is the luminosity of the continuum source in units of the eddington limit , @xmath43 is the half thickness of the disk at radius @xmath40 , @xmath44 is the mass of the central source , @xmath45 is the proton mass , and @xmath46 is the eddington luminosity . in this estimate , we have assumed that @xmath47 . this gives @xmath48 the value : @xmath49 for a typical thin disk , as those present in agn and black hole binaries , one expects @xmath50 . if the luminosity is sub eddington , @xmath51 as in most agns , @xmath48 is of order unity . however , since the disk surface may not be normal to the continuum radiation , @xmath19 may be only a fraction of the value assumed above , which increases @xmath48 by one or two orders of magnitude . on the other hand , as the source approaches the eddington limit , @xmath48 may become smaller than 1 . therefore , we expect @xmath48 to be in the range of 0.1 10 . the exceptional cases of much smaller and larger @xmath48 are discussed in [ sec_summary ] . the temperature profiles and optical depths of the transition layers and possible intermediate stable layer are shown in figures [ fig_transition ] . three labeled curves in solid lines correspond to `` s - curve 1 '' ( in figure [ fig_scurve ] ) with different values of the force factors @xmath48 = 10 , 1 , and 0.1 , respectively . the dashed curve corresponds to `` s - curve 2 '' , where there is no intermediate stable layer . for each of the solid curves , three layers are clearly seen , with two transition layers being connected by an intermediate stable layer , as illustrated for the case of @xmath52 . the smaller @xmath48 produces a more extended intermediate stable layer as expected . because the stable phase with the lowest temperature is almost neutral , and the stable phase at the highest temperature is almost fully ionized , they are not efficient in generating x - ray line emission , except for iron fluorescence lines from the neutral material . only the transition layers and the intermediate stable layer are expected to emit discrete lines in the soft x - ray band . in a photoionized plasma , the temperature is too low for collisional excitation to be an important line formation process . instead , radiative recombination ( rr ) followed by cascades dominates the line emission . the flux of a particular line can be written as @xmath53 where @xmath54 is the density of the ion before recombination , @xmath55 is the line energy , and @xmath56 is the line emissivity defined as in @xcite . in ionization equilibrium where the ionization rate is equal to the recombination rate , we have @xmath57 , where @xmath58 is the recombination coefficient of ion @xmath59 , @xmath60 is the number density of ion @xmath61 , @xmath6 is again the monochromatic incident flux ( differential in energy ) , and @xmath62 is the photoionization cross section of ion @xmath61 . defining the branching ratio @xmath63 , equation ( [ flux_emission ] ) can be rewritten as : @xmath64 where @xmath65 is the fractional abundance of ion @xmath61 with respect to the electron density . as indicated , @xmath66 and @xmath67 depend only on temperature @xmath21 and ionization parameter @xmath17 , which are both functions of optical depth @xmath68 . for convenience , we further define the `` emission equivalent width '' @xmath69 . then if @xmath70 , which depends only on the shape of the incident continuum , @xmath71 can be written as : @xmath72 therefore the emission equivalent width @xmath71 is independent of the density of the medium and the incident flux . it is a unique function of the structure deduced in section [ sec_structure ] . all other variables depend only on @xmath21 and @xmath17 . the numerical values of @xmath73 were provided by d. liedahl ( private communication ) , and were calculated using the models described in @xcite . the values of @xmath74 were computed using xstar @xcite . in figure [ fig_emission ] , we plot the spectra of the recombination emission within the 0.5 1.5 kev band with a spectral resolution @xmath75 ev , which is close to the spectral resolution of the grating spectrometers on _ chandra _ and _ xmm - newton_. the top panel corresponds to the case without an intermediate stable layer , and the bottom panels corresponds to the case with an intermediate layer for @xmath76 and 0.1 , respectively . for clarity , from top to bottom , the flux in each panel is multiplied by a factor indicated in each panel . it appears that the existence of an intermediate stable layer enhances the emission in this energy band . this is not surprising since the ions that are responsible for these lines have peak abundances at temperatures close to that of the intermediate stable layer . in all cases , the equivalent widths ( ews ) of the emission lines are less than 1.0 ev , with respect to the ionizing continuum . the strongest lines are the hydrogen - like and helium - like lines of oxygen , with ews approaching several tenths of an ev . hydrogen - like and helium - like lines from iron outside the plotted energy band are somewhat stronger , with ews reaching a few ev . we note that our low equivalent width values conflicts with those derived by @xcite , who found some lines with equivalent widths as high as 30 ev . however , since they did not consider the appropriate locations for the transition regions in @xmath17-space , their intermediate stable layer subtended a much larger optical depth than we find here . naturally with a thicker layer , they found larger equivalent widths . emission from rr is not the only line formation process in the transition layers and the intermediate stable layer . due to very large cross sections in resonance line scattering , the reflected flux in these lines may be significant . with the computed thermal and ionization structure , the column density in each ion and the optical depth in all resonance lines can be calculated straightforwardly . the cross sections for resonance line scattering depend on the line broadening . we assume thermal doppler effects as the only mechanism . although the gas temperature is a function of depth , we calculate the line width for a temperature where the abundance of each ion peaks as an average , and assume that the resonance scattering cross sections are uniform along the vertical direction . in terms of absorption oscillator strength @xmath77 , this cross section of the resonance line scattering @xmath78 can be written as @xmath79 where @xmath80 is the electron mass , @xmath81 is the electron charge , @xmath82 is the wavelength of the line and @xmath83 is the average line width in wavelength . under the assumption of thermal doppler broadening , @xmath84 where @xmath85 is the temperature at which the ion abundance peaks , and @xmath86 is the ion mass . the resonance scattering optical depth @xmath87 for a line from ion @xmath59 can be estimated as @xmath88 where @xmath89 is the column density of the ion . the radiative transfer in the line is a complicated issue @xcite . a full treatment is beyond the scope of this work . however , since we are only interested in a reasonable estimate of the reflected line flux , a simple approach may be adopted . we assume the resonance line scattering is isotropic and conservative and neglect the polarization dependence . under such conditions , the reflection and transmission contributions by a plane - parallel slab of finite optical depth @xmath90 have been solved by @xcite . for normal incident flux @xmath19 , the angle dependent reflectivity is @xmath91 where @xmath92 , @xmath93 is the reflectivity at @xmath94 , @xmath95 is the reflected intensity , @xmath19 is the incident flux , and @xmath96 is the scattering function defined as @xmath97 where @xmath98 and @xmath99 are two functions that satisfy the following integral equations : @xmath100d\mu^{\prime } \nonumber \\ y(\mu)&= & e^{-\tau /\mu}+\frac{\mu}{2}\int_{0}^{1}\frac{1}{\mu-\mu^{\prime}}[y(\mu)x(\mu^{\prime})-x(\mu)y(\mu^{\prime})]d\mu^{\prime}.\end{aligned}\ ] ] the solutions of these equations may be obtained by an iterative method with the starting point @xmath101 and @xmath102 . the angle integrated reflectivity @xmath103 can be calculated as @xmath104 and is shown in figure [ fig_totref ] as a function of the resonance scattering optical depth @xmath87 . the reflected flux in a line can be written as @xmath105 where @xmath106 is the line width in energy . similarly to the `` emission equivalent width '' @xmath71 , we define a `` reflection equivalent width '' @xmath107 , which results in : @xmath108 this `` reflection equivalent width '' from our numerical results is a few tenths of an ev for strong resonance lines , similar to that of the recombination emission lines . in figure [ fig_reflection ] , we plot the spectra of the resonantly scattered lines in the energy band 0.5 1.5 kev with a spectral resolution @xmath75 ev . the top panel corresponds to the case without an intermediate stable layer , and the bottom panels corresponds to the case with an intermediate layer for @xmath76 and 0.1 , respectively . for clarity , from top to bottom , the flux in each panel is multiplied by a factor of 100 . we see that the equivalent widths of the reflected lines are notably enhanced when there is an intermediate stable layer , but not as significantly as for the recombination emission lines . this is because the optical depths of many strong lines become much larger than unity , and the reflection is saturated . if there are broadening mechanisms other than thermal doppler effects , such as turbulent velocity , the reflected intensity can be further enhanced . in order to gain a crude idea of the relative importance of recombination emission and reflection from the transition layers and intermediate stable layer , we compare them to the `` hump '' produced by compton scattering off a cold surface @xcite . we use the greens function obtained by @xcite to calculate the compton reflection . this method was verified to be accurate with a monte carlo procedure by @xcite . in figure [ fig_spectrum ] , we show the combined spectra including recombination emission and reflection lines from resonance line scattering , and the compton reflection `` hump '' . we now summarize the most important conclusions that can be drawn from the calculations presented in this paper . we also discuss the detectability of the predicted line features . 1 . the unique ionization parameters that characterize the steady - state solutions of the transition layers depend on the shape of the `` s - curve '' . we have shown that two power - law illuminating spectra with different high energy cutoffs produce very different temperature profiles . the harder spectrum only allows one transition layer even though there are two unstable branches in the `` s - curve '' , while the softer one allows two separate transition layers connected by an intermediate stable layer . this is due to the fact that the ionization parameter of the upper transition layer must be larger than that of the lower one , if they are to exist separately in a disk environment . the harder spectrum produces a turnover point of the upper branch of the `` s - curve '' at smaller @xmath17 . therefore the transition layer due to the upper unstable region joins the lower one smoothly without allowing the intermediate stable region to form . the turnover of the upper `` s - curve '' represents the point where compton heating starts to overwhelm bremsstrahlung . the ionization parameter at which this point occurs is related to the compton temperature of the continuum , @xmath109 @xcite . a harder spectrum has larger @xmath110 , therefore the intermediate stable layer tends to disappear for hard incident spectra . although the thomson depths of the transition layers and possible intermediate stable layer are generally negligible , the x - ray emission lines from them may comprise the main observable line features , because the temperatures of these layers are inaccessible to the stable phases , and thus some of the important lines can have appreciable emissivity only in these layers . due to the much larger cross sections for resonance line scattering , reflection due to resonance lines off such transition layers is also important . the strengths of reflected lines are at least comparable with those of the recombination emission lines when there is no intermediate stable layer . because the appearance of the reflected line spectrum is different from that of the recombination emission spectrum , high resolution spectroscopic observations should be able to distinguish these mechanisms . 3 . the justification of the assumption that the ionizing continuum does not scatter in the intermediate layer depends on the magnitude of the parameter @xmath48 . the thomson depth of this layer @xmath68 is given by : @xmath111 for the power - law continuum with high energy cutoff at 150 kev ( `` s curve 1 '' ) , @xmath112 and @xmath113 from our numerical results . therefore , @xmath114 . @xmath68 is much less than unity as long as @xmath48 is greater than 0.1 . for smaller @xmath48 , however , another effect comes into play . @xcite showed that the thomson depth of the coronal layer ( the stable phase with highest temperature ) exceeds unity when the gravity parameter ( identical to @xmath48 defined here when the radiation pressure is neglected ) is @xmath115 0.01 . therefore not much ionizing flux can penetrate this layer , and the reprocessing in the deeper and cooler layers can be neglected completely . as @xmath48 becomes much larger than 10 , the thickness of the intermediate stable layer is negligible compared to the transition layers . therefore , its presence may be ignored . since the recombination rate must equal the photoionization rate in the irradiated gas , recombination radiation is also a form of reflection i.e. the line equivalent widths are independent of the incident flux . they depend only on the structure ( @xmath116 ) deduced from the hydrostatic and energy balance equations . the detectability of these recombination emission and reflection lines depends on whether the primary continuum is viewed directly . when the ionizing continuum is in direct view , our results show that the ews of the strongest lines in the 0.5 1.5 kev band are at most a few tenths of an ev , slightly larger when there is an intermediate stable layer . + the signal to noise ratio ( snr ) in such a line can be written as @xmath117 where @xmath118 is the integration time of the observation , @xmath55 is the energy of the line , @xmath119 is the photon flux in the continuum , @xmath120 and @xmath121 are the effective area and resolving power of the instrument , respectively . for a line at @xmath115 1 kev , with @xmath122 , and with hetgs on board _ chandra _ , we have @xmath123 . a typical seyfert 1 galaxy has a flux of @xmath124 erg @xmath125 s@xmath9 in the energy band of 210 kev . assuming a power law with energy index of 1 , the photon flux at 1 kev would be @xmath126 @xmath125 s@xmath9 kev@xmath9 . for a reasonable integration time of 10 ks , we have snr @xmath127 . when the primary continuum is obscured as in seyfert 2 galaxies , the ews of the emission and reflection lines can be orders of magnitude larger , because the continuum at this energy region is absorbed severely , and the snr can be greatly enhanced , making these lines observable . acknowledgment : smk acknowledges several grants from nasa which partially supported this work , mfg acknowledges the support of a chandra fellowship at mit . we wish to thank m. sako , d. savin and e. behar for several useful discussions . the `` s - curves '' produced by two different incident ionizing spectra . the vertical line indicates the unique solution of @xmath17 which satisfies condition [ eq_y2 ] : two solutions for `` s - curve 1 '' , @xmath35 = 2.82 and @xmath36 = 3.14 ; and only one for `` s - curve 2 '' , @xmath17 = 2.22 . ] the temperature profiles of the transition layers and the intermediate stable layer versus thomson optical depth @xmath68 . the solid curves correspond to `` s - curve 1 '' with different force factors @xmath48 , the dashed line corresponds to the `` s - curve 2 '' . ] the spectra of the emission lines via rr in the transition layers and the intermediate stable layer . the spectral resolution is @xmath128 ev . the top panel corresponds to `` s - curve 2 '' , while bottom panels correspond to `` s - curve 1 '' with different parameter a. ] the spectra of the reflection lines due to resonance scattering in the transition layers and the intermediate stable layer . the spectral resolution is @xmath75 ev . the top panel corresponds to `` s - curve 2 '' , while bottom panels correspond to `` s - curve 1 '' with different parameter a. ] the combined spectrum . emission lines via recombination , red reflection lines due to resonance line scattering , and black the compton reflection hump . the spectral resolution is @xmath129 ev . the top panel corresponds to `` s - curve 2 '' , while bottom panels correspond to `` s - curve 1 '' with different parameter a. ]

we derive a semi - analytic solution for the structure of conduction - mediated transition layers above an x - ray illuminated accretion disk , and calculate explicitly the x - ray line radiation resulting from both resonance line scattering and radiative recombination in these layers . the vertical thermal structure of the illuminated disk is found to depend on the illuminating continuum : for a hard power law continuum , there are two stable phases connected by a single transition layer ; while for a softer continuum , there may exist three stable phases connected by two separate transition layers , with an intermediate stable layer in between . we show that the structure can be written as a function of the electron scattering optical depth through these layers , which leads to unique predictions of the equivalent width of the resulting line radiation from both recombination cascades and resonance line scattering . we find that resonance line scattering plays an important role , especially for the case where there is no intermediate stable layer .

arxiv

polymers with contour length @xmath1 much larger than the persistence length @xmath2 , which is the correlation length for the tangent - tangent correlation function along the polymer and is a quantitative measure of the polymer stiffness , are flexible and are described by using the tools of quantum mechanics and quantum field theory @xcite-@xcite . if the chain length decreases , the chain stiffness becomes an important factor . many polymer molecules have internal stiffness and can not be modeled by the model of flexible polymers developed by edwards @xcite . the standard coarse - graining model of a wormlike polymer was proposed by kratky and porod @xcite . the essential ingredients of this model are the penalty for the bending energy and the local inextensibility . the latter makes the treatment of the model much more difficult . there have been a substantial number of studies of the kratky - porod model in the last half century @xcite-@xcite ( and citations therein ) . in recent years there has been increasing interest in the theoretical description of semiflexible polymers @xcite-@xcite . the reason for this interest is due to potential applications in biology allemand05 ( and citations therein ) and in research on semicrystalline polymers @xcite . it was found in the recent numerical work by lattanzi et al . lattanzi04 , and studied analytically in @xcite within the effective medium approach , that the transverse distribution function of a polymer embedded in two - dimensional space possesses a bimodal shape for short polymers , which is considered to be a manifestation of the semiflexibility . the bimodal shape for the related distribution function of the 2d polymer was also found in recent exact calculations by spakowitz and wang @xcite . in this paper we study the transverse distribution function @xmath3 of the three dimensional wormlike chain with a fixed orientation @xmath4 of one polymer end using the exact representation of the distribution function in terms of the matrix element of the green s function of the quantum rigid rotator in a homogeneous external field @xcite . the exact solution of the green s function made it possible to compute the quantities such as the structure factor , the end - to - end distribution function , etc . practically exact in the definite range of parameters @xcite , @xcite . our practically exact calculations of the transverse distribution function of the 3d wormlike chain demonstrate that it possesses the bimodal shape in the intermediate range of the chain lengths ( @xmath0 ) . in addition , we present analytical results for short and long wormlike chain based on the exact formula ( [ gtkp ] ) , which are in complete agreement with the previous results obtained in different ways @xcite ( wkb method for short polymer ) , @xcite ( perturbation theory for large chain ) . the paper is organized as follows . section [ sect1 ] introduces to the formalism and to analytical considerations for short and large polymers . section [ numer ] contains results of the numerical computation of the distribution function for polymers with different number of monomers . the fourier - laplace transform of the distribution function of the free end of the wormlike chain with a fixed orientation @xmath5 @xmath6 of the second end is expressed , according to @xcite , in a compact form through the matrix elements of the green s function of the quantum rigid rotator in a homogeneous external field @xmath7 as @xmath8where @xmath9 , and @xmath7 is defined by @xmath10with @xmath11 and @xmath12 being the infinite order square matrices given by @xmath13and @xmath14 . the matrix @xmath11 is related to the energy eigenvalues of the free rigid rotator , while @xmath12 gives the matrix elements of the homogeneous external field . since @xmath7 is the infinite order matrix , a truncation is necessary in the performing calculations . the truncation of the infinite order matrix of the green s function by the @xmath15-order matrix contains all moments of the end - to - end chain distance , and describes the first @xmath16 moments exactly . the transverse distribution function we consider , @xmath3 , is obtained from @xmath17 , which is determined by eqs . ( [ gtkp])-([d ] ) , integrating it over the @xmath18-coordinate , and imposing the condition that the free end of the chain stays in the @xmath19 plane . as a result we obtain @xmath20 is the bessel function of the first kind abramowitzstegun . taking the @xmath18-axis to be in the direction of @xmath21 yields @xmath22 , so that the arguments of the legendre polynomials in eq . ( [ gtkp ] ) become zero , and consequently only even @xmath23 will contribute to the distribution function ( [ gyn ] ) . we now will consider the expansion of ( [ gtkp ] ) around the rod limit @xmath24 , which corresponds to the expansion of @xmath25 in inverse powers of @xmath26 . to derive such an expansion , we write @xmath11 in the equivalent form as@xmath27with @xmath28 and @xmath29 . further we introduce the notation @xmath30 with @xmath31 and @xmath32 defined by@xmath33the iteration of @xmath11 and @xmath34 results in the desired expansion of @xmath32 and consequently of @xmath35 in inverse powers of @xmath26 , which corresponds to an expansion of @xmath36 in powers of @xmath37 . the leading order term in the short chain expansion is obtained by replacing @xmath11 by @xmath38 in eq . ( [ gtkp ] ) as @xmath39 _ { 0l}\sqrt{2l+1}p_{l}(\mathbf{t}_{0}\mathbf{n } ) . \label{gtkp0}\]]the latter coincides with the expansion of the plane wave landau - lifshitz3@xmath40where @xmath41 is the angle between the tangent @xmath4 and the wave vector @xmath42 . the connection of @xmath43 with the plane wave expansion is due to the fact that the kratky - porod chain becomes a stiff rod in the limit of small @xmath37 . we have checked the equivalency between the plane wave expansion ( [ plw ] ) and the distribution function ( [ gtkp0 ] ) term by term expanding ( [ gtkp0 ] ) in series in powers of the wave vector @xmath44 . the arc length @xmath37 is equivalent for stiff rod in units under consideration to the chain end - to - end distance @xmath45 . in the @xmath45 space the plane wave ( [ plw ] ) corresponds to the stiff rod distribution function @xmath46 the iteration of @xmath11 in ( [ dit ] ) and @xmath34 in ( [ y ] ) generates an expansion of @xmath35 in inverse powers of @xmath26 . the corrections to the plane wave to order @xmath47 are obtained as@xmath48 _ { 0l}\sqrt{2l+1}% p_{l}(\mathbf{t}_{0}\mathbf{n})+ .... \end{aligned}\ ] ] the above procedure yields for @xmath49 the short chain expansion of the distribution function of the free kratky - porod chain , which was studied recently in @xcite . unfortunately , we did not succeed yet in analytical evaluation of @xmath50 . such computation would be an interesting alternative to the treatment of the short limit of the wormlike chain by yamakawa and fujii @xcite within the wkb method . nevertheless , following the consideration in stepanow05 we succeeded in computing the anisotropic moments @xmath51 for small @xmath37@xmath52the first - order correction coincide with that obtained in @xcite using the wkb method , while the second - order correction is to our knowledge new . the higher - order terms in ( [ rt_0 ] ) can be established in a straightforward way using the present method . note that the computation of @xmath53 does not require the knowledge of the full distribution function @xmath54 . in studying the end - to - end distribution function for large @xmath37 we utilize the following procedure . we expand first the expression @xmath55 in powers of @xmath56 . the structure of this expansion for @xmath49 is presented in table [ table1 ] . the subseries in powers of @xmath56 in the @xmath57th column are denoted by @xmath58 . thus we have @xmath59the series @xmath58 with small values @xmath23 and @xmath57 possess a simple structure and can be summed up . for example @xmath60 and @xmath61 are given by @xmath62@xmath63while @xmath61 corresponds to the distribution function of the gaussian chain , @xmath64 give the @xmath57th correction to the gaussian distribution . the inspection of the series for @xmath65 and @xmath66 shows that they are expressed by @xmath61 and @xmath67 as@xmath68however , it seems that there is no general recursion relation for @xmath69 . the results of computations of @xmath70 for @xmath71 by taking into account @xmath72 ( @xmath49 ) and @xmath73 ( @xmath74 ) are summarized in table [ table2 ] . . [ cols="^,<",options="header " , ] the inverse laplace - fourier transform of @xmath70 given in the table yields the expansion of the end - to - end distribution function @xmath75 to order @xmath76 as @xmath77 , \label{grt}\end{aligned}\]]where @xmath78 , @xmath79 , and @xmath80 is the angle between @xmath81 and @xmath82 . the latter is in accordance with the result by gobush et . @xcite derived in a different way . the expansion of @xmath83 for large @xmath37 can be extended in a straightforward way to include higher - order corrections . the computation of the distribution function of the polymer with the fixed orientation of one end is performed by truncating the infinite order matrices in ( [ gtkp ] ) with the finite ones , and by taking into account the finite number of terms in the summation over @xmath23 . the inverse laplace transform of ( [ gtkp ] ) is carried out with maple . the results of the calculation of the distribution function @xmath3 , for various chain lengths , computed with @xmath84 matrices . the insets show the distribution function at the onset of bimodality , and in the region of its disappearance . ] using the truncations with @xmath84 order matrices and restricting the summation over the quantum number @xmath23 at @xmath85 , are given in fig . fig - doublepeak . the results show that the distribution function possesses the bimodal shape at intermediate chain lengths within the interval @xmath86 . we also find that the distribution function becomes gaussian for very short and very long chains . at the onset there is a tiny maximum at @xmath87 , which we interpret as remnant of the gaussian behavior of short chains . the maximum at @xmath87 for 3d chain is a rather small effect , which is difficult to be explained in a qualitative way . we now will discuss qualitatively the origin of the bimodal behavior of the projected distribution function of the free end of the wormlike chain . the very short wormlike chain behaves similar to a weakly bending stiff rod , so that the distribution function of the free end is gaussian with the maximum at @xmath87 . the typical conformation of the chain in this regime looks like a bending rod with constant sign of curvature along the chain . for larger contour lengths the curvature fluctuations are small and are still controlled by the bending energy , however with varying sign of curvatures along the chain . the typical conformation of the chain can be imagined as undulations along the average conformation of the polymer . the projected distribution function of the free end in this regime is expected to be roughly uniform within some range of @xmath88 . we expect that the inhomogeneities of curvature fluctuations in the vicinity of the clamped end are the reason for the maximum at @xmath89 . the larger curvatures in the vicinity of the fixed end result in larger displacements @xmath88 of the free end , and therefore contribute preferentially to the maximum at @xmath89 . with further increase of @xmath37 the conformations correspond to undulations around the average conformation of the chain , which is now a meandering line . fluctuations become now less controlled by bending energy , which results in weakening of the bimodality . since the difference between 2d and 3d chain is assumed to be marginal for short chains on the projected distribution function , we will compare the onset of the bimodality in both cases . because the transversal displacement is measured in both cases in units of the contour length , we have to recompute the number of segments for 3d and 2d chain at the onset according to @xmath90using the dependence of the persistence length on dimensionality benetatos05 , @xmath91 , we obtain @xmath92 . according to @xcite @xmath93 at the onset , hence we obtain @xmath94 , which is not far from our numerical result , @xmath95 ( see fig . [ fig - doublepeak ] ) . for chain length @xmath96 . squares : truncation with @xmath97 matrices ; continuous line : truncation with @xmath84 matrices . ] we now will address the issues of accuracy of the calculations , which depends on the size of the matrices @xmath11 and @xmath12 , and the maximal @xmath23 at which the summation over @xmath23 is stopped . in order to check our computation we verified that at large @xmath37 ( @xmath98 ) the numerical evaluation of ( gyn ) gives with very high accuracy the gaussian distribution @xmath99 . the general tendency is such that the sufficient level of matrix truncations and the number of terms in the expansion over the legendre polynomials increase with decreasing @xmath37 . in the limit @xmath24 the whole series over @xmath23 should be taken into account . the accuracy of the computations is demonstrated in fig . [ fig - n0 - 5 ] showing the computation of the distribution function for @xmath96 . the squares and the continuous curve correspond to the truncations by @xmath97 matrices and @xmath84 matrices , respectively . in both cases the summation was stopped at @xmath85 . the corrections due to higher @xmath23-s are negligibly small . for example , the corrections associated with @xmath100 and @xmath101 contribute only in 3rd and 5th decimal digits , respectively . thus , the computations depicted in figs . [ fig - doublepeak ] , [ fig - n0 - 5 ] can be considered as exact . figure [ figure3 ] shows the results of the computation of the 3d distribution functions @xmath102 of the free polymer obtained by performing the inverse laplace - fourier transform of the term @xmath49 in eq . ( [ gtkp ] ) for different chain lengths , and its comparison with the monte carlo simulations @xcite . , @xmath103 , @xmath104 , @xmath105 , @xmath106 ( from left to right ) computed with @xmath107 matrices . the symbols are the monte carlo simulation data extracted from fig . 1 in @xcite . ] our results are in excellent agreement with the numerical data . to conclude , we have studied the transverse distribution function of the free end of the three dimensional wormlike chain with fixed orientation and position of the second end using the exact solution for the green s function of the wormlike chain . within the procedure of truncations of the exact formula with finite order matrices we find that the distribution function @xmath108 for intermediate chain lengths , belonging to the interval @xmath0 , possesses the bimodal shape with the maxima at a finite value of the transverse displacement , which is consistent with the recent studies @xcite and @xcite for the two dimensional chain . in contrast to the 2d wormlike chain , the transverse 1d distribution function of the 3d chain shows only a tiny peak at @xmath87 in the vicinity of the onset of bimodality , which however disappears for larger @xmath37 . we present also results of analytical considerations for short and large polymers which are in complete agreement with the classical works @xcite where these limits were investigated using different methods . the computation of the three dimensional distribution function of a free polymer is in excellent agreement with the monte carlo simulations @xcite .

we study the distribution function of the three dimensional wormlike chain with a fixed orientation of one chain end using the exact representation of the distribution function in terms of the green s function of the quantum rigid rotator in a homogeneous external field . the transverse 1d distribution function of the free chain end displays a bimodal shape in the intermediate range of the chain lengths ( @xmath0 ) . we present also analytical results for short and long chains , which are in complete agreement with the results of previous studies obtained using different methods .

arxiv

in this paper , @xmath1 will be a simple graph with vertex set @xmath2 and edge set @xmath3 . [ closeddef ] a _ labeling _ of @xmath1 is a bijection @xmath4 = \{1,\dots , n\}$ ] , and given a labeling , we typically assume @xmath5 $ ] . a labeling is _ closed _ if whenever we have distinct edges @xmath6 with either @xmath7 or @xmath8 , then @xmath9 . finally , a graph is _ closed _ if it has a closed labeling . a labeling of @xmath1 gives a direction to each edge @xmath10 where the arrow points from @xmath11 to @xmath12 when @xmath13 , i.e. , the arrow points to the bigger label . the following picture illustrates what it means for a labeling to be closed : @xmath14 ( n1 ) at ( 2,1 ) { $ i\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ \rule[-2.5pt]{0pt}{10pt}j$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ k\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/n2,n1/n3 } \draw[- > ] ( \from)--(\to ) ; ; \foreach \from/\to in { n2/n3 } \draw[dotted ] ( \from)--(\to ) ; ; \end{tikzpicture}&\hspace{30pt } & \begin{tikzpicture } \node[vertex ] ( n1 ) at ( 2,1 ) { $ i\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ j\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ k\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n2/n1,n3/n1 } \draw[- > ] ( \from)--(\to ) ; ; \foreach \from/\to in { n2/n3 } \draw[dotted ] ( \from)--(\to ) ; ; \end{tikzpicture } \end{array}\ ] ] whenever the arrows point away from @xmath11 ( as on the left ) or towards @xmath11 ( as on the right ) , closed means that @xmath12 and @xmath15 are connected by an edge . closed graphs were first encountered in the study of binomial edge ideals . the _ binomial edge ideal _ of a labeled graph @xmath1 is the ideal @xmath16 in the polynomial ring @xmath17 $ ] ( @xmath18 a field ) generated by the binomials @xmath19 for all @xmath20 such that @xmath10 and @xmath13 . a key result , discovered independently in @xcite and @xcite , is that the above binomials form a grbner basis of @xmath16 for lex order with @xmath21 if and only if the labeling is closed . the name `` closed '' was introduced in @xcite . binomial edge ideals are explored in @xcite and @xcite , and a generalization is studied in @xcite . the paper @xcite characterizes closed graphs using the clique complex of @xmath1 , and closed graphs also appear in @xcite . the goal of this paper is to characterize when a graph @xmath1 has a closed labeling in terms of properties that can be seen directly from the graph . our starting point is the following result proved in @xcite . [ hprop ] every closed graph is chordal and claw - free . `` claw - free '' means that @xmath1 has no induced subgraph of the form @xmath22 ( k ) at ( 3,6 ) { $ \bullet$ } ; \node[vertex ] ( j ) at ( 2.1,3.9){$\bullet$ } ; \node[vertex ] ( l ) at ( 3.9,3.9 ) { $ \bullet$ } ; \node[vertex ] ( i ) at ( 3,5){$\bullet$ } ; \foreach \from/\to in { i / l , i / k , i / j } \draw ( \from ) -- ( \to ) ; \end{tikzpicture } \end{array}\ ] ] besides being chordal and claw - free , closed graphs also have a property called _ narrow_. the _ distance _ @xmath23 between vertices @xmath24 of a connected graph @xmath1 is the length of the shortest path connecting them , and the _ diameter _ of @xmath1 is @xmath25 . given vertices @xmath24 of @xmath1 satisfying @xmath26 , a shortest path connecting @xmath27 and @xmath28 is called a _ longest shortest path _ of @xmath1 . [ narrowdef ] a connected graph @xmath1 is _ narrow _ if for every @xmath29 and every longest shortest path @xmath30 of @xmath1 , either @xmath31 or @xmath32 for some @xmath33 . thus a connected graph is narrow if every vertex is distance at most one from every longest shortest path . here is a graph that is chordal and claw - free but not narrow : @xmath34 ( n1 ) at ( 3,1 ) { $ a\rule[-2pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 2,3 ) { $ b\rule[-2pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 4,3 ) { $ c\rule[-2pt]{0pt}{10pt}$ } ; \node[vertex ] ( n4 ) at ( 3,5){$e\rule[-2pt]{0pt}{10pt}$ } ; \node[vertex ] ( n5 ) at ( 5,5){$f\rule[-2pt]{0pt}{10pt}$ } ; \node[vertex ] ( n6 ) at ( 1,5){$d\rule[-2pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/n2,n1/n3,n2/n3,n2/n4 , n3/n4 , n3/n5 , n4/n5,n2/n6,n4/n6 } \draw ( \from)--(\to ) ; ; \end{tikzpicture } \end{array}\ ] ] narrowness fails because @xmath35 is distance two from the longest shortest path @xmath36 . we can now state the main result of this paper . [ mainthm ] a connected graph is closed if and only if it is chordal , claw - free , and narrow . this theorem is cited in @xcite . since a graph is closed if and only if its connected components are closed @xcite , we get the following corollary of theorem [ mainthm ] . [ cormainthm ] a graph is closed if and only if it is chordal , claw - free , and its connected components are narrow . the independence of the three conditions ( chordal , claw - free , narrow ) is easy to see . the graph is chordal and narrow but not claw - free , and the graph is chordal and claw - free but not narrow . finally , the @xmath37-cycle @xmath38 ( a ) at ( 2,1 ) { $ \bullet$ } ; \node[vertex ] ( b ) at ( 4,1 ) { $ \bullet$ } ; \node[vertex ] ( c ) at ( 4,3 ) { $ \bullet$ } ; \node[vertex ] ( d ) at ( 2,3 ) { $ \bullet$ } ; \foreach \from/\to in { a / b , b / c , c / d , d / a } \draw ( \from)--(\to ) ; ; \end{tikzpicture}\ ] ] is claw - free and narrow but not chordal . the paper is organized as follows . in section [ properties ] we recall some known properties of closed graphs and prove some new ones , and in section [ algorithm ] we introduce an algorithm for labeling connected graphs . section [ characterize ] uses the algorithm to prove theorem [ mainthm ] . in a subsequent paper @xcite we will explore further properties of closed graphs . a path in a graph @xmath1 is @xmath39 where @xmath40 for @xmath41 . a single vertex is regarded as a path of length zero . when @xmath1 is labeled , we assume as usual that @xmath42 $ ] . then a path @xmath43 is _ directed _ if either @xmath44 for all @xmath12 or @xmath45 for all @xmath12 . here is a result from @xcite . [ directed ] a labeling on a graph @xmath1 is closed if and only if for all vertices @xmath46 $ ] , all shortest paths from @xmath11 to @xmath12 are directed . given a vertex @xmath29 , the _ neighborhood _ of @xmath27 in @xmath1 is @xmath47 when @xmath1 is labeled and @xmath48 $ ] , we have a disjoint union @xmath49 where @xmath50 this is the notation used in @xcite , where it is shown that a labeling is closed if and only if @xmath51 and @xmath52 are complete for all @xmath48 $ ] . vertices @xmath53 $ ] with @xmath54 give the _ interval _ @xmath55 = \{k \in [ n ] \mid i \le k \le j\}$ ] . here is a characterization of when a labeling of a connected graph is closed . [ nbdinterval ] a labeling on a connected graph @xmath1 is closed if and only if for all @xmath48 $ ] , @xmath51 is complete and equal to @xmath56 $ ] , @xmath57 . assume that the labeling is closed . then definition [ closeddef ] easily implies that @xmath51 is complete . it remains to show that @xmath51 is an interval of the desired form . pick @xmath58 and @xmath59 $ ] with @xmath60 . a shortest path @xmath61 from @xmath62 to @xmath63 is directed by proposition [ directed ] . since @xmath64 , we have @xmath65 . thus @xmath66 and hence @xmath67 since @xmath68 is complete . since @xmath69 , we have @xmath70 . we now prove by induction that @xmath71 for all @xmath72 . the base case is proved in the previous paragraph . now assume @xmath73 . then @xmath74 since @xmath75 and the labeling is closed . this completes the induction . since @xmath63 , it follows that @xmath76 . then we have @xmath77 with @xmath78 . thus @xmath79 since the labeling is closed , so @xmath80 since @xmath64 . hence @xmath51 is an interval of the desired form . conversely , suppose that @xmath51 is complete and @xmath81 $ ] , @xmath57 , for all @xmath82 . take @xmath6 with @xmath7 or @xmath8 . the former implies @xmath9 since @xmath68 is complete . for the latter , assume @xmath83 . then @xmath84 with @xmath85 . since @xmath86 is an interval containing @xmath87 and @xmath11 , @xmath86 also contains @xmath15 . hence @xmath9 . the following subsets of @xmath2 will play a key role in what follows . [ layerdef ] let @xmath1 be a connected graph labeled so that @xmath88 $ ] . then the _ @xmath89 layer of @xmath1 _ is the set @xmath90 \mid d(i,1 ) = n\}.\ ] ] thus @xmath91 consists of all vertices that are distance @xmath92 from the vertex @xmath93 . note that @xmath94 and @xmath95 . furthermore , since @xmath1 is connected , we have a disjoint union @xmath96 where @xmath97\}$ ] . we omit the easy proof of the following lemma . [ layerlem ] let @xmath1 be a connected graph labeled so that @xmath98 $ ] . then : 1 . if @xmath99 and @xmath10 , then @xmath100 , @xmath91 , or @xmath101 . if @xmath30 is a path in @xmath1 connecting @xmath99 to @xmath102 with @xmath103 , then for every integer @xmath104 , there exists @xmath105 with @xmath106 . [ layerprop ] let @xmath1 be a connected graph with a closed labeling satisfying @xmath42 $ ] . then : 1 . each layer @xmath91 is complete . 2 . if @xmath107 , then @xmath108 . we first show that @xmath109 to see why , take a shortest path from 1 to @xmath110 . this path has length @xmath111 , so appending the edge @xmath112 gives a path of length @xmath113 to @xmath114 . since @xmath115 , this is a shortest path and hence is directed by proposition [ directed ] . thus @xmath116 . for ( 1 ) , we use induction on @xmath117 . the base case is trivial since @xmath118 . now assume @xmath91 is complete and take @xmath119 with @xmath120 . a shortest path @xmath121 from @xmath93 to @xmath122 has a vertex @xmath123 adjacent to @xmath11 , and a shortest path @xmath124 from @xmath93 to @xmath125 has a vertex @xmath126 adjacent to @xmath12 . then @xmath127 and @xmath128 by . if @xmath129 , then @xmath130 , which implies @xmath131 since the labeling is closed . if @xmath132 , then @xmath133 since @xmath91 is complete . assume @xmath134 . then @xmath135 and closed imply @xmath136 . since @xmath137 and @xmath138 , we have @xmath139 by . then @xmath140 and closed imply @xmath131 . hence @xmath101 is complete . we now turn to ( 2 ) . to prove @xmath141 , @xmath142 , take @xmath138 . a shortest path from @xmath93 to @xmath11 will have a vertex @xmath143 such that @xmath144 . then @xmath127 by , hence @xmath145 . also , @xmath146 since @xmath147 . if @xmath148 , then @xmath149 . if @xmath150 , then @xmath151 since @xmath91 is complete . then @xmath152 and closed imply @xmath153 , and then @xmath154 by . thus @xmath149 . to prove the opposite inclusion , take @xmath149 . since @xmath153 and @xmath155 , we have @xmath156 for @xmath157 by lemma [ layerlem ] . if @xmath158 , then would imply @xmath159 , contradicting @xmath160 . if @xmath161 , then @xmath162 , again contradicting @xmath163 . hence @xmath122 . when the labeling of a connected graph is closed , the diameter of the graph determines the number of layers as follows . [ diameters ] let @xmath1 be a connected graph with a closed labeling . then : 1 . @xmath164 is the largest integer @xmath165 such that @xmath166 . if @xmath30 is a longest shortest path of @xmath1 , then one endpoint of @xmath30 is in @xmath167 or @xmath168 and the other is in @xmath169 , where @xmath170 . for ( 1 ) , let @xmath165 be the largest integer with @xmath171 . since points in @xmath169 have distance @xmath165 from @xmath93 , we have @xmath172 . for the opposite inequality , it suffices to show that @xmath173 for all @xmath174 with @xmath175 . we can assume @xmath1 has more than one vertex , so that @xmath176 . suppose @xmath161 and @xmath177 with @xmath178 . if @xmath179 , then @xmath180 and @xmath181 since @xmath102 . also , if @xmath182 , then @xmath183 , so that @xmath184 since @xmath91 is complete by proposition [ layerprop ] . finally , if @xmath185 , let @xmath186 for each integer @xmath187 . by proposition [ layerprop ] , we know that @xmath188 . hence , if @xmath189 , then @xmath190 is a path of length @xmath191 . if @xmath192 , then @xmath193 is a path of length @xmath194 . thus we have a path from @xmath11 to @xmath12 of length at most @xmath191 , so that @xmath195 . for ( 2 ) , let @xmath11 and @xmath12 be the endpoints of the longest shortest path @xmath30 with @xmath161 , @xmath177 and @xmath178 . if @xmath196 , then the previous paragraph implies @xmath197 which forces @xmath198 ( so @xmath199 ) and @xmath200 ( so @xmath201 ) . the remaining cases @xmath202 and @xmath203 are straightforward and are left to the reader . recall from definition [ narrowdef ] that a connected graph @xmath1 is narrow when every vertex is distance at most one from every longest shortest path . narrowness is a key property of connected closed graphs . [ narrowthm ] every connected closed graph is narrow . let @xmath1 be a connected graph with a closed labeling . pick a vertex @xmath204 and a longest shortest path @xmath30 . since @xmath1 is connected , @xmath161 for some integer @xmath92 . by proposition [ diameters ] , the endpoints of @xmath30 lie in @xmath167 or @xmath168 and @xmath169 , @xmath205 . then lemma [ layerlem ] implies that @xmath30 has a vertex @xmath206 in @xmath207 for every @xmath208 . if @xmath209 , then either @xmath210 or @xmath211 , in which case @xmath212 since @xmath91 is complete by proposition [ layerprop ] . on the other hand , if @xmath202 , then @xmath213 , hence @xmath180 . then @xmath214 since @xmath215 . in either case , @xmath11 is distance at most one from @xmath30 . we introduce algorithm [ alg : labeling ] , which labels the vertices of a connected graph . this algorithm will play a key role in the proof of theorem [ mainthm ] . [ alg : input ] [ alg : output ] @xmath216 @xmath217 @xmath218 endpoint of a longest shortest path with minimal degree[alg : possible1 ] label @xmath219 as @xmath11 [ alg : label1 ] @xmath220[alg : function1 ] @xmath221 @xmath222 @xmath223[alg : labelj1 ] the algorithm works as follows . among the endpoints of all longest shortest paths , we select one of minimal degree and label it as @xmath93 . we then go through the vertices in @xmath224 and label them @xmath225 , first labeling vertices with the fewest number of edges connected to unlabeled vertices . this process is repeated for the unlabeled vertices connected to vertex @xmath226 , and vertex @xmath227 , and so on until every vertex is labeled . furthermore , every vertex will be labeled because we first label everything in @xmath228 , then label everything in @xmath229 not already labeled , and so on . since the input graph is connected , this process must eventually reach all of the vertices . hence we get a labeling of @xmath1 . the following lemma explains the function @xmath230 that appears in algorithm [ alg : labeling ] . [ claim : meaningoffunction ] let @xmath1 be a connected graph with the labeling from algorithm [ alg : labeling ] . then : 1 . @xmath231 , and for every @xmath232 $ ] with @xmath233 , @xmath234 . 2 . if @xmath235 , then @xmath236 . algorithm [ alg : labeling ] defines @xmath231 . now assume @xmath233 and let @xmath27 be the vertex assigned the label @xmath11 . by lines [ alg : label2 ] and [ alg : function2 ] of the algorithm , we need to show that when the label @xmath11 is assigned to @xmath27 , the variable @xmath12 equals @xmath237 . this follows because for any smaller value @xmath238 , line [ alg : secondloop ] implies that everything in the neighborhood of @xmath239 is labeled before @xmath239 is incremented . however , lines [ alg : sset][alg : beginsecondloop ] show that @xmath27 is adjacent to @xmath12 and unlabeled at the start of the loop on line [ alg : secondloop ] . hence @xmath27 can not link to any smaller value of @xmath12 , and since @xmath27 has label @xmath11 , @xmath240 follows . \(2 ) suppose that @xmath241 $ ] satisfy @xmath235 . since @xmath242 ( resp . @xmath243 ) is the value of @xmath12 when the label @xmath244 ( resp . @xmath114 ) was assigned in algorithm [ alg : labeling ] , @xmath235 implies that the label @xmath114 was assigned later than @xmath244 in the algorithm . since the labels are assigned in numerical order , we must have @xmath245 . the labeling produced by algorithm [ alg : labeling ] allows us to define the layers @xmath91 . these interact with the function @xmath230 as follows : [ claim : orderingclaim ] let @xmath1 be a connected graph with the labeling from algorithm [ alg : labeling ] . then : 1 . if @xmath246 , then @xmath247 if @xmath248 . 2 . if @xmath246 and @xmath249 with @xmath250 , then @xmath236 . we prove ( 1 ) and ( 2 ) simultaneously by induction on @xmath209 ( the case @xmath202 of ( 2 ) is trivially true ) . the first time algorithm [ alg : labeling ] gets to line [ alg : sset ] , we have @xmath251 . every vertex in @xmath252 , is labeled during the loop starting on line [ alg : firstloop ] , so @xmath253 for all @xmath254 . hence ( 1 ) holds when @xmath255 . also , if @xmath249 with @xmath256 , then the vertex @xmath114 is not labeled at this stage . since labels are assigned in numerical order , we must have @xmath236 for all @xmath254 . hence ( 2 ) holds when @xmath198 . now assume that ( 1 ) and ( 2 ) hold for @xmath111 and every @xmath257 . given @xmath258 , a shortest path from 1 to @xmath244 gives @xmath259 with @xmath260 . since @xmath261 by lemma [ claim : meaningoffunction](1 ) , we have @xmath262 . we have @xmath263 for some @xmath187 . if @xmath264 , then the inductive hypothesis for ( 2 ) would imply @xmath265 , which contradicts @xmath262 . hence @xmath263 for some @xmath266 . but @xmath267 and @xmath268 imply @xmath269 for @xmath270 by lemma [ layerlem](1 ) . hence @xmath271 , proving ( 1 ) for @xmath272 . turning to ( 2 ) , pick @xmath258 and @xmath249 with @xmath273 . we just showed that @xmath274 , and lemma [ layerlem](1 ) implies that @xmath275 , @xmath276 , since @xmath277 . then @xmath278 , so our inductive hypothesis , applied to @xmath279 and @xmath280 , implies @xmath235 . then @xmath236 by lemma [ claim : meaningoffunction](2 ) , proving ( 2 ) for @xmath272 . we now turn to the main result of the paper . theorem [ mainthm ] from the introduction states that a connected graph is closed if and only if it is chordal , claw - free and narrow . one direction is now proved , since closed graphs are chordal and claw - free by proposition [ hprop ] , and connected closed graphs are narrow by theorem [ narrowthm ] . the proof of converse is harder . the key idea that the labeling constructed by algorithm [ alg : labeling ] is closed when the input graph is chordal , claw - free and narrow . thus the proof of theorem [ mainthm ] will be complete once we prove the following result . [ converse ] let @xmath1 be a connected , chordal , claw - free , narrow graph . then the labeling produced by algorithm [ alg : labeling ] is closed . by proposition [ nbdinterval ] , it suffices to show that the labeling produced by algorithm [ alg : labeling ] has the property that for all @xmath281 $ ] , @xmath282 \text { for } r_m = |{n_g^>}(m)|.\ ] ] we will prove this by induction on @xmath283 . in below , we show that holds for @xmath284 , and in below , we show that if holds for all @xmath285 , then it also holds for @xmath283 . thus , we will be done after proving and . after algorithm [ alg : labeling ] runs on a chordal , claw - free and narrow graph @xmath1 , the base case of the induction in the proof of theorem [ converse ] is the following assertion : @xmath286,\ r = |{n_g^>}(1)|,\ \text{and } { n_g^>}(1 ) \text { is complete } .\ ] ] we will first show that @xmath287 $ ] , @xmath288 . the first time through the the loop beginning on line [ alg : firstloop ] in algorithm [ alg : labeling ] , @xmath289 and @xmath290 and @xmath291 . for each vertex in @xmath292 , the loop beginning on line [ alg : secondloop ] labels that vertex @xmath11 , removes it from @xmath292 , and increments @xmath11 . this continues until @xmath293 , at which point every vertex in @xmath292 has been labeled @xmath294 , where @xmath295 is the initial size of @xmath292 . hence @xmath296 $ ] . to prove that @xmath228 is complete , there are several cases to consider . pick distinct vertices @xmath297 and assume that @xmath298 . note that @xmath299 are distance @xmath226 apart and therefore @xmath300 . our choice of vertex @xmath93 guarantees that there is a longest shortest path @xmath30 with 1 as an endpoint . let @xmath301 be the other , so that @xmath302 , @xmath303 and @xmath304 . since @xmath305 is the only vertex of @xmath30 in @xmath168 , @xmath114 and @xmath244 can not both lie on @xmath30 . therefore , either @xmath306 , @xmath307 , or @xmath308 . we will show that each possibility leads to a contradiction , proving that @xmath309 . plus 1 pt minus 1 pt * case 1 . * both @xmath308 . if @xmath114 has distance @xmath310 from @xmath311 , then appending the edge @xmath312 to a shortest path from @xmath114 to @xmath311 gives a longest shortest path @xmath313 from @xmath93 to @xmath311 that contains @xmath114 . replacing @xmath30 with @xmath313 , we get @xmath314 , which is case 2 to be considered below . similarly , if @xmath244 has distance @xmath310 from @xmath311 , then replacing @xmath30 allows us to assume @xmath315 , which is also covered by case 2 below . thus we may assume that neither @xmath114 nor @xmath244 has distance @xmath310 from @xmath311 . since @xmath316 would imply @xmath317 , we conclude that @xmath114 has distance @xmath165 from @xmath311 , and the same holds for @xmath244 . it follows that @xmath318 , since otherwise there is a path shorter than length @xmath165 from @xmath114 or @xmath244 to @xmath311 . since the subgraph induced on vertices @xmath319 can not be a claw , either @xmath320 or @xmath321 or both . we consider each possibility separately . plus 1 pt minus 1 pt * case 1a . * both @xmath322 , @xmath321 , as shown in figure 1(a ) on the next page . then the subgraph induced on @xmath323 is a claw , contradicting our assumption of claw - free . -10pt @xmath324 ( n1 ) at ( 2,1 ) { $ 1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( s ) at ( 3.5,3 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( t ) at ( .5,3 ) { $ t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( v1 ) at ( 2,3 ) { $ v_1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( v2 ) at ( 2,5 ) { $ v_2\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/v1,n1/t , n1/s , v1/v2,s / v1,t / v1 } \draw ( \from)--(\to ) ; \node [ right , font = \large ] at ( 4.2,3 ) { $ l_1 $ } ; \node [ right , font=\large ] at ( 4.2,5 ) { $ l_2 $ } ; \node [ right , font = \large ] at ( 4.2,1 ) { $ l_0 $ } ; \node [ right ] at ( 2.5,.3 ) { ( a ) } ; \end{tikzpicture } & \hspace{25pt}\begin{tikzpicture } \node[vertex ] ( n1 ) at ( 2,1 ) { $ 1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( s ) at ( 3.5,3.2 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( t ) at ( .5,3.2 ) { $ t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( v1 ) at ( 2,3.2 ) { $ v_1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( v2 ) at ( 2,5.4 ) { $ v_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( t2 ) at ( .5,5.4 ) { $ t_2\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/s , n1/t , n1/v1 , v1/v2 , s / v1,t / t2 } \draw ( \from)--(\to ) ; \node [ right , font = \large ] at ( 4.3,3.2 ) { $ l_1 $ } ; \node [ right , font=\large ] at ( 4.3,5.4 ) { $ l_2 $ } ; \node [ right , font = \large ] at ( 4.3,1 ) { $ l_0 $ } ; \node [ right ] at ( 2.5,.3 ) { ( b ) } ; \end{tikzpicture } \end{array}\ ] ] -8pt plus 1 pt minus 1 pt * case 1b . * exactly one of @xmath325 is in @xmath3 . without loss of generality , we may assume @xmath326 and @xmath327 , as shown in figure 1(b ) . recall that @xmath328 and @xmath244 is distance @xmath165 to @xmath311 . since @xmath244 and @xmath93 are both endpoints of longest shortest paths , line [ alg : possible1 ] of algorithm [ alg : labeling ] implies that @xmath329 . since @xmath330 is adjacent to @xmath93 but not @xmath244 , there must be at least one @xmath331 adjacent to @xmath244 but not @xmath93 , i.e. , @xmath332 with @xmath333 . for this @xmath331 , it follows that @xmath334 . we also have @xmath335 since @xmath328 . furthermore , @xmath336 , since otherwise we would have the @xmath37-cycle @xmath337 with no chords as @xmath338 , @xmath339 . similarly , @xmath340 or else we would have the @xmath37-cycle @xmath341 with no chords since @xmath338 , @xmath327 . note also that @xmath342 , since otherwise we would have the @xmath343-cycle @xmath344 with no chords as @xmath345 , @xmath346 , @xmath347 , @xmath348 , @xmath340 , contradicting chordal . hence @xmath331 gives figure 1(b ) as an induced subgraph . since @xmath1 is narrow , either @xmath349 or @xmath331 is adjacent to a vertex of @xmath30 . however , @xmath349 would imply @xmath350 since both lie in @xmath351 , contradicting @xmath352 . thus @xmath353 for some @xmath354 . since @xmath355 and @xmath356 , we have @xmath357 by lemma [ layerlem](1 ) . we just proved @xmath342 , so we must have @xmath358 . this gives the @xmath359-cycle @xmath360 . since figure 1(b ) is an induced subgraph , the only possible chords are @xmath361 , @xmath362 , @xmath363 , but by lemma [ layerlem](1 ) none of these are in @xmath3 since @xmath364 and @xmath365 . hence the @xmath359-cycle has no chords , contradicting chordal . * case 2 . * @xmath314 or @xmath366 . we may assume @xmath367 . arguing as in case 1b , there is @xmath332 with @xmath333 and @xmath334 . we also have @xmath368 , since otherwise the @xmath37-cycle @xmath369 has no chords as @xmath370 , @xmath371 . since @xmath1 is narrow , @xmath331 must either be in @xmath30 or be adjacent to a vertex in @xmath30 . however , @xmath372 would imply @xmath373 since @xmath374 , and the latter would give @xmath375 , which we just showed to be impossible . hence @xmath376 , so that @xmath377 for some @xmath187 . note that @xmath378 by lemma [ layerlem](1 ) . we claim that @xmath379 . to see why , first note that @xmath380 , since otherwise we would have the @xmath37-cycle @xmath381 with no chords as @xmath370 , @xmath382 . we also know that @xmath342 , as otherwise we would have the @xmath343-cycle @xmath383 with no chords since @xmath384 , @xmath338 , @xmath385 , @xmath347 , @xmath386 . see figure 2(a ) . thus we must have @xmath387 . however , this gives a @xmath359-cycle @xmath388 with the same impossible chords as before along with @xmath362 , @xmath361 , @xmath389 , @xmath390 , as in figure 2(b ) . this contradicts chordal , and follows . @xmath391 ( n1 ) at ( 2,1 ) { $ 1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n4 ) at ( 1,4.5){$t_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n5 ) at ( 3,4.5){$v_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n6 ) at ( 3,6){$v_3\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/n2,n1/n3,n2/n4 , n3/n5 , n6/n5 } \draw ( \from)--(\to ) ; ; \foreach \from/\to in { n4/n5 } \draw[dotted ] ( \from)--(\to ) ; \node [ right , font = \large ] at ( 3.7,3 ) { $ l_1 $ } ; \node [ right , font=\large ] at ( 3.7,4.5 ) { $ l_2 $ } ; \node [ right , font = \large ] at ( 3.7,1 ) { $ l_0 $ } ; \node [ right , font = \large ] at ( 3.7,6 ) { $ l_3 $ } ; \node [ right ] at ( 2.5,.3 ) { ( a ) } ; \end{tikzpicture}&\hspace{25pt } & \begin{tikzpicture } \node[vertex ] ( n1 ) at ( 2,1 ) { $ 1\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n4 ) at ( 1,4.5){$t_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n5 ) at ( 3,4.5){$v_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n6 ) at ( 3,6){$v_3\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/n2,n1/n3,n2/n4 , n3/n5 , n6/n5 } \draw ( \from)--(\to ) ; ; \foreach \from/\to in { n4/n6 } \draw[dotted ] ( \from)--(\to ) ; \node [ right , font = \large ] at ( 3.7,3 ) { $ l_1 $ } ; \node [ right , font=\large ] at ( 3.7,4.5 ) { $ l_2 $ } ; \node [ right , font = \large ] at ( 3.7,1 ) { $ l_0 $ } ; \node [ right , font = \large ] at ( 3.7,6 ) { $ l_3 $ } ; \node [ right ] at ( 2.5,.3 ) { ( b ) } ; \end{tikzpicture } \end{array}\ ] ] -8pt after algorithm [ alg : labeling ] runs on a chordal , claw - free and narrow graph @xmath1 , we now prove that the resulting labeling satsifies the inductive step in the proof of theorem [ converse ] : @xmath392 $ , $ r_u = |{n_g^>}(u)|$ , $ { n_g^>}(u)$ is complete , $ 1\leq u < m$ , } \\ & \text{then $ { n_g^>}(m ) = [ m+1,m+r_m]$ , $ r_m = |{n_g^>}(m)|$ , and $ { n_g^>}(m)$ is complete . } \end{aligned}\ ] ] for the first assertion of , we know that @xmath393 $ ] is complete , which implies that @xmath394 . by analyzing the loop beginning on line [ alg : secondloop ] at this stage of algorithm [ alg : labeling ] , one finds that every vertex in @xmath292 will be labeled with consecutive integers , starting at @xmath395 and continuing until the final vertex in @xmath396 is labeled @xmath397 , where @xmath295 is the original size of @xmath292 . it follows that @xmath396 is an interval of the desired form . to show that @xmath398 is complete , pick @xmath399 in @xmath398 . let @xmath400 be a shortest path from @xmath303 to @xmath401 , with @xmath356 for all @xmath187 . lemmas [ layerlem](1 ) and [ claim : orderingclaim](2 ) imply that @xmath402 . hence , @xmath114 and @xmath244 are either both distance @xmath403 from 1 , both distance @xmath404 from 1 , or one of @xmath114 and @xmath244 is distance @xmath404 from 1 and the other is distance @xmath403 from 1 . we consider each case separately . plus3pt minus2pt * case 1 . * @xmath405 . then @xmath406 , @xmath407 by lemma [ layerlem](1 ) . since the subgraph induced on @xmath408 can not be a claw , we must have @xmath409 . plus3pt minus2pt * case 2 . * @xmath410 . we can assume @xmath411 and choose a shortest path @xmath412 from @xmath413 to @xmath414 with @xmath415 . then @xmath416 by lemma [ claim : orderingclaim](2 ) , giving @xmath417 . since @xmath418 and @xmath419 is an interval by hypothesis , we have @xmath420 . but then @xmath309 since we are also assuming that @xmath419 is complete . plus3pt minus2pt * case 3 . * we can assume @xmath421 and @xmath422 , so @xmath411 by lemma [ claim : orderingclaim](2 ) . we also have @xmath423 by lemma [ claim : meaningoffunction](2 ) since @xmath424 . we will consider separately the two possibilities that @xmath425 and @xmath426 . plus3pt minus2pt * case 3a . * suppose that @xmath427 . then @xmath428 since @xmath429 by lemma [ claim : meaningoffunction](1 ) . we also have @xmath430 , for otherwise we would have @xmath431 since @xmath261 . then @xmath432 , which implies @xmath236 by lemma [ claim : meaningoffunction](2 ) , contradicting @xmath411 . since the subgraph induced on @xmath433 can not be a claw , we must have @xmath409 . plus3pt minus2pt * case 3b . * suppose that @xmath426 . we will assume @xmath434 and derive a contradiction . the equality @xmath426 means that @xmath283 and @xmath114 were both labeled when @xmath435 in the loop starting on line [ alg : firstloop ] of algorithm [ alg : labeling ] . consider the moment in the algorithm when the label @xmath283 is assigned . since @xmath436 and @xmath435 , this happens during an iteration of the loop on line [ alg : secondloop ] for which @xmath437 . line [ alg : beginsecondloop ] guarantees that the vertices assigned the labels @xmath283 and @xmath114 satisfy @xmath438 . since @xmath114 is not yet labeled at this point and @xmath411 , @xmath244 is also not yet labeled and therefore @xmath439 . it follows that @xmath440 and @xmath441 . but , in order for @xmath442 to hold , there must be @xmath443 with @xmath444 and @xmath445 and @xmath446 . let us study @xmath447 . if @xmath448 , then @xmath449 . but we also have @xmath450 . since @xmath451 , @xmath452 is complete by the hypothesis of , so we would have @xmath453 . this contradicts our choice of @xmath447 . hence @xmath454 . we also have @xmath455 , since otherwise the @xmath37-cycle @xmath456 would have no chords as @xmath457 . also , since @xmath458 , lemma [ claim : orderingclaim](1 ) implies that @xmath459 . we claim that @xmath460 . lemma [ layerlem](1 ) , @xmath461 , and @xmath462 imply that @xmath463 or @xmath464 . if @xmath463 , then @xmath465 by lemma [ claim : orderingclaim](2 ) . from here , @xmath466 implies @xmath467 by lemma [ claim : orderingclaim](2 ) . hence we have @xmath468 . the hypothesis of implies that @xmath469 is complete and is an interval . since @xmath470 , it follows that @xmath471 , which contradicts our choice of @xmath447 . hence @xmath460 and we have figure 3(a ) . @xmath391 ( n1 ) at ( 2,1 ) { $ j\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ m\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n4 ) at ( 1,4.5){$t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n5 ) at ( 3,4.5){$s_2\rule[-2.5pt]{0pt}{10pt}$ } ; \foreach \from/\to in { n1/n2,n1/n3,n2/n3,n2/n4 , n3/n5 } \draw ( \from)--(\to ) ; ; \node [ vertex , font = \large ] at ( 4,3 ) { $ l_q$ } ; \node [ vertex , font=\large ] at ( 4,4.5 ) { $ l_{q+1}$ } ; \node [ vertex , font = \large ] at ( 4,1 ) { $ l_{q-1}$ } ; \node [ right ] at ( 2.5,.3 ) { ( a ) } ; \end{tikzpicture}&\hspace{25pt } & \begin{tikzpicture } \node[vertex ] ( n1 ) at ( 2,1 ) { $ j\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n2 ) at ( 1,3 ) { $ m\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n3 ) at ( 3,3 ) { $ s\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n4 ) at ( 1,4.5){$t\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n5 ) at ( 3,4.5){$s_2\rule[-2.5pt]{0pt}{10pt}$ } ; \node[vertex ] ( n6 ) at ( 1,6){$w_{q+2}$ } ; \foreach \from/\to in { n1/n2,n1/n3,n2/n3,n2/n4 , n3/n5 , n4/n6 , n6/n5 } \draw ( \from)--(\to ) ; ; \node [ right , font = \large ] at ( 4,3 ) { $ l_q$ } ; \node [ right , font=\large ] at ( 4,4.5 ) { $ l_{q+1}$ } ; \node [ right , font = \large ] at ( 4,1 ) { $ l_{q-1}$ } ; \node [ right , font = \large ] at ( 4,6 ) { $ l_{q+2}$ } ; \node [ right ] at ( 2.5,.3 ) { ( b ) } ; \end{tikzpicture } \end{array}\ ] ] -10pt let @xmath311 be a vertex of distance @xmath170 from @xmath93 and pick a longest shortest path @xmath472 from @xmath413 to @xmath473 , so @xmath415 . since @xmath1 is narrow , @xmath244 and @xmath447 must each either be in @xmath124 or be adjacent to a vertex in @xmath124 . we will consider each of these cases . _ first _ , suppose that @xmath474 . then @xmath422 implies that @xmath475 . since @xmath476 , there is a path of length @xmath477 connecting @xmath93 to @xmath478 . using @xmath479 , it follows that @xmath480 is a path of length @xmath170 . since @xmath1 is narrow , @xmath447 must be adjacent to some vertex @xmath481 . then @xmath482 , @xmath483 and lemma [ layerlem](1 ) imply that @xmath484 . this gives the @xmath343-cycle @xmath485 with no chords since @xmath482 , @xmath486 , @xmath298 and @xmath487 , @xmath488 since @xmath489 but @xmath490 . see figure 3(b ) . hence we have a contradiction since @xmath1 is chordal . _ second _ , suppose that @xmath491 . then @xmath492 . arguing as in the _ first _ , we arrive at figure 3(b ) with the same @xmath343-cycle with no chords , again a contradiction . _ third _ , suppose that @xmath493 . first note that @xmath124 was an arbitrary longest shortest path starting at @xmath93 . thus the above _ first _ and _ second _ give a contradiction whenever @xmath447 or @xmath244 are on _ any _ longest shortest path starting at @xmath93 . hence we may assume that @xmath447 and @xmath244 are not on any shortest path of length @xmath165 starting at @xmath93 . since @xmath1 is narrow , @xmath460 is adjacent to a vertex of @xmath124 , which must be @xmath494 , @xmath495 , or @xmath496 by lemma [ layerlem](1 ) . however , if @xmath484 , then we would get a path of length @xmath165 from @xmath93 to @xmath311 by taking any shortest path from 1 to @xmath447 , followed by @xmath497 , and then continuing along @xmath124 from @xmath496 to @xmath311 . this longest shortest path starts at @xmath93 and contains @xmath447 , contradicting the previous paragraph . hence @xmath498 and @xmath447 must be adjacent to @xmath494 or @xmath495 , and the same is true for @xmath244 by a similar argument . in fact , we must have @xmath499 , since otherwise @xmath500 and the subgraph induced on @xmath501 would be a claw . a similar argument shows that @xmath502 . since @xmath503 and @xmath504 , this implies that the subgraph induced on @xmath505 is a claw , again contradicting claw - free . this final contradiction completes the proof of , and theorem [ converse ] is proved . [ remark : ching ] in and , the chordal hypothesis is applied only to cycles of length @xmath37 , @xmath343 , or @xmath359 . hence , in theorem [ mainthm ] and corollary [ cormainthm ] , we can replace chordal with the weaker hypothesis that all cycles of length @xmath37 , @xmath343 , or @xmath359 have a chord . theorem [ mainthm ] is based on the senior honors thesis of the second author , written under the direction of the first author . we are grateful to amherst college for the post - baccalaureate summer research fellowship that supported the writing of this paper . thanks also to michael ching for remark [ remark : ching ] . m. crupi and g. rinaldo , _ binomial edge ideals with quadratic grbner bases _ , electron . j. combin . * 18 * ( 2011 ) , paper 211 , 13pp . v. ene , j. herzog and t. hibi , _ cohen - macaulay binomial edge ideals _ , nagoya math . j. * 204 * ( 2011 ) , 5768 . v. ene , j , herzog and t. hibi , _ koszul binomial edge ideals _ , arxiv:1310.6426 [ math.ac ] . v. ene , j. herzog and t. hibi , _ linear flags and koszul filtrations _ , arxiv:1312.2190 [ math.ac ] . v. ene and a. zarojanu , _ on the regularity of binomial edge ideals _ , arxiv:1307.2141 [ math.ac ] . j. herzog , t. hibi , f. hreinsdttir , t. kahle and j. rauh , _ binomial edge ideals and conditional independence statements _ , adv . in appl . * 45 * ( 2010 ) , 317333 . m. ohtani , _ graphs and ideals generated by some 2-minors _ , commun . algebra * 39 * ( 2011 ) , 905917 . j. rauh , _ generalized binomial edge ideals _ , adv . in appl . * 50 * ( 2013 ) , 409414 . s. saeedi madani and d. kiani , _ binomial edge ideals of graphs _ , electron . j. combin . * 19 * ( 2012 ) , paper 44 , 6pp .

a graph is closed when its vertices have a labeling by @xmath0 $ ] with a certain property first discovered in the study of binomial edge ideals . in this article , we prove that a connected graph has a closed labeling if and only if it is chordal , claw - free , and has a property we call _ narrow _ , which holds when every vertex is distance at most one from all longest shortest paths of the graph .

arxiv

cosmic rays are elementary particles arriving at the earth from outside that were discovered in the beginning of the 20th century as one of the main sources of natural radiation . the cosmic ray spectrum has been observed as a continuum at all energies since their discovery . throughout this period cosmic rays have always been the source of the highest energy elementary particles known to mankind , and for this reason they have given birth to particle physics . the high energy tail of the spectrum as it is known today corresponds to energies up to 3 @xmath0ev and rates of a few particles per km@xmath1 per century . it is remarkable that the cosmic rays have a quite featureless power law energy spectrum which decreases as approximately the cube of the primary energy . for energies above the few hundred tev the observed flux necessarily requires techniques that take advantage of the extensive air showers that the arriving particles develop as successive secondary particles cascade down into the atmosphere . shower measurements allow the reconstruction of the arrival directions and the shower energy but the nature of the primary particle is extracted by a number of indirect methods . for energies above few tens of gev the detected particles , mainly protons , have arrival directions with a remarkably isotropic distribution . this is understood in terms of diffusive propagation in the galactic magnetic fields . as the energy rises above a given value that depends on the charge of the particle , propagation in the galaxy should cease to be diffusive . such high energy particles are expected to be extragalactic . the observation of high energy cosmic rays has been recently reviewed by nagano and watson @xcite who have shown that there is very good agreement between different experiments including the low and high energy regions of the spectrum . there is increasing evidence for a different component of the high energy end of the cosmic ray spectrum @xcite . combining data of five different experiments , agasa , akeno , haverah park , stereo fly s eye and yakutsk , nagano and watson conclude that there is a clear signal of a change of the spectral slope in the region just above @xmath2ev @xcite . composition studies have also given indications that there is a change to light element composition for energies above @xmath3ev @xcite although this conclusion is model dependent to some extent @xcite . also the small anisotropy ( @xmath4 ) of @xmath2ev cosmic rays in the direction of the galactic anticenter detected with agasa disappears at higher energies @xcite . the highest energy events detected present a serious challenge to theory and little is known about their origin . if they are protons they should attenuate in the cosmic microwave background ( cmb ) over distances of order 50 mpc . such attenuation was predicted to appear in the cosmic ray spectrum as a cutoff , the greisen - zatsepin - kuzmin ( gzk ) cutoff , just above 4 @xmath5gev @xcite . if they are photons or iron nuclei it turns out that interactions with the radio and the infrared backgrounds are respectively responsible for attenuations over similar or even shorter distances . no such features are seen in the observed cosmic ray spectrum . if they are produced sufficiently close to us to avoid the cutoff then the arriving particles should be pointing to their sources . this seems difficult to accommodate because there are very few known astrophysical sources capable of reaching the observed energies and on the other hand there is little evidence for the anisotropy that would result . this article firstly discuses the problem presented by the high energy end of the cosmic ray spectrum with emphasis in the role of composition . then it outlines new progress made in understanding different features of inclined showers illustrating how these showers can contribute to the composition issue reviewing the results obtained by a recent analysis of the inclined data in haverah park . the discovery of events with energies above @xmath6 ev ( 100 eev ) dates back to the 1960 s , to the early days of air shower detection experiments @xcite . since then they have been slowly but steadily detected by different experiments as illustrated in fig . [ uptonow ] . now there is little doubt about the non observation of a gzk cutoff , with over 17 published events above @xmath6 ev and five preliminary new events from hires @xcite . on the contrary the data suggests that the spectrum continues smoothly within the statistical errors , possibly with a change of slope . on the other hand the data show no firm evidence of anisotropy but the significance of such studies is even more limited by the poor statistics . both the details of the spectrum at the cutoff region and the extent to which the arrival directions of these particles cluster in the direction of their sources are very dependent on a number of unestablished issues . these include the source distribution , the distance of the nearest sources , their emission spectra , the intervening magnetic fields and of course on the nature of the the cosmic rays themselves or composition . if these particles are nuclei or photons the observational evidence is suggesting that these are coming from relatively nearby sources compared to the 50 mpc scale . the conclusive power of observations is however strongly limited by both the poor statistics and a complex interrelation of hypotheses , but the situation is bound to change in the immediate future with a new generation of large aperture experiments , some like hires @xcite already in operation , others in construction @xcite and many others in planning @xcite . the complex puzzle that connects particle physics , magnetic fields , and cosmic rays has attracted the attention of many fields in physics . in a conventional approach these particles would be nuclei as the bulk of the cosmic ray spectrum which are accelerated through stochastic acceleration as suggested by fermi in 1949 . this happens every time charged particles cross interfaces between regions that have astrophysical plasmas with different bulk motions , such as shock fronts . transport is assumed to be diffusive in the plasma s magnetic field and on average in these processes a very small fraction of the bulk plasma kinetic energy is transferred as a boost to the individual particles , that typically end up with a power like spectrum . acceleration of a particle of charge @xmath7 to an energy @xmath8 is strictly limited by dimensional arguments to objects that are sufficiently large or have sufficiently large magnetic fields . basically for a particle with momentum @xmath9 to be able to undergo such a boost , propagation must be diffusive , or equivalently the accelerator region @xmath10 must be larger than the larmor radius of the particle , @xmath11 , in its characteristic magnetic field @xmath12 : @xmath13 the requirement is well known by accelerator designers and is the ultimate reason for their high cost . it turns out that few of the known astrophysical objects satisfy the minimum requirements to accelerate particles to @xmath0ev . this is conveniently illustrated in a plot first conceived by michael hillas @xcite which is reproduced in fig . [ hillasplot ] . a number of possible scenarios are being discussed ; they imply acceleration in some objects including young pulsars , gamma ray bursts ( grb ) , our own galaxy , active galaxies and the local group of galaxies @xcite . the power supply needed to keep the observed cosmic rays at the highest energies is consistent with the known power and distributions of these objects @xcite . it is difficult to explain the observed flux spectrum in this conventional approach . a solution in which particles are accelerated nearby has difficulties because there are very few objects which are capable of accelerating particles to the maximum observed energies . moreover many such objects are either too large or too distant for the cosmic ray spectrum detected at the earth not to show the predicted gzk cutoff . if the sources were to be galactic no absorption cutoff would be expected but some spectral features are predicted for primary protons that are produced at a distance of more than a few mpc . on the other hand the non observation of anisotropy complicates the puzzle , because the location of the possible accelerators in our vicinity is pretty well known . primary protons having energies in the @xmath6 ev range are expected to be little deviated in the galactic magnetic fields . our knowledge of extragalactic magnetic fields is poor but bounds on extragalactic magnetic fields also imply that the deviations of protons produced in the few mpc range are not large . there are however possible configurations of the extragalactic magnetic fields that could explain many of the ultrahigh energy events as coming from a single source @xcite . the issue is far from being resolved and knowledge about composition is bound to play a crucial role for future progress in understanding . motivated by particle physics beyond the standard model , many alternatives have been proposed that avoid acceleration and others that postulate different particles or different interactions . these include annihilation of topological defects created in the early universe , heavy relics that survive from the primeval bath , non thermal particles that couple to gravity , or wimpzillas and annihilation of relic neutrinos with messenger neutrinos coming from remote places @xcite . as regards composition two large categories of possible scenarios can be made namely those in which the observed particles are accelerated and those in which they are decay products of other particles . these two classes differ greatly in composition . a knowledge of composition is doubly important because firstly it may decide between these two classes of solutions and secondly because it would simplify the task of interpreting anisotropy measurements . the models that depend on acceleration can reach higher energies if the accelerated particles have large charge @xmath14 . this shows as a different restriction line in fig [ hillasplot ] . the relative composition of different nuclei resulting from such a scenario will depend on the local abundances of the different nuclei and on the energy . depending on distance to the source and the surrounding environment there may be energy losses , absorption and the production of secondary particle fluxes . for instance in active galactic nuclei ( agn ) models the accelerated protons are expected to interact with ambient light or matter to produce pions that decay into photons and neutrinos . the neutrinos can reach the earth unattenuated and provide a signature of proton acceleration . unless the environment becomes opaque to protons the relative fluxes of neutrinos and protons that reach the earth should be a number of order one or smaller , just because neutrinos are secondaries with repect to protons . the relative fluxes of photons and protons would be similar to neutrinos or smaller depending on the photon absorption both at the source and during transport to earth . ratios of the same order of magnitude would apply to most acceleration models . most of the non accelerating alternatives postulate the cosmic rays are products of the decay of other more massive particles produced by different mechanisms . typically these particles of mass of @xmath15ev ( often an @xmath16 particle ) decay into standard model particles which eventually fragment into hadrons , mostly pions and a small fraction of order @xmath17 of nucleons . while neutral pions decay into photons charged pions decay into neutrinos . fragmentation processes , known form accelerator experiments and extrapolated to the high energies , become the common reference point for these mechanisms . for this reason all these models share a very similar composition dominated by photons and neutrinos which typically are about ten times more numerous than nucleons at the production site . depending on the source distribution the relative fluxes of these particles are modified through their interactions with the background radiation fields . the neutrinos are the particles that preserve their production spectrum without being attenuated . protons get attenuated in few tens of mpc in the cosmic microwave background , ( the gzk cutoff ) , while photons are attenuated already in few mps mainly through pair production in the radio background . as a result the ratio of neutrinos to protons can in principle become higher at the earth than when they are produced if the sources are quite distant or cosmologically distributed . many of the proposed mechanisms are expected to cluster in our galactic halo . this possibility is receiving a lot of attention because it would provide a relatively natural explanation for the absence of the gzk cutoff . in that case however the sources will be quite near and the ratio of photons to nucleons should be expected to be of order 10 , close to its value at production . other sources are not expected to cluster and hence the photon to nucleon ratio is expected to drop to values close to one . the ratio of photons to nucleons depends on the source distribution and is rather sensitive to clustering . most air shower detectors in existence consist on arrays of particle detectors that sample the extensive air shower front as it reaches the ground . multiple particle production takes place in the successive high energy interactions produced as the shower penetrates the medium . as a result the number of particles in the shower front increases exponentially . when the average particle energy in the front becomes too low for multiple particle production the shower reaches it maximum number of particles . the development of these showers is typically governed by the radiation length in the material which is of order 36 g @xmath18 in air and shower maximum , which is only logarithmically dependent on the primary particle energy , occurs at a couple of thousand meters for vertical showers of energies of order @xmath6 ev . vertical showers are close to shower maximum when reaching the earth s surface , have pretty good circular symmetry and are less affected by the earth s magnetic field . it is thus not surprising that air showers have traditionally been studied at close to vertical incidence , typically for zenith angles below @xmath19 , in summary because it is much simpler . moreover in most extensive air shower arrays the particle detectors are oriented to have maximum collection area for vertical incidence . since these detectors are often scintillator sheets , they tend to become very inefficient for very inclined showers . as the zenith angle increases the traversed atmospheric depth rises from 1000 to close to 36000 g @xmath18 . as a result the shower maximum is reached in the upper layers of the atmosphere and most of the shower is absorbed before reaching the ground . it has been known for a long time that weakly interacting particles such as neutrinos can induce close to horizontal air showers deep in the atmosphere with particle distributions that are quite similar to vertical showers @xcite . air shower array detectors looking in the close to horizontal direction can thus be sensitive to high energy neutrino fluxes @xcite . in fact most bounds on neutrino fluxes have already been obtained from air shower experiments @xcite . the original motivation of studying inclined showers was to understand the cosmic ray background to the neutrino induced showers . although the electromagnetic part of the air shower induced by an inclined cosmic ray is indeed absorbed before reaching ground level , the shower front however also contains muons which are mainly produced by charge pion decay when the primary particle is a hadron . these muons do travel practically unattenuated all the slant atmospheric depth and produce density patterns on the ground that are much affected by the earth s magnetic field . it has recently become quite clear that such inclined showers can be analysed . this not only nearly doubles the aperture of any air shower array but , when combined with vertical measurements , it has a remarkable potential for the study of primary composition @xcite . much development in this field has been possible by the modelling of the muon density patterns produced by inclined showers under the influence of the earth s magnetic field @xcite . the lateral distributions of muons in inclined showers can be understood in terms of a simple model @xcite in which the magnetic field is firstly neglected . the model stresses two important facts that have been extensively checked with simulations in the absence of a magnetic field @xcite : most of the muons in an inclined shower are produced in a well defined region of shower development which is quite distant from the ground and the lateral deviation of a muon is inversely correlated with its energy . indeed most of the fundamental properties of these inclined showers are governed by the distance and depth travelled by the muons . it is remarkable that the average slant distance travelled by the muons is of order 4 km for vertical showers , becomes 16 km at 60@xmath20 and continues to rise as the zenith angle rises to reach 300 km for a completely horizontal shower . this distance plays a crucial role as a low energy smooth cutoff for the muon energy distribution . for inclined showers the muons must have much more energy at production to reach ground level without decaying than in the vertical case . both the travel time and the muon energy loss become relevant . the model simply assumes that all muons are produced at a given altitude @xmath21 with a fixed transverse momentum @xmath22 that is uniquely responsible for the muon deviation from shower axis . in the transverse plane to the shower at ground level the muon deviation , @xmath23 , is inversely related to muon momentum @xmath9 . the density pattern has full circular symmetry when there is no magnetic field . when the magnetic field effects are considered the muons deviate a further distance @xmath24 in the perpendicular direction to the magnetic field projected onto the transverse plane @xmath25 , given by : @xmath26 where in the last equation @xmath27 is to be expressed in tesla , @xmath21 in m and @xmath28 in gev . as the muon deviations are small compared to @xmath21 they can be added as vectors in the transverse plane and the muon density pattern is a relatively simple transform of the circularly symmetry pattern . the muon patterns in the transverse plane can be projected onto the ground plane to compare with data as well as standard simulation programs . eq . [ alpha ] is telling us that all positive ( negative ) muons that in the absence of a magnetic field would fall in a circle of radius @xmath23 around shower axis , are translated a distance @xmath24 to the right ( left ) of the @xmath29 direction . the dimensionless parameter @xmath30 measures the relative effect of the translation . for small zenith angles @xmath21 is relatively small and @xmath31 so that the magnetic effects are also small , and results into slight elliptical shape of the isodensity curves . for high zeniths however @xmath32 the magnetic translation exceeds the deviation the muons have due to their @xmath28 . in this case _ shadow _ regions with no muons are expected in the muon density profiles . for an approximate @xmath33 mev and @xmath34 t this happens when @xmath21 exceeds a distance of order 30 km , that is for zeniths above @xmath35 . these shadow regions in the transverse plane are indeed an outstanding feature of the ground density profiles at high zeniths as seen in the simulations . the simple model can be actually generalized to account for muon energy distributions as a function of distance to shower axis , and improved using the correlation between the average muon energy and the distance to shower axis as obtained in dedicated simulations . when all this is done the obtained muon density patterns are shown to be accurately reflect those obtained with simulations and this proves to be a very useful tool for the study of inclined showers . for each zenith angle the primary particle energy sets the normalization of the particle densities . for proton primaries the total number of muons in the shower scales with the proton energy @xmath8 as : @xmath36 where @xmath37 is a constant . it is remarkable that the shape of the lateral distribution of the muons does not significantly change for showers of energy spanning over three orders of magnitude . the same happens for heavier nuclei with slightly different parameters . the results are slightly model dependent . two alternative hadronic interaction models have been compared , the quark gluon string model ( qgsm ) and sibyll to give also the same behaviour with also different parameters . table [ nmu.tab ] illustrates these effects . .relationship between muon number and primary energy for proton and irons in two hadronic models ( see equation [ escaling ] ) . [ cols="<,>,^,^",options="header " , ] as a final result the muon distributions can be represented by continuous functions which are analytically obtained once we know the main features of a shower in the absence of magnetic field . in practice this implies that only different zenith angles have to be simulated . different azimuths are obtained by adequate transformations of the showers without magnetic deflections . the algorithm is fast and allows detector simulation and also event by event reconstruction of data obtained by air shower experiments . this powerful technique has been used to analyse the inclined shower data obtained in the haverah park array . the haverah park detector was a 12 km@xmath1 air shower array using 1.2 m deep water erenkov tanks that was running from 1974 until 1987 in northern england which has been described elsewhere @xcite . it is possibly the most appropriate detector for this study because the water erenkov tanks have a uniquely large cross section to sample shower fronts of horizontal air showers . moreover the erenkov technique gives larger signals for muons than for electrons simply because the muons have typically larger energies and travel through the whole detector . a careful study has been made of the energy deposition of signal in water erenkov tanks by horizontal muons using conventional simulation programs for this purpose @xcite . a number of effects have to be considered to interpret the observed data . inclined particles can produce light that falls directly into the phototubes without being reflected in the tank walls . horizontal muons produce more signal through delta rays because on average they have higher energies than in vertical showers . there is a significant signal deposited by electromagnetic particles that arise mainly through muon decay . finally the higher energy muons are more likely to deposit more energy in the tanks because of catastrophic energy losses . the event rate as a function of zenith angle has been simulated with careful treatment of all these effects using the muon distributions obtained as described in the previous section . the qualitative behaviour of the registered rate is well described in the simulation and the normalization is also shown to agree with data to better than @xmath38 using the measured cosmic ray spectrum for vertical incidence , assuming proton primaries and using the qgsm model @xcite . more impressive are the results of fits of the models for muon densities to the observed particle densities sampled by the different detectors on an event by event basis . the nearly 10,000 events recorded with zenith angles above @xmath39 have been analysed for arrival directions , impact point and primary energy in the assumption the primaries are protons . a complex sequence of arrival direction and density fits is performed to minimize the effect of correlations between energy and arrival directions . the analysed date is subject to a set of quality cuts : the shower is contained in the detector ( distance to core less than 2 km ) , the @xmath40 probability of the event is greater than @xmath41 and the downward error in the reconstructed energy is less than @xmath42 . these cuts ensure that the events are correctly reconstructed and exclude all events detected above @xmath43 . examples of reconstructed events compared to predictions are illustrated in fig . [ events.fig ] . two new events with energy exceeding @xmath6 ev have been revealed . the results have been compared to a simulation that reproduces the same fitting procedure and cuts using the cosmic ray spectrum deduced from vertical air shower measurements in reference @xcite . the agreement between the integral rate above @xmath44 ev measured and that obtained with simulation is striking when the qgsjet model is used . sibyll leads to a slight underestimate @xcite . the universality of the muon lateral distribution function is very powerful and once the equivalent proton energy is determined for all events , the corresponding energies in the assumption that the primaries are iron nuclei ( photons ) can be obtained multiplying the proton energy by a factor which is @xmath45 ( 6 ) for @xmath44 ev . as a result when a photon primary spectrum is assumed the simulated rate seriously underestimates the observed data by a factor between 10 and 20 . a fairly robust bound on the photon composition at ultra high energies can be established assuming a two component proton photon scenario . the photon component of the integral spectrum above @xmath46ev ( 4 @xmath44 ev ) must be less than @xmath47 ( @xmath48 ) at the @xmath49 confidence level . details of the analysis are presented in @xcite . the results of this method when applied to a first analysis of inclined showers produced by cosmic rays above @xmath46ev demonstrates that the study of inclined showers not only can double the acceptance of air shower arrays but it can be a very useful tool for the study of photon composition . 9 m. nagano and a.a . watson , _ rev phys . _ 72 ( 2000 ) 689 . t.k gaisser in _ proc . of the int . workshop on observing ultra high eenergy cosmic rays from space and earth _ , ( 2000 ) metepec , puebla , mexico , to be published by aip . d.j . bird _ et al . _ , _ phys . * 71 * ( 1993 ) 3401 . n. hayashida _ et al . _ , _ astropart . * 10 * ( 1999 ) 303 . k. greisen ; _ phys . _ , * 16 * ( 1966 ) 748 . zatsepin and v.a . kuzmin , _ jetp lett . _ , * 4 * ( 1966 ) 78 . j. linsley , _ phys . _ , * 10 * ( 1963 ) 146 . d. kieda _ et al . _ , _ proc . xxvi icrc _ salt lake city ( 1999 ) . design report , auger collab . fermilab - pub-96 - 024 , jan 1996 . for a recent review see _ proc . of the int . workshop on observing ultra high eenergy cosmic rays from space and earth _ , ( 2000 ) metepec , puebla , mexico , to be published by aip . a.m. hillas _ _ , _ proc . of xi icrc _ , budapest ( 1969 ) , acta physica academiae scietiarum hungaricae 29 , suppl . . for recent reviews see for instance , a.v . olinto , _ phys . * 333 - 334 * ( 2000 ) 329 ; g. sigl , lectures at mexican school of astrophysics ( 1999 ) , guanajuato , e - print archive : * astro - ph/0008364*. e. ahn , g. medina - tanco , p.l . biermann , and t. stanev preprint archive : astro - ph/9911123 . berezinsky and g.t . zatsepin , _ yad . fiz . _ * 10 * ( 1969 ) 1228 . [ _ sov . j. nucl . phys _ * 10 * ( 1969 ) 696 ] . e. zas , f. halzen and r.a . vzquez , _ astropart . phys . _ * 1 * ( 1993 ) 297 . j. capelle , j.w . cronin , g. parente , and e. zas , _ astropart . * 8 * ( 1998 ) 321 blanco - pillado , r.a . vzquez , and e. zas , phys . * 78 * ( 1997 ) 3614 . et al . _ , d 31 * , 2192 ( 1985 ) . m. ave , j.a . hinton , r.a . vazquez , a.a . watson , and e. zas , phys . lett . * 85 * , ( 2000 ) 2244 . m. ave , j.a . hinton , r.a . vazquez , a.a . watson , and e. zas , astropart . ( 2000 ) 109 . m. ave , r.a . vzquez , and e. zas , _ astropart . _ 14 ( 2000 ) 91 . tennent , _ proc phys soc _ * 92 * ( 1967 ) 622 . lawrence , r.j.o . reid , and a.a . watson , _ j phys g _ * 17 * ( 1991 ) 733 . j.r.t . de mello neto , wtank : a geant surface array simulation program gap note 1998 - 020 .

in this article i review the main theoretical problems that are posed by the highest energy end of the observed cosmic ray spectrum , stressing the importance of establishing their composition in order to decide between proposed scenarios . i then discuss the possibilities that are opened by the detection of inclined showers with extensive air shower arrays . recent progress in modelling magnetic deviations for these showers has allowed the analysis of inclined showers that were detected by the haverah park experiment . this analysis disfavours models that predict a large proportion of photons in the highest energy cosmic rays and open up new possibilities for future shower array detectors particularly those , like the pierre auger observatory , using water erenkov detectors .

arxiv

cold atoms in optical lattices is the application of two formerly distinct aspects of physics : quantum gases from atomic physics @xcite and laser theory from quantum optics @xcite . the optical lattices are artificial crystals of light , that is , a periodic intensity pattern formed by interference of two or more laser beams . as an insight , a pair of these laser beams in opposite directions ( that is , two orthogonal standing waves with orthogonal polarization ) will give a one - dimensional ( 1d ) lattice , two pairs in two opposite directions can be used to create a 2d lattice and a similar three pairs in opposite directions will give a 3d lattice . atoms can be cooled and trapped in these optical lattices . thus in simple form , an optical lattice looks effectively like an egg carton where the atoms , like eggs , can be be arranged one per well to form a crystal of quantum matter @xcite . though the cold atoms in optical lattices was initially used to investigate quantum behaviour such as bloch oscillations , wannier - stark ladders and tunneling phenomena usually associated with crystals in a crystalline solid @xcite , it is the theoretical proposal @xcite and consequent experimental realization @xcite of the superfluid to mott insulator ( sf - mi ) transition which is an important phenomenon in condensed matter physics that has given rise to the possibility of using it as a test laboratory for phenomena in condensed matter physics . the success of the sf - mi transition in turn emanates from the laboratory observation of bose einstein condensation ( bec ) . the history of bec began in 1924 when satyendra nath bose first gave the rules governing the behaviour of photon which is the commonest boson . excited by this work , einstein in the same year extended the rules to other bosons and thereby gave birth to the bose - einstein distribution ( bed ) @xcite . while doing this , einstein found that not only is it possible for two bosons to share the same quantum state at the same time , but that they actually prefer doing so . he therefore predicted that when the temperature goes down , almost all the particles in a bosonic system would congregate in the ground state even at a finite temperature . it is this physical state that is called bose - einstein condensation . thus it has always been considered a consequence of quantum effects from statistical mechanics in many textbooks as the phase transition is achieved without interactions @xcite . the einstein s prediction , however , was considered a mathematical artifact for sometime until fritz london in 1938 while investigating superfluid liquid helium , realized that the phase transition could be accounted for in terms of bec . this analysis , however , suffered a major set back because the helium atoms in the liquid interacted quite strongly . this was why scientists had to move ahead in search of bec in less complicated systems that would be close to the free boson gas model . fortunately , the breakthrough came in 1995 when the first bec was observed in rubidium atoms and this was followed by similar observations in some other cold alkali atoms such as those of lithium and sodium ( see more details in ref . ( @xcite and a guideline to the literature of bec in dilute gases in ref . as stated above , the observation of bec led to the observation of ( sf - mi ) transition and thereby open the possibility to investigate various phenomena in condensed matter physics by mimicking them with ultracold atoms in optical lattices . this possibility has led to a deluge of studies ( see @xcite for a recent review ) as it brings together atomic physicists , quantum opticians and condensed matter physicists . one draw back is that even when there have been theoretical papers investigating these phenomena with fermionic cold atoms @xcite , cold bosons are used in the actual experiments @xcite for testing spin ordering . this has been overcome by the recent observation of the mi with fermonic atoms @xcite . it follows then that the possibility to use cold atoms in optical lattices as a test laboratory for condensed matter physics is no longer a speculative physics . rather , it has become an aspect of physics with its own methods and approaches . therefore , it has reach a stage when it should start having some introductory impact on our curriculum , possibly as applications of optics , atomic physics and simulation of spin ordering hamiltonians @xcite in condensed matter physics . the purpose of this current study is to present a pedagogical study of investigating spin ordering in an isolated double - well - type potential ( simply double well ( dw ) ) which can be adopted for instructional purposes . for the dw is one of the simplest experimental set ups of optical lattices to study spin hamiltonians @xcite . this is because the system can be completely controlled and measured in an arbitrary two - spin basis by dynamically changing the lattice parameters @xcite . on the theoretical side , the dw can be considered as two localized spatial modes separated by a barrier and consequently be investigated as a two - mode approximation @xcite . the dw is a 1d optical lattice in which the transverse directions are in strong confinement and thus the motions of an atom in these directions are frozen out . to create the dw , we start with a standing wave of period @xmath0 ( long lattice ) so that the potential seen by the atoms trapped in it is @xmath1 where @xmath2 is the lattice depth , which is a key parameter for a special lattice potential . + next we superpose a second standing wave with period @xmath3 and depth @xmath4 ( short lattice ) on the first one as in fig . this will lead to a symmetric double - well superlattice ( fig 1c ) with a total optical lattice @xcite @xmath5.\ ] ] the configuration and varying of the parameter space ( i.e. various parameters ) of a hamiltonian to be tested in this superlattice is achieved by manipulating and controlling the depths of the short and long lattices . for example , by increasing the lattice depth of long - lattice @xmath2 , we could reach from superfluid to mott - insulator regime , which is convenient for studying the few particles phenomena in a local double - well cell . and the barrier height of the double - well is controlled by the lattice depth of short - lattice , @xmath4 . the effective double - well is reached if @xmath6 . otherwise the minimal points of the optical lattice are the bottom of the long - lattice . this could be seen clearly from eq . ( [ topotential ] ) after we expand the @xmath7 term , @xmath8 ^ 2+\frac{v_1}{4}\left(2-\frac{v_1}{4v_2}\right),\ ] ] so that @xmath9 results in @xmath10 , that is , the minimum is one of the long - lattice with a lift of @xmath4 . on the other hand , if @xmath11 , an effective double - well is created . the minimum can be found at @xmath12/2 $ ] . making a power series expansion around the potential minimum , then a single atom of mass @xmath13 trapped initially in any of the well will freely tunneling back and forth with the oscillation frequency of @xmath14 thus the frequency depends on not only the lattice depths @xmath2 and @xmath4 , but also on the lattice spacing @xmath0 . usually , the small lattice spacing @xmath0 is preferred as it leads to a large frequency though this could also be restricted by changing the ratio @xmath15 . this preference also lead to the use of the recoil energy of the short lattice as the unit of the depths of the optical lattice @xmath16 where @xmath17 is the wave length of the short lattice . + for example , in the experiment @xcite , the depth of the long lattice is @xmath18 while the depth of the short lattice is about @xmath19 . this gives the oscillation frequency in a range @xmath20 for @xmath21rb and @xmath22 , we can get @xmath23 , which gives @xmath24 . lets define the harmonic oscillator length @xmath25 , then we can readily get @xmath26 . comparing to the period of short lattice @xmath27 , it implies the ground state wave function is rather localized , which ensures the validity of the two - mode approximation . + finally , it is pertinent to describe how to create an asymmetric dw ( fig 1d ) . the potential bias or the tilt @xmath28 of the double - well is introduced by changing the relative phase of the two potentials ( i.e. short and long lattices ) and this can be realized by applying a magnetic field gradient of @xmath29 @xcite . consequently , tuning @xmath30 gives the potential difference between the two potential minima of the dw . we can realize the adiabatic and diabatic operations on the tilt of the dw by controlling the increasing speed of @xmath30 @xcite . and @xmath3 resulting in ( b ) a chain of double wells from which we can study ( c ) an isolated symmetric double well or ( d ) asymmetric double well ] . [ double well ] within the above consideration and assuming atoms confined in an isolated dw , we reach the two - mode approximation represented as a two - site version of the hubbard model @xcite @xmath31+u(n_{\uparrow l}n_{\downarrow l}+n_{\uparrow r}n_{\downarrow r } ) \label{h1}\ ] ] where @xmath32 is the creation operator ( annihilation operator ) for an atom with spin @xmath33 , @xmath34 is the corresponding number operator , @xmath35 ( both j and t are used in the literature though the cold matter community seems to prefer j ) describes the tunneling rate between the two wells , @xmath28 is the potential bias for the double - well and @xmath36 is the two - body interaction when two atoms occupy the same site . + basically , the hubbard model is a single band lattice model supporting a single atomic state which can hold up to two particles . if we consider that these particles have two internal spins , @xmath37 , then the hamiltonian will consist of the superposition of six fock basis states @xcite denoted by @xmath38 and @xmath39 , with a basis state @xmath40 denoting the @xmath41 and @xmath42 wells . the use of the superposition principle which is a fundamental concept in quantum theory @xcite is consistent with both the model hamiltonian and experiment in which the time evolution of the initial states produces coherent superposition of states . thus the wavefunction of the system will be the superposition of all possible states @xmath43 where for convenience , @xmath44 , @xmath45 and @xmath46 , with i , j denoting the sites while @xmath47 and @xmath48 denote singlet and triplet states respectively . from basic physics , the orientation of the two spins in a state can either be singlet @xmath47 if @xmath49 or triplet @xmath48 if @xmath50 @xcite . by tuning the potential bias @xmath51 , we can obtain all the eigenenergies and corresponding eigenstates analytically . this is achieved by directly diagonalizing the hamiltonian in eq . ( [ h1 ] ) to obtain eigenenergies and eigenstates as shown in table ( [ tabeigen1 ] ) . in the weak interacting case , @xmath52 , the state @xmath53 and state @xmath54 have lower energy so that the doubly occupied singlet state , @xmath55 will be the ground state of the system . this can be considered as the signature for a superfluid state for bosonic atom in a double well . however , the strong interaction regime is more interesting to study for spin ordering . in this regime , @xmath56 , the ground state will be singly occupied as the large atomic repulsion energetically suppress the double occupancy . here it is the @xmath57 that are occupied while the @xmath58 are unpopulated @xcite . the populated @xmath47 and @xmath48 of @xmath57 are nearly degenerate because the energy difference between them is about @xmath59 , which is a small quantity . however , when @xmath60 , the ground state approaches @xmath47 while the first excited state is @xmath48 . if we prepare the initial state as antiferromagnetic , @xmath61 , the dynamical evolution involves two frequencies @xcite @xmath62 from the above frequencies , one could get the tunneling rate @xmath63 and the interaction strength @xmath64 respectively . these two frequencies can be obtained from exerimental data and then used to test the validity of the simple two - mode model . this has been done for bosonic atoms in experiment @xcite . the extension to fermionic atoms may be different but the eigenenergies and corresponding eigenstates are the same , which means we can get similar dynamics as long as the interaction between atoms satisfy @xmath65 @xcite . on the other hand , the interaction of fermion could be also attractive generally . it is interesting to identify the ground state in this situation . table ( [ tabeigen1 ] ) works here too . .eigenstates and eigenenergies of hamiltonian ( [ h1 ] ) [ cols="^,^,^,^,^,^,^",options="header " , ] [ tabeigen1 ] when @xmath66 , the ground state does not change too much at weak tunneling . however , the first excited state is not the triplet state @xmath67 anymore but the state @xmath68 . it is interesting that the energy difference is the same @xmath69 although the interaction is attractive and the first excited state is changed . this analysis shows that @xmath68 could be involved in the dynamics if we start from the antiferromagnetic initial state @xmath70 . the two frequencies that can be observed in experiment are @xmath71 here the two - body interaction strength directly relates to @xmath72 , which can be extracted from the measured experimental data . the results show that ultracold atoms trapped in the superlattice not only could be used to simulate the phenomena in condensed matter physics , but also offer the possibility to compare the results with theoretical calculation of model hamiltonians . . left , @xmath74 in the strong interaction regime . middle @xmath75 in the weak interaction regime . right , attractive interaction @xmath76 . , title="fig : " ] . left , @xmath74 in the strong interaction regime . middle @xmath75 in the weak interaction regime . right , attractive interaction @xmath76 . , title="fig : " ] . left , @xmath74 in the strong interaction regime . middle @xmath75 in the weak interaction regime . right , attractive interaction @xmath76 . , title="fig : " ] one of the advantages of the trapped ultracold atoms is that they can be controlled precisely and easily . by tuning the two optical lattices , we can change the bias of the double - well superlattice . the ground state of the two fermions are trapped in the same site in the large potential bias . by slowly reducing the bias , the ground state is followed adiabatically to the singlet state . in this way , we can prepare the initial states either in the state @xmath53 or @xmath54 . so it is interesting to investigate how the bias influences the states . when the potential bias @xmath28 is included , the eigenstates and the eigenenergies have complicated expressions . three of the eigenenergies are @xmath77 always . the others are the roots of the algebra equation @xmath78 we numerically solve the equation and plot the eigen spectra in fig . ( [ spectra ] ) . when @xmath79 , the ground state energy is negative and modified by the presence of potential bias . when @xmath28 is not too big , the ground state energy is close to the one without potential bias . also , the energy difference between the ground state and first excited state is small . at large potential bias @xmath80 , however , the approximate ground state energy reads @xmath81 the two atoms are in the right well and the ground state reaches @xmath54 . on the contrary , the system is degenerate further in the weak interaction regime @xmath82 . if the interaction is attractive , the energy spectra is reversed . the energy difference between the ground state and the first excited state is bigger except at @xmath51 , i.e. an anti - crossing appears . this is shown obviously in fig . ( [ spectra ] ) . the observation from this analysis is that the potential bias can be used to control the energy difference between the singlet and triplet states @xcite . this is why an attempt was made in @xcite to use it to drive eq . [ h1 ] into superexchange interaction observed experimentally . the outcome , however , is that inter - well interactions have to be included to eq . [ h1 ] to get close to the experimental data . thus in the next section , we will consider such an extension . it is obvious from the preceding section that a hamiltonian to study spin ordering in the cold atoms in optical lattices needs to contain long range interactions . it is important to point out that the overlapping of different electronic orbitals gives rise to the interaction between spins in condensed matter but this overlapping is very small in optical lattices @xcite . however , the possibility of the atoms to tunnel through the barrier in quantum mechanics enables the inter - site interactions @xcite . two natural candidates are the inter - site coulombic interaction @xmath83 and exchange interaction @xmath84 . interestingly , the inclusion of these interactions as means of going beyond the standard hubbard hamiltonian to account for ferromagnetism in metals have been proposed @xcite . furthermore , we do not need a potential bias since the spin ordering is induced by these interactions . within these considerations , the extended form of eq . [ h1 ] with @xmath85 is @xmath86 . \label{h2}\ ] ] it is then easy to obtain the ground state energy and wavefunction using the highly simplified correlated variational approach ( hscva ) in @xcite . the beauty of this pedagogical approach is that the ground state energy clearly depicts the physics of the model as one vary the parameter space as in experiments with optical lattices . interestingly , the method allows the decoupling of the kinetic part from the interaction parts so that we can observe the effects of including each of them to the kinetic part . thus the combination of these two factors makes the hscva very suitable to investigate the spin hamiltonian to be tested using cold atoms in optical lattices . + we start with the variational ground state energy @xmath87 where the h is the model hamiltonian and the ket in the hilbert space is the trial wave function ( cf . ( [ wavefunction1 ] ) ) defined as + @xmath88 the x and y in eq . ( [ wavefunction2 ] ) are the variational parameters . it is straightforward to show @xcite that eq . ( [ variational ] ) leads to a 3 x 3 blocked matrix of 2 x 2 and 1 x 1 resulting in the lowest state energies @xcite , @xmath89 for the singlet states @xmath55 or @xmath90 depending on u , @xmath91\ ] ] and @xmath92 for the triplet state @xmath93 , @xmath94 the smallest of these two energies will be the ground state energy of the system . the corresponding eigenvectors are then substituted as the variational parameters in eq . ( [ wavefunction2 ] ) to give the corresponding ground state wavefunctions . thus when @xmath95 , the system will be antiferromagnetic while it will be ferromagnetic otherwise . taking into account this condition and eqs . ( [ singlete ] ) and ( [ triplete ] ) , the critical value of @xmath84 at which there is transition from one state to another is + @xmath96.\ ] ] now to test this hamiltonian in a double well , we need to know how the atomic positions and spin orientations varies with the parameter space . for example , as demonstrated in subsection ( a ) , the ground state of the system will be a mott insulator when the u is very strong . this generally accepted property of the half - filled standard hubbard hamiltonian ( i.e. @xmath97 ) is already achieved with ultracold fermionic atoms @xcite . one of the signatures of the mi state is the decrease in doubly occupied states in the ground state as @xmath36 is increased . this is demonstrated in fig . 3 showing the level of occupation of the states denoted by the variational parameters of the ground state wavefunction with increase in @xmath36 . the inclusion of @xmath83 , however , enhances the double occupancy and is therefore expected to suppress the observation of the mi especially for low values of u. it follows then that when we switch on the @xmath84 , the @xmath36 is likely to drive the system into more singly occupied states and thereby enhancing the transition to a ferromagnetic state while the @xmath83 will suppress it . this is demonstrated in eqs . ( [ singlete ] ) - ( [ criticalj ] ) and then depicted in fig . 4 showing the variation of the antiferromagnetic - ferromagnetic transition critical point of @xmath84 with @xmath36 at various values of @xmath83 . + the above theoretical @xmath83 and @xmath84 can also be compared with the ones obtained from extracted data from the experiments as was done for j and u in the standard hubbard hamiltonian . this is by expressing the possible dynamic evolution frequencies from for the singlet states and triplet states as @xmath98\ ] ] @xmath99 we see immediately that we can recover eq . ( [ frequency ] ) from eq . ( [ freqsinglete ] ) when @xmath97 . taking into account eqs . ( [ frequency ] ) , ( [ freqsinglete ] and ( [ freqtriplete ] ) , we can then estimate @xmath100 $ ] and @xmath101 $ ] . thus we can also obtain the inter - site interaction parameters from the data extracted from the experiments . the increasing advancement on how to prepare , manipulate and detect phenomena in condensed matter physics using cold atoms in optical lattices has reached a stage when it can be used as instructional means . the fact that laser cooling and trapping are now widely used in atomic physics laboratory @xcite means the realization of the double wells experiment can also be achieved . the first investigation is to mimic the mott insulator state . by extracting @xmath102 and @xmath103 from the experiment , the experimental values of j and u can be compared with the ones from their theoretical values . the experiment can then be advanced to determine @xmath83 and @xmath84 and then compare them with their theoretical values as depicted in fig . interestingly , the model hamiltonian studied here has been proposed to account for spin ordering in transition metals . it is hoped therefore , that the testing of this extended hubbard model can easily be compared to available data for the transition metals @xcite after some refining of the approach here . this will also include extending the study to dynamic properties of the model . while there is increase in inter - site states @xmath104 as the on - site coulomb interactions u increases . ] with the on - site coulomb interactions @xmath36 at various values of the inter - site coulomb interactions @xmath83 . a similar graph was obtained by ref . 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laser cooling and trapping are now widely used in atomic physics laboratory . interestingly , cold atoms in optical lattices are now used in advanced research to mimic phenomena in condensed matter physics and also as a test laboratory for the models of these phenomena . it follows then that it is now possible and necessary to advance the atomic physics laboratory by including the use of ultracold atoms in optical lattices for instructional contents of phenomena in condensed matter physics . in this paper , we have proposed how to introduce into the atomic physics laboratory the study of quantum magnetism with cold atoms in a double well optical lattice . in particular , we demonstrates how to compare the theoretical parameters of a spin hamiltonian model with those extracted from spin ordering experiment .

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